7 - Energy Method
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Transcript of 7 - Energy Method
7/23/2019 7 - Energy Method
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Mechanical & Aerospace Engineering
West Virginia University
Energy Method for Beam Deflection- Castigliano’s heorem
7/23/2019 7 - Energy Method
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Mechanical & Aerospace Engineering
West Virginia University
Differential equations of the deflection curve
• Prismatic beams
BENDING – MOMENTEQUATION SEA! – "O!#EEQUATION $OAD EQUATION
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Mechanical & Aerospace Engineering
West Virginia University
Castigiliano’s heorem
he strain Energy in Beam
EI
L P U
6
32
=
a!e the derivative of "train Energy
#ith respect to the load $
EI
PL
EI
L P
dP
d
dP
dU
3)
6(
332
==
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Mechanical & Aerospace Engineering
West Virginia University
Castigiliano’s heorem
The partial derivative of the strain energy
of a structure with respect to any load to
any load is equal to the displacement
corresponding to that load %
i
i
P
U
∂
∂=δ
Load and corresponding displacement are
sed in a generali'ed sense% hey can (e)
•*orce vs% ranslation•Cople vs% rotation•+ther pairs
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Mechanical & Aerospace Engineering
West Virginia University
Application of Castigiliano’s heorem
o find deflection ,
A and angle of rotation ,
A at point A)
EI
L M
EI
PL
P
U A
23
2
0
3
+=∂
∂=δ
0
2 22 2 32 0 0
00 0
(0 )
1
( )2 2 6 2 2
L L
M Px M x L
PM L M L M dx P L
U Px M dx EI EI EI EI EI
= − − ≤ ≤
= = − − = + +∫ ∫ Therefore,
EI
L M
EI
PL
M
U A
0
2
0 2
+=∂
∂=θ
and
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Mechanical & Aerospace Engineering
West Virginia University
Dmmy-.oad Method
o find deflection ,
C at center point C/ assme a dmmy-load 0 at C)
EI
QL
EI
L M
EI
PL
Q
U
U U U
EI
LQ
EI
LQM
EI
PQL
EI
L M
EI
L PM
EI
L P
EI
dx M U
EI
L M
EI
L PM
EI
L P dx M Px
EI EI
dx M U
C
CB AC
L
LCB
L L
AC
24848
5
48848
5
48
3
48
7
2
4848)(
2
1
2
32
0
3
322
0
32
0
2
0
32
2/
2
20
20
322/
0
2
0
2/
0
2
++=∂
∂=
+=
+++++==
++=−−==
∫
∫ ∫
δ
)2/()2/(
)2/0(
0
0
L x L L xQ M Px M
L x M Px M
≤≤−−−−=
≤≤−−=
hen
When 0 1 2 EI
L M
EI
PLC
848
5 2
0
3
+=δ
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Mechanical & Aerospace Engineering
West Virginia University
Modified Castigiliano’s heorem
We can differentiate the integral (y differentiating nder the integral
sign)
∫ ∂
∂=
∂
∂=
EI
dx M
P P
U
ii
i2
2
δ
∫ ∫ ∂
∂
=∂
∂
= dx P
M
EI
M
EI
dx M
P iii ))((2
2
δ
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Mechanical & Aerospace Engineering
West Virginia University
E3ample: A simple beam AB supports a uniform load of intensity q = 1.5
k/ft and a concentrated load P = 5k. Te load P acts at te midpoint ! of
te beam. Te beam as len"t # = $.% ft& modulus of elasticity ' =
(%×1%) psi& and moment of inertia * = +5.% in,.
-etermine te donard deflection at midpoint of te beam by usin" 10
!asti"ilianos teorem2 and 30 modified form of !astili"ilianos teorem.
