6 Inelastic Dynamic Responce

PERGAMON

Computers and Structures 69 (1998) 685±693

Inelastic dynamic response of reinforced concrete in®lled frames Harpal Singh a, *, D. K. Paul b, V. V. Sastry a a

Thapar Institute of Engineering and Technology, Patiala, India Department of EQ. Engineering, University of Roorkee, Roorkee, India

b

Received 9 January 1997; received in revised form 28 April 1998

Abstract An inelastic ®nite element model to simulate the behaviour of reinforced concrete frames in®lled with masonry panels subjected to static load and earthquake excitation has been presented. Under the loads, the mortar may crack causing sliding and separation at the interface between the frame and the in®ll. Further, the in®ll may get cracked and/or crushed which changes its structural behaviour and may render the in®ll ine€ective, leaving the bare frame to take all the load which may lead to the failure of the framing system itself. In this study, a mathematical model to incorporate this behaviour has been presented. # 1998 Elsevier Science Ltd. All rights reserved.

1. Introduction Considerable e€ort has been made on the analysis of the in®lled frame systems. Holmes [1] and Smith [2] proposed the concept of in®ll as an equivalent diagonal compression strut. Mallick and Severn [3], Liauw and Kwan [4] and May and Ma [5] used the ®nite element model for the analysis of 2D in®lled frames. Papia [6] used boundary elements to model the behaviour at the frame and the in®ll interface. Haddad [7] analysed cracked frames with masonry in®ll using the ®nite element and fracture mechanics. May and Naji [8] carried out nonlinear analysis of in®lled frames under monotonic and cyclic loadings using the ®nite element method. The skeletal frame was modelled with a 3-noded frame element and the panel was modelled with an 8-noded isoparametric element. A six-noded interface element was used to model the interface between the frame and the in®ll. Choubey and Sinha [9] carried out the experimental investigation into the behaviour of reinforced concrete frames in®lled with brick masonry under lateral cyclic loading. Singh [10] investigated the inelastic response

of three dimensional reinforced concrete frames subjected to earthquake excitation using the ®nite element method.

2. Finite Element Idealization In the present study, reinforced concrete in®lled frames have been analysed using the ®nite element method. The skeleton frame, the panel and the interface between the frame and the panel have been modelled by a 3-noded frame element, 8-noded isoparametric element and 6-noded interface element, respectively, as shown in Fig. 1. 2.1. Reinforced Concrete Frame Element A 3-noded beam±column element as shown in Fig. 1 has been used to model the skeletal frame [11]. Inelastic behaviour of the element is governed by the interaction of the axial force, two ¯exural moments and a torsional moment. The yield surface: 1

f ˆ ‰…Mx =Mxu †2 ‡ …My =Myu †2 ‡ …Mz =Mzu †2 Š2 ‡ ‰Fx =Fxu Šn * Corresponding author. 0045-7949/98/$ - see front matter # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 9 8 ) 0 0 1 2 4 - 2

…1†

686

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Fig. 1. Di€erent elements used for modelling the in®lled frame.

proposed by Powell and Chen [12] has been used. Here M x, My, M z are the moments about the X, Y and Z axes, respectively, F x is the axial force, and M xu, M yu, M zu are the corresponding yield moments; F xu is the axial yield force; and the exponent n has been taken as 1.6. Concrete is not purely an elastic material. Plastic ¯ow (creep) has been observed in it. The modulus of elasticity varies with the stress rate and magnitude of the stress. The e€ective reinforced concrete section also varies with the stress level. Both the modulus of elasticity and the e€ective cross section decrease with the increase in stress level. In the present study, 50% reduction in the short term value of the static modulus of elasticity of concrete [13] and e€ective sectional properties, both calculated as per Ref. [14] have been assumed for the entire `elastic' range prior to the development of the ultimate yield surface. 2.2. Brick Masonry In®ll The eight-noded isoparametric element as shown in Fig. 1 has been used to model the in®ll panels. Masonry is a complex material consisting of an assemblage of bricks and mortar joints, each with di€ering properties. Its behaviour is made more complex by the mortar joints acting as planes of weakness due to their low tensile, shear and bond strengths. The out-of-plane

sti€ness of the unreinforced masonry panels is very low as compared to its in-plane sti€ness. In the present study only in-plane sti€ness has been taken into consideration. The material has been assumed to be linearly elastic till failure. To predict the cracking and crushing type of failure, Von Mises failure criterion with a tension cut o€ as shown in Fig. 2 has been adopted [15]. In compression, upon crushing, the sti€ness and all stresses are reduced to zero. In tension, upon cracking (see Fig. 3), the sti€ness normal to crack is reduced to zero but along the crack partial, shear sti€ness is maintained. The stress normal to the crack is reduced to zero, however, a partial shear transfer due to interlocking between the particles is maintained. The normal sti€ness and stresses along the crack are also maintained.

