Flexural Strength-ductility Performance Of Flanged Beam Sections Cast Of High-strength Concrete (p 29-43)

THE STRUCTURAL DESIGN OF TALL AND SPECIAL BUILDINGS Struct. Design Tall Spec. Build. 13, 29–43 (2004) Published online 9 June 2004 in Wiley Interscience (www.interscience.wiley.com). DOI:10.1002/tal.231

FLEXURAL STRENGTH–DUCTILITY PERFORMANCE OF FLANGED BEAM SECTIONS CAST OF HIGH-STRENGTH CONCRETE A. K. H. KWAN* AND F. T. K. AU Department of Civil Engineering, University of Hong Kong, Hong Kong

SUMMARY Flanged sections are often used for long-span concrete beams to maximize their structural efficiency. However, although for the same sectional area a flanged section could render a higher flexural strength, it would also lead to a lower flexural ductility, especially when heavily reinforced. Thus, when evaluating the flexural performance of a beam section, both the flexural strength and ductility need to be considered. In this study, the post-peak flexural behaviour of flanged sections is evaluated by means of an analytical method that uses the actual stress–strain curves of the materials and takes into account strain reversal of the tension reinforcement. From the numerical results, the flexural strength–ductility performance of flanged sections is investigated by plotting the strength and ductility that could be simultaneously achieved in the form of design graphs. It is found that (1) at the same overall dimensions and with the same amount of reinforcement provided, a flanged section has lower flexural ductility than a rectangular section; (2) at the same overall dimensions, a flanged section has inferior strength– ductility performance compared to a rectangular section; and (3) at the same sectional area, a flanged section has better strength–ductility performance compared to a rectangular section. Copyright © 2004 John Wiley & Sons, Ltd.

1.

INTRODUCTION

When a reinforced concrete beam section is subjected to flexure, the applied bending moment is resisted jointly by the compressive force developed in the concrete and the tensile force developed in the steel reinforcement, which are equal and opposite to each other and together form a couple, as illustrated in Figure 1(a). Let the compressive force in the concrete and the tensile force in the reinforcement be denoted by C and T respectively and the lever arm, i.e. the distance between C and T, be denoted by a. Since the applied bending moment is always equal to C or T times a, for given C and T, the flexural strength of the beam section increases with the lever arm a. Hence, a larger lever arm would lead to a better structural efficiency of the beam section. One way to increase the lever arm is to increase the depth of the beam section, but very often the depth of a beam section is limited because of the headroom or other geometric requirements and cannot be increased without affecting the overall layout of the structure. If the depth of the beam section cannot be increased, an alternative is to change the beam section to a flanged section, as shown in Figure 1(b). In a flanged section, such as a T- or box-shaped section, most of the concrete is located near the extreme compression fibre and thus the line of action of C is at a larger distance from that of T. Because of the larger lever arm in a flanged section than in a rectangular section, the structural efficiency of a flanged section is generally larger than that of a rectangular section. * Correspondence to: Professor A. K. H. Kwan, Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong. E-mail: [email protected]

Copyright © 2004 John Wiley & Sons, Ltd.

Received December 2002 Accepted February 2003

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A. K. H. KWAN AND F. T. K. AU

C

la

T Strain distribution

Section

Stress distribution

(a) A rectangular section subjected to flexure

C

la

T Section

Strain distribution

Stress distribution

(b) A flanged section subjected to flexure Figure 1. Beam sections subjected to flexure

It is sometimes argued that the concrete near the geometric centre of the section is of little use because it develops small compressive stresses and contributes little to the flexural strength, and that the higher structural efficiency of a flanged section is attained mainly by relocating the concrete near the centre of the section to the flange area near the extreme compression fibre where much higher compressive stresses would be developed (Ferguson, 1981). In actual fact, apart from contributing to the shear and torsional strengths of the beam section, the concrete near the centre also contributes significantly to the flexural ductility of the beam section. At the post-peak stage, when the peak bending moment has already been reached and the moment resisting capacity of the beam section is decreasing, the concrete near the extreme compression fibre would gradually lose its strength and the neutral axis of the section would move towards the tension reinforcement. Consequently, the concrete near the centre would at the post-peak stage develop much higher compressive stresses than before and contribute significantly to the residual moment resisting capacity of the section. Because of this, a rectangular section, which has relatively more concrete near the centre, is generally more ductile than a

Copyright © 2004 John Wiley & Sons, Ltd.

