• 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 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.
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,
• 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,
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
T2 2 U dA dx dA dx 2 2 2GJ 2GJ A 0A 0
L
T 2 2
L
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.
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
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,
• 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
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
MECHANICS OF MATERIALS
Strain-Energy Density
Beer • Johnston • DeWolf
• The strain energy density resulting from setting 1 R is the modulus of toughness.
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
© 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
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.
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
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.
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
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
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
11 - 3
x1
Third Edition