in which

After solving {Ax8} from Equation 3.100 as

{Ax8} = [^]-1{AF} (3.103) {Ax} is determined by the following formula:

The incremental velocity vector, {Ax:}, and displacement vector, {Ax}, are obtained from

At2 At 2

Total displacement, velocity, and acceleration vectors are then determined from

3.6.4 Stability Condition and Selection of Time Interval Stability

Stability of a numerical integration method requires that any error in displacement, velocity, and acceleration at time t does not grow for different incremental time intervals used in the integration. Therefore, the response of an undamped system subjected to an initial condition should be a harmonic motion with constant amplitude that must not be amplified when different Ats are employed in the analysis. Stability of an integration method can thus be determined by examining the behavior of the numerical solution for arbitrary initial conditions based on the following recursive relationship of a single d.o.f. motion:

[33(t + At), x (t + At), x(t + At )]T = [A][33(t), x (t), x(t)]T (3.107b)

where [A] is an integration approximation matrix. If we start at time t, and take n time steps, Equation 3.107 may be expressed as

To investigate the stability of an integration method, we use the decomposed form of matrix [A] in Equation 3.108 as

where [ln] is a diagonal matrix with eigenvalues in, I2, l" in the diagonal position; and is a modal matrix with eigenvectors F1, F2, and F3. Now, define the spectral radius of matrix [A] as r(A)=max|li |; i = 1,2,3 (3.110)

in order to keep [A]n in Equation 3.108 from growing without bound. The condition of Equation 3.111 is known as the stability criterion for a given method.

Numerical results for the Newmark method (linear acceleration and constant acceleration) and the Wilson-0 method from At/T = 0.001 to At/T = 100 are plotted in Figure 3.30. It can be seen that the spectral radius for linear acceleration is stable (r(A) < 1) at approximately At/T< 0.55 and becomes unstable (r(A) > 1) at At/T > 0.55. The stability of this method depends on the magnitude of At, and is called the conditional stability method. However, the spectral radii for the constant acceleration method in the range of At/T = 0.001 to 100 are all less than or equal to 1 (r(A) < 1); this case is called the unconditional stability because it does not depend on the magnitude of At. The Wilson-0 method with 0 = 1.4 is unconditionally stable and it becomes conditionally stable with 0 = 1.36. For unconditional stability, the solution is not divergent even if time increment At is large. Selection of At

Numerical error, sometimes referred to as computational error, is due to the incremental time-step expressed in terms of At/T. Such errors result not from the stability behavior from two other sources: (1) externally applied force or excitation and (2) number of d.o.f. assigned to a vibrating system. A forcing function, particularly an irregular one such as earthquake ground motion, is composed of a number of forcing periods (or frequencies). A larger At may exclude a significant part of a forcing function. That part is associated with smaller periods, a forcing function's higher modes. This error may occur for both single- and multiple-d.o.f. vibrating systems. Therefore, At must be selected small enough to ensure solution accuracy by including the first several significant vibrating modes in the analysis.

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