Hysteretic beam element with degrading bouc-wen models

In this work a beam element based on the finite element method, suitable for the inelastic dynamic analysis of structures is presented. The hysteretic beam element proposed by Triantafyllou and Koumousis [1] is extended to account for stiffness degradation, strength deterioration and pinching phenomena. The behavior of the element is governed by the BoucWen model of hysteresis while stiffness and strength degradation are based on Baber and Wen model [2] and pinching on Foliente’s model [3]. The case of non-symmetrical yielding, important for concrete members, is also taken into account. The proposed formulation is based on additional hysteretic degrees of freedom which herein are considered as hysteretic curvatures and hysteretic axial deformations of the crosssections. The elements are assembled using the direct stiffness method to determine the mass and viscous damping matrices, as well as the elastic stiffness and the hysteretic matrix of the structure. The entire set of governing equations of the structure is solved simultaneously. This consists of the linear global equations of motion and the nonlinear local constitutive evolutionary equations for every element. The system is converted into a state space form and the numerical solution is obtained implementing a variable-order solver based on numerical differentiation formulas (NDFs). In this way linearization at the global structural level is avoided facilitating considerably the solution. Furthermore, degradation phenomena are easily controlled through the model parameters at the element level and not in a macroscopic way which requires a computationally demanding bookkeeping mechanism. Numerical results are presented that validate the proposed formulation and verify its computational efficiency as compared to the standard elastoplastic finite element method and existing experimental data

Material point method for deteriorating inelastic structures

The material point method (MPM) is one of the latest developments in particle in cell methods (PIC). The structure is discretized into a number of material points that hold all the state variables of the system [1] such as stress, strain, velocity, displacement etc. These properties are then mapped to a temporary background grid and the governing equations are solved. The momentum conservation equations (together with energy and mass conservation considerations) are solved at the grid nodes. The state variables of the particles are then updated by transferring the solutions from the grid nodes back to the material points. Since the background grid is used only to solve the governing equations at the end of each computational step it can be reset to its undistorted form and thus mesh distortion and element entanglement are avoided. In this work an explicit MPM accounting for elastoplastic material behavior with degradations is proposed. The stress tensor is decomposed into an elastic and a hysteretic – plastic part [5] where the hysteretic part of the stresses evolves according to a Bouc-Wen type hysteretic rule [2]. The inelastic constitutive material law provides a smooth transition from the elastic to the inelastic regime and accounts for the different phases during elastic loading, unloading, yielding and stiffness and strength degradation. Heaviside type functions are introduced that act as switches, incorporate the yield criterion and the terms for stiffness and strength degradation as in the Bouc-Wen model of hysteresis [2]. The resulting constitutive law relates stresses and strains with the use of the tangent modulus of elasticity, which now includes the Heaviside functions and gathers all of the governing inelastic degrading behavior