Msc_plastic Design Of Steel-concrete Composite Bridges.pdf

Plastic Design of Steel-Concrete Composite Girder Bridges

by

Supervisors:

Árpád Rózsás

Dr. Nauzika Kovács Dr. László Dunai Dr. Theodore V. Galambos

Master of Science Thesis Budapest, Hungary, 2011

Budapest University of Technology and Economics Faculty of Civil Engineering Department of Structural Engineering

Plastic Design of Steel-Concrete Composite Girder Bridges Master of Science Thesis

by Árpád Rózsás

Supervisors: Dr. Nauzika Kovács Dr. László Dunai Dr. Theodore V. Galambos

Budapest, Hungary, 2011

i

Abstract The primary purpose of this thesis is to investigate the plastic reserve of composite plate girder bridges. These structures are suitable for this due to the synergetic combination of the concrete and steel. The former provides the “cheap” stiffness and strength in compression while the steel in tension ensures the ductility. However, the theoretical and experimental aspects of plastic design are well established only in the US provisions are available for the designers. The aim was to inquiry the plastic design in the framework of the Eurocode through an existing elastically designed bridge. In the first part of the study the necessary theoretical background is overviewed, the related literature is examined. The main emphasis is placed on the ultimate load bearing capacity, which is determined using various limit states, such as first hinge, incremental collapse and plastic collapse. The safety levels of these limit states were also investigated. To ensure the ductility of the pier-sections innovative structural solutions gathered and evaluated. The selected bridge is a composite, plate girder, continuous structure formed by three spans (30,0-40,0-30,0m). This was redesigned following plastic principles, the relevant provisions and the findings of the researchers. The calculations showed that − for the original structure − the traffic load could be increased by ~30 and ~60% over the first yield in case of using first hinge and shakedown limit states, respectively. It was found that the safety levels of these limit states at least reach or exceed that of the first yield or first hinge. It should be noted that these results reflect only one example; nevertheless, they are in good agreement with the American results. The redesign yielded to a structure with cleaner lines with considerably less section transition and about 25% structural steel saving. Based on the calculations and international data the plastic design of girder bridges appears to be a promising way, at the same time more research required.

iii

Összefoglalás Jelen diplomamunka fő célja az öszvér szerkezetű gerendahidak képlékeny tartalékainak vizsgálata. Ezen szerkezetek a beton és acél szinergikus kapcsolata miatt különösen alkalmasak erre. Az előbbi viszonylag alacsony költséggel biztosítja a szükséges merevséget és teherbírást a nyomott, míg az acél a szükséges duktilitást a húzott zónákban. Habár a képlékeny tervezés elméleti és gyakorlati vonatkozásai jól kidolgozattak, tervezési előírások kizárólag az Egyesült Államokban állnak a mérnökök rendelkezésére. A vizsgálódás arra irányult, hogy egy megépült, rugalmasan méretezett hídon keresztül elemezzük a képlékeny tervezést az Eurocode keretein belül. A tanulmány első fele a szükséges elméleti hátteret és a kapcsolódó irodalmat tekinti át. A hangsúly a tartó különféle teherbírásainak meghatározásán volt, mint a(z): első folyás, első képlékeny csukló, halmozódó képlékeny alakváltozások, képlékeny törés. Szintén megvizsgáltuk az ezen határállapotokhoz tartozó biztonsági szinteket. A közbenső támasz környéki szelvények duktilitásának biztosítása érdekében újító megoldásokat is összegyűjtöttünk és elemeztünk. A kiválasztott szerkezet egy háromtámaszú (30,0-40,0-30,0m), öszvér szerkezetű, folytatólagos gerendahíd. Ezt a képlékenységtani elvek, szabványos elírások és kutatási eredmények alapján átterveztük. A számítások azt mutatták (az eredeti hídra vonatkozóan), hogy a forgalmi teher az első képlékeny csukló, mint határállapot választásakor ~30%-kal, míg a beállási határállapot esetén ~60%-kal növelhető az első folyáshoz viszonyítva. A megbízhatósági analízis megmutatta, hogy a fenti határállapotokhoz tartozó biztonsági szintek legalább elérik vagy meghaladják az első folyáshoz vagy első képlékeny csuklóhoz tartozó értéket. Ugyanakkor meg kell jegyezni, hogy ezek az eredmények egyetlen példára vonatkoznak; mindazonáltal jól egyeznek az amerikai eredményekkel. Az áttervezés egy tisztább vonalú, jelentősen kevesebb keresztmetszetváltást tartalmazó szerkezetet eredményezett, ~25% szerkezeti acél megtakarítással. Az elvégzett számítások és nemzetközi eredmények tükrében a gerendahidak képlékeny méretezése egy ígéretes módszernek tűnik, ugyanakkor még további kutatást igényel.

iv

Acknowledgement I owe my deepest gratitude to Associate Professor Nauzika Kovács and to Professor László Dunai for their guidance and hints throughout the numerous consultations during the semester. I highly appreciate their effort reading, correcting and commenting on the raw material amid their various occupations. I am also grateful for the assistance and advices of Professor Theodore V. Galambos; for helping outline the subject of the project, reading and amending the study and for answering the emerged questions even via email. Furthermore, I would like to thank the useful comments and consultations on the reliability of structures, to Assistant Professor Tamás Kovács. Finally yet importantly, I would like to thank the valuable conversations for István Hegedűs structural engineer, the designer of the 142/k bridge and Gábor Pál structural engineer for let at my disposal the static calculation and drawings of the structure.

v

CONTENTS ABSTRACT ............................................................................................................................. iii  ÖSSZEFOGLALÁS ................................................................................................................ iv  ACKNOWLEDGEMENT ....................................................................................................... v  1. 

INTRODUCTION ............................................................................................................ 1 

2. 

THEORETICAL BACKGROUND ................................................................................ 2  2.1.  Brief Review of the Theorems of Plasticity ................................................................ 2  2.1.1. 

Plastic Collapse Theorems/ Theorems of Plastic Limit Analysis ........................ 2 

2.1.2. 

Shakedown Theorems .......................................................................................... 7 

2.2.  Reliability Analysis of Structures .............................................................................. 13 

3. 

2.2.1. 

Measures of Reliability ...................................................................................... 14 

2.2.2. 

Methods to Evaluate the Reliability Index ( ................................................... 18 

REVIEW OF PREVIOUS INVESTIGATIONS ......................................................... 24  3.1.  Historical Overview ................................................................................................... 24  3.2.  State of the Art in Plastic Design of Bridges ............................................................. 27  3.2.1. 

Ultimate Limit States ......................................................................................... 27 

3.2.2. 

Safety Concern, the Reliability of Plastic Design .............................................. 29 

3.2.3. 

Cost-Saving ........................................................................................................ 30 

3.2.4. 

Experimental Verification .................................................................................. 32 

3.2.5. 

Rotational Capacity of the Cross-section ........................................................... 33 

3.3.  Worked-out Design Methods..................................................................................... 35 

4. 

3.3.1. 

United States ...................................................................................................... 35 

3.3.2. 

Contributions From Outside of the US .............................................................. 41 

BRIDGE DESIGN PROBLEM..................................................................................... 47  4.1.  Problem Statement ..................................................................................................... 47  4.2.  Introduction of the Bridge to be Studied ................................................................... 47  4.3.  Solution Strategy, Extent of the Thesis ..................................................................... 48 

5. 

GLOBAL STRUCTURAL ANALYSIS ....................................................................... 49  5.1.  Elastic Check According to Eurocode ....................................................................... 49  5.1.1. 

Finite Element Model ......................................................................................... 49 

5.1.2. 

Loads and Load Combinations ........................................................................... 51 

5.2.  Investigation of Plastic Reserves ............................................................................... 53  5.2.1. 

First-Hinge Limit Analysis................................................................................. 56 

5.2.2. 

Shakedown Limit Analysis ................................................................................ 56  vi

5.2.3. 

Single-Girder Plastic Collapse Limit Analysis .................................................. 58 

5.2.4. 

System Plastic Collapse Limit Analysis............................................................. 59 

5.3.  Summarization of the Rating Factors ........................................................................ 60  5.4.  The Effect of Shear Force .......................................................................................... 61  5.5.  Conclusions ............................................................................................................... 64  6. 

STRUCTURAL SOLUTIONS TO MEET THE DUCTILITY DEMAND .............. 66  6.1.  Concrete Filled Closed and Open Sections ............................................................... 67  6.1.1. 

Concrete Filled Tubular (CFT) Girder ............................................................... 67 

6.1.2. 

Concrete Filled Narrow-width Steel Box-girder ................................................ 68 

6.1.3. 

Partially Encased Rolled and Welded Sections.................................................. 69 

6.2.  Double Composite Action ......................................................................................... 72  6.3.  Reinforcing the Web .................................................................................................. 75  6.3.1. 

Bolted Longitudinal Plate or Stiffener ............................................................... 75 

6.3.2. 

Welded Longitudinal Stiffeners ......................................................................... 78 

6.4.  Conclusions ............................................................................................................... 79  7. 

RELIABILITY ANALYSIS ACCORDING TO EUROCODE ................................. 80  7.1.  Principles, Methods ................................................................................................... 80  7.1.1. 

Eurocode Recommendations .............................................................................. 80 

7.1.2. 

Reliability Analysis ............................................................................................ 82 

7.2.  Results of the Analysis on the Studied Bridge .......................................................... 87  7.3.  Conclusions ............................................................................................................... 90  8. 

REDESIGN OF THE BRIDGE BASED ON PLASTIC PRINCIPLES ................... 91  8.1.  Considerations, Eurocode Principles ......................................................................... 91  8.2.  Proposed Method for Plastic Design ......................................................................... 92  8.3.  Introduction of the Proposed Method through a Trial Design................................... 94  8.3.1. 

The Redesigned Structure .................................................................................. 94 

8.3.2. 

Verification of the Trial Plastic Design.............................................................. 96 

8.3.3. 

Comparison of the Findings to the American Results...................................... 104 

8.4.  Conclusions ............................................................................................................. 106  9. 

SUMMARY AND CONCLUSIONS........................................................................... 107  9.1.  Summary .................................................................................................................. 107  9.2.  Conclusions ............................................................................................................. 107  9.3.  Further Research/ Future Work ............................................................................... 110 

REFERENCES ..................................................................................................................... 111  vii

ANNEX A – DESIGN CHECK TO EUROCODE ..................................................................   ANNEX B – LIMIT STATE ANALYSIS ................................................................................   ANNEX C – RELIABILITY ANALYSIS ................................................................................   ANNEX D − USED PROGRAMS ............................................................................................  

viii

Abbreviations and acronyms: AASHTO AR CC CDF CF CFT EN FEM FORM FOSM GMNIA GU LM LN LRFD LRFR LSF LTB MPP MSZ NA ND PDF PF PNA RC RCA RF RR SLS SND SORM SRC UF ULS

American Association of State Highway and Transportation Officials available rotation consequence class cumulative distribution function concrete filled concrete filled tube European Norm finite element analysis first order reliability method first order second moment method geometric and material nonlinear analysis with imperfections Gumbel distribution load model lognormal distribution load and resistance factor design load and resistance factor rating limit state function lateral torsional buckling most probable point Hungarian Standard (Magyar Szabvány) National Annex normal distribution probability density function probability of failure plastic neutral axis reliability class/ reinforced concrete rotation compatibility approach (AASHTO) rating factor required rotation serviceability limit state standard normal distribution second order reliability method steel and reinforced concrete utilization factor ultimate limit state

Symbols  g(.) ν 

relative slenderness limit state function coefficient of variation cumulative distribution function of the standard normal distribution

ix

    

reliability index standard normal density function mean value/ load factor Pearson correlation standard deviation

x

Introduction

1.

Introduction

The current structural design is prevalently based on the theory of elasticity. However, the plastic capacity of the materials and structures are extensively investigated and verified long time ago. Typically, the plastic reserve is used only indirectly, for example, through the moment redistribution in structural engineering. The bridge engineers all around the world advocate a much more conservative principle, they consider solely the elasticity. Even in the US which is currently the only country in the world where standards, guides in the topic of plastic design are accessible for bridge designers, the engineers follow the conservative elastic methods [Haiyan and Fangfang, 2010]. Barker and his fellow researchers mention four main reasons for this: (1) perceived difficulty of application compared to elastic design provisions, (2) lack of training, (3) safety concern with the inelastic limit states, and (4) insufficient experimental verification [Barker et al., 2000]. Later, I will show that all of these concerns have been resolved, still the inelastic design concepts are not widely accepted. In the US since 1973, the researchers are working out improved design procedures. In particular cases − compared to the typically used elastic methods − around 20% load-bearing capacity increase can be achieved with the proposed provisions [Barker and Galambos, 1992; Barker and Zacher, 1997]. These methods cannot only be used for the design of new cost-efficient, competitive structures, but for the revision of old deficient bridges as well. In many European countries, like in Hungary the national standards did not allow the exploitation of the plastic reserve of structures. However, the new European normative permits it, solely a method how to determine the plastic resistance of the cross-section is provided, but it does not give in the designers' hand an applicable tool how to carry out the global structural analysis. The aim of this study is to investigate the possibility of inelastic design of bridge structures in the frame of Eurocode, using the available provisions and research data. Furthermore, to compare the design procedure to the conventional elastic methods, and to assess the accessible cost-saving. I restrict my attention exclusively to steel-concrete composite1 girder bridges. Since, if one would like to study and apply the plasticity theory to bridge design, some in the conventionally curriculum not fully covered fields have to be acquired, the next chapter will sum up this necessary knowledge.

1

Thereinafter composite will refer to the steel-concrete composite.

1

Theoretical Background

2.

Theoretical Background

If we allow a structure to overpass the elastic limit, new failure modes will occur, such as the plastic collapse or deflection instability. These ultimate limits are presented here with the conventional modes, according to [Halász and Platthy, 1989]: a) loss of equilibrium of the structure or any part of it, considered as a rigid body; b) rupture: - plastic rupture; - brittle rupture; - high-cycle fatigue; - low-cycle (plastic) fatigue; c) plastic collapse (transformation of the structure or any part of it into a mechanism); d) unrestrained accumulation of the plastic strains; e) yield of material (first yield), failure by excessive deformation of the structure or the connections; f) failure by loss of stability of the structure or any part of it. The next section will summarize the basic theorems, which are required to evaluate the ultimate load to the b), c) and d) failure modes.

2.1.

Brief Review of the Theorems of Plasticity

2.1.1. Plastic Collapse Theorems/ Theorems of Plastic Limit Analysis Assumptions: - the load is proportional, that means it can be described by one parameter: (load factor);

f    f0

(2.1)

The entire load history can be represented by the f0 basic load, which is stationary, and by the load factor (), which is monotonically increasing during the loading process. - the deformations are relatively small (small deformation theory is valid). One important theorem should be highlighted before the main extremum theorems, the constancy of curvatures during plastic collapse (constant stress) theorem. This states that during the plastic collapse, when the structure cannot bear more loads and turns into a mechanism, the stresses and elastic strains are stationary. The mechanism means that the whole structure or part of it undergoes increasing displacements while the load is unchanging (Figure 2.1). For a global mechanism, on a structure with statically indeterminacy to nth degree, the formulations of n+1 plastic hinges are required. During the proportional increase of the displacements and plastic strains, the other variables are constant. This theorem has some very important consequences such as: - the true collapse load can be determined considering a rigid-ideally plastic material law; 2

Theoretical Background - the collapse load is not affected by the previous load history.

Figure 2.1: Global and local collapse mechanisms of a frame. Static (Lower Bound) Theorem Under a load (f), computed on the basis of an arbitrary statically admissible and stable internal force field (s), the structure will not collapse. The s internal force field should be statically admissible, which means that it satisfies the equilibrium equations (2.4). Stable means that s obeys to the yield criteria (2.5). From the theorem it can be seen that the only requirements are to be satisfied the equilibrium and constitutive equations. With using the load factor the theorem can be rewritten as: Any statically admissible and stable load factor (s) is less than or equal to the true collapse load factor (p).

s   p

(2.2)

Mathematically speaking the determination of the collapse load factor can be formulated as an optimization problem:

 p  max(s )

(2.3)

BT  s  s  f

(2.4)

s0  s  s0

(2.5)

Where s is the independent variable (for example si describes one possible (statically admissible) internal force field), if sj maximizes the load factor we have found the true internal force distribution at collapse. The s vectors are containing internal forces and resistances. Each element corresponds to a cross-section.

3

Theoretical Background

s

internal forces in the cross-sections;

s0

the plastic resistance of the cross-sections;

f

external force vector;

BT

static matrix2, transpose of the kinematic matrix (B);

In finite element analysis (FEM) the matrix with the same meaning denoted with the same letter ( K e 

B

T

 D  B  d  ), called strain-displacement matrix.

e

As it used to be in the science of mechanics, with the swap of the force and displacement variables, the pair of one theorem can be attained. This static-kinematic duality exists in the case of plastic extremum theorems as well. Kinematic (Upper Bound) Theorem Under a load (f), computed on the basis of an arbitrary kinematically admissible and unstable displacement field ( d ), the structure will collapse. The kinematically admissible condition expresses the continuity requirement (geometric equations) (2.6). The vectors d and e denote the displacement velocity and plastic strain rate respectively. The unstable means that while a collapse mechanism formulate, the power of external loads ( W ) should be greater than or equal to the rate of dissipation (Dint) (2.7), ext

which is given by the sum of the product of the stresses and corresponding plastic strain rates. W is the power of external loads on the displacement velocity field. During the plastic ext

collapse the power of external loads is dissipated by irreversible plastic process in material, mainly as heat. B  d  e

(2.6)

Wext  Dint

(2.7)

It is necessary to deal with velocities and power, since there is no unique correspondence between the stresses and strains, solely between the increments of these variables. The time is simply chosen as a monotonic increasing parameter to form these. From the theorem it can be seen that the only requirements are to be satisfied the kinematic and constitutive equations. With using the load factor the theorem can be rewritten as: Any kinematcally admissible and unstable load factor (k) is greater than or equal to the true collapse load factor (p).

 p  k 2

Describes the connection between the internal forces and external loads, equilibrium matrix.

4

(2.8)

Theoretical Background The kinematically admissible unstable load factor means, that it is obtained by equating the power of external forces on kinematically possible displacement velocities ( d ) with the corresponding rate of work (power) of stresses on the plastic strain rates ( e ). f(x)  ·

d(x) L/2

L/2

Figure 2.2: One possible collapse mechanism of a continuous beam. In simple cases it is sufficient to deal only with the internal and external works. Figure 2.2 and the following equations show the calculation of the kinematic load factor for a given collapse mechanism.



k

 f ( x)  d ( x)  dx  M pl  2   M pl 

l

k 

M pl  2   M pl 

 f ( x)  d ( x)  dx l

In the same manner as the static theorem, here is the extremum formulation of the upper bound theorem:

 p  min( k )

(2.9)

B  d  e

(2.10)

k  f T  d  sT0  e

(2.11)

In this case, d is the independent variable (for example d i describes the the i-th possible (kinematically admissible) mechanism), if d j minimizes the load factor we call this j-th mechanism as the true collapse mechanism. These theorems were first proved by Gvozdev (1897-1986) in 1936 for beams, frames and plates. However, his works were published in Russian, therefore were unnoticed in the West until 1960. Independently from Gvozdev the theorems were discovered in 1949 by Horne and in 1951 by Greenberg and Prager (1903-1980). In 1952 Drucker (1918-2001), Greenberg and Prager have generalized the theorems for bodies with arbitrary triaxial stresses [Kaliszky, 1975; Bažant and Jirásek, 2001].

5

Theoretical Background Unicity

Since the collapse load is the result of a global maximum and global minimum search, it has to be unique (Figure 2.3).

 s   p  k k

(2.12)

s

min k max s

sij ijk Figure 2.3: Symbolical interpretation of the uniqueness of collapse load. If, both statically and kinematically admissible mechanism has found, it means that the corresponding load factor is the true collapse factor. If we have two or more proportional loads, which can act together, the problem can be handled in the following way. For example, with two loads, after we choose a ratio m 

1 2

the problem is reduced to a one-load case. For this particular instance, the collapse load factor can be determined. With more ratio and load factor, the collapse surface can be approximated with a desired accuracy (Figure 2.4).

1

1

PC

PC

EP

EP

E m

1,0

approximation from 3 points

E p

2

2

Figure 2.4: Interpretation of the collapse load factor and the load-bearing domains.

6

Theoretical Background With relatively low loads, the response of the structure is purely elastic (E). With the increase of the load one or more points of the structure will yield. This is represented by the yield surface (E-EP). This separates the elastic and elasto-plastic (EP) domains. In the latter region the structure is partially in a plastic state, however it is still able to carry the loads. Moreover, any loads lower than the before-applied maximum are carried in an elastic manner. With further raising of the load, the structure will reach the surface of plastic collapse (PC). Inside this surface, any design points considered safe. Points outside the surface represent the loads under which the structure will collapse. The one parametric case in Figure 2.4 is represented by a straight line and the collapse surface is shrunk to a point. It can be seen from Figure 2.4 that the collapse load factor (p) can be considered as the measure of safety against the plastic collapse. 2.1.2. Shakedown Theorems

In the field of buildings, the loads can be considered non-variable, in contrast to bridge’s live load. This is an important distinction because in the presence of variable load, a “new” failure mode can occur, namely the shakedown, or to be more precise the absence of shakedown. This phenomenon can happen under loads lower than the plastic collapse load, in the elastoplastic (EP) region. Shakedown consists of two failure modes (Figure 2.5): A) If we consider the successive application, for example, two loads. Supposing that one load always destroys the residual force field developed by the previous load, in every cycle it has to be restored and consequently new residual rotations and deflections occur. In this way the displacements will accumulate to an unacceptable range or even the material will fail, due to the reach of the ultimate strain. This failure mode is called incremental collapse (deflection instability, elastic shakedown, ratcheting). B) If a point of a cross-section, due to a variable load, undergoes yielding in both directions (tension and compression as well), the material will fail after relatively small number of cycles in a brittle way. This failure mode is called alternating plasticity (low-cycle fatigue, plastic shakedown). If a structure subjected to cyclic load, which generates stresses beyond the elastic limit, and after successive cycles it withstands any subsequent loads in a purely elastic manner, then the structure has shaken down. The word was introduced by Prager. This elastic response can only be achieved with the development of a favorable self-equilibrated residual force field. Two criteria have to be fulfilled to reach the purely elastic load bearing.

s r  semax  s0   s r  semin  s0 

(2.13)

s emax  s emin  2  s y

(2.14)

BT  sr  0

(2.15)

7

Theoretical Background where:

sr

represents the residual forces;

semax , semin the maximum and minimum values of the force envelopes at given points, determined on a ideally elastic structure;

sy

the elastic resistance of the cross-sections.

The first condition (2.13) has to be hold to prevent the unrestrained accumulation of plastic strains (incremental collapse). The second one (2.14) expresses the alternating plasticity criteria. At typical engineering structures the latter one is usually not governing. According to experiment’s results the material will fail by alternating plasticity, after relatively few number of load cycles (never more than about 100) [Bažant and Jirásek, 2001]. Eq.(2.15) simply represents the requirement that the residual stresses should be self-equilibrated. 







max

max

max



·y

max







min min

min min

Purely elastic

Plastic fatigue

Shakedown

Incremental collapse

Figure 2.5: Illustration of various structural behaviors and failures. The illustration of the shakedown can be seen in Figure 2.6 (considering only the incremental collapse), neglecting the effect of the shear forces. Figure a) represents the elastic maximal moment diagram from the external loads ( semax , semin ). Figure b) shows the distribution of the residual moments which develop after the unloading of the structure which was subjected to a load over its elastic capacity. After the formulation of the first hinge over the internal support, the structure carries its loads as two simply supported beams. Since the unloading process is taken place in a purely elastic manner, during the removal of loads the structure acts as the original continuous beam. This leads to the development of a self-equilibrated residual forces. If there exists a residual internal force filed in which presence the loads are carried elastically the structure has shaken down (Figure c). Some general comments can be added, which are well illustrated on the figures as well. It can be seen from Figure c) that after the favorable residual moment develops, any subsequent loads are carried elastically. The post-shakedown envelopes are more uniformly distributed. On the place of the first hinge a residual rotation takes place (d). Typically, this kink and the residual deflections are not significant.

8

Theoretical Background

a

e

b

r

pl c

e+r pl

d

wr

Figure 2.6: Shakedown of a two-span-beam. About the shakedown of a structure, two questions arise. First, does the required residual stress exist? Moreover, if there is a residual stress field, which fulfils the shakedown conditions, will this even develop in the structure? The first question can be answered relatively easily by calculation. The second is a tougher one. The first time Melan (18901963) answered it in 1936 for beams and frames. He proved that if the needed residual moment exists, that will develop in the structure after a certain number of load cycles. Melan’s theorem is also called the lower bound theorem of shakedown analysis. Melan’s (Lower Bound) Shakedown Theorem

If, for a given load history, there exist shakedown forces sr such that the conditions of plastic admissibility (2.13) are satisfied as strict inequality at any time t, then the structure shakes down [Bažant and Jirásek, 2001]. The incremental collapse condition can be rephrased with the load factor, applied to the maximum and minimum forces generated by the variable load, in the following way: The safety factor is the largest statically admissible and stable multiplier.

sh  max(s )

(2.16)

s r   s  s emax  s 0   s r   s  s emin  s 0 

(2.17)

BT  sr  0

(2.18)

9

Theoretical Background The geometric interpretation of this and the load-bearing regions can be seen in Figure 2.9. It should be noted that this theorem was stated before the static theorem of plastic collapse (1938, Gvozdev). It was recognized later, that Melan’s theorem comprises the latter. Therefore the shakedown theory is more general and includes the elastic and plastic collapse theories as well [Bažant and Jirásek, 2001]. Koiter’s (Upper Bound) Shakedown Theorem

If there is an admissible plastic deformation cycle ek (t) such that T

T

0

0

T  f (t )  d k (t )  dt   Dint (e k (t ))  dt

(2.19)

Then the structure cannot shake down under the cyclic loading history f(t) [Bažant and Jirásek, 2001]. With using the load multiplier: The safety factor is the smallest kinematically admissible and unstable multiplier.

sh  min(k )

(2.20)

B  d k  e k

(2.21)

T

D

int

k 

0 T

f

T

(e k (t ))  dt (2.22)

(t )  d k (t )  dt

0

In the above form Eq.(2.22) cannot really used to determine the load factor. Since it contains the loading history (f(t)) and the evolution of plastic strain rate in time. With some considerations, like the maximization of the denominator, the best upper bound of the safety factor can be derived, for details see [Bažant and Jirásek, 2001]. In this way, the practical reformulation of the general expression:

k 

sT0  (ek  ek ) (semax )T  ek  (semin )T  ek

where:

e k

the total positive plastic strain increment;

e k

the total negative plastic strain increment.

10

(2.23)

Theoretical Background These are defined the following way: T

e   e (t )  dt  k

T

e   e k (t )  dt

 k

 k

0

0

e k and e k are the negative and positive part of the e k plastic strain rate respectively. e k   e k  e k  2

e k   e k  e k  2

e k  e k  e k

These are illustrated on a simple beam example in Figure 2.7.  

B

A

Me,min B

Me,max A

Figure 2.7: Upper bound load factor calculation of a continuous beam. Substituting the moments and rotations into Eq.(2.23) we get:

k 

M pl     M pl    M Ae ,max     M Be ,min   

Based on geometric considerations, knowing the locations of plastic hinges, + can be expressed by  - and after that the equation can be simplified with the rotation. As mentioned before, the shakedown theory is more general than the plastic collapse theory. If we look Eq.(2.13)-(2.15) and consider a non-variable, proportional loading the residual force field (sr) will became zero, and thus the equations are simplified to the static limit analysis conditions (2.4)-(2.5). Furthermore, in plastic collapse equations setting the crosssection capacities to the elastic resistance, the load factor corresponding to elastic loadbearing can be calculated.

11

Theoretical Background

Shakedown theory Limit analysis theory Elastic theory

Figure 2.8: Symbolical interpretation of the theories for structural analysis. Of course the shakedown theory has to be applied only if we dealing with variable loads and would like to exploit the plastic reserves. Unicity

The same can be told about the uniqueness of the shakedown limit load as about the plastic collapse load.

s  sh  k sh  max(s )  min(k )

(2.24)

In the space spanned by the load factors (Figure 2.9) region S means the domain where the structure will shake down. The shakedown and incremental collapse, alternating plasticity (IC, A) regions are separated by the shakedown limit surface. The shakedown load factor (safety factor) can be achieved with the radial scaling of load point to the shakedown limit surface. The points outside this surface represent loading under which the structure will collapse. 1

1

PC

PC IC, A

IC, A

S

S E

E

1, 0  sh

2

2

Figure 2.9: Interpretation of the shakedown load factor and the load-bearing domains.

12

Theoretical Background As mentioned before the shakedown limit load is between the elastic and the plastic collapse loads. Therefore, the safe region is reduced compared to the non-variable loading case (Figure 2.4). Nevertheless, the domain bounded by the shakedown limit surface is larger than the conservative elastic regime. One important difference should be emphasized that in the case of variable loading to make the structure collapse certain number of load cycles are required, while for the plastic collapse one load is sufficient. 1

1

PC IC, A

p sh

S

e

L

E 2

2

Figure 2.10:Limit load factors and the illustration of the loading region (L). In Figure 2.10 the blue region illustrates the load domain (L), the linear combination of possible loadings. During design it should be verified that all possible loading points are inside the safe domain. To obtain the limit load factors an extreme value search has to be carried out (see e.g., Eq.(2.20)-(2.22)) To solve the general case with arbitrary nonlinear constraints advanced mathematical methods are required. However, analytical solution still can be achieved only in very limited cases. With discrete variables, such as the available sections, the optimization became more complex, in these cases usually metaheuristics are applied (e.g. evolutionary strategies) [Rizzo et al., 2000]. The advantage of these methods is that they provide rather good solutions when the analytical methods are not applicable. Even if the best solution cannot be guaranteed, they usually give considerably good candidates. The optimization problem without additional constraints can be simplified with a linearized yield criterion to a linear programming problem. If one would like to use this knowledge to design of bridges, he or she has to apply it on such a way that it yields to a safe structure. Since there is no standardized method in Eurocode for inelastic design of bridges, to carry out this, it is requisite to overview the basics of the reliability analysis of structures.

2.2.

Reliability Analysis of Structures

Since engineering structures are serving many people, represent a high value and the consequences of collapse are significant, they must be designed to satisfy a prescribed

13

Theoretical Background reliability. This can be achieved by using the methods, rules and partial factors provided by the standards or by direct reliability analysis to verify that the structure meets the safety criterion. 2.2.1. Measures of Reliability

Survival probability (Ps) To be able to judge the adequacy of reliability, a numerical value is necessary to measure it. The most obvious choice is the survival probability (Ps). This can be expressed with the probability of its complement event, the failure probability (Pf):

Ps  1  Pf

(2.25)

To calculate this probability, a function, which determines the failure criteria, is required. Corresponding to the examined phenomena a limit state function (LSF) has to be formulated. This function can be created from the basic design inequality: RE

(2.26)

where: R

resistance (more generally: capacity), random variable;

E

effect (more generally: demand), random variable.

Rearranging it the performance or limit state function: g ( R, E )  R  E

(2.27)

Both abovementioned variables can contain many random variables, for the sake of simplicity a limit state function with only two variables will be used herein. The failure probability can be calculated the following way: Pf  P ( R  E )  0  P  g  0 



f X ( X )  dX

(2.28)

g ( X )0

where f X (X) is the joint probability density function (joint PDF) of the elements of the X vector, which are random variables. Joint probability expresses the probability that two or more random events will happen simultaneously. In this case X   R E  . These variables are called the state variables, which are used to formulate the limit-state function. The geometric explanation of the failure probability is shown in Figure 2.11.

14

Theoretical Background f g<0 failure region

f (g)

g=0 g>0 safe region

probability of failure

g

Figure 2.11: Probability density of the limit state function. g 0

safe region,

g 0

border between safe and unsafe domains,

g0

failure region.

These regions are illustrated in Figure 2.11 and in Figure 2.12 as well.

Figure 2.12: Joint probability density function and regions [Du, 2005]. Although Eq.(2.28) is a straightforward definition of the failure probability, apart from very simple cases, the integral cannot be calculated or it requires special numerical techniques which accuracy may not be adequate [Nowak and Collins, 2000]. Therefore, in practice, the reliability is expressed by other measures.

