Dynamic & Seismic Analysis in Software

Equation of motion

For a structure discretised into finite elements, the dynamic equilibrium is

$[M]$ = mass matrix, $[C]$ = damping matrix, $[K]$ = stiffness matrix; $\{u\}$, $\{\dot{u}\}$, $\{\ddot{u}\}$ = displacement, velocity and acceleration. In earthquake analysis the force is the ground acceleration acting through the mass:

with $\{r\}$ the influence vector (ones in the direction of the ground motion).

Mass and damping

• Mass comes from the self-weight of elements, superimposed dead load and, for seismic analysis, a fraction of the live load (the code gives it). It can be lumped (a diagonal matrix, translational masses at the nodes) or consistent (built from the same shape functions, better for beams). Units must be consistent: if forces are in kN and lengths in m, mass is in tonnes (kN·s²/m).
• Damping is not derived from geometry; it is assumed as a fraction of critical damping $\xi$ — about 2–5 % for steel and bridges, about 5 % for reinforced concrete buildings at the design level, and 0.5–2 % for cables and long-span bridges at working level. Rayleigh damping builds $[C]$ from mass and stiffness:

with $\alpha$ and $\beta$ set so that the damping is correct at two chosen frequencies.

Modal analysis

Free vibration without damping gives the eigenvalue problem

whose solutions are the natural circular frequencies ${\omega }_n$ (period $T_n=2\pi /{\omega }_n$) and mode shapes $\{{\varphi }_n\}$. Mode shapes are orthogonal with respect to $[M]$ and $[K]$, so the equations decouple: each mode behaves like a single-degree-of-freedom oscillator.

Modal quantities

• Modal participation factor — how strongly a mode is excited by the ground motion in a given direction.
• Effective modal mass — the part of the total mass that participates in a mode; codes require the included modes to capture at least 90 % of the total mass in each direction. If not, add modes or use a residual-mass correction.
• Mode shapes are checked visually: the first modes should be the global modes (sway, torsion, bending), not local vibrations of a single element — local modes signal a mesh or mass modelling problem.