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Direct integration algorithms are typically used to solve temporally discretized equations of motion to evaluate the performance of structures under dynamic loading. The stability of these direct integration algorithms are usually investigated for linear elastic structures. However integration algorithms are often applied to structures with nonlinear behavior. This paper presents a procedure based on discrete control theory to investigate the stability of direct integration algorithms applied to multidegree-of-freedom (MDOF) nonlinear structures. The discrete root locus approach is used to investigate properties of the poles of the discrete transfer function matrix representing the nonlinear structural dynamics and to assess the stability of the integration algorithm. The procedure is illustrated using a nonlinear shear building MDOF system to investigate the stability of popular direct integration algorithms, including the Newmark family of integration algorithms, the Hilber-Hughes-Taylor [alpha]-method, and two newly developed explicit integration algorithms. Stability limits are derived for the direct integration algorithms that are found to be conditionally stable. DOI: 10.1061/(ASCE)EM.1943-7889.0000083 CE Database subject headings: Algorithms; Transfer functions; Stability; Structural dynamics; Dynamic loads. Author keywords: Algorithm; Dynamics; MDOF; Transfer function; Stability; Nonlinear.

Direct integration algorithms are often used to solve the temporally discretized equations of motion for structural dynamic problems. Numerous studies have been conducted to investigate the stability of integration algorithms for linear elastic structures. Studies involving the stability analysis of integration algorithms for nonlinear structures are limited. This paper utilizes discrete control theory to investigate the stability of direct integration algorithms for nonlinear structural dynamics. The direct integration algorithms are represented by a closed-loop block diagram, where the nonlinear restoring force of the structure is related to a varying feedback gain. The root locus method is used to analyze the stability of the closed-loop system for various degrees of nonlinear structural behavior. The well-known methods of the Newmark family of integration algorithms and the Hilber-Hughes-Taylor a method, as well as a newly developed integration algorithm, referred to as the CR integration algorithm, are analyzed using the proposed method. It is shown that the stability of an integration algorithm under nonlinear structural behavior is dependent on the poles and zeros of its open-loop discrete transfer function. An unconditionally stable integration algorithm for linear elastic structures is shown not to always remain stable under nonlinear structural behavior. DOI: 10.1061/(ASCE)0733-9399(2008)134:9(703) CE Database subject headings: Algorithms; Dynamics; Transfer functions; Stability; Structural dynamics; Structural behavior.

In structural dynamics, integration algorithms are often used to obtain the solution of temporally discretized equations of motion at selected time steps. Various time integration algorithms have been developed in the time domain using different methods. In order for an integration algorithm to be reliable it must be stable and accurate. A discrete transfer function is used to study the properties of integration algorithms. A pole mapping rule from control theory in conjunction with a discrete transfer function is used to develop new integration algorithms for obtaining solutions to structural dynamics problems. A new explicit integration algorithm, called the CR (Chert and Ricles) algorithm, is subsequently developed based on the proposed method. The properties of the algorithm are investigated and compared with other well established algorithms such as the Newmark family of integration algorithms. By assigning proper stable poles to the discrete transfer function the newly developed CR explicit algorithm is unconditionally stable and has the same accuracy as the Newmark method with constant acceleration. In addition, the CR algorithm is based on expressions for displacement and velocity that are both explicit in form, making it an appealing integration algorithm for solving structural dynamics problems. DOI: 10.1061/(ASCE)0733-9399(2008) 134:8(676) CE Database subject headings: Algorithms; Structural dynamics; Transfer functions; Stability; Discrete elements.

Rate-independent and rate-dependent models are presented for the hysteretic shear stress-strain behavior of elastomeric damping materials. A rate-independent hysteretic model, called the general asymptote and power function (GAPF) model, is presented that simulates different types of hysteretic behavior depending on the selected asymptote function. A rate-dependent hysteretic model, formed from a parallel combination of the GAPF model and a dashpot, is also presented which simulates loading frequency dependent behavior in addition to strain amplitude dependent behavior. Closed-form expressions for the shear stress as a function of shear strain are provided for each model. The models are calibrated for three different damping materials, and good correlation between experimental and analytical hysteretic behavior is observed. The models are investigated under variable cyclic loading. To prevent unrealistic stress values (overshooting) after a small strain reversal followed by reloading, a sequential asymptote model is introduced, based on the GAPF model. The hysteretic models were incorporated into a finite-element program within an elastomeric damper element, and the results of nonlinear time history analyses of a building structure with elastomeric dampers under simulated earthquake loading are presented to illustrate behavior of the hysteretic models under several loading histories. DOI: 10.1061/(ASCE)0733-9399(2007)133:11(1162) CE Database subject headings: Hysteresis; Energy; Damping; Elastomer; Loading history.

Direct integration algorithms are typically used to solve temporally discretized equations of motion to evaluate the performance of structures under dynamic loading. The stability of these direct integration algorithms are usually investigated for linear elastic structures. However integration algorithms are often applied to structures with nonlinear behavior. This paper presents a procedure based on discrete control theory to investigate the stability of direct integration algorithms applied to multidegree-of-freedom (MDOF) nonlinear structures. The discrete root locus approach is used to investigate properties of the poles of the discrete transfer function matrix representing the nonlinear structural dynamics and to assess the stability of the integration algorithm. The procedure is illustrated using a nonlinear shear building MDOF system to investigate the stability of popular direct integration algorithms, including the Newmark family of integration algorithms, the Hilber-Hughes-Taylor α-method, and two newly developed explicit integration algorithms. Stability limits are derived for the direct integration algorithms that are found to be conditionally stable. [ABSTRACT FROM AUTHOR]