15

Theoretical Background Reliability index ( he reliability index multiplied by the standard deviation (g) shows the distance from the mean value (g) to the most probable failure point (MPP). Thus in a space of variables normalized with standard deviation,  expresses the distance to the MPP, this way the reliability is a unitless quantity, which can be used to describe the reliability.



g g

(2.29)

Based on this definition it can be considered as the inverse of the coefficient of variation  ( Vg  g ). g f

 (g) g=0 g<0 failure region

 ·g

g>0 safe region

g Figure 2.13: Explanation of the geometrical meaning of the reliability index for normal distribution. In case of normally distributed variables a direct relation can be found between  and Pf (probability of failure):

Pf  ( )

(2.30)

Whereis the cumulative distribution function (CDF) of the standard normal distribution (SND). SND is a special normal distribution where the mean value (g) is zero and the standard deviation (g) is unit. Table 2.1 shows some calculated values. Table 2.1: Relation between Pf and  Pf [-]

1,0E-01

1,0E-02

1,0E-03

1,0E-04

1,0E-05

1,0E-06

1,0E-07

 [-]

1,282

2,326

3,090

3,719

4,265

4,753

5,199

Generally there is no explicit relation between  and Pf. In a special case, if we have normally distributed uncorrelated variables then with Eq.(2.30) we would get the true failure probability [Nowak and Collins, 2000]. In every other case the obtained probabilities cannot 16

Theoretical Background considered as true values, rather appropriate measures to compare the reliability levels of structures. It is convenient to transform the joint PDF to the space spanned by the reduced state variables (Ui), Eq.(2.31). This means that the variables are converted to standard normal distribution (SND). The transformation equations: UR 

R  R

R E  E UE  E

(2.31)

Figure 2.14:Joint PDF transformed to the U space [Du, 2005]. The result of the conversation is illustrated in Figure 2.14 and in Figure 2.15. In the space of reduced variables, because the standard variation is unit, the reliability index is simply the distance to the MPP. Moreover, if we take into account the rotational symmetry of the SND, the reliability index can be defined as the shortest distance from origin to the limit-state function (g=0). This definition, which was introduced by Hasofer and Lind (1974), is illustrated in Figure 2.14 and in Figure 2.15. The  is often called Hasofer-Lind reliability index as well. The advantage of this conversation is that the search for the distance to the most probable failure point on g=0 is reduced to the search of shortest distance to the failure limit (g=0) in U space.

17

Theoretical Background E- u E=  E E g=0



E·

MPP

R- u R=  R R

R·

Figure 2.15: Top view of a normal distribution transformed into the U space. 2.2.2. Methods to Evaluate the Reliability Index (

As stated in the previous section, in practice, the  index is used to measure the reliability, thus in this section I will show how to determine it. There are many methods to accomplish this; however, only the method recommended by the Eurocode 0 and used in this study will be presented in details. FORM (First Order Reliability Method) Its name is from the order of approximation of the limit-state function. It approximates the LSF in points of the g=0 hyperline, the points of this line are called limit points. From these points we have to find the most probable, which is simultaneously the closest to the origin. The basic formulation of it only applicable to normally distributed uncorrelated variables. Later it will be shown how can it be expanded to general cases. It can be shown that in case of linear limit-state function the reliability index can be obtained by the following equation: n

g ( X 1 , X 2 ,..., X n )  a0   ai  X i

(2.32)

i 1

n



a0   ai   X i i 1

n

a i 1

18

i

2

 Xi

2

(2.33)

Theoretical Background If the LSF is nonlinear we can still apply this method to get an approximate value by linearizing the function using Taylor series expansion at point X*i : n

g ( X 1 , X 2 ,..., X n )  g ( X 1 , X 2 ,..., X n )  g ( X *1 , X 2* ,..., X n* )    X i  X i*  i 1

g X i

(2.34) evaluated  at  X i*

Now applying the derived expression for linear case, the approximate reliability index: n

g ( X *1 , X 2* ,..., X n* )   i 1

 n

g X i

g

 X i 1

n

X i*

 X i*   i 1

g X i

2

  Xi X i*

(2.35)

 Xi 2

i X i*

Writing in a shorter form with vectors:



g ( X * )   g ( X * )T  X *   g ( X * )T  μ X  g ( X* )   g ( X* ))T  (σ X  σ X )

(2.36)

If the limit-state function is nonlinear, an iterative process has to be carried out. The so called matrix procedure, to calculate , according to [Nowak and Collins, 2000], consists the following steps: 1) Formulate the limit state function and appropriate parameters for all random variables Xi (1=1,2,…,n) involved. 2) Obtain an initial design point X i* by assuming values for n-1 of the random variables Xi. (Mean values are often reasonable choice.) Solve the limit-state equation g=0 for the remaining variable. This ensures that the design point is on the failure boundary line. 3) Determination of the reduced values ( Ui* ) for the corresponding design point ( X i* ), using Eq.(2.31). 4) Determine the partial derivatives of the limit-state function respect to the reduced variables (Ui). Because the LSF is expressed as the function of the original variables (Xi) to obtain the aforementioned derivatives the chain rule has to be applied:

g g X i   U i X i U i

(2.37)

The second multiplier can be derived from the connection of the original and reduced variables (2.31). Using that, the partial derivatives are the following:

g g   Xi U i X i

19

(2.38)

Theoretical Background For convenience, define a column vector G, containing the partial derivatives, by the following way: Gi  

g X i

(2.39) evaluated at  X i*

5) Calculate the reliability index using the following formula, based on (2.36):



GT  U*

(2.40)

GT  G

6) Calculate the vector containing the sensitivity factors ():

GT

α

(2.41)

GT  G

If we consider a vector showing from the origin to the design point, the length of this vector is the . The sensitivity factors are representing the direction cosines of this vector respectively to the reduced variables. 7) Obtain the new design point Ui* by calculating values with using the following equation:

U*  α  

(2.42)

8) Determine the corresponding design values in the original space for n-1 of the variables, using Eq.(2.31). Solve the limit-state equation g=0 for the remaining variable. 9) Repeat steps 3 to 8 until  and the design point (Xi) converge. The geometric interpretation of this process is shown in Figure 2.16. E

g=0

2 3 1

   

uR

Figure 2.16: Geometric interpretation of the FORM iteration process. 20

Theoretical Background Since this method uses the assumption that the random variables are normally distributed, some modification is required to apply it to other distributions. The expansion to non-normal distributions can be done by the Rackwitz-Fiessler procedure. The basic idea behind it is the calculation of equivalent normal distribution values (mean and standard coefficient) for every variables. To obtain these equivalent normal mean (  Xe ) and standard deviation (  Xe ), we require that the original variable’s CDF and PDF be equal to the normal distribution CDF and PDF values respectively, at the design point ( X* ). With the addition of this step to matrix procedure the  value can be determined. From the definition of equivalency, it can be seen that it is necessary in every iteration cycle to calculate the equivalent values. f xi(xi) non-normal distribution

f xi(x*i)=f exi(x*i)

F xi(x*i)=F exi(x*i)

equivalent normal distribution

 exi  xi

xi

Figure 2.17: The equivalent normal distribution [Choi et al., 2007]. The method can be extended to correlated random variables as well. The Pearson correlation coefficient3 () describes the degree of linear dependency between two variables. It can take values from -1 to 1. Figure 2.18 shows the meaning of the coefficient.

Figure 2.18: The meaning of the Pearson correlation coefficient [Wikipedia01]. If two variables are independent, than  = 0, but if = 0 it does not indicate that the variables are independent, it solely means that there is no linear relation at all between those variables, other relations are possible.

3

Thereinafter the correlation will refer to the Pearson correlation.

21

Theoretical Background If the variables are correlated the steps to calculate the reliability index essentially remain the same. The only difference is that the correlation matrix () will appear in Eq.(2.40) and in Eq.(2.41). This changes these equations in the following way:



α

G T  U* GT  ρ  G GT GT  ρ  G

(2.43)

(2.44)

The Eurocode considers the FORM method sufficiently accurate. However, since the program being used has more advanced capabilities, these are also being briefly introduced in the following. SORM (Second Order Reliability Method) In some cases e.g., with highly nonlinear failure surface (g=0), the failure probability estimated by FORM is inaccurate. In these cases the SORM can be used which applies second-order Taylor series approximation of the limit-state function [Choi et al., 2007]. Simulation methods - Monte Carlo method Any method which solves a problem by generating suitable random numbers and observing that fraction of the numbers obeying some property or properties is called Monte Carlo method. The method is useful for obtaining numerical solutions to problems which are too complicated to solve analytically. It was coined in the 1940s by John von Neumann (19031957), Stanislaw Ulam (1909-1984) and Nicholas Metropolis (1915-1999), while they were working on nuclear weapon projects in Los Alamos [Wikipedia02; WolframMathworld]. In this thesis, I will apply only the crude Monte Carlo method where the distribution-based samples are generated without any special selection rule, simply by random process. In reliability analysis, the method is used to calculate the integral which expresses the failure probability, Eq.(2.28). The calculation starts with the generation of values of basic variables following their distribution. These are used to obtain the values of compound variables, which compose the limit state function (g). The failure probability is calculated as the ratio of the number of points violate the limit state function and the total number of points. The concept of the Monte Carlo method is illustrated in Figure 2.19.

22

Theoretical Background

g=0

E

R Figure 2.19: Illustration of the Monte Carlo method in reliability analysis.

23

Review of Previous Investigations

3.

Review of Previous Investigations

After going through the fundamental necessary knowledge, the brief overview of the historical development and the state of the art in plastic design follows, to show the place of the current study in their context. This chapter is focusing on the practical application of plasticity to engineering structures.

3.1.

Historical Overview

The roots of the plasticity can be traced back to Galileo Galilei (1564-1642). In his work Discourses and Mathematical Demonstrations Relating to Two New Sciences (1638) in the discourse about the load-bearing capacity of a cantilever beam (Figure 3.1) one can identify the basic concept of kinetic approach of limit analysis [Kaliszky, 1975; Bažant and Jirásek, 2001; Kurrer, 2008]. It should be mentioned, that notwithstanding Galileo has made a prominent contribution to the science, apparently he has made some mistakes as well. For example, in the case of the cantilever beam, he considered only the rotational (moment) equation, which led him to a wrong result. Now this can be construed, that Galileo assumed that in cross-section A uniform tensile stress field would develop (albeit he never did this explicitly) [Kurrer, 2008].

Figure 3.1: Galileo’s cantilever beam [Kurrer, 2008]. The theories of limit analysis can be rephrased in a much simple form than as stated in section 2.1.2, in which they seem to be intuitively obvious: “A body will not collapse under a given loading if a possible stress field can be found that is in equilibrium with a loading greater than the given loading.

24

Review of Previous Investigations A body will collapse under a given loading if a velocity field obeying the constraints (or a mechanism) can be found that so that the internal dissipation is less than the rate of work of the given loading.” [Lubliner, 2008] After Galileo, great architects4 are intuitively applied the principle − which now we call the static approach to limit analysis − to carry out the calculation of some big domes, arches and vaults. De La Hire in his book − first published in 1695 − deals with the design of vaults, applies this approach. About a century after Galileo in 1742, three mathematicians examined the safety of the dome of Saint Peter’s cathedral in a manner which can remind us of the upper bound theorem of limit analysis. Based on their analysis they recommended installing a second tension ring [Kurrer, 2008; Lubliner, 2008]. Figure 3.2 shows the drawing of the dome with the observed damages from the report of the mathematicians.

Figure 3.2: The dome of Saint Peter’s in Rome with the observed cracks in 1742 [Kurrer, 2008]. The first realistic and almost complete static analysis of failure, along with the concept of the plastic slip and yield condition, is found in Coulomb’s (1776) study of earth retaining walls for military fortifications [Bažant and Jirásek, 2001]. In the next decades many researchers contributed to the theory of plasticity in the field of structural mechanics and geotechnics as well. It was an outstanding step forward when Gábor von Kazinczy (1889-1964) in 1914 conducted his experiments with clamped steel beams encased in concrete [Kazinczy, 1914]. Based on

4

Refers to the meaning used before the modern times

25

Review of Previous Investigations that research he pointed out that the ultimate bearing capacity of a statically indeterminate structure cannot be determined with the theory of elasticity.

Figure 3.3: Kazinczy’s test beam [Kurrer, 2008]. Another probably more influential contribution of Kazinczy was the introduction of the concept of plastic hinge. In 1917, N. C. Kist proposed the ideal-elastic and ideal plastic material law for mild steel, which is still applied in steel design to determine the collapse load. Soon after many scientists and engineers accepted the plastic design, which was being spread widely across Europe [Kurrer, 2008]. In spite of the theoretical and experimental results numerous scientists were against the plastic theory. Sharp debate has started between the two opposite groups, which finally led to a paradigm shift. To picture the atmosphere, a piece one of these acrimonious discussions is presented here: “A statically indeterminate structure remains statically indeterminate also if the limit of proportionality or the yield stress of the material is exceeded in particular cross-sections. This means that besides the equilibrium conditions, the deformation conditions also remain valid even in the post-elastic loading range. The inadequacy of the ultimate load method is based on the fact that it treats this fundamental fact wrongly and upon closer inspection its ‘simplicity’ is revealed as unacceptable primitiveness. … If, however, favouring the ultimate load method is intended to placate those people who cannot master, and given normal talents cannot learn, the normal methods of calculating statically indeterminate structures, then the introduction of such a ‘theory of structures for idiots’ should certainly be rejected.”[Stüssi, 1962] It should be noted that the quoted opinion belongs to Fritz Stüssi (1901-1981) who was an outstanding professor of his century. The focal point of the debates was the paradox of the plastic hinge, which was resolved by Neal and Symonds in 1952 [Kurrer, 2008]. Since the theoretical, experimental and practical (standards) conditions are satisfied in the US, maybe “only” a paradigm shift is required amongst the designers. In other countries the practical part is missing as well. In the next decades, the basic extremum theorems of plasticity have been formulated and proved as mentioned in Section 2.1. In the following section the state of the art practical side of the plastic design of bridges is summarized.

26

Review of Previous Investigations

3.2.

State of the Art in Plastic Design of Bridges

Even in the building construction where it is allowed to overpass the elastic limit, typically, like in Hungary, only the first-plastic-hinge method is applied. The bridge engineering community is even more conservative. Standards for plastic design of bridges are only available in the US, nevertheless, not widely applied. When we are dealing with bridges, as mentioned in Section 2.1.2, due to the relatively high live to dead load ratio and to the variable loading, the phenomenon of shakedown has to be considered. This has a crucial importance in the formulation of the design procedures. In the following section I will go through the problems arise on the plastic design of bridge structures and show some possible answers. At the end, the most recent design procedures from the US (AASHTO) and Europe will be reviewed in details. 3.2.1. Ultimate Limit States

The first step in a design procedure is to choose an appropriate failure mode for the ultimate limit state (not considering now the stability and fatigue problems). A) Elastic limit Traditional ultimate limit state, related to the first yield of the material. B) Moment redistribution Indirect consideration of partially or fully formed plastic hinges, many of the following more complex methods are simplified to an easy-to-apply moment redistribution approach. The limitation is the ratio of redistribution, which is derived from the assumed failure mode and other limitations. C) First plastic hinge concept The bridge is loaded until the formulation of first plastic hinge, this state is considered as the ultimate limit. The basic procedure is that we allow to formulate a plastic hinge in the most loaded cross-section and the redistribution of the moment towards the less loaded regions, until the formulation of the next hinge. The methods differ in that manner how accurately they determine the maximum applicable moment redistribution, this depends mainly on the rotation capacity of the section. D) Single-girder shakedown Only the most loaded girder is examined. The load on this beam is determined through the lateral load distribution. This way only the longitudinal structural reserve is taken into account. If the shakedown is chosen as ultimate limit, it has a great importance to know the number of load cycles required to failure. Since it determines the probability of failure for a given load distribution.

27

Review of Previous Investigations How many successive cycles are required to reach the stability of deflection? Based on the experimental research of Barker and his fellow researchers with a third-scale steel-concrete composite bridge, the needed cycles are around twenty to thirty [Barker et al., 1996]. Neal suggests 10 successive load cycles to be chosen as appropriate failure criterion for design [Neal, 1977]. Buildings are also subjected to variable loads. However, it is very unlikely to have the shakedown as the governing failure mode, because rather high live load to dead load ratio is necessary to experience the deflection instability. Typically live load higher than two-thirds to three-fourth of dead load is requisite to get this problem [Bruneau et al., 1998]. Neal also suggests that for typical buildings the plastic collapse is the governing. In other cases if the ratio of live load to dead load is high, the decision should be based on the comparison of the probability of the occurrence of incremental and plastic collapse [Neal, 1977]. In AASHTO for inelastic design methods of bridges, the shakedown is applied as ultimate limit state. E) System shakedown The entire bridge system is modeled, the first plastic hinge will form in the most loaded girder. After that the reserve, both in longitudinal and in transverse directions are mobilized as well. This can be reduced to an equivalent single girder shakedown analysis (in case of global mechanism, which is typical), where the equivalent elastic-moment envelopes and resistances are the sum of corresponding the individual girders’ values. Compared to the single girder shakedown analysis the load capacity is increased about 15%, due to the transverse redundancy [Barker and Zacher, 1997]. The rating factors can be seen in Table 3.1 and in Table 3.3. The rating factor (RF) is used in the US to classify the performance of bridges.

RF 

  R    D  D   ( L I )  L( L I )

(3.1)

n

where:



resistance factor;

R



resistance (capacity); load factor;

D

dead load coefficient;

D

dead load effect in member; live load coefficient;

( L I ) L( L I )

n

live load effect in member including impact.

RF is a scale factor of the live load effect required to reach the failure limit. According to this RF ≥ 1 means that the structure fulfils the requirements. The theorems of shakedown analysis can directly applied to determine the shakedown load, nevertheless the rotation capacity of

28

Review of Previous Investigations cross-sections with plastic hinges and the permanent deflections should be checked supplementary. For normal two way bridges, Grundy showed that the global mechanisms govern the plastic collapse, the local mechanisms − which are convenient to avoid − can become ruling only in case of rather wide bridges [Grundy, 1987]. F) Plastic collapse Only one high load is sufficient to reach this limit state, under which a mechanism forms and the structure collapses. Since for a structure statically indeterminate to nth degree, to fail with a global mechanism, n+1 plastic hinges are required, we can conclude that higher redundancy yield to higher load capacity. It is valid for the incremental collapse as well. The order of ultimate limit states represents the order of accessible load capacity as well. 3.2.2. Safety Concern, the Reliability of Plastic Design

Another important question to be answered is the reliability of the aforementioned methods. Table 3.1 summarizes the results of calculations based on different limit states for a simply supported (13,4m) and a three-span (12,5 x 16,2 x 12,5m) composite bridges. The  value represents the correlations between the corresponding probability variables e.g., R is the correlation of the cross-section resistances in the plastic hinge locations. Table 3.1: Reliability indices () and rating factors (RF) for first-hinge RF=1 [Barker and Zacher, 1997]. One-span bridge

Limit state First-hinge Single-girder shakedown: R = D = L = 0% Single-girder shakedown: R = D = L = 50% Single-girder shakedown: R = D = L = 100% Single-girder shakedown: R = 70%; D = 50%; L = 80% System shakedown: R = D = L = 0% System shakedown: R = D = L = 50% System shakedown: R = D = L = 100% System shakedown: R = 70%; D = 50%; L = 80%

Three-span bridge

RF



RF



1,00 1,00 1,00 1,00 1,00 1,16 1,16 1,16 1,16

3,09 3,09 3,09 3,09 3,09 6,91 4,37 3,46 3,88

1,00 1,20 1,20 1,20 1,20 1,36 1,36 1,36 1,36

2,76 5,34 4,17 3,50 3,83 12,00 5,18 3,84 4,47

The reliability indices determined for the load level corresponds to the single girder shakedown, presented in Table 3.2.

29

Review of Previous Investigations Table 3.2: Reliability indices () and rating factors (RF) for single-girder shakedown RF=1 [Barker and Zacher, 1997]. Three-span bridge RF 

Limit state First-hinge Single-girder shakedown: R = D = L = 0% Single-girder shakedown: R = D = L = 50% Single-girder shakedown: R = D = L = 100% Single-girder shakedown: R = 70%; D = 50%; L = 80% System shakedown: R = D = L = 0% System shakedown: R = D = L = 50% System shakedown: R = D = L = 100% System shakedown: R = 70%; D = 50%; L = 80%

0,83 1,00 1,00 1,00 1,00 1,13 1,13 1,13 1,13

2,07 4,42 3,41 2,86 3,13 9,88 4,24 3,14 3,66

It is important to emphasize that the values are determined for particular composite girder bridges, as suggested by the authors more calculations are required to generalize the results. The target reliability index is for the redundant three-span structure is 2,5 according to the applied standard (AASHTO). It can be seen from the data that every case at least reach or exceed the level of reliability of the reference first hinge method. In Table 3.3 one can see the rating factors of a steel, two-span (2x16,75m), girder bridge with the achieved increase in load bearing capacity over the AASHTO Guide Specification for Strength Evaluation of Existing Steel and Concrete Bridges, thereinafter called the “guide spec”, first hinge method. Table 3.3: Rating factors for different limit states [Barker and Galambos, 1992]. Limit state

Rating factor (RF)

Increase over guide spec first hinge [%]

0,944 1,013 1,127 1,631

7,3% 19,4% 72,8%

Guide spec first hinge System model first hinge System model shakedown System model collapse

3.2.3. Cost-Saving

As demonstrated in previous sections, with the application of inelastic design procedures significant load capacity increase can be achieved. In new designs, it results reduced amount of material and minimizes the number of section transitions, compared to the elastic method. It has been shown that savings are mainly manifested due to the reduction of fabrication cost and not due to the material savings. On the other hand, the costly reinforcement of existing

30

Review of Previous Investigations bridges, which are classified as deficient by traditional methods, can be avoided [Barth et al., 2004]. This second advantage of plastic design methods is really significant. According to a survey (1987) involving all highway bridges in the US, more than 40% of the existing bridges were either structurally deficient or functionally obsolete using the contemporary (1987) design-based rating procedures [Barker and Galambos, 1992]. In the previously referred paper, Barker and Galambos showed that with allowing the structure to enter the plastic regime and with shakedown as the ultimate limit state, many of these deficient bridges could be qualified as adequate. The aging of bridges is a universal problem in every country, e.g., in China nearly 80% of railway steel bridges have served for more than 40 years. Due to the general increase in traffic weights and traffic densities, the loads those old bridges received are greater than they were designed for [Haiyan and Fangfang, 2010]. Leon and Flemming also mention the poor maintenance and increased traffic over the expectations as reasons for the high percentage of deficient bridges [Leon and Flemming, 1997]. The costly reinforcing or reconstruction could be avoided by plastic design methods. Because the allowance to overpass the elastic limits yields to a more uniformly distributed moment through the girders (Figure 2.6), application of rolled girders could be considered. Since the contribution of the fabrication and welding cost to the entire cost is significant (Figure 3.4), it may lead to a more economic design. Erection 11-25%

Material 24-50%

Transportation 3-7%

Paint 11-18% Fabrication 19-29%

Figure 3.4: Distribution of total cost for a bare steel girder of a conventional composite plate girder bridge [Collings, 2005]. From the above figure it can be seen that the savings in the fabrication plus erection side has almost the same weight as the material saving, respect to the total cost. To make a more accurate assessment, life-cycle cost comparison should be carried out. It should be mentioned that due to the reduced sectional dimensions and consequently the diminished flexural stiffness the serviceability limit states might governing. The American results showed that the utilization of SLS are slightly increased compared to the elastic design but still far from being governing [Barth and White, 2000]. The before-mentioned publication

31

Review of Previous Investigations does not check the crack widths; however, it can be found for example in AASHTO 2010, Section 5.7.3.4. Nevertheless, there is no indication in the standard whether the proposed method is applicable to plastic design. 3.2.4. Experimental Verification

Experimental verifications of the analytical and developed numerical methods were conducted [Flemming, 1994; Barker et al., 1996; Leon and Flemming, 1997; Grundy, 2004]. For low load levels the results were in good agreement with measured values, nevertheless, in the regime of loads close to the theoretical shakedown limit, some discrepancies were found. It should be noted that there is a still unsolved problem about the shakedown of steel-concrete composite beams constructed with shear studs. Many experiments show that under a cyclic load, close to the shakedown limit, the structure tends to behave like a steel structure. The measured deflections converge to the values calculated considering solely the bare steel (Figure 3.5). The suspected reason was the degradation of the concrete slab and the deterioration of the studs due to concentrated applied force by the hydraulic jacks [Galambos, 2007]. However, the same phenomenon was observed by Flemming who conducted the shakedown test of a composite bridge using trucks for loading [Flemming, 1994]; Grundy also experienced the same [Grundy, 2004].

Figure 3.5: Experimental and analytical shakedown residual deflections [Barker et al., 1996]. The problem was partially answered by Leon and Flemming in 1997. They carried out an experimental research with a half-scale, two-span, two-girder composite bridge subjected to actual moving load. The aforementioned strength reduction was observed, notwithstanding they pointed out that, with composite action5 80% or higher the structure is able to reach the theoretically calculated shakedown limit, due to the strain hardening and the redistribution of forces. They observed 30% load capacity reduction with 50% composite action. They also pointed out that even with 80% composite action the section capacity cannot be maintained

5

The ratio of normal force that can be transmitted by the studs and the normal force required to reach the full plastic resistance in the concrete flange; denoted by  in Eurocode.

32

Review of Previous Investigations for large number of loading cycles in the inelastic range [Leon and Flemming, 1997]. This question still requires further research to get fully resolved. 3.2.5. Rotational Capacity of the Cross-section

As mentioned before, in some cross-sections increased rotations occur due to the plastification. These rotations are necessary to reach the desired internal force distribution and load capacity. Therefore, it is very important to know the capacity and demand of rotations. A moment−rotation curve is shown in Figure 3.6 with the illustration of available rotation at certain moment levels. This determined as the difference between the rotation on descending part of the curve and the elastic rotation at the same moment level. These curves are required solely for composite sections under negative bending, since in the sagging region almost the whole steel section is under tension.

Figure 3.6: Available rotation [McConelli et al., 2010]. Researchers derived moment−rotation (M−θ) characteristic curves based on experimental results. Then FEM models calibrated with the measured data were used to extend the application limits. This way the rotation capacity of plate girders’ cross-sections have been determined. In Figure 3.7 a standardized M−θ curve can be seen, θRL and Mn are the rotation capacity and nominal resistance of the cross section, for the determination of these values see the detailed introduction of the rotation compatibility approach in the next section.

Figure 3.7: Moment versus plastic rotation model [McConelli et al., 2010]. 33

Review of Previous Investigations Two M−θ models are presented herein, both can be used to determine the rotation capacity of cross-sections in any class with some limitations. Since in this thesis results obtained by using the philosophy of the US and European standards will be used as well, it is necessary to compare some part of it. The cross-section classes according to the AASHTO and Eurocode are illustrated in Figure 3.8.

Figure 3.8: Cross-section classes [Gupta, 2006]. The resistances Mp, My, Mr are the plastic, elastic and reduced elastic − due to the local stability loss − cross-section resistances, respectively. The M−θ curves applied in the rotational compatibility approach, presented at the end of the chapter, proposed by Righman and Barth (2005) are applicable for rather wide range of sections including slender ones as well. In case of non-compact and slender sections the nominal moment capacity (Figure 3.7) is the elastic resistance. [Lääne and Lebet, 2005] investigated the rotation capacity of slender composite plate girders under negative moment. They derived a rotation capacity−plate slenderness curve (Figure 3.9) based on experimental and FEM models with cross-sections in Class 4. Considering 63 mrad required rotation as sufficient amount for plastic design and as the limit for Class 1, the authors extrapolated the data for the other section classes. Experiments with sections in Class

3 and 2 were verified this extension. θav and  p' are the rotation capacity and modified plate slenderness, for the determination of these values see the detailed introduction of Lebet’s method in the next section. It is important to note that in case of slender (Class 4 and 3) sections the reference moment which can be maintained during the development of plastic rotations is 0,9  M el , Rd , 0,9 takes into account the effect of the sequence of the construction in a simplified manner.

34

Review of Previous Investigations

Figure 3.9: The minimum available rotation capacity (Eq.(3.13)) vs. modified plate slenderness (Eq.(3.14)) [Lääne and Lebet, 2005].

3.3.

Worked-out Design Methods

3.3.1. United States

The first standard, which allowed the yield of the material of bridge structures was the AASHTO 1973. It contained two provisions to take into account the plastic reserve. First, the moment resistance of the compact cross-sections was increased to the plastic strength. Second, a limited 10% redistribution of hogging moment to the sagging region was allowed. 1986 is the year of the introduction of the first comprehensive inelastic design procedures, called autostress design. This is utilizing enhanced limit states which allow for inelastic load distribution for continuous structures. Under the autostress approach, the bridge is overloaded by an initial live loading of the structure. This overload has a prestressing effect on the bridge inducing stresses over the yield point in the negative moment region and relieving some residual stresses. The name autostress is derived from automatic load redistribution which occurs. This was applicable only for highway bridges constructed with compact sections. It was primary intended to eliminate the need for: (1) additional cover plates on rolled beam sections and (2) multiple flange thickness transitions in welded beams. Inelastic rating procedures were proposed in 1993 by Galambos et al. (Inelastic rating procedures for steel beam and girder bridges), this defines the strength limit state either as the incremental collapse or as the specified maximum permanent deflection. Shilling et al. (1996) recognized that this approach is too complicated to apply in everyday design and did not apply to all possible cross-section configurations. Therefore, they developed a simplified inelastic design procedure based on the shakedown limit state. Later essentially the main direction of the research was to provide the engineers an accurate and easy-to-use method. Thanks to this process the methods are more accurate and the limitations are reduced, the field of applicability is significantly extended [Barth and White, 2000].

35

Review of Previous Investigations The most recent design procedure is the rotational compatibility approach [McConelli et al., 2010], which can be found in the current AASHTO appendix as an alternative inelastic design method for steel and composite bridges. Basically, this and other standardized methods are simplified shakedown approaches, which based on the shakedown as ultimate limit state reduce the design to a moment redistribution. The differences are lying in the accuracy, the number of assumptions and consequently the limitations. In the following this new rotation compatibility approach for inelastic design of steel girder bridges is introduced based on the work of [McConelli et al., 2010]. Rotation compatibility approach The basic idea behind it is that the available rotation must be greater than the required one. Scope of application: -

-

-

straight, continuous steel or composite girder bridges with I-section; the moment is redistributed from the interior-pier section; not skewed more than 10 degrees from radial; cross-sections throughout the unbraced lengths immediately adjacent to interior-pier sections from which moments are redistributed shall have a specified minimum yield strength not exceeding 480 MPa (70ksi); holes or staggered cross-frames shall not be placed within the tension flange over a distance of two times the web depth on either side of the interior-pier sections from which moments are redistributed; rule for holes in tension flange of other sections can be found in AASHTO 2010. Article 6.10.1.8; the length over which the girder is exempt from satisfying the elastic strength requirements equals to one unbraced length on each side of the pier; there are no section transitions or longitudinal stiffeners within this exempt region; shear and bearing stiffness requirements, these are applicable to all I-girders.

The main improvement of this new rotation compatibility approach compared to the previous methods that it contains no limitations to the cross-section or compression flange bracing outside the requirements for cross-section without longitudinal stiffeners (6.10.2 section in AASHTO). Thus, this method is applicable for wider range of bridges. These 6.10.2 requirements are as follows [AASHTO, 2010]: – for the web

D  150 tw

(3.2)

where D is the depth and tw is the thickness of the web, respectively.

36

Review of Previous Investigations – for the compression and tension flanges: bf

2t f bf 

 12, 0

(3.3)

D 6

(3.4)

t f  1,1 tw 0,1 

I yc I yt

(3.5)

 10

(3.6)

Where bf and tf are the width and thickness of the flange, respectively, and Iyc and Iyt are the moment of inertia of the compression and tension flanges about the vertical axis in the plane of the web, respectively. The following flowchart (Figure 3.10) shows the design procedure of a continuous bridge using the rotation compatibility specifications. The explanation of the variables can be found after the flowchart and in Figure 3.11.

37

Review of Previous Investigations

Preliminary design or existing structure.

Check the applicability conditions.

No

All fulfilled?

Yes

Conventional linear elastic analysis, Mu and Mn determined in the same manner as for elastic design.

- Perform an elastic analysis of the structure to determine the maximal internal forces (Mu,). - Determine section properties, resistances (Mn).

RL

Calculate RL using Eq.(3.7). Determine CR using Eq.(3.9) or Eq.(3.10)

CR

Negative bending region check  RL C R redistributed moment ratio.

M u  (1   RL C R )  M n

No

Check this section for the other requirements (service, fatigue, constructibility).

No

available rotation capacity, necessary to determine the required rotation.

The same procedures as in the elastic design.

All fulfilled?

Yes

Positive bending region check (Figure 3.11).

M u  M rdL  ( x L)  (M rdR  M rdL )  M n

No

Yes Check this section for the other requirements (service, fatigue, constructibility).

No

All fulfilled?

The same procedures as in the elastic design.

Yes

End

Figure 3.10: Flowchart of the rotation compatibility approach.

38

Review of Previous Investigations As appears from the flowchart this method applies the lower bound (Melan’s) theorem of shakedown. The applied loads are identical to those used in elastic methods. This method does not take into account transverse redundancy. Explanation of the symbols used in negative bending moment check: Mu Mn

moment due to factored load (based on elastic analysis); nominal moment capacity;

Mrd

redistribution moment, M rd  Mu  M n ;

MrdL MrdR

redistribution moment at left end of a span; redistribution moment at right end of a span;

The calculation of Mn can be found in AASHTO 6.10.7.1.2 and detailed below. These variables are illustrated in Figure 3.11.

u L

rdR rdR rdL rdL

rdL

x

rd rdR x rd(x) = rdL+(rdR -rdL)·L -n

urd

+n

Figure 3.11: Illustration of the rotation capacity method in respect to internal forces.

39

Review of Previous Investigations The available plastic rotation:

 RL  128  143  (b fc t fc )  Fyc E  43.2  Dcp b fc  48.2  (b fc t fc ) 

D

cp

(3.7)

b fc   Fyc E  max  0, 0.5  Lb rt   30    5

where: bfc, tfc Fyc E Dcp Lb rt

width and thickness of compression flange, respectively; compression flange yield strength; modulus of elasticity; web depth in compression when the moment is equal to the plastic moment capacity; distance between compression flange bracing locations; radius of gyration of the portion of the cross-section in compression about the vertical axis.

