Soil-Structure System Behavior - Inertial Interaction

Inertial Interaction

2.1 Soil-Structure System Behavior

A rigid base refers to soil supports with infinite stiffness (i.e., without soil springs).

A rigid foundation refers to foundation elements with infinite stiffness (i.e., not deformable). A fixed base refers to a combination of a rigid foundation elements on a rigid base. A flexible base analysis considers the compliance (i.e., deformability) of both the foundation elements and the soil.

Consider a single degree-of-freedom structure with stiffness, k, and mass, m, resting on a fixed base, as depicted in Figure 2-1a. A static force, F, causes deflection, :

 F

k (2-1)

From structural dynamics, the undamped natural vibration frequency, , and period, T, of the structure are given by Clough and Penzien (1993) as:

, 2 2

k m

m T k

  

    (2-2)

By substituting Equation 2-1 into Equation 2-2, an expression for the square of period is obtained as:

     

² ² Figure 2-1 Schematic illustration of deflections caused by force applied to: (a)

fixed-base structure; and (b) structure with vertical, horizontal, and rotational flexibility at its base.

Now consider the same structure with vertical, horizontal, and rotational springs at its base, representing the effects of soil flexibility against a rigid foundation, as depicted in Figure 2-1b. The vertical spring stiffness in the z direction is denoted kz, the horizontal spring stiffness in the x direction is denoted kx, and the rotational spring is denoted k_yy, representing rotation in the x-z plane (about the y-y axis). If a force, F, is applied to the mass in the x direction, the structure deflects, as it does in the fixed-base system, but the fixed-base shear (F) deflects the horizontal spring by uf , and the base moment(F h ) deflects the rotational spring by . Accordingly, the total deflection with respect to the free-field at the top of the structure,, is:

    

If Equation 2-4 is substituted into Equation 2-3, an expression for flexible base period, ,T is obtained as:

 

² ²

Combining expressions in Equation 2-5 and Equation 2-2 results in:

2 1 1 2

Equation 2-6 simplifies into a classical period lengthening expression (Veletsos and Meek, 1974):

Equation 2-7 can be applied to multi-degree-of-freedom structures by taking the height, h, as the height of the center of mass for the first-mode shape. This is commonly referred to as the effective modal height, which is approximately two-thirds of the overall structure height, and taken as 0.7 times the height in

ASCE/SEI 7-10 (ASCE, 2010). In such cases, period lengthening applies to only the first-mode period.

In previous work by Veletsos and Nair (1975) and Bielak (1975), it has been shown that the dimensionless parameters controlling period lengthening are:

, , , , and

s s4

h h B m

V T B L BLh 

 ^(2-8)

where h is the structure height (or height to the center of mass of the first mode shape), B and L refer to the half-width and half-length of the foundation,m is the mass (or effective modal mass),s is the soil mass density, and  is the Poisson’s ratio of the soil. Previous work was applicable to circular foundations, and has been adapted here for rectangular shapes considering the ratio, B/L.

To the extent that h/T quantifies the stiffness of the superstructure, the term h/(VsT) in Equation 2-8 represents the structure-to-soil stiffness ratio. The term h/T has units of velocity, and will be larger for stiff lateral force resisting systems, such as shear walls, and smaller for flexible systems, such as moment frames. The shear wave velocity, Vs, is closely related to soil shear modulus, G, computed as:

s / s

V  G  (2-9)

For typical building structures on soil and weathered rock sites, h/(VsT) is less than 0.1 for moment frame structures, and between approximately 0.1 and 0.5 for shear wall and braced frame structures (Stewart et al., 1999b). Period lengthening

increases markedly with structure-to-soil stiffness ratio, which is the most important parameter controlling inertial SSI effects.

