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MECHANICS OF MATERIALS
Third Edition
Ferdinand P. Beer
E. Russell Johnston, Jr.
John T. DeWolf
Lecture Notes:
J. Walt Oler
Texas Tech University
CHAPTER
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
11Energy Methods
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALS
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Energy Methods
Strain Energy
Strain Energy Density
Elastic Strain Energy for Normal Stresses
Strain Energy For Shearing Stresses
Sample Problem 11.2
Strain Energy for a General State of Stress
Impact Loading
Example 11.06
Example 11.07
Design for Impact Loads
Work and Energy Under a Single Load
Deflection Under a Single Load
Sample Problem 11.4
Work and Energy Under Several Loads
Castigliano’s Theorem
Deflections by Castigliano’s Theorem
Sample Problem 11.5
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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• A uniform rod is subjected to a slowly increasing load
• The elementary work done by the load P as the rod elongates by a small dx is
which is equal to the area of width dx under the load-deformation diagram.
workelementarydxPdU
• The total work done by the load for a deformation x1,
which results in an increase of strain energy in the rod.
energystrainworktotaldxPUx
1
0
11212
121
0
1
xPkxdxkxUx
• In the case of a linear elastic deformation,
Strain Energy
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Strain Energy Density• To eliminate the effects of size, evaluate the strain-
energy per unit volume,
densityenergy straindu
L
dx
A
P
V
U
x
x
1
1
0
0
• As the material is unloaded, the stress returns to zero but there is a permanent deformation. Only the strain energy represented by the triangular area is recovered.
• Remainder of the energy spent in deforming the material is dissipated as heat.
• The total strain energy density resulting from the deformation is equal to the area under the curve to
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Strain-Energy Density
• The strain energy density resulting from setting R is the modulus of toughness.
• The energy per unit volume required to cause the material to rupture is related to its ductility as well as its ultimate strength.
• If the stress remains within the proportional limit,
E
EdEu x 22
21
21
01
1
• The strain energy density resulting from setting Y is the modulus of resilience.
resilience of modulusE
u YY
2
2
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Elastic Strain Energy for Normal Stresses
• In an element with a nonuniform stress distribution,
energystrain totallim0
dVuU
dV
dU
V
Uu
V
• For values of u < uY , i.e., below the proportional limit,
energy strainelasticdVE
U x 2
2
• Under axial loading, dxAdVAPx
L
dxAE
PU
0
2
2
AE
LPU
2
2
• For a rod of uniform cross-section,
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Elastic Strain Energy for Normal Stresses
I
yMx
• For a beam subjected to a bending load,
dVEI
yMdV
EU x
2
222
22
• Setting dV = dA dx,
dxEI
M
dxdAyEI
MdxdA
EI
yMU
L
L
A
L
A
0
2
0
22
2
02
22
2
22
• For an end-loaded cantilever beam,
EI
LPdx
EI
xPU
PxM
L
62
32
0
22
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Strain Energy For Shearing Stresses
• For a material subjected to plane shearing stresses,
xy
xyxy du
0
• For values of xy within the proportional limit,
GGu xy
xyxyxy 2
2
212
21
• The total strain energy is found from
dVG
dVuU
xy
2
2
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Strain Energy For Shearing Stresses
J
Txy
dVGJ
TdV
GU xy
2
222
22
• For a shaft subjected to a torsional load,
• Setting dV = dA dx,
L
L
A
L
A
dxGJ
T
dxdAGJ
TdxdA
GJ
TU
0
2
0
22
2
02
22
2
22
• In the case of a uniform shaft,
GJ
LTU
2
2
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 11.2
a) Taking into account only the normal stresses due to bending, determine the strain energy of the beam for the loading shown.
b) Evaluate the strain energy knowing that the beam is a W10x45, P = 40 kips, L = 12 ft, a = 3 ft, b = 9 ft, and E = 29x106 psi.
SOLUTION:
• Determine the reactions at A and B from a free-body diagram of the complete beam.
• Integrate over the volume of the beam to find the strain energy.
• Apply the particular given conditions to evaluate the strain energy.
• Develop a diagram of the bending moment distribution.
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 11.2
SOLUTION:
• Determine the reactions at A and B from a free-body diagram of the complete beam.