In new design, it is convenient to take the last term in the equation equal to zero. Since that term takes into account the effect of the bracing distance. This way it can be seen if it is possible to redistribute the required amount of moment with the given cross-section. The required rotation capacity calculated the following way:

 pR  C R  ( M rd / M u ) C R  80  20  n  mrads

(3.8)

for homogenous pier sections

C R  90  22,5  n  mrads 

for hybrid pier sections 6

(3.9) (3.10)

where n   0 1 2 is the number of adjacent interior piers to the examined pier where moment redistribution is used. The nominal moment capacity for compact sections is as follows: If Dp  0,1 Dt

Mn  M p

(3.11)

Dp   M n  M p   1, 07  0, 7   Dt  

(3.12)

otherwise:

6

Hybrid means that the steel section is made up from different class of steels, e.g., S355 flange, S275 web.

40

Review of Previous Investigations where: Dp Dt Mp

distance from the top of the concrete deck to the neutral axis of the composite section at the plastic moment; total depth of the composite section; plastic moment capacity of the cross-section.

Other specifications for various cases can be found in the standard, but essentially it can be seen that in case of compact sections the section capacity should be taken as the plastic resistance. Noncompact sections In case of noncompact sections since stress verification is prescribed the capacity can be taken as the elastic resistance. 3.3.2. Contributions From Outside of the US

It is interesting that a method with very similar concept as the rotational capacity approach was developed independently by Lebet [Lebet, 2011]. Both based on the comparison of the required and available rotation of the cross-sections. The main difference is that this based on the plastic collapse limit state, considering the limited rotation capacity of the section. Since it uses the default live load level with 1000-year7 return period [EN 1991-2 Table 2.1] it seems to be reasonable. Nevertheless, the reliability of this method should be verified pondering subsequent loadings as well. In Lebet’s method, span sections can be loaded up to the plastic bending resistance and the intermediate support regions up to the limit of the rotation capacity, θav, that they can offer. This rotation yields to a moment redistribution. The method is worked-out for slender (Class 3 and 4) sections which are typical in case of composite plate girder bridges. The available rotation as mentioned above in Section 3.2.5 was derived based on experimental and FEM models by Lääne and Lebet (2005). The curve presented in Figure 3.9 can be described by the following formula:  15, 75 2 cv   av  min    p'    63 mrad

(3.13)

 h 1, 05 f y   2     p if   0,5  2   w  t E k  w with  p'    hw 1, 05 f y   p if   0,5 t  E k  w

(3.14)

7

Corresponds to ~10% probability of exceedence in 100 years, using extreme distribution. “Briefly, the value of the return period has been selected in order to limit the probability for any irreversible limit state to be exceeded during the period of reference and it is rational to think that the loads will increase in the future.” [Calgaro et al., 2010]

41

Review of Previous Investigations where: 

relative position of the plastic neutral axis, zpl to the bottom flange, to the web height,   z pl hw ;

cv

coefficient to take into account the effect of shear force according to Eq.(3.15);

 p'

modified relative plate slenderness;

p

relative plate slenderness;

k

local buckling coefficient;

fy

yield strength of the structural steel;

E

modulus of elasticity of the steel;

hw, tw height thickness of the web, respectively.

1,3 if VEd VRd  0, 7   cv   1, 0 if 0, 7  VEd VRd  0,8 method  not  applicable if VEd VRd  0,8 

(3.15)

Another applicability condition beside the maximum shear force is the maximum compression flange bracing distance in the negative hinge region, expressed by Eq.(3.16), where ic denotes the radius of gyration. LD  0, 2    ic 

E fy

(3.16)

The overview of the method can be seen in the following flowchart (Figure 3.12). The flowchart shows solely the check when the traffic load in a position to induce maximum negative moments. The loading corresponding to the maximum positive moment should be checked conventionally, using the plastic resistance.

42

Review of Previous Investigations

Preliminary design or existing structure.

- Perform an elastic analysis of the structure to determine the maximal internal forces (MEd,). - Determine section properties, resistances (MRd).

Check the applicability conditions for max shear Eq.(3.15) and for bracing Eq.(3.16)

All fulfilled?

No

Yes

For the pier-section.

Calculate the reference bending moment, M ref  0, 9  M el , Rd

Check the maximum moment redistribution max=0,3

 M ref  (1, 0   max )  M Ed

No

Yes 

M ar , Ed Capacity

Demand

Determine the redistribution ratio in the

Calculate the modified plate slenderness,

p

'

- moment after

redistribution, Figure 3.11



Eq.(3.14)

hogging area:  

→θreq,1

No Determine the rotation capacity, θav Eq.(3.13) Figure 3.9

Figure 3.13 left chart

Decrease 



M Ed  M ar , Ed 

M Ed

 av   req ,1  req , 2  max  

0

based on θav=θreq pier-section ‘plastification’

 req ,1   av

No

midspan-section plastification

Increase Mel,Rd or decrease 

Yes Plastic moment utilization θreq→ from charts, θav = θreq = θreq,1+ θreq,2

Check the positive region Eq.(3.17) and Eq.(3.18)

End

Yes

All fulfilled?

No

Increase Mpl,Rd or decrease 

Figure 3.12: Flowchart of the Swiss method. 43

Review of Previous Investigations The mentioned charts to determine the required rotation and plastification factor for various spans are available in [Lebet and Nissile, 2010]. It is the designer’s choice whether he or she wants to utilize the negative or positive rotation capacity (plastic reserve) or both of them. The flowchart presented in Figure 3.12 shows the design process when the moment is redistributed form the interior support to span. Moreover, the plastification of the span-section generates additional rotation demand at the pier-section (θreq,2). The positive bending check contains the check of the cross-section to Eq.(3.17) and for the maximal moment Eq.(3.18)

M ar , Ed  M el , Ed  M r    M pl , Rd

(3.17)

where:

M ar , Ed

the bending moment at midspan after redistribution for the max negative

loading;

M el , Ed

bending moment determined by elastic analysis at midspan for the max negative

loading; M r

residual moment at midspan.  M Ed  M pl , Rd

(3.18)

The 0,9 multiplier approximately takes into account the effect of the construction method; applicable when the designer would like to avoid the tedious calculation related to the accurate consideration of it. The maximum moment redistribution ratio (max) is limited to 0,3, this ensures that the structure remain elastic under the service load (characteristic combination). In comparison, the AASHTO procedure does not contain such a limitation. req,1 is the required rotation for the redistribution of the moment from support to span, and req,2 is the required rotation due the plasticization in span. These values are illustrated in Figure 3.13 as the function of the span and the amount of redistribution.

44

Review of Previous Investigations  Q·Q·Qk

 Q· Q·Qk  Q·Q·qk

 Q·Q·qk

 G·gc,k  G·ga,k

 G·gc,k  G·ga,k M -Ed

M -Ed

M ref

M -r,Ed

M ref M el,Rd M pl,Rd

M+r,Ed

 req,1

 req,2  pl,span

 req,1mrad  

 req,2mrad

Two-span bridge



Continuous bridge



 = 0.95  = 0.85  = 0.75

 

 = 0.3  = 0.2  = 0.1



 = 0.90  = 0.80  = 0.70

 





 

Plastification ratio





lspan [m]

















 

lspan [m]



















Figure 3.13: Rotation requirements of the interior pier cross-section [Lebet, 2011]. Lebet (2011) also provides a formula to take into account the contribution of an appropriately placed longitudinal stiffener to the rotation capacity of the cross-section. In contrast, the rotation compatibility approach in AASHTO is applicable only for sections without longitudinal stiffeners. The stiffener should be placed in an optimal distance, as illustrated in Figure 3.14, from the bottom flange, h1  0,1  0,3  hw .

Figure 3.14: Location of the additional longitudinal stiffener [Lebet, 2011].

45

Review of Previous Investigations With the application of the stiffener the available rotation can be increased with the following term:

av,sup  40  45,5 

VEd  mrad VRd

(3.19)

It appears to be rather arbitrary since does not contain any information about the stiffener.

46

Bridge Design Problem

4.

Bridge Design Problem

4.1.

Problem Statement

As illustrated in previous sections the plastic reserve of bridges is exploitable and significant compared to the elastic limit. Nevertheless, standards for plastic design of bridges are solely available in the US. In Europe some progress can be seen in this field, however still not on a standardized level. Among the Hungarian bridge engineers the elastic methods are solely applied as well. Based on the American and European results the aim of this thesis is to investigate the possibility of the inelastic design of bridge structures in the philosophy of Eurocode. For this purpose, a typical three-span composite plate girder bridge is chosen to be studied. This highway bridge is located on the M6-M0 motorways in Hungary and designed by Speciálterv Ltd. according to the Hungarian standard (MSZ Út), following elastic methods.

4.2.

Introduction of the Bridge to be Studied

The bridge is located on the M0-M6 highways in Hungary, denoted as 142/k. The flyover carries a 3-lane single carriageway highway road over another road. It is a continuous steelconcrete composite bridge formed by three spans of 30,0 - 40,0 - 30,0 m (Figure 4.1) and with a 13,47 m wide deck. The cross-section is composed of two constant depth I-girders with a reinforced concrete slab on top of them, in total about 1,85 m height, illustrated in Figure 4.2. The distance between the main girders is 7,5 m, they connected in a 5,0 m raster with a crossbracing formed of rolled HEA200 sections. The deck is haunched at the top of the girders, its average thickness is around 28 cm.

Figure 4.1: Elevation of the 142/k bridge.

Figure 4.2: Cross-section at region of the internal supports.

47

Bridge Design Problem The bridge is designed to MSZ Út Hungarian pre-Eurocode national standard. The steel girders are symmetrical; their geometry is presented in Figure 4.3; the length and color of each plate are proportional to their real length and thickness, respectively. The asymmetry is coming from the construction process. The temporary supports were placed asymmetrically due to geometric restrictions. The strength class of the materials: – – –

structural steel: S355 concrete: C30/37 reinforcement: S500B

Figure 4.3: Layout of the bare steel girder.

4.3.

Solution Strategy, Extent of the Thesis

The following tasks were proposed to complete: – Calculation and comparison of the ultimate load level − assuming sufficient rotation capacity in the plastic hinge region to reach the desired moment redistribution − for: – first yield; – first plastic hinge; – single girder shakedown; – system shakedown; – single girder plastic collapse; – system plastic collapse. – Study the effect of the shear force on the above-mentioned ULSs. – Evaluation the reliability of the examined ultimate limit states. – Investigation the rotation capacity of the critical cross-sections, structural solutions to increase the ductility. – Redesign of the structure and investigation of the economic aspects of it.

Based on the American results presented in Chapter 3 the plastic capacity of composite plate girders appears an interesting and promising way to obtain structures that are more economical. Due to the limited available time, this thesis lays emphasis mainly on the investigation of the theoretically available reserve of a composite girder bridge. I am aware that there are plenty of other questions to answer, like the residual deflections, possible mechanisms, limited rotation capacity, etc. These will be mentioned briefly in the related sections and summarized in the further work chapter. Since the topic is currently an intensively researched area some of these questions are still not yet fully answered.

48

Global Structural Analysis

5.

Global Structural Analysis

The aim of this chapter is to perform the global structural analysis and check the selected bridge using the Eurocodes and to investigate its plastic capacities. It was designed according to the Hungarian standard (MSZ); the details of the bridge are given in Section 4.2. The most important difference between the MSZ and EC standard is that the former adopts the allowable stress method and limit state concept as well, while the European Norm solely applies the limit state concept. For this particular bridge the source of the difference is that the structure was designed by the allowable stress method. Another important dissimilarity is the level of traffic load, which is about 15% lower for this road line in MSZ (load class “A”). The following considerations were taken into account during the analysis: –

Where the standard offers the designer options (like how accurately consider the effective widths for shear-lag), I always chose the possibility which was closer to the original design, to establish a more or less solid base to the comparison. Nevertheless, even keeping this in mind due to the sometimes significant differences, the results are representing rather a qualitative than quantitative comparison. – Where the Eurocode is used the relevant values are taken as the default recommended ones (not considering the NAs). – Since our bridge is a flyover (overpass) which does not serve for pedestrian traffic and in the original design it was also neglected the pedestrian, cycle load on the sidewalks not considered to occur simultaneously with the governing traffic load. Moreover, with disregarding the abnormal loads which require permission, only LM1 load is considered. Solely the gr1a load group is taken into account in the calculation as traffic action.

5.1.

Elastic Check According to Eurocode

The detailed calculation can be found in Annex A. The documentation is organized in such a way that it can be read independently from this chapter, this eventuates some redundancy. 5.1.1. Finite Element Model

The analysis was carried out in a FEM program called midas Civil8. The aim was to choose a similar modeling level to the original design, to establish a profound base of comparison. Single-girder beam model was used in the original calculation, since it is a relatively simple system which does not demand higher modeling level. According to this a grillage model is used to perform the verification. It can be considered equivalent to the simple, single beam model applied by the designer. The only difference is the lateral load distribution, which is an inherent property of the grillage model and can take into account automatically the 8

South-Korean developed FEM software, part of the midas IT’s software package with various analysis and design features mainly for bridge engineering purposes.

49

Global Structural Analysis longitudinal change of that function. By the way, this does not make too much difference since the grillage model verified the lateral distribution used in the simple model. About the model: – – – – – – –

Grillage model was chosen, which build up from the main girders, concrete deck, and cross bracings. Every structural element is modeled as beam element following the Bernoulli-Navier beam theory9. The elements are inserted with eccentricity in respect to their centroid, in order to reach the same level for the top surface of the concrete slab. The main girders are beam elements composed of a steel and concrete part, the latter modeled with its effective width. The shear studs are modeled as perfectly rigid connections. The fix supports are rigid restraints in respect of the “fixed” degrees of freedom, the hinges are perfect hinges without rotational stiffness. The beams representing the deck are placed at 2,5m distance in longitudinal direction.

The grillage model is illustrated in Figure 5.1.

Figure 5.1: Illustration of the grillage model. The slab’s reinforcement is considered for the calculation of element’s stiffness. Due to the chosen modeling level the shear lag is taken into account “manually”. The cracking of the concrete is contemplated in a simplified manner, by neglecting it at both sides of internal piers in 15% of each span’s length. The smoothness of the finite element mesh is adequate by inspection, there is a beam element between every deck-beam element. Therefore, no needs for convergence check. The global model was verified against various partial models, like simple hand-calculations of a continuous beam with constant flexural stiffness and other FEM-model, using a single beam model for particular stages of construction. The results are in good agreement.

9

The analysis was performed by using Timoshenko elements as well, which takes into account the work done by the shear forces. The results differed less than 0,1%, hence the simpler element was used for the further analyses.

50

Global Structural Analysis Further details about the calculations can be found on the attached disc storage, it contains the model files and matlab m-files (executable, input files) as well. 5.1.2. Loads and Load Combinations

Actions taken into account: – – – – – – –

Dead loads (G); Shrinkage (S); Creep (S); Traffic loads (Lt); Wind load, with and without traffic; Thermal actions (T); Construction loads.

The live load’s adjustment factor:  qi   Qi  1, 0 , this corresponds to a road for which a heavy industrial international traffic is expected. The traffic load is placed in the most unfavorable positions using the Moving Load Analysis feature of the software, which loads the structure based on its influence lines. The consideration of time-dependent effect such as shrinkage and creep are essential for composite structures and also complicates the analysis. The software is capable of contemplate the course of time as an additional dimension. These effects were taken into account by using the software’s special feature, which divides the design life into time intervals and calculating the time-dependent kinematic loads in every step and adding to the previous ones while reflecting the effect of boundary, element and/or load changes. The convergence check of the time-dependent effect calculation was carried out by modifying the internal step size and convergence limit. The time-course of shrinkage and creep are taken as recommended in EN 1992-1:2004. These functions depend on many factors like the type of cement, age at first loading, relative humidity, etc. The following considerations are taken: –

The loads from shrinkage are calculated with using the concrete area of the composite section. Since the area of the slab is reduced to take into account the effect of shearlag, the internal forces from the shrinkage are multiplied by 1,15 to approximate its real effect. The value was determined to get an upper bound on the primary and secondary effects in every section. The construction stages were also considered in the same model. During the construction the structure was propped by one shoring at each span. The additional dimension is great help, since there is no need to build numerous models for different construction stages and the effect of varying boundary conditions and loads can hardly be followed by “hand” using the common Fritz-method.

More comments and details on these loads can be found in the documentation of the static calculation, Annex A. Combination factors are summarized in Table 5.1. They highlighted here because they will be used in the reliability analysis as well.

51

Global Structural Analysis Table 5.1: Combination factors.

traffic loads

TS ULS Fwk persistent design situation execution * Fw

wind forces

thermal actions 3)

Tk







0,75 0,4

0,75 0,4

0 0

0,6 0,8 1,0

0,2 0

0 0

0,63)

0,6

0,5

in most cases it may be reduced to 0 in ultimate limit states EQU, STR and GEO

According to EN 1994-2:2005 5.4.2.5 (2) Temperature effects may normally be neglected in analysis for the ultimate limit states other than fatigue, for composite members where all cross-sections are in Class 1 or Class 2 and in which no allowance for lateral-torsional buckling is necessary.

The load combinations contemplated in the design presented in Table 5.2. Table 5.2: Post-construction stage load combinations. Permanent

Variable Traffic

Dead load

Shrinkage

Creep

LC1

1,1475

1,150

1,000

LC2 LC3 LC4 LC5 LC6 LC7

1,350 1,1475 1,350 1,1475 1,1475 1,1475

1,150 1,150 1,150 1,150 1,150 1,150

LC8

1,350

1,150

1 2

1

Thermal

1

Wind

Top warmer

Bottom warmer

With traffic2

Without traffic2

1,350 1,350

0,900

-

-

-

1,000 1,000 1,000 1,000 1,000 1,000

1,013 1,350 1,013 1,013 1,013 1,350

0,540 1,350 0,540 0,540 0,540 1,350

0,900 1,500 -

0,900 0,900 1,500 -

1,500

-

1,000

-

-

-

-

-

1,500

TS

UDL

contains the primary and secondary effects as well contains the left and right side winds as well in separate combinations

The shrinkage’s partial factor is 1,0; the 1,15 multiplier compensates the reduced concrete flange due to shear-lag. The governing combination for maximum positive bending is LC1 and for maximum negative bending is LC3. The verification is focused on the ultimate limit states. As in the original design, for the elastic check to EC also the first yield was used as the limit state. The corresponding rating and utilization factors are presented in Table 5.3, with the plastic limit states’ values.

52

Global Structural Analysis

5.2.

Investigation of Plastic Reserves

The source of the plastic reserve is in one hand the plastic capacity of the cross-section over the elastic resistance and on the other hand, the plastic reserve of the structure due to its redundancy. Every structure has the first type cross-sectional reserve, which can be described with the shape factor (c) Eq.(5.1).

c

Wpl Wel

(5.1)

where: Wpl

plastic section modulus;

Wel

elastic section modulus.

The second type of reserve is only exploitable in case of globally statically indeterminate structures. In most cases the higher degree of redundancy yields to higher reserve. The restrain of the steel by the concrete and vice versa is established by mechanical connections, this can be considered as the source of local statically indeterminacy. The strains cannot develop freely in the concrete part. This induces self-equilibrated stresses at the level of cross-section. The Eurocode calls this effect primary, and the internal forces form the statically indeterminacy (global) are called secondary effect. This classification is not universally accepted, in South Korea − where the midas Civil was developed − both, the cross-section level and global effects are called secondary, and the fictitious axial force is the primary. In my judgment, the local and global denotations are more convenient, firstly because these words contain more information about the actual effects, secondary due to that the words also refer to the way how to relieve these effects. Reduction of local indeterminacy for example by plastification leads to the relieve of the self-equilibrated stresses locked into the structure by the particular restrain. The same is true for the global properties, the release of the global statically indeterminacy relieves the global residual forces. These are summed up in Figure 5.2.

53

Global Structural Analysis

First hinge

Elastic

Shakedown, plastic collapse

static static static plastification plastification plastification indeterminacy indeterminacy indeterminacy local global self-equilibrated forces local







relieved



relieved

relieved

global

Figure 5.2: Static indeterminacy and self-equilibrated forces induced by shrinkage, creep and/or nonlinear temperature difference in respect of various limit states. The nonlinear temperature difference induces primary (local) stresses due to the difference of the heat-transfer coefficient and/or thermal inertia of the build-up materials. The moment diagrams necessary to evaluate the limit-states in the following sections are presented in Figure 5.3 for t=100 day at the opening time to traffic and in Figure 5.4 for t=100 years at the end of the design life. The G, Lt, T and S letter indicates the effect of dead loads, traffic loads, thermal actions and shrinkage and creep, respectively. The maximum positive bending moment occurs at the opening to the traffic, when the global internal forces from shrinkage are lower. According to this, the maximal negative bending moment develops at the end of the design life.

54

Global Structural Analysis

12023

11778

[kNm] G

8095 7236

7325

Lt 9609

T 2571

2601

2616

1837

1853

1867

S

Figure 5.3: Moment diagrams for one girder at the day of opening to traffic (t=100 days); for maximal positive moment. 3142

3135

3124

[kNm] T

4276

4310

4337

S

Figure 5.4: Moment diagrams for one girder (t=100 years), the MG and MLt diagrams are identical; for maximal negative moment. In the following, the term of rating factor (RF) and utilization factor (UF) will be used frequently to measure the performance of the bridge. RF - rating factor, the multiplier applied to the live load to reach the particular limit state; UF - utilization factor, the ratio of the demand and capacity; The next subsections give details about the particular calculations, the final section summarizes and compares the results.

55

Global Structural Analysis 5.2.1. First-Hinge Limit Analysis

The first hinge resistance is obtained by increasing the live load till the formulation of the first plastic hinge, till reaching the plastic resistance of one section. The basic design inequality Eq.(2.26) for this particular limit state as follows:

 G  G   S  S   Qt  Lt  0   Q  T      E E

R pl

R

(5.2)

From this, the multiplier of the traffic load to reach the particular limit state: RF 

R pl  R   G  G   S  S  0   Q  T

 Qt  Lt

(5.3)

The variable S consists of the effect of shrinkage and creep. For this limit state this variable contains only the global (secondary) effect of these phenomena, because simultaneously with the plastification of the cross-section the local residual stresses vanish (Figure 5.2). The critical section for this limit state is the middle point of the internal span, at t=100 days. Using Eq.(5.3) the rating factor is the following (Annex A):

RF 

36401 1, 0  1,148  8095  1,0  (1853)  0, 6 1,5  2601  2, 053 1,35  9609

The utilization factor is critical in section over the pier under negative moment. This is due to that the traffic and dead load induces maximum moments in different sections. 5.2.2. Shakedown Limit Analysis

As mentioned earlier (Section 2.1.2) in case of typical bridges the incremental collapse failure mode is governing, since the alternating plasticity can be verified by inspection its numerical check is omitted. The simplified formula derived from Koiter’s theorem (Section 2.1.2) is used to calculate the rating factor (RF). It should be noted that the obtained load factor is always greater than or equal to the true load factor. Only if the true collapse mechanism has found − where the static admissibility is fulfilled and the internal forces do not violate the plastic criterion − provides the formula the true value. In every other case we are not on the safe side. For this bridge there is no difference in the ultimate load capacity between the single-girder and system shakedown limit states. Because the bridge’s girders are symmetrical and their moment envelopes are identical. It can be seen from the rating factor formula too, where the nominator and denominator are increased by two, Eq.(5.6). If we rigorously examine the question there is a difference, since the same loading is necessary on both girders to induce their incremental collapse. This requires let’s say 10 subsequent loading cycles on each girder. This means that for the system incremental collapse twice as many load crossings are requisite, which has lower probability of occurrence. Therefore, there is a difference on the

56

Global Structural Analysis ultimate load if we compare their values corresponding to the same failure probability. This is neglected in this study. In this case the critical mechanism is likely to be the one illustrated in Figure 5.5. Since the hinges assumed at locations where most utilized cross-sections are and the structure is more or less symmetrical, so it is quite safe to take that mechanism as critical. If the side span sections are more exploited, for example we have a two span bridge, the choose of the correct mechanism is not so straightforward. The positive hinge is expected to being formed close to the corresponding plastic collapse hinge (see in the next section).

1



2



3 



· Figure 5.5: The assumed critical mechanism. In case of global plastification the stresses from time-dependent actions are relieved. This considerably simplifies the calculation. It should be noted that this is valid only for the calculation of ultimate load. The basic design equation for this limit state using Koiter’s theorem as follows:  max  max   G   G1    G2    G3      Qt   Lmax t ,1    Lt ,2    Lt ,3   



RF 

1

R

  R pl ,1     R pl ,2     R pl ,3    

1  R   R pl ,1    R pl ,2    R pl ,3       G   G1    G2    G3       max max  Qt   Lmax t ,1    Lt ,2    Lt ,3   

(5.4)

(5.5)

Substituting the values into the formula (5.5), the rating factor: RF 

1 1, 0   46423  (1)  36401  (2)  46112  (1)   1,148  12023  (1)  8095  (2)  11778  (1)  1, 35   7325  (1)  9609  (2)  7236  (1) 

 2, 626

The rating factor was determined by Melan’s theorem as well (Section 2.1.2). The basic concept is that according to the lower bound theorem we are searching for the largest load factor for which a favorable residual force field still can be found. The favorable means that adding the residual force field to the maximal forces from external loads − determined on the elastic structure − the plastic limits are not violated. This is an optimization problem which in this case with the particular assumptions shrinks to a linear programming problem. It can be solved by any computational software or even by hand for small tasks. For this example the Matlab program was used, as expected it yielded to the same result as the kinematic theorem.

57

Global Structural Analysis For the system shakedown limit the rating factor can be calculated the following way:

RF 

1 R 

girders

R i

pl ,1



 i   i R pl ,2  i   i R pl ,3  i    G 

i 1

 Qt 

girders

L i

max t ,1

  L i



i

max t ,2

girders

G i

i



1

 i G2  i   i G3  i 

i 1

  L i



i

max t ,3



i 1

i







(5.6)

The summations to the number of girders are partitioned in such a way to show that the calculation is equivalent to a single-girder analysis where the corresponding resistances and effects are summed up. As has already mentioned this gives the same rating factor as the single-girder incremental collapse analysis. 5.2.3. Single-Girder Plastic Collapse Limit Analysis

The following two limit states (single girder and system plastic collapses) are calculated solely to grasp the ultimate capacity of the structure. As mentioned in Section 2.1.2 and illustrated in Figure 2.10, since the moving load is significant, the incremental collapse failure would occur before the plastic collapse. The same collapse mechanism is assumed as in case of the shakedown analysis.







 G  gtot 2     lm 2  lm 2    Qt  Qsingle     lm 2   qsingle     lm 2  lm 2  

RF 

1

R

(5.7)

  R pl ,1     R pl ,2    R pl ,3    



1  R   R pl ,1     R pl ,2     R pl ,3       G  gtot 2     lm 2  lm 2 



 Qt  Qsingle    lm 2   qsingle    lm 2  lm 2  









(5.8)

With the actual values the rating factor (see Annex A): RF 

1 1, 0   46423  (1)  36401  (2)  46112  (1)   1,148   201 2  1  40 2  40 2   1, 35   873, 3  1  40 2   32, 66  1  40 2  40 2  

 2, 900

This value was also checked with its dual theorem, the results are in good agreement. There is about 2% difference between the values. It is addressed to the facts that in the kinematic theorem the loads were calculated by simply using the midspan lateral load distribution value and it cannot reflect the effect of construction sequence. The latter is detailed in Section 5.5. Comments on the mechanism for plastic collapse: For beams with constant negative and positive ultimate resistance along their length and loaded with a concentrated force (Q) and with uniformly distributed load (q) the critical mechanism can be determined by applying the kinematic theorem for a general mechanism and searching the maxim value of the load.

58

Global Structural Analysis



l

 ·l



 ·( l l

Figure 5.6: General collapse mechanism. Using the kinematic theorem: (l   )   1 (l   )   (l   )    l   q     Q  M R     M R    M R    l 2 l l l

With introducing the ratio of the total distributed load and concentrated force (rq) and the ratio of negative and positive resistances (rR), moreover with isolation of the expression for q, the following function can be obtained:

rq 

M Q ; rR  R q l MR

  M R   1  rR   l  q ( )   l       1  rq  This q() function takes its minimum value at point:



1  rR  1 l rR

(5.9)

With a similar calculation it can be shown that for an internal span the critical positive hingelocation is in midspan, as expected. This is not only independent of the load ratio but of the resistance ratio as well. For shakedown the same principle could be applied, however since that requires the elastic moment envelopes, the calculation is more complicated. 5.2.4. System Plastic Collapse Limit Analysis

The global mechanism corresponding to the system plastic collapse is illustrated in Figure 5.7. 32 22 12 11

21





31

·

Figure 5.7: System collapse mechanism. 59

Global Structural Analysis The rating factor can be determined by applying the kinematic theorem to every girder participates in the mechanism. RF 



1  R   R pl ,1     R pl ,2     R pl ,3       G  gtot     lm 2  lm 2 



 Qt  Qtot     lm 2   qtot     lm 2  lm 2 





(5.10)

With the actual load values: RF 

2  1 1, 0   46423  (1)  36401  (2)  46112  (1)   1,148   201  1  40 2  40 2   1, 35  1200  1  40 2   47  1  40 2  40 2  

 4,137

For this limit state the ~2% difference also observed between the static and kinematic theorem, the summarizing tables in Section 5.5 contain the lower values, which is in this case the 4,052 obtained by using the static method.

5.3.

Summarization of the Rating Factors

The results of the previous sections’ calculation are summed up in Table 5.3. The rating factors denote the multiplier of the traffic load required to reach the particular limit states. It can be seen from the table that with allowing the formulation of the first plastic hinge ~37% traffic load increase can be achieved over the first yield. By adopting the incremental collapse limit state further ~28% increase can be mobilized over the first plastic hinge. The table also comprises the plastic collapse limit states; these correspond to one load application and advantageous to access the true capacity of the structure. The utilization factor denotes the ratio of the effect and resistance at the most loaded section; therefore, calculated only in cases where the ultimate limit state corresponds to one cross-section. The inverse of the UF for the first plastic hinge expresses the shape factor. These values are determined considering the effect of creep, shrinkage and thermal actions as well, where relevant. Table 5.3: Rating and utilization factors for various limit states. RF MSZ Út

UF

-

1/UF

0,926 h

1

1,080

RF

UF

base

base

RF

Eurocode elastic

1,496

h 0,863

h

1,159

EN first plastic hinge

2,053

s

h

1,508

EN single girder shakedown EN system shakedown EN single girder collapse

2,626 2,626 2,900

-

-

75,53% 75,53% 93,85%

-

27,94% 27,94% 41,28%

EN system collapse

4,052

-

-

170,9%

-

97,41%

0,663

37,21% 30,17%

base

1

the letters 'h' and 's' indicates the critical sections − where the particular limit state and corresponding measure is governing − hogging and sagging area, respectively

In order to solely show the amount of the plastic reserve the self-equilibrated stresses from shrinkage, creep and thermal action are detached from the other effects. This way the results

60

Global Structural Analysis also better comparable to the American results (Table 3.1 and Table 3.2). The results are presented in Table 5.4. Table 5.4: Rating and utilization factors for various limit states without the effect of shrinkage, creep and thermal action.

-

-

RF UF base base 33,38% 22,88% 67,58% 67,58% 85,07% -

-

-

158,6%

RF

UF

1/UF

MSZ Út Eurocode elastic EN first plastic hinge EN single girder shakedown EN system shakedown EN single girder collapse

1,567 2,090 2,626 2,626 2,900

n/a 0,752 0,612

1,330 1,634

EN system collapse

4,052

s s

s s

-

RF

base 25,65% 25,65% 38,76% 93,88%

The system collapse ultimate load is rather high. Its rating factor can be regarded as a theoretical value for two reasons. Firstly, the shakedown limit state will occur before this could happen in case of moving loads. Secondly, if we assume a static (not moving) load the rating factor is still unrealistic, since the global mechanism requires so big lateral load distribution demand that is not likely to have the structure.

5.4.

The Effect of Shear Force

In the above calculations the effect of increased shear force due to raised traffic load is not considered. The bending resistances are calculated with the basic load level corresponding to RF=1,0. Since the shear force−moment interaction diagram is not linear it requires an iterative process to obtain the true rating factor, if one would like to take it into account. In case of global plastification the shear forces corresponding to the residual moments also should be considered if they significant. For this particular case the shear forces due to the residual moment field is negligible. Their magnitude is relatively small compared to the governing shear force. On the other hand, the bigger value is added to the shear forces from the side span directions, which are the smaller ones at the pier from the external loads (Figure 5.8).

