1、 Lateral stiffness estimation in frames and its implementation to continuum models for linear and nonlinear static analysis Tuba Eroglu Sinan Akkar Abstract： Continuum model is a useful tool for approximate analysis of tall structures including moment-resisting frames and shear wall-frame systems. I
2、n continuum model, discrete buildings are simplified such that their overall behavior is described through the contributions of flexural and shear stiffnesses at the story levels. Therefore, accurate determination of these lateral stiffness components constitutes one of the major issues in establish
3、ing reliable continuum models even if the proposed solution is an approximation to actual structural behavior. This study first examines the previous literature on the calculation of lateral stiffness components (i.e. flexural and shear stiffnesses) through comparisons with exact results obtained fr
4、om discrete models. A new methodology for adapting the heightwise variation of lateral stiffness to continuum model is presented based on these comparisons. The proposed methodology is then extended for estimating the nonlinear global capacity of moment resisting frames. The verifications that compa
5、re the nonlinear behavior of real systems with those estimated from the proposed procedure suggest its effective use for the performance assessment of large building stocks that exhibit similar structural features. This conclusion is further justified by comparing nonlinear response history analyses
6、 of single-degree-of-freedom (sdof) systems that are obtained from the global capacity curves of actual systems and their approximations computed by the proposed procedure. Key words: Approximate nonlinear methods Continuum model Global capacity Nonlinear response Frames and dual systems 1 Introduct
7、ion Reliable estimation of structural response is essential in the seismic performance assessment and design because it provides the major input while describing the global capacity of structures under strong ground motions.With the advent of computer technology and sophisticated structural analysis
8、 programs, the analysts are now able to refine their structural models to compute more accurate structural response. However, at the expense of capturing detailed structural behavior, the increased unknowns in modeling parameters, when combined with the uncertainty in ground motions, make the interp
9、retations of analysis results cumbersome and time consuming. Complex structural modeling and response history analysis can also be overwhelming for performance assessment of large building stocks or the preliminary design of new buildings. The continuum model, in this sense, is an accomplished appro
10、ximate tool for estimating the overall dynamic behavior of moment resisting frames (MRFs) and shear wall-frame (dual) systems. Continuum model, as an approximation to complex discrete models, has been used extensively in the literature. Westergaard (1933) used equivalent undamped shear beam concept
11、for modeling tall buildings under earthquake induced shocks through the implementation of shear waves propagating in the continuum media. Later, the continuous shear beam model has been implemented by many researchers (e.g. Iwan 1997； Glkan and Akkar 2002； Akkar et al. 2005； Chopra and Chintanapakde
12、e 2001) to approximate the earthquake induced deformation demands on frame systems. The idea of using equivalent shear beams was extended to the combination of continuous shear and flexural beams by Khan and Sbarounis (1964). Heidebrecht and Stafford Smith (1973) defined a continuum model (hereinaft
13、er HS73) for approximating tall shear wall-frame type structures that is based on the solution of a fourthorder partial differential equation (PDE). Miranda (1999) presented the solution of this PDE under a set of lateral static loading cases to approximate the maximum roof and interstory drift dema
14、nds on first-mode dominant structures. Later, Heidebrecht and Rutenberg (2000) showed a different version of HS73 method to draw the upper and lower bounds of interstory drift demands on frame systems. Miranda and Taghavi (2005) used the HS73 model to acquire the approximate structural behavior up t
15、o 3 modes. As a follow up study, Miranda and Akkar (2006) extended the use of HS73 to compute generalized drift spectrum with higher mode effects. Continuum model is also used for estimating the fundamental periods of high-rise buildings (e.g. Dym and Williams 2007). More recently, Gengshu et al. (2
16、008) studied the second order and buckling effects on buildings through the closed form solutions of continuous systems. While the theoretical applications of continuum model are abundant as briefly addressed above, its practical implementation is rather limited as the determination of equivalent fl
17、exural (EI) and shear (GA) stiffnesses to represent the actual lateral stiffness variation in discrete systems have not been fully addressed in the literature. This flaw has also restricted the efficient use of continuum model beyond elastic limits because the nonlinear behavior of continuum models
18、is dictated by the changes in EI and GA in the post-yielding stage This paper focuses on the realistic determination of lateral stiffness for continuum models. EI and GA defined in discrete systems are adapted to continuum models through an analytical expression that considers the heightwise variati
19、on of boundary conditions in discrete systems. The HS73 model is used as the base continuum model since it is capable of representing the structural response between pure flexure and shear behavior. The proposed analytical expression is evaluated by comparing the deformation patterns of continuum mo
20、del and actual discrete systems under the first-mode compatible loading pattern. The improvements on the determination of EI and GA are combined with a second procedure that is based on limit state analysis to describe the global capacity of structures responding beyond their elastic limits. Illustr
21、ative case studies indicate that the continuum model, when used together with the proposed methodologies, can be a useful tool for linear and nonlinear static analysis. 2 Continuum model characteristics The HS73 model is composed of a flexural and shear beam to define the flexural (EI) and shear (GA
22、) stiffness contributions to the overall lateral stiffness. Themajor model parameters EI and GA are related to each other through the coefficient (Eq.1). As goes to infinity the model would exhibit pure shear deformation whereas = 0 indicates pure flexural deformation. Note that it is essential to i
23、dentify the structural members of discrete buildings for their flexural and shear beam contributions because the overall behavior of continuum model is governed by the changes in EI and GA. Equation 2 shows the computation of GA for a single column member in HS73. The variables Ic and h denote the column moment of inertia and story height, respectively. The inertia terms Ib1 and Ib2 that are divided by the total lengths l1 and l2, respectively, define the relative rigidities of beams adjoining to the column from top (see Fig. 3 in the referred paper).