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FLEXURAL STRENGTH–DUCTILITY PERFORMANCE

31

flanged section. In other words, although a flanged section has a higher structural efficiency in terms of flexural strength, it might at the same time have a somewhat lower flexural ductility. Since ductility may also be a governing factor determining the safety of a structure, when evaluating the structural performance of a beam both the flexural strength and ductility need to be considered. Maximizing the flexural strength or the structural efficiency in terms of flexural strength without paying proper attention to the flexural ductility may not produce the best overall structural performance. The best way to evaluate the flexural strength–ductility performance of a beam section is to evaluate the flexural strength and flexural ductility that could be simultaneously achieved at various levels of reinforcement, as will be illustrated in this paper. However, while the flexural strength of a beam section can be determined quite easily using the ordinary beam bending theory, it is not possible to evaluate the flexural ductility using any simple analytical method. To evaluate the flexural ductility of a beam section, it is necessary to carry out non-linear flexural analysis, extended well into the post-peak range so that the ability of the beam section in withstanding inelastic curvature without excessive loss in flexural strength may be determined. Such kind of analysis is highly non-linear and involves stress path dependence of the constitutive materials during strain reversal at the post-peak stage. In most books on reinforced concrete design, the analysis of flanged beams is just treated as a straightforward extension of that for rectangular beams. That is probably the reason why, compared with investigations on the structural behaviour of rectangular beams, much less work has been done on flanged beams. Moreover, the previous work on flanged beams generally concentrated on aspects other than the post-peak flexural behaviour. For example, Swamy et al. (1973) studied the shear resistance of T-beams with varying flange widths and found that the shear resistance of T-beams was significantly increased by the flange width, the percentage of tension steel and the amount of web reinforcement. Desayi et al. (1978) conducted tests of reinforced concrete T-beams and rectangular beams to study the influence of the flange on the torsional strength of reinforced concrete T-beams. The results indicated that the torsional strength contribution of the flange may be estimated by the plastic theory of torsion. Prakash Rao (1982) presented a comprehensive summary of the research done in Europe on the design of webs and web–flange junctions under combined bending and shear, taking into account the interaction between various forces acting on the section. Subedi et al. (1992) and Subedi (1993) carried out experimental work focusing on the failure behaviour of thin-walled reinforced concrete flanged beams and observed that the major modes of failure were flexure, diagonal splitting and web crushing. So far, there has been little research on the post-peak flexural behaviour of flanged beams. At the University of Hong Kong, a new method of non-linear flexural analysis that uses the actual stress–strain curves of the materials and takes into account stress path dependence of the tension reinforcement has recently been developed and applied to study the post-peak flexural behaviour of singly and doubly reinforced rectangular beam sections (Ho et al., 2003; Pam et al., 2001). It has been found from these studies that at the post-peak stage both the line of action of C and the neutral axis of the section would gradually shift towards the tension reinforcement and then at a certain point the axial strain of the tension reinforcement would start to decrease, causing strain reversal in the tension reinforcement and consequently stress path dependence of the tensile stress developed therein. Using this newly developed method, the effects of various structural parameters including the concrete grade, the tension and compression steel yield strengths and the tension and compression steel area ratios on the flexural ductility of rectangular beam sections have been quite thoroughly studied (Kwan et al., 2003). In the present study, the aforementioned method of analysis has been extended to deal with flanged sections. Using the extended analysis method, the non-linear flexural behaviours of typical flanged

Copyright © 2004 John Wiley & Sons, Ltd.

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A. K. H. KWAN AND F. T. K. AU

sections have been analysed and compared to those of rectangular sections with the same overall dimensions. After gaining a better understanding of the non-linear flexural behaviour of flanged sections, the flexural strength and ductility that could be simultaneously achieved by flanged sections of different shapes have been evaluated. From the numerical results obtained, the flexural strength–ductility performance of flanged sections has been studied by plotting the flexural ductility against the flexural strength in the form of graphs that may in fact be used as design charts. The results revealed that in the design of flanged beam sections more attention to the provision of sufficient flexural ductility is generally needed. 2.

METHOD OF ANALYSIS

The constitutive model for unconfined concrete developed by Attard and Setunge (1996), which has been shown to be applicable to a broad range of concrete strength from 20 to 130 MPa, is adopted in the moment–curvature analysis. The stress–strain curve of the constitutive model is given by

s c fco =

K1 (e c e co ) + K2 (e c e co )

2

1 + ( K1 - 2)(e c e co ) + ( K2 + 1)(e c e co )

2

(1)

in which sc and ec are the compressive stress and strain at any point on the stress–strain curve, fco and eco are the compressive stress and strain at the peak of the stress–strain curve, and K1 and K2 are coefficients dependent on the concrete grade. It should be noted that fco is actually the in situ compressive strength, which may be estimated from the cylinder or cube compressive strength using appropriate conversion factors. Figure 2(a) shows some typical stress–strain curves so derived. For the steel reinforcement, a linearly elastic–perfectly plastic stress–strain curve is adopted. Since there could be strain reversal in the steel reinforcement at the post-peak stage despite monotonic increase of curvature (Ho et al., 2003; Pam et al., 2001), the stress–strain curve of the steel is stresspath dependent. It is assumed that when strain reversal occurs, the unloading path of the stress–strain curve is linear and has the same slope as the initial elastic portion of the stress–strain curve. Figure 2(b) shows the resulting stress–strain curve of the steel reinforcement. Only three other basic assumptions have been made in the analysis: (1) plane sections before bending remain plane after bending, (2) the tensile strength of concrete is negligible, and (3) there is no bond-slip between concrete and steel. These assumptions are widely accepted in the literature (Park and Paulay, 1975). The moment–curvature behaviour of the beam section is analysed by applying prescribed curvatures to the beam section incrementally starting from zero. At a prescribed curvature, the strain profile in the section is first evaluated based on the above assumptions. From the strain profile so obtained, the stresses developed in the concrete and the steel reinforcement are determined from their respective stress–strain curves. The stresses developed have to satisfy the axial equilibrium condition, from which the neutral axis depth is evaluated by iteration. Having determined the neutral axis depth, the resisting moment is calculated from the moment equilibrium condition. The above procedure is repeated until the curvature is large enough for the resisting moment to increase to the peak and then decrease to 50% of the peak moment. Details of the analysis procedure have been presented in Ho et al. (2003) and Pam et al. (2001). The method of analysis previously applied to solid rectangular sections is extended to deal with flanged sections by taking the width of the section as a variable instead of a constant. For instance, in the following equations governing the axial and moment equilibrium conditions of the flanged beam section shown in Figure 3: Copyright © 2004 John Wiley & Sons, Ltd.