2.3. Concrete Mortar Interface Element The behaviour of an in®lled frame depends upon the interaction between the in®ll and the frame. There can be separation, closing of gap and slipping between the frame and the in®ll. A six noded interface element as shown in Fig. 1 has been used to model this behaviour between the frame element and the panel element. Two in-plane translational degrees of freedom per node have been considered. The displacement vector is

H. Singh et al. / Computers and Structures 69 (1998) 685±693

Fig. 2. Yield surface for the masonry panel.

d ˆ ‰u vŠT :

Fig. 3. Crack and principle axes directions.

…2†

The strains are the relative displacements at the top and bottom of the element. The strain vector is de®ned as T

E ˆ ‰Du DvŠ :

s ˆ ‰su

sv Š :

The material modulus matrix is de®ned as   ks 0 Dˆ 0 kn

e

Ke ˆ TT K T

…8†

where T is the transformation matrix. 3. Inelastic Analysis

…4†

…5†

where ks and kn are the shear and the normal sti€ness coecients, respectively. The values of sti€ness coecients for di€erent interface conditions are listed in Table 1. The strain matrix is de®ned as B ˆ ‰ÿIN1 ÿ IN2 ÿ IN3 IN3 IN2 IN1 Š

to calculate the sti€ness matrix. The sti€ness matrix in the global coordinate system has been calculated as

…3†

The relevant stress vector is T

687

…6†

where I is identity matrix of order 2  2 and N i are the shape functions. The sti€ness matrix is calculated as … Ke ˆ BT DBdx: …7† The three point Gauss quadrature rule has been used

For the inelastic static analysis, an incremental iterative procedure has been adopted. For inelastic dynamic analysis a predictor±corrector form of the Newmark method [16] has been used. Initially, element forces or stresses are calculated assuming an elastic behaviour for each element. The stress components and/or strains at Gauss points are examined and cracking, yielding, and separation is checked. When any of the above events has occurred, the forces/stresses are reduced to the yield surfaces and the equivalent nodal forces for the element are calculated. The solution has been assumed to be converged when the ratio of the norms of unbalanced load to the norms of the total load is within the permissible limits. 4. Numerical Examples To study the behaviour of reinforced concrete frames in®lled with masonry panels subjected to static

Table 1 Selection of interface sti€ness coecients Interface conditions

Sti€ness coecients ks

Sti€ness coecients kn

Firm contact vsuv < vmsvv Contact with slip Dv ÿ compressive vsuv r vsvv Separation or initial lack of ®t Dv-tensile

Experimental Very low value Very low value

Very low value Very high value Very low value

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and dynamic loads, two such structures have been analysed using the proposed algorithm.

4.1. A Single Reinforced Concrete In®lled Frame A single storey one-bay in®lled reinforced concrete frame shown in Fig. 4(a), previously investigated experimentally and analytically by Choubey [17], has been analysed using the proposed model. The structure consists of a reinforced concrete frame and masonry in®ll. The physical and material properties and other details of the structure are given in the ®gure. The structure has been discretised as shown in Fig. 4(b). The load de¯ection curve obtained by using the proposed model has been compared with that reported by Choubey [17] in Fig. 4(c). A good agreement with the

experimental results has been observed. The failure load of 180 kN as predicted by the proposed model is close to that obtained experimentally (175.38 kN) by Choubey [17]. The hinges in the frame and cracks in the in®ll at the failure predicted by the proposed model, as well as those obtained experimentally by Choubey [17] have been presented in the Fig. 5. A good comparison between the predicted and the reported results [8] has been obtained. The separation coecients, de®ned as the ratio of separation length to the dimension of the in®ll, have been plotted in Fig. 6. The maximum value of the separation coecient on each side as estimated and those obtained experimentally by Choubey [17] are shown in the ®gure. The strut width observed at the centre is 0.627 L whereas that proposed by Liauw and Kwan [4] is 0.707 L. Where L is the lateral dimension of the in®ll.

Fig. 4. Load de¯ection behaviour of the in®lled reinforced concrete frame-1.

H. Singh et al. / Computers and Structures 69 (1998) 685±693

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Fig. 5. Location of hings and crack pattern for the in®lled reinforced concrete frame 1.

