"A Beam Element for Seismic Damage Analysis"
by E. Spacone, V. Ciampi and F.C. Filippou

Abstract: This study proposes a beam finite element model with distributed inelasticity and two nonlinear end rotational springs for the nonlinear dynamic analysis of frame structures under earthquake excitations. The beam element is based on the assumption that deformations are small and shear deformations are neglected. The axial behavior is assumed linear elastic and is uncoupled from the flexural behavior. The element is derived with the mixed method of finite element theory. The force distribution within the element is based on interpolation functions that satisfy equilibrium. The relation between element forces and corresponding deformations is derived from the weighted integral of the constitutive force-deformation relation. While the element can also be derived with the virtual force principle, the mixed method approach has the advantage of pointing the way to the consistent numerical implementation of the element state determination.

The constitutive force-deformation relation of the control sections of the beam and of the end rotational springs has the form of a differential relation that is derived by extending the simple standard solid model according to the endochronic theory. This constitutive model can describe a wide range of hysteretic behaviors, such as strain hardening, "pinching" and degradation of mechanical properties due to cycles of deformation reversals. The deterioration of the mechanical properties of structural elements due to incurred damage is an evolutionary process that can be readily accounted for in the proposed incremental force-deformation relation. Damage is defined as the weighted sum of the dissipated plastic work and the maximum previous deformation excursions. Several examples highlight the effect of the various parameters of the proposed constitutive law.

A special nonlinear algorithm for the state determination is proposed that yields the stiffness matrix and the resisting forces of the flexibility based beam element. The proposed algorithm is general and can be used with any nonlinear section force-deformation relation. The procedure involves an element iteration scheme that converges to a state that satisfies the constitutive relations within the specified tolerance. During the element iterations element equilibrium and compatibility are always satisfied in a strict sense. The proposed method is computationally stable and robust and can trace the complex hysteretic behavior of structural members, such as strain hardening, "pinching" and softening under cyclic nodal and element loads.

The study concludes with a demonstration of the ability of the proposed model to trace the softening response of a cantilever beam without numerical difficulties and with correlation studies of the response of the model with the experimental behavior of two reinforced concrete cantilever beams that highlight the flexibility of the constitutive law in the description of the hysteretic behavior of structural members.

If you are interested in a copy of this report, please contact the EERC Library at eerclib@shake.berkeley.edu or send e-mail to Professor Filippou