Abstract Summary
Scalar constant coefficients are generally found in conventional integration methods and it has been shown that there is no conventional integration method that can have explicit formulation and unconditional stability simultaneously. Alternatively, a novel type of solution methods is proposed for solving equation of motion. A concept of mode can give a fundamental basis for the successful development of this type of methods. The profile of the methodology contains three major stages: (1) To decouple a coupled system of n equations of motion (i.e., the problem under analysis) into a set of n uncoupled equations of motion, (2) To develop an eigen-dependent method for solving each uncoupled equation of motion, and (3) To convert all the n eigen-dependent methods into a problem-dependent method by using a reverse procedure of an eigen-decomposition technique. Although the novel type of integration methods is derived from a concept of mode, it does not involve solving any eigenvalue problem during the step-by-step integration procedure. This type of integration methods is characterized by problem dependen-cy since the coefficients can be functions of the initial physical properties (i.e., the mass matrix, damping coefficient matrix and stiffness matrix) for defining the problem under analysis. A fully explicit method is developed and it has an explicit displacement difference equation and an ex-plicit velocity difference equation. Notice that the well-known Newmark family method has an implicit displacement difference equation and an implicit velocity difference equation. The de-velopment details are presented and each major stage is theoretically justified by a concept of mode. Numerical properties of stability and accuracy for the proposed fully explicit method are ana-lytically explored. As a result, it is concluded that it can have unconditional stability and second order accuracy. An unconditional stability allows it select a step size without stability considera-tion and a second order accuracy allows it choose an appropriate step size for conducting the step-by-step integration procedure. Since the proposed fully explicit method has explicit dis-placement and velocity difference equations, both the displacement and velocity of the next step can be directly calculated without involving any nonlinear iterations. This method is simple in logic and structure and thus coding is much easier than implicit methods. Consequently, it can easily handle complex nonlinearities. Numerical tests for solving a coupled system of 250, 500 and 1000 equations of motion have been performed and the CPU demand for the proposed fully explicit method is only about 3.3%, 2.2% and 1.4% of that consumed by the Newmark family method ( beta = 0.25 and gamma = 0.5), respectively. As a consequence, it is affirmed that the proposed fully explicit method is of high computational efficiency in the solution of general structural dynamic problems in contrast to conventional integration methods, such as the Newmark family method.