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FLEXURAL STRENGTH–DUCTILITY PERFORMANCE

Stress (MPa)

80 70

fco=30MPa fco

60

ffco=50MPa co

50

fco=70MPa fco

40 30 20 10 0 0

0.001

0.002

0.003

0.004

0.005

0.006

Strain

Stress

(a) Concrete

ss = fy

fy

unloading

ss =Es(es-ep) ep

Es

Es Strain

(b) Steel reinforcement Figure 2. Stress–strain curves of concrete and steel reinforcement

dn

P = Ú s c bdx + Â Ascs sc - Â Asts st

(2)

M = Ú s c bxdx + Â Ascs sc (dn - D1 ) + Â Asts st ( D - dn )

(3)

0

dn

0

where P is the applied axial load (compressive force taken as positive), M is the resisting moment (sagging moment taken as positive) and b is the width of the section at x from the neutral axis, the width b is taken as a variable during the numerical integration. Because of the variable width of the flanged section, a more sophisticated numerical integration technique has to be used when integrating over the concrete section to determine the axial force and the resisting moment contributed by the stresses developed in the concrete. In the present study, Romberg integration (Gerald and Wheatley, 1999), which can significantly improve the accuracy of the simple trapezoidal rule when the integrand is known at equispaced intervals, has been adopted. Copyright © 2004 John Wiley & Sons, Ltd.

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A. K. H. KWAN AND F. T. K. AU

B Asc D1 Df x

dn

b

Neutral axis

ssc

D

Ast

sst Bw Stress distribution

Section

Figure 3. Analysis of flanged section

B =1200mm

Bw =400mm

Bw =800mm

Section A

Section B

B =1200mm

D =1500mm

Df =400mm

B =1200mm

Section C

(Note: In all sections, Ast = 40,000 mm2) Figure 4. Typical flanged sections analysed

3. 3.1

FLEXURAL BEHAVIOUR OF TYPICAL FLANGED SECTIONS

Sections analysed

For the sake of comparing the non-linear flexural behaviour of flanged beam sections to that of a solid rectangular beam section with the same overall dimensions, three beam sections have been analysed, as shown in Figure 4. They have the same overall dimensions of B = 1200 mm and D = 1500 mm and Copyright © 2004 John Wiley & Sons, Ltd.

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FLEXURAL STRENGTH–DUCTILITY PERFORMANCE

are named Section A, Section B and Section C. Section A is a T-shaped section with web breadth and flange depth given by Bw = 400 mm and Df = 400 mm respectively, while Section B is a T-shaped section with web breadth and flange depth given by Bw = 800 mm and Df = 400 mm respectively. On the other hand, Section C is a solid rectangular section. In each section, the same amount of tension reinforcement given by Ast = 40,000 mm2 is provided. However, no compression reinforcement is provided in any of the three sections. For a preliminary study, the in situ concrete compressive strength fco is fixed at 50 MPa, while the yield strength fyt and Young’s modulus Es of the steel reinforcement are fixed at 460 MPa and 200 GPa respectively. 3.2

Complete moment–curvature curves

The complete moment–curvature curves obtained for the three beam sections are shown in Figure 5. It is seen that the three sections, each provided with the same amount of tension reinforcement, have similar peak resisting moments. They all fail in tension (i.e. the tension reinforcement yields before the concrete fails) and are therefore under-reinforced. That is why all the three sections fail in a ductile manner, as evidenced by the presence of a flat yield plateau in each of their moment–curvature curves. Nevertheless, it is obvious that the flexural ductility of Section A is lower than that of Section B, while the flexural ductility of Section B is lower than that of Section C. Bearing in mind that the three sections have the same overall dimensions and the same amount of reinforcement provided and that they differ from one other only in the web breadth, it is evident that although the concrete in the web contributes little to the peak resisting moment, it does contribute to the residual resisting moment at the post-peak stage and thus the flexural ductility of the section. 4. 4.1

FLEXURAL DUCTILITY OF FLANGED SECTIONS

Flexural ductility evaluation

The flexural ductility of the beam section may be evaluated in terms of a curvature ductility factor m defined by 30

Bending moment (MNm)

25

20

Section C

15

Section B

10

Section A

5

0 0

2

4

6

8

10

12

Curvature ( X10-3 radian/m)

14

16

18

20

Figure 5. Moment–curvature curves of beam sections analysed Copyright © 2004 John Wiley & Sons, Ltd.

