DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS IN LAYERED MEDIA

Abstract

Foundations for reciprocating machines, which are also known as Block Foundations, are usually heavy blocks and they are rigid compared to the supporting soil media. They are designed to fulfill the following requirements: (1) to limit the amplitudes within permissible values, and (2) to keep the natural frequencies of the block-soil system away from the operating frequencies in order to avoid resonance. Block foundations are usually embedded fully or in part, into the supporting soil, which consists of, in most ofthe cases, layers comprising ofdifferent soil types. The need for embedding the block arises chiefly because of two reasons, viz., (1) to rest the block on a good bearing stratum, and (2) to provide the block top at a level convenient for operating the machine(s) it supports. To meet the above design requirements, it is necessary to determine the response of the block to prescribed dynamic loading. Determination of response of the block involves finding the equivalent stiffness and damping parameters, i.e., thedynamic impedance/compliance functions of the soilfoundation^ system. Various methods like Finite Elements (Lysmer and Kuhlemeyer, 1969; Song and Wolf, 1994; Borja et. al, 1994), Integral Equation Approach (Karasudhi, et. al., 1968; Apsel and Luco, 1987 ; Luco and Mita, 1987.), Boundary Element Method (Dominguez and Roesset, 1978; Wong and Luco, 1986) and Baranov- Novak Approach (Novak and Beredugo, 1971, 1972; Beredugo and Novak, 1972) etc. have been used to determine impedance functions of soil-foundation systems. Finite element methods need the use of special wave absorbing boundaries to avoid the "box effect", which refers to the trapping of energy of the system and distorting its dynamic characteristics. Various types ofboundaries have been proposed. However, all these boundaries have their own limitations in reproducing the dynamic behavior of infinite or semi-infinite media, as discussed by Wolf and Song (1996). In li addition, Most of the frequently used finite element codes need arbitrary rigid stratum below the soil layer(s), which may not be present in reality (Gazetas, 1983). On the other hand, both integral equation approach and boundary element methods are associated with considerable mathematical and numerical difficulty. They also obscure the physical insight to the problem and belong more to the discipline of applied mechanics, than to CIVILENGINEERING (Wolf, 1994a). In boundary element method a considerable amount of expertise in idealising the actual dynamic system is necessary (Gazetas, 1987). In addition, it involves a significant amount of data preparation, making it difficult, from an economical point of view, to perform the necessary parametric studies and to investigate alternative design schemes. Thus, their use in routine projects may provide a false sense of security to the user (Wolf, 1994a). The fundamental solutions are very complicated and are not available for all types of problems of practical interest (Wolf and Song, 1996). The Baranov-Novak semianalytical approach, though simple, requires foundations to be deeply embedded, for better results. Therefore, relatively simpler, physically motivated models, which retain the accuracy of above methods as much as possible, are welcome in this field. Wolf (1994a) has discussed the necessity, importance and advantages of such simplified methods. Cone Models (Meek and Wolf, 1992a; Gazetas, 1987) and Consistent Lumped Parameter Models (Wolf, 1991a, 1991b) are both simple and physically motivated. Cone models have been developed to determine Impedance functions of foundations on or embedded in homogeneous halfspace (Dobry and Gazetas, 1986; Gazetas, 1987; Meek and Wolf, 1991; 1992a), and in layered halfspace (Meek and Wolf, 1992b; Wolf and Meek, 1993; Wolf and Meek, 1994; Wolf, 1994a) However, in the cone models (Meek and Wolf, 1992a), the stiffness and damping coefficients are frequency independent. The results of these cone models are rigorously correct at static limit and at very high frequency limit. In the practically important range of operating frequencies of machine foundations, these models overiii estimate the stiffness and damping. Wolf (1994a) has indicated the importance and " advantages of using frequency dependent impedance of surface foundations to determine (a) impedance of embedded foundations in homogeneous half-space and (b) impedance of surface and embedded foundations in layered half-space. Sung (1953) has shown the importance of contact pressure distribution below the block on its dynamic response. The same has been re-emphasised by Novak (1970), for realistic prediction of dynamic response in cohesionless soil. It is well known that, in cohesionless deposit, the contact pressure below the block becomes parabolic and thus affect the dynamic response of block foundations on or embedded in such soils. This variation of contact pressure occurs because of the variation of elastic moduli across the length or width of the foundation, the latter being caused by the variation of confining pressure. Most ofthe rigorous and simplified methods, including cone model, assume either rigid contact pressure or constant elastic modulus of soil across the width or length of block foundations. Such assumptions usually do not apply in case of surface or embedded blocks in cohesionless soils. Experimental evidences (Ho and Burwash, 1969; Vijayvargiya, 1980) suggest that contact pressure in such soils is more close to parabolic distribution. In the light of the above, a study has been undertaken to (1) develop modified cone model for finding frequency dependent stiffness and damping, (2) use the model to find impedance functions and dynamic response of surface and embedded