L
PaR
L
PbR BA
• Develop a diagram of the bending moment distribution.
vL
PaMx
L
PbM 21
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 11.2
vL
PaM
xL
PbM
2
1
BD,portion Over the
AD,portion Over the
43 in 248ksi1029
in. 108in. 36a
in. 144kips45
IE
b
LP
• Integrate over the volume of the beam to find the strain energy.
baEIL
baPbaab
L
P
EI
dxxL
Pa
EIdxx
L
Pb
EI
dvEI
Mdx
EI
MU
ba
ba
2
2223232
2
2
0
2
0
2
0
22
0
21
6332
1
2
1
2
1
22
EIL
baPU
6
222
in 144in 248ksi 10296
in 108in 36kips4043
222
U
kipsin 89.3 U
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Strain Energy for a General State of Stress
• Previously found strain energy due to uniaxial stress and plane shearing stress. For a general state of stress,
zxzxyzyzxyxyzzyyxxu 21
• With respect to the principal axes for an elastic, isotropic body,
distortion todue 12
1
change volume todue 6
21
22
1
222
2
222
accbbad
cbav
dv
accbbacba
Gu
E
vu
uu
Eu
• Basis for the maximum distortion energy failure criteria,
specimen test tensileafor 6
2
Guu Y
Ydd
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Impact Loading
• Consider a rod which is hit at its end with a body of mass m moving with a velocity v0.
• Rod deforms under impact. Stresses reach a maximum value m and then disappear.
• To determine the maximum stress m
- Assume that the kinetic energy is transferred entirely to the structure,
202
1 mvUm
- Assume that the stress-strain diagram obtained from a static test is also valid under impact loading.
dVE
U mm 2
2
• Maximum value of the strain energy,
• For the case of a uniform rod,
V
Emv
V
EUmm
202
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Example 11.06
Body of mass m with velocity v0 hits the end of the nonuniform rod BCD. Knowing that the diameter of the portion BC is twice the diameter of portion CD, determine the maximum value of the normal stress in the rod.
SOLUTION:
• Due to the change in diameter, the normal stress distribution is nonuniform.
• Find the static load Pm which produces the same strain energy as the impact.
• Evaluate the maximum stress resulting from the static load Pm
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Example 11.06
SOLUTION:
• Due to the change in diameter, the normal stress distribution is nonuniform.
E
VdV
E
mvU
mm
m
22
22
202
1
• Find the static load Pm which produces the same strain energy as the impact.
L
AEUP
AE
LP
AE
LP
AE
LPU
mm
mmmm
5
16
16
5
4
22 222
• Evaluate the maximum stress resulting from the static load Pm
AL
Emv
AL
EU
A
P
m
mm
20
5
8
5
16
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Example 11.07
A block of weight W is dropped from a height h onto the free end of the cantilever beam. Determine the maximum value of the stresses in the beam.
SOLUTION:
• The normal stress varies linearly along the length of the beam as across a transverse section.
• Find the static load Pm which produces the same strain energy as the impact.
• Evaluate the maximum stress resulting from the static load Pm
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Example 11.07
SOLUTION:
• The normal stress varies linearly along the length of the beam as across a transverse section.
E
VdV
E
WhU
mm
m
22
22
• Find the static load Pm which produces the same strain energy as the impact.
For an end-loaded cantilever beam,
3
32
6
6
L
EIUP
EI
LPU
mm
mm
• Evaluate the maximum stress resulting from the static load Pm
2266
cIL
WhE
cIL
EU
I
LcP
I
cM
m
mmm
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Design for Impact Loads• For the case of a uniform rod,
V
EUmm
2
V
EU
VLcccLcIL
cIL
EU
mm
mm
24
//
6
412
4124
412
2
• For the case of the cantilever beam
Maximum stress reduced by:
• uniformity of stress• low modulus of elasticity with
high yield strength• high volume
• For the case of the nonuniform rod,
V
EU
ALLALAV
AL
EU
mm
mm
8
2/52/2/4
5
16
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Work and Energy Under a Single Load
• Previously, we found the strain energy by integrating the energy density over the volume. For a uniform rod,
AE
LPdxA
E
AP
dVE
dVuU
L
22
2
21
0
21
2
• Strain energy may also be found from the work of the single load P1,
1
0
x
dxPU
• For an elastic deformation,
11212
121
00
11
xPxkdxkxdxPUxx
• Knowing the relationship between force and displacement,
AE
LP
AE
LPPU
AE
LPx
2
211
121
11
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Work and Energy Under a Single Load
• Strain energy may be found from the work of other types of single concentrated loads.