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A. K. H. KWAN AND F. T. K. AU

m = fu f y

(4)

where fu and fy are the ultimate curvature and yield curvature respectively. The ultimate curvature fu is taken as the curvature of the beam section when the resisting moment of the beam section has, after reaching the peak value of Mp, dropped to 0·8 Mp. On the other hand, the yield curvature fy is taken as the curvature at the hypothetical yield point of an equivalent linearly elastic–perfectly plastic system with an elastic stiffness equal to the secant stiffness of the section at 0·75 Mp and a yield moment equal to Mp. 4.2

Flexural ductility of flanged sections

When comparing the flexural ductility of a flanged section to that of a rectangular section with the same overall dimensions, it is necessary to take into account also the other structural parameters such as the concrete grade and the amount of tension reinforcement provided because the flexural ductility varies significantly with these parameters. The flanged and rectangular sections shown in Figure 4 are reanalysed using different values of fco and Ast. To study the effect of the concrete grade, fco is set equal to 30, 50 or 70 MPa. To study the effect of the amount of tension reinforcement provided, Ast is varied from 10,000 to 80,000 mm2. Figure 6 shows the variation of the curvature ductility factor m with the tension steel area Ast for the beam sections analysed at different concrete grades. It is seen that the flexural ductility of a beam section, regardless of the sectional shape, decreases as the tension steel area increases. In general, at the same concrete grade and the same tension steel area, the flexural ductility of a flanged section is lower than that of a rectangular section with the same overall dimensions. The difference in flexural ductility is relatively small when the beam sections are lightly reinforced but could be quite significant when the beam sections are heavily reinforced. In fact, when the beam sections are heavily reinforced, even though the sections are provided with the same amount of reinforcement, it could happen that a flanged section is already over-reinforced while a rectangular section still remains under-reinforced. So when the beam sections are heavily reinforced with

25

Curvature ductility factor m

fco=30MPa. Section A fco 20

fco=30MPa, Section B fco

15

fco=70MPa, Section A fco

fco=30MPa, Section C fco fco=70MPa, Section B fco fco=70MPa, Section C fco

10

5

0 0

20,000

40,000

60,000

80,000

100,000

120,000

Tension steel area, Ast (mm2)

Figure 6. Variation of curvature ductility factor m with tension steel area Ast Copyright © 2004 John Wiley & Sons, Ltd.

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FLEXURAL STRENGTH–DUCTILITY PERFORMANCE

the same amount of reinforcement, a change in sectional shape from rectangular to flanged shape could lead to a change of failure mode from ductile tension failure to brittle compression failure. This is because in general a flanged section has a smaller balanced steel area (the tension steel area at which balanced failure occurs) than a rectangular section. It is also evident from Figure 6 that at the same tension steel area the flexural ductility increases slightly with the concrete grade, albeit a higher-grade concrete should be less ductile. This may also be explained by looking at the degree of the beam section being under- or over-reinforced. The balance steel area is larger when the concrete grade is higher. Thus, relatively, at the same tension steel area, the tension to balanced steel ratio (the ratio of the tension steel area to the balanced steel area) is smaller when a higher-grade concrete is used. The tension to balanced steel ratio may be interpreted as a measure of the degree of the beam section being under/over-reinforced. When the tension to balanced steel ratio is smaller, the degree of the beam section being under-reinforced is higher and the degree of the beam section being over-reinforced is lower. Therefore, at the same tension steel area, the degree of the beam section being under-reinforced increases with the concrete grade and as a result the flexural ductility increases slightly with the concrete grade. From the above, it is obvious that one major structural parameter determining the flexural ductility is the tension to balanced steel ratio, which is denoted hereafter by l. Figure 7 shows the variation of the curvature ductility factor m with the tension to balanced steel ratio l for the beam sections analysed at different concrete grades. It is seen that regardless of the concrete grade and the shape of the beam section, m decreases as l increases and then remains roughly constant when l > 1·0. In general, at the same concrete grade and the same tension to balanced steel ratio, the m-value of a flanged section is slightly higher than that of a rectangular section with the same overall dimensions when l < 0·85 (when the sections are lightly reinforced) and the m-value of a flanged section is slightly lower than that of a rectangular section with the same overall dimensions when l > 0·85 (when the sections are heavily reinforced). Nevertheless, since a flanged section has a smaller balanced steel area compared to that of a rectangular section, at the same tension steel area, a flanged section has a higher tension to balanced steel ratio and therefore a lower flexural ductility, especially when the section is heavily reinforced. Figure 7 also reveals that at the same tension to balanced steel ratio, regardless of the sec-

Curvature ductility factor m

25

Section A ffco=30MPa. co Section B ffco=30MPa, co Section C ffco=30MPa, co Section A ffco=70MPa, co ffco=70MPa, Section B co ffco=70MPa, Section C co

20

15

10

5

0 0.0

0.2

0.4

0.6

0.8

Tension to balanced steel ratio l

1.0

1.2

Figure 7. Variation of curvature ductility factor m with tension to balanced steel ratio l Copyright © 2004 John Wiley & Sons, Ltd.