61

Global Structural Analysis

Vext

r

Vr

Figure 5.8: Illustration of the internal forces from residual moments and external loads. With the above considerations determined rating factors are presented in Table 5.5. In the calculation, the actions were taken into account as illustrated in Figure 5.2; therefore, it should be compared to the values of Table 5.3. In the shear buckling resistance the contribution of the flanges is neglected (typically relatively small). Table 5.5: Rating factors determined by considering the effect of increased shear force. Table 5.3

0,00% -6,82% -6,82% -9,34% -19,92%

RF MSZ Út Eurocode elastic EN first plastic hinge EN single girder shakedown EN system shakedown EN single girder collapse EN system collapse

1,496 2,053 2,447 2,447

h s

2,629 3,245

1/UF

RF

UF

RF

1,159 1,410 -

base 37,21% 63,56%

base 21,72% -

base 23,71% 23,71%

-

-

75,75%

-

32,93%

-

-

116,9%

-

64,07%

UF

2

0,863 0,709 -

h h h

2

the "hinge" is formed when the shear force reached the shear resistance of the pier-section, the load level corresponds to the shear buckling failure of the web.

The characteristic line of the plastic resistance for the pier-section is illustrated in Figure 5.9. The blue marks representing the plastic bending resistances at the first-hinge, incremental collapse, single-girder collapse and system collapse limit states, the grow of the shear force follows the list. Mpl,Rd,0 represents the plastic resistance without any reduction due to shear force.

62

Global Structural Analysis 1.1

M pl . Rd M pl . Rd .0

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

VEd VRd Figure 5.9: The shear−moment interaction curve with the resistances corresponding to particular limit states. The value at 1,0 corresponds to the residual resistance of the section without the web (only the flanges carrying the bending moment), which is fully utilized for shear. In this case the longitudinal stiffener has a significant role to ensure the shear resistance of the web. To illustrate this, the rating factors were determined without the stiffeners as well. It is only an approximate comparison since the lack of the stiffener was taken into account only in the shear resistance. The results of the calculations presented in Table 5.6. Table 5.6: Rating factors determined by considering the effect of increased shear force without longitudinal stiffener. Table 5.3

-23,37% -15,21% -15,21% -18,55% -39,66% 1

RF MSZ Út Eurocode elastic EN first plastic hinge EN single girder shakedown EN system shakedown EN single girder collapse EN system collapse

UF

1,496 1,978 2,227 2,227

h h1

2,362 2,445

2

1/UF

RF

UF

0,926

h

1,080

-

-

0,863 0,761

h h

1,159 1,314

base 32,21%

base 13,40%

-

-

48,83%

-

-

-

57,89%

-

-

63,44%

RF

base 12,57% 12,57% 19,43%

-

23,62%

due to the increased shear force the pier-section became the critical section

2

the "hinge" is formed when the shear force reached the shear resistance of the pier-section the load level is correspond to the shear buckling failure of the web

From the tables it can be concluded that the effect of shear force should be considered and its effect greatly depend on the particular structural solution.

63

Global Structural Analysis

5.5.

Conclusions

Consequences of the application of plastic principles in ULS: Advantages/ Pros In many aspects the analysis is considerably simplified. Since for the limit states involve global plastification the shrinkage, creep and non-uniform thermal actions do not influence the ultimate load value. Of course, in other limit states they should be considered, but for dimensioning the sections based on ULS it is a major simplification. Allowing the formulation of the first plastic hinge the rating factor increased about 30% and with permitting more hinges and choosing the shakedown as the ultimate limit state, additionally around 25% increase can be achieved. The American results are in a similar range. More examples would require to generalize the founding. The consideration of the sequence of the construction could be neglected also, since the stresses locked into the steel section can equalize during the plastification process. Nevertheless, due to the fact that the relative inertia ratios are changing during the construction and therefore, some loads applied to a “different” structure, there is some effect of the stages. The magnitude of this effect was investigated by a simple calculation using the 142/k bridge. The stages taken into account: the loading of the propped steel girders and the removing of the shorings after the composite action established. The results are presented in Table 5.7. Comments on the data illustrated in the table: step-by-step (accurate) - with subsequent elastic analyses and summing the moments acting on the steel and composite sections; cracked comp. - ignoring the concrete in the 15% regions; uncracked comp. - assuming that the concrete is sound at the given moment level; Ma, Mc - moments acting on the bare steel and on the composite sections, respectively; M1, M2 - the total maximum negative and positive moments, respectively; On the left and right part of the table, the maximal negative and positive bending moments are presented, respectively. In the first rows, the accurate values with direct consideration of the sequence of the construction can be seen. While in the second rows, the numbers are determined using only the final, composite structural model considering cracked and sound concrete deck at the pier region.

64

Global Structural Analysis Table 5.7: The effect of omitting the construction stages Ma+Mc

M1

M1

Ma+Mc

M2

M2

step-by-step cracked

-3273,8 - 5052,5

-8326

-

558,5 + 4856,3

5415

-

cracked comp.

-

-8080

-2,96%

-

5585

3,13%

step-by-step uncracked

-3273,8 - 5721,0

-8995

-

558,5 + 4086,7

4645

-

uncracked comp.

-

-9042

0,53%

-

4631

-0,32%

At the highest negative moment the tensile stress in the concrete is about 32% higher than the mean value of the tensile strength. Based only on this one example it seems to be reasonable to apply a 1,05 amplification factor to the moments calculated ignoring the construction sequence and considering only the composite section. This multiplier should be applied to the moments carried by the bare steel girders. It can be approximated or without any a priori knowledge of the construction sequence, the relevant loads could be multiplied with this factor. As suggested by the results, the multiplier actual value depends on the degree of relative inertia change, more examples would be needed to generalize the results. Disadvantages/ Contras The increased deflections due to the residual forces may govern the design, but certainly demand more calculation. The rotation at the hinge locations could induce serviceability problems, like extensive crack width. There is not too much existing structure designed applying plastic principles, no real full-scale real life verification. The principles of plasticity of bars have to be acquired, which requires additional time and work. This point can be counted as an advantage as well, since this knowledge places in wider perspective the behavior of the structures. Moreover, even if it does not applied in the design the engineer can fully aware and asses the ultimate capacity of the structure. In this chapter, based on an example, it was illustrated that with plastic design principles significant, hitherto ignored reserves are mobilizable. The source of this reserve on one hand is that with allowing local and global plastification some internal forces − to which the elastically designed structures are subjected − relieve. The more significant contributor however, is simply the local and global plastic reserve. For this particular structure the former yields to 33% rating factor increase over the elastic, first yield limit. Furthermore, the RF corresponding to the incremental collapse limit-state is 25% higher than that of the first hinge. It was also shown that the shear forces could have considerable effect on the ultimate capacity of the structure. The structural solutions introduced in Chapter 6 offer some alternative ways to increase the shear resistance of the web or even fully restrain the buckling of it in order to reduce this effect.

65

Structural Solutions to Meet the Ductility Demand

6.

Structural Solutions to Meet the Ductility Demand

In the previous section it is assumed that the sections have the required rotation capacity to shake down or to maintain plastic moment till the formulation of all hinges. In reality, the sections’ rotation capacity is limited by stability loss of its various parts. This chapter deals with solutions to issues which are restricting the global plastic performance of bridges. Generally, the bridges, which are allowed to enter the inelastic range do not differ significantly from the elastically designed ones. The main distinctness is in the hogging moment area. Structures with elastically designed sections also have some reserves over the first yield. Moreover, with relative minor reinforcing it can be made adequate to exploit more plastic resources. The global stability problem − lateral torsional buckling − can be prevented by sufficient bracing. In most cases the plastically designed bridges do not require more cross-bracing than the conventionally designed ones, only the rearrangement of them [Barth and White, 2000]. To avoid this failure mode the American and Swiss methods also contain provisions, these are expounded in Section 3.3. The other issue is the local ductility. This cannot be solved as easily as the global stability question. This chapter focuses on the local, cross-sectional level problem: how to ensure the section to be able to undergo plastic rotations. One approach is to “upgrade” the conventional sections by applying more stocky plates or the second one is to somehow increase the ductility of the sections by changing their build-up. The first approach would require considerable increase of the plates’ thickness in order to reach the 2nd or 1st Class. Based on many research the second approach seems to be more practical and feasible. As mentioned before the sections in sagging zone are not susceptible to local buckling, since typically almost the entire steel section is under tension due to the large concrete flange. Nevertheless, there is one problem related to the positive moment region which should be examined. Namely, the locked-in stresses of the steel member due to the construction process. The neutral axis of the before-composite section is somewhere in the mid of the web and the composite plastic neutral axis (PNA) is typically in the concrete or in the upper flange of the steel section. During the plastification of the section the neutral axis (NA) converges to the PNA. Before reaching the “safe” PNA the steel web should be checked against plate buckling. Austrian researchers have studied this for slender plate girders reinforced by longitudinal stiffeners using GMNI analysis. They found that the utilization of nearly the full plastic moment capacity of the composite section is possible, also for composite sections with slender webs and slender longitudinal stiffener that are highly stressed and susceptible to local buckling due to preloads acting on the structural steel section [Unterweger et al., 2011]. The rotation capacity of sections in hogging area is the key question, since this determines the amount of redistribution in plastic design. In the following some ingenious solutions are being presented, how to economically solve this ductility problem without significantly increase the amount of structural steel. These solutions can be used during either the design of new structures or reinforcing existing ones. The used publications are dealing with innovative solutions and dominantly applying the concept of elastic design. Herein the ones which are

66

Structural Solutions to Meet the Ductility Demand relevant to plastic design are introduced; emphasizing the parts which are important to this study.

6.1.

Concrete Filled Closed and Open Sections

The strength of the steel compared to the concrete’s compression strength is high. Therefore, it is applied with relatively small sectional dimensions. Under tension this can be fully utilized, however in compression these slender parts are susceptible to buckling. By filling or encasing the steel its stability loss is prevented and can be considered automatically as Class 1 section. Moreover, there is no need for additional stiffeners and the costly and labor-intensive weldings can be avoided. Considering the distribution of the costs of typical composite plate girders (Figure 2.1) this has a significant economic impact. 6.1.1. Concrete Filled Tubular (CFT) Girder

The CFT girder bridges are great examples how to reduce the fabrication costs. There is fewer welding required to assembly the section and the stiffeners can be entirely avoided. Currently, the advantages of this structure type are mainly utilized/mobilized in the field of railway bridges, where the stiffness requirements are stricter. One built railway bridge from Japan is illustrated in Figure 6.1 and in Figure 6.2.

Figure 6.1: Illustration of the bullet train bridge [Nakamura et al., 2002]. Albeit the tubular form does not follow the mechanical demand as the I-shape, it is superior in many aspects. In the following, based on [Nakamura et al., 2002] research, the benefits of the structure are presented.

Figure 6.2: Concrete filled tubular girder bridge on the Japanese Shinkansen railway line, finished in 2000 [Nakamura et al., 2002]. 67

Structural Solutions to Meet the Ductility Demand The resistance of a particular zone can be increased with higher strength filler material as in case of the hogging region (Figure 6.1). In order to reduce the self-weight, air mortar filling was used in the sagging region. One important finding was that the concrete filling and steel tube are acting as an integral section without any additional mechanical connector. Hence, the cost of welding shear connectors can be avoided as well (inside the tube). In respect to plastic design probably the most important results are presented in Figure 6.3. The researchers studied only the filled tubular sections without the concrete flange. The strength and ductility10 of the sections are increased significantly due to the filling.

Figure 6.3: Test specimens and load−deflection curves [Nakamura et al., 2002]. Moreover, the researchers have found that the noise and vibration generated by the filled tubes are considerably favorable than that of the conventional I-girders. In the particular project steel fibers were mixed to the concrete slab in the negative moment region to minimize the cracks and to increase ductility. The area of fiber-reinforced concretes (FRC) is extensively researched and its superior properties are well-proven. It should be considered to apply in the hogging region in case of plastic design as well, where the additional rotations demand more ductility. 6.1.2. Concrete Filled Narrow-width Steel Box-girder

Another possible solution − similar to the CFT girder − is the concrete filled box birder illustrated in Figure 6.4.

10

Herein the term ductility refers to the area under the M−θ curve.

68

Structural Solutions to Meet the Ductility Demand

Figure 6.4: Illustration of a CF steel box-girder bridge [Nakamura and Morishita, 2008] The arrangement was studied by Nakamura and Moroshita for fully, partially filled and void box girders. The main conclusions relevant to plastic design are as follow: – – – –

–

the steel plates and the filler concrete behave as one piece, and no shear connector is necessary at the interface; the ultimate bending moment of the fully concrete-filled girder was 40% larger than that of the steel girder model; the ductility11 also increased about 8 times; the half concrete-filled model showed that the ultimate bending strength was 25% larger than the steel box girder and the ductility was over 6,5 times larger; the half concrete-filled model without vertical stiffeners had the same ultimate bending strength as that of the model with vertical stiffeners, but its ductility was about half; the authors worked out a simple calculation method, which was in good agreement with the test results [Nakamura and Morishita, 2008].

*to the experiments, mortar with compression strength 30

ே ௠௠మ

was used.

Moreover, trial designs were conducted on a three-span girder bridge with spans 65 x 85 x 65m. These showed that the estimated construction cost of the narrow-width partially concrete filled steel box girder is 10% lower than that of narrow-width steel box-girder and 19% lower than that of normal-width box girder bridge. The calculation covers only the cost of the superstructure. Additionally, in case of concrete filled sections the stress concentrations are decreased therefore the risk of fatigue is reduced. Special attention should be paid to avoid the corrosion and fatigue cracks inside the tube, since they cannot be discovered by inspection and can be repaired only by removing the filling. 6.1.3. Partially Encased Rolled and Welded Sections

In this chapter the properties of partially concrete encased rolled and welded girders are introduced (Figure 6.5 and Figure 6.6).

11

The ductility now - according to the authors - measured by the ratio of the ultimate- and yield curvatures.

69

Structural Solutions to Meet the Ductility Demand

Figure 6.5: Partially encased rolled H-girder bridge [Nakamura et al., 2002]. Japanese researchers examined the potential of this structural solution by using rolled Hgirders with tensile strength 500MPa and 900mm maximum height, illustrated in Figure 6.5. The hogging area is strengthened by steel and reinforced concrete (SRC). They concluded the following: – – – – – –

– – –

–

bridges composed of rolled girder require minimal amount of welding; the web is compact, not susceptible to plate buckling, thus there is no need for expensive welding of stiffeners; the sections available from a discrete range, with more limitations than the welded ones; in the zone with concrete encasement the ductility of the section is significantly increases, due to the confinement of the web; the ultimate resistance of the SRC section is 1,5 times higher than that of the bare Hgirder, due to the additional concrete and the restrained web; the feasible span with the maximum height girder, concrete filling, continuous global structural model and using the plastic sectional resistance12 can remarkably extended up to 50 m over the previously typical simply supported, elastically designed 25 m; structures with rolled girders can be built with very low structural depth (~ L/35); the bridges are considerably stiff, the maximum deflection from the live load is only the half of the limit value; since the H-girder has a low web height and the slab is relatively stiff, steel crossbeams are eliminated by assuming that the concrete slab could contribute as crossbeams; due to there is no cross-bracing, a potentially fatigue hot spot is removed from the structure [Nakamura et al., 2002].

The same principles could be applied to welded sections. The plate girder bridges are typically heavily stiffened in the internal pier location due to the concentrate force application from the bearing and the high internal forces. The required horizontal and longitudinal 12

The Japanese standard does not allow the plastification of the section, in the framework of Eurocodes this reserve is exploitable.

70

Structural Solutions to Meet the Ductility Demand stiffeners can be avoided by the concrete filling. Steel bars are welded vertically to connect both flanges and steel bars are also set horizontally to restrain the filled concrete from falling off (Figure 6.6).

Figure 6.6: Illustration of a partially concrete-encased composite plate girder bridge [Nakamura et al., 2002]. In this case experiments were also carried out with encased and void steel girders. The arrangement and results are summarized in Figure 6.7.

Figure 6.7: Experimental arrangements and results of the bending (top) and shear (bottom) tests [Nakamura et al., 2002]. The increase of the moment and shear capacity is remarkable. This is because the filled concrete not only contributing to the bending strength but preventing local buckling of compressive flanges as well. The ductility increases considerably too, moreover the flexural stiffness is also higher in case of the encased specimens. These tests showed that the partially concrete filled I-girder has superb bending and shear strength. As the structural detail is simple and the erection procedure is easy, this seems to be 71

Structural Solutions to Meet the Ductility Demand feasible and practical. This method is useful not only for newly constructed bridges but also for strengthening and repairing the existing plate girders.

6.2.

Double Composite Action

Double composite sections are composed with an additional bottom concrete flange. The idea is not new, nevertheless it is not widely used in practice, albeit it has numerous advantages. The first such structure the Ciérvana bridge was built in Basque Country, Spain in 1978 [Sen and Stroh, 2010]. The advantageous properties of the double composite action are: – –

– – – – –

increases span lengths to values which were previously the domain of steel bridges with orthotropic plates, arch bridges or cable-stayed bridges; lower costs deriving from the use of concrete instead of steel and reinforcement instead of prestressing, which may be a decisive factor in favor of construct a steel composite bridge, especially in developing countries (Table 6.1); due to the closed section the torsional stiffness is significantly increased, it yields to favorable lateral load distribution; enables acceptable deformation in railway bridges to be achieved economically with the concrete bottom slab; it fits well to the advance of the rapid trains and the increased stiffness requirements, also could be used to reinforce existing structures; thick on-site welds are avoided, with their corresponding residual stresses and deformations; instability problems in the ultimate limit state are avoided, not only the bottom flanges but the webs are compact as well due to the low position of neutral axis at the ultimate limit state [Brozetti, 2000; Saul, 2000; Kim and Shim, 2009]. Table 6.1: Stiffness and costs compared for steel and concrete for an arbitrary normal force of 100 MN [Saul, 2000].

72

Structural Solutions to Meet the Ductility Demand In order to show the effect of the additional bottom concrete flange, the sectional properties of the hogging zone of the 142/k bridge were calculated with and without the second flange. 1,5% rebar ratio was assumed in the bottom concrete slab, which width is also reduced due to shear lag. The sections are illustrated in Figure 6.8. 4815

4815 2407,5

2407,5

2407,5

250

100 1830

1830

100

292,3

2407,5

2500

Figure 6.8: Dimensions of the single and double composite sections. The results of the calculation are summarized in Table 6.2. One important advantage of the bottom flange that it lowers the neutral axis in such extent that the web falls into Class 1, thus the composite section is in Class 1 and fulfills the plastic design criteria. Therefore, no additional longitudinal stiffener is required like in case of the single flange section. Table 6.2: Comparison of the single and double composite sections. Composite sectional properties, resistances

single comp. action

double comp. action



1510,2

2566,5

69,94%

Height of elastic NA [mm]

1074,9

708,6

-34,08%

Height of plastic NA1 [mm]

1444,6

605,3

-58,10%

Inertia about NA [cm4]

1,185E+07

1,682E+07

41,97%

Elastic modulus, top flange [cm3]

1,569E+05

1,500E+05

-4,40%

Elastic modulus, bottom flange [cm3]

1,102E+05

2,374E+05

115,34%

Elastic resistance2 [MNm]

35,693

44,500

24,67%

Plastic resistance [MNm]

46,112

58,192

26,20%

Area [cm2] 1

1

from the very bottom surface, for the composite section

2

depends on the sequence of construction

It should be noted that solely the single flange girder without the stiffener would have smaller plastic resistance, due to the reduced web. These results are showing the same trend as Cornejo and Raoul’s calculation. They also investigated the effect of the second concrete flange for a particular girder bridge. They have found that even with halving the thickness of the bottom flange and applying a 50 cm thick concrete bottom flange the plastic resistance is increased by 30,6% over the conventional one’s. They concluded that significant strength and

73

Structural Solutions to Meet the Ductility Demand stiffness increase can be achieved while reducing the amount of structural steel [Cornejo and Raoul, 2010]. In my B.Sc. final project, I compared double and single composite alternatives for a railway bridge formed by three spans 45,0−45,0−22,9 m. Around 7,5% structural steel saving could be materialized over the conventional plate girder solution while providing the same maximum deflection [Rózsás, 2010]. A twin girder double composite bridge’s cross-sections are illustrated in Figure 6.9. The shear connection is solved with standing and laying studs. Kim and Shim Korean researchers investigated its ultimate behavior with advanced FEM analysis. For the particular arrangement they did not observe any local buckling They also stated that the flexural strength of the double composite section can be evaluated by rigid-plastic analysis when the full shear connection and the compact section requirements are achieved [Kim and Shim, 2009].

Figure 6.9: Cross-sections of the double composite bridge; a) sagging region; b) hogging region [Kim and Shim, 2009]. The solution is advantageous for bridges in middle and longer span range as well. For the former constant structural depth with twin plate girder is typical while for the latter, due to economic reasons, tapered sections are applied. One long-span example with varying structural depth is presented in Figure 6.10. The increased stiffness and load bearing capacity are mainly utilized for railway bridges.

74

Structural Solutions to Meet the Ductility Demand

Figure 6.10: Double composite railway bridge in Nantenbach over the river Maine, Germany [Saul, 2000].

6.3.

Reinforcing the Web

6.3.1. Bolted Longitudinal Plate or Stiffener

Iranian researcher Vasseghi proposed and investigated two arrangements to enhance slender (for the classification see Figure 3.8) sections with plastic capacity. These are illustrated in Figure 6.11. He examined these reinforcing methods especially to improve the performance of continuous composite plate girder bridges. His advocated aim was to enhance the noncompact, slender sections with ductility while sustaining the maximum moment level. This goal was achieved by a daedal solution: bolting plates to the compressed part of the web. These elements are providing elastic support and bracing the web against local buckling, thus changing the failure mechanism and allows the section to formulate a plastic hinge in it.

Figure 6.11: Bracing of the web with bolted plate (left) and with bolted stiffeners (right) [Vasseghi, 2009].

75

Structural Solutions to Meet the Ductility Demand The main advantage of the bolted reinforcement over welded stiffeners is coming from the fact that the plates are not rigidly connected to the web. Therefore, when the structure undergoes deformations there are reduced/limited stresses developing in the plates. Due to the manner of the connection the bolted bracing does not yield while the main-section already reached the plastic state. This way the plates provide more effective restrains of the web than the welded stiffeners. This connection can be achieved by oversized holes for bolts, nevertheless some axial force will develop in the elements due to the friction between the interfaces (this can be reduced by treating the surfaces) and to the limited size of the holes. Actually, the design formulas are constructed in order to ensure the adequate bracing of the web while taking into account the axial force and therefore the buckling of the plates. One interesting consequence of this bracing method is that the reinforcing element may be made of any kind of engineering material, e.g., timber, steel. To investigate the behavior of the bracings nonlinear finite element analyses were carried out. The basic model and section dimensions are illustrated in Figure 6.12.

Figure 6.12: The FEM model arrangement and geometry [Vasseghi, 2009]. The performances of various bracings based on FEM analyses compared to the unbraced, slender section are presented in Figure 6.13. The numbers after the material are the thickness and length of the element, respectively.

76

Structural Solutions to Meet the Ductility Demand

Figure 6.13:Moment−deflection curves for various reinforcing and the behavior of the slender section with (left) and without (right) additional bracing [Vasseghi, 2009]. As can be seen in Figure 6.13, the plates raised the performance of the section to the level of the compact section, sometimes even increased over that. The bottom images of Figure 6.13 also illustrate how the structural behavior has changed. Without bracing the local buckling of the web was followed by the crippling of the flange into the web, since it has lost its support. With bracing, because the web-buckling is prevented, the section can develop higher resistance and more importantly can sustain it in the realm of bigger deformations. Thus allowing the second hinge to form in midspan. The optimal location of the bracing is 0,2∙hw from the compression flange [Vasseghi, 2009]. Vasseghi also proposed formulas to design and verify the bracing, derived from basic mechanical considerations. These equations for bolted plates and bolted stiffeners can be found in the following publication [Vasseghi, 2009]. The advantages of the proposed method over applying compact sections or welded stiffeners according to Vasseghi are the following: –

it does not require any welding and could be used for improving existing structures as well; – it does not change the inertia ratios along the structure like the welded stiffener, therefore the moment distribution is the same as without the bracing;

77

Structural Solutions to Meet the Ductility Demand –

the reinforcement is localized near the interior supports and the associated fabrication work is not very costly; – the proposed method improves the fatigue performance of girders near interior supports because the bracing elements greatly reduce web out of plane deflection at service load. This reduces the possibility of fatigue cracking due to oil canning of the web. The bolted connections of the bracing elements also have a better fatigue performance than the welded ones mainly because they do not contain problematic weld details. The bolt holes in the web are in the compression zone and are generally at locations where stress due to service load is not very high. These holes are not expected to cause significant fatigue problem [Vasseghi, 2009]. The reinforcing plates are placed in the compression zone, where no additional requirements for the holes prescribed by the AASHTO rotation compatibility method. Based on the presented results it can be concluded that with using bolted bracing the strength and ductility of slender sections can be significantly increased. With relatively minor additional cost the cross-section can be upgraded to compact class which plastic reserves are exploitable. 6.3.2. Welded Longitudinal Stiffeners

By welding longitudinal stiffeners − with appropriate stiffness − to the web it can be divided into smaller panels, which have higher resistance against local buckling. Therefore, the class of the section can be raised. Vasseghi’s research has shown that in many cases the bolted plates are superior to the welded stiffeners. Unlike the bolted bracing the welded stiffeners are rigidly working together with the section, hence they go under plastic strains as well. The global buckling (Figure 6.14) of the stiffened plate also should be considered. This solution is more susceptible to this failure more than the bolted bracing since the stiffeners are subjected to the same loading as the section.

Figure 6.14: Global (left) and local (right) buckling of the stiffened web, the two thicker lines represent the webs of the trapezoidal stiffener (EBPlate13). The EN 1993-5 contains comprehensive provisions and methods how to handle these kinds of structures. Lääne and Lebet (2005) not only showed that the welded longitudinal stiffener can

13

Freeware software developed by CTICM with partial funding from the European Research Fund for Coal and Steel (RFCS). It assesses the critical stresses associated to the elastic buckling of plates loaded in their plan.

78

Structural Solutions to Meet the Ductility Demand increase the section’s rotational capacity but also proposed a formula to determine it, Eq.(3.19).

6.4.

Conclusions

This chapter introduced some conventional and innovative solutions, which can enhance the composite sections in the negative bending region with significant ductility. This increased rotational capacity is essential for plastic design, since it increases the ultimate load of the structure. Nevertheless, many methods are available only for the welded longitudinal stiffeners are formulas attainable to assess its contribution to the rotational capacity. The performance of these solutions in respect to rotation capacity-increase requires further research.

79

Reliability Analysis According to Eurocode

7.

Reliability Analysis According to Eurocode

The reliability analysis of the same structure was also carried out in order to investigate whether the particular limit states with the default partial factors fulfill the prescribed safety level or not. Herein, only the principles, main results and conclusions are presented. For further details, see Annex C – Reliability Analysis.

7.1.

Principles, Methods

The reliability analysis was elaborated in conformity with Eurocodes. FORM and SORM methods were used to obtain the reliability indices, and then they were compared to the target value prescribed by the standard. Basic variables: -

Actions (E) Resistances (R) Geometric properties (a)

By way of introduction, it should be noted that there are no clear, solid guidelines how to perform reliability analysis and assume some basic input variables. It appears that the results of any reliability study depend significantly on the assumed theoretical models used to describe the basic variables. Moreover, these models are not yet unified and have not been used systematically [Gulvanessian et al., 2002]. 7.1.1. Eurocode Recommendations

One of the fundamental questions in reliability analysis is the appropriate choice of the probability distribution function (PDF). Many recommendations can be found in the literature and in standards as well. Recommendations by EN 1990:2001 Annex C6: –

Lognormal or Weibull distributions have usually been used for material and structural resistance parameters and model uncertainties; – Normal distributions have usually been used for self-weight; – For simplicity, when considering non-fatigue verifications, Normal distributions have been used for variable actions. Extreme value distributions would be more appropriate. These provisions were adopted in the current reliability analysis. The target reliability of a structure depends on their importance and on the consequences of its failure. Reliability classes in the European Norm are presented in Table 7.1.

80

Reliability Analysis According to Eurocode Table 7.1: Consequence and reliability classes according to EN 1990:2001. Consequence Class

Description Related to Consequence

Reliability Class

CC1

Low consequence for loss of human life; economic, social, or environmental consequences small or negligible

RC1

CC2

Moderate consequence for loss of human life; economic, social, or environmental consequences considerable

RC2

CC3

Serious consequences for loss of human life or for economic, social, or environmental concerns

RC3

The base of the Eurocode reliability management is that it prescribes a yearly safety level for every consequence class. Regardless of the design life, this annual probability of failure should be ensured. This required safety is expressed by the reliability index. This is 4,70 for buildings and 4,75 for bridges in CC2 class in respect to one year, according to the standard and the corresponding literature. This 0,05 difference means that the probability of failure is 1,30 times higher in case of buildings. Handling the yearly failures as independent events the reliability index for arbitrary design life can be determined by the following equation:    n      1  

n

(7.1)

where:

n

is the reliability index for a reference period of n years;

1

is the reliability index for one year.

Using Eq.(7.1) the reliability index for a bridge with 100-year design life is:



100   1   4, 75  

100

  3, 715 .

Hereinafter, this will be used as the target value in the reliability analysis. The reliability indices to Eurocode and AASHTO LRFD Bridge Design Specification are summarized in Table 7.2 for different design lives and consequence classes.

81

Reliability Analysis According to Eurocode Table 7.2: Inherent probabilities of failure (PF) and corresponding reliability indices (ß).[Hida et al., 2010]. Reference Period [Years]

Code CC2 Eurocode CC3

LRFD

Typical bridges Important bridges

1

50

75

100

120

1,00E-06 4,75 1,00E-07 5,20 2,67E-06

5,00E-05 3,89 5,00E-06 4,42 1,33E-04

7,50E-05 3,79 7,50E-06 4,33 2,00E-04

1,00E-04 3,72 1,00E-05 4,26 2,67E-04

1,20E-04 3,67 1,20E-05 4,22 3,20E-04

4,55

3,65

3,50

3,46

3,41

9,60E-06

4,80E-05

7,20E-05

9,60E-05

1,15E-04

4,76

3,90

3,80

3,73

3,68

The typical design life for bridges in Hungary, USA and UK are 100, 75 and 120 years respectively. From Table 7.2 it can be seen that the CC2 consequence class can be classified as an important bridge according to LRFD. 7.1.2. Reliability Analysis

The reliability analysis is elaborated in a simplified manner. The first major simplification is that the time is taken into account indirectly, by using combination factors to model the simultaneous occurrence of actions. The second is that the variation of the geometry is considered only on the resistance side, or it can be interpreted that its effect on the global behavior is assumed negligible. In order to take into account this effect a global Monte Carlo simulation would have had to be carried out and connect the finite element analysis with the reliability calculation. By neglecting the geometry’s variation on the effect side and taking into account the time indirectly the complexity of the model is reducing substantially. Because the effect of shear force significantly complicates the calculation, the reliability indices determined assuming the same plastic moment resistance regardless of the level of shear force. If we have the input variables − means () and standard deviations () − the calculation is rather straightforward, as presented in Section 2.2.2. Unfortunately, no unambiguous recommendations can be found in the literature (according to my knowledge) how to assume these values. This seems to be reasonable since plenty of factors influence these actions. Based on some publications related to buildings (50 years design life) [Gulvanessian and Holický, 2005; Honfi and Mårtensson, 2009] and comparing the values to the American reliability analysis conducted by Barker and Zacher (1997), moreover, with basic assumptions the coefficient of variations and mean values were determined. The values are slightly modified in order to reach the Eurocode prescribed safety level (= 3,715). In my judgment, this has no significant effect and the main emphasis is on the comparison of different limit

82

Reliability Analysis According to Eurocode states not on the actual absolute numeric values14. In my view for this purpose, assumptions close to literature values are satisfactory, and the results should be handled as qualitative rather than quantitative measures. The coefficient of variations for variables are taken from numerous publications [Sedlacek; Sørensen; Gulvanessian and Holický, 2005; Honfi and Mårtensson, 2009]. The variables in the reliability analysis are inherently time-dependent. It was transformed to a time-invariant problem by using Turkstra’s rule.

max( E1 )  E2  Emax,T  max    E1  max( E2 ) 

(7.2)

where: Emax,T

the maximum value of the combined effect in the reference period T;

max( Ei ) maximum value of the ith action; Ej

the value of the accompanying action at arbitrary-point-of-time, in practical situations taken as the mean value of the action [Ghosn et al., 2003].

The Eurocode uses the same combination rule for design situations. Since the above definition refers to the accompanying action as mean value, and the EN forms this with the combination factor, the actions’ mean values are determined using this multiplier. The mean values approximated this way are in good agreement with numbers found in the literature. It should be noted that this rule is considered over-simplification by some authors and often yields to unconservative result [Melchers, 1999]. Nevertheless, in this study this method is applied since in many papers, dealing with reliability analysis this method can be found and the following publications are also recommending to EC reliability analysis [ISO 2394, 1998; JCSS Probabilistic Model Code, 2000]. The effect of traffic load is significantly higher than the thermal effect; therefore, only the preceding will be used as a leading action (corresponds to the partial factor based design). According to these considerations and recommendations, the chosen distributions, mean values and coefficient of variations are summarized in Table 7.3.

14

EN 1990:2001 Annex B6: The ‘probability of failure’ and its corresponding reliability index are only notional values that do not necessarily represent the actual failure rates but are used as operational values for code calibration purposes and comparison of reliability levels of structures.