The closeness between the experimentally observed and estimated load de¯ection behaviour, failure load, central strut width, location of hinges, crack pattern and mode of failure establishes the reliability of the proposed model to simulate the behaviour.

4.2. Two Storey In®lled Frame A two-storey single bay in®lled reinforced concrete frame shown in Fig. 7 has been studied. The dimensions of the in®lled frame along with member proper-

Fig. 6. Separation of frame with in®ll using program NIFAP.

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Fig. 7. Geometry and X-sectional details of reinforced concrete in®lled frame-2.

ties are given in the ®gure. The discretisation of the structure and loads are shown in Fig. 8. Both inelastic static and inelastic dynamic analysis have been performed.

4.2.1. Inelastic Static Analysis The inelastic analysis of the structure has been carried out using the proposed model. The load±de¯ection curve at the roof level has been shown in the Fig. 9. The sequence of formation of plastic hinges in the frame and the cracks in the in®ll along with the corresponding de¯ections at the roof level have been listed in the Table 2 and are shown in Fig. 10. The im-

portant features of the response of the problem are discussed below: At a load factor, de®ned as the current load divided by the load at the ®rst increment (as shown in Fig. 8), of about 6.0, the cracking in the lower panel starts at the ends of the tension diagonal (tension±tension zone) with some cracks at the centre of in®lls 4 and 5. With the increase in load, the cracking spreads from the end of the tension diagonal to the centre, and from the centre to the ends of both the tension and compression diagonals. At a load factor of 10.5, the ®rst hinge forms at the bottom of the load ward column and it progresses upwards with the increase in load. At load factor of 11.85, six more cracks develop and the

Fig. 8. Discretisation of in®lled frame-2.

H. Singh et al. / Computers and Structures 69 (1998) 685±693

691

Fig. 9. Load de¯ection curves for in®lled frame-2.

structure becomes too ¯exible to take any additional load. The structure fails at a load factor of about 12.0.

Fig. 10. Crack pattern for in®lled frame-2.

4.2.2. Earthquake Response The in®lled frame shown in Fig. 7, subjected to the EL±Centro earthquake, has been analysed. In addition to the mass of the structure itself, four concentrated masses have been considered to be attached at points A, B, C and D as shown in Fig. 7. The structure has been subjected to the S-O-E component of the 1940. EL±Centro, May 18, 1940, earthquake in the lateral direction. The earthquake had a peak acceleration of 3417 mm/s2 at 2.14 s. Both elastic and inelastic responses have been studied for 20 s. The entire time history response of the structure at selected point has been observed. The elastic and inelastic responses in terms of de¯ection at the roof level (point A) have been plotted in Fig. 11. The elastic and inelastic acceleration responses at the roof level (point A) have been shown in Fig. 12. The elastic and

inelastic variation of bending moment at the base of the left column has been shown in Fig. 13. The sequence of formation of plastic hinges in the frame and that of cracks in the in®ll are presented in Fig. 14. The structure has been found to remain elastic up to 2.25 s. Both the elastic and inelastic responses in terms of de¯ection, acceleration and bending moment are the same up to 2.25 s and no hinge in the frame or crack in the in®ll are developed. At a time of 2.5 s, both the bottom storey columns exhibit a number of plastic hinges in the lower half portion and the in®ll panels of both the storeys exhibit cracking along the tension diagonal as shown in Fig. 14(a). The roof de¯ection at this stage is ÿ10.0 mm. At time of 3.0 s, the bottom storey columns are completely plasticised, but most of the cracks in the panels `disappear' since the state of

Table 2 Sequence of formation of plastic hinges/cracks in the in®lled frame-2

Load factor 6.00 7.00 8.00 9.00 10.00 10.50 11.00 11.40 11.50 11.85

Location and sequence of appearance of cracks in the in®ll panels 1 to 8 9 and 10 11 and 12 13±16 17 and 18 19 20 and 21 22 and 23 24 25±30

Location and sequence of appearance of hinges in frames De¯ection at roof level (mm) Ð 7.09 Ð 8.61 Ð 10.28 Ð 12.03 Ð 14.04 1 15.30 2 16.92 Ð 18.80 Ð 19.19 Ð 30.17

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Fig. 13. Elastic and inelastic variation of bending moment of section F.