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A. K. H. KWAN AND F. T. K. AU

tional shape, the flexural ductility decreases as the concrete grade increases, for the simple reason that a higher-grade concrete is generally less ductile. 5. 5.1

FLEXURAL STRENGTH–DUCTILITY PERFORMANCE OF FLANGED SECTIONS

Parametric study

In order to study the effect of sectional shape on the flexural strength–ductility performance, a number of beam sections with different Bw/B and Df/D ratios have been analysed using the method developed herein. All together, 16 flanged sections with Bw/B and Df/D ratios ranging from 0·1 to 0·4 and one rectangular section with Bw/B = 1 and Df/D = 1 have been analysed. All the beam sections analysed have the same overall dimensions of B = 1200 mm and D = 1500 mm. For each beam section, the concrete strength fco is set equal to 30, 50 or 70 MPa and the tension to balanced steel ratio is varied from 0·3 to 1·2. However, to reduce the number of variables, the tension steel yield strength fyt is fixed at 460 MPa. 5.2

Flexural strength–ductility performance at same overall dimensions

For a given beam section, regardless of whether it is a rectangular section or a flanged section, the use of a higher-tension steel area leads to a higher flexural strength but a lower flexural ductility whereas the use of a lower-tension steel area leads to a higher flexural ductility but a lower flexural strength. Hence, the increase in flexural strength obtained by using a higher-tension steel area is achieved at the expense of a lower flexural ductility and the increase in flexural ductility obtained by using a lower tension steel area is achieved at the expense of a lower flexural strength. The achievement of both high flexural strength and high flexural ductility is not easy. If a beam section can attain high flexural strength and high flexural ductility at the same time, it is said to have a good flexural strength–ductility performance. The flexural strength–ductility performance of a beam section is best revealed by plotting the flexural strength and the flexural ductility that could be simultaneously achieved in the form of graphs. Figure 8 shows the flexural strength–ductility graphs so obtained for the beam sections analysed. In the figure, the x-ordinate is the flexural strength, expressed in terms of Mp/BD2, while the y-ordinate is the curvature ductility factor m. The purpose of expressing the flexural strength in terms of Mp/BD2 is to render it independent of the actual dimensions and allow comparison of the flexural strength– ductility performances of beam sections having the same overall dimensions but different shapes. From the figure, it can be seen by comparing the curves for flanged sections to those of rectangular sections that in general the flexural strength–ductility curve of a flanged section is lower than that of a rectangular section. Hence, at the same overall dimensions, the flexural strength–ductility performance of a flanged section is inferior to that of a rectangular section. In other words, although at the same overall dimensions, a flanged section has a smaller sectional area and is thus lighter than a rectangular section, its flexural strength–ductility performance is not as good as that of a rectangular section. Comparing the curves for flanged sections with different Bw /B and Df /D ratios, it is evident that when the sections are lightly reinforced the ratio Df /D has greater influence on the flexural strength–ductility performance, while the Bw /B ratio has relatively little influence, but when the sections are heavily reinforced both the Bw /B and Df /D ratios have significant influences on the flexural strength–ductility performance. It can also be seen by comparing the curves for beam sections cast of concrete of different grades that for the same sectional shape and dimensions those beam sections cast of a higher-grade concrete

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 29–43 (2004)

39

FLEXURAL STRENGTH–DUCTILITY PERFORMANCE fco = 30MPa

25

Df/D=0.1, Bw/B=0.1 Df /D Bw /B Df/D=0.1, Bw/B=0.2 D Bw /B f /D Df/D=0.1, Bw/B=0.3 D Bw /B f /D Df/D=0.1, Bw/B=0.4 D Bw /B f /D Df/D=0.2, Bw/B=0.1 Bw /B Df /D Df/D=0.2, Bw/B=0.2 B w /B Df /D Df/D=0.2, Bw/B=0.3 B w /B Df /D Df/D=0.2, Bw/B=0.4 Bw /B D f /D Rectangular

m

15

10

Df/D=0.3, Bw/B=0.1 D B f /D w /B Df/D=0.3, Bw/B=0.2 D Bw /B f /D Df/D=0.3, Bw/B=0.3 D Bw /B f /D Df/D=0.3, Bw/B=0.4 D Bw /B f /D Df/D=0.4, Bw/B=0.1 Bw /B Df /D Df/D=0.4, Bw/B=0.2 Bw /B Df /D Df/D=0.4, Bw/B=0.3 Bw /B D f /D Df/D=0.4, Bw/B=0.4 Bw /B D f /D Rectangular

20

15

m

20

fco = 30MPa

25

10

5

5

0

0 0.0

2.0

4.0

6.0

8.0

10.0

12.0

Mp /BD 2 (MPa)

14.0

16.0

18.0

20.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

Mp /BD 2 (MPa)

fco = 50MPa

fco = 50MPa 25

25

Df/D=0.1, Bw/B=0.1 D Bw /B f /D Df/D=0.1, Bw/B=0.2 D Bw /B f /D Df/D=0.1, Bw/B=0.3 D Bw /B f /D Df/D=0.1, Bw/B=0.4 Bw /B Df /D Df/D=0.2, Bw/B=0.1 Bw /B Df /D Df/D=0.2, Bw/B=0.2 Bw /B D f /D Df/D=0.2, Bw/B=0.3 Bw /B D f /D Df/D=0.2, Bw/B=0.4 Bw /B D f /D Rectangular

m

15

10

Df/D=0.3, Bw/B=0.1 B D w /B f /D Df/D=0.3, Bw/B=0.2 D Bw /B f /D Df/D=0.3, Bw/B=0.3 D Bw /B f /D Df/D=0.3, Bw/B=0.4 D B f /D w /B Df/D=0.4, Bw/B=0.1 Bw /B Df /D Df/D=0.4, Bw/B=0.2 B D /D w /B f Df/D=0.4, Bw/B=0.3 Bw /B D f /D Df/D=0.4, Bw/B=0.4 Bw /B D f /D Rectangular