83

Reliability Analysis According to Eurocode Table 7.3: Statistical properties of the input variables to reliability analysis. coefficient of variation (ν)

partial factor

distribution type

Concrete (fc)

1,50

LN

15,00%

Reinforcing steel (fs)

1,15

LN

8,00%

Resistance (R)

1,00

1

2,48%2

‐ 

LN

Concrete (ac)

-

N

5,00%

Reinforcement (as)

-

N

5,00%

Structural steel (aa) Effect/action Permanent actions Dead load (D)

‐ 

N

3,00%

1,1475 1,00

N N

Traffic 100-year (L)

1,35

GU

Thermal 100-year (T)

1,50

GU

0,6∙Tk

30,00%

Thermal 1-2-year (Ta)

-

GU

0,5∙Tk

50,00%

Uncertainty (ΘE)

-

LN

1,00

5,00%

mean ()

Resistance/material

Uncertainty (ΘR)

1,00

4,00%

Geometry3

Shrinkage, creep (S) Variable actions

(1-1,645∙D)·Dk 1,0∙Sk

8,00% 0,00%

0,75∙Qk

20,00%

0,4∙qk

LN - lognormal; N - normal; GU - Gumbel distribution The variation of the structural steel strength (fa) is taken as 0, since its partial factor is 1,0. 1

the distribution is generated by Monte Carlo simulation thus the distribution cannot be classified as a pure type (approximately lognormal) 2

the result of the input variables uncertainty, therefore slightly differs from section to section

3

the uncertainties from geometry are taken into account only on the resistance's side

 The mean value of the thermal action for the shorter reference period is used as an accompanying action. In conformity with Eurocode, the partial factors comprise two sources of uncertainty. For instance, the variation of the action itself and the uncertainty of the model established to represent the action. The unified partial factor is actually a simplification, in a more sophisticated calculation the particular partial factors should be applied separately. These uncertainties on both the resistance (ΘR) and effect side (ΘE) are also indicated in Table 7.3. In case of many variables, non-normal distributions and correlation between these variables the “hand-calculation” becomes tedious and time consuming. Therefore, an open-source

84

Reliability Analysis According to Eurocode Matlab toolbox called FERUM15 was used to carry out the analysis. It has many options such as FORM, SORM and simulation techniques. The program was tested with basic two-variable problems and with the results found in the literature from [Barker and Zacher, 1997]. In Annex C, for the first hinge limit state the “handcalculated” values are compared to ones provided by FERUM. The results are in excellent agreement. For first approximation, FORM method was used, and then SORM was applied to obtain a more accurate result. The geometry and material properties are considered solely as elementary variables, which used to produce compound variables like the cross-section resistance. The variation of the elastic modulus is taken into account through the variation of the mean value of the concrete’s compression strength, this affects only the elastic resistance. The resistances are determined by Monte Carlo simulation, with geometry and strength as elementary variables. In geometrical deviation of plates only the variations of the thicknesses are considered. The result one of the Monte Carlo simulations for the plastic resistance of midspan section is presented in Figure 7.1. With increasing the “population size” the histogram approaches to a lognormal distribution. 0.0005

frequency [-]

0.0004 0.0003 0.0002 0.0001 0

36000

38000

40000

resistance [kNm]

Figure 7.1: The distribution of the plastic moment resistance of the midspan section with a 10’000 elements sample. The mean values and standard deviations obtained from the simulations were used as input data for the reliability analysis. To get authentic values for further calculation an approximate convergence analysis was carried out. In Figure 7.2 the results of one simulation for each sample size are presented, this makes possible a rough estimation of the effect of the number of generated elements.

15

The development of FERUM (Finite Element Reliability Using Matlab®) as an open-source Matlab® toolbox was initiated in 1999 under Armen Der Kiureghian’s leadership at the University of California at Berkeley (UCB). This general-purpose structural reliability code was developed and maintained by Terje Haukaas until 2003, with the contributions of many researchers at UCB [FERUM].

85

Reliability Analysis According to Eurocode

standard deviation mean

MplRd,neg [kNm]

37950

945 940

37900

935

37850

930 925

37800

920

37750

915 910

37700

905

Standard deviation [kNm]

950

38000

900

37650 10

100

1000

10000

100000

log(n) - sample size

Figure 7.2: Convergence of the Monte Carlo method with various “population sizes”. The coefficient of variation of the mean and standard deviation was estimated with a 10element sample for each “population size”. This is still an approximate check but more sophisticated than the previous one. The results presented in Figure 7.3.

Coefficient of variation

18,0%

standard deviation mean

16,0% 14,0% 12,0% 10,0% 8,0% 6,0% 4,0% 2,0% 0,0% 10

100

1000

10000

100000

log(n) - sample size

Figure 7.3: Coefficient of variation of the mean and standard deviation of the positive plastic resistance. For the hogging area and elastic resistances the same trend was observed as illustrated above. Based on the results population size 10’000 was chosen for the further calculations. The reliability of a particular limit state is worth examining only if we are on the failure line with factored variables (Rd=Ed). This can be achieved by scaling the live load by rating factors determined in Section 5.2 or by scaling the resistances to get RF=1,0 while keeping the live load at the same level.

86

Reliability Analysis According to Eurocode Scaling the live load to reach the particular limit state It can be considered as the deficient bridge subjected to an increased traffic load scenario. The multiplier applied to the live load to reach the failure point is the rating factor. Scaling the resistances to reach the particular limit state It can be considered as the new bridge design scenario. It assumed that the scaling is done such a way that the inertia ratios do not change. Therefore, the moment envelopes determined on the original structure can be used. The decrease of the dead load due to section’s dimension change is estimated as minor and neglected. Nevertheless, it is a safe side approximation. If two or more sections involved in the limit state equation their resistances scaled keeping their original ratio. The same equations are used as limit state functions (g) which were used for the calculation of ultimate load capacity (design equations) in Section 5.2. The only difference that the variables are represented with their mean values and therefore, they are not multiplied by partial factors. The general form of this expression illustrated by Eq.(7.3).

g  X   R  X   E ( X)   R  R0  X    R  E0 ( X)

(7.3)

For single-girder shakedown the limit state function is as follows:

g (...)   R   R1,m    R2,m    R3,m        max max   E   G1,m    G2,m    G3,m      L1,max m    L2, m    L3, m    

(7.4)

Different limit-states may have different governing load combination than that of the elastic, it should be checked every time. In this particular case, the load combination with reduced dead load and leading traffic accompanied with thermal action was the governing load combination in every case.

7.2.

Results of the Analysis on the Studied Bridge

This section summarizes the safety level of the studied 142/k bridge. The reliability indices corresponding to different limit states with live load scaling are presented in Table 7.4.  represents the difference between the reliability indexes of a particular limit state and of the lowest value of the first yield or first hinge. The  values are expressing the degree of correlation between the variables. For instance R is the correlation between the resistances at location 1,2 and 3, see Figure 5.5. R=1,0 means that the three variables taking the same values, and R=0 means that their values are changing completely (linearly) independently from each other. The correlation between the dead load effect and section resistance is neglected. As mentioned in Section 7.1.1 the target reliability level is 3,715 for the 100-year design life. However, it gives more information if we compare the safety levels to the limit states for which the standard gives methods and applied in practical design; these are the first yield and first hinge limit states. It is interesting that the correlations have such a significant 87

Reliability Analysis According to Eurocode effect on the safety levels; nevertheless, the same trend was observed by the American researchers, Table 3.1 and Table 3.2 [Barker and Zacher, 1997]. They recommend the 70-5080% correlation combination to be accepted as representing the real cases.

Table 7.4: Reliability indices () for various limit states with live load scaling ( correlation). bridge 142/k

Limit state





First yield

4,163

7,06%

First-hinge

3,889

0,00%

Single-girder shakedown: R = D = L = 0%

5,212

34,02%

Single-girder shakedown: R = D = L = 50%

4,577

17,71%

Single-girder shakedown: R = D = L = 100%

4,029

3,61%

Single-girder shakedown: R = 70%; D = 50%; L = 80%

4,231

8,80%

System shakedown: R = D = L = 0%

5,790

48,89%

System shakedown: R = D = L = 50%

4,589

18,00%

System shakedown: R = D = L = 100%

3,860

-0,75%

System shakedown: R = 70%; D = 50%; L = 80%

4,110

5,69%

Single-girder plastic collapse: R = D = L = 0%

4,678

20,30%

Single-girder plastic collapse: R = D = L = 50%

4,283

10,14%

Single-girder plastic collapse: R = D = L = 100%

3,926

0,95%

Single-girder plastic collapse: R = 70%; D = 50%; L = 80%

4,062

4,46%

System plastic collapse: R = D = L = 0%

4,779

22,89%

System plastic collapse: R = D = L = 50%

4,405

13,27%

System plastic collapse: R = D = L = 100%

4,025

3,49%

System plastic collapse: R = 70%; D = 50%; L = 80%

4,170

7,22%

every  value corresponds to the load level required to reach the particular limit state 1

The reliability indices were determined for the resistance scaling as well, the results are summarized in Table 7.5.

88

Reliability Analysis According to Eurocode

Table 7.5: Reliability indices () for various limit states with resistance scaling ( correlation). bridge 142/k

Limit state





First yield

3,950

0,00%

First-hinge

4,471

13,20%

Single-girder shakedown: R = D = L = 0%

5,543

40,34%

Single-girder shakedown: R = D = L = 50%

4,899

24,02%

Single-girder shakedown: R = D = L = 100%

4,029

2,00%

Single-girder shakedown: R = 70%; D = 50%; L = 80%

4,569

15,66%

System shakedown: R = D = L = 0%

6,098

54,37%

System shakedown: R = D = L = 50%

4,921

24,59%

System shakedown: R = D = L = 100%

4,183

5,90%

System shakedown: R = 70%; D = 50%; L = 80%

4,464

13,02%

Single-girder plastic collapse: R = D = L = 0%

4,961

25,60%

Single-girder plastic collapse: R = D = L = 50%

4,624

17,06%

Single-girder plastic collapse: R = D = L = 100%

4,285

8,47%

Single-girder plastic collapse: R = 70%; D = 50%; L = 80%

4,418

11,86%

System plastic collapse: R = D = L = 0%

5,140

30,12%

System plastic collapse: R = D = L = 50%

4,814

21,88%

System plastic collapse: R = D = L = 100%

4,489

13,66%

System plastic collapse: R = 70%; D = 50%; L = 80%

4,622

18,85%

every  value corresponds to the same load level, the difference lays in the resistances, which are scaled to reach the particular limit states 1

Albeit, the ultimate load is identical to single and system shakedown limit states, their reliability indices differ, since the system incremental collapse failure mode involves more hinges. This way, the comparison is a bit distorted, because as mentioned before the number of required load passing is not contemplated in the calculation, which strongly influences the reliability. Actually, the same is true for the single girder shakedown as well with a lower degree. However, it should be noted that although the values for shakedown limit states do not represent a “real” value, they are on the safe side. This distortion could be eliminated from the system by determining a load level with the same occurrence probability as the “basic” load to the let’s say 10 passing number. Comments on the reduced live load: The possibility to reduce the characteristic value of the standardized load model was also numerically investigated. Based on the fact that many subsequent load applications are necessary to experience the incremental collapse failure, an approximate calculation was carried out using the scant data available in the following EN background publication

89

Reliability Analysis According to Eurocode

[Sedlacek et al., 2008]. The same method was applied as in the mentioned reference to obtain the value corresponding to the same reliability level as the LM1 load model has. A load value was determined to shakedown whereat greater than or equal to load occurrence probability in 10 or more times is 10% in a 100-year reference period. This value is ~86% of the characteristic value of the basic load model LM1. This number is obtained after applying a reduction to take into account the effect of the dynamic factor in a same manner as used in the referred publication. Nevertheless, it should be mentioned that in case of using the reduced load to shakedown limit, the check of the plastic collapse of the structure subjected to the basic standardized load level is not negligible. If in case of plastic collapse the rating factor increase over the shakedown limit is greater than 1/0,86 = 1,163 then the shakedown is governing. Otherwise, the possibility to reach the plastic collapse is higher. Based on the one available example on steel bridges and the one composite bridge examined herein this is not likely to have the plastic collapse as the governing limit state. However, it should be kept in mind that this reserve is currently not exploitable due to shear connection degradation mentioned in Section 3.2.4. This effect was taken into account in one of the AASHTO inelastic procedure with a 1,10 multiplier applied to the moment resistance [Barth et al., 2004]. This is not part of the mostrecent rotation compatibility approach.

7.3.

Conclusions

In this chapter, the safety level of a particular bridge (142/k) was examined. Through a reliability analysis it has been found that the safety of the structure at least fulfills or even exceeds that of the first hinge or first yield limit state, in case of every plastic limit state. From a standardized reliability viewpoint, the safety levels related to the plastic limit states are typically higher than that of the conventional first hinge or first yield. Of course, the elastically designed structures have the plastic reserve over the first yield, but it currently ignored in design. The American results are showing very similar trend; the researchers also concluded that plastic limit states provide the prescribed safety level [Barker and Zacher, 1997]. As they pointed out more analytical studies on a range of bridge types need to be executed to generalize the findings.

90

Redesign of the Bridge Based on Plastic Principles

8.

Redesign of the Bridge Based on Plastic Principles

In order to show the economic aspect of the plastic design the 142/k bridge was redesigned and compared to the elastic results. This chapter introduces the Eurocode based calculation and the American results as well.

8.1.

Considerations, Eurocode Principles

Since the 142/k bridge is not fully utilized under the default level of loads, it should be modified to reach the yield stress at the most loaded point. This achieved by scaling the dead and traffic loads by the same value in order to reach the first yield while keeping their original ratio. It should be kept in mind that this way the ratios of different load types are broken. To get a more reliable comparison the bridge firstly should be redesigned following elastic principles. Nevertheless, for the sake of simplicity the original arrangement with proportionally increased loads was considered as the basis for the further calculations. The scaling factor applied to the traffic and dead loads was 1,1580 to reach the first yield. Since this thesis is mainly focusing on the ultimate load bearing capacity of plate girder bridges, the redesign will be performed in that philosophy and only a few other serviceability limit states will be checked. About the adopted methods and considerations: – The only provisions that can be found in the Eurocode in respect of non-linear global analysis (EN 1994-2 2006 5.4.3) are the following:

(1)P Non-linear analysis may be used. No application rules are given. (2)P The behavior of the shear connection shall be taken into account. (3)P Effects of the deformed geometry of the structure shall be taken into account.

These are principles (P) and must be followed; nevertheless, these would significantly complicate the analysis. Therefore, the principles and provisions of the American standard and literature were adopted, the fact that they working on this topic for a couple of decades also supports this. It would be interesting to check the effect of the above-mentioned Eurocode principles, but this is not part of this study. – One role of the rotations is to redistribute the moments towards the elastic regions. The other role is to relieve the residual stresses; these can happen simultaneously. The question is that whether these rotations sufficient to relieve the residual forces. Moreover, if for instance the global (secondary) effect of thermal actions relieved it can occur another time and “destroy” the favorable residual moment field, consequently, induces more rotation to reach the shaken down state again. Precisely, it should be examined from a reliability aspect, what is the probability that the thermal action with its characteristic value will happen again? Based on the above considerations it was decided to use the total moment envelope comprising all effects, as a conservative approximation. It should be noted that

91

Redesign of the Bridge Based on Plastic Principles applying the total elastic envelop with the default traffic load level is the worst-case scenario. In respect of the residual displacements, it can be considered as an upper bound. The American methods neglect these (creep, shrinkage, nonlinear temperature difference) actions even in the first plastic-hinge limit state. – The most-recent AASHTO method appears to handle only the cases where the first hinge formulates in the negative zone. Nevertheless, it seems to be reasonable to have higher plastic resistance in the positive region where the “cheap stiffness” of the concrete can be mobilized. Moreover, the moment−rotation curves are developed for sections without longitudinal stiffeners. It could be neglected; however, the section would become very slender (Class 4) and the resistance would decrease significantly. In my judgment, in this level the mixture of standards also can lead to errors; therefore it is not used. The composite plate girder bridges in Hungary typically build up from two main girders or more but with relatively high girder distance. On the contrary, in the US the multiple main girders with smaller distances are typical. This could be the answer why the stiffened sections are not considered. – The Swiss method uses the plastic collapse with the rotation limit as the ultimate limit state. In my view, the shakedown approach is more promising. If the degradation problem of the shear connectors was solved, the live load level could be reduced and structures that are more economical could be designed. Therefore, I decided to adopt the shakedown concept. If data on real traffic flow were available, it could be calculated which limit state has higher probability of failure. – Since the mixture of different standards should be avoided, I will use the American results, principles solely where they represent clear mechanical considerations and independent of the philosophy of the norm. The adopted step is the calculation of the required plastic rotations to redistribution, presented in Section 3.3.1.

8.2.

Proposed Method for Plastic Design

The following flowchart (Figure 8.1) is derived from the methods and considerations can be found in the literature and mentioned above and would require more study to fully verify it. However, this exceeds the range of this study. Nevertheless, the assumptions and approximations will be summarized. It should be noted that this chart describes only the firstnegative-hinge scenario. The method is being introduced is mainly based on the newest inelastic AASHTO method [McConelli et al., 2010] and the Swiss method [Lebet, 2011].

92

Redesign of the Bridge Based on Plastic Principles

Preliminary design or existing structure.

- Perform an elastic analysis of the structure to maxim

determine the maximal internal forces ( M el , Ed ). - Determine section properties, resistances (MRd).

Check the applicability conditions for max shear Eq.(3.15) and for bracing Eq.(3.16)

All fulfilled?

No

Yes Capacity

Demand

Calculate the modified plate slenderness:



' p

   Eq.(3.14),  p

p

f





Determine the total rotation required to redistribute the full elastic moment: θtot Eq.(3.9) or Eq.(3.10)

y

cr

Determine the rotation capacity,

 

 ar  p Eq.(3.13) Figure 3.9 '

M

No

max   (1   ar el , Ed

 tot )  M

Negative bending region check  av  tot redistributed moment ratio.

 Rd

Yes

M

No

max  el , Ed

 x  M

resi , Ed

 x



 M Rd  x 

Positive bending region check (Figure 8.2).

Yes

The same procedures as in the elastic design.

Check other limit states and criteria

No

All fulfilled?

Yes

End

Figure 8.1: Flowchart to the shakedown design, derived from the AASHTO and Swiss methods. 93

Redesign of the Bridge Based on Plastic Principles Actually, the most time-consuming and complicated part of the calculation is still the elastic analysis. The redistribution ratio is the following:   ar tot ; of course if we have high rotation capacity ( ar ) the ratio can be lower than the maximum, in order to fulfill other requirements. The symbols used in the flowchart illustrated in Figure 8.2. -pl,Rd maxim el,Ed

L

+pl,Rd

x

resi,Ed

Lresi,Ed

Rresi,Ed x resi,Ed(x) = Lresi,Ed+(Rresi,Ed-Lresi,Ed)·L

-pl,Rd maxim pl,Ed maxim el,Ed resi,Ed

+pl,Rd

Figure 8.2: Concept of moment redistribution and illustration of the symbols. The applicability conditions are identical as in the Swiss method, since the same M−θ curve is applied and the criteria are related to that.

8.3.

Introduction of the Proposed Method through a Trial Design

The aim of the redesign was not to utilize every bit of the material, rather to get clean lines for the structure. More material could be removed from the region of end-supports, sidespan and even from pier-sections by reducing the length where the thicker web is applied (Figure 8.3). In this case, the sections were modified in order to have the first plastic hinges over the piers. This is a reasonable decision since it is easier and cheaper to provide relatively high resistance in the sagging zone. The proposed method also follows this logic; however the first positive plastic hinge scenario should be additionally included. This could be part of the further research. 8.3.1. The Redesigned Structure

The design was performed using the above-mentioned method. To obtain the elastic internal force envelopes midas Civil was used. The finite element model has the same properties as 94

Redesign of the Bridge Based on Plastic Principles the model introduced in Section 5.1.1; moreover, the same considerations were taken into account both on resistance and load side as well. The outlines of the bridges after redesign are illustrated in Figure 8.3. The colors and section lengths are proportional with the actual plate thickness and length, respectively. The numbers in the plates representing their thicknesses and the values before the girders are the widths.

Figure 8.3: The build-up of steel girders following elastic (top) and plastic (bottom) principles. The plastic resistances of the redesigned sections are 30484, 24005 and 25971 kNm in the positive center-span, positive side-span and negative region, respectively. The effect of shear force is taken into account in the negative plastic resistance. These sections with the original elastically designed ones are presented in Figure 8.4 and in Figure 8.5. 2 layers; O20/110

2 layers; O25/100

1750-20

1225

600-40

525

400-20

100

800-60

292,3

1750-25

1225

4815

525

600-20

100

4815

Figure 8.4: Cross-sections at pier, designed following elastic (left) and plastic (right) principles.

95

Redesign of the Bridge Based on Plastic Principles 2 layers; O16/200

2 layers; O16/200

400-20

1750-15

1750-15

800-40

600-40

100

6735

292,3

600-20

100

6735

Figure 8.5: Cross-sections at the middle of the center-span, designed following elastic (left) and plastic (right) principles. The thicker web is provided in the length of the cracked concrete. This means 15% of the adjacent spans at each side of the pier. The consequences of the redesign are listed here; they will be recapitulated in the last section with additional comments:

– –

– – – – –

cleaner outline of the structure, the number of section transitions reduced from 10 to 4, not counting the on-site weldings; the steel girder is symmetric; the asymmetry of the temporary support placing does not break the symmetry. As illustrated in Section 5.5 it has minor effect on the internal force distribution and could be neglected; nevertheless, the sequence of construction is taken into account in the calculation; 4 different plate thicknesses instead of 8; 25% structural steel saving in respect of the bare steel girders; 41% reinforcement saving in the pier region; the bracings rearranged, they placed with higher density in the region of negative plastic hinge, while keeping their original number; more uniform moment distribution;

8.3.2. Verification of the Trial Plastic Design

Hereinafter, I will go through the steps of the verification procedure presented in Figure 8.1. 8.3.2.1. Available Rotation

The M−curves proposed by Lebet and Lääne is used. It is derived for very slender plate girders without any longitudinal stiffeners. The results were extrapolated and checked in the realm of more stocky plates as well. Lebet (2011) suggests a formula to take into account the effect of a longitudinal stiffener in the compression zone, Eq.(3.19) and Figure 3.14. This way, the method is extended to girders with one stiffener in a specified location. Unfortunately, this still covers only limited type of plate girders. Lääne and Lebet (2005) also mention that the contribution of the stiffener is materializing in the increase of critical elastic stress. Following this idea it seems to be more reasonable to use the general expression for the calculation of the relative plate slenderness (EN 1993-1-5:2006 10 (10.2)), Eq.(8.1). 96

Redesign of the Bridge Based on Plastic Principles

p 

fy  ult ,k   cr  cr

(8.1)

This way, the relative slenderness of plates with arbitrary stiffness configuration can be determined calculation of the critical elastic stress in case of orthotropic plates can be complicated. However, using the provisions of EN 1993-1-5 and for example a FEM program to determine the critical stresses it can be done easily. The modified plate slenderness then can be calculated by Eq.(8.2). The 2∙ multiplier takes into account the favorable effect of the plastification of the tension zone, where relevant.  2     p if   0, 5 if   0,5  p

 p'  

(8.2)

With the actual section build-up in the negative zone (Figure 8.4) the subpanels of the web are in at least Class 2 according to EN 1993-1-1. The “global” buckling of the stiffened web was checked in EBPlate, this predicted the local buckling of the subpanels as first eigenshape. According to this, the buckling of the orthotropic plate is not governing. Determination of the relative slenderness

hw·

hw

Since the EBPlate cannot handle that type of loading when the PNA is located in the examined plate, it was assumed that the subpanel under uniform compression with hinged supports at the PNA has the same slenderness as the whole arrangement (Figure 8.6).

Figure 8.6: The original (left) and simplified (right) arrangements. This assumption was checked for the case of unstiffened plate, where the formulas are available and easier to handle. Using the Class 1 limits for unstiffened plate under uniform compression (f1()) and under compression and bending (f2()) the Class limits are the following, in respect of the place of PNA:

f1 ( ) 

 396   13    1 if   0,5 f 2 ( )    36   if   0,5  

33  



97

Redesign of the Bridge Based on Plastic Principles

Figure 8.7 illustrates the functions and their difference in respect of the compressed part () ratio. 3

1.510

0

3

1.210

2

900



600

6

300 0

5%

4

f

8 0

0.25

0.5



0.75

1

 10

0

0.25

0.5



0.75

1

Figure 8.7: Comparison of the simplified (f1(), red) and accurate slenderness limits (f2(), blue, left) and their difference in percentage (right). The very same trend was observed for other Class limits. Identical behavior was assumed for the stiffened plate. The compressed portion of the web in this particular case is = 0,772. The critical elastic stress of the partially compressed panel was determined by EBPlate; the corresponding first eigenshape is presented in Figure 8.8.

Figure 8.8: The first eigenshape of the stiffened web (EBPlate). The related relative plate slenderness:

 p ,1 

355  0,527    ar ,1  63 mrad 847,1

It also expected to have rotation capacity corresponding to Class 1 since the compressed subpanels are in Class 1 according to EC and the EBPlate predicts the local subpanel buckling as first eigenshape.

98

Redesign of the Bridge Based on Plastic Principles The calculation was done for the fully compressed web, to measure the effect of this approximation.

 p ,2 

355  0,527    ar ,2  29, 2 mrad 334,8

As later turned out, even this lower rotation capacity − determined by rough estimation − is sufficient to redistribute the desired amount of moment towards the sagging regions. These rotation capacities with their slenderness are illustrated in Figure 8.9. 70 52.5

 ar mrad

35

17.5 0

0

0.33

0.67

1

 p' 

1.33

1.67

2

Figure 8.9:  p'  ar curve with the partially (blue) and fully compressed web (green). 8.3.2.2. Required Rotation

The value of required rotation is taken from the AASHTO rotation compatibility approach, Eq.(3.9) and Eq.(3.10). Since only basic mechanical considerations were used to derive them, they can be applied here as well. 8.3.2.3. Incremental Collapse Check

For the verification against incremental collapse Melan’s theorem is adopted. This ensures that if the maximal moment envelope added to an arbitrary statically admissible selfequilibrated moment field does not violate the plastic resistances at any point than the true incremental collapse load is greater than or equal to the actual loads. The check of the plastic fatigue limit state is not necessary, can be verified by inspection. With the actual values, for the particular 142/k bridge the moment envelopes are presented in Figure 8.10.

99

Redesign of the Bridge Based on Plastic Principles

30426

30270

kNm

maxim el,Ed

20710

25997

resi,Ed 4299

4455

25971

25971

maxim pl,Ed maxim el,Ed resi,Ed

30374

Figure 8.10: Various moment envelopes for one girder of the examined bridge (t=100 years). The above reduction corresponds to a 14,6 % moment redistribution ratio. Higher redistribution would demand higher resistance in the positive zone. The residual moment  equals to   M elmax , Ed . Using the formula (Eq.(3.9)) from AASHTO rotation compatibility approach to determine the required rotation at the pier we get:  80  20 1  0,146  14, 6mrad .

 ar  63 mrad  29, 2 mrad    rr  14, 6 mrad The negative section has the required rotation capacity to redistribute 15% of its moments. The verification of the section in the center span under maximum positive moment as follows:



 L R M elmax l 2   M resi  M resi , Ed  , Ed , Ed



2   25997   4455  4299  2  30374 kNm  M pl , Rd  30484kNm

The verification of the sidespan-section under sagging moment was performed assuming that the positive hinge will form at the most unfavorable location corresponding to plastic collapse. This section is at the following distance from the abutment, using Eq.(5.9): 25971 1 1  rR  1 30484  l   30m  12, 71m 25971 30484 rR 1

100

Redesign of the Bridge Based on Plastic Principles At this point the positive moment:  L M elmax   M resi , Ed   , Ed 

 l

 20270  4455 

12, 71  22157 kNm  M pl , Rd  24005kNm 30, 0

With this we verified that the structure can shake down under the particular loads. The residual displacements and moments developed in the structure in order to shake down are presented in Figure 8.11. The maximum deflection is about 1/1000 of the corresponding span. This is close to the deflection form the characteristic value of the traffic load, which is 48,9 mm. 30000

40000

30000

resi,Ed 5300

5072

4,25+2,59=6,84 mrad

4,19+2,42=6,61 mrad

[kNm]

eresi 13,4

35,9

12,7

[mm]

Figure 8.11: The maximum residual moment and displacement field after shakedown. After the favorable residual moment field has developed the structure carries its loads in a purely elastic manner. This means that the deflections can be determined by adding the elastically calculated deflections to the residual ones. The deformation is an interesting issue since due to the shrinkage and creep it is a challenge to “set” the shape of composite structures even without plastic deformations. This about 1/1000 residual deflections could be eliminated by pre-cambering the deck. It should be noted that probably the structure would get lower loads than those considered in the design. Therefore, the actual residual moments and deflections will be lower or even zero. This makes more complicated to obtain the desired shape of the structure. It should be pointed out that the main problem with the negative moment is the plate buckling; this has a quite significant effect. Since the PNA is close to the top flange a considerable part of the web is under compression. The solutions introduced in Chapter 6 can effectively tackle this problem, for example, as supported by calculation, the double composite section would significantly reduce the compressed portion of the web, susceptible to buckling. It is vital to provide the ductility of the sections in the hogging area. 8.3.2.4. Stress and Crack Control at SLS

Since the cracks and stresses suspected to govern the design, these were checked. It should be noted that these methods are not able to contemplate the effect of plastic rotations, which may significantly increase the crack widths. The stresses are not influenced by this effect, since where it seemed to be more unfavorable the residual moments were taken into account. 101

Redesign of the Bridge Based on Plastic Principles Verification of crack control at SLS (EN 1994-2:2005 7.4 Simplified method) 0,4mm crack width limit was chosen, keeping the total area of the reinforcement and reducing its diameter; stricter limits could be verified as well. The maximum tension stress in the reinforcement in quasi-permanent load combination (the thermal action is accompanied in it) immediately after cracking, considering the effect of tension stiffening as well:

 qpmax,s  221, 4MPa The applied reinforcement as illustrated in Figure 8.4 is Ø20/110 in two layers.

1) Minimum reinforcement This minimum reinforcement requirement ensures that the rebars will be able to take the tension forces relieved form the concrete after its cracking.

as ,min  2758

mm2 mm2  as ,applied  5712 m m

2) Control of cracking due to direct loading

This criterion ensures that the crack width does not exceed the limit. Prescribes a maximum max bar spacing distance to diameters and reinforcement stresses (  qp , s ). smax  250mm  sapplied  110mm

Limiting stresses at SLS

1) Characteristic combination Structural steel Criterion: The maximum stress in the structural steel should not exceed its yield strength (EN 1993-2:2006 7.3 (1)). In characteristic combination the maximum stresses over the pier from elastic calculation well exceeds the yield stress. It does not seem to be a problem since the steel will not yield every time the characteristic load combination occurs. It will yields once and then carries every further load less than or equal to that in a purely elastic manner; the structure has shaken down in a lower load level. Moreover, the section was designed to able undergo large plastic deformations. Furthermore, with the plastification and small additional rotations some self-equilibrated forces would be relieved. Actually, due to the construction sequence this yielding seems to be inevitable, since one of our aims is to equate, smear and relieve these very inefficiently developed stresses.

102

Redesign of the Bridge Based on Plastic Principles Since this stress limit does not related to cracking and prescribed to elastic design, it appears to me that it is not applicable to plastic design. It should be noted that the characteristic value of the traffic load corresponds to a heavy load with 1000-year return period. Reinforcement The reinforcement is in a “better situation” than the structural steel, since it has higher yield limit and gets stresses only after the establishment of the connection between the concrete and steel. Unacceptable cracking or deformation may be assumed to be avoided if, under the characteristic combination of loads, the tensile stress in the reinforcement does not exceed the following value (EN 1992-1-1:2004 7.2 (5)), without taking into account the favorable effect of the residual field due to shakedown: max  char , s  354 MPa  k3  f yk  0,8  500  400 MPa

Without taking into account the favorable effect of the residual field due to shakedown. Concrete

Compression stress limitation to avoid longitudinal cracks and consequently the reduction of durability (EN 1992-1-1:2004 7.2 (2)): max  char , c  12,3MPa  k1  f ck  0, 6  30  18MPa

2) Quasi-permanent combination The limitation of maximum compression stresses in concrete should be fulfilled to avoid microcracks and nonlinear creep. If the stresses do not exceed the 45% of the characteristic value of compressive strength (cylinder) the linear creep model is valid. It is the verification of the assumption made at the beginning of the analysis (EN 1992-1-1:2004 7.2 (3)):

 qpmax,c  6,1MPa  k2  f ck  0, 45  30  13,5MPa According to the Eurocode the thermal action should be considered as quasi-permanent load as well, with 2=0,5. The more unfavorable thermal action was taken into account every location. During the determination of the stresses some safe-side approximations have been made in order to simplify the calculation, even with these the previous limits are not violated. The calculation above is only a rough estimation, since the method cannot contemplate the effect of plastic rotations. It should be noted that there are other SLS limits which should be satisfied to avoid extensive cracking, like the maximum tension stress in the reinforcement or concrete in characteristic load combination. As expected, the verification with methods developed for elastic analysis (do not take into account the plastic rotations) all criteria are fulfilled. Due to the reduced section dimensions the stresses are slightly increasing. As it has already mentioned only the stress limits can be 103

Redesign of the Bridge Based on Plastic Principles considered valid; the crack widths due to the plastic deformations are certainly higher. However, no procedure exists to check it, requires further research. 8.3.3. Comparison of the Findings to the American Results

American researchers, Barth and White investigated the effect of inelastic design methods to different limit states compared to elastic design. Albeit these calculations follow the AASHTO provisions, the results can be used to assess and approximately judge the effect of inelastic design. The plastic design is based on a previous inelastic provision, not on the rotation compatibility approach. This method involves the calculation of the inelastic rotation capacity of the section and based on this the amount of redistribution is determined. The examined structure was a tangent, three-span, continuous composite, I-girder bridge with spans of 43,0 x 53,0 x 43,0m. A typical cross-section of it presented in Figure 8.12.

