Reinforced concrete membrane elements subjected to reversed cyclic in-plane shear stress

Nuclear Engineering and Design 115 (1989) 61-72 North-Holland, Amsterdam

61

R E I N F O R C E D C O N C R E T E M E M B R A N E E L E M E N T S S U B J E C T E D T O REVERSED CYCLIC IN-PLANE SHEAR STRESS N. O H O M O R I , H. T S U B O T A , N. I N O U E , K. K U R I H A R A a n d S. W A T A N A B E Kajima Institute of Construction Technology, 19-1 Tobitakyu 2-Chome, Chofu-shi, Tokyo 182, Japan Received 17 September 1987

A macroscopic model has become noticeable instead of a microscopic model by FEM analysis for concrete structures. Among them the model proposed by Vecchio and Collins is considered to be one of the powerful tools because of its rationalility and simplicity. The essential point of this model is a modified compression theory of cracked concrete and the stress-strain relationship for cracked concrete. This paper follows basically Collins' theory but is revised so as to be able to predict the response of reinforced concrete elements subjected to alternate reversed cyclic in-plane shear simulating earthquake forces. The overall characteristics of the new model exist in the tensile residual strains accumulated by cycles in cracked concrete, which is different from the monotonic loading.

1. Introduction In order to confirm the safety of large-scale, complex reinforced concrete structures such as containment structures and reactor buildings for nuclear power plants, it is neccessary to develop an analytical model which can predict how such structures will respond under various external loads, Up to now, many analytical models based on the nonlinear finite element method have been proposed and most of these models are microscopic models, in which concrete, steel and bond are treated separately and their characteristics are defined in precise microscopic terms. Recently, instead of these microscopic models, macroscopic models in which a cracked reinforced concrete element is formulated on the basis of average stress-strain conditions, have become noticeable. These macroscopic models are considered to be more practical than the microscopic models when evaluating the overall behaviour of the structure. Especially, the analytical model proposed by Vecchio and Collins [1] is considered to be one of the most powerful tools because of its rationality and simplicity. However, this model is applicable only in predicting the response of reinforced concrete elements subjected to monotonic membrane stresses and, in considering the safety of structures against earthquake loads, it is necessary to develop a practical model capable of predicting the responses under general reversed cyclic loadings. In

this paper, a new analytical macroscopic model is proposed that is able to predict the responses of reinforced concrete elements subjected to reversed cyclic in-plane shear stresses.

2. Approach to the problem In establishing a new analytical model that is capable of predicting the response of reinforced concrete elements subjected to reversed cyclic in-plane shear stresses, the same analytical approach as proposed by Vecchio and Collins was essentially adopted. The analytical model of Vecchio and Collins is schematically summarized in fig. 1 (reprinted from ref. [1]). This analytical model separately treats the conditions in the reinforcement and the conditions in the cracked concrete in terms of average stresses and average strains. The following three conditions are used for solving the problem: (1) Compatibility requirements; the average strains in the reinforcement must equal average strains in the concrete. (2) Equilibrium conditions; the average stresses in the reinforcement plus the average stresses in the c o n crete equal the total stresses acting on the reinforced concrete element. (3) Stress-strain relationships; a simple bi-linear relationship is adopted for evaluating the average

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stress-strain relationship for the reinforcement. The average stress-strain relationships for concrete were determined from test results of an extensive experimental program. The principal compressive stress in the concrete was found to be a function not only of the principal compressive strain but also of the co-existing principal tensile strain. The principal tensile stress in the concrete after cracking was found to depend on the principal tensile strain in the concrete and a decreasing function of the tensile stress with the increasing tensile strain was suggested. However, this model is applicable only in predicting the response of reinforced concrete elements subjected to monotonic loading and in order to establish a new model applicable to reversed cyclic loading it is necessary to construct new stress-strain relationships of concrete and steel including the hysteretic loop of unloading and reloading stages. But, the compatibility requirements and the equilibrium conditions formulated in Vecchio and Collins' model are also applicable to reversed cyclic loadings.

