Structural dynamics · response spectrum

Reading a Response Spectrum

Relate oscillator period and damping to peak earthquake response.

Approx. 32 min

Learning objectives

By the end of this lesson

Define response spectrum in physical engineering terms.
Use pseudo-acceleration 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

What the formula is for

Pseudo-acceleration is derived from spectral displacement and natural frequency for the selected damping ratio.

Sₐ = ωₙ²S_d

Where an engineer uses it

Preliminary seismic demand for a building mode: This application turns response spectrum into a real modelling decision.

Estimate period and damping, then read the matching spectral ordinate without treating it as a time history.

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

response spectrum

A curve of peak response values from a family of oscillators subjected to the same ground-motion record.

Why it mattersIt provides the central modelling idea for reading a response spectrum and connects observed motion to a solvable engineering representation.

Build the physical picture

Relate oscillator period and damping to peak earthquake response. 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

Pseudo-acceleration

Sₐ = ωₙ²S_d

Pseudo-acceleration is derived from spectral displacement and natural frequency for the selected damping ratio.

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

Force-balance lab

Explore response spectrum

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

Displacement, u 20 mmVelocity, u̇ 0.12 m/sAcceleration, ü 0.60 m/s²Educational force limit 3.0 kN

Illustrative linear SDOF: m = 1000 kg, k = 20 kN/m, c = 5 kN·s/m

/\/\/\/

Fᵢ 0.60 kN ←→ Fₛ + Fᴅ

Stiffness forceFₛ = ku = 0.40 kN

Damping forceFᴅ = cu̇ = 0.60 kN

Inertia forceFᵢ = mü = 0.60 kN

Educational demand/limit ratio0.53

Calculated component-force sum is within the limit you entered.

Not a structural safety verdict. The entered limit is only a learning comparison. Real safety requires code-defined load combinations, strength and serviceability checks, geometry, materials, detailing, and professional review.

Real-world application

Preliminary seismic demand for a building mode

This application turns response spectrum into a real modelling decision.

Modelling move: Estimate period and damping, then read the matching spectral ordinate without treating it as a time history.
Engineering question: Which period range controls the selected structural system?

Calculate, compare, interpret

Editable formula question

Experiment with period: Reading a Response Spectrum

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

Tₙ = 2π√(m/k)

Generalized mass m = 1000 kgMin 100 · Textbook example 1000 · Max 10000 kgLateral stiffness k = 40000 N/mMin 5000 · Textbook example 40000 · Max 200000 N/m

Substitute your current valuesTₙ = 2π√(1000 ÷ 40000)Answer format: enter the numerical result only in s. Example: 12, not “answer = 12”.

Your calculated answer

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

Model demand D = 4 unitsMin 0.1 · Textbook example 4 · Max 20 unitsEntered criterion L = 5 unitsMin 0.1 · Textbook example 5 · Max 20 units

Substitute your current valuesratio = 4 ÷ 5Answer format: enter the numerical result only (no unit for this dimensionless result). Example: 12, not “answer = 12”.

Your calculated answer

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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.