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\\ *Direct-integration* [time-history|kb:Time-history analysis] analysis is a [nonlinear|kb:Nonlinear], dynamic analysis method in which the equilibrium equations of motion are fully integrated as a structure is subjected to dynamic loading. Analysis involves the integration of structural properties and behaviors at a series of time steps which are small relative to loading duration. The equation of motion under evaluation is given as |
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NOTE: Integration is performed at every time step of the input record, regardless of the output increment. |
See Also
- Mass and stiffness-proportional damping – Damping of stiff elements with inelastic softening
- Output accuracy – Time-history output-acceleration accuracy
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!Figure 1.png|align=center,border=0!
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*NOTE:* Integration is performed at every time step of the input record, regardless of the output increment.
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h1. Stability conditions
For well-behaved problems, we recommend using the Newmark method with {math}beta{math} = {^}1{^}/{~}4{~}, which yields the constant average acceleration method, and for poorly converging nonlinear [time-history|kb:Time-history analysis] cases, the Hilber-Hughes-Taylor ([HHT|kb:Time-history output-acceleration accuracy]) method with 0 < {math}alpha{math} ≤ -{^}1{^}/{~}3{~}. During HHT application, when {math}alpha{math} = 0, formulation is identical to the average acceleration method, so HHT will actually suffice for all problems. Note that all nodes should have [mass|kb:Mass], and rotational inertia may be added to improve dynamic response.
The Newmark method is unconditionally stable only when 2 {math}beta{math} ≥ {math}gamma{math} ≥ {^}1{^}/{~}2{~}, which is why we recommend using {math}beta{math} = {^}1{^}/{~}4{~}. When {math}beta{math} = {^}1{^}/{~}6{~}, which yields the linear acceleration method, formulation is only conditionally stable, and can become unstable when {^}dt{^}/{~}T{~} > {^}2√ 3{^}/{~}(2 {math}pi{math}){~}, 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|kb:Modal analysis] analysis such that time step may be coordinated to ensure convergence.
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*Related Incident:*
* {incident:no= 47043|comment=Questions about numerical error on SAP2000 Program}
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h1. See Also
* *Mass and stiffness-proportional damping* -- [kb:Damping of stiff elements with inelastic softening]
* *Output accuracy* -- [Time-history output-acceleration accuracy|kb:Time-history output-acceleration accuracy]
* *Ground motion at specific supports* -- [Manual multi-support excitation|kb:Manual multi-support excitation] |