Figure 8.12: Typical cross-section of the examined bridge [Barth and White, 2000]. Comparison of different limit states for elastic (first yield, based on AASHTO LRFD) and inelastic design are presented in Table 8.1. The performance ratios mean the ratio of the particular capacities and demands. The highlighted entries are the ratios which are influenced by the inelastic design. Differences to elastic design: – –

bracings are rearranged in favor of the internal pier zone; only three plate thicknesses are used (14, 20, 30 mm) versus the five plate thicknesses for elastic design (14, 20, 25, 30, 45 mm); – structural steel saving in favor of inelastic design (133 → 128 kg/m2) [Barth and White, 2000]. As can be seen from the values, in this particular case, the amount of structural steel is not reduced considerably (~4%), conversely the fabrication cost, due to the less section-transitionsection, may reduced significantly. The researchers performed serviceability check as well, however no crack width verification; probably this SLS limit was added later to the AASHTO. Based on data summarized in Table 8.1 it can be concluded that the application of inelastic principles does not change the performance notably, and the critical limit states are identical in both cases.

104

Redesign of the Bridge Based on Plastic Principles

Table 8.1: Performance ratios for elastic and inelastic designs [Barth and White, 2000]. POSITIVE MOMENT SECTIONS STRENGTH I Limit States Flexure, end spans, compact section (1996 Iterims, 1,1My limit) Flexure, interior span, compact section (1996 Iterims, 1,1My limit) Ductility (1996 Interims) Shear, stiffened web end bearing Fatigue and Fracture Limit States Base metal at connection-plate weld to bottom flange (at cross-frame closest to mid-span) Maximum concrete tensile stress/cracking stress (at above location) Web requirements, flexure Web requirements, shear and bearing Service Limit States Live-load deflection, end span Live-load deflection, center span Permanent deflection, tension flange (SERVICE II) Constructability Web slenderness Compression flange slenderness Compression flange bracing Shear in second panel from end bearing INTERIOR PIER SECTION STRENGTH I Limit States, noncompact section Web slenderness Compression flange slenderness Compression flange bracing Flexure, compression flange Flexure tension flange Fatigue and Fracture Limit States Shear conn. weld to top flange Bearing stiffener/connection weld plate to top flange Web requirements, flexure Web requirements, shear Service Limit States θp/θRL Elastic reinforceing steel stress Constructability Flexure, tension flange 1

Elastic

Inelastic

0,779

0,943

0,721

1,038

0,274 0,890

0,274 0,873

0,968

0,980 1

(1,108)

(1,083)1

0,162

0,068

0,337 0,492

0,422 0,479

0,328 0,345

0,351 0,359

0,746

0,794

0,692 0,698 0,815 0,274

0,657 0,730 0,760 0,267

0,924 0,544 0,924 0,912 0,953

0,871 0,952 0,930 NA NA

0,607

0,805

0,485

0,659

0,626 0,888

0,720 0,590

NA NA

0,547 0,445

− 

0,661

Values in parentheses are based on the steel section only for calculation of stresses due to negative moment.

105

Redesign of the Bridge Based on Plastic Principles

8.4.

Conclusions

In case of the Eurocode plastic design, the structural steel saving, due to choosing the incremental collapse as the ultimate limit instead of first yield, is about 25%. The number of the section transitions is reduced significantly as well, not counting the on-site weldings, the transitions changed from 10 to 4 and the applied plate thicknesses from 8 to 4. If we consider the composition of the total cost of a steel girder (Figure 3.4) it can be seen that the fabrication costs are in the same magnitude as the material costs. Another important consequence is the much cleaner outline of the structure. The steel girder is symmetric; the asymmetry of the temporary support placing does not break the symmetry. As illustrated in Section 5.5 it has minor effect on the internal force distribution and could be neglected; nevertheless, the sequence of construction was taken into account in the calculation. The material saving corresponds to the stiffness reduction and consequently the slight increase of the deflections. The redistribution yielded to a more uniform moment distribution. The bracings were rearranged; they placed with higher density in the region of negative plastic hinge, while keeping their original number. The utilization of the most loaded cross-section is 1,0 in case of the original structure for first yield; due to the scaled loads. The same, full utilization was achieved in case of the redesigned structure for the most critical collapse mechanism. Albeit these utilizations are identical, the other sections are not fully exploited in both cases. Still considerable amount of material is in the webs especially in hogging region. The bigger thickness is required to avoid the plate buckling not for the higher resistance, this could be lowered by using one of the solutions presented in Chapter 6. The amount of reinforcement in the negative zone was also reduced to avoid a highly asymmetrical cross-section, this lowered the PNA and therefore, the web could be classified as Class 2. This reduction is yielded to ~41% saving in the negative region. Although the sections obtained by inelastic design are not optimal solutions it does not yield to any problem since in the original design for example the sections in sagging region are not fully utilized. In serviceability limit states the stresses and crack width were checked by using the methods proposed to elastic design, since no other available. These calculations showed that the redesigned structure fulfills every relevant condition. However, because the methods cannot take into account the effect of plastic rotations, it requires further investigation.

106

Summary and Conclusions

9.

Summary and Conclusions

9.1.

Summary

This thesis investigated the plastic capacities of composite plate girder bridges in the philosophy of Eurocodes. Its goal was to determine the magnitude of reserves currently ignored in the design and to examine the safety level related to them. In order achieve to this goal the theoretical background and related literature was overviewed (Chapter 2-3). The latter comprises the only available AASHTO procedures and the reports of completed and ongoing researches on the topic. Moreover, since there is no standardized method exists in the Eurocode for plastic design, the reliability analysis of structures was reviewed as well, in order to establish a sound base to conduct the reliability analysis. To study the consequences of plastic design an existing structure was chosen. It is a threespan continuous composite plate girder bridge, introduced in Chapter 4. To this, the rating factors of various limit states such as: first yield, first plastic hinge, incremental collapse, single girder and system plastic collapse − assuming sufficient cross-sectional rotation capacity − were determined in Chapter 5. All of the calculations were conducted in the framework of Eurocode. The rating factors were evaluated with the upper and lower bound theorems as well. Furthermore, the effect of the shear force was also investigated. Innovative structural solutions were introduced in Chapter 6 to show some possibilities how to enhance the ductility of the sections. These also increase the shear resistance; therefore, the reduction of moment resistance due to the interaction may be lowered or even neglected. The safety levels of the before-mentioned limit states were evaluated in Chapter 7 using FORM and SORM methods. These calculations are based on publications available on the topic and compatible with the Eurocodes. In Chapter 8, based on the American and Swiss methods, the redesign of the structure was performed following plastic principles. The incremental collapse limit state was adopted incorporating the limited rotation capacity of the sections. The cost savings and consequences of the new design were also investigated.

9.2.

Conclusions

During the recent decades intensive research have been performed on the plastic design concepts of steel and composite bridges. Researchers from the United States played and still playing leading role in this topic. They adopted the incremental collapse as ultimate limit state and worked out design provisions to support the practical application. Few other researchers are also contributed to the topic from all over the world; maybe the results of Swiss researchers should be highlighted. They accepted the plastic collapse ULS and, as the Americans considered as well, the limited rotation capacity of the sections. Based on these researches some fundamental methods and considerations are adopted to evaluate the ultimate

107

Summary and Conclusions capacity of an existing bridge (142/k), a typical two-girder, composite highway overpass with three spans. Since the Eurocodes does not contain method for global plastic analysis and design of bridge structures, an EC based method is developed to evaluate the ultimate loadbearing limit of the bridge using the incremental collapse ULS. This procedure is applied to redesign the selected structure. The conclusions and finding of this thesis are partitioned into two portions. The first one could be referred as general and the second as special conclusions. The former contains the generic findings while the latter summarizes the numerical results corresponding to the particular bridge. These are not entirely separable, so some overlappings are inevitable. After reviewing the relevant literature and completing this study the following general conclusions are deducted: On the plastic reserves of girder bridges: – On one hand, every structure has the cross-sectional reserve; this can be described by the shape factor and depends on the particular build-up. The other source is coming from the global redundancy. Independently from the actual structure, it can be said that these reserves are worth to exploit. Due to the high live to dead load ratio and moving load, the application of the shakedown limit state seems to be reasonable.

On the plastic design: – If we neglect the stresses induced by shrinkage, creep and thermal actions − since they are relieved during the successive reduction of the degree of redundancy − the global structural analysis becomes considerably simpler. Moreover, the effect of the sequence of the construction may be neglected as well. The American methods do not take into account these global (secondary) effects even in the first plastic limit state. – Based on the data available in the literature, a method compatible with Eurocode was suggested to perform the incremental collapse verification. This is mainly based on the rotation compatibility method (AASHTO) and on the Swiss method [Lebet, 2011]. – The formulations of plastic hinges are allowed and desirable in ULS for buildings and under seismic loads for any structures. Therefore, these provisions should be checked and try to implement where it is relevant and possible to bridge structures for shakedown limit state. In seismic design the capacity design is widely accepted and used. This states that the intended plastic hinge locations should be designed to be able to form the hinge. Moreover, the structure should be designed in such a way to ensure the formulation of plastic hinges in the planned locations. This achieved by applying overstrength factors. Due to economic reasons the girder bridges are constructed with many section-transitions, hence it often requires tedious work to check every possible mechanism and hinge-locations in case of plastic design. This could be simplified by applying of the overstrength principle of capacity design. Furthermore, this would ensure that the plastic hinges formulate at the locations where the designer imagined.

108

Summary and Conclusions – During the plastic design the limited rotational capacity of the cross-section should be taken into account. This available rotation determines the amount of maximum redistributable moment. – It is expected that the plastic design yield to a structure with considerably less section transitions and structural steel saving over the elastic design. Moreover, due to the plastic rotations the moment envelope becomes more uniformly distributed.

About ensuring the ductility of the cross-sections: – The structural solutions introduced in Chapter 6 showed that the performance of the sections could be significantly increased. Their resistance and ductility, both under bending and shear, are superior over the conventional solutions. Therefore, they seem to be an appropriate choice for plastic design.

Among the still unsolved problems the degradation of the shear connectors and the verification of serviceability limit states should be highlighted. To prevent the former issue it appears to be sufficient to use the default, standardized traffic load level. The SLS limit states, especially the crack control would require more research, since only methods related to elastic design are available. On the bases of the evaluated results on the selected bridge the following main conclusions are deducted: – The ultimate load levels corresponding to various ULS showed that there are significant reserves in the composite plate girder bridges. The rating factors increased over the first yield with 37% and 75% using the first plastic hinge and incremental collapse limits, respectively. – The results related to plastic collapse show even higher capacities. With the single-girder and system collapse over the incremental limit state additional 10% and 40% reserves can be achieved respectively. These are not exploitable; however, give an idea about the true strength of our structures. – The calculations contemplating the shear force showed that it has significant influence on the ultimate load level. For the particular bridge about 6-18% reduction was experienced compared to the case where it was neglected. The amount of reduction significantly depends on the structural arrangement. – The reliability analysis showed that the safety levels of the above-mentioned limit states are at least reach or exceed that of the first yield or first hinge. It should be noted that these contain many safe side approximations, and the actual values of the reliability indices are considerably higher in case of shakedown limit states. – The redesign of the structure − based on suggested method − yielded to the following: – 25% structural steel saving in respect of the steel girders; – 41% reinforcement saving at the pier region; – cleaner outline; 4 different cross-sections instead of 10, and 4 different plate thicknesses instead of 8. – The relevant serviceability states were verified with the methods prescribed for elastic design in EC. It should be noted that albeit these do no predict extensive cracking of the

109

Summary and Conclusions negative zone, it is likely to occur due to the plastic rotations in ULS. No provisions or research papers are available on this topic. On the grounds of the results, it can be concluded that the plastic methods are a promising way to design structures that are more economical and to extend the range of applicability of composite bridges. The plastic principles are also utilizable to verify the load bearing capacities of structurally deficient bridges and thus extend their working life. The knowledge of the inelastic capacity of our bridges could be advantageous in case of catastrophic or war events as well. When − for the move of heavy equipments − the quick estimation of the true load bearing capacity of the structures is vital. Nevertheless, many other aspects of it still should be cleared; the last chapter summarizing some of these and pointing out the problems which demand further research.

9.3.

Further Research/ Future Work

This section summarizes the issues and questions raised during the completion of this thesis; to these I did not find any research data or solution. The following list contains the problems that are in judgment worth or should be examined: – Extension of the reliability analysis to a higher modeling level, with direct consideration of the time and involving every relevant variable and their correlation. – More calculated examples would require to generalize the findings of this study. Furthermore, the study should be extended to every EC prescribed limit states and to lifecycle cost assessment, taking into account the cost of fabrication, erection, demolition, etc. through the entire life of the structure as well. – Checking other global structural concepts or bridge types such as reinforced concrete or integral bridges. Since the latter own higher degree of redundancy their plastic reserve worth investigating. – Investigation of the phenomenon of shear connection degradation, and as related to this the applicability of a reduced live load level or increased resistance. – Because the theorems of shakedown analysis lead to an optimization problem, metaheuristic algorithms could be applied to find quasi-optimal solutions to the highly complex design problems. – Detailed, deep investigation of the deflections, plastification taking into account the nonlinear behavior of the shear studs as well. – Extension of the moment−rotation curves; investigation of the effect of special structural solutions, introduced in Chapter 6, on the available rotation. – A design procedure to incremental collapse in the philosophy of Eurocodes should be worked out involving the first-positive-hinge scenario as well; verification of the method. – Investigation of the possibility to apply fiber reinforced concrete in the negative hinge region to increase the ductility, to involve the concrete to the load bearing and to reduce the crack widths. Moreover, the effect of plastic rotations on the cracking of the negative zone should be studied.

110

References

References AASHTO. (2010). AASHTO LRFD Bridge Design Specifications, 5th Ed. Washington, D.C.

AASHTO. 978-1-56051-451-0 Barker, M.G., Bergson, P.M., French, C.M., Galambos, T.V. and Klaiber, F.W. (1996).

Shakedown Test of One-Third-Scale Composite Bridge. Journal of Bridge Engineering. Vol. 1. No. 1. February. 1996. Barker, M.G. and Galambos, T.V. (1992). Shakedown Limit State of Compact Steel Girder

Bridges. Journal of Structural Engineering. Vol. 118. No. 4. April. 1992. Barker, M.G., Hartnagel, B.A., Schilling, C.G. and Dishong, B.E. (2000). Simplified

Inelastic Design of Steel Girder Bridges. Journal of Bridge Engineering. Vol. 5. No. 1. February. 2000. Barker, M.G. and Zacher, J.A. (1997). Reliability of Inelastic Load Capacity Rating Limits

for Steel Bridges. Journal of Bridge Engineering. Vol. 2. No. 2. May. 1997. Barth, K.E., Hartnagel, B.A. and White, D.W. (2004). Recommended Procedures for

Simpliefied Inelastic Design of Steel I-girder Bridges. Journal of Bridge Engineering. Vol. 9. No. 3. May. 2004. Barth, K.E. and White, D.W. (2000). Inelastic Desing of Steel I-girder Bridges. Journal of

Bridge Engineering. Vol. 5. No. 3. August. 2000. Bažant, Z.P. and Jirásek, M. (2001). Inealistic Analysis of Structures. Chichester: John

Wiley & Sons, Ltd. 0-471-98716-6. Brozetti, J. (2000). Design Development of Steel-concrete Composite Bridges in France.

Journal of Constructional Steel Research. Vol. 55. No. 2000. 229-243. Bruneau, M., Uang, C.-M. and Whittaker, A. (1998). Ductile Design of Steel Structures.

USA: McGraw-Hill Comp. 0-07-008580-3. Choi, S.-K., Grandhi, R.V. and Canfield, R.A. (2007). Reliability-based Structural

Engineering. London: Springer-Verlag. 978-1-84628-444-1. Collings, D. (2005). Steel-concrete Composite Bridges. London: Thomas Telford. 0-7277-

3342-7. Cornejo, M.O. and Raoul, J. (2010). Composite Bridge Design (EN1994-2) - illustration of

Basic Element Design. Workshop on "Bridge Design to Eurocodes" with worked examples. Vienna, Austria. Davies, J.M. and Brown, B.A. (1996). Plastic Design to BS 5950. Bodmin: Blackwell

Science Ltd. 0-632-04088-2.

111

References Du, X. (2005). Probabilistic Engineering Design. Rolla: Universtiy of Missouri. EN 1990. (2002). Eurocode 0: Basis of Structural Design. EN 1991. (2003). Eurocode 1: Actions on Structures. EN 1992. (2004). Eurocode 2: Design of Concrete Structures. EN 1993. (2005). Eurocode 3: Design of Steel Structures. EN 1994. (2005). Eurocode 4: Design of Composite Steel and Concrete Structures. Ferum.

(02

October,

2011).

FERUM

homepage

[online].

http://www.ifma.fr/Recherche/Labos/FERUM [Accessed: 02 October, 2011]. Flemming, D.J. (1994). Experimental Verification of Shakedown Loads for Composite

Bridges. Minneapolis: MSc thesis, University of Minnesota. Galambos, T.V. (2007). Introduction to the Shakedown Behavior of Steel Frame Srtuctures.

North American Steel Construction Conference, AISC, 2007. New Orleans. Galambos, T.V., Leon, R.T., French, C.M., Barker, M.G. and Dishong, B.E. (1993).

Inelastic Rating Procedures for Steel Beam and Girder Bridges. Washington, D.C.: Transportation Research Board. ISO 2394. (1998). General Principles on Reliability for Structures. Ghosn, M., Moses, F. and Wang, J. (2003). NCHRP Report 489. Design of Highway

Bridges for Extreme Events. Washington, DC: Transportation Research Board. 0-30908750-3. Grundy, P. (1987). Shakedown Design of Bridges. 1st Nat. Struct. Engrg. Conf. Australia. Grundy, P. (2004). Incremental Collapse of Continuous Composite Beams under Moving

Load. 7th Pacific Steel Structures Conference. Long Beach, California. Gulvanessian, H., Calgaro, J.-A. and Holický, M. (2002). Designer's Guide to EN 1990,

Eurocode: Basis of Structural Design. London: Thomas Telford. 0 7277 3011 8. Gulvanessian, H. and Holický, M. (2005). Eurocodes: Using Reliability Analysis to

Combine Action Effects. Structures & Buildings. Vol. No. SB4. 2005. 243-252. Gupta, V.K. (2006). Development of Section Classification Criterion and Ultimate Flexural

Equation for Composite I-girders. Saitama, Japan. Doctoral thesis. Haiyan, Y. and Fangfang, L. (2010). Inelastic Load Capacity Analyzing of Steel I-Girder

Bridges with Shakedown Method. Mechanic Automation and Control Engineering (MACE),

International

Conference

2010.

Wuhan,

PRC.

946-949.

10.1109/MACE.2010.5536250 Halász, O. and Platthy, P. (1989). Acélszerkezetek. Budapest, Hungary: Tankönyvkiadó. 963

18 2268 0. 112

References Hida, S., Ibrahim, F.I.S., Capers, H.A., Bailey, G.L., Friedland, I.M., Kapur, J., Martin, B.T., Mertz, J., Dennis R., Perfetti, G.R., Saad, T. and Sivakumar, B., (2010).

Assuring Bridge Safety and Serviceability in Europe. FHWA-PL-10-014. Honfi, D. and Mårtensson, A. (2009). Reliability of Steel Flexural Members According to

EC in Serviceability Limit State. NSCC2009. Iles, D.C. (2011). Composite Highway Bridge Design: Worked Examples. In Accordance with

Eurocodes and UK National Annexes. Berkshire. UK: The Steel Construction Institute. 978-1-85942-195-6. Georgia Institute of Technology. (2006). Inelastic Design. Lecture notes by Leon R.T. JCSS Probabilistic Model Code. (2000). Jones, R.M. (2009). Deformation Theory of Plasticity. Blacksburg: Bull Ridge Corporation.

9780978722319. Kachanov, L.M. (1971). Foundations of the Theory of Plasticity. Amsterdam: North-Holland

Publishing Comp. 0-7204-2363-5. Kaliszky, S. (1975). Képlékenységtan. Elmélet és Mérnöki Alkalmazások. Budapest:

Akadémiai Kiadó. 963-05-0652-1. Kaliszky, S. (2007). Képlékenységtani Kutatások Magyarországon a Tartószerkezetek

Mechanikája Területén. Építés – Építészettudomány. Vol. 35. No. 2. 2007. 141-158. Kaliszky, S., Lógó, J. and Nédli, P. (1998). Rúdszerkezetek Teherbírásának Számítása és

Optimális Tervezése. Kazinczy von, G. (1914). Kisérletek Befalazott Tartókkal. Betonszemle. Vol. 2. No. 1914.

68-71, 83-87 & 101-104. Kim, H.-H. and Shim, C.-S. (2009). Experimental Investigation of Double Composite Twin-

girder Railway Bridges. Journal of Constructional Steel Research. Vol. 65. No. 2009. 1355-1365. Kurrer, K.-E. (2008). The History of the Theory of Structures. From Arch Analysis to

Computational Mechanics. Berlin: Ernst & Sohn. 9783433018389. Lääne, A. and Lebet, J.-P. (2005). Available Rotation Capacity of Composite Bridge Plate

Girders Under Negative Moment and Shear. Journal of Constructional Steel Research. Vol. 61. No. 3. 2005. 305-327. Lebet, J.-P. (2011). Innovative Design Method, Steel-Concrete Composite Girder Bridges.

Eurosteel 2011. Budapest, Hungary. 1275-1280. Lebet, J.-P. and Nissile, R. (2010). New Method for Design of Concrete-Steel Composite

Plate Girder Bridges. Bern, Switzerland. Publication VSS 640. In French. 113

References Leon, R.T. and Flemming, D.J. (1997). Experimental Verification of Shakedown

Approaches for Bridge Design. Transportation Research Record: Journal of the Transportation Research Board. Vol. 1597 / 1997. No. January. 1997. 50-56. Lubliner, J. (2008). Theory of Plasticity. USA: Dover Publications. 9780486462905. McConelli, J.R., Barth, K.E. and Barker, M.G. (2010). Rotation Compatibility Approach

to Moment Redistribution for Design and Rating of Steel I-Girder Bridges. Journal of Bridge Engineering. Vol. 15. No. 1. January. 2010. Melchers, R.E. (1999). Structural Reliability Analysis and Prediction. Chichester, UK: John

Wiley and Sons Ltd. 0471987719. Mitsuro, I. and Galambos, T.V. (1993). Minimum Weight Design of Continuos Composite

Girders. Journal of Strucural Engineering. Vol. 119. No. 4. April. 1993. Nakamura, S.-I., Momiyama, Y., Hosaka, T. and Homma, K. (2002). New Technologies of

Steel/concrete Composite Bridges. Journal of Constructional Steel Research. Vol. 58. No. 2002. 99-130. Nakamura, S.-I. and Morishita, H. (2008). Bending Strength of Concrete-filled Narrow-

width Steel Box Girder. Journal of Constructional Steel Research. Vol. 64. No. 2008. 128-133. Neal, B.G. (1977). The Plastic Methods in Structural Analysis. London: Chapman and Hall.

0412214504. Nowak, A.S. and Collins, K.R. (2000). Reliability of Structures. USA: The McGraw-Hill

Companies, Inc. 0-07-048163-6. Rizzo, S., Spallino, R. and Giambanco, G. (2000). Shakedown Optimal Design of

Reinforced Concrete Structures by Evolution Strategies. Engineering Computations. Vol. 17. No. 4. 2000. 440-458. Rózsás, Á. (2010). Design of a Steel-concrete Composite Railway Bridge According to

Eurocodes: B.Sc. Thesis. In Hungarian. Saul, R. (2000). Cost Efficient Design and Construction of Major Steel Composite Bridges.

ATS International Steelmaking Conference. Paris, France. Schilling, C.G., Barker, M.G. and Dishong, B.E. (1996). Simplified Inelastic Design of

Steel Girder Bridges. Final Rep., Nat. Sci. Found. Study on Devel. and Experimental Verification of Inelastic Des. Procedures for Steel Bridges Comprising Noncompact Girder Sections. University of Missuri, Columbia, Mo. Sedlacek, G. Traffic Loads on Road Bridges. European Development of EN 1991 – Eurocode

1 – Part 2. 114

References Sedlacek, G., Merzenich, G., Paschen, M., Bruls, A., Sanpaolesi, L., Croce, P., Calgaro, J.-A., Pratt, M., Jacob, Leendertz, M., V. De Boer, Vrouwenfelder, A. and Hanswille, G. (2008). Background Document to EN 1991 - Part 2 - Traffic Loads for

Road Bridges - and Consequences for the Design: JRC European Commission. Sen, R. and Stroh, S. (2010). Design and Evaluation of Steel Bridges Double Composite

Action. Final Report: University of South Florida. Contract No. # BD544-18. Sørensen, J.D. Calibration of Partial Safety Factors in Danish Structural Codes. JCSS

Workshop on Reliability Based Code Calibration. Stüssi, F. (1962). Gegen das Traglastverfahren. Schweizerische Bauzeitung. Vol. 80. No. 4.

1962. 53-57. Unterweger, H., Lechner, A. and Greiner, R. (2011). Plastic Load Bearing Capacity of

Multispan Composite Highway Bridges with Longitudinally Stiffened Webs. Steel and Composite Structures. Vol. 11. No. 1. January. 2011. Vasseghi, A. (2009). Improving Strength and Ductility of Continuous Composite Plate Girder

Bridges. Journal of Constructional Steel Research. Vol. 65. No. 2009. 479-488. Wikipedia01.

(02

October,

2011).

Correlation

and

Dependence

[online].

http://en.wikipedia.org/wiki/Correlation_and_dependence [Accessed: 02 October, 2011]. Wikipedia02.

(14

November,

2011).

Monte

Carlo

Method

[online].

http://en.wikipedia.org/wiki/Monte_Carlo_method [Accessed: 14 November, 2011]. WolframMathworld.

(14

November,

2011).

Monte

Carlo

Method

[online].

http://mathworld.wolfram.com/MonteCarloMethod.html [Accessed: 14 November, 2011]. Wong, M.B. (2001). Plastic Analysis and Design of Steel Structures. Burlington: Elsevier

Ltd. 978-0-7506-8298-5.

115

Design Check to Eurocodes

Annex A

Annex A - Design Check to Eurocode Three-span, composite, plate girder, highway bridge check The following publications are frequently used in the calculation, they also highlighted among the main references: [A1] Iles, D.C. (2011). Composite highway bridge design: Worked examples. In accordance with Eurocodes and UK National Annexes. Berkshire. UK: The Steel Construction Institute. 978-1-85942-195-6. [A2] Hendy, C.R.J., R. P. (2006). Designers' Guide to EN 1994-2. Eurocode 4: Design of Steel and Composite Structures. Part 2: General rules and rules for bridges. London. UK: Thomas Telford Ltd. 0-7277-3161-0.

A-1

Design Check to Eurocodes

Annex A

Contents

I. Elastic design according to the Eurocodes 1

Structural arrangement

2

Design basis 2.1 Partial factors on actions 2.2 Factors for combination values 2.3 Factors on materials 2.4 Factors on resistances 2.5 Structural material properties

3

Actions on the bridge 3.1 Permanent actions 3.1.1 Self weight of structural and non-structural elements 3.1.2 Creep 3.1.3 Shrinkage 3.1.4 Support lifting 3.1.5 Uneven settlement 3.2 Variable loads and actions 3.2.1 Traffic loads 3.2.2 Traffic load groups 3.2.3 Thermal actions 3.2.4 Wind actions 3.2.5 Construction loads 3.3 Accidental loads

4

The sequence of the construction

5

Girder make up and slab reinforcement 5.1 Main girders 5.2 Cross-bracing 5.3 Reinforcement

6

Beam cross-sections 6.1 Section properties - main girders 6.1.1 Section properties and resistances; sagging 6.1.2 Section properties and resistances; hogging

A-2

Design Check to Eurocodes

Annex A

I. Design according to the Eurocodes 1 Structural arrangement The bridge is located on the M0-M6 motorways in Hungary. The flyover carries a 3-lane single carriageway highway road over another road. Its a continuous steel-concrete composite bridge formed by three spans of 30,0 - 40,0 - 30,0 m and with a 13,47 m wide deck. The cross-section is composed of two constant depth I-girders with a reinforced concrete slab on top of them, in total about 1,85 m height. The main girders distance is 7,5 m, they connected in a 5,0 m raster with a cross-bracing formed of rolled HEA sections. The deck is haunched at the top of the girders, its average thickness is around 28 cm.

Elevation

Cross section (at a cross girder in the range of internal support)

2 Design basis The bridge is checked in accordance with the Eurocodes, applying the generally recommended values for partial factors and other variables where relevant. The basis of design set out in EN 1990 is verification by partial safety factor method. Where the standard offers the designer options (like how accurately consider the effective widths) I always chose the possibility which was closer to the original design, to establish a more or less solid base to the comparison. Nevertheless, even keeping this in mind due to the sometimes significant differences, the results are representing rather a qualitative than quantitative comparison. A-3

Design Check to Eurocodes

Annex A

The most prominent difference is that the MSZ ÚT standard applies the allowable stress method (the safety is lumped into one safety factor which is applied to the yield stress) while the Eurocodes adopts the partial safety factor method (the safety factors are distributed to various effects and resistances). The EC is based on the concept of limit state design. During the calculation the following differences have found: - significant difference in the live load; - small differences in the dead and construction loads; - creep and shrinkage, the standardized model seems to be very similar, however the; approach applied in the finite element model is slightly overestimates the stresses from these effects; - the lateral load distributions slightly differs, the single beam model of the original design with the simply supported lateral load distribution overestimates the loads on one girder. The design check is carried out in a simplified manner, since the inelastic design is emphasized; some local checks are ignored and simply accepted the results of the original as adequate.

