Dynamics of Building - Soil Interaction

In this study of the dynamics of building-soil interaction, the soil is modeled by a linear elastic half-space, and the building structure by an n-degree-of-freedom oscillator. Both earthquake response and steady-state response to sinusoidal excitation are examined. By assuming that the interaction system possesses n + 2 significant resonant frequencies, the response of the system is reduced to the superposition of the responses of damped linear oscillators subjected to modified excitations. The results are valid even though the interaction systems do not possess classical normal modes. For the special cases of single-story systems and the first modes of n-story systems, simplified approximate formulas are developed for the modified natural frequency and damping ratio and for the modified excitation. Example calculations are carried out by the approximate and more e x a c t analysis for one-story, two-story and ten-story interaction systems. The results show that interaction tends to decrease all resonant frequencies, but that the effects are often significant only for the fundamental mode for many n-story structures and are more pronounced for rocking than for translation. If the fixed-base structure has damping, the effects of interaction on the earthquake responses are not always conservative, and an increase or decrease in the response can occur, depending on the parameters of the system. INTRODUCTION There are two aspects of building-foundation interaction during earthquakes which are of primary importance to earthquake engineering. First, the response to earthquake motion of a structure founded on a deformable soil can be significantly different from that of a structure supported on a rigid foundation. Second, the motion recorded at the base of a structure or in the immediate vicinity can be different in important details from that which would have been recorded had there been no building. Observations of the response of buildings during earthquakes have shown that the response of typical structures can be markedly influenced by the soil properties if the soils are sufficiently soft (Ishizaki and Hatakeyama, 1960). Furthermore, for relatively rigid structures such as nuclear reactor containment structures, interaction effects can be important even for relatively firm soils because the important parameter apparently is not the stiffness of the soil, per se, but the relative stiffness of the building and its foundation. From the point of view of engineering, it is important to determine the conditions under which soilstructure interaction is practically significant, and to develop methods that can be used in design for calculating interaction effects. In terms of the dynamic properties of the building foundation system, past studies have shown that interaction will, in general, reduce the fundamental frequency of systems from that of the structure on a rigid base, dissipate part of the vibrational energy of the building by wave radiation into the foundation medium (there will also be energy losses from internal friction in the soil), and modify the base motion of the structure in comparison to the free-field motion. Although all of these effects may be present in some degree for every structure, the important point is to determine the conditions under which the effects are of practical significance. 9 10 PAUL C. JENNINGS AND JACOBO BIELAK The complex material properties of soils, the involved geometries of building foundations, and the complicated nature of earthquake ground motions combine to make the soil-structure interaction problem extremely complicated, and it is necessary, in general, to make major simplifying assumptions in all these aspects of the problem before calculations can be made. In most studies, the soil is idealized as a linear, homogeneous, isotropic, elastic half-space (Sato and Yamaguchi, 1960; Parmelee, 1967; Sarrazin, 1970; Scavuzzo et al., 1971), and in many instances the dynamic properties of the half-space are further approximated by discrete springs and dashpots. A still further approximation often made is that the discrete elements have properties that do not vary with frequency (Merritt and Housner, 1954; Thomson, 1960; Parmelee et al., 1969). The building foundation is usually simplified by assuming that the building-soil interface is at the ground surface and that the cross-section of the contact area can be represented by a circle (Thomson, 1960). The earthquake excitation is typically idealized as vertically propagating, horizontally incident, planar motion (Hradilek and Luco, 1970). In addition to earthquake motion, studied among others by Housner (1957), Parmelee et al. (1969) and Castellani (1970), steady-state response to sinusoidal excitation has been studied extensively to clarify the basic features of the problem (Sato and Yamaguchi, 1960; Thomson, 1960; Parmelee, 1967). The application of the finite element method to the problem can avoid some of the above assumptions which are primarily geometrical, but simplified models of the soil and excitation are still required and, unless a threedimensional approach is used, it is necessary to make a two-dimensional idealization of the problem. Thus, using a finite element formulation for plane strain problems, Isenberg (1970) has studied the effects of interaction for elastic buildings embedded into elastic/perfectly plastic soils. The representation of the foundation system by springs and dashpots is an attractive approach for design because the resulting system is similar to the usual representation of a fixed-base