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For well-behaved problems, we recommend using the Newmark method with
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{math}beta{math} |
β =
1/
4, which yields the constant average acceleration method, and for poorly converging nonlinear
time-history cases, the Hilber-Hughes-Taylor (
HHT) method with 0 <
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{math}alpha{math} |
α ≤ -
1/
3. During HHT application, when
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{math}alpha{math} |
α = 0, formulation is identical to the average acceleration method, so HHT will actually suffice for all problems. Note that all nodes should have
mass, and rotational inertia may be added to improve dynamic response.
The Newmark method is unconditionally stable only when 2
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{math}beta{math} |
≥ | Wiki Markup |
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{math}gamma{math} |
β ≥ γ ≥
1/
2, which is why we recommend using
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{math}beta{math} |
β =
1/
4. When
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{math}beta{math} |
β =
1/
6, which yields the linear acceleration method, formulation is only conditionally stable, and can become unstable when
dt/
T >
2√ 3/
(2 | Wiki Markup |
|---|
{math}pi{math} |
2π), where dt is the time step and T is the shortest structural period which is excited by loading. The shortest period may be obtained during
modal analysis such that time step may be coordinated to ensure convergence.
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Related Incidents: - Incident 47043: Questions about numerical error on SAP2000 Program
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See Also
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