Shuenn-Yih Chang

Improved numerical dissipation for explicit methods in pseudodynamic tests

There is no second-order accurate, dissipative, explicit method in the currently available step-by-step integration algorithms. Two new families of second-order accurate, dissipative, explicit methods have been successfully developed for the direct integration of equations of motion in structural dynamics. These two families of methods are numerically equivalent and possess the desired numerical dissipation which can be continuously controlled. These two families of algorithms are very useful for pseudodynamic tests since the favourable numerical damping can be used to suppress the spurious growth of high-frequency modes due to the presence of numerical and/or experimental errors in performing a pseudodynamic test.

Improved time integration for pseudodynamic tests

Converting the second-order differential equation to a first-order equation by integrating it with respect to time once as the governing equation of motion for a structural system can be very promising in the pseudodynamic testing. This was originally found and developed by Chang. The application of this time-integration technique to the Newmark explicit method is implimented and investigated in this paper. The main advantages of using the integral form of Newmark explicit method instead of the commonly used Newmark explicit method in a pseudodynamic test are: a less-error propagation effect, a better capability in capturing the rapid changes of dynamic loading and in eliminating the adverse linearization errors. All these improvements have been verified by theoretical studies and experimental tests. Consequently, for a same time step this time-integration technique may result in less-error propagation and achieve more accurate test results than applying the original form of Newmark explicit method in a pseudodynamic test due to these significant improvements. Thus, the incorporation of this proposed time-integration technique into the direct integration method for pseudodynamic testings is strongly recommended.

THE γ-FUNCTION PSEUDODYNAMIC ALGORITHM

Practical applications of the γ-function dissipative explicit method to pseudodynamic tests are thoroughly investigated herein. Detailed implementation of this pseudodynamic algorithm is schematically sketched. Numerical experiments and verification tests strongly indicate that the γ-function dissipative explicit method can effectively filter out the spurious participation of high frequency responses while the lower mode responses can be obtained very accurately. In addition, error propagation analysis also shows that the γ-function dissipative explicit method possesses much better error propagation characteristics when compared to the Newmark explicit method. Thus, this pseudodynamic algorithm is very suitable for the test system where the responses are dominated by the low frequency modes and the high frequency responses are of no interest.

A series of energy conserving algorithms for structural dynamics

Abstract A series of step‐by‐step integration methods has been effectively developed which does not increase the total number of equations of motion and avoids the use of the derivatives of external force. The well‐known Newmark β method [16] with β = 1/4 is the lowest order of accuracy of this series of methods. All the algorithms of this series are unconditionally stable, without overshoot in displacement or in velocity, and they do not possess any numerical dissipations. The rapid changes of dynamic loading can be automatically overcome. It is also verified that the higher the order of the integration method, the more accurate. Consequently, the higher‐order algorithms of this series allow the use of a large time step in step‐by‐step dynamic analysis. Thus, they are competitive in dynamic analysis, especially when the response of a long duration is of interest.

Studies of Newmark method for solving nonlinear systems: (I) basic analysis

Abstract The performance of the Newmark method in the solution of linear elastic systems can be reliably predicted by the currently available evaluation techniques. However, its performance for solving nonlinear systems is still not clear. Therefore, its numerical characteristics in the solution of nonlinear systems are explored in this study by using a newly developed technique. Only a specific implementation of the Newmark method is studied in this paper although there are several possible implementations for this integration method. It seems numerical properties of the Newmark method in the solution of linear elastic and nonlinear systems can be entirely captured by the newly developed technique. Although this technique is only aimed at a specific time step, it is still indicative for the whole step‐by‐step integration procedure since this procedure consists of each time step.

Two new implicit algorithms of pseudodynamic test methods

Abstract Either measured displacements or calculated displacements are required in the available implicit algorithms. Two new formulations of the implicit algorithms will be presented herein. Neither measured displacements nor calculated displacements are used in the new algorithms. Hence, propagation errors can be significantly reduced in performing pseudodynamic tests. A series of simulations have been performed to study the error propagation characteristics of the algorithms. For the sake of comparisons, the error propagation characteristics of the implicit algorithm originally proposed by Thewalt and Mahin and that suggested by Shing and Manivanna are also considered in this study.

AN UNCONDITIONALLY STABLE EXPLICIT METHOD FOR STRUCTURAL DYNAMICS

An explicit integration method with unconditional stability was proposed and presented in this paper. Numerical characteristics of this explicit method for linear elastic sys-tems are almost the same as those of the constant average acceleration method while for a nonlinear system it is more efficient in computing than for the constant average acceleration method. This explicit method integrates the most promising advantages possessed by the explicit and implicit methods. No limitation on the time step in satisfying the stability limit, which is the most important property for an implicit method, is theoretically proved for this explicit method. Furthermore, the avoidance of solving any implicit system or using any iterative procedure, which usually brings considerable simplification in practical applications for explicit methods, leads to the very low cost of one explicit step. Thus, the computational effort can be significantly reduced when compared to implicit methods in each time step. Consequently, this explicit method can be used to solve dynamic problems efficiently due to the unconditional stability and the very low cost of explicit steps. Rough guidelines with regards to the selection of a time step in achieving accurate solutions for the constant average acceleration method are also appropriate for the proposed explicit method.

Explicit Pseudodynamic Algorithm with Improved Stability Properties

An explicit pseudodynamic algorithm with an improved stability property is proposed herein. This algorithm is shown to be unconditionally stable for any linear elastic systems and any instantaneous stiffness softening systems. The most attracting stability property is that it can have unconditional stability for the instantaneous hardening systems with the instantaneous degree of nonlinearity less than or equal to 2. This property has never been found among the currently available explicit algorithms. Hence, it may be applied to perform a general pseudodynamic test without considering the stability problem since it is rare for a civil engineering structure whose instantaneous degree of nonlinearity is greater than 2. This explicit algorithm can be implemented as a common explicit pseudodynamic algorithm, such as the use of the Newmark explicit method, since it does not involve any iteration procedure. In addition, it possesses comparable accuracy as that of a general second-order accurate integration method such as the Newmark explicit method. Both numerical and error propagation properties are analytically studied and numerical experiments are used to confirm these properties. Actual pseudodynamic tests attested to the feasibility of this proposed explicit pseudodynamic algorithm.

An improved on-line dynamic testing method

This paper applies a new developed integration method to solve the momentum equations of motion in performing an on-line dynamic test. This integration method is unconditionally stable and explicit. Thus, the difficulty arising from the presence of high frequency modes can be easily overcome and the on-line dynamic tests can still have an explicit implementation, which is much simpler than for an implicit implementation. Error propagation analysis for this explicit on-line dynamic testing method is performed and the improved characteristics in error propagation are thoroughly verified. In addition, both numerical examples and verification tests are conducted to verify the superiority of this on-line dynamic testing method.

Analytical study of the superiority of the momentum equations of motion for impulsive loads

Abstract In the step-by-step computing the response to an impulsive loading, the application of the Newmark explicit method to solve the momentum equations of motion is much better than to solve the force equations of motion. The main cause to this effect is theoretically verified and confirmed by a series of numerical illustrations. The original form of the Newmark explicit method is generally require to select an appropriate step size largely dependent upon the shape of an impulse, especially the ending value of the impulse. On the other hand, it is almost shape independent for the use of the integral form of the Newmark explicit method. In fact, a much larger time step can be employed by the integral form of the Newmark explicit method to have the same accuracy of solutions when compared to the original form. Numerical experiments show that the integral form of the other members of the Newmark method also has this improved property.