Structural dynamics

Dynamic loads include people, wind, waves, traffic, earthquakes, and blasts.

This action can be in the form of load due to the weight of things such as people, furniture, wind, snow, etc.

or some other kind of excitation such as an earthquake, shaking of the ground due to a blast nearby, etc.

The distinction is made between the dynamic and the static analysis on the basis of whether the applied action has enough acceleration in comparison to the structure's natural frequency.

A dynamic load is one which changes with time fairly quickly in comparison to the structure's natural frequency.

If it changes slowly, the structure's response may be determined with static analysis, but if it varies quickly (relative to the structure's ability to respond), the response must be determined with a dynamic analysis.

A simple single degree of freedom system (a mass, M, on a spring of stiffness k, for example) has the following equation of motion: where

The static deflection of a single degree of freedom system is: so we can write, by combining the above formulae: This gives the (theoretical) time history of the structure due to a load F(t), where the false assumption is made that there is no damping.

Although this is too simplistic to apply to a real structure, the Heaviside step function is a reasonable model for the application of many real loads, such as the sudden addition of a piece of furniture, or the removal of a prop to a newly cast concrete floor.

However, in reality loads are never applied instantaneously – they build up over a period of time (this may be very short indeed).

As the number of degrees of freedom of a structure increases it very quickly becomes too difficult to calculate the time history manually – real structures are analysed using non-linear finite element analysis software.

For a given mode shape of a multiple degree of freedom system you can find an "equivalent" mass, stiffness and applied force for a single degree of freedom system.

For simple structures the basic mode shapes can be found by inspection, but it is not a conservative method.

, of a structural system with mass M; bending stiffness, EI (Young's modulus, E, multiplied by the second moment of area, I); and applied force, F(x): then, as above: The complete modal response to a given load F(x,t) is

The summation can be carried out by one of three common methods: To superpose the individual modal responses manually, having calculated them by the energy method: Assuming that the rise time tr is known (T = 2π/ω), it is possible to read the DAF from a standard graph.