## Analysis of Structure Supported By Elastic Foundation

### Soil Structure Interaction

- Soil is a very complex material for the modeling.
- It is very difficult to model the soil-structure interaction problem.
- In RCC buildings slab on grade is a very common construction system e.g. mat footing
- Very heavy slab loads occur in these structures.

- For safe and economical design, compute plate displacement and stresses accurately.
- Difficult to obtain samples for testing producing results in accordance with ground behavior.
- Necessary to make simplifying assumptions.

**Scope Of Study**

- To develop a workable approach for analysis of plates on elastic foundations.
- Structural Engineers go for simplified assumptions of rigid foundation
- STAAD Pro is used to incorporate the elasticity of soil that will provide approximate solutions as close to the exact solutions.

**Types of Foundation Models**

- The plate-foundation system is idealized as a thin elastic plate resting on a linearly elastic foundation.
- Various foundation models were given by the investigators which are discussed ahead.

## WINKLER MODEL

- Winkler first studied beam on elastic springs
- Model based on the pure bending beam theory.

p = Kw

Here, w = vertical translations of the soil, p = contact pressure, K = modulus of subgrade reaction

- Plates based on Winkler model involve fourth order differential equation:

D ▼4 w+ Kw = q

Here D is the plate flexural rigidity, q is the pressure on the plate and▼ is the Laplace operator.

- The deformations outside the loaded area were neglected and taken as zero.
- Winkler foundation model has two major limitations:
- No interaction between springs is considered.
- The spring constant may depend on a number of parameters, such as stiffness of beam, geometry of beam, soil profile, and behavior.

## Filonenko Borodich Model

- Top ends of springs connected to a elastic membrane stretched to constant tension T.
- It was done to achieve some degree of interaction between the spring elements,
- Modulus of subgrade reaction is given by

**p = Kw – T ▼2 w**

## HETENYI MODEL

- Embedded a plate in the three-dimensional case in the material of the Winkler foundation to accomplish interaction among springs.
- Assumed that the plate deforms in bending only.

p = Kw + D▼2 ▼2 w

Here, p = load, w = vertical translation, D = flexural rigidity of plate.

## PASTERNAK FOUNDATION MODEL

- Pasternak assumed shear interactions between spring elements.
- Connecting the ends of springs to a beam or plate consisting of incompressible vertical elements, which can deform only by transverse shear.

p = Kw – G ▼2 w

## TIMOSHENKO MODEL

- This model is based on Timoshenko beam theory
- Plane sections still remain plane after bending but are no longer normal to the longitudinal axis.
- This model considers both the bending and shear deformations.

### Modulus Of Subgrade Reaction

- Pressure sustained per unit deformation of subgrade at specified deformation or pressure level.
- Calculated from plate load test from the plot of q versus δ

K = q/δ

Here , q = mean bearing pressure, K = modulus of subgrade reaction, δ = mean settlement

For Analytical Study refer PPT