# Analysis of Structure Supported By Elastic Foundation – Foundation Models

## 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

## 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.

• 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