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Chapter 9Defections using Energy Methods
Reading Assignment:
Chapters 9.1 - 9.5
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Externa !or"
#sed $or more compicated oadingsor $or structures such as
trusses and
$rames.
Conser%ation o$ Energy &rincipe
#e' #i
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Externa !or" (
)orce*Moment
- +inear Eastic Response
2
2
2
00
0
=
=
==
=
=
PU
xPxdx
PFdxU
xP
F
FdxU
e
XX
e
X
e
MU
MdU
e
e
2
1
0
=
=
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Strain Energy Axial Force
,oo"es +a / ' E0
orma 2tress / ' *A
A'Constant Cross-2ectiona Area
)ina 2train 0'3*+
+'+ength
3 ' +*AE
#i' 4+*4AE
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2train Energy ( ending
2train Energy
d#i'M4dx*4E6 2train Energy o%er eam +ength
=
L
e
EI
dxMU
0
2
2
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Exampe 71
8russ Exampe
Rea !or" Method
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&rincipe o$ !or" and Energy
Appication o$ this method is imitedny one oad may e appied to
the structureny the dispacement under the $orce can eotained.
( ) =
==
L
i
EI
LP
EI
dxPx
EI
dxMU
0
3222
6
1
22
2
= PUe
EI
PL
EI
LPPUU
ie362
332
==
=
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&rincipe o$ ;irtua !or"
Aso "non as the c ocation on astructure.
2ee )igure 9-? in textoo".
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&rincipe o$ ;irtua !or"
Externa ;irtua !or" &3
6nterna ;irtua !or"
@ud+ @&3 ' @uB
& 3 ' @ud+
Moment & ' @uC d+
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Method o$ ;irtua !or":8russes
Externa +oading
8emperature
)arication Error and Camer
=AE
nNLP'
= TLnP '
= LnP'
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Method o$ ;irtua !or"&rocedure
1 &ace unit oad on the truss at the oint herethe desired
dispacement is to e determinedand remo%e a rea oads.
8E: +oad shoud e in the same direction o$
dispacement.4 Cacuate the interna $orces in each truss
memer. FG ' tensionH - ' compression
I Remo%e unit oad and appy rea oads to truss.
J Cacuate the interna $orces in each trussmemer. FG ' tensionH -
' compression
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Method o$ ;irtua !or" 6mportantotes
1 Kou must retain the ageraic sign $or each o$ the n and %aues
hen pacing them into the %irtua or" eLuation.
4 6$ the %irtua or" eLuation is positi%eH the dispacementis in
the same direction as the unit oad. 6$ it is negati%eit is in the
opposite direction.
I )or 8emperature ELuation:1 38 ' &ositi%e ith temperature
increaseH egati%e ith
temperature decrease.
J )or )arication ELuation:
1 3+ ' &ositi%e ith increase in engthH egati%e ith
decreasein ength.
5 MAE 2#RE K# &AK A88E86 8 K#R #682.
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Exampe 74
8russ Exampe
;irtua !or" Method
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Exampe 7I
8russ Exampe
;irtua !or" Method