The structure-height-to-foundation-width ratio, h/B, and foundation-width-to-length ratio, B/L, in Equation 2-8 are aspect ratios describing the geometry of the soil-structure system. The mass ratio, m/s4BLh, is the ratio of structure mass to the mass of soil in a volume extending to a depth equal to the structure height, h, below the foundation. In Equation 2-7, it can be seen that period lengthening has no

fundamental dependence on mass. The mass ratio term was introduced so that period lengthening could be related to easily recognizable characteristics such as structural first mode period, T, and soil shear wave velocity, Vs, rather than structural stiffness, k, and soil shear modulus, G. The effect of mass ratio is modest, and it is commonly taken as 0.15 (Veletsos and Meek, 1974). The Poisson’s ratio of the soil, , affects the stiffness and damping characteristics of the foundation.

Using models for the stiffness of rectangular foundations (of width, B; half-length, L; and L ≥ B) resting on a homogeneous isotropic half-space with shear wave velocity, Vs, period lengthening ratios can be calculated with the results shown in Figure 2-2a, which is plotted for the special case of a square footing (L = B).

Figure 2-2 Plot of period lengthening ratio (T~ T ) and foundation damping (f ) versus structure-to-soil-stiffness ratio for square foundations (L = B) and varying ratios of h/B. In this plot,  = 0.33, B/L = 1.0, hysteretic soil damping s = 0, mass ratio= 0.15, and exponent n = 2.

All other factors being equal, period lengthening increases with the structure-height-to-foundation-width ratio, h/B, due to increased overturning moment and foundation rotation, . This implies that inertial SSI effects would be more significant in tall buildings, but this is not the case. Tall buildings typically have low h/(VsT) ratios, which is more important for controlling inertial SSI effects. Hence period

lengthening in tall buildings is near unity (i.e., little or no period lengthening). For a fixed ratio of h/B, period lengthening is observed to decrease modestly with

foundation-width-to-length ratio, B/L, due to increased foundation size (and therefore stiffness) normal to the direction of loading.

In addition to period lengthening, system behavior is also affected by damping associated with soil-foundation interaction, referred to as foundation damping, _f. This damping is composed of two parts: (1) contributions from soil hysteresis (hysteretic damping); and (2) radiation of energy away, in the form of stress waves, from the foundation (radiation damping). Foundation damping is a direct contributor to the flexible-base system damping, _₀:

where _i is the structural dampingin the superstructure assuming a fixed base, which is generally taken as 5% for typical structural systems. More refined estimates of i

are possible based on structural system type and configuration, as described in PEER/ATC-72-1, Modeling and Acceptance Criteria for Seismic Design and Analysis of Tall Buildings (ATC, 2010). Observations from case studies (Stewart et al., 1999b) have shown that _f ranges from approximately 0% to 25%. The exponent, n, on the period lengthening term in Equation 2-10 is taken as 3 for linearly viscous structural damping, and 2 otherwise (e.g., for hysteretic damping) (Givens, 2013).

Analytical models for foundation damping have been presented by Veletsos and Nair (1975), Bielak (1975 and 1976), Roesset (1980), Wolf (1985), Aviles and Perez-Rocha (1996), Maravas et al. (2007), and Givens (2013), among others. The classical solution of Veletsos and Nair accounts for the frequency dependence of foundation damping terms. It assumes structural damping to be purely viscous, and applies for a circular foundation resting on a half-space. The equation for f provided by Veletsos and others is complex-valued (i.e., composed of real plus imaginary values), which complicates the interpretation of its physical meaning. Bielak’s work utilizes the same conditions except that the foundation is assumed to be a cylinder penetrating a half-space to an embedment depth, D, and the resulting expressions are real-valued.

The value of exponent n in Equation 2-10 is taken as 3 for the Veletsos and Bielak solutions because structural damping is assumed to be viscous.

The procedure given by Wolf (1985) neglects the frequency dependence of

foundation stiffness terms, and assumes foundation radiation damping to be linearly viscous (i.e., constant dashpot coefficients for translation and rotation, c_x and c_yy), and applies for a circular foundation resting on a half-space. Considering frequency dependence, the form of Wolf’s damping expression (similar to Roesset, 1980) can be re-written as: damping from translational and rotational modes (described further in Section 2.2), and T_x and T_yy are fictitious vibration periods, calculated as if the only source of the vibration was foundation translation or rotation, as follows:

Exponents n_s, n_x, and n_yy depend on the specific form of damping associated with the respective components of the foundation damping, and all other terms are as

previously defined. However, because none of these terms would be expected to be linearly viscous, it is recommended to take these exponents as 2 (Givens, 2013).

