Document Type



Doctor of Philosophy


Civil Engineering

First Adviser

Ricles, James M.

Other advisers/committee members

Wilson, John L.; Sause, Richard; Pakzad, Shamim N.; Bursi, Oreste S.


Dynamic response of linear and nonlinear structural systems subjected to any arbitrary excitation is often determined by solving the equations of motion using a direct integration algorithm. Numerous direct integration algorithms have been developed in the past, which are generally classified as either explicit or implicit. Explicit algorithms are generally only conditionally stable, whereas implicit algorithms can provide unconditional stability. Implicit algorithms that are unconditionally stable and have some form of numerical dissipation are preferred for inertial problems where only a small number of low-frequency modes dominate the response. Nevertheless, implicit algorithms require an iterative solution procedure for nonlinear systems and can be computationally intense. Because explicit algorithms are non-iterative, they are preferred for hybrid simulation (HS) in earthquake engineering, an experimental method where the dynamic response of a structural system is simulated from coupled domains of physical and analytical substructures. Explicit algorithms are even more preferred for HS performed at the true time scale, known as real-time hybrid simulation (RTHS). For such simulations involving a large number of degrees of freedom, the need for unconditional stability and numerical dissipation within an explicit formulation is well recognized. Consequently, a new class of ‘model-based’ explicit methods evolved which can achieve unconditional stability through the use of model-based integration parameters. However, limited studies were conducted to assess the accuracy of model-based algorithms under nonlinear structural response. Furthermore, the studies on dissipative model-based algorithms and assessment of their efficacy in eliminating spurious participation of higher modes through actual tests are also limited. This research is focused on developing model-based algorithms for application to numerical simulation and RTHS of inertial problems. Two new families of model-based algorithms, namely, the semi-explicit-α (SE-α) and explicit-α (E-α) methods, are developed where the former uses an explicit displacement and implicit velocity formulation, and the latter uses explicit formulations for both displacement and velocity. These two methods are further analyzed and four single-parameter subfamilies of algorithms having second-order accuracy, unconditional stability, and controllable numerical dissipation with an optimal combination of high-frequency and low-frequency dissipation are developed. In particular, the single-parameter semi-explicit-α-1 (SSE-α-1) and single-parameter semi-explicit-α-2 (SSE-α-2) methods from the SE-α method, and Kolay-Ricles-α (KR-α) and modified-Kolay-Ricles-α (MKR-α) methods from the E-α method are developed. Numerical characteristics of these four methods are studied for free and force vibrations of linear systems and the advantages and limitations of these methods are presented. The results show that the controllable numerical dissipation provided by these method negligibly influences the low-frequency mode response while providing sufficient high-frequency dissipation to eliminate spurious participation of higher modes. The analysis further show that the SSE-α-1 method possesses the best numerical characteristics for linear systems compared with the other three methods. When no numerical dissipation is used, the KR-α method shows some unusual tendency to overshoot for higher modes which is however controlled with numerical dissipation. The MKR-α method, which is designed to address this issue, further improves the overshoot characteristics of the KR-α method. Stability characteristics of the proposed methods applied to nonlinear systems are investigated using the concept of linearized stability and the necessary stability conditions are derived. The results show that a stiffness softening-type response is a necessary (may not be sufficient) condition for unconditional stability to be achieved. The SSE-α-2 method compared with the SSE-α-1 method, and the MKR-α method compared with the KR-α method are found to have enhanced stability characteristics for nonlinear systems. The enhanced stability characteristics of the SSE-α-2 method is achieved at the cost of increased overshoot for higher frequencies. Efficient implementation procedures are presented for linear and nonlinear dynamic analysis using the proposed methods. Representative numerical examples of linear and nonlinear systems are presented to complement the analytical findings on the numerical characteristics of the proposed methods. The results show that the SSE-α-1 method produces large damping forces for inelastic seismic response analysis of frame structures, which lead to an inaccurate solution. The reason behind this is found to be associated with the semi-explicit formulation of the method. The KR-α method, however, produces an accurate solution for this type of problem. Application of the KR-α method for structural collapse simulation is presented. The results indicate that the KR-α method is a computationally efficient and accurate method for such applications. Using the KR-α method, RTHS of a three-story 0.6-scale prototype steel building with nonlinear elastomeric dampers are conducted with a ground motion scaled to the design basis and maximum considered earthquake hazard levels. The RTHS configuration consists of a moment resisting frame, gravity system, and seismic tributary masses modeled as the analytical substructure, and a damped-braced frame modeled as the experimental substructure. Inherent damping in the analytical substructure is defined using a form of nonproportional damping model. Through numerical simulation using an implicit algorithm it is found that the nonproportional damping model produces an accurate result that is comparable with that obtained using mass and tangent stiffness proportional damping. However, the nonproportional damping model when used with explicit integration algorithms can require a small time step to achieve the desired accuracy in an RTHS involving a structure with a large number of degrees of freedom. Restrictions on the minimum time step exist in an RTHS that are associated with the computational demand. Integrating the equations of motion in an RTHS with too large of a time step can result in spurious high-frequency oscillations in the member forces for elements of the structural model that undergo inelastic deformations. The problem is circumvented by introducing the controllable numerical energy dissipation provided by the KR-α method. The results show that controllable numerical energy dissipation can significantly eliminate spurious participation of higher modes and produce exceptional RTHS results. Using the KR-α and MKR-α methods, RTHS of a two-story reinforced concrete (RC) special moment resisting frame (SMRF) with a nonlinear viscous damper in the second story are conducted with a ground motion scaled to the maximum considered earthquake hazard level. The RC SMRF and the seismic masses are modeled analytically and the nonlinear viscous damper is modeled physically in the laboratory. To better model the complex hysteretic behavior of RC members, flexibility-based elements are considered. A new implementation scheme for the state determination of flexibility-based elements is developed based on a fixed-number of iterations for application to RTHS using explicit algorithms. The influence of unbalanced section forces which exist because of the limited number of iterations are studied numerically. The results show that the carrying over of the unbalanced section forces to the next integration time step and applying the necessary corrections can lead to an accurate solution with a small number of element level iterations. Inherent damping in the analytical substructure is modeled using a combination of mass, initial stiffness, and tangent stiffness proportional damping, where tangent stiffness is used for all flexibility-based elements. The equivalent stiffness and damping coefficients of the experimental substructure, which are required to determine the model-based integration parameters, are estimated based on a nonlinear Maxwell damper model and found to be frequency dependent. The parameters of the Maxwell model are identified from a suit of predefined sinusoidal characterization tests conducted at various excitation frequencies. The influence of the frequency dependency of the model-based parameters, which is due to the experimental substructure, on the stability and accuracy of RTHS results are investigated based on the test results. The test data shows that controllable numerical energy dissipation provided by the KR-α and MKR-α methods plays an important role on the stability characteristics of an RTHS. Accuracy of RTHS results with respect to this frequency dependency and numerical energy dissipation are assessed and found to be not sensitive. The influence of fixed-number of element iterations are assessed using RTHS results. The investigation shows that the proposed element implementation is efficient and accurate for application to RTHS.