Ch05 Lecture (C) Copy

8/10/2019 Ch05 Lecture (C) Copy

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Determining Elastic Deflections Castiglianos Method

Work = Energy = Fs and Work = Energy = T

Force Fis gradually applied to an elastic body

The force does work as the body deforms

This work can be calculated from

is the work- absorbing displacement of the point ofapplication of F

is the displacement component in the direction of F

1

0

dFU


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Castiglianos Method

2

If the stress is below the proportional limit, then Fis

proportional to , as shown

then

FdFU2

1

0


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Castiglianos Method

3

Elastic load deection curve for the completely

general case The load Qcan be anyforce or moment The displacement being the corresponding

linear or angular displacement

Stored ElasticEnergy

ComplementaryEnergy

dQUddUQUU and

2

dQ

dU


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Castiglianos Theorem

The deection in the direction of Qand at thepoint where Qis applied is found by taing thederivative while all other loads are held constant!

When a body is elastically defected by anycombination o loads, the defection at any point

and in any direction is equal to the partialderivative o strain energy (computed with all

loads acting) with respect to a load located atthat point and acting in that direction.

Q

U


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Load Types Q E!i"alent E!i"alent

#$ial %orce &

'ending M

Torsion T

(hear )

!

P

U

M

U

T

U

V

U

#pplication of Castiglianos Theorem

"ote# $or %&'T problems, the e(ects of'hear will be negligible

Q

U


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"

Deflection %orm!las&age *+,


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#

Consider the #$ial Load

$%eflection&'ial$%eflection&'ialinEnergy 2(1(AE

PLPUQU axial

2

)

2

)Elasticityof*odulus+arying,ossibly

section)-cross.arying

//engthof0ar

$onubstituti (3( dxAE

PU

LE

AL!AE

LPU

L

0

22

2*+

2


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Table ./ 0p. 1213(!mmary of Energy and Deflection E!ations for 4se with Castigliano5s

Method.


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6ow to 4se Castiglianos Theorem

1( btain the proper e'pression for all componentsof energy

4se the e5uations for Uin Table ./

2( Taking the appropriate partial deri+ati+e to

obtain deflection

or

3( 4se the techni5ue of differentiating under the

integral sign) represented by the deflection

e5uations in the final column of Table !(3(

6

Q

U


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17


In general, the strain energy of a bar subected to combinedloading is obtained by superimposing the contributions of a-ialloading, torsion, and bending#

The deflection i . / $i is best evaluated by differentiating

inside the integral signs before integrating! This procedure ispermissible because $i is not a function of -!

1-ialoadType#

Torsion

3ending


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11


)! If no load acts at the point where thedeection is desired 1 dummy load in the direction of the

desired deection must be added at thatpoint

2! 1fter di(erentiating but before integrating

'et the dummy load e4ual to 5ero+this avoids integration of terms that will eventually be set

e4ual to 5ero*

6! 7enote the dummy load by 8 The displacement in the direction of 8 thus

is


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12


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13

6andbook 7nformation for Comparison


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1

Q

U

3ending%oment

Transverse 'hear

$or thisproblem, Q

is aconcentrate

d load, P


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1!


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1"


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1#


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1


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16


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27


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21


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22


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$or thisproblem, Q

is a

concentrated load, P23


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2

8eometry of the (ample &roblem


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2!

P = F cos

M = F (R-Rcos ) = FR(1 -cos )

= F sin


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2"

$or thisproblem, Q

is a

concentrated load, F


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2#

!o to

"a#le $%&


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2


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26

!o to"a#le $%&


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37


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9ed!ndant 9eactions by Castiglianos Method

31

8edundant 8eaction & supporting force or moment that is not necessary for

e5uilibrium

&s the magnitude of a redundant reaction is +aried) deflections

change but e5uilibrium remains

9astigliano:s Theorem The deflection associated with any reaction ;or applied load

Ch05 Lecture (C) Copy

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