Note that for n = n_s, the period lengthening terms in front of the _i term in Equation 2-10 and the s term in Equation 2-11a are weight factors that together sum to unity.

Accordingly, Equation 2-11a can be viewed as a “mixing rule” for damping in different vibration modes and sources. Because Wolf’s results were produced neglecting the frequency dependence of foundation stiffness terms, Equation 2-11a can provide more accurate results if those effects are included in the period

lengthening calculation.

Soil hysteretic damping, s, is strain-dependent, and can typically be evaluated from information in the literature. Classical models are summarized in Kramer (1996).

More contemporary empirical models by Darendeli (2001) and Menq (2003) account for overburden pressure and shear strain in a consistent manner across multiple soil types.

The Wolf solution for foundation damping in Equation 2-11a, along with the classical Veletsos, Bielak, and Roesset solutions, neglect contributions from terms involving the product of two damping ratios. Maravas et al. (2007) presents exact solutions in which those terms are included. Like Wolf, Maravas et al. (2007) utilizes hysteretic damping so exponents n = 2, and if terms involving the product of two damping ratios are excluded, Equation 2-11a is recovered.

As was the case for period lengthening shown in Figure 2-2a, Figure 2-2b shows that foundation damping _f increases strongly with structure-to-soil-stiffness ratio, h/(V_sT). In Figure 2-2b, all exponents were taken as 2. Damping f decreases with increasing values of h/B, indicating that lateral movements of the foundation (which dominate at low h/B ratios) dissipate energy into soil more efficiently than foundation rocking (which dominates at high h/B ratios). Radiation damping terms (_x and _yy) are reduced significantly when a stiff bedrock layer is encountered at moderate or shallow depths, as described further in Section 2.2.2.

Analysis procedures for /T T and f similar to those described above have been validated relative to observations from instrumented buildings shaken by earthquakes (Stewart et al., 1999a; 1999b). These studies show that the single most important parameter controlling the significance of inertial interaction is h/(V_sT), and that inertial SSI effects are generally negligible for h/(V_sT) < 0.1, which occurs in flexible structures (e.g., moment frame buildings) located on competent soil or rock.

Conversely, inertial SSI effects tend to be significant for stiff structures, such as shear wall or braced frame buildings, located on softer soils.

The effect of inertial SSI on the base shear of a building is illustrated in Figure 2-3.

Because base shear for elastic response is commonly computed based on spectral acceleration in the first mode, the figure depicts the variation in pseudo-spectral acceleration versus period in both linear and log scales. The pseudo-pseudo-spectral acceleration for a flexible-base structure,S_a, is obtained by entering the spectrum drawn for effective damping ratio, ₀, at the corresponding elongated period, T.

Figure 2-3 Illustration of inertial SSI effects on spectral acceleration (base shear) associated with period lengthening and change in damping.

The effect of SSI on base shear is related to the slope of the spectrum. Base shear tends to increase when the slope is positive and decrease when the slope is negative.

For the common case of buildings with relatively long periods on the descending portion of the spectrum, use of S_a(flexible base) in lieu of S_a (fixed base) typically results in reduced base shear demand. Conversely, inertial SSI can increase the base shear in relatively short-period structures.

The period at which the spectral peak occurs, referred to as the predominant period of ground motion, Tp, is generally controlled by the tectonic regime, earthquake

magnitude, and site-source distance (Rathje et al., 2004), and will only match the site period in certain cases involving large impedance contrasts due to soil layering. In the absence of unusual site effects, typical values of T_p range from approximately 0.2 to 0.5 seconds for shallow crustal earthquakes in tectonically active regions, such as California.

2.2 Equations for Shallow Foundation Stiffness and Damping

In document Soil-Structure Interaction for Building Structures (Page 33-40)