Module 1 · Lesson 1.1

What makes a load dynamic

Not how large it is, and not how quickly it changes on its own — but how quickly it changes compared with the structure it is applied to.

Why this matters

A footbridge that carries its crowd loading with a comfortable margin can be closed on its opening day because people cannot walk on it. A floor designed for twice the imposed load it will ever see can be unusable because a photocopier makes the desks shake. In both cases every static check passed. Something else was being asked of the structure, and the static calculation had no way to see it.

That something else is the structure's own mass. The moment a load changes quickly enough that the structure has to be ACCELERATED to follow it, the mass starts generating forces of its own — and those forces are not in any static calculation.

By the end of this lesson you should be able to

Define a dynamic load in terms of the structure, not the load
Explain why the natural period is the reference against which 'quickly' is measured
State D'Alembert's principle and what it lets us do
Give the order of magnitude of natural periods for common structures

What you should already know

Force and moment equilibrium
Newton's second law, F = ma
The stiffness of a beam or column — force per unit deflection

The question is not about the load

Ask an engineer what makes a load dynamic and the usual answer is that it varies with time. That is not enough. The self-weight of a building varies with time — over the years, as fit-out changes — and nobody would call it dynamic. A tide varies with time, and a quay wall is designed for it statically.

The useful definition is this:

A load is dynamic if it changes quickly enough that the structure's own inertia produces forces comparable with the applied load.

Notice what that definition contains. It contains the load, because how fast the load changes matters. And it contains the STRUCTURE, because whether that speed counts as fast depends on how quickly the structure can respond — which is set by its mass and its stiffness together.

The same load can therefore be static for one structure and dynamic for another. A door slamming is a static event for the wall it is set into and a dynamic one for the glass panel beside it.

The reference: the natural period

Every structure has a natural period, T_n — the time it takes to complete one cycle of free vibration if you displace it and let go. It is a property of the structure alone, not of any load.

That period is the yardstick. A load that takes many natural periods to be applied is static; one applied in a fraction of a natural period is emphatically dynamic. Everything in between is the interesting region, and Module 6 makes the boundary precise.

Some orders of magnitude worth carrying:

Structure	Typical natural period
A concrete floor slab, one bay	0.05 – 0.15 s
A long-span composite floor	0.15 – 0.30 s
A single-storey portal frame	0.3 – 0.8 s
A five-storey building	0.4 – 0.7 s
A twenty-storey building	1.5 – 2.5 s
A hundred-storey tower	6 – 10 s
A long-span suspension bridge	5 – 20 s

A useful first estimate for a building is T_n ≈ 0.1N seconds for N storeys. It is crude, it is not a design value, and it is the right order — which is exactly what you want when the software has just told you the first period is 47 seconds.

Note

Inertia is a force, and this is the trick that lets us keep using equilibrium

Newton's second law says F = ma. Every dynamics course you will ever meet rearranges it as

F − ma = 0

and then says something that sounds like sleight of hand: treat the quantity −ma as a FORCE, called the inertia force, and the problem becomes one of equilibrium again.

This is D'Alembert's principle, and it is not a trick — it is a change of viewpoint with a real physical content. Stand on a bus that brakes hard and you feel a force pushing you forward. There is no such force in the ground frame; you are simply continuing at the speed you had. But in YOUR frame, which is decelerating, the force is entirely real and you have to brace against it.

D'Alembert's principle says: adopt the structure's own accelerating frame, add the inertia force −ma, and everything you know about equilibrium works again. Free-body diagrams, force resolution, moment equilibrium — all of it, unchanged.

That is why the equation of motion is usually written as a statement of equilibrium rather than as F = ma:

inertia force + damping force + stiffness force = applied force

Every term is a force. They balance at every instant. Module 3 develops this properly.

Try it

The same peak load, five different ways

Every case below applies the same peak force to the same structure. Only the way it arrives in time is different.

Natural period T_ns

Damping ratio ζ

Peak force F₀kN

Show which cases

Slowly applied
Suddenly applied
Harmonic at resonance

Applied force against Time. Slowly applied reaches a peak magnitude of 50 kN. Suddenly applied reaches a peak magnitude of 50 kN. Harmonic at resonance reaches a peak magnitude of 50 kN.

Slowly applied
Suddenly applied
Harmonic at resonance

Displacement against Time. Slowly applied reaches a peak magnitude of 22.7 mm. Suddenly applied reaches a peak magnitude of 42.3 mm. Harmonic at resonance reaches a peak magnitude of 218 mm.