20

15

m

20

10

5

5

0

0 0.0

2.0

4.0

6.0

8.0

10.0 2

12.0

14.0

16.0

18.0

0.0

20.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

M p /BD 2 (MPa)

Mp /BD (MPa)

fco = 70MPa

fco = 70MPa

25

25

Df/D=0.1, Bw/B=0.1 D Bw /B f /D Df/D=0.1, Bw/B=0.2 D Bw /B f /D Df/D=0.1, Bw/B=0.3 B D /D w /B f Df/D=0.1, Bw/B=0.4 D Bw /B f /D Df/D=0.2, Bw/B=0.1 Bw /B D f /D Df/D=0.2, Bw/B=0.2 Bw /B Df /D Df/D=0.2, Bw/B=0.3 B Df /D w /B Df/D=0.2, Bw/B=0.4 Bw /B D f /D Rectangular

m

15

10

Df/D=0.3, Bw/B=0.1 D Bw /B f /D Df/D=0.3, Bw/B=0.2 Bw /B D f /D Df/D=0.3, Bw/B=0.3 D B f /D w /B Df/D=0.3, Bw/B=0.4 D B /D f w /B Df/D=0.4, Bw/B=0.1 Bw /B Df /D Df/D=0.4, Bw/B=0.2 Bw /B D f /D Df/D=0.4, Bw/B=0.3 Bw /B D f /D Df/D=0.4, Bw/B=0.4 Bw /B Df /D Rectangular

20

15

m

20

10

5

5

0

0 0.0

2.0

4.0

6.0

8.0

10.0

12.0

2

M p /BD (MPa)

14.0

16.0

18.0

20.0

0.0

2.0

4.0

6.0

8.0

10.0 2

M p /BD

12.0

14.0

16.0

18.0

20.0

(MPa)

Figure 8. Flexural strength–ductility performance at same overall dimensions

generally have much better flexural strength–ductility performances. The advantage of using highstrength concrete is thus obvious. Although a high-strength concrete is more brittle than a normalstrength concrete, its higher compressive strength, which leads to a larger balanced steel area, does allow a higher flexural strength–ductility performance of the beam section to be achieved. In other words, the use of high-strength concrete could allow a higher flexural strength to be achieved at the same flexural ductility, or a higher flexural ductility to be achieved at the same flexural strength, or both a slightly higher flexural strength and a slightly higher flexural ductility to be achieved at the same time.

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 29–43 (2004)

40 5.3

A. K. H. KWAN AND F. T. K. AU

Flexural strength–ductility performance at same sectional area

The above comparison of the flexural strength–ductility performances of flanged and rectangular sections at the same overall dimensions which leads to the conclusion that the flexural strength–ductility performance of a flanged section is inferior to that of a rectangular section may not be fair, because at the same overall dimensions a flanged section has a smaller sectional area and is thus lighter than a rectangular section. It may therefore be argued that to be fair the comparison should be made at the same sectional area. Such comparison could be made by expressing the flexural strength of the beam section in terms of Mp /AcD rather than Mp /BD2, in which Ac is the sectional area above the centroid of tension steel. This would allow direct comparison of the flexural strengths of beam sections having the same sectional area and the same effective depth to be made. Figure 9 shows the flexural strength–ductility graphs of the beam sections plotted with the flexural strength, expressed in terms of Mp /AcD, as the x-ordinate and the curvature ductility factor m as the yordinate. It can be seen from this figure that at the same sectional area the flexural strength–ductility performance of a flanged section is better than that of a rectangular section. In other words, even with the lower flexural ductility discounted, the change of sectional shape from rectangular to flanged shape would at the same sectional area improve the flexural strength–ductility performance of the beam section. 5.4

Use of flexural strength–ductility graphs as design charts

Apart from revealing the flexural strength–ductility performances of beam sections of various shapes and cast of concrete of different grades, Figures 8 and 9 may also be used as design charts for the preliminary design of flanged beam sections, even before the shape of the beam section and the concrete grade are decided, by following the procedures explained in the following. Similar design charts and procedures have been developed and successfully applied to the concurrent flexural strength and ductility design of rectangular beams (Kwan et al., 2003). In the design of a beam section, it needs to be decided at an early stage what the overall dimensions B and D of the beam section should be because the overall dimensions are often restricted by the general layout of the structure and could not be increased without affecting the other parts of the structure. Assuming that the overall dimensions are already given and that the required flexural strength and ductility have been specified as Mp and m respectively, the sectional shape and the concrete grade to be used may be determined using Figure 8. First of all, the value of Mp /BD2 needs to be evaluated. Then by plotting the point (Mp /BD2, m) on the various charts in Figure 8, several alternative designs may be obtained. This is best illustrated by considering the following example. Example. A beam with B = 2000 mm, D = 2500 mm, Mp = 60 MNm and m = 5·0 is to be designed. The value of Mp/BD2 is evaluated as 4·8 MPa. Plotting the point (4·8, 5·0) on the charts in Figure 8, the following design options are obtained: Design option 1: fco = 30 MPa, Df /D = 0·2 and Bw /B = 0·2 Design option 2: fco = 50 MPa, Df /D = 0·1 and Bw /B = 0·2 Design option 3: fco = 70 MPa, Df /D = 0·1 and Bw /B = 0·1 Design option 1 does not require the use of any high-strength concrete but the sectional area would be larger and therefore the beam section would be heavier. On the other hand, design options 2 and 3 require the use of a high-strength concrete but the resulting beam section would have a smaller sectional area and therefore would be lighter. The choice among the above options is left to the design engineer taking into account the grade of the concrete used in other parts of the building structure and the effect of the dead weight of the beam on the total loading. Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 29–43 (2004)

