Elastic Foundations - Beams - Materials - Engineering Reference With Worked Examples

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.

Elastic force is a force arising from the

deformation of a solid body which

depends only on the body's instantaneous

deformation and not on its previous

history, and which is conservative.

The deflection of beams where the foundations are not rigid.

Engineering › Materials › Beams ›

Contents

Beams On Elastic

Foundations.

1.

(a) Long Beam Carrying A

Central Load.

2.

(b) Short Beam Carrying A

Central Load W

3.

(c) An Infinite Beam Carrying

A Load.

4.

Page Comments5.

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 - .

Using the theory shown in "Bending of Beams Part 1"

Or

(2)

where

Three standard are now considered:

(a) Long Beam Carrying A Central Load.This example assumes that the foundations can only exert an upward force. The length of beam in contact with the foundations is and

the Origin is at the left-hand end.

A solution to can be written as:-

At

And

Also

Giving:

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Deflection is a term that is used to

describe the degree to which a structural

element is displaced under a load.

Thus,

The equation is now reduced to:

At

The least solution of this is which determines the length in contact with the ground. The value of the constant is obtained

from the condition that the Shear Force at the centre is since by symmetry it must be numerically the same on either side of the

load and it must change by an amount on passing through the load.

Hence,

Or

The maximum Deflection and Bending Moment are at the centre,

(b) Short Beam Carrying A Central Load W

If in case (a) above, the Beam will sink below the unstressed level of the foundations at all points. Again taking the origin at the

left hand end and the overall length of the Beam as , then the following conditions are obtained for the constants of integration of the

general solution shown in the previous paragraph.

At

And And

At

And

Thus

Solving for and

The complete solution for is now known, the maximum deflection and Bending Moment occur under the load.

(c) An Infinite Beam Carrying A Load.Assuming that the Support can exert pressure either upwards or downwards and taking the axis through the load and the axis at the

undeformed level, a solution to equation (2) can be written in the form:


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Example: 1

Last Modified: 4 Oct 11 @ 14:14 Page Rendered: 2013-06-16 16:55:44

For the length to the right of since as

At

And

Giving

And

The distance from the load at which is given by:

(3)

And the least solution is

The maximum deflection and Bending Moment are at

[imperial]Example - Example 1

Problem

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


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Elastic Foundations - Beams - Materials - Engineering Reference With Worked Examples

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