3. Experimental program A series of loading tests on reinforced concrete panels subjected to reversed and cyclic in-plane stresses has

63

been planned and is now under study at the Kajima Institute of Construction Technology. The main object of the experimental program is to obtain useful information for developing a new hysteresis loop model of the concrete and the reinforcement. Several reinforced concrete panels have already been tested. The test specimens were 2500 mm square and 140 mm thick and deformed rebars were arranged in two layers in orthogonal directions as shown in fig. 2. In the test panels, the percentage of reinforcing steel was varied from 0.8% to 2.0%, but always with the same reinforcement ratio of Psx and psy in orthogonal directions. Wire strain gages were attached to the orthogonal reinforcing bars in order to confirm steel yield. These data were not used for establishing analytical models but were very useful for checking the orthogonal strains of cx and Cy obtained from LVDTs' measurements shown in fig. 2. The test panels were loaded by a newly developed testing facility as shown in figs. 3 and 4. This facility can apply any combination of in-plane shear and normal forces to the test panels by automatically controlled 24 closed loop hydrauhc actuators and a network of links. The maximum capacity of load and stroke of each actuator are + 125 ton and + 125 mm respectively. All tests were conducted in reversed cyclic pure in-plane shear.

Fig. 3. Overall view of testing facility.

64

N. Ohomori et al. / Cyclic in-plane shear stress

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4. Analysis of test data In the test, the total average shear stress V applied to the test panel was evaluated from measured data of 24

load cells attached to the hydraulic actuators. For evaluating the average strains of the test panel, 6 LVDTs, X1, X2 etc., were attached to the surface of the specimen as shown in fig. 2(a). The measured data from 6 LVDTs were reduced to average strains in the X, Y, and two diagonal directions; %,, ~y, c45 and El35 respectively. These strains were then used to define a Mohr's circle of average stratus for each load stage and finally one optimum circle passing as close as possible through all four measured points was determined. Having defined a strain circle, the remaining strain parameters such as principal strains (El, E2)' and shear strain (y) could be determined.

N. Ohomori et aL / Cyclic in-plane shear stress

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Fig. 5. Hysteretic loop model of the principal stress-strain relationship for (a) concrete; (b) stress-strain relationship for reinforcement. stress-strain relationship of concrete is most important and from this relationship some significant characteristics of cracked concrete subjected to reversed cyclic in-plane shear stresses can be found. Namely, decrease of stiffness during the unloading stage, successive slip phenomena and restoration of compressive stiffness at reloading stage are obvious. In these figures, the predicted responses by Vecchio and Collins' model for monotonic loading are also indicated for reference. Here, the cracking strength of concrete was assumed to be the observed values.

Average stresses of reinforcement (fsx, fsy) were determined from the measured strains in the X- and Y-directions. Using these reinforcement stresses, the average concrete stresses in the X and Y directions (fcx, fcy) were calculated from equilibrium equations. Knowing the applied shear stress (V) acting on the test panels, the remaining concrete stress parameters could be determined. Thus, it was possible to determine concrete strain and stress circles at each load stage. For example, the shear stress-strain relationship, the shear stress versus orthogonal strain and the principal stress-strain relationship of concrete obtained from a typical panel (KP6) are shown in fig. 11 (a), fig. 12(a) and fig. 13(a), respectively. The reinforcement ratio of this specimen was 2.0% and its failure mode was a sliding shear failure of concrete prior to yielding of reinforcement. Among these test results, the principal

5. Analytical model

Based on the stress-strain relationships obtained from the test results, new hysteresis models of cracked concrete and steel were formulated.