41

FLEXURAL STRENGTH–DUCTILITY PERFORMANCE fco = 30MPa

25

fco = 30MPa

25

Df/D=0.1, Bw/B D Bw /B=0.1 f /D Df/D=0.1, Bw/B Df /D Bw /B=0.2 Df/D=0.1, Bw/B Df /D Bw /B=0.3 Df/D=0.1, Bw/B Bw /B=0.4 D f /D Df/D=0.2, Bw/B D Bw /B=0.1 f /D Df/D=0.2, Bw/B Bw /B=0.2 Df /D Df/D=0.2, Bw/B Bw /B=0.3 Df /D Df/D=0.2, Bw/B Bw /B=0.4 Df /D Rectangular

m

15

10

Df/D=0.3, Bw/B D B f /D w /B=0.1 Df/D=0.3, Bw/B D Bw /B=0.2 f /D Df/D=0.3, Bw/B B D /D w /B=0.3 f Df/D=0.3, Bw/B D Bw /B=0.4 f /D Df/D=0.4, Bw/B Bw /B=0.1 Df /D Df/D=0.4, Bw/B Bw /B=0.2 Df /D Df/D=0.4, Bw/B Bw /B=0.3 Df /D Df/D=0.4, Bw/B Bw /B=0.4 D f /D Rectangular

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m

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35.0

40.0

Mp /A c D (MPa)

fco = 50MPa

fco = 50MPa

25

25

Df/D=0.1, Bw/B=0.1 Df /D Bw /B Df/D=0.1, Bw/B=0.2 Df /D Bw /B Df/D=0.1, Bw/B=0.3 Df /D Bw /B Df/D=0.1, Bw/B=0.4 Bw /B Df /D Df/D=0.2, Bw/B=0.1 B w /B Df /D Df/D=0.2, Bw/B=0.2 Df /D Bw /B Df/D=0.2, Bw/B=0.3 Df /D Bw /B Df/D=0.2, Bw/B=0.4 D Bw /B f /D Rectangular

m

15

10

Df/D=0.3, Bw/B=0.1 D B f /D w /B Df/D=0.3, Bw/B=0.2 D Bw /B f /D Df/D=0.3, Bw/B=0.3 B D w /B f /D Df/D=0.3, Bw/B=0.4 D B f /D w /B Df/D=0.4, Bw/B=0.1 Bw /B D f /D Df/D=0.4, Bw/B=0.2 D Bw /B f /D Df/D=0.4, Bw/B=0.3 Bw /B Df /D Df/D=0.4, Bw/B=0.4 Bw /B D f /D Rectangular

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15

m

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0

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Mp /A c D (MPa)

M p /A c D (MPa)

fco = 70MPa

fco = 70MPa

25

25

Df/D=0.1, Bw/B=0.1 D B f /D w /B Df/D=0.1, Bw/B=0.2 D B f /D w /B Df/D=0.1, Bw/B=0.3 Bw /B D f /D Df/D=0.1, Bw/B=0.4 D B f /D w /B Df/D=0.2, Bw/B=0.1 Bw /B D f /D Df/D=0.2, Bw/B=0.2 B w /B Df /D Df/D=0.2, Bw/B=0.3 Bw /B D f /D Df/D=0.2, Bw/B=0.4 D Bw /B f /D Rectangular

m

15

10

Df/D=0.3, Bw/B=0.1 D B f /D w /B Df/D=0.3, Bw/B=0.2 D B f /D w /B Df/D=0.3, Bw/B=0.3 Bw /B Df /D Df/D=0.3, Bw/B=0.4 D B f /D w /B Df/D=0.4, Bw/B=0.1 Bw /B D f /D Df/D=0.4, Bw/B=0.2 Bw /B D f /D Df/D=0.4, Bw/B=0.3 Bw /B D f /D Df/D=0.4, Bw/B=0.4 Bw /B D f /D Rectangular

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15

m

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10

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5

0

0 0.0

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Mp /Ac D (MPa)

10.0

15.0

20.0

25.0

30.0

35.0

40.0

Mp /A c D (MPa)

Figure 9. Flexural strength–ductility performance at same sectional area

If, however, not just the overall dimensions, but also the sectional shape of the beam section (i.e. Bw /B and Df /D) have already been fixed, then Figure 9 may also be used for the design. The procedures are the same as before except that instead of plotting the point (Mp /BD2, m) on the charts, the point (Mp /AcD, m) is to be plotted on the various charts in Figure 9. 6.

CONCLUSIONS

The post-peak flexural behaviour of flanged beam sections cast of normal- and high-strength concrete has been studied by an analytical method that uses the actual stress–strain curves of the materials and Copyright © 2004 John Wiley & Sons, Ltd.

