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Research Papers: Design and Analysis

# An Efficient Computing Explicit Method for Structural Dynamics

[+] Author and Article Information
Shuenn-Yih Chang

Department of Civil Engineering, National Taipei University of Technology, NTUT Box 2653, Taipei, Taiwan 106, Republic of Chinachangsy@ntut.edu.tw

J. Pressure Vessel Technol 131(2), 021211 (Jan 22, 2009) (12 pages) doi:10.1115/1.3066850 History: Received January 02, 2008; Revised April 08, 2008; Published January 22, 2009

## Abstract

An integration algorithm, which integrates the most important advantage of explicit methods of the explicitness of each time step and that of implicit methods of the possibility of unconditional stability, is presented herein. This algorithm is analytically shown to be unconditionally stable for any linear elastic and nonlinear systems except for the instantaneous stiffness hardening systems with the instantaneous degree of nonlinearity larger than $43$ based on a linearized stability analysis. Hence, its stability property is better than the previously published algorithm (Chang, 2007, “Improved Explicit Method for Structural Dynamics  ,” J. Eng. Mech., 133(7), pp. 748–760), which is only conditionally stable for instantaneous stiffness hardening systems although it also possesses unconditional stability for linear elastic and any instantaneous stiffness softening systems. Due to the explicitness of each time step, the possibility of unconditional stability, and comparable accuracy, the proposed algorithm is very promising for a general structural dynamic problem, where only the low frequency responses are of interest since it consumes much less computational efforts when compared with explicit methods, such as the Newmark explicit method, and implicit methods, such as the constant average acceleration method.

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## Figures

Figure 10

Shock response to a descending triangular impulse

Figure 11

Free vibration response with viscous damping of 2%

Figure 12

Displacement responses for a 100DOF spring-mass system

Figure 13

Displacement responses for a 200DOF spring-mass system

Figure 14

Displacement responses for a 500DOF spring-mass system

Figure 1

Comparison of variation in upper stability limit with δi+1

Figure 6

Responses of two-story shear building with instantaneous stiffness softening

Figure 7

Response time histories of two-story shear building with instantaneous stiffness softening

Figure 8

Responses of two-story shear building with instantaneous stiffness hardening

Figure 9

Response time histories of two-story shear building with instantaneous stiffness hardening

Figure 2

Variation in relative period error versus Δt/T0

Figure 3

Numerical solutions of Duffing’s equation

Figure 4

Response time histories of Ωi+1, δi+1, and Pi+1 for a single degree of freedom system

Figure 5

Responses of two-story shear building with linear elastic stiffness

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