blocks subjected to rigid contact pressure, (3) compare the resulting response with the rigorous solutions for practically important range of frequency, embedment and mass/ inertia ratios of foundations, in order to validate the modified cone model, (4) find the impedance functions and dynamic response ofsurface and embedded blocks subjected to parabolic contact pressure in (a) homogeneous and (b) layered soils, (5) compare the analytical dynamic response with measured dynamic response in (a) vertical vibration, (b) coupled sliding and rocking vibration, and (6) study the effects of foundation parameter (mass ratio) and soil profile parameters (layer depths, embedment depths, iv and contrast in shear moduli among layers) on impedance functions and dynamic response of block foundations. In this study, the basic translational and rotational cone models (Meek and Wolf, 1992a) have been modified to include frequency dependent stiffness, and frequency dependent damping co-efficients in terms of Analog Velocity Ratio, the latter being based on the concept of cone model (Meek and Wolf, 1992a) and analog velocity (Dobry and Gazetas, 1986). One of the advantages of using frequency dependent analog velocity ratio is that, the effect of three-dimensional wave propagation on radiation damping can be incorporated within the framework of cone model, which is otherwise based on one-dimensional wave propagation. The modified cone model is used to determine the approximate Green's functions in translation and rotation, which are then used to find the impedance functions of surface foundations on layered half-space and embedded foundations in homogeneous or layered half-space, following the generalised procedures (Meek and Wolf, 1994; Wolf and Meek, 1994; Wolf, 1994a). The model has been used initially to find the dynamic response of embedded circular, square and rectangular foundations in homogeneous half-space in vertical, pure sliding and pure rocking vibration subjected to rigid contact pressure. The results are then compared with available rigorous results (Gazetas, 1991; Mita and Luco, 1989). It is found that the model gives reasonably good match over a wide range of frequency and mass and inertia ratios. Parabolic contact pressure has been taken care of by using an equivalent elastic modulus, which is obtained by assuming a parabolic variation of the elastic modulus across the width of block. This is then used to determine an equivalent velocity of onedimensional waves to be utilised by the modified cone model. The rigorous vertical dynamic stiffness of surface blocks in parabolic contact pressure (Sung, 1953) is used for deriving the frequency dependent impedance functions of (a) embedded block foundations in homogeneous and (b) surface or embedded block foundations in layered half-space, subjected to parabolic contact pressure in vertical mode. In addition, approximate frequency dependent dynamic impedance functions of surface block in pure sliding vibration and static stiffness in pure sliding and rocking vibration, subjected to parabolic contact pressure are proposed. The model is then used to determine the dynamic response of surface and embedded block in homogeneous and layered half-space, subjected to parabolic contact pressure. Finally, the response is compared with measured field response (Vijayvargiya, 1980) of surface and embedded block foundations in cohesionless deposit in vertical and coupled sliding and pitching vibration, which covers wide range of embedment depths and dynamic force levels. The theoretical response is computed for both rigid and parabolic contact pressure for the purpose of comparison. It is found that, the model, with incorporation of parabolic contact pressure, gives very good match in vertical vibration and satisfactory match in coupled vibration, over a wide range of frequency, excitation force level, and embedment depths. Based on these, practical recommendations have been made to select the strain compatible shear modulus in response analysis. Parametric studies have been made to analyse the effects of (1) layer depth, (2) embedment depths on impedance functions and dynamic response, and, (3) contrast between the shear moduli of layers, and for vertical, pure sliding and pure rocking vibration of surface and embedded block foundations, subjected to parabolic contact pressure in layered media. The following conclusions are drawn from the study: (1) It is found that in addition to other features of a layered halfspace, the modified cone model is able to predict the fundamental frequencies of the layers resting on rigid rock halfspace. (2) In general, presence of layering decreases the damping and increases the stiffness when layer rests on stiffer halfspace. (3) The embedment depth increases the stiffness and damping of the system and thus increases the resonant frequency and decreases the resonant amplitudes. However, when the foundation is heavy, the response is mainly dominated by its own inertia and the effects of layering on dynamic response are vi minimal. (4) The increased contrast of shear moduli between the layers increases the effects of layering on system stiffness and damping, and in turn, on the dynamic response. (5) Contact pressure, or the effect of confining pressure on elastic moduli affects the response to a great extent. In cohesionless soil deposit, parabolic contact pressure gives the closest match between computed and measured response, particularly in vertical vibration. (6) In coupled vibration, the response of surface foundations, predicted by modified cone model, shows satisfactory match with measured response. In embedded foundation, the match is not as good as in surface foundations. After analysing the possible causes, it is concluded that further studies in this case are required. VI1