Load combinations Ultimate limit state (ULS) Ultimate limit state other than fatigue The ultimate limit state STR is verified for persistent and transient design situation with the following combination formula: EN-1990:2002 (6.10a and b)

Fatigue EN-1992:2004 6.8.3 (6.14b)

non-cyclic combination - in bracket - combined with the cyclic load Serviceability limit state (SLS) Characteristic combination EN-1990:2002 (6.14b)

For verification of stresses in concrete, structural steel and reinforcement. To verify the deflections due to the live-load. A-4

Design Check to Eurocodes

Annex A

Frequent combination EN-1990:2002 (6.15b)

Quasi-permanent combination EN-1990:2002 (6.16b)

For the verification of crack widths in the deck. To verify the applicability of linear creep to the concrete. Comments to the combinations: Wind actions and thermal actions need not be taken into account simultaneously unless EN-1990:2005 otherwise specified for local climatic conditions. A2.2.2 (6) The effects of creep and shrinkage of concrete, temperature, uneven settlement effects EN-1994-2:2004 5.4.2.2 (7) and the effect of sequence of construction may be neglected in analysis for 5.4.2.5 (2) verifications of ultimate limit states other than fatigue, for composite members with all 5.1.3 (2) cross-sections in Class 1 or 2 and in which no allowance for lateral-torsional buckling 5.4.2.4 (2) is necessary. There is no reasoning for this in the standard, but probably it is due to the plastification. 2.1 Partial factors on actions Permanent actions self weight of materials

EN-1990:2005 Annex A2 Table A2.4(B)

γG.sup  1.35

sufficient to consider only the upper bound value shrinkage

γsh  1.0

creep

γcr  1.0

uneven settlement

γGset  1.2

EN-1992-1-1:2004 2.4.2.1

in case of elastic analysis

EN-1990:2005 Annex A2 Table A2.4(B)

If non-linear analysis is carried out γ Gset=1,35 should be applied. Variable actions road traffic actions

γQ.t  1.35

other variable actions (wind actions, thermal actions)

γQ  1.5

partial factor for equivalent constant amplitude stress range

γF.f  1.0

EN-1990:2005 Annex A2 Table A2.4(B)

No values are given for transient situations (such as during construction) but it is assumed that the above factors for permanent actions may be used. A-5

Design Check to Eurocodes

Annex A

2.2 Factors for combination values Traffic loads: gr1a, LM1-TS

Ψ 0.LM1.ts  0.75

Ψ 1.LM1.ts  0.75

Ψ 2.LM71.ts  0

gr1a, LM1-UDL

Ψ 0.LM1.udl  0.4

Ψ 1.LM1.udl  0.4

Ψ 2.LM1.udl  0

gr1a, pedestrian+cycle

Ψ 0.ped  0.4

Ψ 1.ped  0.4

Ψ 2.ped  0

persistent design situationΨ0.w.p  0.6

Ψ 1.w.p  0.2

Ψ 1.w.p  0

EN-1990:2005 Annex A2 Table A2.1

Wind forces: FWk

execution F*W

Ψ 0.w.p  0.8

-

Ψ 1.w.p  0

Ψ 0.w.p  1.0

-

-

Thermal actions:

Ψ 0.th  0.6

Construction loads:

Ψ 0.con  1.0

Ψ 1.th  0.6

-

Ψ 2.th  0.5

Ψ 2.con  1.0

Where FWk is the characteristic wind force and F *W is the wind force compatible with the road traffic. 2.3 Factors on materials structural steel

γa  1.0

reinforcing steel

γs  1.15

reinforced concrete

γc  1.5

EN-1993-1-1:2005 2.4.1 (1)P EN-1992-1-1:2004 Table 2.1N

2.4 Factors on resistances for resistance of cross-sections whatever the class is

γM0  1.0

for resistance of members to instability assessed by member checks

γM1  1.10

for resistance of cross-sections in tension to fracture

γM2  1.25

for fatigue strength

γMf

A-6

Design Check to Eurocodes

Annex A

for fatigue strength of studs in shear γMf.s for design shear resistance of a headed stud

γv  1.25

EN-1994-2:2005 2.4.1.2 (5)

2.5 Structural material properties concrete - C30/37 N

characteristic value of the compression strength

fck  30.00

design value of the compression strength

fck N fcd   20.00  γc 2 mm

2

mm

2 3

mean value of the tensile strength

 mm2  N N   fctm  0.3  fck  2.896  N  2 2  mm mm

2   mm fck 8   N mean value of the elastic modulus Ecm  22   10 

0.3



νc  0.18

mean value of the shear modulus

Ecm Gc   13.91  GPa 2  1  νc

strain at crushing (in bending)

ε cu  3.5‰

stress reduction factor

α  0.85

2

 32.84 

kN 2

mm

EN-1992-1-1:2004 Table 3.1



kN

kN

kN

volumetric weight of the reinforced γrc  24 1  25 3 3 3 concrete m m m structural steel - S355 to EN 10025-2 N

design value of the yield stress (t<40mm)

fy.1  355

design value of the yield stress (80mm
fy.2  335

modulus of elasticity

Ea  210GPa

Poisson coefficient

νa  0.3

shear modulus

Ea Ga   80.77  GPa 2  1  νa



kN mm

Poisson coefficient



EN-1992-1-1:2004 Table 3.1

2

mm N

2

mm



A-7

EN 1991-1-1:2002 Table A.1

Design Check to Eurocodes

volumetric weight

Annex A γav  78.5

coefficient of thermal expansion

kN 3

m

5

αT  1.2 10

1 /°C

reinforcing steel S500B N

characteristic value of the yield stress

fyk  500

design value of the yield stress

fyk N fyd   434.8  γs 2 mm

modulus of elasticity

Es  210GPa

2

mm

EN 1994-2:2005 3.2 (2)

3 Actions on the bridge 3.1 Permanent actions 3.1.1 Self weight of structural and non structural elements The self weights are based on nominal dimensions. Calculation for a half-section. a) structural steel: The software automatically calculates the weight of the girders, for the verification and comparison to other loads for one selected section the self weight is calculated. tf.b  40mm

b f.b  800mm

tf.t  20mm

b f.t  600mm

tw  15mm

h w  1750mm

Aa  tw h w  tf.t b f.t  tf.b b f.b  702.5  cm

f.t

tf.t tw hw

2

tf.b

kN g a.girder  Aa γav  5.515  m

bf.b

ag  7500mm

distance between the main girders

am  5m

distance between two adjacent cross-bracings

The section of the cross bracing is HEA200, the area of this cross-section is: AHEA200  53.83cm

2

Cross-bracings are smeared on the a m length:

A-8

Design Check to Eurocodes

Annex A

GcrossB  3  ag  AHEA200  γav  9.508  kN GcrossB

g crossB 

am

 1.902 

kN m

Weight of other steel elements such as the inspection footway and studs: g a.other  1

kN m

With a safe-side approximation the weight of the temporary, top cross-bracing is considered during the whole lifespan of the structure. b) Reinforced concrete (RC) deck: equivalent thickness of the deck

v c.eq  29.233cm

total width of the deck

b deck  13470mm

area of the half of it

Ac  v c.eq

load on one girder

kN g rc  Ac γrc  49.22  m

b deck 2

 19688.4  cm

2

c) Surfacing Layers 4 cm wearing course 7 cm binder course 4 cm protection coating 0,2 cm waterproofing

3

Volume weight [kN/m ] 24,0 24,0 24,0 10,0

average length of one surfacing layer lsurf  11013mm g surf 

lsurf 2

kN  γaszf  ( 4cm  7cm  4cm)  γszig 0.2cm  20.088   m

EN-1991-1-1:2001 According to the EN 1991-1-1 the characteristic value of the surfacing weight should be calculated by multiplying the value - obtained by the nominal dimensions 5.2.3 (3) of the layers- by 1,2. It takes into account the uncertainties of the thickness of the pavement. kN g surf  1.2 g surf  24.11  m

d) Raised sidewalk, kerb On the two sides of the cross-section slightly different sidewalks are applied. cross-sectional area of sidewalk 1

As.1  6621.8cm

A-9

2

Design Check to Eurocodes

Annex A

cross-sectional area of sidewalk 2

As.2  5662.2cm

2

kN g s.1  As.1 γrc  16.55  m kN g s.2  As.2 γrc  14.16  m

These loads are placed with eccentricity to the main girders. To be on the safe-side the heavier kerb is placed on both side. e) Parapet: g p  1.0

kN m

f) other (cables, lighting, etc.): g other  1.0

kN m

The permanent loads on one girder are summarized in the following table.

loads on one girder girder's self weight cross-bracing other steel elements RC deck surfacing sidewalk parapet other sum

line load [kN/m] 5,51 1,90 1,00 49,22 24,11 16,55 1,00 1,00 100,30

3.1.2 Creep and shrinkage The necessary properties to determine the shrinkage and creep: N - normal type cement 70% - relative ambient humidity h0=280 mm - notional size (h0=2*Ac/u); Ac-area; u- perimeter exposed to air u - the perimeter of that part which is exposed to drying

A-10

Design Check to Eurocodes

Annex A

It should be noted that the moisture loss is sealed with the application of waterproofing, therefore the notional thickness doubles. However, until the placing of waterproofing the notional thickness can be considered as the nominal thickness of the slab. Also pondering that significant part of the creep is taken place at the first months, it is a safe side approximation to use the smaller value [A2]. Moreover the effect of this notional thickness doubling is investigated calculating the modular ratios for this particular case, the results are summarized in the following table: Creep induced by permanent load h0 = 280 mm t0 day at first

t = 100 years h0 = 560 mm

loading

fi

nLp

fi

7 28

2,424 1,866

23,45 19,52

2,265 1,744

Creep induced by shrinkage h0 = 280 mm t0 day at first -4

loading

eps x10

1

3,274

nLp

dnLp

22,33 4,78% 18,66 4,41%

based on: EN 1992-1:2004 Annex B and EN 1994-2:2004

t = 100 years h0 = 560 mm -4

nLsh

eps x10

18,61

2,945

nLsh

dnLsh

17,81 4,30%

From the table above it can be seen that the notional thickness has no significant effect on the modular ratios, to be on the safe side I will calculate with the thinner one. In the worked example in [A1] (with waterproofing) also the smaller value were applied. For the easier calculation the time-dependent phenomena of concrete are taken into account by using the special capability of midas Civil. It can model the time course of the creep and shrinkage. For the concrete the age at the beginning of shrinkage, 1 day is used. Consequently, to consider in an appropriate way the creep induced by the shrinkage, the concrete elements are activated at age 1 day. Additional comparison was carried out to check results provided by the software, and to verify them. Since the program using the user specified creep and shrinkage functions which are in this case the ones provided by the Eurocode, it is not surprising that the end values of creep coefficient shrinkage strains and the corresponding loads are equal. It is more interesting the compare the actual results of the calculations in respect of stresses and deflections considering the composite section. The Eurocode takes into account the effect of the creep by a modified modular ratio, where the modification depends on the type of the action. For example in case of shrinkage ψsh  0.55 should be applied. The standard distinguishes three long-term actions based on their effects on the creep. These are illustrated in the following figure:

A-11

EN 1994-2:2004 5.4.2.2 (4)

Design Check to Eurocodes

Annex A

The figure illustrates well why a reduced modular ratio should be applied for the shrinkage induced creep. The shrinkage reaches its final value many years after the pouring, therefore the creep induced by it should be less than that of the permanent load. Less creep corresponds to a lower modular ratio. Of course the software takes these effects automatically into consideration. For the comparison a simple composite column (solely considering axial loads) and plate girder (axial load and bending) were used. In case of creep induced by permanent (long-term in time unchanging) load the difference between the deflections and stresses are around 6-7% compared to the standardized method using ψp  1.1, The software provides higher values. A possible reason of this difference can be that the software applies infinitely rigid connection between the steel and concrete parts. This stiffer connection means more restrain and so more stress in the concrete and consequently more creep. In case of creep induced by shrinkage ( ψsh  0.55 for the standardized method) the same trend was experienced, the stresses provided by the software are about 5-6% higher than the values determined by the standardized method. This again probably can be explained by the flexibility of real shear connections. For the sake of simplicity in the global analysis the composite beam element with rigid shear connection were used. Based on the abovementioned results this assumptions have not too much effect, moreover we are on the safe side with it. The primary effect of the shrinkage should not be taken into account in the cracked sections.

EN-1994-2:2004 6.2.1.5 (5)

The limit of linear creep limit should be checked in SLS quasi-permanent combination. 3.1.4 Support lifting With the removal of the shoring the reaction forces in opposite direction are loading the structure. 3.1.5 Uneven settlement Since it was not considered in the original design, I neglect it.

A-12

Design Check to Eurocodes

Annex A

3.2 Variable loads and actions

3.2.1 Traffic loads Number and width of notional lanes Carriageway width w [m]

Number of notional lanes

7,00

2

Width of a notional Width of the lane w l [m] remaining area [m] 3,00

1,00

The bridge belongs to the I. traffic class, the adjustment factors related to this class are the following: αQ.i  1.0 αq.i  1.0 αq.r  1.0

Vertical loads

1. Load Model 1 (LM1): Its characteristic value corresponds to a traffic load 1000 year return period - it means a probability of exceedance 10% in 100 years - it describes the main roads in Europe. These values are mainly determined from traffic-measurements in Auxerre. The frequent value corresponds to a traffic load with 1 week return period on the main roads of Europe. The dynamic amplification is included in the models.

EN-1991-2:2003 Table 2.1 EN-1991-2:2003 4.2 (1)

For global and local analysis. TS (Q)

UDL (q )

Lane 1 Lane 2 Lane 3 Other lanes

Q ik axle loads [kN] 300 200 100 0

q ik (or q rk ) [kN/m ] 9,0 2,5 2,5 2,5

remaining area (q rk )

0

2,5

Location

2

A-13

EN-1991-2:2003 Table 4.2

Design Check to Eurocodes

Annex A

The details of LM1 are illustrated on the following figure.

EN-1991-2:2003 Figure 4.2a

2. Load Model 2 (LM2): For local verification. αQ.1 Qa.k Qa.k  400kN EN-1991-2:2003 Figure 4.2b

Where relevant only one wheel may be taken into account. Vertical loads on footways and cycle tracks characteristic value: q k  5.0

kN EN-1991-2:2003 5.3.2.1 (1)

2

m

A-14

Design Check to Eurocodes

Annex A

combination value for the gr1a load group: q k.comb  3.0

kN

EN-1991-2:2003 Table 4.4a

2

m

In this particular case due to the location of the structure the footway loads are not combined with carriageway loads in the load groups. This also corresponds to the consideration applied through the original design. Fatigue load Fatigue Load Model 3 (FLM3) is used to carry out the fatigue verification. Horizontal loads

3.2.2 Traffic load groups

Only gr1a and gr1b are considered. 3.2.3 Thermal actions

3.2.3.1 Uniform temperature component Temperature during the construction =>

T0  20 °C

Due to the boundary conditions the uniform temperature load does not induce stresses in the structure. 3.2.3.2 Temperature difference component To take into account the effect of the temperature difference the Approach 1 method is used. Linear temperature distribution in the whole depth of the section. The temperature difference between the top and bottom surfaces: A-15

EN 1991-1-5:2003 6.1.4.1

Design Check to Eurocodes

Annex A

Top surface warmer: ΔTM.cool  18 °C EN 1991-1-5:2003 Table 6.1

Bottom surface warmer: ΔTM.heat  15 °C

In case of composite superstructures these values should be modified with a factor (ksur) which takes into account the effect of the surfacing thickness. In current case this value for both cases (top- and bottom surface warmer) is equal to 1,0.

EN 1991-1-5:2003 Table 6.2

Assuming the same heat transfer coefficient for the steel and concrete, there will not develop primary stresses from this effect. 3.2.4 Wind actions

Dynamic response procedure is not needed for the current bridge, since the longest span less than or equal to 40 m and can be considered as a normal bridge.

EN 1991-1-4:2004 8.2 (1)

The wind force in x direction (parallel to the deck width, perpendicular to the span) is EN 1991-1-4:2004 8.3.2 taken into account by the simplified method. a) without live load: m

fundamental value of basic velocity v b0  40 s This action may be governing for the transverse bending or horizontal reactions, in this case is not considered in the global analysis. b) simultaneously with live load: m

fundamental value of basic velocity v b0  23 s

EN 1991-1-4:2004 8.1 (4)

Basic wind velocity:

EN 1991-1-4:2004 4.2 (2)P

v b = cd  cs v b0

where: cd directional factor cs season factor

cd  1.0 cs  1.0

Wind pressure: 1 2 fw =  ρ v b  c 2

where: ρ c

kg

ρ  1.25 air density 3 m wind load factor (from the table below)

A-16

Design Check to Eurocodes

b/dtot

Annex A

ze < 20m ze = 50m

< 0,5 > 4,0

5,7 3,1

7,1 3,8

b  b deck  2  0.205 m  13.88 m d tot  2.71m  2.0m  4.71 m b d tot

EN 1991-1-4:2004 8.3.1 (5a)

 2.947

ze  10m

height above the ground level

From the above table with linear interpolation: c  3.882

With this the wind pressure on the structure and vehicles: 1 kN 2 fw   ρ v b  c  1.284  2 2 m

The reference area subjected to the wind pressure is determined by its height: d tot and its length which corresponds to the loaded length The software is able to apply this pressure load to a beam element and takes into account its eccentricity. The wind load is not likely to be the governing action in case of small and medium span bridges, moreover it should not be combined with the thermal effects as pointed out in Section 2. Nevertheless, it is considered in the global analysis and turned out that is not governing. 3.2.5 Construction loads

a) Personnel and hand tools: Working personnel, staff and visitors, possibly with hand tools or other small site equipment q ca  0.7

kN

EN 1991-1-6:2005 Table 4.1

2

m

b) storage of movable items: e.g.: equipment, building and construction materials q cb  0.1

kN 2

m

Fcb.k  100kN

EN 1991-1-6:2005 Table 4.1

A-17

Design Check to Eurocodes

Annex A

c) nonpermanent equipments: e.g.: formwork panels, scaffolding, falsework,... q cc  0.5

EN 1991-1-6:2005 Table 4.1

kN 2

m

d) additional weight of the unhardened concrete q c.add  1

kN 3

m

 v c.eq  0.292 

kN

EN 1991-1-1:2002 Table A.1

2

m

The additional load on one girders due to the water inside the unhardened concrete and formwork:

qc.add  qcc

b deck 2

 5.336 

kN m

The load on one girder due to the construction loads, without the formwork.





q cs  q ca  q cb 

b deck 2

 5.388 

kN m

A-18

Design Check to Eurocodes

Annex A

4 The sequence of the construction In the original design the construction sequence was assumed in a simplified manner, in respect to no detailed information was a priori available about the construction. This makes an approximation that the whole deck is concreted in one step and their weight is carried by the bare steel girders. Based on the assumptions applied in the original design the following construction sequence was adopted in the FEM model:

A-19

Design Check to Eurocodes

Annex A

5 Girder make-up and slab reinforcement 5.1 Main girders

The numbers in the plates are representing the thicknesses and the numbers before the girder are the widths. 5.2 Bracing arrangements

The bracings are placed in equal distance of 5.0 m in all three of spans. 5.3 Slab reinforcement

The longitudinal reinforcement in regions where the concrete is sound: D16/200 in two layers in the cracked zone: D25 /100 in two layers

6 Beam cross-sections 6.1 Section properties - main girders

Since the geometry of the bridge is rather simple a grillage model is adequate for global analysis. Because in grillage analysis the girders and deck are modeled as beams the shear lag effect is not an inherent property of the system; it has to be taken into account 'manually'.

4,50

6,00

6,00

4,50

When elastic global analysis is used, a constant effective width may be assumed over EN 1994-2:2005 the whole of each span. This value may be taken as the value beff,1 at mid-span for a 5.4.1.2 (4) span supported at both ends, or the value b eff,2 at the support for a cantilever. Since in the original design the change of effective width was considered in the region of supports, I take it into account as well. Effect of the shear lag, calculation of the effective s cross-section properties

at midspan or internal support: A-20

Design Check to Eurocodes

Annex A

2

b eff = b 0 



b ei

EN 1994-2:2005 5.4.1.2 (5.3)

 βibei

EN 1994-2:2005 5.4.1.2 (5.4)

i 1

at end support: 2

b eff = b 0 

i 1

b 0  440mm

distance between the centerline of the outer studs

The illustration of the regions with different effective widths can be seen in the following figure. Le is the approximate distance between points of zero bending moments, which can be assumed as illustrated on the figure in case of typical continuous composite beams.

EN 1994-2:2005 5.4.1.2 (5)

For the ease of modeling the linear change in the effective width is not considered, rather only the constant values are used with sudden change in the midpoint of linear sections. The sections with different effective width are presented in the following figure:

4,00

20,00 beff.0

b'eff.1

6,00 8,00

24,00

beff.2 beff.2

b''eff.1

8,00 6,00

20,00

beff.2 beff.2

b'eff.1

4,00 beff.0

The geometrically available concrete flange, this is the same all over the entire structure:

A-21

Design Check to Eurocodes

Annex A

2985 2765

3750 440

3530

a) End span (l = 30,0 m), b'eff.1: Le  0.85 30m  25.5 m b e.1  b e.2 

Le 8 Le 8

 3.188 m

>

2765 mm (geometrically possible)

→

b e.1  2765mm

 3.188 m

<

3530 mm (geometrically possible)

→

b e.2  3187.5 mm

b´eff.1  b 0  b e.1  b e.2  6392.5 mm

- beff.0: Le  0.85 30m  25.5 m Le    β1  min 0.55  0.025   1.0  0.781 b e.1   

Le    β2  min 0.55  0.025   1.0  0.75 b e.2   

b e.1  β1  b e.1  2158.25  mm

b e.2  β2  b e.2  2390.6 mm

b eff.0  b 0  b e.1  b e.2  4988.9 mm

- beff.2: Le  0.25 ( 30m  40m)  17.5 m b e.1  b e.2 

Le 8 Le 8

 2.188 m

<

2765 mm (geometrically possible)

→

b e.1  2187.5 mm

 2.188 m

<

3530 mm (geometrically possible)

→

b e.2  2187.5 mm

b eff.2  b 0  b e.1  b e.2  4815 mm

b) Midspan (l = 40m), b''eff.1: Le  0.7 40m  28 m

A-22

Design Check to Eurocodes b e.1  b e.2 

Le 8 Le 8

Annex A

 3.5 m

>

2765 mm (geometrically possible)

→

b e.1  2765mm

 3.5 m

>

3530 mm (geometrically possible)

→

b e.2  3530mm

b´´eff.1  b 0  b e.1  b e.2  6735 mm

The structure is symmetric in respect of spans and the concrete slab, thus there is no need to determine the remaining effective widths. EN 1994-2:2005 The effect of cracking of concrete is taken into account in a simplified manner, neglecting the concrete in 15% of the span on each side of each internal support. A 5.4.2.3 (3) similar method with fully neglecting of concrete in pier region was used in the original design. Just to investigate the effect of tension stiffening global analysis was performed with 1/10 modulus of elasticity for concrete in the cracked region.

The section properties calculated by homogenization of the composite section to steel. t=0 open to the traffic t=∞ 100 years, end of design life Only the two sections' properties with the highest negative and positive moment are calculated, to verify the software results. 6.1.1. Section properites and resistances; sagging

a) Cross-section #1 (middle of the center-span) b eff

z

Si

yi

ya Sa

ya

yi

za

zi

zc

hc

Sc

z n0 

Ea Ecm

 6.395

A-23

Design Check to Eurocodes

Annex A

aa) Bare steel: section dimensions: tw  15 mm

h w  1750 mm

tf.t  20 mm

b f.t  600  mm

tf.b  40 mm

b f.b  800  mm

area: Aa.s  h w tw  b f.t tf.t  b f.b tf.b  702.5  cm

2

first moment of area about the bottom surface line tf.b  hw   tf.t  3 Sa.s  tw h w   tf.b  tf.t b f.t   h w  tf.b  tf.b b f.b  46259  cm 2 2 2    

height of centroid from the bottom surface za.s 

Sa.s Aa.s

 658.49 mm

moment of inertia about the neutral axis: 3

2

3

tf.t tw h w   6 4 Ia.s   b f.t tf.t  tf.b  h w   za.s   3.711  10  cm  12 12 2   2  3 2 hw tf.b      tf.b  bf.b   tw h w  tf.b   za.s    tf.b b f.b  za.s   2 2     12  b f.t tf.t

elastic modulus to the top flange: Ia.s 3 Wel.tf.s   32230  cm h w  tf.b  tf.t  za.s

elastic modulus to the bottom flange: Ia.s 3 Wel.bf.s   56361  cm za.s

Plastic sectional properties: Determination of the height of the plastic neutral axis from the bottom surface with a horizontal equilibrium equation: Fx b f.t tf.t   h w  x   tw = bf.b tf.b  x  tw solving it the place of neutral axis (z pl): x  208.3  mm zpl  x  tf.b  248.3  mm

A-24

Design Check to Eurocodes

Annex A

arms of the tension and compression forces 2  zpl  tf.b   b f.b tf.b zpl  tw  2    235.507  mm d1  A a.s

2 2  h w  tf.b  zpl   b f.t tf.t  h w  tf.b  tf.t  zpl  tw  2    1041.011 mm d2  A a.s

2

d  d 1  d 2  1.277 m

plastic sectional modulus: Aa.s 3 Wpl.s  d   44838  cm 2

plastic reserve of the cross-section, shape factor: cs 

Wpl.s



min Wel.bf.s Wel.tf.s



 1  39.1 %

ab) effective rc deck: equivalent thickness of the deck

v c.eq  292.33 mm

effective width

b´´eff.1  6.735 m

height of haunch

h h  100mm

area

Ac.s  v c.eq b´´eff.1  19688.4  cm

2

v c.eq height of centroid from the bottom zc.s  tf.b  h w  tf.t  h h   205.6  cm surface 2 3 b´´eff.1 v c.eq moment of inertia about the neutral 6 4 Ic.s   1.402  10  cm axis of the deck 12

reinforcement: Φ16/200 in two layers ( 16mm) as.s 

4

2

 π 2

200mm

2

 2011

mm m

Its centroid coincidences with the centroid of the the concrete slab. A-25

Design Check to Eurocodes

Annex A

ac) homogenized section properties: The composite section is converted to a homogenous steel cross-section. 1 1  2 Ai.s.0  Aa.s  A  1   a  b´´  3895 cm n 0 c.s  n 0  s.s eff.1   1 1  5 3 Si.s.0  Aa.s za.s  A z  1   a  b´´  z  7.028  10  cm n 0 c.s c.s  n 0  s.s eff.1 c.s   Si.s.0 zi.s.0   1804.1 mm tf.b  h w  tf.t  1810 mm Ai.s.0



1

2 

Ii.s.0  Ia.s  Aa.s zi.s.0  za.s   1 



n0

 Ic.s  Ac.s zc.s  zi.s.0 



2   1.5179  107 cm4

2   a  b´´  z  z   s.s eff.1 c.s i.s.0  n0 

1

increase over neglecting the reinforcement under compression: 7

Ii.s.0.2  1.510362 10 cm Ii.s.0  Ii.s.0.2 Ii.s.0

4

 0.496  %

Ii.s.0 7 3 Wi.el.tf.s   2.573  10  cm h w  tf.b  tf.t  zi.s.0





Ii.s.0 3 Wi.el.bf.s   84136  cm zi.s.0

These properties are summarized in the following table: Sectional properties, n0 2

Area [cm ] *

Height of NA [mm] 4

Inertia about NA [cm ] 3

Elastic modulus, top flange [cm ] 3

Elastic modulus, bottom flange [cm ] *

bare steel

composite

702,5

3895,3

658,5

1804,1

3,711E+06

1,518E+07

3,223E+04

2,573E+07

5,636E+04

8,414E+04

from the very bottom surface

For the check of midas calculated stresses it is necessary to determine the cross-sectional properties for shrinkage induced creep, since the software cannot separate the primary and secondary effects.

A-26

Design Check to Eurocodes

Annex A

Repeating the above calculation with modified modulus ratio the results: composite

Sectional properties, nL.sh 2

2336,2

Area [cm ] *

1635,9

Height of NA [mm] 4

1,342E+07

Inertia about NA [cm ] 3

7,705E+05

Elastic modulus, top flange [cm ] 3

8,201E+04

Elastic modulus, bottom flange [cm ] *

from the very bottom surface

Cross-section resistance

elastic resistance EN 1994-2:2005 6.2.1.4 Eq. (6.4)

M el.Rd = M a  k  M c.Ed

As it can be seen from the expression it depends on the moment locked-in the bare steel due to the construction sequences. It also depends on the time, since the primary effect of shrinkage and creep varies in time. t = 100 days moment in the steel, determined from the global analysis: M a.s  504kN m

the reason for such a low value is the favorable position of the section and temporary supports

stresses induced in the steel girder: M a.s σa.1.tf   15.64  MPa Wel.tf.s

M a.s σa.1.bf    8.942  MPa Wel.bf.s

shrinkage primary effect: n L.sh.o  13.05

ε sh.o  139.610  10

6

n0 Nsh  Ecm A ε  4423.0 kN n L.sh.o c.s sh.o





M sh  Nsh  zc.s  zi.s.L.sh  1858.9 kN m Nsh σc.t.sh.1   2.246  MPa Ac.s

Nsh σc.b.sh.1   2.246  MPa Ac.s

M sh  v c.eq   Nsh 1 σc.t.sh.2      zc.s  zi.s.L.sh     2.052  MPa 2  n L.sh.o  Ai.s.L.sh Ii.s.L.sh  

A-27

Design Check to Eurocodes

Annex A

M sh  v c.eq  Nsh 1 σc.b.sh.2      zc.s  zi.s.L.sh     1.742  MPa Ii.s.L.sh  2  n L.sh.o Ai.s.L.sh  





M sh Nsh σa.t.sh    21.34  MPa Wi.el.tf.s.L.sh Ai.s.L.sh M sh Nsh σa.b.sh    3.74 MPa Wi.el.bf.s.L.sh Ai.s.L.sh σc.t.sh  σc.t.sh.1  σc.t.sh.2  0.194  MPa σc.b.sh  σc.b.sh.1  σc.b.sh.2  0.505  MPa

Moment acting on the composite section: M c.1  24060kN m  M a.s M c.1 σa.2.tf   0.915  MPa Wi.el.tf.s

M c.1 σa.2.bf    280  MPa Wi.el.bf.s

M c.1 σs   z  zi.s.0  39.12  MPa Ii.s.0 c.s





M c.1  v c.eq  1 σc.t    zc.s   zi.s.0   9.664  MPa Ii.s.0  2  n0 M c.1  v c.eq  1 σc.b    zc.s   zi.s.0   2.57 MPa Ii.s.0  2  n0

multipliers (k) required to reach the yield stress in particular points:

k a.t 

ks 

k c.t 

fy.1  σa.1.tf  σa.t.sh σa.2.tf fyd

 347.4

k a.b 

fy.1  σa.1.bf  σa.b.sh

 11.115

σs

fcd  σc.t.sh σc.t



 2.09

k c.b 

fcd  σc.b.sh σc.b

 7.979



k ss  min k a.t k a.b k s k c.t k c.b  1.223

A-28

σa.2.bf

 1.223

Design Check to Eurocodes

M c.el.Rd.s 

k ss M c.1 γM0

Annex A

 28802  kN m

highest stress: σa.bf  σa.1.bf  σa.b.sh  σa.2.bf  292.7  MPa

at the bottom flange of the steel girder

the value obtained from midas: σa.bf.midas  295.3 MPa σa.bf.midas  σa.bf σa.bf

 0.904  %

The reason of difference is already mentioned in Section 3.1.2 Creep and Shrinkage. M el.Rd.s  M c.el.Rd.s  M a.s  29306  kN m

Plastic resistance: plastic resistance of the web

Rw  h w tw fy.1  9319 kN

plastic resistance of the top flange

Rf.t  b f.t tf.t fy.1  4260 kN

plastic resistance of the bottom flange Rf.b  b f.b tf.b fy.1  11360 kN plastic resistance of the concrete





Rc  0.85 fcd Ac.s  as.s b´´eff.1  33240  kN

plastic resistance of the reinforcement Rs.s  as.s b´´eff.1 fyd  5888 kN Rw  Rf.t  Rf.b  24939  kN

<

Rc  Rs.s  39128  kN

the plastic neutral axis is in the concrete slab The whole steel section is under tension, therefore it belongs to Class 1. 

