Beams On Elastic Foundations.There are many problems in which a beam is supported on a compressible

foundation which exerts a distributive reaction on the Beam of intensity

proportional to the compressibility.

In some cases the foundations can only exert upward forces and the beam may,

if it is sufficiently long, lose contact with the foundation. In other cases pressure

may be exerted either way. Again the support may not be truly continuous( eg.

the support of railway lines) but can be replaced by an equivalent distributed

support.

If is the upward deflection of the foundation at any point, then the rate of upward

reaction is - . A beam is a horizontal structural element that is capable of withstanding load primarily

by resisting bending. The bending force induced into the material of the beam as a result

of the external loads, own weight, span and external reactions to these loads is called a

bending moment.Using the theory shown in "Bending of Beams Part 1"

Or

(1)

where

Three standard are now considered:

Example - Example 1Problem

A steel railway track is supported on timber sleepers which exert an equivalent

load of 400lb./in. length of rail per inch deflection from its unloaded position. For

each rail and .

If a point load of 10 tons acts on each rail, find the length of rail over which the

sleepers are depressed and the maximum Bending Stress in the rail. Workings

Each rail can be considered to be an infinitely long beam for which the length

over which the downward deflection occurs is given by equation (2)

And the Maximum Bending Moment is given by:

And the maximum Stress is:

Solution

← The length is

← The maximum Bending Stress in the rail is