Structural Analysis of a RC Shear Wall by Use of a Truss Model

Purpose of present work is to develop a reliable and simple method for structural analysis of RC Shear Walls. The shear wall is simulated by a truss model as the bar of a truss is the simplest finite element. An iterative method is used. Initially, there are only concrete bars. Repeated structural analyses are per-formed. After each structural analysis, every concrete bar exceeding tensile strength is replaced by a steel bar. For every concrete bar exceeding compressive strength, first its section area is increased. If this is not enough, a steel bar is placed at the side of it. For every steel bar exceeding tensile or compressive strength, its section area is increased. After the end of every structural analysis, if all concrete and steel bars fall within tensile and compressive strengths, the output data are written and the analysis is terminated. Otherwise, the structural analysis is repeated. As all the necessary conditions (static, elastic, linearized geometric) are satisfied and the stresses of ALL concrete and steel bars fall within the tensile and compressive strengths, the results are accepta-ble. Usually, the proposed method exhibits a fast convergence in 4 - 5 repeats of structural analysis of the RC Shear Wall.


Introduction
The study of behavior of structural shear walls is recently of great interest for Civil Engineers [1] [2]. The structural analysis of a RC shear wall can be per-

Discretization
The shear wall is discretized to a grid of 8 × 6 = 48 equal square elements with dimensions 50 cm × 50 cm × 30 cm, as shown in Figure 3(a). One of these square elements is shown enlarged in Figure 3(b).

Truss Model
Every continuum square element is simulated by an elementary square plane truss, as shown in Figure 4. By combining the stress-strain equations of the continuum square element of Figure 4(a), with the force-displacement equations of the elementary square plane truss model of Figure 4

Numbering of Nodes and Bars
For the structural analysis of the truss model of the shear wall, the 9 × 7 = 63 nodes and 8 (7 + 6 + 2 × 6) = 200 bars, of the whole truss model of the shear wall, are systematically numbered, as shown in Appendix A. In the same figure, are also shown the external loads , x z P P at the top nodes of the wall, as well as the reference axes system Oxz.

Computer Program
The computer program, developed here for the 2D structural analysis of the truss model of a RC (reinforced concrete) shear wall, will be documented, in the following, step-by-step, by applying it on the specific example under consideration. only for the support nodes 1 up to 7. Then, the initial co-ordinates x, z in m of the node, with respect to global reference axes system Oxz, are read, and the external loads , Within subroutine STIF, for every concrete bar, its axial elastic stiffness is determined initial Young (Elasticity) modulus of concrete (later, when steel bars will be considered, 2 0 7000 kN cm E = will be used, too, as initial Young (Elasticity) modulus of steel). A is section area in cm 2 of the bar and 0 l initial undeformed length in m of the bar. The initial projections of bar axis are found and its initial direction cosines are determined and the 2 × 2 stiffness matrix of the bar is formed which, with positive or negative sign, is summed to appropriate submatrices 2 × 2 of the global stiffness matrix K, as shown in Figure 7, where , l r numbers of nodes that the bar connects, left and right. When this procedure is completed for  For every node, its displacements , x z u u in mm and its new co-ordinates , x z x u z u + + in m, are written as output in Appendix B. Then, the main program MAIN calls subroutine NONL, which applies the geometrically nonlinear equations of the problem.
For every bar, first the present projections of its axis , Then, the quantities , x z u u of first run of Appendix B, the deformed configuration of the structure has been drawn in Figure 9, by using a   And, it is observed that they are verified with high accuracy, with errors only 0.542‰ up to 2.667‰ (per thousand), as demonstrated in Figure 9.
Based also on Appendix C, Figure 10 shows, for the first run, the stresses in kN/cm 2 of the 66 concrete bars exceeding tensile strength,