EI
LP
EI
LPP
yPdyPUy
63
321
31
121
1121
0
1
• Transverse load
EI
LM
EI
LMM
MdMU
2
211
121
1121
0
1
• Bending couple
JG
LT
JG
LTT
TdTU
2
211
121
1121
0
1
• Torsional couple
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Deflection Under a Single Load• If the strain energy of a structure due to a
single concentrated load is known, then the equality between the work of the load and energy may be used to find the deflection.
lLlL BDBC 8.06.0
From statics,PFPF BDBC 8.06.0
From the given geometry,
• Strain energy of the structure,
AE
lP
AE
lP
AE
LF
AE
LFU BDBDBCBC
2332
22
364.02
8.06.0
22
• Equating work and strain energy,
AE
Ply
yPAE
LPU
B
B
728.0
364.021
2
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 11.4
Members of the truss shown consist of sections of aluminum pipe with the cross-sectional areas indicated. Using E = 73 GPa, determine the vertical deflection of the point E caused by the load P.
SOLUTION:
• Find the reactions at A and B from a free-body diagram of the entire truss.
• Apply the method of joints to determine the axial force in each member.
• Evaluate the strain energy of the truss due to the load P.
• Equate the strain energy to the work of P and solve for the displacement.
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 11.4SOLUTION:
• Find the reactions at A and B from a free-body diagram of the entire truss.
821821 PBPAPA yx
• Apply the method of joints to determine the axial force in each member.
PF
PF
CE
DE
815
817
0
815
CD
AC
F
PF
PF
PF
CE
DE
821
45
0ABF
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Sample Problem 11.4
• Evaluate the strain energy of the truss due to the load P.
2
22
297002
1
2
1
2
PE
A
LF
EEA
LFU
i
ii
i
ii
• Equate the strain energy to the work by P and solve for the displacement.
9
33
2
21
1073
1040107.29
2
2970022
E
E
E
y
E
P
PP
Uy
UPy
mm27.16Ey
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Work and Energy Under Several Loads• Deflections of an elastic beam subjected to
two concentrated loads,
22212122212
21211112111
PPxxx
PPxxx
• Reversing the application sequence yields
21111221
22222
1 2 PPPPU
• Strain energy expressions must be equivalent. It follows that (Maxwell’s reciprocal theorem).
22222112
21112
1 2 PPPPU
• Compute the strain energy in the beam by evaluating the work done by slowly applying P1 followed by P2,
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Castigliano’s Theorem
22222112
21112
1 2 PPPPU
• Strain energy for any elastic structure subjected to two concentrated loads,
• Differentiating with respect to the loads,
22221122
12121111
xPPP
U
xPPP
U
• Castigliano’s theorem: For an elastic structure subjected to n loads, the deflection xj of the point of application of Pj can be expressed as
and j
jj
jj
j T
U
M
U
P
Ux
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Deflections by Castigliano’s Theorem
• Application of Castigliano’s theorem is simplified if the differentiation with respect to the load Pj is performed before the integration or summation to obtain the strain energy U.
• In the case of a beam,
L
jjj
L
dxP
M
EI
M
P
Uxdx
EI
MU
00
2
2
• For a truss,
j
in
i i
ii
jj
n
i i
iiP
F
EA
LF
P
Ux
EA
LFU
11
2
2
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
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Sample Problem 11.5
Members of the truss shown consist of sections of aluminum pipe with the cross-sectional areas indicated. Using E = 73 GPa, determine the vertical deflection of the joint C caused by the load P.
• Apply the method of joints to determine the axial force in each member due to Q.
• Combine with the results of Sample Problem 11.4 to evaluate the derivative with respect to Q of the strain energy of the truss due to the loads P and Q.
• Setting Q = 0, evaluate the derivative which is equivalent to the desired displacement at C.
SOLUTION:
• For application of Castigliano’s theorem, introduce a dummy vertical load Q at C. Find the reactions at A and B due to the dummy load from a free-body diagram of the entire truss.
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Sample Problem 11.5
SOLUTION:
• Find the reactions at A and B due to a dummy load Q at C from a free-body diagram of the entire truss.
QBQAQA yx 43
43
• Apply the method of joints to determine the axial force in each member due to Q.
QFF
QFF
FF
BDAB
CDAC
DECE
43;0
;0
0
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Sample Problem 11.5
• Combine with the results of Sample Problem 11.4 to evaluate the derivative with respect to Q of the strain energy of the truss due to the loads P and Q.
QPEQ
F
EA
LFy i
i
iiC 42634306
1
• Setting Q = 0, evaluate the derivative which is equivalent to the desired displacement at C.
Pa1073
104043069
3
NyC mm 36.2Cy