上海交通大学考研专业课材料力学讲义11_energy_methods
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© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
11 - 3
x1
Third Edition
MECHANICS OF MATERIALS
Strain Energy Density
Beer • Johnston • DeWolf
• To eliminate the effects of size, evaluate the strainenergy per unit volume,
U V
x1 0
P dx A L
u x d strain energy density
0
1
• The total strain energy density resulting from the deformation is equal to the area under the curve to 1.
• Remainder of the energy spent in deforming the material is dissipated as heat.
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
11 - 4
Third Edition
• 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,
2 Y
2E
modulus of resilience
11 - 5
Third Edition
MECHANICS OF MATERIALS
Elastic Strain Energy for Normal Stresses
U dU dV V 0 V
Beer • Johnston • DeWolf
dU P dx elementary work
which is equal to the area of width dx under the loaddeformation diagram. • The total work done by the load for a deformation x1,
• The total strain energy is found from
U u dV
2 xy
2G
dV
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
11 - 8
Third Edition
MECHANICS OF MATERIALS
• In an element with a nonuniform stress distribution,
u lim U u dV total strain energy
• For values of u < uY , i.e., below the proportional limit,
Beer • Johnston • DeWolf
Sample Problem 11.4 Work and Energy Under Several Loads Castigliano’s Theorem Deflections by Castigliano’s Theorem Sample Problem 11.5
11 - 6
Third Edition
MECHANICS OF MATERIALS
Elastic Strain Energy for Normal Stresses
Beer • Johnston • DeWolf
• For a beam subjected to a bending load, 2 x M 2 y2
2 2 E1 1 u E1 d x 2 2E 0
1
• The strain energy density resulting from setting 1 Y is the modulus of resilience.
uY
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
U 2E dV 2 EI
2
dV
• Setting dV = dA dx,
x
My I
M2 2 U dA dx y dA dx 2 2 2 EI 2 EI A 0 A 0
L
M 2 y2
L
0
L
M2 dx 2 EI
• For an end-loaded cantilever beam,
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
• Integrate over the volume of the beam to find the strain energy. • Apply the particular given conditions to evaluate the strain energy.
MECHANICS OF MATERIALS
Strain-Energy Density
Beer • Johnston • DeWolf
• The strain energy density resulting from setting 1 R is the modulus of toughness.
U
2 x
2E
dV elastic strain energy
dV A dx
• Under axial loading, x P A
P2 U dx 2 AE
0 L
• For a rod of uniform cross-section,
P2L U 2 AE
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
11 - 9
Third Edition
MECHANICS OF MATERIALS
Sample Problem 11.2
SOLUTION:
Beer • Johnston • DeWolf
• Determine the reactions at A and B from a free-body diagram of the complete beam. • Develop a diagram of the bending moment distribution. 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.
U P dx total work strain energy
x1 0
which results in an increase of strain energy in the rod.
• In the case of a linear elastic deformation,
2 1 U kx dx 1 kx 2P 1 1x1 2 0
Beer • Johnston • DeWolf
• For a material subjected to plane shearing stresses,
xy
u
xy d xy
0
• For values of xy within the proportional limit,
2 xy 2 1 u1 G xy 2 2 xy xy 2G
Energy Methods
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
Third Edition
MECHANICS OF MATERIALS
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
M Px P2 x2 P 2 L3 U dx 2 EI 6 EI
0
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
11 - 7
L
Third Edition
MECHANICS OF MATERIALS
Strain Energy For Shearing Stresses
Third Edition
CHAPTER
11
MECHANICS OF MATERIALS
Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University
T2 2 U dA dx dA dx 2 2 2GJ 2GJ A 0A 0
L
T 2 2
L
ห้องสมุดไป่ตู้
xy
T J
T2 dx 2GJ
0
L
• In the case of a uniform shaft,
T 2L U 2GJ
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
11 - 2
Third Edition
MECHANICS OF MATERIALS
Strain Energy
Beer • Johnston • DeWolf
• 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
Strain Energy For Shearing Stresses
Beer • Johnston • DeWolf
• For a shaft subjected to a torsional load,
U
2 xy
2G
dV
T 2 2 2GJ
2
dV
• Setting dV = dA dx,
• 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.