Fig. 11. Elastic and inelastic displacement response at point A.

stresses at these points now lies inside the yield surface, however, the plastic strains continue to be present. At a time of 5.0 s some new cracks `appear; and some old cracks `disappear', but the bottom storey columns remain plasticised. Subsequently, almost all the cracks in the panels and hinges in the frame `disappear'. The maximum elastic de¯ection of 32.0 mm occurs at 4.9 s whereas the maximum inelastic de¯ection of 22.0 mm occurs at 9.0 s. The inelastic de¯ection is 68.8% to that of elastic de¯ection. Fig. 12 indicates the variation of the elastic and inelastic accelerations at a roof level point A. The maximum acceleration observed for the elastic and the inelastic analyses are 14.7 and 7.4 m/s2, respectively. The inelastic acceleration being 50.3% of that obtained for the elastic analysis. Fig. 13 shows the variation of the moment at section F near the base. The maximum elastic moment observed is 49.0 kN m, while the maximum inelastic moment is 28.0 kN m, which is 42.8% less than the elastic moment. The above analysis demonstrates the necessity of using an inelastic analysis for the realistic prediction of the response of the in®lled framed structures.

5. Conclusions Fig. 12. Elastic and inelastic acceleration response of point A.

Based on the above investigation the following observations are made:

Fig. 14. Crack pattern at di€erent times for in®lled frame-2.

H. Singh et al. / Computers and Structures 69 (1998) 685±693

1. The proposed model is able to simulate the experimentally observed load de¯ection behaviour, separation of the in®ll from the frame, central strut width, failure mode and failure load. The inelastic algorithms are able to predict the sequence of formation of the plastic hinges in the frame members and the cracks in the in®lls. 2. For inelastic dynamic analysis, the proposed model can predict the entire time history response of the in®lled frame systems. The plastic hinges and the cracks `disappear' on the reversal of loads and on reduction of the magnitude of the exciting force, however, the state of stresses at these points now lies inside the yield surface, but plastic strains continue to be present. The inelastic response quantities have been found to di€er a lot form the elastic response quantities. So elastic analysis is not adequate, and inelastic analysis is required to simulate the realistic behaviour of the in®lled frame systems.

References [1] Holmes M. Steel frames with brick work and concrete in-®lling. Institute of Civil Engineers, London 1961;19:473±98. [2] Smith BS. Lateral sti€ness of in®lled frames. J Struct Div, ASCE 1962;88(ST6):183±99. [3] Mallick DV, Severn RT. The behaviour of in®lled frames under static loading. Proc of ICE 1967;38 (Sept/ Dec):639±56. [4] Liauw TC, Kwan KH. Nonlinear behaviour of non-integral in®lled frames. Comp Struct 1984;18(3):.

693

[5] May IM, Ma SYA. 1984. Computer aided analysis and design of shear wall panels in frames using ®nite element method. Proc of Int Conf on Computer Aided Analysis and Design of Concrete Structs. Yugoslavia. [6] Papia M. Analysis of in®lled frames using a coupled ®nite element and boundary element solution scheme. Int J Num Meth Engng 1988;26:731±42. [7] Haddad MHE. Finite element analysis of in®lled frames considering cracking and separation phenomena. Comp Struct 1991;41(3):439±47. [8] May IM, Naji JH. Nonlinear analysis of in®lled frames under monotonic and cyclic loading. Comp Struct 1991;38(2):149±60. [9] Choubey UB, Sinha SN. Cyclic response of in®lled frames. J Struct Engng 1994;21(3):203±11. [10] Singh H, 1995. Response of reinforced concrete frames with in®lled panels under earthquake excitation, Ph.D. thesis, submitted to Thapar Institute of Engineering and Technology, Deemed University, Patiala, India. [11] Hughes TJR. The ®nite element method, linear static and dynamic ®nite element analysis. Englewood Cli€s, NJ: Prentice-Hall, 1987. [12] Powell GH, Fu-song Chen Paul. 3D beam±column element with generalized plastic hinges. J Engng, Mech, ASCE 1986;112(7):. [13] Anderson JC, Townsend WH. Models for RC frames with degrading sti€ness. J Struct Engng Div, ASCE 1977;103(ST12):2361±76. [14] IS: 456-1978. Code of practice for plain and reinforced concrete. Bureau of Indian Standards, New Delhi, India. [15] Page AW, Ali SS. Finite element model for masonary subjected to concentrated loads. J Struct Engng 1988;14(8):1761±83. [16] Hughes TJR, Pister KS, Taylor RL. Implicit±explicit ®nite elements in nonlinear transient analysis. Comp Meth App Mech Engng 1979;17/18:59±82. [17] Choubey UB. Behaviour of in®lled frames under cyclic loads. Ph.D. thesis. submitted to IIT Delhi, 1990.