structure. It is important to realize, however, that the representation of the foundation by constant springs and dashpots is not consistent with using an elastic half-space as an idealization of the soil. If the springs and dashpots are to be equivalent to the elastic half-space, their properties must be frequency-dependent (Hsieh, 1962). Fortunately, it seems possible in many instances to approximate the frequencydependence reasonably well by representative constant values, an approximation that results in a system of linear differential equations with constant coefficients. Thus, some of the standard methods of analysis can be applied to interaction systems when the foundation medium is modeled this way (Parmelee et al., 1969). Normal mode methods of structural dynamics cannot be applied to such systems, however, because the foundation dashpots are such that the building-foundation model does not possess classical normal modes. Unlike these methods, operational methods of analysis can be used even when the properties of the springs and dashpots are frequency-dependent (Sandi, 1960; Rosenberg, 1965). There are two major efforts required to make the theory of soil-structure interaction a better tool for use in earthquake-resistant design. First, methods of calculation must be developed which are accurate within the framework of simplifying assumptiorls, and second, more experimental studies and earthquake-response measurements are needed to establish the range of validity of the various methods of simplifying the problem. The present study is directed toward the first of these efforts and is performed under the scope of the assumptions outlined above. As is common in structural analysis, the building itself is modeled by a linear, viscously damped, multi-degree-of-freedom oscillator. In the first part of the paper an analytical examination of the n-story buildingsoil interaction problem is made to show that under the assumption that n + 2 resonant DYNAMICS OF BUILDING-SOIL INTERACTION 11 frequencies exist, the earthquake response of the interaction system reduces to the linear superposition of the responses of damped, linear, single-degree-of-freedom oscillators subjected to modified excitations. This result is shown to be valid even for systems that do not possess classical normal modes. The major advantages of the approach are that it makes the calculations involved equivalent to those for simple, rigid-base structures, and that it gives physical insight into the dynamics of building-foundation systems. The second portion of the study is devoted to the examination of the effects of the important parameters on the dynamics of singleand multi-degree-of-freedom soilstructure systems, and to the presentation of examples of earthquake response and steady-state response to sinusoidal excitation. Simplified formulas for natural frequency changes, radiation damping values, and other response parameters of interest in design are developed from the analysis and the examples. ANALYSIS OF THE SYSTEM General information. The system under investigation is shown in Figure 1. It consists of a linear, viscously damped n-story structure with one degree of freedom per floor, resting on the surface of an elastic half-space with density p, shear modulus /~, and Poisson's ratio, a. For fixed-base response, the superstructure has a stiffness matrix K, mass matrix M, and damping matrix C, satisfying the condition M 1 K M I C = M 1 C M 1 K . O'Kelly (1964) has shown this to be a necessary and sufficient condition for the superstructure to admit decomposition into classical normal modes. (The assumption of classical normal modes for the superstructure can be removed, but because buildings seem to possess such modes over significant range of amplitudes, this simplifying assumption is retained.) The structural base is assumed to be a rigid plate of radius a and negligible thickness, and no slippage is allowed between the base and the soil. Formulated this way, the building-foundation system has n + 2 significant degrees of freedom, namely, horizontal translation of each floor mass, horizontal translation of the base mass, and rotation of the system in the plane of motion. The system, initially at rest, will be subjected to seismic motion or harmonic excitation represented by plane, horizontal shear waves traveling vertically upward. No scattering will result as the waves are normally incident on the flat foundation. In this idealization of the excitation, the free-field acceleration at the surface is twice the amplitude of the incoming wave, and the motion at depth is the sum of the incident and reflected waves. The model for the building-foundation system shown in Figure 1 has also been studied by Tajimi 0967), Parmelee et al. (1969) and others. Because the superstructure by itself has classical normal modes, there is a simple physical model which is equivalent to the building-foundation system under study. This model, shown in Figure 2, consists of n simple, damped oscillators attached to a base identical to that of the system shown in Figure 1. Each oscillator is described by its natural frequency co~, critical damping ratio, ~/j, mass Mj and height H~ defined by the corresponding modal quantities (given in the Appendix). In addition, the sum of the centroidal moments of inertia of the n masses is the same for both systems. Assuming small displacements, the equations of motion of the building-foundation model shown in Figure 1 are M V + CC¢+Kv = 0 (la) ~ mjig'J t +mo(Vo +va) +P(t) = 0 (lb) j = l m / ~ p / + 6 ~ + Q(t) = O. (Ic) j = l 12 PAUL C. JENNINGS AND JACOBO BIELAK