strain energy - beams - materials - engineering reference with worked examples

7/28/2019 Strain Energy - Beams - Materials - Engineering Reference With Worked Examples

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5/5/12 Strain Energy - Beams - Materials - Engineering Reference with Worked Examples

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Contents

1. Strain Energy Due To

Bending.

2. Application To Impact

3. Deflection By Calculus

4. Page Comments

Strain energy due to bending and deflection calculated using C alculus

Engineering Materials Beams

Strain Energy

Strain Energy Due To Bending.

Consider a short length of beam under the action of a Bending Moment M. If f is the Bending Stress on an element of the cross

sect ion of area at a distance y from the Neutral Axis, then the Strain energy of the length is given by:-

(1)

(2)

(3)

(4)

(5)

For the whole beam

(6)

The product EI is called the flexural Rigidity of the beam

Example 1

A simply supported beam of length l carries a concentrated load W at distances of a and b from the two ends. Find expressions

for the total strain energy of the beam and the deflection under load.

The integration for strain energy can only be applied over a length of beam for which a continuous expression for M can be

obtained. This usually implies a separate integration for each section between two concentrated loads or reactions.

For the section AB.

(7)

(8)

(9)

(10)

Similarly by taking a variable X measured from C

(11)

Total

(12)

(13)

But if is the deflection under the load, the strain energy must be equal to the work done by the load if it is gradually applied.

(14)

(15)


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For a Central Load

(16)

Hence

(17)

It should be noted that this method of finding deflection is limited to cases where only one concentrated load is applied ( i.e.

doing work)and then only gives the deflection under the load A. For a more general application of strain energy to deflection see

Castigliano's Theorem. This can be found under Engineering Materials Curved Beams.

Example 2

Compare the strain energy of a beam, simply supported at its ends and loaded with a uniformly distributed load, with that of the

same beam centrally loaded and having the same value of maximum bending moment. (U.L.)

If l is the span and EI the Flexural Rigidity, then for a uniformly distributed load w, the end reactions are and at a distance x

from one end:-

(18)

Using Equation (6)

(19)

(20)

(21)

(22)

Using Equation (13) fro a central load W

(23)

The Maximum Bending Stress and for a given beam depends upon the maximum Bending Moment. ( See Engineering

materials Shear Force and Bending Moments)

Equating maximum Bending Moments:-

(24)

(25)

Using Equations (22) and (23), the ratio

(26)

(27)

Using Equation (25)

(28)

Application To Impact

Example 3

A concentrated load W is gradually applied to a horizontal beam simply supported at its ends, produces a deflection y at the

load point. If this falls through a distance h onto the beam find an expression for the maximum deflection produced.

In a given beam, for a load W, y = 0.2 in. and the maximum stress is 4 tons/sq.in.. Find the greatest height from which a load of

0.1 W can be dropped without exceeding the elastic limit of 18 tons/sq.in. (U.L.)

The loss ofPotential Energy by the load = The gain in Strain Energy by the beam

i.e.


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(29)

where is the maximum deflection caused by dropping the load W onto the beam and P is the equivalent gradually applied

load which would produce the same deflection.

But a gradually applied load of W would produce a deflection of y and hence by proportion :-

(30)

or

(31)

Substituting in equation(29)

(32)

or

(33)

(34)

The Energy equation for dropping 0.1W is:-

(35)

But the equivalent gradually applied load and hence the deflection is proportional to the maximum stress i.e.

(36)

(37)

Substituting in equation (34)

(38)

(39)

Thus

(40)

Deflection By Calculus

In "Bending Stress" equation (3) it the general equation on bending was written. From this it can be seen that:-

(41)

And that in terms of the co-ordinates x and y

(42)

The sign depends upon the convention for axes. For beams met with in normal

engineering practice the slope is everywhere very small and may be neglected in

comparison to 1 in the denominator.

Taking y as positive upwards, under the action of a positive Bending Moment, the

curvature of the beam is shown in the diagram. It can be seen that dy/dx is

increasing as x increases.

Hence

(43)

or

(44)

Thus provided that M can be expressed as a function of x equation(43) can be integrated to give the slope dy/dx and the deflection y

can be found for any value of x. Two constants of integration will be involved and these can be found by substituting known values of

slope or deflection at particular points. A mathematical expression is thus obtained for the form of the deflected beam. ( Also known

as The Elastic Line


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Notes on Application

Take the X axis through the level of the supports.

Take the origin at one end of the beam or at a point of zero slope.

For built in or fixed end beams or when the deflection is a maximum. the slope dy/dx=0

For points on the X axis( Usually the supports) the deflection y = 0

For those working in Imperial Units.

It is convenient to use the following units

E in lb./sq.in. ( or tons/sq.in.

I in

y in in.

M in lb.ft (or tons-ft.)

x in ft.

After the integration, one side of the Equation has units and the other side has units . Hence in numerical

questions, the right hand side has to be multiplied by 144. After the second integration ELy has units and the corresponding

right-hand side must be multiplied by 1728

It is possible to differentiate equation (43) and in which case:-

(45)

(46)

(See the paragraph on the relationship between F M and w in Engineering Materials Shearing Force and Bending Moment.)

These forms are of use in some cases although generally the Bending Moment relationship is the most convenient.

Example 4

Obtain expressions for the maximum slope and deflection of a cantilever of length l carrying (a) a concentrate4d load W at its

free end and (b) a uniformly distributed load w along its whole length.

(a) If the origin is taken through the free end and the X axis through the fixed end then at a distance x from the origin:-

(47)

And using equation (43)

(48)

Integrating:-

(49)

but

(50)

(51)

Integrating again:-

(52)

At x = l y = 0

(53)

The slope and deflection at the free end ( Where they are at a maximum) are given by the values of dy/dx and y when x = 0

(54)

And the deflection (Note the negative sign indicating downwards)

b)

(55)

Integrating


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Page Comments

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(56)

When x = l dy/dx = 0

(57)

Integrating again:-

(58)

When x = l y = 0

(59)

Putting x = 0

The Maximum Deflection

The Maximum Deflection

strain energy - beams - materials - engineering reference with worked examples

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