Structural Dynamics / What structural dynamics is

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.

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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.

Comparing the responses

For a linear SDOF system, a force applied sufficiently slowly gives a response close to the static displacement F₀/k. The remaining dynamic contribution depends on the actual application history.

An instantaneous constant step force applied from rest gives a peak displacement of 2F₀/k in the undamped model. The mass passes the static equilibrium position with velocity and continues beyond it. Only quantities proportional to that same coordinate inherit the displacement factor; it is not a universal multiplier for all responses of a multi-mode structure.

Under sustained harmonic force at the undamped natural frequency, the steady-state displacement magnification is 1/(2ζ) for linear viscous damping. This gives 10 at 5% damping and 50 at 1%. The dimensional displacement also depends on force and stiffness, and a finite loading duration may not allow the steady state to develop.

For a pulse much shorter than the relevant natural period, an impulse approximation uses the area under the force-time curve. Pulse shape still affects the exact finite-duration response. At fixed peak force, a shorter pulse has a smaller impulse.

Ground motion is imposed at the supports, which transmit forces to the structure. In coordinates relative to a uniformly moving base, it appears as an effective inertial forcing term. Module 13 develops that formulation.

Predict first

A crane places a 200 kN load onto a beam. It can either lower it gently over ten seconds, or transfer its weight instantaneously while it is already in contact with the beam, with no drop height or impact velocity. 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

An idealised step load on a 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 displacement and elastic restoring-force equivalent for an instantaneous, zero-velocity load transfer in an equivalent SDOF model.

Assumptions

The floor stays elastic
The crate lands and stays — it does not bounce
There is no drop height or initial velocity; the prescribed force is an idealised step. Use compatible effective mass and stiffness for the SDOF model

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. Treat the beam as a linear SDOF model and apply the same force as an instantaneous step from rest, with no drop height or impact velocity. 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 needs dynamic treatment when inertia materially affects the response being checked
'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