Four Steps for the Strengthening of the Shear Wall
After every run of the program, the following four steps are performed, for the strengthening of the shear wall, as described by the flow-chart of Figure 11.
Step 1 Every concrete bar exceeding tensile strength, Step 2 For every concrete bar exceeding compressive strength, 2 3.0 kN cm σ < − , first its section area is increased, by increasing the corresponding local width w of the shear wall, so that to achieve, as far as possible, a reduction of compressive stress σ of concrete bar.
Step 3 If the above step 2 is not enough to allow a sufficient reduction of concrete bar compressive stress, the compressed concrete bar is maintained by receiving a part of the axial compression 0 compressive strength of steel. So, for the limit axial compressive deformation 0.002 y ε = − , for both compressed bars, the initial concrete bar and the additional steel bar at the side of it, the total axial compressive force of two bars tends to 0 N N N + ∆ = , as is required. If any of two bars slightly exceeds the corresponding compressive strength, the steel bar section area is increased.
Step 4 For every steel bar exceeding tensile or compressive strength, its section area is increased.
At the right side of the shear wall, compressed due to overturning moments of horizontal seismic loading, it is here advised that the concrete section is increased, near the base, in order to avoid stress concentrations. Open Journal of Civil Engineering Usually, the performing of above four steps takes only 3 -4 runs of the computer program, after which, ALL stresses σ, of concrete and steel bars, fall within the permissible limits of corresponding tensile and compressive strengths: 2 2 3.0 kN cm 0.3 kN cm σ − < < + for concrete bars and 2 2 14 kN cm 14 kN cm σ − < < + for steel bars. So, the proposed iterative method, for strengthening of a shear wall, exhibits a rapid convergence, in 4 -5 runs of the computer program.
And, as all the necessary conditions (static, elastic, linearized geometric) are satisfied, according to what mentioned in previous sections, the results of the proposed here iterative method, for the strengthening of a shear wall, are acceptable.

Fourth and Final Run of the Computer Program
In the fourth and final run of the computer program, for the present application, the number of nodes is 66 n ν = , that is 63 initial nodes and 3 additional ones, of the truss model for right base enlargement, as shown in Figure 13. Whereas, the number of bars is 212 b ν = , that is the 200 initial bars, plus 4 new compressed vertical steel bars at the right lower side of the shear wall and 2 new compressed horizontal steel bars at the top left corner of the shear wall, as well as the 6 new bars of the truss model, for right base enlargement, as shown in Figure 13.
The numbering of 3 additional nodes and 12 additional bars, for the fourth and final run of the computer program, is shown in Figure 13.

Conclusions
An iterative method is proposed for the nonlinear 2D structural analysis of a truss model of a RC shear wall, which is applied on a typical RC shear wall.
A relevant computer program has been developed, which is documented, stepby-step, by applying it on the specific example under consideration.
Initially, the truss model of the shear wall is assumed consisting only of concrete bars obeying a linear elastic axial σ-ε law.
After every structural analysis of truss model, the following four steps are performed: 1) Every concrete bar exceeding tensile strength is replaced by a steel bar.
2) For every concrete bar exceeding compressive strength, its section area is increased.
3) If the above is not enough,the concrete bar is maintained, by receiving part of axial compression and a steel bar is added at the side of it. 4) For every steel bar exceeding tensile or compressive strength, its section area is increased.
At the lower part of side of shear wall, compressed due to overturning moments from horizontal seismic loading, the concrete section is enlarged, in order to avoid stress concentration.
Usually, in 4 -5 runs of the computer program, ALL stresses of concrete and steel bars fall within tensile and compressive strengths. So, the proposed here iterative method exhibits a rapid convergence.
And, as all necessary conditions (static, elastic, linearized geometric) are satisfied, the results of proposed here iterative method are acceptable.
Because of alternating nature, in direction of the horizontal seismic loading, the results of the application have to be symmetrically extended to both sides of the shear wall.
The results of the application confirm the need for boundary columns, at the two sides of the shear wall, and a boundary horizontal beam at the top, as well as the need for a grid of ascending diagonal and horizontal steel bars, receiving tension and shear, in the main body of the shear wall. Trondheim, Tapir. [