2. Benchmark solution

2.1. Calculation method

We use the coherence function proposed by Luco and Wong (1986) [bib1]:

where \(>\) refers to the distance between two points \(>\) and \(>\) on the foundation, \(>\) is the frequency and \(>\) the apparent surface propagation speed of the wave \(>\). The parameter \(>\) may vary from \(>\) to \(>\) depending on the case but is generally taken to be equal to \(>\).

2.2. Reference quantities and results

Covariance coefficients obtained by Mita and Luco for \(>\) [bib2]:

\(>\)	\(>\)	\(>\)
1.0	0.732	0.730
2.0	0.402	0.416
3.0	0.251	0.270

a0 refers to the non-dimensional frequency \(>\)

For \(>\), we get the case without spatial variability, for this case we know the solution (analytical). Since the foundation is rigid without weight, the response to a unit excitation is equal to \(>\), regardless of the calculation frequency.

2.3. Uncertainties about the solution

No uncertainties.

2.4. Bibliographical references

[bib1] Luco J.E and Wong H.L.: Response of a rigid foundation to a spatially random ground motion. Earthquake Engrg. Structure. Dyn. 14, 1986, pp.891-908.

[bib2] Luco J.E and Mita A.: Response of a circular foundation to spatially random ground motion. J. Engrg.Mech. 113, 1987, pp.1-15.