Rc  0.85 fcd  Ac.s 

as.s b´´eff.1 



2

  33355  kN 

Rw  Rf.t  Rf.b = ξ Rc ξ  0.748

concrete under compression

ξ  v c.eq  218.567  mm

<

v c.eq  40mm  252.33 mm

zpl.c.s  tf.b  h w  tf.t  h h  ( 1  ξ )  v c.eq  1984 mm

A-29

bottom layer of reinforcement is under tension

Design Check to Eurocodes

Annex A







M c.pl.Rk.s  Rw  Rf.t  Rf.b  zpl.c.s  za.s 

ξ  v c.eq 2

 ξ Rc 

 36401  kN m

Rs.s Rs.s  ξ  v c.eq  40mm   ( 1  ξ )  v c.eq  40mm    2 2





M c.pl.Rk.s

M c.pl.Rd.s 

γM0

 36401  kN m

shear resistance: VEd  903kN

from midas analysis

η  1.20 h w tw Vpl.a.Rd  η

EN 1993-1-5:2006 5.1 (2)

fy.1

γM0

3

EN 1993-1-1:2005 6.2.6 (2) (3)

 6456.2 kN

shear buckling resistance:

Contribution from the web: 235

ε 

 0.814

fy.1

EN 1993-1-5:2006 5.3 (3)

MPa

distance between the transverse stiffeners

a  2500mm

In order to take into account the contribution of the stiffener it is assumed that the bigger subpanel shear buckling is governing. h wi  1070mm α 

a h wi

 2.336

minimum shear buckling coefficient: kτ 

4

5.34

if α  1.0

2

 6.073

α 5.34 

4 2

if α  1.0

α

relative web slenderness: λ´w 

hw 37.4 tw ε  k τ

 1.556

EN 1993-1-5:2006 5.3 (5.6)

A-30

Design Check to Eurocodes

Annex A

shear buckling factor: with rigid end post χw 

0.83 η if λ´w  η 0.83 λ´w

if

0.83 η

1.37 0.7  λ´w

Vbw.Rd 

 0.607

EN 1993-1-5:2006 5.3 Table 5.1

 λ´w  1.08

if 1.08  λ´w

χw fy.1 h w tw 3  γM1

 2970.4 kN

EN 1993-1-5:2006 5.3 (5.2)

Contribution from flanges: neglected Vbf.Rd  0kN Vb.Rd  Vbw.Rd  Vbf.Rd  2970.4 kN





VRd  min Vpl.a.Rd Vb.Rd  2970.42  kN

reduction due to the shear force ρ 

0 if VEd  0.5 VRd

  VEd  1  2  VRd 

0

2

otherwise

1 ρ 1 M c.pl.Rd.s  36401  kN m

A-31

Design Check to Eurocodes

Annex A

6.1.2 Section properties and resistances; hogging b) Cross-section #2 (over the internal support) b eff

z

Si

yi

ya Sa

ya

yi

za

zi

s

hc

Ss

z tw  25mm

h w  1750 mm

tf.t  20 mm

b f.t  600  mm

tf.b  60mm

b f.b  800  mm

Aa.h  h w tw  tf.t b f.t  tf.b b f.b  1037.5 cm

2

tf.b  hw   tf.t  3 Sa.h  tw h w   tf.b  tf.t b f.t   h w  tf.b  tf.b b f.b  64186.25 cm 2 2 2    

neutral axis (NA) form the very bottom surface: za.h 

Sa.h Aa.h

 618.66 mm

Inertia about the NA: 3

Ia.h 

b f.t tf.t 12

2

3

tf.t tw h w    b f.t tf.t  tf.b  h w   za.h   12 2  

2 3 2 hw tf.b  tf.b  b f.b     tw h w  tf.b   za.h   tf.b b f.b  za.h   2 2  12   

Ia.h 3 Wel.tf.h   40872  cm h w  tf.b  tf.t  za.h



Ia.h 3 Wel.bf.h   80027  cm za.h



M a.el.h  min Wel.tf.h Wel.bf.h  fy.1  14510  kN m

A-32

6

 4.951  10  cm

4

Design Check to Eurocodes

Annex A

plastic modulus of the section determination of the location of the plastic neutral axis (PNA)





b f.t tf.t  h w  x  tw = b f.b tf.b  x  tw

solving: x  155  mm zpl.h  x  tf.b  215  mm Aa.h 2

 518.75 cm

from the very bottom surface

2

 zpl.h  tf.b2 b f.b tf.b zpl.h  tw  2    204.729  mm d1  A a.h

2

 hw  tf.b  zpl.h2 b f.t tf.t  h w  tf.b  tf.t  zpl.h  tw  2    986.608  mm d2  A a.h

2

d  d 1  d 2  1.191 m Aa.h 3 Wpl.h  d   61800.625  cm 2 M pl.a.h  Wpl.h fy.1  21939  kN m

shape factor ch 



Wpl.h



min Wel.bf.h Wel.tf.h

 1  51.2 %

cs  39.119 %

bb) reinforced concrete deck: The contribution of the concrete to the flexural stiffness through the tension stiffening is neglected. equivalent thickness of the deck

v c.eq  292.33 mm

effective width

b eff.2  4.815 m

height of haunch

h h  100mm

A-33

Design Check to Eurocodes

Annex A

area

Ac.h  v c.eq b eff.2  14075.7  cm

height of centroid from the bottom zc.h  tf.b  hw  tf.t  h h  surface

2

v c.eq 2

 207.6  cm

3 b eff.2 v c.eq moment of inertia about the neutral 6 4 Ic.h   1.002  10  cm axis of the deck 12

reinforcement: Φ25/100 two layers ( 25mm) as.h 

4

2

 π 2

2

 9817

100mm

mm m

2

Ai.h.0  Aa.h  as.h b eff.2  1510 cm

5

Si.h.0  Aa.h  za.h  as.h b eff.2 zc.h  1.623  10  cm

3

composite NA: zi.h.0 

Si.h.0 Ai.h.0

 1074.9 mm



2   1.18497  107 cm4 2  as.h b eff.2  zc.h  zi.h.0

Ii.h.0  Ia.h  Aa.h  zi.h.0  za.h

Ii.h.0 5 3 Wi.el.tf.h   1.569  10  cm h w  tf.b  tf.t  zi.h.0





Ii.h.0 5 3 Wi.el.bf.h   1.102  10  cm zi.h.0

M el.appr  Wi.el.bf.h fy.2  36931.094  kN m

Summation of the values: Sectional properties 2

Area [cm ] *

Height of NA [mm] 4

Inertia about NA [cm ] 3

Elastic modulus, top flange [cm ] 3

Elastic modulus, bottom flange [cm ] *

bare steel

composite

1037,5

1510,2

618,7

1074,9

4,951E+06

1,185E+07

4,087E+04

1,569E+05

8,003E+04

1,102E+05

from the very bottom surface

A-34

Design Check to Eurocodes

Annex A

Cross-section resistance

elastic resistance M el.Rd = M a  k  M c.Ed

as it can be seen from the expression it depends on the moment locked-in the bare steel due to the construction sequences with the use of the value determined from the global analysis: t = 100 years M a.h  3279kN m

stresses locked into the bare steel, during construction: M a.h σa.1.tf   80.23  MPa Wel.tf.h

M a.h σa.1.bf    40.974 MPa Wel.bf.h

determination of the bending resistance of the composite section: M c.1  32922 kN m  M a.h  29643  kN m M c.1 σa.2.tf   188.901  MPa Wi.el.tf.h

M c.1 σa.2.bf    268.9  MPa Wi.el.bf.h

M c.1 σs   z  zi.h.0  250.48 MPa Ii.h.0 c.h



k a.t 

ks 



fy.1  σa.1.tf

 1.5

σa.2.tf fyd σs

k a.b 

fy.2  σa.1.bf σa.2.bf

 1.093

 1.736





k h  min k a.t k a.b k s  1.093 M c.el.Rd.h 

k h  M c.1 γM0

 32414  kN m

M el.Rd.h  M c.el.Rd.h  M a.h  35693  kN m

Plastic resistance: plastic resistance of the web

Rw  h w tw fy.1  15531  kN

A-35

Design Check to Eurocodes

Annex A

plastic resistance of the top flange

Rf.t  b f.t tf.t fy.1  4260 kN

plastic resistance of the bottom flange

Rf.b  b f.b tf.b fy.2  16080  kN

plastic resistance of the reinforcement

Rs.h  as.h b eff.2 fyd  20553  kN

Rw  Rf.b  31611  kN

>

Rs.h  Rf.t  24813  kN

the plastic neutral axis is in the web ξ  Rw  Rf.b = Rs.h  Rf.t  ( 1  ξ )  Rw

web portion under compression

ξ  0.781

Cross section classification for bending: 235

ε 

fy.1

 0.814

MPa

Flanges: The top flange is under tension -> Class1. topFlange  1

The bottom flange: b f.b  tw 2  tf.b

 6.458

bottomFlange 

3 if

2 if

1 if

b f.b  tw 2  tf.b b f.b  tw 2  tf.b b f.b  tw 2  tf.b

 14 ε

EN 1993-1-1:2005 Table 5.2 sheet 2

 10 ε

 9 ε

4 otherwise bottomFlange  1

Web: hw tw

 70

A-36

Design Check to Eurocodes

web 

2 if ξ  0.5 

2 if ξ  0.5 

1 if ξ  0.5 

1 if ξ  0.5 

Annex A

hw tw hw tw hw tw hw tw









456  ε

EN 1993-1-1:2005 Table 5.2 sheet 1

13 ξ  1 41.5 ε ξ 396  ε 13 ξ  1 36 ε ξ

"Class 3 or 4; check the elastic stress distribution" otherwise web  "Class 3 or 4; check the elastic stress distribution"

The elastic stress distribution is required to classify the web, these stresses depend on the construction sequence, they induced by the Ma and Mc moments. Therefore, for the classification the sequence of the construction should be known a priori, which is not typical. In this particular case the primary effects are not effecting the stress distribution. The following way the classification can be done without knowing the actual Ma and Mc moments. The neutral axis for arbitrary Ma-Mc pair is between the following values: za.h  618.7  mm

ψ1  

zi.h.0  1075 mm

ψ2  

web1 

tf.b  h w  tf.t  za.h za.h

 1.958

tf.b  h w  tf.t  zi.h.0 zi.h.0

hw 42 ε 3 if ψ1  1   tw 0.67  0.33 ψ1

 0.703

web2 

hw 3 if ψ1  1   62 ε  1  ψ1  tw



 ψ1

hw 42 ε 3 if ψ2  1   tw 0.67  0.33 ψ2 hw  62 ε  1  ψ2  3 if ψ2  1  tw



web otherwise

web otherwise

web1  3

web2  3

From the above web classification it can be seen that the web is in Class 3 regardless of the moment locked-in by the construction sequence. In the given load combination with the particular construction sequence ψ 

σa.1.tf  σa.2.tf σa.1.bf  σa.2.bf

 0.869

A-37

 ψ2

Design Check to Eurocodes

web 

3 if ψ  1 

3 if ψ  1 

Annex A

hw tw hw tw



42 ε 0.67  0.33 ψ

 62 ε  ( 1  ψ)  ( ψ)

web otherwise web  3 crossSection 

2 if web = 3  bottomFlange  2  topFlange  2 max( web topFlange bottomFlange) otherwise

crossSection  2

Class

If the web is in class 3 and the flanges at least in class 2, the section may be treated as class 2 where the web is taken into account with its effective dimensions in accordance with EN 1993-1-1:2005 6.2.2.4. It means that in the elastic regime the whole section is working, while for plastic resistance a reduced web should be considered. Considering the robust, longitudinal trapezoidal stiffener in the compressed web-zone it can be seen by inspection that the web is at least in section class 2. Therefore no reduction is adopted for the web (also checked with EBPlate). The plastic reserves of the cross-section can be exploited, however the rotation is limited. zpl.c.h  tf.b  ξ  h w  1427 mm tf.b d f.b  zpl.c.h   1.397 m 2 tf.t d f.t  tf.b  h w   zpl.c.h  0.393 m 2 tf.t v c.eq d s.h  d f.t   hh   0.649 m 2 2 ξ hw ( 1  ξ)  hw M c.pl.Rk.h  Rf.b d f.b  ξ Rw  ( 1  ξ)  Rw   46423  kN m 2 2  Rf.t d f.t  Rs.h d s.h

M c.pl.Rd.h 

M c.pl.Rk.h γM0

 46423  kN m

A-38

EN 1994-2:2004 5.5.2 (3)

Design Check to Eurocodes

Annex A

shear resistance: VEd  4629kN η  1.20 h w tw Vpl.a.Rd  η

fy.1 3

γM0

 10760.4  kN

shear buckling resistance:

Contribution from the web: 235

ε 

 0.814

fy.1 MPa

distance between the transverse stiffeners

a  2500mm α 

a h wi

 2.336

minimum shear buckling coefficient: kτ 

4

5.34

if α  1.0

2

EN 1993-1-5:2006 5.3 (3)

 6.073

α 5.34 

4 2

if α  1.0

α

relative web slenderness: λ´w 

h wi 37.4 tw ε  k τ

 0.571

EN 1993-1-5:2006 5.3 (5.6)

The stiffener is rather robust, it is not expected to get a "global" ortotrop plate buckling as first eigenshape, by the way it was checked with EBPlate. the relative slenderness for the longitudinally stiffened web: by using the general formula for stability problems fy.1

λ´w.int 

3

722.91 MPa

 0.532

A-39

Design Check to Eurocodes

Annex A

the critical stress is determined with EBPlate using hinged supports at every edge It is slightly lower than the stability loss of the upper subpanel, but it is also predict the buckling of the bigger subpanel. The reason of the difference that in the standardized calculation the bottom restrain of the plate is assumed to be hinged, while in EBPlate its connected to the next subpanel

shear buckling factor: with rigid end post χw 

0.83 η if λ´w  η 0.83 λ´w

if

0.83 η

1.37 0.7  λ´w

Vbw.Rd 

 1.2

EN 1993-1-5:2006 5.3 Table 5.1

 λ´w  1.08

if 1.08  λ´w

χw fy.1 h w tw 3  γM1

 9782.2 kN

EN 1993-1-5:2006 5.3 (5.2)

Contribution from flanges: neglected Vbf.Rd  0kN Vb.Rd  Vbw.Rd  Vbf.Rd  9782.2 kN





VRd  min Vpl.Rd Vb.Rd  9782.151 kN

A-40

Design Check to Eurocodes

Annex A

reduction due to the shear force ρ 

0 if VEd  0.5 VRd

 VEd   1  2  VRd 

0

VEd VRd

2

 0.473

otherwise

1 ρ 1

M c.pl.Rd.h 

M c.pl.Rk.h γM0

 46423  kN m

M c.pl.Rd.h  46423  kN m

Composite sectional properties, resistances

internal pier

midspan

3895,3

1510,2

1804,1

1074,9

1,518E+07

1,185E+07

2,573E+07

1,569E+05

8,414E+04

1,102E+05

36401

46423

2

Area [cm ] *

Height of NA [mm] 4

Inertia about NA [cm ] 3

Elastic modulus, top flange [cm ] 3

Elastic modulus, bottom flange [cm ] Plastic resistance [kNm] *

from the very bottom surface

Resistances: plastic

elastic

Sagging:

only composite

M c.pl.Rd.s  36401  kN m

total

M c.el.Rd.s  28802  kN m

M el.Rd.s  29306  kN m

t = 100 days

M c.el.Rd.h  32414  kN m

M el.Rd.h  35693  kN m

t = 100 years

Hogging: M c.pl.Rd.h  46423  kN m

plastic reserve of the examined cross-sections, in case of the particular construction sequence: cc.s 

M c.pl.Rd.s M c.el.Rd.s

 1  26.39  %

cc.h 

M c.pl.Rd.h M c.el.Rd.h

 1  43.22  %

These high values are the result of the very asymmetrical sections. A-41

Limit State Analysis

Annex B

Annex B - Limit State Analysis Evaluation of the rating and utilization factors for different ultimate limit states. This Annex uses the results of Annex A. Resistances: Summarization of the section resistances determined in Annex A: plastic

elastic

Sagging:

only composite

M c.pl.Rd.s  36401  kN m

M c.el.Rd.s  28802  kN m

total t = 100 days

M el.Rd.s  29306  kN m

Hogging: M c.pl.Rd.h  46423  kN m

M c.el.Rd.h  32414  kN m

M el.Rd.h  35693  kN m

t = 100 years

plastic reserve of the examined cross-sections, in case of the particular construction sequence: cc.s 

M c.pl.Rd.s M c.el.Rd.s

 1  26.39  %

cc.h 

M c.pl.Rd.h M c.el.Rd.h

 1  43.22  %

These high values are the result of the very asymmetric sections. Considering only the three assumed critical cross-sections.

1 First Yield Since the superstructure contains various sections and with taking into account the cracking and shear lag this number at least doubles - based on the original elastic calculation - in case of the check of the ULS of main girders only the critical, most exploited cross sections will be checked. The maximal internal forces at the application of the first life load (day 100, the concrete is 94 days old): These values are obtained by summing the forces locked-in the bare steel and forces acting on the composite section e.g. Ma,Ed + Mc,Ed. max positive moment

with tension stiffening effect

M Ed.s.0  22747kN m

M´Ed.s.0  22117kN m

B-1

M´Ed.s.0  M Ed.s.0 M Ed.s.0

 2.77 %

Limit State Analysis

Annex B

max negative moment M Ed.h.0  28350.5 kN m

M´Ed.h.0  M Ed.h.0

M´Ed.h.0  29365 kN m

M Ed.h.0

 3.578  %

The maximal internal forces at the end of the design life (100 years): max positive moment

with tension stiffening effect

M Ed.s.t  20291kN m

M´Ed.s.t  19560kN m

M´Ed.s.t  M Ed.s.t M Ed.s.t

 3.603  %

max negative moment M Ed.h.t  30789 kN m

M´Ed.h.t  31906 kN m

M´Ed.h.t  M Ed.h.t M Ed.h.t

 3.628  %

As it expected the maximum positive bending moment appears at the opening of the structure to traffic due to the secondary effect of shrinkage which induces negative moments in the entire structure. According to this the maximum negative moment develops at the end of the design life. UF - utilization factor RF - rating factor, is the multiplier applied to the live load to reach the particular limit state

UF el.s 

M Ed.s.0 M el.Rd.s

 0.776

UF el.h 

M Ed.h.t M el.Rd.h

B-2

 0.863

Limit State Analysis

Annex B

Decomposition of the maximal moments: The numbers in the tables are multiplied with the relevant partial factors. Permanent

Variable

M [kNm]

Traffic Dead load Shrinkage1 Creep 1 9289

LC1

-1765

sum: 22747

-89

nontraffic: 9776

Thermal

TS UDL

Top warmer

12971

2340

Wind

Without Bottom 2 2 warmer traffic traffic With

trafic: 12971

The composition of the maxium negative bending moment (t = 100 years) Permanent

Variable

M [kNm]

Traffic Dead load Shrinkage

LC3

-13797

sum: -30789

1

-4404

Creep

1

129

nontraffic: -20901

Wind

Thermal

TS UDL

Top warmer

-9889

-

Bottom With Without warmer traffic2 traffic2 -2828

-

-

taffic: -9889

1

implicitly these moments are only from the secondary (global) effect of the shrinkage and creep, since only these effects induce internal forces the values are multiplied with the partial factors

M Ed.s.0  22747  kN m

RFel.s 

M el.Rd.s  M nontraffic.s M traffic.s

1

 1.506

UF el.s

 1.288

M Ed.h.t  30789  kN m

RFel.h 

M el.Rd.h  M nontraffic.h M traffic.h

1

 1.496

UF el.h

 1.159

To make more realistic comparison to the incremental and plastic collapse limits the rating factors and utilization ratios are calculated with only considering the dead and traffic loads as well. Without the primary (local) and secondary (global) effect of shrinkage and creep the elastic resistance in sagging: M el.Rd.s.2  29620kN m M dead.s  9289 kN m

M el.Rd.h  35693.1  kN m M dead.h  13797  kN m

B-3

Limit State Analysis UF el.s.2 

RFel.s.2 

RFel.h.2 

Annex B

M traffic.s  M dead.s M el.Rd.s.2

 0.752

M el.Rd.s.2  M dead.s

 1.567

M traffic.s

M el.Rd.h  M dead.h M traffic.h

UF el.h.2 

1 UF el.s.2

1

 2.214

UF el.h.2

M traffic.h  M dead.h M el.Rd.h

 0.664

 1.331

 1.507

2 First Plastic Hinge It is assumed that the plastic hinge formulates when the moment reaches the plastic resistance in a particular cross-section. This means that the analysis is identical to the first yield check with the only difference, that the plastic resistance is used. Redistribution is not considered due to the partial plastification of the cross sections. The primary (local) effect of the shrinkage, creep, thermal actions can be neglected since they are equilibrated during the plastification of the cross section. UF pl.s 

RFpl.s 

RFpl.h 

M Ed.s.0 M c.pl.Rd.s

 0.625

UF pl.h 

M c.pl.Rd.s  M nontraffic.s M traffic.s

 2.0526

1 UF pl.s

M c.pl.Rd.h  M nontraffic.h M traffic.h

 2.581

1 UF pl.h

M Ed.h.t M c.pl.Rd.h

 0.663

 1.6

 1.508

without the secondary (global) effects: UF pl.s.2 

RFpl.s.2 

RFpl.h.2 

M traffic.s  M dead.s M c.pl.Rd.s

 0.612

M c.pl.Rd.s  M dead.s M traffic.s M c.pl.Rd.h  M dead.h M traffic.h

 2.090

UF pl.h.2 

1 UF pl.s.2 1

 3.299

UF pl.h.2

M traffic.h  M dead.h M c.pl.Rd.h

 0.51

 1.635

 1.96

If the shakedown or plastic collapse limit states are applied then there is no effect to the load bearing capacity of the following actions: thermal actions, shrinkage, creep, uneven settlement B-4

Limit State Analysis

Annex B

3 Single girder shakedown The dead load and traffic load moment envelopes (without partial factors): ‐15000 ‐12023

‐10000 ‐5000

0

‐11778

50

100

0 5000 8095

10000 ‐15000 ‐10000 ‐7325 ‐5000

0

‐7235,7 50

100

0 5000 9609,4

10000

Assuming a collapse mechanism with hinges over the internal supports and in the middle of midspan: kinematic method: Mc.pl.Rd.s ( 2 )  Mc.pl.Rd.h  ( 1  1 )  RFsi.sh.k.1 

RFsi.sh.k.1 

 γG.sup ξ  [ 8095kN m ( 2 )  12023kN m ( 1 )  11778kN m ( 1 ) ] γQ.t [ 9609kN m ( 2 )  7325kN m ( 1 )  7236kN m ( 1 ) ]

[ 36401kN m ( 2 )  46423  ( 1  1 )  ( kN m) ]   1.1475 [ 8095kN m ( 2 )  12023kN m ( 1 )  11778kN m ( 1 ) ] 1.35 [ 9609kN m ( 2 )  7325kN m ( 1 )  7236kN m ( 1 ) ]

 2.626

 2.626

static method: The rating factor was determined by linear programing. The calculation is performed in Matlab, the m-files to this and other limit states can be found on attached storage disc.

B-5

Limit State Analysis

Annex B

RFsi.sh.s.1  2.6262

matlab

RFsi.sh.s.1  RFsi.sh.k.1 RFsi.sh.s.1

 0.001  %

RFsi.sh.s.1  RFsi.sh.k.1 2

 2.626

4 System Shakedown Since this particular bridge consists only two girders which are assumed to be equally loaded, their maximal moment envelopes are identical; there is no difference compared to the system shakedown.

5 System Plastic Collapse live load: concentrated load:





Qtot  2 Q1.k  Q2.k  Q3.k  1200 kN w  11.0m

deck width for traffic:

uniformly distributed load:





kN q tot  q 1.k  q 2.k  q 3.k  3.0m  q r.k 2.0m  47.00  m

Comparison to the MSZ ÚT standard: This standard adopts one uniformly distributed load all over the lanes and only one concentrated load with maximum value: 800 kN and axle distance 2,70 m. Moreover the dynamic factor is not included in the load model. This traffic load represents the heaviest traffic and it should be applied to the busiest roads. p  3.525

kN 2

m

μ  1.161

dynamic factor

p tot  μ p  11.0m  45

kN

p tot

m

q tot

Ptot  μ 0.91 100 kN 8  845.2  kN

Ptot Qtot

 95.8 %

 70.4 %

dead load:





kN g tot  2  g gir.tot  g crossB  g a.other  g rc  g surf  g p  g other  g s.1  201  m

ξ  0.85

B-6

ÚT 2-3-401:2004 2.2.1.

Limit State Analysis

Annex B

Assuming a collapse mechanism with hinges over the internal supports and in the middle of midspan: kinematic method:

RFpc.k.1 

2 M c.pl.Rd.s 2  M c.pl.Rd.h  ( 1  1 )  γG.sup ξ   g tot  



γQ.t  20m Qtot 

40m 20m



40m 20m  2

 q tot

   4.137



2

static method: Using the results of the elastic analysis RFpc.s.1  4.052 RFpc.s.1  RFpc.k.1 RFpc.s.1

matlab

RFpc.s.1  RFpc.k.1

 2.108  %

2

 4.095

6 Single Girder Plastic Collapse: Loads on one girder: q single 

7.875

q tot  47

kN

Qsingle 

7.5

 q 1.k 3 m 

4.875 7.5

 q 2.k 3 m 

1.875 7.5

 q 3.k 3 m 

0.1875 7.5

 q r.k ( 0.25m  0.621m)  35.154

kN m

m

 7.875  Q  4.875  Q  1.875  Q   2  940  kN  7.5 1.k 2.k 3.k 7.5 7.5  

considering that the lateral distribution is slightly differs from the simply supported beam influence line, the 0,947 multiplier takes into account this effect. kN q single  0.929  q single  32.658 m

neglecting the increased loaded area under in the remaining area lane.

Qsingle  0.929  Qsingle  873.26 kN

Comparison to the MSZ ÚT standard p single  ( μ 11.0m p )  0.5 0.929  20.91 

kN

p single

m

q single

 64 %

According to the ÚT standard the tandem load should be placed in a way that the ÚT 2-3-401:2004 outer surface of the wheel is 50cm from the safety barrier (in sum from the tire 2.2.1.1 centerline 90cm). This is the bigger distance than implemented by the Eurocode using the 3,0 m width nominal lanes and 2,0 m axle distance. B-7

Limit State Analysis

Annex B

Psingle 7.125 Psingle  ( μ 0.91 100 kN 8 )   0.929  745.9  kN  85.4 % 7.5 Qsingle

It is interesting that there is a significant difference in either the total and in the to-girder-reduced traffic loads. Since the concentrated load induce higher positive bending moments this increase the exploitage of sections in the sagging region against the original design. Determination of the rating factor assuming a collapse mechanism with hinges over the internal supports and in the middle of midspan: kinematic method:  g tot 40m 20m   Mc.pl.Rd.s 2  Mc.pl.Rd.h  ( 1  1 )  γG.sup ξ  2  2    2.900 RFspc.k.1  40m  20 m γQ.t  20m Qsingle   q single 2   RFspc.k.1  2.900

static method: Using the results of the elastic analysis RFspc.s.1  2.939 RFspc.s.1  RFspc.k.1 RFspc.s.1

matlab

 1.318  %

RFspc.s.1  RFspc.k.1 2

 2.92

It should be noted that the increased shear force and its effect on the plastic resistance is omitted in the above calculations. Therefore the actual values are smaller than the determined ones. Since the M-V interaction curve is not linear it requires some advanced method or an iterative procedure to obtain these values. The iteration process was used to determine the rating factors reflecting the effect of shear force. The calculations are basically identical, hence not repeated in the documentation.

B-8

Reliability analysis

Annex C

Annex C - Reliability Analysis First-hinge reliability analysis The critical section is the one loaded with the highest positive moment, t=100 days. The reliability analysis conducted by using the First Order Reliability Method, considering the model uncertainties as well. The limit state has reached by scaling the live load.

1 Basic statistical variables R - resistance; LN D - dead load; N L - livel load; GU T - thermal action; GU S - shrinkage, creep;N θR- res. model unc. LN θE- act. model unc. LN

log normal normal Gumbel Gumbel normal lognormal lognormal

EN-1990:2001 Annex C6

Rating factor: RF  2.053

mean values and standard deviations: R  37847

νR  0.02461

σR  R νR  931.415

D  7029.7

νD  0.08

σD  D νD  562.376

L  5888.2

νL  0.20

σL  L νL  1177.64

Ta  1300.5

νT.a  0.5

σT.a  Ta νT.a  650.25

S  1896

νS  0

σS  S νS  0

θR  1

νθ.R  0.04

σθ.R  θR νθ.R  0.04

θE  1

νθ.E  0.05

σθ.E  θE νθ.E  0.05

2 Reliability analysis The algorithm follows the steps introduced in Section 2.2.2. The numbers herein correspond to the last iteration. The results of the iterations are summarized in a table at the end of the calculation.

C-1

Reliability analysis

Annex C

2.1 Formulation of the limit state function









g R D L Ta S θR θE  θR R  θE D  L  Ta  S

2.2 Initial design point Rˇ  37439.644 Dˇ  7187.337 Tˇa  1395.762 θˇR  0.971 θˇE  1.043 Sˇ  S

Lˇ determined to be on the failure line

Lˇ 





θˇR Rˇ  θˇE Dˇ  Tˇa  Sˇ RF θˇE

 13720.4

2.3 Equivalent normal means and standard deviations

Resistance; lognormal lognormal parameters: 2  σR   σLN.R  ln 1   0.0246  2  R   2

μLN.R  ln( R)  0.5 σLN.R  10.541

lognormal CDF and PDF evaluated at point Rˇ: C-2

Reliability analysis

Annex C









FR( x )  plnorm x μLN.R σLN.R fR( x )  dlnorm x μLN.R σLN.R

the equivalent normal distribution parameters: 1 σRˇ.eq   dnorm qnorm FR( Rˇ) 0 1 0 1  921.3 fR( Rˇ)













μRˇ.eq  Rˇ  σRˇ.eq  qnorm FR( Rˇ) 0 1  37833

Resistance model uncertainty; lognormal lognormal parameters: 2  σθ.R     0.04 σLN.θ.R  ln 1   2  θR  

 

2

μLN.θ.R  ln θR  0.5 σLN.θ.R  0.001

lognormal CDF and PDF :









Fθ.R( x )  plnorm x μLN.θ.R σLN.θ.R fθ.R( x )  dlnorm x μLN.θ.R σLN.θ.R

the equivalent normal distribution parameters: 1 σθˇ.R.eq   dnorm qnorm Fθ.R θˇR 0 1 0 1  0.039 fθ.R θˇR

 







 

 







μθˇ.R.eq  θˇR  σθˇ.R.eq qnorm Fθ.R θˇR 0 1  0.999

Action model uncertainty; lognormal lognormal parameters: 2  σθ.E     0.05 σLN.θ.E  ln 1   2  θE  

 

2

μLN.θ.E  ln θE  0.5 σLN.θ.E  0.001

C-3

Reliability analysis

Annex C

lognormal CDF and PDF:









Fθ.E( x )  plnorm x μLN.θ.E σLN.θ.E fθ.E( x )  dlnorm x μLN.θ.E σLN.θ.E

the equivalent normal distribution parameters: 1 σθˇ.E.eq   dnorm qnorm Fθ.E θˇE 0 1 0 1  0.052 fθ.E θˇE



 





 

 







μθˇ.E.eq  θˇE  σθˇ.E.eq qnorm Fθ.E θˇE 0 1  0.998

Dead load; normal no need for transformation μDˇ.eq  D  7029.7 σDˇ.eq  σD  562.376

Live load; Gumbel Gumbel parameters: a 

π 6  σL

u  L

γ a

 0.00109

 5358

Gumbel CDF and PDF: FL( x )  e

 ( x u)  a

e

fL( x )  a e

 ( x u)  a

 ( x u)  a e

the equivalent normal distribution parameters: 1 σLˇ.eq   dnorm qnorm FL( Lˇ) 0 1 0 1  3612.147 fL( Lˇ)













μLˇ.eq  Lˇ  σLˇ.eq qnorm FL( Lˇ) 0 1  381.175

C-4

Reliability analysis

Annex C

Thermal; Gumbel Gumbel parameters: a 

π 6  σT.a

 0.00197

γ u  Ta   1008 a

Gumbel CDF and PDF:

FT( x )  e

 ( x u)  a

e

fT( x )  a e

 ( x u)  a

 ( x u)  a e

the equivalent normal distribution parameters: 1 σTˇ.a.eq   dnorm qnorm FT Tˇa 0 1 0 1  656.338 fT Tˇa

 



  



  





μTˇ.a.eq  Tˇa  σTˇ.a.eq qnorm FT Tˇa 0 1  1181.501

Eq. normal CDF and PDF:





PDF





CDF

n T( x )  dnorm x μTˇ.a.eq σTˇ.a.eq

NT( x )  pnorm x μTˇ.a.eq σTˇ.a.eq

C-5

Reliability analysis

Annex C

Illustration of the PDF and CDF of Gumbel and equivalent normal distributions.

pdf

4

8 10

4

6.4 10

4

4.8 10

4

3.2 10

4

1.6 10

0 650.25

3

1.3 10

975.375

1.626 10

3

1.951 10

3

2.276 10

3

3

2.189 10

3

2.601 10

3

cdf 1 0.8 0.6 0.4 0.2 0 130.05

541.875

953.7

1.366 10

3

1.777 10

Shrinkage, creep; normal no need for equivalent normal distribution μSˇ.eq  S σSˇ.eq  σS

2.4 Transformation to U space URˇ 

UDˇ 

USˇ 

Rˇ  μRˇ.eq σRˇ.eq Dˇ  μDˇ.eq σDˇ.eq Sˇ  μSˇ.eq σSˇ.eq

Lˇ  μLˇ.eq

 0.427

ULˇ 

 0.28

UTˇ.a 

0

Uθˇ.E 

σLˇ.eq

 3.693

Tˇa  μTˇ.a.eq σTˇ.a.eq θˇE  μθˇ.E.eq σθˇ.E.eq

C-6

 0.326

 0.868

2.601 10

3

Reliability analysis

Uθˇ.R 

Annex C

θˇR  μθˇ.R.eq σθˇ.R.eq

 0.716

Uˇ  URˇ 0

Uˇ  UDˇ 1

Uˇ  USˇ 4

Uˇ  Uθˇ.R 5

Uˇ  ULˇ 2

Uˇ  UTˇ.a 3

Uˇ  Uθˇ.E 6

 0.4275     0.2803   3.6929  Uˇ   0.3264     0.0000   0.7160   0.8675    2.5 Partial derivatives of the limit state function

 

G   θˇR  σRˇ.eq 0









G   1  θˇE  σSˇ.eq  0 4

G   θˇE  σDˇ.eq 1

G  ( Rˇ)  σθˇ.R.eq

  G   1  θˇE  σTˇ.a.eq 3

G   Dˇ  RF Lˇ  Tˇa  Sˇ   σθˇ.E.eq 6  

G   RF θˇE  σLˇ.eq 2

5





 894.534     586.558   7734.615  G   684.561    0    1453.575   1816.56    2.6 Estimation of the reliability index (β)

β 

G Uˇ T

 3.9077

G G

C-7

Reliability analysis

Annex C

2.7 Sensitivity factors (α)

α 

 0.109     0.072   0.946  G   0.084    T G G  0   0.178   0.222   

2.8 New design point Uˇ  α  β  0.428 0

0

Uˇ  α  β  0.28 1

1

Uˇ  α  β  0.327 3

3

Uˇ  α  β  0 4

4

Uˇ  α  β  0.695 5

5

Uˇ  α  β  0.868 6

6

2.9 Determination of the new design point in the original space Rˇ  Uˇ  σRˇ.eq  μRˇ.eq  37439.603 0 Dˇ  Uˇ  σDˇ.eq  μDˇ.eq  7187.355 1 Tˇa  Uˇ  σTˇ.a.eq  μTˇ.a.eq  1396.24 3 Sˇ  Uˇ  σSˇ.eq  μSˇ.eq  1896 4 θˇR  Uˇ  σθˇ.R.eq  μθˇ.E.eq  0.971 5 θˇE  Uˇ  σθˇ.E.eq  μθˇ.E.eq  1.043 6

Lˇ is determined in order to be on the failure line

Lˇ 





θˇR Rˇ  θˇE Dˇ  Tˇa  Sˇ RF θˇE

 13716.348

C-8

Reliability analysis

Annex C

Repeat the iteration until the convergence of the design point or β! The iterations are summarized in the following table: iteration number

beta

L

1

-

15300,9

2

3,9112

13768,2

3

3,9077

13721,1

4

3,9077

13716,7

5

3,9077

13716,3

The value provided by the FORM analysis of FERUM: βFERUM  3.9077

After refinement with SORM analysis, the reliability index: βSORM  3.8887

C-9

Used Programs

Annex D

Annex D - Used Programs Used applications, only the engineering, scientific ones are listed: -

AutoCAD 2009 C.56.0 Axis VM10 rls.3j. EBPlate 2.01 FERUM 4.1 Matlab toolbox Mathcad v.14.0.0.163 MATLAB 7.1 v.7.1.0.246 (R14) Service Pack 3 midas Civil 2011 v.2.1

D-1