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Structural dynamics · modal combination

Combining Peak Seismic Effects

Estimate demand when modal peaks do not occur simultaneously.

Approx. 32 min

Learning objectives

By the end of this lesson

  • Define modal combination in physical engineering terms.
  • Use srss combination with consistent quantities and units.
  • Change model inputs and explain the resulting trend.
  • Separate an educational model result from a code-based safety decision.

Before you calculate

What this means in the real world

01

What the formula is for

Square-root-of-sum-of-squares is appropriate only under its modal-separation assumptions; closely spaced modes need a correlation-aware method.

R = √(ΣRᵣ²)
02

Where an engineer uses it

Seismic analysis of a regular multi-storey frame: This application turns modal combination into a real modelling decision.

Read modal spectral responses, apply participation factors, and combine compatible response quantities.

03

What a wrong value can do

A wrong unit, sign, scale, boundary condition, or property can produce a believable number with the wrong physical meaning. It may understate demand, exaggerate it, or lead to the wrong failure check.

Definition

modal combination

A statistical procedure for estimating a total peak effect from separate modal peak values.

Why it mattersIt provides the central modelling idea for combining peak seismic effects and connects observed motion to a solvable engineering representation.

Build the physical picture

Estimate demand when modal peaks do not occur simultaneously. Start by identifying the system boundary, response coordinates, applied action, and the properties that store or dissipate energy. A useful model is the simplest one that preserves the behaviour needed for the engineering question.

Formula

SRSS combination

R = √(ΣRᵣ²)

Square-root-of-sum-of-squares is appropriate only under its modal-separation assumptions; closely spaced modes need a correlation-aware method.

u
response displacement or displacement vectorm
t
times
ω
angular frequency where usedrad/s

Interactive concept

Explore modal combination

Change the available model inputs, watch the motion or force relationship, and describe the trend before reading the numerical result.

How is the load applied?
How does the structure behave?
LoadTime →
Load⇣
Lower dynamic tendency

A gradual action on a relatively rigid structure is more likely to remain close to static behaviour.

24

Real-world application

Seismic analysis of a regular multi-storey frame

This application turns modal combination into a real modelling decision.

Modelling move
Read modal spectral responses, apply participation factors, and combine compatible response quantities.
Engineering question
Are any important modes close enough to require correlated combination?

Calculate, compare, interpret

Editable formula question

Experiment with period: Combining Peak Seismic Effects

Change mass and stiffness, predict the period, and check your result.

Tₙ = 2π√(m/k)
Substitute your current valuesTₙ = 2π√(1000 ÷ 40000)Answer format: enter the numerical result only in s. Example: 12, not “answer = 12”.
s
Round only at the final step. The accepted tolerance for this exercise is ±0.01.

The displayed minimum, maximum, and textbook values define this learning exercise only. They are not universal design limits.

Editable formula question

Compare a model demand with your entered criterion

Enter a demand and a comparison value chosen for this exercise, then interpret the ratio.

utilization indicator = model demand / entered criterion
Substitute your current valuesratio = 4 ÷ 5Answer format: enter the numerical result only (no unit for this dimensionless result). Example: 12, not “answer = 12”.
—
Round only at the final step. The accepted tolerance for this exercise is ±0.01.

The displayed minimum, maximum, and textbook values define this learning exercise only. They are not universal design limits.