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.

This dissertation presents a family of new beam-column element models which are based on damage-plasticity and are suitable for the damage assessment and the collapse simulation of structures.

First, a new 1d hysteretic damage model based on damage mechanics is developed that relates any two work-conjugate response variables such as force-displacement, moment-rotation or stress-strain. The strength and stiffness deterioration is described by a damage variable with continuous evolution. The formulation uses a criterion based on the hysteretic energy and the maximum absolute deformation value for the damage initiation with a cumulative probability distribution function for the damage evolution.

The damage evolution function is extended to accommodate the sudden strength and stiffness degradation of the force-deformation relation due to brittle fracture. The model shows excellent agreement with the hysteretic response of an extensive set of reinforced concrete, steel, plywood, and masonry specimens. In this context, it is possible to relate the model's damage variable to the Park-Ang damage index so as to benefit from the extensive calibration of the latter against experimental evidence.

The 1d damage model is then extended to the development of beam-column elements based on damage-plasticity. In these models, the non-degrading force-deformation relation in the effective space is described by a linear elastic element in series with two rigid-plastic springs with linear kinematic and isotropic hardening behavior. The first model, the series beam element, assumes that the axial response is linear elastic and uncoupled from the flexural response. The second model, the NMYS column element, uses an axial-flexure interaction surface for the springs to account for the inelastic axial response and capture the effect of a variable axial load on the flexural response. A novel aspect of the beam-column formulation is that the inelastic response is monitored at two locations that are offset from the element ends to account for the spread of inelasticity for hardening response and the size of the damage zones for softening response. The plastic hinge offsets account for the response coupling between the two element ends.

The implementation of the damage-plasticity elements with the return-mapping algorithm ensures excellent convergence characteristics for the state determination. The proposed elements compare favorably in terms of computational efficiency with more sophisticated models with fiber discretization of the cross section while achieving excellent agreement in the response description for homogeneous metallic structural components. The excellent accuracy is also confirmed by the agreement with experimental results from more than 50 steel specimens under monotonic and cyclic loading. The models are able to describe accurately the main characteristics of steel members, including the accumulation of plastic deformations, the cyclic strength hardening in early cycles, the low-cycle fatigue behavior, and the different deterioration rates in primary and follower half cycles. With the plastic axial energy dissipation accounted for in the damage loading function, the damage-plasticity column model captures the effect of a variable axial force on the strength and stiffness deterioration in flexure, the severe deterioration under high axial compression, the nonsymmetric response under a variable axial force, and the very large plastic axial and flexural deformations before column failure. The validation studies point out the dependence of the strength and stiffness deterioration on the section compactness, the element slenderness, the axial force history, and the axial shortening of the columns. A regression analysis is then used to establish guidelines for the damage parameter selection in relation to the geometry and the boundary conditions of the structural member.

The proposed damage-plasticity frame elements are deployed in an analysis framework for the large-scale simulation and collapse assessment of structural systems. The capabilities of the modeling approach are demonstrated with the

case study of an 8-story 3-bay special moment-resisting steel frame that investigates various aspects of the structural collapse behavior, including the global and local response under strength and stiffness deterioration, the magnitude and distribution of the local damage variables, and the different types of collapse mechanism. The study proposes new local and global damage indices, which are better suited for the collapse assessment of structures than existing engineering demand parameters like the maximum story drift. The incremental dynamic analysis of the 8-story moment frame under a suite of earthquake ground motions confirms the benefits of the proposed damage indices for the collapse assessment of structures. The study shows that an aftershock as strong as the main shock increases the collapse margin ratio by as much as 30\% and requires more stringent design criteria for protecting the building from collapse that currently specified.

The study compares different modeling aspects for the archetype building to assess the benefits of the proposed beam-column elements, such as the ability to account for the member damage, the offset location of the plastic hinges, the inelastic axial response, the axial-flexure interaction, and the sudden strength and stiffness deterioration due to brittle fracture of the structural member.

The study concludes that the proposed family of beam-column elements holds great promise for the large scale seismic response simulation of structural systems with strength and stiffness deterioration, because of their computational efficiency and excellent accuracy. Consequently, the proposed models should prove very useful for the damage assessment and the collapse simulation of structures under extreme loading conditions.

