Statistical Linearization of Frames With the Bouc-Wen Model and Axial Force-Bending Moment Interaction

Abstract eng:
Considering the probabilistic methods for seismic inelastic structural analysis, Monte Carlo simulation is a reference point where generality and accuracy are of concern. Nevertheless, simulation requires a great amount of computation that makes any extensive analysis of multi-degree-of-freedom systems almost unfeasible. Then the statistical equivalent linearization may be a sound alternative to Monte Carlo simulation. Assuming the Bouc-Wen hysteresis model in particular, may yield the equivalent linear system in analytic, closed form, which is fundamental to any efficient computation. Furthermore, such model is quite versatile because a number of extensions in the literature cover the cyclic deterioration, pinching, biaxial bending, isotropic hardening, and asymmetric strength. A novel extension is proposed in order to introduce a dependency between the resisting axial force and bending moment of a beam element to be used in seismic statistical linearization analysis of framed structures. The lumped plasticity idealization consisting in linearly elastic beam elements and hysteretic rotational springs at their ends is considered; the Bouc-Wen model is assumed as the moment-rotation law of each spring. A second-degree parabola depending on the axial force between pure tensile and compressive strength, approximates the actual interaction diagram with the bending moment. The axial force is expressed depending on the axial stiffness and deformation of the linearly elastic beam element between the springs, which involves the translational degrees of freedom of the frame joints at the beam extremities. The resisting bending moment, i.e. the yield moment of the springs, is controlled by the parameters of the Bouc-Wen model with asymmetric strength as well as by the proposed parabola. The formulation for expected values and covariance matrix of the response quantities of the statistically equivalent linear frame is developed following the usual technique for the non-zero mean stationary problem with Gaussian processes. Non-zero mean values may come from deterministic load, asymmetric strength or both; dispersion is due to randomness of the ground motion idealized as doubly filtered white noise. A preliminary, partial validation of the proposed model is presented, limited to the zero-mean case where the deterministic load is absent and the flexural strength is symmetric. Two portal frames are analyzed with increasing seismic intensity in the horizontal direction. These frames differ in their aspect ratio that causes different variation of the axial force in the columns under a same seismic action. Consequently, the importance of interaction between axial force and bending moment is different as well. A number of response quantities are compared: the joint translations, hysteretic rotations, dissipated energies, and several coefficients of correlation. In the frame where the effect of interaction is expected to be more important, this appears to be stressed in fact. All results from the proposed formulation are reasonable. Furthermore, Monte Carlo simulation is carried out with an elasto-plastic model provided with interaction. In this case the interaction curve is piecewise linear, as implemented in the well-known computer program DRAIN. The results from statistical linearization are consistent with the results from Monte Carlo simulation. All of these encourage further validation of the proposed model aiming at future application to actual structures.

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