Seismic Input Theory
Effective seismic input
We consider two possible states of the soil domain during an earthquake: one with the structure, and one without.
Consider first the structure in an earthquake, for which the equations of motion of the free body are:
where is the total motion of the system, and is the reaction force on the structure from the soil.
For the associated soil domain, the free-body equations of motion are:
where and is the earthquake force.
The same soil domain in the absence of the structure will be governed by the following equation:
where is the free-field ground motion.
The scattered motion in the soil domain is obtained by taking the difference between the two, on all nodes other than those on the interface with the structure:
When put together with the equation of motion for the structure, the equations for the whole system become:
wherein the right-hand side give the effective earthquake forces that are equivalent to —- these depend only on the free-field ground motion at the interface, and because of the sparsity of the mass and stiffness matrices, are confined to one layer of elements around the soil-structure interface.
In other words, using the scattered motion in the soil domain creates a discontinuity at the interface with the structure, where the total motion is used, and this discontinuity creates effective forces at the interface. The discontinuity is exactly the free-field ground motion at the interface, and thus effective forces depend solely on that free-field ground motion.
This is the effective seismic input method developed by Bielak and co-workers - it directly uses the free-field earthquake ground motions at the soil-structure and does not require their deconvolution down to depth.
Non-linear analysis of the structure
Since the goal of transient soil-structure interaction analysis is to predict the non-linear behaviour of the structure, the transient analysis needs to start from a static state of the structure. Furthermore, the soil itself may behave non-linearly, and this needs to be accounted for in the analysis. However, the soil domain itself is
- linear by assumption, in order to allow calculating the scattered motion by subtraction, and
- incapable of carrying any static load, because (i) the static state is eliminated in calculating the scattered motion, and (ii) the PML is meant to absorb only wave motion and cannot support static loads.
This conflict may be resolved as follows:
1. Assume that all the non-linearity in the soil is limited to a region near the structure, and define the generalized structure to be the physical structure itself along with this non-linear part of the soil. The rest of the soil domain is then linear and can be taken to be the soil domain for the purpose of the interaction analysis.
2. For the analysis, first calculate the static reactions at the base of the generalized structure by a static analysis, and apply those reactions at the base during the transient analysis to support the weight of the structure and non-linear soil.