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5.1. Principal stress-strain relationship for cracked concrete

Fig. 5(a) shows a schematic hysteresis loop model of the principal stress-strain relationship of cracked concrete (fd, f c 2 - Cl, c2)- The points C, F, I, L and Q correspond to the maximum applied shear stress stages in each loading cycle and the points A, D, G, J, M and R correspond to the loading step when applied shear stress is equal to zero. The outlines of the proposed hysteresis loop model for cracked concrete are as follows: (1) At first, residual stresses and strains of cracked concrete are defined as the stresses and strains when the applied shear stress is equal to zero (D, G, J, M, and R in fig. 5(a)). The magnitude of the residual strains (0~1, 0c2) is considered to depend on the maximum strains (pq, p ~ 2 , p ~ x , pCy) at the start of unloading. In fig. 6 and fig. 7, relationships between the measured residual strains and measured maximum strains are shown. Generally speaking, there can be noticed two groups of plots, i.e. steel yielded ([3) group and not yielded (©). But the data from steel are not yet sufficient to be formulated separately from non-yield ones. Therefore it was decided to assume approximate functions to define the magnitude of the residual strains used in the proposed analytical model as the curves indicated in figs. 6 and 7. Since the residual stress and strain are known, the unloading path (CD, FG, I J, LM, and QR in fig. 5(a)) can be defined. (2) Next, the stiffness of cracked concrete in the slip region must be defined. The measured shear stress-strain relationships as shown in fig. 11(a) can be simplified to a schematic model as illustrated in fig. 8.

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N. Ohomori et aL / Cyclic in-plane shear stress

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cracked concrete in the slip region (Eslip) is formulated as a function of the residual principal strain. Fig. 9 shows the relationship between the decreasing ratio (fl) of the slip stiffness to the elastic stiffness of concrete and the principal residual strain. Since the behaviour in this slip region is unstable, the measured data display considerable scatter. An approximate function to define the stiffness of cracked concrete in the slip region is also represented in fig. 9. (3) The restoring tensile stiffness (Et) of cracked concrete from the residual stress state to the reloading a (×10-2)

tensile stress state is defined as a function of the maximum tensile strain experienced in the former loading cycle. Fig. 10 shows the relationship between the ratio ( a ) of restoring tensile stiffness to elastic stiffness and maximum tensile strain (p~l, pC2). From these measured data, an approximate function to define the restoring stiffness ratio of the cracked concrete used in the analytical model is proposed as indicated in fig. 10. (4) The original principal compressive and tensile stress-strain relationships of cracked concrete proposed by Vecchio and Collins were adopted for evaluating the

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envelope stiffness of cracked concrete during the reloading stage (EF and K L in compressive region and BC, HI and PQ in tensile region shown in fig. 5(a)). In the proposed analytical model, however, an appropriate coordinate transformation of the original stress-strain relationships are performed in order to satisfy the continuous conditions of the hysteresis loop at the junction points (E and K in fig. 5(a)).

5.2. Stress-strain relationship for steel

As for the stress-strain relationship for reinforcing steel, the usual bilinear uni-axial stress-strain relationship shown in fig. 5(b) is adopted. In the model, both the unloading and reloading stiffness after the yielding of steel are equal to the elastic stiffness. It is assumed that steel does not go into compression and that there is no strain hardening.

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N. Ohomori et al. / Cyclic in-plane shear stress

69

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5.3. Numerical solution technique

The solutions were continuously obtained step by step at each loading stage and each loading cycle.

By using the aforementioned stress-strain relationships of concrete and steel, equilibrium equations and compatibility requirements, nonlinear simultaneous equations of unknown average stress and strain parameters were derived. The Newton-Raphson iteration technique was adopted to solve these nonlinear equations.

6.

