# Influence of Multiple Openings on Reinforced Concrete Outrigger Walls in a Tall Building
## 3 Evaluation of the Influence of Multiple Openings
### Finite Element Modelling  
In this study, Abaqus [20], which is a commercial nonlinear finite element analysis program, was used. Several researchers have used Abaqus in analyzing the nonlinear behavior of reinforced concrete structures [21; 22; 23; 24]. The program provides three material models for concrete. These are the smeared crack concrete, brittle crack concrete, and concrete damaged plasticity models. In this paper, the concrete damaged plasticity (CDP) model was used because it can represent the complete inelastic behavior of concrete, both in compression and tension [25; 26]. The CDP is a continuum, plasticity-based, damage model for concrete. It assumes that the main two failure mechanisms of concrete material are tensile cracking and compressive crushing, which are represented by the uniaxial tension and compression behavior. Under uniaxial tension, the stress-strain response follows a linear elastic relationship until the value of the failure stress is reached. The failure stress corresponds to the onset of micro-cracking in concrete material. Beyond the failure stress, the formation of micro-cracks is represented macroscopically with a softening stress-strain response, which induces strain localization in concrete structures. Under uniaxial compression, the response is linear up to the value of the initial yield. In the plastic regime, the response is typically characterized by stress hardening, which is followed by strain softening beyond ultimate stress. The CDP model in Abaqus is a plasticity material model that requires the definition of the yield surface and requires hardening and flow rules. The model makes use of the yield function proposed by [PERSON] and [PERSON] [27] to account for varying degrees of evolution of concrete strength under tension and compression. The CDP model assumes a non-associated potential plastic flow. The flow potential is the Drucker-Prager hyperbolic function, which requires the dilation angle (\(\psi\)), flow potential eccentricity (\(\epsilon\)), ratio of the initial biaxial compressive yield stress to the initial uniaxial compressive yield stress (\(\sigma_{b0}/\sigma_{c0}\)), and the ratio of the second stress invariant on the tensile meridian to that on the compressive meridian (\(K_{c}\)). The viscosity parameter should be entered to overcome some of the convergence difficulties caused by material models exhibiting softening behavior and stiffness degradation. The values of the parameters for the plasticity model used in this study are shown in Table 2 below. CDPs require a stress-strain relationship of concrete in uniaxial compression and tension. The relationship suggested by [PERSON] and [PERSON] [28], as shown in Figure 7, was used for the compression model. For the tension model, the stress-strain relationship proposed by [PERSON] et al. [29] was used. The tension model as shown in Figure 8 includes the tension stiffening to simplify the post cracking behavior for concrete and the interaction between reinforcement and concrete. The CDP model was used in predicting the shear behavior of reinforced and prestressed concrete deep beams and confirmed that the shear strengths predicted by the finite element analysis agreed well with those obtained by experiments [30]. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline \(\psi\) & 38 \\ \(\epsilon\) & 0.1 \\ \(\sigma_{b0}/\sigma_{c0}\) & 1.16 \\ \(K_{c}\) & 0.667 \\ Viscosity parameter & 0.0001 \\ \hline \end{tabular}
\end{table} 
Table 2: Plasticity parameter of concrete damaged plasticity (CDP) specified for this study. When applying downward force on the loading plate for the analysis model used in this study, the reinforcements with cross-sectional areas \(A_{s1}\) and \(A_{53}\) were subjected to compression. On the other hand, the reinforcements with cross-sectional areas \(A_{s2}\) and \(A_{s4}\) were subjected to tensile forces.