By Ray Delaforce
Software Engineer, Intergraph CADWorx & Analysis Solutions
In last month’s newsletter, we discussed the basis for seismic design and how an earthquake does not deliver constant force (acceleration). An actual earthquake acts in this manner:
This complicates things. Everything is moving in a chaotic fashion. We have a further complication because the tower has its own natural frequency of vibration. If we consider the basic equation of natural frequency and put our equation f = m x a in a more formal setting, it looks like this:
Notice the 0 on the right, which indicates there is no force providing energy to the tower.
This would give us a tower that vibrates like this:
According to the above equation, the tower would vibrate forever. As in the case of a child’s swing, the vibrations die away in time. This effect, known as damping, is one we must consider. Damping is a sort of resistance to the vibration which causes the amplitude (furthest distance from the neutral position) to diminish over time. We get this sort of damped vibration:
We have to modify the previous equation to account for damping. Notice that the damping factor (k) is proportional to the instantaneous velocity of the tower movement. That damping is very important when we come to consider seismic analysis.
Notice that zero to the right of the equation. That tells us that the vibrations will die away in time. This is the point; the ground motion tells us that energy is being fed into the system.
Remember, from our first statement, all the terms of the above equation are force terms. So, if energy is being fed into the system, it too must be a force term. This force term must be a function of time, because the seismic event occurs over time. So we again modify the equation:
The initial condition at time zero is represented by ‘c’, and -f(t) represents the energy or forcing input from the ground motion. Let us now put all these elements together and step back to see what is going on:
In this graph, the red curve represents the natural frequency of the tower, the blue curve represents the energy input from the ground motion, and the thick black curve represents the response or the final mode of vibration of the tower. It is the response of the tower that induces the stresses.
In the next article, we shall consider how the response can be derived using a simplified method.