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CD02-003 MODELING OF NONLINEAR BEHAVIOR OF RC SHEAR WALLS UNDER COMBINED AXIAL, SHEAR AND FLEXURAL LOADING

B. Ghiassi1, M. Soltani2, A. A. Tasnimi3

1M.Sc. Student, School of Engineering, Tarbiat Modares University, Tehran, Iran 2Assistant Professor, School of Engineering Tarbiat Modares University, Tehran, Iran

3Professor, School of Engineering, Tarbiat Modares University, Tehran, Iran ABSTRACT Predicting the behavior of RC shear walls under bending moment without existence of interaction of any other kind of loadings like shear or axial load is simple and can be conducted with good accuracy. But what is a great concern, is predicting their behavior under the interaction of shear, axial and flexural loadings. In this research, there is an effort to investigate the behavior of RC shear walls under this condition of loading with a novel approach. A general but simple macro model is proposed that can include flexural and shear behavior of the wall by considering the effects of pull put and slippage of reinforcing bars as well as concrete tension softening, stiffening and confinement. This simple model is applicable to different wall shapes with different reinforcement ratios and its prediction has good agreement with experimental results. The predicted behavior of the walls is compared with some available experimental results to show the accuracy of the proposed method. Keywords: RC shear wall, behavior, nonlinear analysis, P-M-V interaction, modeling

1. INTRODUCTION Reinforced concrete walls are very effective in resistance of lateral loads imposed by earthquakes. They provide high strength and stiffness and if truly designed, can also provide good ductility for structures. So many analytical and experimental researches have been carried out to study the seismic behavior of RC walls and RC frame-wall systems. The response of these elements is complex and their overall behavior is influenced by a combination of flexural, shear, and axial deformations. Prediction of the exact inelastic response of RC walls requires accurate analytical material models that consider the important characteristics and behavioral response features such as concrete tension-stiffening, nonlinear shear behavior and effects of loading condition, confinement, transverse reinforcement, and reinforcing bars slippage on strength, stiffness and deformation capacity. Analytical modeling of the inelastic behavior of RC wall systems can be accomplished either by using microscopic finite element models or macroscopic models. Various analytical models have been proposed for predicting the inelastic

214 / Modeling of Nonlinear Behavior of RC Shear…. ––––––––––––––––––––––––––––

behavior response of RC walls through a microscopic or macroscopic approach. Although microscopic finite element models can provide a refined and detailed definition of the local response, their efficiency, practicality, and reliability are questionable due to complexities involved in developing the model and interpreting the results. Macroscopic models, on the other hand, are practical and efficient, although their application is restricted, based on the simplifying assumptions upon which the model is based[1]. A common macro modeling approach is using a beam-column element model. This model consists of an elastic flexural element with a nonlinear rotational spring at each end to account for the inelastic behavior of critical regions (Error! Reference source not found.). To model the RC walls more realistically, improvements, such as multiple spring representation [2], varying inelastic zones [3], and specific inelastic shear behavior [4] have been introduced into simple beam column models. However, inelastic response of structural walls subjected to horizontal loads is dominated by large tensile strains and fixed end rotation due to bond slip effects, associated with shifting of the neutral axis. This feature cannot directly be modeled by a beam-column model, which assumes that rotations occur around points on the centroidal axis of the wall and this method disregards the important features of the experimentally observed behavior, including the variation of the neutral axis of the wall cross section, rocking and reinforcing bars slippage [1].

Figure 1. Beam-column element model[1]

Kabeyasawa et al. [5] proposed a new macroscopic three-vertical-line element model (TVLEM) to account for the features that beam-column model cannot capture. In this model, the wall was idealized as three vertical line elements with rigid beams at the top and bottom levels (Figure 1). In this model, shear stiffness degradation was incorporated, but was assumed to be independent of the axial load and bending moment. This method is improved by different authors such as Vulcano et el. [6], [7], Kabeyasawa [8] and Colotti [9].

–––––––––––––––––––––––– 3rd International Conference on Concrete & Development / 215

Figure 1. Three-vertical-line element model (TVLEM)[1]

In this paper a novel macro model is proposed that can include flexural and shear behavior of the wall by considering the effects of reinforcing bars pull put and slippage, concrete tension softening and stiffening and confinement. This simple model is applicable to different wall shapes and reinforcement ratios and shows good agreement with experimental results. 2. ANALYTICAL MODEL The adopted method for nonlinear analysis of RC walls in this paper can take into account the effects of flexural and shear behavior. Flexural behavior is computed by considering a macro fiber model for the wall with including the effects of confinement and reinforcing bars pull-out. Shear behavior is predicted according to a nonlinear analysis of RC elements under in-plane stresses through a fixed smeared crack analysis approach. In this method, as is shown in Figure 3, the force-displacement curve of the flexural behavior, reinforcement pull out and shear behavior of the wall is computed separately and will be combined to obtain the total nonlinear behavior of the wall. The total behavior of the wall is computed by adding the displacements caused by each of the three behavioral modes that have been mentioned above for any shear value (Figure ). If one of the behavioral modes has lower strength than the others, it will be the controlling behavior (e.g. Shear behavior in Figure ).

