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Basic superposition

Today there exist three approaches to SSI problems, namely “direct”, “impedance”, and “combined” ones. To understand them all, let us start from common general approach based on the superposition of the wave fields.
Let us call the problem with seismic wave, soil and structure “problem A” and start with completely linear soil-structure model. Let Q be some surface surrounding the basement in the soil and dividing the soil-structure model into two parts: the “external” part Vext and internal part Vint. Let (-F) be additional external loads distributed over Vint and specially tuned so, to provide zero displacements in Vint. Then “problem A” can be split in the sum of two wave pictures: “problem A1”, including seismic excitation and loads (-F), and “problem A2” including only loads (F) without seismic wave – see Fig.1.
This simple superposition leads to a number of important conclusions.
1. As in “problem A1” all displacements in the internal volume Vint are zero, the motion of Vint in “problem A2” is the same as in “problem A”. Hence, if we are interested in the motion of Vint only, we can substitute “problem A” with “problem A2”.
2. As in “problem A1” all displacements in the internal volume Vint are zero, all the strains and internal forces in the internal volume Vint are zero, and the external loads (-F) must be zero everywhere in Vint, except surface Q.
3. As in “problem A1” all displacements, strains and internal forces in the internal volume Vint are zero, no forces are impacting Q from Vint (i.e., forces impacting Q from Vext are balanced by loads (-F)). Hence, Vint can be withdrawn or replaced by another medium (with zero displacements) without changing Vext, seismic excitation, or loads (-F). In particular, Vint can be replaced by initial soil without structure.

4. We can withdraw structure from Fig.1 and call the problem with initial soil in Vint “problem B”. This problem can be also split in “problem B1” and “problem B2” in the same manner as “problem A”. Wave fields in Vext and loads (-F) are equal in “problem A1” and “problem B1”. However, in “problem A2” and “problem B2” wave fields are different in spite of similar loads F and similar Vext in both problems. Generally, the motions of Q in “problem B2” and “problem A2” are different due to the waves, radiating from the structure in “problem A2”.
5. Wave field in volume Vint in “problem B2” is the same as in “problem B” (see conclusion 1 above). Very often this field is known apriori or easily calculated. This creates a powerful tool to verify models suggested for “problem A2”. Each of these models contains some description of the internal part Vint, external part Vext and the loads F. It is useful to take the same Vext and F and substitute the internal part Vint by the initial soil, thus coming from “problem A2” to “problem B2”. The suggested Vext and F must provide adequate solution for “problem B2”; otherwise they cannot be applied to “problem A2”.
6. Loads (–F) can be obtained from the wave field U0 in “problem B” and dynamic stiffness operator G0 for the initial unbounded soil as follows

Formula (1) uses operator G0 in the time domain. This operator is applied to the displacement field in the volume Vint and provides the loads, distributed over the volume (this formula can be applied to the whole volume Vint, but for the internal nodes the result will be zero). For linear initial soil this operator in the frequency domain will turn into complex frequency-dependent dynamic stiffness function. Note that for the given surface Q the loads F in (1) can be split in two parts: loads Fint acting from Vint and loads Fext acting from Vext. Physical meaning is illustrated in Fig.2.

With given free-field wave field U0 one can easily obtain Fint just as surface forces at Q corresponding to the internal stress field.
Theoretically we can change the content of the volume Vint in “problem B” (e.g., withdraw the medium completely, taking loads Fint to zero). This change will change operator G0, change wave field U0, but the left part of (1) will stay the same, as it is in fact fully determined by soil properties in Vext and initial seismic wave.
Even if there exists some physical non-linearity in the model, and this non-linearity is localized inside Vint, initial “problem A” can still be substituted by “problem A2” without changing the loads F, as compared to the linear case. This is a consequence of the logic of the previous point: the loads are fully determined by soil in Vext and initial seismic wave.