UC Berkeley

The objective of this study is the development of analytical capabilities for the simulation of the inelastic response of structures under the strength and stiffness deterioration they experience when subjected to extreme events. The study addresses the development of such an analytical capability for steel frames. To this end, a family of 2d and 3d frame element models is proposed based on damage-plasticity. The strength and stiffness of these models degrade continuously as a function of one or more damage indices making them suitable for the damage assessment of steel frames up to incipient collapse.

The study extends an existing damage model to cover the damage evolution of the constitutive relation of the frame element under multiple, interacting stresses or stress resultants. The formulation uses several damage indices that evolve continuously with the weighted sum of the plastic energy dissipation of the stress resultants with the work-conjugate deformation variables. The damage evolution function accounts for low-cycle fatigue and the different rate of damage accumulation in primary and follower deformation cycles. The function also accounts for the fact that the behavior in one loading direction may be affected by the damage accumulated in the opposite direction.

The damage model operates as an independent wrapper of the effective force-deformation relation of the element, section or material and returns the true forces or stress resultants and the true tangent stiffness of the force-deformation relation under damage. With this modular formulation it is possible to use the damage wrapper with a material stress-strain relation, with a section force-deformation relation, or with the constitutive relation between the element basic forces and the work-conjugate deformations. Consequently, the study investigates the following three modelling alternatives for steel frame members without damage: a plasticity-based frame element with the basic forces and the work-conjugate deformations in the role of stress resultants and generalized strains, and a frame element that integrates the section force-deformation relation over the element length, with the section model based on plasticity theory for stress-resultants and generalized strains, or on the integration of the material stress-strain relation over the cross-section, a model commonly referred to as fiber section model. With the introduction of the damage wrapper at the element, or at the section, or at the material level, six modeling alternatives for steel frame members under damage result.

Before embarking on the evaluation of the damage plasticity formulations, this study assesses the accuracy of the section model for stress-resultants by comparing its response with the response of the section model that integrates the material stress-strain relation over the cross section. To this end, an existing formulation is extended to accommodate the kinematic and isotropic hardening of the stress-resultants and the numerical implementation is enhanced with the scaling of the state determination variables to minimize the risk for an ill-conditioning of the Jacobian for the return-mapping algorithm of the section state determination.

The same process is repeated for two existing stress-resultant frame elements: a 2d beam-element with linear elastic axial response, and a 3d beam-column element with axial force-biaxial flexure interaction of the basic forces in the role of stress-resultants with linear elastic torsional response. The former is suitable for steel girders experiencing small to negligible axial forces, while the latter is suitable for steel columns under any level of axial force, including variable axial forces due to the overturning effect of steel frames under lateral loads. The existing elements are extended to accommodate the kinematic and isotropic hardening of the stress-resultants and the numerical implementation is again enhanced with the scaling of the state determination variables to minimize the risk for an ill-conditioning of the Jacobian for the return-mapping algorithm of the element state determination. To account for the spread of inelasticity at the ends of steel beams and columns under strain hardening, both elements allow for the plastic hinges to be offset from the element ends. This feature requires the careful determination of the equivalent kinematic and isotropic hardening ratio for the element to match the moment-rotation relation of steel members under symmetric or anti-symmetric flexure. The study derives the necessary analytical expressions for this calibration, which are exact for beams and approximate for columns under axial force-flexure interaction. Correlation studies are conducted to assess the quality of the approximation for typical load-deformation scenarios of a steel member.

After completing the evaluation of the resultant plasticity formulations, the study compares the response of four alternatives for a frame element under damage against available experimental data from the hysteretic uniaxial and biaxial bending response of steel columns under constant and variable axial force. These comparisons lead to recommendations on a consistent set of damage parameter values for typical steel members.

The study concludes with the seismic response analysis of an irregular six-story steel frame under a strong ground acceleration in both principal directions at the base. The inelastic response history evaluates the effect of the damage evolution on the collapse risk of the frame and assesses the effect of nonlinear geometry and ground motion intensity on its global and local response.