11 - 3
x1
Third Edition
MECHANICS OF MATERIALS
Strain Energy Density
Beer • Johnston • DeWolf
• To eliminate the effects of size, evaluate the strainenergy per unit volume,
U V
x1 0
P dx A L
u x d strain energy density
0
1
• The total strain energy density resulting from the deformation is equal to the area under the curve to 1.
• Remainder of the energy spent in deforming the material is dissipated as heat.
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
11 - 4
Third Edition
• 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,
2 Y
2E
modulus of resilience
11 - 5
Third Edition
MECHANICS OF MATERIALS
Elastic Strain Energy for Normal Stresses
U dU dV V 0 V
Beer • Johnston • DeWolf
dU P dx elementary work
which is equal to the area of width dx under the loaddeformation diagram. • The total work done by the load for a deformation x1,
• The total strain energy is found from
U u dV
2 xy
2G
dV
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
11 - 8
Third Edition
MECHANICS OF MATERIALS
• In an element with a nonuniform stress distribution,
u lim U u dV total strain energy
• For values of u < uY , i.e., below the proportional limit,
Beer • Johnston • DeWolf
Sample Problem 11.4 Work and Energy Under Several Loads Castigliano’s Theorem Deflections by Castigliano’s Theorem Sample Problem 11.5
11 - 6
Third Edition
MECHANICS OF MATERIALS
Elastic Strain Energy for Normal Stresses
Beer • Johnston • DeWolf
• For a beam subjected to a bending load, 2 x M 2 y2
2 2 E1 1 u E1 d x 2 2E 0
1
• The strain energy density resulting from setting 1 Y is the modulus of resilience.
uY
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
U 2E dV 2 EI
2
dV
• Setting dV = dA dx,
x
My I
M2 2 U dA dx y dA dx 2 2 2 EI 2 EI A 0 A 0
L
M 2 y2
L
0
L
M2 dx 2 EI
• For an end-loaded cantilever beam,
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
• Integrate over the volume of the beam to find the strain energy. • Apply the particular given conditions to evaluate the strain energy.
MECHANICS OF MATERIALS
Strain-Energy Density
Beer • Johnston • DeWolf
• The strain energy density resulting from setting 1 R is the modulus of toughness.
U
2 x
2E
dV elastic strain energy
dV A dx
• Under axial loading, x P A
P2 U dx 2 AE
0 L
• For a rod of uniform cross-section,
P2L U 2 AE
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
11 - 9
Third Edition
MECHANICS OF MATERIALS
Sample Problem 11.2
SOLUTION:
Beer • Johnston • DeWolf
• Determine the reactions at A and B from a free-body diagram of the complete beam. • Develop a diagram of the bending moment distribution. 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.
U P dx total work strain energy
x1 0
which results in an increase of strain energy in the rod.
• In the case of a linear elastic deformation,
2 1 U kx dx 1 kx 2P 1 1x1 2 0
Beer • Johnston • DeWolf
• For a material subjected to plane shearing stresses,
xy
u
xy d xy
0
• For values of xy within the proportional limit,
2 xy 2 1 u1 G xy 2 2 xy xy 2G
Energy Methods
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
Third Edition
MECHANICS OF MATERIALS
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
M Px P2 x2 P 2 L3 U dx 2 EI 6 EI
0
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
11 - 7
L
Third Edition
MECHANICS OF MATERIALS
Strain Energy For Shearing Stresses
Third Edition
CHAPTER
11
MECHANICS OF MATERIALS
Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University
T2 2 U dA dx dA dx 2 2 2GJ 2GJ A 0A 0
L
T 2 2
L
ห้องสมุดไป่ตู้
xy
T J
T2 dx 2GJ
0
L
• In the case of a uniform shaft,
T 2L U 2GJ
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
11 - 2
Third Edition
MECHANICS OF MATERIALS
Strain Energy
Beer • Johnston • DeWolf
• 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
Strain Energy For Shearing Stresses
Beer • Johnston • DeWolf
• For a shaft subjected to a torsional load,
U
2 xy
2G
dV
T 2 2 2GJ
2
dV
• Setting dV = dA dx,
• 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.