Figure 2. Adopted method for computing the behavior of RC walls


3. FLEXURAL MODELING 3.1. Moment Curvature Analysis Adopted method here for flexural analysis of RC wall resembles a macro fiber model. In this model, the wall is divided into a series of uniaxial elements (Error! Reference source not found.) and then by considering the appropriate material uniaxial nonlinear models, the moment-curvature analysis of the wall will be done with considering the effects of confinement and reinforcing bars pull out from the foundation. Each fiber in the model can have different material properties and steel ratio. Maekawa's [10] material models are used for modeling the uniaxial behavior of the concrete and reinforcing bars.

Figure 4. RC wall geometry and fibers definition

Knowing the applied axial force on the wall, the moment curvature analysis is done by assuming a linear strain distribution across the section and calculating the stresses in each fiber (Error! Reference source not found.) and controlling if Eq.(1) is satisfied. If this equation is satisfied, the moment and curvature in the section can be computed according to Eq.(2) and Eq.(3) in that step, and if it is not, then the assumed strain distribution should be corrected in an iterative procedure. This procedure is repeated in several steps until the failure of the steels or crushing of the concrete occurs.

Figure 5. Assumed strain distribution in the section


∑ = NAiiσ (1)

∑ = MyA iiiσ (2)

lct εε

κ+

= (3)

where, iσ is the stress in each fiber, iA is the area of each fiber, N is the constant

axial force applied to the wall, iy is the fiber distance to the neutral axis of the

section, κ is the curvature of the section, tε is the first layer strain, cε is the last layer strain and l is the length of the section.

Figure 6. Moment-curvature behavior of a RC wall

3.2. Shear Displacement Curve Due to Flexural Behavior By using the Eq. (4) the shear corresponding to the moment of the wall can be computed in each step and the curvature is convertible to the wall base rotation,θ , by using the Eq. (5). The top displacement of the wall can also be calculated by using Eq. (6).

hMV = (4)

∫κ

=θh

0

dxhx.

(5)

hl

θ=δ (6)

where h is the wall height. So by computing the shear and the corresponding displacement in each step, the shear displacement curve of the wall due to flexural behavior is attained.


Figure 7. Determination of the wall rotation Figure

4. REINFORCMENT PULL OUT In reinforced concrete members, local discontinuities, such as pulling out of reinforcing bars from the thicker element and sinking the thinner element to the thicker one, tend to take place as a result of abrupt changes in the section stiffness at the joint planes connecting two components of different thickness [11]. This phenomenon has an important effect on the displacements of the wall that should be considered in the analytical methods to obtain good results in comparison to experimental results. Here, the Maekawa's pull out model [11] is used to consider this important effect. This model describes a relation between steel strain and loaded end slip or relative displacement of steel bar to concrete and is applicable to both elastic and plastic stress states. This model is capable of giving a unique strain-slip relation for a bar that has a long embedded length and that has slip at the free-end prevented (Eq. (7)).

⎪⎩

⎪⎨

⎧

−−+=

<<=

<+=

))((047.0

)35002(

shsyuy

shyy

yss

ffss

forss

fors

εε

εεε

εεεε

(7) where sε is the bar strain, ys is the normalized slip when bar strain is equal to

yield strain, uf is the tensile strength of steel bar, yf is yield strength of the bar and s is normalized slip that is related to bar diameter and concrete compressive strength as follows:

67.0)

200'( cf

DSlips =

(8) where D is the bar diameter, cf ' is the concrete compressive strength and s is normalized slip as defined in Eq. (8). Using this model, the moment rotation curve of the wall due to slipping can be computed (Error! Reference source not found.).


Figure 8. Slip calculation flowchart

5. SHEAR MODELING Nonlinear analysis of the wall in shear is done through a fixed smeared crack approach by considering the wall as a RC element.