Analytical

result

The proposed analytical model was adopted to predict the response of the specimens tested in this investi-

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70

N. Ohomori et al. / Cyclic in-plane shear stress

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gation. Among the obtained analytical results, the shear stress-strain relationship for specimen KP6, is presented in fig. ll(b). As shown in this figure, the predicted envelope curve represents the observed behaviour well. However, the predicted ultimate shear strength is larger than the observed one. Such a discrepancy is also found in the predicted result of monotonic loading as shown in fig. ll(a). This means that more precise studies are necessary to evaluate more accurate envelope curves under not only reversed cyclic loading but also monotonic loading. As to the hysteresis loop of the shear stress-strain relationship, the following characteristic behaviour under reversed cyclic shear loading are well predicted by

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the proposed method such as (a) decrease of concrete stiffness during unloading stage, (b) slip stiffness depending on residual strain and (c) keeping a constant value of shear stress at shear strain equal to zero. The predicted and observed shear stress versus longitudinal strain is compared in figs. 12(a) and 12(b). The ability of the model to predict the deformation of the panel reasonably well at all unloading and reloading stages should be noted. Especially, the residual strains after cracking are well evaluated in the proposed analytical model. Next, the predicted principal stress-strain relationship of concrete for specimen KP6 is presented in fig. 13(b). Comparing the test result, the analytical result

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71

N. Ohomori et al. / Cyclic in-plane shear stress

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represents well the dominant decrease of stiffness in the unloading stage from the tensile stress state, successive slip phenomena and restoration of compressive stiffness at the reloading stage. Furthermore, the analytical model was used to predict the response of the panel tested in another experimental study[2]. This panel (No. 5) was 2500 mm square and 167 mm thick and the reinforcement ratio was 0.85%. Fig. 14 shows the observed shear stress-strain relationship of this panel reprinted from ref. [2]. Since this specimen contained small amounts of reinforcement, the observed response was ductile and its failure mode was a yielding of reinforcement. In this figure, the predicted response by Vecchio and Collins' model for monotonic loading is also indicated for reference and the predicted response agrees well with the envelope curve of the observed shear stress-shear strain relationship. The predicted hysteretic response of the shear stress-shear strain relationship by the proposed model is shown in fig. 15. As shown in this figure, the analytical result agrees quite well with the observed hysteresis loop including the unloading stages and the reloading stages. In fig. 16 the predicted shear stress versus orthogonal strain response by the proposed model for this panel is shown. Fig. 17 shows the predicted principal stress-strain relationship for the concrete. From these analytical results, it is found that the characteristic behaviour of the accumulation of the tensile residual strain after the yielding of reinforcement can be predicted by the proposed analytical model. Thus, it is confirmed that the proposed model can predict the

response of the reinforced concrete element even after the yielding of reinforcement.

7. Conclusions

On the basis of the analytical and experimental work discussed in this paper the following conclusions can be made. (1) The response of reinforced concrete elements subjected to reversed cyclic in-plane shear stresses can be predicted by the proposed analytical model, which considers equilibrium, compatibility and stress-strain relationships including the hysteresis loop of unloading and reloading stages all expressed in terms of average stresses and average strains. (2) The analytical results obtained by the proposed model show that the dominant hysteretic behaviour with regard to the decrease of stiffness during unloading, successive slip phenomena and restoration of compressive stiffness at the reloading stages are well simulated analytically. And, these analytical results agree quite well with the observed behaviour. (3) However, as for the envelope curve of the hysteretic response there remain the discrepancies that the stiffness and ultimate strength are a bit larger than the observed results, especially in the case of a panel with a large reinforcement ratio. Such descrepancies are also found in the predicted results of monotonic loading and more precise studies are necessary to evaluate more accurate envelope curves under not

72

N. Ohomori et al. / Cyclic in-plane shear stress

only reversed cyclic loading but also monotonic loading. (4) Furthermore, it is necessary to obtain more data about panel tests with various parameters for improving the accuracy of the proposed analytical model. Such parametric tests will be performed in the near future at Kajima Institute.

References [1] FJ. Vecchio and M.P. Collins. The modified compressionfield theory for reinforced concrete elements subjected to shear, ACI J. 83-22 (1986) 219-231. [2] M. Saito et al, Experimental investigations on reinforced concrete panels for RCCV under in-plane shear loading. Summaries of technical papers of annual meeting Architectural Institute of Japan. Structures 1 (1985) 817-818 (in Japanese).