Base-slab averaging effect

When a surface foundation is subjected to incoherent and or inclined seismic waves, seismic motions arrives at different points of the foundation footprint with various amplitudes and phase. As discussed, the foundation would move according to an average of these motions. This effect is called base slab averaging (Luco & Wong, 1986). The base slab averaging of surface foundations is experimentally investigated in this research.

Previous research has shown significant variabilities in ground motions recorded even over short distances in seismograph arrays during seismic events (Bolt et al., 1982; Hoshiya & Ishii, 1983; Luco & Wong, 1986; Abrahamson et al., 1991; Zerva & Zervas, 2002). Variations of the free-field motion from point to point happens as a result of inclined body waves, surface-waves, waves coming from different points along with an extended seismic source, and waves traveling through inhomogeneous materials (Luco & Wong, 1986). The wave passage effect causes deterministic incoherence since it can be calculated by having an angle αy between the propagation direction of the motion and the vertical axes. The remaining incoherence after removing the influence of wave passage effect is stochastic.

It is essential to calculate the kinematic transfer function to analyze the dynamic response of a foundation subjected to spatially varying seismic ground motions. This transfer function is defined for massless foundations, while harmonically excited. The function indicates the amplitude ratio of the components of a steady-state motion of a massless foundation to the free-field ground motion at a reference point.

Mita and Luco (1986) developed nearly exact integral expressions for transfer functions of a rigid, circular foundation, placed on an elastic half-space, by considering the wave passage effect and the stochastic incoherence. They modeled the stochastic incoherence using a dimensionless ground

motion incoherence parameter, 𝜅. 𝜅 is a real positive number. As the stochastic incoherence increases, the 𝜅 value increases.

Figure 2-3 shows the coordinate system, which Mita and Luco (1986) used in their study with some modifications. It should be noted that notations used by Mita and Luco (1986) in their study were modified to be consistent with the notations used in this dissertation. Mita and Luco (1986) also numerically solved their integral expressions for a few specific 𝜅 values and calculated kinematic transfer functions, as shown in Figure 2-4. The transfer functions are plotted as functions of the dimensionless frequency, 𝑎₀ = 𝜔𝑟 𝑉⁄ _𝑠; where 𝜔 is the circular frequency, 𝑟 is the radius of the foundation, and 𝑉_𝑠 is the shear wave velocity of the soil layer.

Figure 2-3. Coordinate system used by Mita and Luco (1986) to calculate transfer functions of a rigid circular foundation, placed on an elastic half-space [redrawn from (Mita & Luco, 1986)].

Figure 2-4. Transfer functions of foundation input motion for circular foundations, excited by vertically incident incoherent waves: (a) lateral transfer function; (b) vertical transfer function; (c) torsional transfer function; (d) rocking transfer function (Mita & Luco, 1986).

Figure 2-4 (a) demonstrates lateral transfer functions between the lateral component of the foundation input motion and the lateral component of the free field motion. Vertical transfer functions between the vertical component of the foundation input motion and the vertical component of the free field motion are depicted in Figure 2-4 (b). Figure 2-4 (c) illustrates torsional transfer function between the torsional component of the foundation input motion (around the vertical axis shown in Figure 2-3 ) to the lateral component of the free field motion. Furthermore, Figure 2-4 (d) depicts rocking transfer functions between the rocking component of the foundation input motion (around the lateral axis shown in Figure 2-3) to the vertical component of the free field motion. While the general trend of the lateral transfer functions is similar to the one for the

vertical transfer function, the overall trend of the torsional transfer functions is analogous to the one for the rocking transfer functions. Furthermore, for a given incoherence parameter, as the frequency of the motion increases, while the lateral and vertical transfer functions continuously decrease, the rocking and torsional transfer functions increase up to a peak value, then decrease. Thus, it is possible that the amplitude of the rocking and torsional transfer functions for a higher incoherence parameter would be smaller than the one for a smaller incoherence parameter in some frequencies.

Later, Luco and Wong (1986) determined similar integral expressions for a rectangular foundation in the same conditions. The coordinate system, which they used in their study with some modifications, is displayed in Figure 2-5. Luco and Wong (1986) also numerically solved the integral expressions for several incoherence parameters and estimated a variety of transfer functions between components (3 translational and 3 rotational) of the foundation input motion and components (3 transitional) of the free field motions.

Figure 2-5. Coordinate system used by Luco and Wong (1986) to calculate transfer functions of a rigid rectangular foundation, placed on an elastic half-space [redrawn from (Luco & Wong, 1986)].

Figure 2-6 and Figure 2-7 show transfer functions, estimated by Luco and Wong (1986), for square foundations subjected to vertically incident incoherent waves. While Figure 2-6 illustrates transfer functions between the lateral and rocking components of the foundation input motion to the lateral and vertical components of the free field motions, Figure 2-7 depicts transfer functions between the rocking and torsional components of the foundation input motion to the lateral and vertical component of the free field motions. The transfer functions are plotted in these figures as a function of the dimensionless frequency parameter for square foundation, 𝑏₀ = 𝜔𝑏 𝑉⁄ _𝑠; where 𝑏 is the half width of the foundation. Transfer functions, shown in Figure 2-6 and Figure 2-7, can be used to estimate the foundation input motion for a structure with a square surface foundation when the structure is subjected to spatially variable ground motions.

Veletsos and Prasad (1989) simplified the integral expressions and developed closed-form solutions for lateral and torsional transfer functions for a circular foundation. Finally, Veletsos et

al. (1997) expanded the same closed-form expressions for a rectangular foundation. Since the focus of this dissertation is on surface square foundations, the closed-form solutions, developed by Veletsos et al. (1997), are discussed in more details in the following sections.

Figure 2-6. Transfer functions between the lateral and vertical components of the foundation input motion for square foundations to the lateral and vertical components of the free field motion, when the foundations are

Figure 2-7. Transfer functions between the rocking and torsional components of the foundation input motion for square foundations to the lateral and vertical components of the free field motion when the foundations are excited by vertically incident incoherent waves (Luco & Wong, 1986).