This study concerns the formulation and validation of a 3d frame element that accounts for the inelastic response under the interaction of axial, flexural, shear and torsional effects. The proposed finite element addresses the need for a general beam element capable of accurately simulating the global as well as the local response of slender structural elements.

The new element accounts for the warping of arbitrary cross sections due to shear and torsion by introducing warping degrees of freedom at each section in order to accommodate higher order strain kinematics. The warping profile at a section uses the necessary number of Lagrange interpolation functions for the desired accuracy of local response. The warping distribution along the element axis is described by either Lagrange polynomials or spline interpolation functions. The number of interpolation parameters can be adjusted to control the accuracy of the local and global response.

The 3d beam element formulation is derived from a mixed Hu-Washizu variational potential, with the inclusion of the warping displacements as independent variables. In the proposed formulation the section response is coupled through the interpolation of the stress resultants and the warping displacements along the element axis. Because the stress resultants satisfy the element equilibrium exactly with the use of suitable force interpolation functions, the element is free of shear-locking. Non-uniform warping is accommodated by the warping displacement distributions, so that the proposed element can represent the stresses arising from local warping constraints. The element is incorporated in a general purpose finite element analysis program with the consistent linearization of the governing equations for warping force equilibrium and for element compatibility resulting in a robust algorithm for the element state determination.

The element is validated with several examples of linear and nonlinear material response of steel members. The linear elastic response is validated with analytical results, finite element models and some available experimental measurements. The inelastic response under monotonic and cyclic load conditions involves shear link specimens with wide flange and box section under high shear. The accuracy and computational efficiency of the proposed element is demonstrated by comparing the numerical results with available experimental measurements from eccentrically braced steel frames and with local response results from solid and shell finite element models.

The study concludes with the investigation of the axial, flexural and torsion interaction under large displacements. The corotational formulation is derived from a general hypothesis that establishes the accuracy and scope of application of this method. Several numerical examples are used for the illustration of key nonlinear geometry aspects of slender elements under the interaction of axial, flexural and torsional effects.

This dissertation presents the formulation and validation of a new 3d mixed cable element that is based on a two-field variational formulation and is suitable for the nonlinear static and dynamic analysis of cable structures.

The new cable element is derived in general curvilinear coordinates under finite deformations, and identifies conjugate strain and stress measures for the nonlinear catenary problem. The formulation uses the weak form of the strain-displacement relation and the principle of virtual work to propose two numerical implementations of the element, one with a continuous axial force distribution and one with a discontinuous one.

This dissertation also proposes a new filtered energy-momentum conserving algorithm for obtaining the dynamic response of cable structures in time. This new algorithm exactly conserves the Hamiltonian structure of the two-field mixed catenary problem for any {non\-linear} elastic complementary energy, and provides a consistent time integration in the case of inelastic material response. A postprocessing Savitzky-Golay filter is included to address the high-frequency contributions that appear in cables with a large sag-to-span ratio, without jeopardizing the desired conserving properties.

The new element and the consistent time integration scheme are first validated under nonlinear elastic material response with several benchmark problems from the literature. In these examples, the mixed cable element obtains very accurate results for coarse meshes, and displays especially accurate axial force distributions compared to other models. For cables with a small sag-to-span ratio, the energy-conserving scheme and the Newmark method yield nearly identical results, while in the case of cables with a large sag-to-span ratio, for which the Newmark time integrator diverges, the new scheme gives accurate results and exactly conserves the total energy of the system.

The proposed formulations are also validated under viscoelastic material response. In the case of small viscoelastic strains, the new element behaves robustly and gives excellent results. A new finite viscoelastic material model is formulated for large viscoelastic strains, and results show that it reduces to the infinitesimal model when small deformations are considered. Simple benchmark problems involving the free vibration and the earthquake response of simply-supported cables demonstrate that small relaxation times can reproduce the internal physical mechanisms that dissipate the high-frequency waves in the axial force field, while long relaxation times account for the decay of the dynamic response.

The study concludes with the structural analysis of three-dimensional cable nets using the proposed 3d mixed cable elements. First, numerical joints are introduced to accommodate the discontinuities in the axial force field that appear in physical cable joints. These complex structures show excellent results for displacements and axial forces. For available experimental results, the proposed formulations give the smallest relative error compared to other models in the literature.