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المستخلص
Intense research has been dedicated to develop beam-column element models for cyclic analysis of building structures in order to achieve high levels of accuracy and efficiency. The reference models for the nonlinear analysis of reinforced concrete members are the flexibility-based Finite Element models (fiber models). In these models the element is subdivided into segments distributed along the member length, and the cross section of each segment is subdivided into fibers. The cross section constitutive relation is derived by integrating of the fiber responses, which follow the uniaxial stress-strain relation of the particular material. The fiber models require relatively large amount of computations and computer storage requirements for monitoring the responses of the concrete and steel fibers of the various segments located along the member length.
The objective of this study is to develop a reliable and computationally efficient beam-column element for the analysis of reinforced concrete members subjected to seismic loading. The developed model strikes a good balance between accuracy and simplicity. It is flexibility-based and relies on using force interpolation functions that satisfy equilibrium of moment and axial forces along the element length. The simplicity of the proposed modeling approach is achieved by monitoring only the responses of the end sections of the element and eliminating the need for considering many segments along the length of the element, which results in a significant reduction in computations.
In the proposed model, the end sections are descretized into steel and concrete fibers and the section force-deformation relation is derived by integrating the stress-strain relations of the fibers. The selected material model used for representing the stress-strain relationship of concrete is simple to programming and consider the various behavioral characteristics of concrete under cyclic loading conditions. The effect of confinement by transverse steel on concrete behavior is incorporated into the uniaxial stress-strain relation. The fiber strains can be obtained as an output of the model. These strains can be used for performing seismic damage evaluation of the frame members.
The inelastic length at the ends of the RC member is divided into two inelastic zones; cracking and yielding. The length of each zone varies according to the loading history. In the proposed modeling approach, the lengths of the cracking and yielding zones are calculated at the member ends at every load increment. The overall response of the member is estimated using a preset flexibility distribution functions along the element length. The preset flexibility distribution functions are selected to fit the actual flexibility distributions. A flexibility factor (η) is utilized to facilitates selecting the proper flexibility distribution shape along the element length. It enables the user to select any suitable flexibility distribution shape by determining the value of a flexibility shape factor (η).
The developed model eliminates the need to formulate the element flexibility matrix each time there is a change in the flexibility distribution along the beam-column length. Also it facilitates calibrating the simplified beam-column elements with the exact finite elements to find a proper selection of the flexibility shape factor (η) that produces flexibility distribution shapes with good match with the complex actual flexibility distributions of the finite elements.
The proposed beam-column element model presented in this study is implemented in the general purpose computer program DRAIN-2DX (Prakash and Powell [1993]) as a separate element to be one of the program element library. This element model is developed for performing inelastic analysis of RC structures under the effect of various loading conditions (monotonic, cyclic and earthquake).
The results of the proposed element are compared with the outcomes of the flexibility-based finite element model (fiber model). The comparison is conducted on the one element level of a beam-column element and on the structure level of a three-story frame under monotonic, quasi static (cyclic) and dynamic loadings. The analysis conducted indicates that the proposed model is capable of describing, with satisfactory accuracy and computational efficiency, the response of reinforced concrete frame structures under different loading conditions. The model performance is evaluated under the effect of various levels of axial loading and steel strain hardening ratios.
The reduction in the solution time when using the proposed model has been found approximately to be equal to 84% of the solution time of the flexibility-based finite element model in the case of the three story frame. Also, the computer storage requirement for the proposed model has been found less than that of the flexibility-based finite element model. The number of integer and real variables required for one element of the proposed modeling approach is approximately 16% of the corresponding number of the fiber modeling technique.