Kinematic Interaction
3.1 Shallow Foundations at the Ground Surface
Base-slab averaging results from adjustment of spatially variable ground motions that would be present within the envelope of the foundation, which are averaged within the foundation footprint due to the stiffness and strength of the foundation system.
Base-slab averaging can be understood by recognizing that the motion that would have occurred in the absence of the structure is spatially variable. Placement of a foundation slab across these variations produces an averaging effect in which the foundation motion is less than the localized maxima that would have occurred in the free-field. Torsional rotations, referred to as the “tau effect” (Newmark, 1969), can also be introduced.
Motions of surface foundations are modified relative to the free-field when seismic waves are incoherent. Incoherence of the incident waves at two different points means that they have variations in their phase angle. Some incoherence is
deterministic (i.e., predictable), because it results from wave passage. For example, as illustrated in Figure 3-1a, the presence of a non-zero vertical angle causes waves to arrive at different points along the foundation of a building at different times. This is
referred to as the wave passage effect. Investigation of wave passage effects in dense seismic arrays at soil sites indicates apparent propagation velocities, Vapp, of
approximately 2.0 km/sec to 3.5 km/sec, which appears to be controlled by wave propagation in crustal rock beneath the soil column (Ancheta et al., 2011).
Incoherence that remains when waves are aligned to have common arrival times is stochastic, and is quantified by lagged coherency models. Stochastic incoherence results from source-to-site heterogeneities in the seismic path of travel, which scatters seismic waves. Lagged coherency is also well-documented in array studies (e.g., Abrahamson et al., 1991; Ancheta et al., 2011). As a practical matter, incoherence from wave passage and lagged coherency is always present in earthquake ground motions to some degree.
Figure 3-1 Illustration of foundation subjected to inclined shear waves: (a) schematic geometry; (b) transfer functions between FIM and free-field motion for wave passage using a semi-empirical model for incoherent waves (parameter a is defined in Equation 3-5).
In the presence of incoherent wave fields, translational base-slab motions are reduced relative to the free-field, and rotational motions are introduced. The reduction in translational motion is generally the more important result. Reductions of base-slab translation and the introduction of torsion and rotation in the vertical plane are effects that tend to become more significant with increasing frequency. The frequency-dependence of these effects is primarily associated with: (1) the increased effective size of the foundation relative to the seismic wavelengths at high frequencies; and (2) significant reductions in lagged coherency with increasing frequency (Abrahamson et al., 1991).
There are numerous theoretical models for predicting the relationship between foundation input and free field ground motions in the presence of inclined, but
otherwise coherent, shear waves (i.e., wave passage effects). Mylonakis et al. (2006) synthesized these models with the following expressions:
FIM u g
In the above expressions, a0k is similar to a0 defined in Equation 2-15, except that the foundation dimension is related to base contact area, as follows:
0 expression in Equation 3-1b becomes indeterminate, and Hu should be taken as unity.
If, as indicated in array studies, Vapp ranges from approximately 2.0 km/s to 3.5 km/s, then for a typical soil site, a reasonable estimate of the velocity ratio, Vapp/Vs, is approximately 10. In Figure 3-1b, the result labeled “wave passage only” shows the transfer function between uFIM and ug based on Equations 3-1. Using this model, wave passage alone causes relatively modest base-slab reductions in ground motion across the frequency range of engineering interest.
Transfer functions of recorded foundation input and free-field motions are generally significantly lower at high frequencies than predicted by wave passage models. This occurs because wave passage is a relatively modest contributor to the spatial
variation in ground motion that drives base-slab averaging. Additional sources of variability include stochastic phase variability (quantified by lagged coherency) and stochastic variations in Fourier amplitudes. Two approaches for capturing these effects in the analysis of transfer functions are: (1) continuum modeling of the soil and foundation system subject to input motions with a defined coherency function (Computer Program SASSI2000, Lysmer et al., 1999; Ostadan et al., 2005); and (2) application of a semi-empirical simplified model (Veletsos et al., 1997; Kim and Stewart, 2003).
In SASSI, a site-specific and foundation-specific model is generated in three dimensions. The foundation and soil material properties are equivalent-linear without iteration on strain-dependent properties. Empirical coherency models can be
used that include wave passage and lagged coherency (e.g., Ancheta et al., 2011, for soil sites; EPRI, 2007, for hard rock sites).
The semi-empirical model is based on a theoretical formulation of the kinematic interaction problem by Veletsos and Prasad (1989) and Veletsos et al. (1997), who apply spatially variable ground motions to a rigid foundation perfectly bonded to the soil. Models evaluate the response of rigid, massless, circular and rectangular foundations on the surface of an elastic half-space to incoherent S-waves propagating either vertically or at an angle v to the vertical, as shown in Figure 3-1a. The results are a relationship between transfer function amplitude and a0k. This relationship is essentially independent of foundation shape, but is strongly dependent upon a parameter, a, related to lagged coherency and wave inclination that scales the frequency axis of the theoretical transfer function. For vertically propagating waves this transfer function (adapted from Veletsos and Prasad, 1989) can be written as:
02
0 02 1 02
1 2 order, respectively. Equation 3-3 was developed for circular foundations. The4 term adapts a0k(for rectangles) to a0 defined for an equivalent radius that preserves foundation area. For small and large values of the argument (2b02), the Bessel function summation in Equation 3-3 can be written in terms of power series and exponential functions, respectively (Watson, 1995); for routine application, these approximations can be expressed as:
Note that the exponential terms in Equation 3-3 and Equation 3-4 cancel for b0 > 1.
The two functions in Equation 3-4 have a misfit of 0.0073 at b0 = 1, which is accurate enough for practical purposes.
By matching model predictions to observed variations between foundation input and free-field ground motions from instrumented buildings, Kim and Stewart (2003) developed a semi-empirical model for a that can be written as:
0.00065 , 200 500 /
a Vs Vs m s
(3-5)
where Vs is a representative small-strain shear wave velocity for the soil beneath the foundation, which can be calculated as described in Section 2.2.2.
Values of a identified through calibration reflect the combined effects of incoherence from wave passage and stochastic processes, as well as Fourier amplitude variability. Figure 3-1b shows transfer function Hu calculated using the semi-empirical approach near the upper and lower limits of a. Two key
observations from this figure are: (1) as a increases (indicating increasingly spatially variable motions), Hu decreases significantly; and (2) for the range of a supported by case history data, Hu from the semi-empirical procedure is much lower than from wave passage models. This can be attributed to significant contributions of stochastic phase and amplitude variability to base-slab averaging.
The data set considered by Kim and Stewart (2003) consists of buildings with mat foundations, footing and grade beam foundations, and grade beam and friction pile foundations, generally with base dimensions,B , in the range of 15 m to 40 m. eA Although the Veletsos models strictly apply to rigid foundations, the semi-empirical model applies to the more realistic foundation conditions present in the calibration data set.
Errors could occur, however, when the model is applied to conditions beyond the calibration data set. In particular, the effects of incoherence in the Veletsos models is taken as proportional to wavelength, thus implying strong scaling with frequency and distance. Array data indicate that distance scaling is much weaker than the frequency scaling (Abrahamson et al., 1991; Ancheta et al., 2011), so the model would be expected to over-predict the effects of incoherence (under-predict Hu) for very large foundations. The opposite would be true for small foundations. Even within the parameter range of the calibration data set, it should be recognized that the empirical model fits the data in an average sense, and should not be expected to match any particular observation.