Show the numbers behind this plot

Loading	Peak x	x / x_static
Slowly applied	22.7 mm	1.00
Suddenly applied	42.3 mm	1.85
Harmonic at resonance	218 mm	9.56
Short impulse	6.06 mm	0.27
Ground acceleration	218 mm	9.56

Static displacement is 22.8 mm. The worst case here — harmonic at resonance — reaches 9.6 times that. A static calculation would have reported the same number for all five.

What to look for

The slowly applied load gives exactly the static answer. That is what 'static' means: applied slowly enough that inertia never enters.
The suddenly applied load reaches almost exactly twice the static displacement, and nothing about that requires resonance or unusual loading.
The harmonic load at resonance builds up over many cycles. Reduce the damping and watch how much further it goes and how much longer it takes.
The short impulse produces a large velocity and a modest displacement. Shorten it further and the peak force stops mattering — only the area under the force curve does.
Ground acceleration produces a response even though no force is applied to the structure at all.

What this shows: A load is not described by its magnitude alone. How quickly it arrives, relative to the structure's natural period, changes the response by more than a factor of ten.

What the comparison shows

Five loadings, one peak force, one structure.

Applied slowly. The response is exactly the static deflection F₀/k. Nothing else happens, because at no point is the structure being accelerated appreciably. This is what 'static' means, and it is a limiting case rather than a separate theory.

Applied suddenly. The peak is twice the static deflection. Not approximately twice, and not because of any resonance — exactly twice, for an undamped system, for any suddenly applied constant load. The structure overshoots because it arrives at the static position with velocity, and the velocity carries it on.

Applied harmonically at the natural frequency. The response builds over many cycles to something limited only by damping. At 5% damping it reaches ten times the static value; at 1% damping, fifty times. This is resonance, and its size is a property of the damping and nothing else.

Applied as a short pulse. The peak is small, and — this is the part that surprises people — it does not depend on the shape of the pulse at all. Only on the area under the force–time curve. A very short blow is described by its impulse, not its magnitude.

Applied as ground acceleration. There is no force on the structure at all. The ground moves, the structure is dragged along by its supports, and the response comes entirely from the structure's own inertia resisting that movement. Module 13 develops this.

Predict first

A crane places a 200 kN load onto a beam. It can either lower it gently over ten seconds, or release it suddenly from just above the beam. The beam's natural period is 0.4 s. How does the peak deflection compare?

From first principles

Why a suddenly applied load gives exactly twice the static deflection

We want to show: Show that a constant load applied instantaneously to an undamped structure produces a peak deflection of exactly 2F₀/k — and understand where the factor of two comes from physically.

Think about where the energy goes. The load F₀ acts through whatever distance the structure moves. The spring stores energy as it deflects. At the very bottom of the movement the structure is momentarily still, so all the work done by the load has become stored strain energy — and because the load is constant while the spring force builds up from zero, the load has done twice as much work as the spring has absorbed at the static position. The structure must therefore go past the static position, and by exactly the same amount again.

Worked example

Does a dropped load matter on this floor?

Given

A composite floor bay, natural frequency 4.2 Hz
Static deflection under the load in question: 3.1 mm
The load is a 12 kN packing crate
Damping about 3%, typical for a composite floor with light fit-out

Find

The peak deflection and the peak load effect if the crate is dropped rather than placed.

Assumptions

The floor stays elastic
The crate lands and stays — it does not bounce
The drop height is small, so the crate arrives with negligible velocity and this is a sudden application rather than an impact

Practice

A structure has a natural frequency of 2.5 Hz. What is its natural period, in seconds?

Practice

A beam deflects 8 mm under a slowly applied load. The same load is dropped onto it instead. Ignoring damping, what is the peak deflection in mm?

Practice

Estimate the natural period, in seconds, of a 14-storey building using the rule T_n ≈ 0.1N.

Check yourself

Which statement best defines a dynamic load?

Summary

A load is dynamic when it changes quickly enough that the structure's inertia matters
'Quickly' is measured against the natural period, which is a property of the structure alone
D'Alembert's principle lets us treat −ma as a force and keep using equilibrium
A suddenly applied load gives exactly twice the static response, undamped
That factor of 2 needs no resonance and no unusual loading
T_n ≈ 0.1N seconds for an N-storey building is a sanity check, not a design value

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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