INTERACTION OF FRAME STRUCTURES ON VISCO-ELASTIC FOUNDATIONS

Abstract

1.0 INTRODUCTION The interaction between structures, their foundations and the subsoil covers a broad and complex field in geotechnical engineering. The term "Soil-structure interaction", has largely been used for the mechanics of interaction between super-structure, foundation and subsoil or its part buried in soil. Conventional design procedure for frame structures involves the assumption of fixity of base, thus neglecting the flexibility of the foundation and compressibility of the sub soil. Consequently, the effect of the foundation deformations results in the re distribution of forces in the super-structure. It is, therefore, essential to consi der the mutual interaction between super-structures, foundation and subsoil. 2.0 PROBLEM IDENTIFICATION Review of literature indicated that considerable work has been done in the field of soil-structure interaction considering the elastic behaviour of soil. For such cases, the analytical procedure for carrying out the interactive analysis is more or less established. On the contrary, the literature available on inter active analysis with time-dependent behaviour of the soil is scanty. The time dependent behaviour includes the consolidation and creep characteristics of the soil. Due to these characteristics, large deformations can occur which may be damaging to the structure, particularly if the total and differential settlements are excessive. It is, therefore, essential to study the effect of time-dependent behaviour of soils on super-structure. Thus, the problem of interaction of frame structures founded on clay soils considering the frame, its footing and soil as a single compatible unit has been taken up for investigation in the present study. (iv) 3.0 SCOPE OF THE WORK (a) It has been seen that the super-structure, foundation and subsoil can best be physically, represented and analysed as a single compatible unit using finite element technique. In view of this, viscoelastic interactive analysis using above technique has been proposed. The approach involves the following components : (i) Development of the constitutive laws for time - dependent soil behaviour considering soil as visco-elastic material. (ii) Development of finite element formulation. (iii) Evaluation of visco-elastic material constants, capable enough to reflect the time-dependent behaviour of soil and (iv) Development of computer software. (b) However, the solution mentioned in (ii) above, is uneconomical and time consuming especially when the problem is three dimensional and iterative in nature. Therefore, a need for simplified approach was realised and the matrix displacement method has been proposed for use in the interactive analysis of the problem, for its convenient use in design practice. This method involves the following components : (i) Development of coupled visco-elastic equilibrium equations for the super structure visco-elastic subsoil system using suitable mechanical (rheological) model. (ii) Evaluation of model constants, capable to reflect the consolidation charac teristics of the soil and (iii) Development of the computer software. (v) 4.0 THEORETICAL ANALYSIS 4.1 Visco-Elastic Finite Element Approach (i) Constitutive law for visco-elastic material The applied load imposes a three dimensional stress situation at any point in the underlying soil mass. This stress situation can be separated into its hydro static and deviatoric components. The constitutive laws for hydrostatic and deviatoric components were expressed in differential forms (Flugge, 1967 and Findley, 1976). Further, to represent the time-dependent behaviour of visco-elastic material (here clay soil), a rheological model i.e. Kelvin model has been used. The constitutive laws for Kelvin body under both the components of stress have also been obtained. Equating these two sets of constitutive laws, the relationships between time operators and model constants (spring and dashpot constants) have been established. Further, the expressions for the time-dependent modulus of elasticity and poisson's ratio of clay have been obtained in terms of time opera tors. Analogous to elasticity matrix, a visco-elasticity matrix has been formulated. Knowing the relationships between the time operators and model constants, the visco-elasticity matrix in terms of model constants has been written and the time-dependent stress-strain relationship is established for general, three dimen sional stress condition. This visco-elasticity matrix could also be reduced to two dimensional stress situation. (ii) Finite element formulation The problem under investigation consists of three components, namely super structure, foundation and subsoil. The subsoil may be stratified in nature having clay (i.e. visco-elastic) and or sand (i.e. elastic) layers. The structure and sandy soil have been assumed to be elastic and their finite element formulation, available in standard text books (Zienkiewicz, 1981) has been used. The finite (vi) element formulation for visco-elastic material has been developed, expressing its time-dependent stress-strain relationship in finite difference form. (iii) Development of computer software A suitable computer software has been developed to carry out the interactive analysis accomodating both the elastic and visco-elastic behaviour of the various components of the system. 4.2 Simplified Visco-Elastic Approach (i) Visco-elastic equilibrium equations The approach is based on the Matrix Displacement Method (MDM) for the interactive analysis of the system. Mechanical model analogy representing the time-dependent soil behaviour has been used to couple the structure to the subsoil. Using such an arrangement, the generalised coupled visco-elastic equilibrium equations have been obtained for the system. The mechanical model consists of the spring and dashpot in parallel (i.e. Kelvinean) in which spring repre sents the elastic effect and dashpot simulates the time-delaying or viscous proper ties of the soil mass. (ii) Development of computer software For this approach also, a suitable computer software has been developed to take into account the visco-elastic response of the soil. 5.0 EVALUATION OF MATERIAL CONSTANTS Material constants are evaluated utilising the consolidation test data. Fur ther, a more general approach for their evaluation on the basis of triaxial test data has been suggested. (vii) 5.1 Material Constant Using Consolidation Test Data (i) Visco-elastic finite element approach The three dimensional constitutive law for visco-elastic material warrants the determination of visco-elastic constants (i.e. spring and dashpot constants) separately for both the hydrostatic and deviatoric stress components of a stress system. Therefore, the above constants have been evaluated for clay soil separa tely for both the stress components using their vertical component of stress and strain obtained on the basis of three dimensional compression of the soil mass. For the evaluation of hydrostatic and deviatoric strains, it is assumed that both components of strain occur simultaneously in an element of soil when it is subjected to a generalised system of stress with drainage being permitted. For such a stress condition, the vertical component of strain, Ae has been computed using Skempton and Bjerrum (1957) concept for the settlement under three dimen sional compression of soil mass. The volumetric strains, Ae has been obtained v using the coefficient of volume compressibility, m . under isotropic compression of soil mass (Eq. 1). Consequently the vertical component of deviatoric strain, Ae, has been computed (Eq. 2). Ae = m . Aa v vi v (1) A£d = A£ " JAev (2) (ii) Simplified visco-elastic approach In this approach, the constitutive law for visco-elastic material under uniaxial stress system has been utilised. The visco-elastic constants (i.e. spring constant and viscosity coefficient) for clay soil are evaluated on the basis of the assumption that the deformation occurs due to one-dimensional consolidation (viii) of the soil layer. Using this deformation and time factor of the primary consoli dation, the visco-elastic constants are evaluated. 5.2 Material Constants Using Triaxial Test Data An attempt has also been made to evaluate the visco-elastic constants sepa rately for hydrostatic and deviatoric components of a stress system on the basis of triaxial test data. For the evaluation of the constants, the principle presented in art 5.1.1 is utilised. The test data indicate that the Kelvin model is as such nonlinear and therefore the hyperbolic fit has been found to represent the test data in the best manner. The methodology has been suggested to evaluate the constants of the hyperbola and therefore the rheological constants. 6.0 APPLICATION OF PROPOSED APPROACHES 6.1 Visco-Elastic Finite Element Approach To establish the validity of the approach, a footing problem resting on clay layer of a stratified soil medium was analysed using the approach. Once it was realised that the approach works well in predicting the time-settlement behaviour of the footing, it was then employed to solve a problem of single storey orthogonal space frame (two bay by two bay) to study its time-dependent interactive behaviour. The frame was considered to rest on clay layer underlain by a sand layer. 6.2 Simplified Visco-Elastic Approach To establish the validity of this approach based on Matrix Displacement Method, a single storey orthogonal space frame (two bay by two bay), founded on stratified soil medium consisting of clay layer and analysed by Morris (1966) was solved. In this case too, when it was realised that the approach works (ix) satisfactorily, then it was employed to analyse a six storeyed space frame (three bay by three bay) resting on stratified soil medium consisting of clay layer to show the general applicability of the approach and to study the time-dependent interactive behaviour of the frame. The effect of the variation of dashpot constant with time has also been accounted for in both the approaches. 7.0 CONCLUSIONS (i) The two proposed approaches namely visco-elastic finite element approach and simplified visco-elastic approach are found to be capable of analysing the structure-foundation-soil system successfully while taking into account the elastic and visco-elastic behaviour of the elements of the system. (ii) Due to the visco-elastic behaviour of the soil, all the deformations, forces/ stresses, and moments in the structure-foundation-soil system are timedependent and approach their final values rapidly due to the settlement in early period of consolidation. (iii) The severe stresses are noted in outer columns of the frame during consoli dation period which may not be observed by a time-independent analysis. (iv) Simplified visco-elastic approach, suggested for ease in use, gives sub stantial saving in computational time and effort.