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A. K. H. KWAN AND F. T. K. AU

takes into account strain reversal of the tension reinforcement. It was revealed that although the concrete in the web contributes little to the peak resisting moment, it does contribute to the residual resisting moment at the post-peak stage and thus the flexural ductility of the section. In other words, at the same overall dimensions and with the same materials used, a flanged section would generally have a lower flexural ductility compared to that of a rectangular section. It was also found that regardless of the sectional shape, at the same tension steel area, the flexural ductility increases slightly with the concrete grade but at the same tension to balanced steel ratio the flexural ductility decreases as the concrete grade increases. A parametric study has been carried out to evaluate the effects of various structural parameters on the flexural strength–ductility performance of flanged sections. Based on the numerical results, the flexural strength–ductility performance of different beam sections was compared by plotting the flexural strength and flexural ductility that could be simultaneously achieved in the form of graphs. The graphs showed that at the same overall dimensions the flexural strength–ductility performance of a flanged section is inferior to that of a rectangular section. Hence, although at the same overall dimensions a flanged section has a smaller sectional area than a rectangular section, its flexural strength–ductility performance is not as good. However, at the same sectional area, the flexural strength–ductility performance of a flanged section is better than that of a rectangular section. Hence, even with the lower flexural ductility discounted, the change from a rectangular to a flanged section would at the same sectional area improve the flexural strength–ductility performance. Lastly, by comparing the performance curves for beam sections cast of different grades of concrete, it was shown that for the same sectional shape and dimensions a beam section cast of a higher-strength concrete would have better flexural strength–ductility performance. Hence, although a high-strength concrete is generally more brittle, its use could at the same flexural strength lead to an increase in flexural ductility. The flexural strength–ductility performance curves developed may also be used as design charts for the preliminary design of beam sections even before the shape of the section and the concrete grade are decided. A numerical example illustrating the use of these charts has been presented.

ACKNOWLEDGEMENTS

The work described in this paper was carried out with financial support provided by the Croucher Foundation of Hong Kong.

REFERENCES

Attard MM, Setunge S. 1996. The stress strain relationship of confined and unconfined concrete. ACI Materials Journal 93(5): 432–444. Desayi P, Sundara Raja Iyengar KT, Venkatakrishnaiah M. 1978. Influence of flange on the torsional strength of reinforced concrete T-beams. Journal of the Institution of Engineers (India), Part CV: Civil Engineering Division 58(CI 5): 243–246. Ferguson PM. 1981. Reinforced Concrete Fundamentals (4th edn). Wiley: New York. Gerald CF, Wheatley PO. 1999. Applied Numerical Analysis (6th edn). Addison-Wesley: Reading, MA. Ho JCM, Kwan AKH, Pam HJ. 2003. Theoretical analysis of post-peak flexural behavior of normal- and highstrength concrete beams. Structural Design of Tall and Spec Buildings 12: 1–17. Kwan AKH, Ho JCM, Pam HJ. 2003. Effects of concrete grade and steel yield strength on flexural ductility of reinforced concrete beams. Computers and Structures (in press). Kwan AKH, Ho JCM, Pam HJ. 2003. Flexural strength and ductility of reinforced concrete beams. Proceedings of the Institution of Civil Engineers, Structures and Buildings 152: 361–369. Pam HJ, Kwan AKH, Ho JCM. 2001. Post-peak behavior and flexural ductility of doubly reinforced normal- and high-strength concrete beams. Structural Engineering and Mechanics 12(5): 459–474. Park R, Paulay T. 1975. Reinforced Concrete Structures. Wiley: New York. Copyright © 2004 John Wiley & Sons, Ltd.

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Prakash Rao DS. 1982. Design of webs and web flange connections in concrete beams under combined bending and shear. Journal of the American Concrete Institute 79(1): 28–35. Subedi NK. 1993. Reinforced concrete flanged beams: test and analysis. ACI Materials Journal 90(6): 601–615. Subedi NK, Rahem A, Abdel-Rahman GT. 1992. Slender reinforced concrete flanged beams: experimental work. Proceedings of the Institution of Civil Engineers, Structures and Buildings 94(4): 439–468. Swamy RN, Bandyopadhyay AK, Erikitola MK. 1973. Influence of flange width on the shear behaviour of reinforced concrete T beams. Proceedings of the Institution of Civil Engineers, Part 1 55(2): 167–190.

NOTATION Asc, Ast B Bw b D Df D1 dn Es fco fyc, fyt M Mp P x e ec eco esc, est f fu, fy l m sc ssc, sst

= = = = = = = = = = = = = = = = = = = = = = = = =

areas of compression and tension steel reinforcement breadth of beam section breadth of the web or total breadth of the webs breadth at x from neutral axis effective depth of beam section depth of compression flange depth of compression reinforcement depth of neutral axis Young’s modulus of steel reinforcement in situ uniaxial compressive strength of concrete yield strengths of compression and tension steel reinforcement moment acting on beam section peak resisting moment of beam section axial load acting on beam section distance from neutral axis strain in beam section strain in concrete strain in concrete at peak stress strains in compression and tension reinforcement curvature of beam section ultimate and yield curvatures of beam section tension to balanced steel ratio curvature ductility factor stress developed in concrete stresses developed in compression and tension reinforcement

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 29–43 (2004)