Figure 9. Shear modeling of RC walls


In the smeared crack approaches the cracks and reinforcing bars are idealized as being smeared over the element. The cracks, once generated, are not modeled directly but their effects will be considered by changing the material constitutive models. Generally, the smeared crack approach is conducted by two methods. Rotating crack approach and fixed crack approach. The rotating crack approach assumes that the crack direction coincides with the principal direction of average strain. Accordingly, it can be changed or rotated following the stress condition (Error! Reference source not found.), and because the shear stress vanishes on the continually updated principle planes, no shear model is needed in this method. In the step-by-step computation, one crack is considered and the previous ones are erased. So the rotating crack approach does not explicitly account for shear slip and shear stress transfer due to aggregate interlock. In the fixed crack approach the crack direction once generated will not change during the analysis until the direction changes more than a specific value. So there will be shear stress in the assumed crack direction and the shear transfer due to aggregate interlock will be considered. In this paper the fixed smeared crack approach is used to model the shear behavior of RC wall in an iterative procedure as shown in.

Figure 10. Rotating smeared crack approach

Figure 11. Fixed smeared crack approach


xyγ

oo xyxyxyy γγσσσ === ,0,

1221 ,, γεε pp

xyxy σστσσ ,,,, 21

0,01 == xyy σσσ

0,0, === xyxyy γσσσ o

θ

Figure 12. Fixed smeared crack approach flow chart for analysis of RC element

6. VERIFICATION To control the accuracy of the adopted method, the analysis results are compared with the experimental behavior of some RC walls. The experimental data for the first example are taken from (Lanker and Mang[12]). The wall shape and reinforcement arrangement are shown in. This wall is analyzed with the adopted method and the results are compared in Figure 13. The second RC wall is taken from Oesterle et al. [13]. This wall is also analyzed with the adopted method and the results are compared in Figure 15.


Figure 13. Experimental and analytical behavior of the first wall

Figure 14. Geometry and reinforcement arrangement of the second wall[14]


Figure 15. Experimental and analytical results of the second wall

7. CONCLUSION In this paper a general macro modeling method is proposed that can include flexural and shear behavior of the wall by considering the effects of reinforcing bars pull put and slippage, concrete tension softening, stiffening and confinement. This method can be used to predict the nonlinear response of RC walls by considering all important characteristics and behavioral response features, and is applicable to different wall shapes and the amount of reinforcement. To study the accuracy of the proposed method, the results are compared with some experimental results and their good agreement is shown. REFERENCES 1. Orakal, K., L.M. Massone, and W. Wallace, Analytical Modeling of Reinforced

Concrete walls for Predicting Flexural and Coupled-Shear-Flexural Responses. 2006, Pacific Earthquake Engineering Research Center.

2. Takayanagi, T. and W.C. Schnobrich, Computed behavior of reinforced concrete coupled shear walls. 1976, University of Illinois.

3. Keshavarzian, M. and W.C. Schnobrich, Computed nonlinear response of reinforced concrete wall-frame structures. 1984, University of Illinois.

4. Aristazabal-Ochoa, J.D., Cracking and shear effects of structural walls. ASCE Journal of Structural Engineering, 1983. 109(5): p. 1267-1275.

5. Kabeyasawa, T., et al., analysis of full-scale seven-story concrete test structure. Journal of Faculty of Engineering, The University of Tokyo, 1983. 37(2): p. 731-478.

6. Vulcano, A. and V.V. Bertero. Nonlinear analysis of RC structural walls in 8th European Conference on EQ Engineering 1986. Lisbon, Portugal.

7. Vulcano, A., V.V. Bertero, and V. Colotti. Analytical modeling of RC structural walls. in 9th World Conference on Earthquake Engineering. 1988. Tokyo-Kyoto, Japan.

8. Kabeyasawa, T. Design of RC Shear Walls in Hybrid Wall System. 1997. 9. Colotti, V., Shear behavior of RC structural walls. ASCE Journal of Structural


Engineering, 1993. 119(3): p. 728-746. 10. Maekawa, K., A. Pimanmas, and H. Okamura, Nonlinear Mechanics of

Reinforced Concrete. 2003: Spon Press. 11. Okamura, H. and K. Maekawa, Nonlinear Analysis and Constitutive Models of

Reinforced Concrete. 1991. 12. Lancker, R. and H.A. Mang, Adaptive FE Analysis pf RC Shells,

II:Applications. Journal of Engineering Mechanics, 2001. 127(12): p. 1213-1222.

13. Oesterle, R.G., et al., Eartquake-Resistant Structural walls-Tests of Isolated Walls. 1976, National Science Foundation, Constructuion Technology Laboratories, Portlan Cement Association: Skokie.

14. Palermo, D. and F.J. Vecchio, Compression Field Modeling of Reinforced Concrete Subjected to Reversed Loading: Verification. ACI Structural Journal, 2004. 101(2): p. 155-164.

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