Module 6 · Lesson 6.2

Pulses, and why duration is everything

Step, ramp, rectangular, triangular and half-sine — and the single ratio that decides which of them you need to care about.

Why this matters

Module 1 showed that a suddenly applied load doubles the static response. That was one point on a curve. This lesson draws the whole curve, and the curve has a shape worth knowing by heart: it rises from 1, peaks near 1.8 to 2 somewhere around t_d = T_n/2, and falls away on both sides for quite different reasons.

Knowing where a given loading sits on that curve tells you immediately whether to do a static calculation, an impulse calculation, or a real dynamic analysis.

By the end of this lesson you should be able to

Compute the response to the standard idealised pulses
Explain the two branches of a shock spectrum and what divides them
Say why the peak response can occur after the load has gone
Show that pulse SHAPE stops mattering once the pulse is short enough

One ratio governs

For any pulse, define

t_d / T_n = (how long the load lasts) / (how long the structure takes to respond)

Three regions follow, and each has a different physical character.

t_d/T_n above about 4 — quasi-static. The load rises and falls slowly compared with the structure, which follows it. The peak response is essentially the peak load divided by the stiffness. Inertia never becomes significant.

t_d/T_n around 0.5 to 1 — genuinely dynamic. The load and the structure are on comparable timescales. The structure is still accelerating when the load changes, and the dynamic load factor is at its largest — typically 1.5 to 2 depending on shape.

t_d/T_n below about 0.5 — impulsive. The load has come and gone before the structure has moved appreciably. The peak occurs afterwards, in free vibration, and it depends only on the AREA under the force–time curve.

The shape stops mattering

That last point deserves emphasis because it is genuinely surprising. In the impulsive region, a rectangular pulse and a triangular pulse of the same area give the same answer. So does a half-sine, and so does an irregular measured trace.

The reason is the derivation of the last lesson: when the pulse is short, only ∫F dt survives the integration of the equation of motion. Every other detail of the force history integrates to something negligible.

This is why blast and impact loading are characterised by their impulse, and why measuring the exact shape of a very short force history is often not worth the effort.

The shock spectrum

Plotting the dynamic load factor against t_d/T_n for a given pulse shape gives a shock spectrum — a curve that answers, for every possible structure, what that pulse does to it.

For a rectangular pulse on an undamped system the answer has an exact closed form worth knowing:

t_d/T_n	Dynamic load factor
≥ 0.5	2 exactly
< 0.5	2 sin(π t_d/T_n)

The two branches meet at t_d/T_n = 0.5, where the sine reaches 1. Above that the pulse lasts long enough for the structure to complete its first half-cycle while the load is still on, so the answer is the step-load result of 2. Below it, the load ends first and the structure carries less.

Shock spectrum, rectangular pulse, undamped

DLF = ⎩ ⎨ ⎧ 2 sin (π \frac{t _{d}}{T _{n}}) 2 t_{d} / T_{n} < 0.5 t_{d} / T_{n} \geq 0.5

What it calculates: Peak response as a multiple of the static deflection under the peak force

t_d: Pulse duration (s)
T_n: Natural period of the structure (s)
DLF: Dynamic load factor, x_max/(F₀/k) (—)

This assumes

Rectangular pulse of constant magnitude F₀
No damping — this is the upper bound
Linear elastic single-degree-of-freedom response

In plain terms: The factor never exceeds 2, whatever the pulse duration. A rectangular pulse cannot do worse than a step, because a step is what it becomes when it lasts long enough. Short pulses do progressively less, and the transition is at exactly half a natural period.

Try it

Build a force history and see what it does

Choose a shape and change how long it lasts, measured against the structure's own natural period.

Pulse shape

Pulse duration t_d / T_n

The single most important parameter on this panel.

Natural period T_ns

Damping ratio ζ

Peak force F₀kN

Force against Time. Applied force reaches a peak magnitude of 60 kN.

Response
Static deflection F₀/k

Displacement against Time. Response reaches a peak magnitude of 57.4 mm. Static deflection F₀/k reaches a peak magnitude of 36.5 mm.

Show the numbers behind this plot

t = 0.00 s

Animation time — drag to examine any instant

Static displacement: 36.5 mm
Peak displacement: 57.4 mm
Dynamic load factor: 1.57
Peak occurs at: 0.24 s
Total impulse ∫F dt: 10.9 kN·s
Impulsive estimate I/(mω): 69.1 mm
Rectangular shock spectrum: 1.62

Around t_d ≈ T_n/2 the pulse and the structure are in step, and this is where the dynamic load factor is largest. It is 1.57 here.

What this shows: What decides the response to a pulse is not its magnitude or even its shape, but its DURATION measured against the natural period. Below about half a period only the area under the curve matters.

Worked example

A blast-resistant panel, checked three ways

Given

A façade panel spanning 3.5 m, idealised as SDOF
Natural frequency 14 Hz
A pressure pulse, idealised as triangular, peak 45 kPa, duration 18 ms
Panel area 3.5 × 1.2 m; damping neglected, which is conservative

Find

The peak response, treated as static, as impulsive, and properly.

Assumptions

The panel stays elastic — a real blast design would not assume this, and the point here is the comparison of methods
Damping neglected: over a fraction of one cycle it removes almost nothing

Predict first

A rectangular pulse and a triangular pulse have the same peak force and the same duration, which is very short compared with the natural period. Which produces the larger response?

Practice

A rectangular pulse lasts 0.06 s on a structure with a natural period of 0.4 s. Using the undamped shock spectrum, what is the dynamic load factor?

Practice

For the same structure, what pulse duration in seconds would give the maximum dynamic load factor of 2?

Practice

A triangular pulse has a peak of 80 kN and lasts 0.025 s. What is its impulse, in kN·s?

Check yourself

For a pulse with t_d/T_n = 0.2, when does the peak response occur?

Worked example

The same impulse, delivered three ways

Given

A structure with a natural period of 0.5 s
A rectangular pulse of the same total impulse, delivered over 0.05 s, 0.25 s and 2.0 s

Find

How the peak response differs

Summary

t_d/T_n is the single ratio that decides the treatment
Above about 4: quasi-static. Below about 0.5: impulsive. Between: genuinely dynamic
Rectangular shock spectrum: DLF = 2 sin(π t_d/T_n) below 0.5, and 2 above it
In the impulsive region only the AREA under the force curve matters
The impulsive DLF can be written neatly as Iω_n/F₀
For a short pulse the peak occurs AFTER the load has gone
'DLF = 2 to be safe' is neither safe nor conservative outside the step-load case

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This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint