# Methods of Soil Structure System Analysis

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In the direct method the soil, structure and foundation is modelled together using finite element method (FEM). The whole model is then analysed in a single step. The ground motion is specified as free field motion. This motion of the ground is applied at all boundaries. The soil domain with some material damping is limited by a fictitious exterior boundary. It is placed very far away from the structure that during the total earthquake excitation, the waves generated along the soil-structure interface does not reach it. The nodes along the soil-structure interface has denoted by subscript ‘f’ (foundation). The nodes of the structure has denoted by ‘st’. The nodes along the interior foundation medium/soil are denoted by ‘s’.

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In the above fig 1.6, the soil is modelled as an assemblage of rectangular plane strain elements. It is having two translational degrees of freedom at each node. The structure is modelled as an assemblage of beam elements. It is assumed that kinematic interaction is insignificant and the foundation block will move with free field ground motion. The inertia forces acting on the structure produces the vibration of structure, foundation and soil at the soil-structure interface and at the soil below it. The equation of motion in time domain can be written as:

Mü+Ců+ku=-M_st Iü_g

Where,

M = Mass matrix for the entire Structure, foundation and the soil

C = Damping matrix (Material) of the structure and the soil

k = Stiffness matrix of total system, which can be generated using standard assembling procedure.

Mst = Mass matrix having non-zero masses for the structural degree of freedom

I = Mass matrix having non-zero masses for the structural degree of freedom

ü_g = Free field ground acceleration (can be calculated by doing one simple one-dimensional analysis of site model, prior to the soil-structure analysis)

u = Vector of the relative displacement with respect to the base / foundation.

The right hand side of equation shows the inertia force. It tends to deform the soil at the soil-structure interface, when transferred to the base (foundation). Inertia force is applied in the form of shear force and moment. The material damping of soil contributes the response reduction of the structure soil on system. It is very insignificant and can be neglected. The deformation of soil due to inertia forces at the interface propagates in the form of radiation waves. Radiation waves are giving radiation damping which mostly affects the structure-soil foundation response. If the radiation damping will not die out or reflect back from the boundary, some error in the solution may introduce and the problem may become very large. The concept of absorbing boundaries is introduced in the FEM in order to reduce the size of the problem.

By using the direct method of analysis, like time domain method problem can also be solved in frequency domain method using Fourier transform function. It is for a specific free field ground motion. If the time histories of the ground motion are different at different supports, then problem can be solved by modifying the influence coefficient vector {I}, used in above equation.

The direct method is well suited for non-linear material laws of the soil to be taken into account. To solve the dynamic SSI problem by direct method, computer programs can be used. There are few shortcomings of the direct method of analysis; some of them are listed below.

– The good representation of damping matrix is difficult.

– If the superstructure is modeled as 3D system, the problem size becomes very large and the modeling of soil/foundation – structure interface becomes complex.

Sub-Structure Method

Sub-structure method is computationally more efficient than the direct method. By using this method most of the disadvantages of the direct method can be removed. In this method, initially the effective input motion is expressed in terms of free-field motions of the soil layer. In continuation to this step, the soil medium and the structure are represented as two independent mathematical models or substructures as shown in Fig 1.6.

The connection between them is provided by interaction forces of equal amplitude. It is acting in opposite directions of the two sub-structures. The total motions developed at the interface are the sum of the free-field motions at the interface of the soil. The total motions do not include the structure and the additional motions resulting from the interaction. The substructure method is advantageous as it allows to break down the complicated soil-structure system into more simplified form. It can be easily solved and checked without any ambiguity. The stiffness and damping properties of the soil are dependent on the frequency of the earthquake. It is most convenient to carry out the earthquake response analysis in the frequency domain, then to obtain the response history and transform it in the time domain.

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