Direct-integration time-history analysis is a 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 follows:

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The Newmark method is unconditionally stable only when 2 β ≥ γ ≥ ¹/₂, which is why we recommend using β = ¹/₄. When β = ¹/₆, which yields the linear acceleration method, formulation is only conditionally stable, and can become unstable when ^dt/_T > ^{2√ 3}/_(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.

See Also

• Mass and stiffness-proportional damping – Damping of stiff elements with inelastic softening

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