This document provides a summary of key concepts relating to stress, strain, and Hooke's law:
1) It defines stress and strain, describing normal stress, shear stress, normal strain, and shear strain.
2) It explains that stress and strain are second-order tensors with six independent components that define their state at a point, and how their values depend on the coordinate system used.
3) It describes plane stress and plane strain conditions, as well as transformations between coordinate systems.
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Review of Stress, Strain, and Hooke's Law Concepts
1. Review of the Concepts of Stress,
Strain, and Hooke's Law
by:
Prof. Mark E. Tuttle
Dept. Mechanical Engineering
Box 352600
University of Washington
Seattle, WA 98195
1
2. Prof. M. E. Tuttle
University of Washington
Force Vectors
• A force, F , is a vector (also called a "1st-order tensor")
• The description of any vector (or any tensor) depends on
the coordinate system used to describe the vector
100 lbf 100 lbf 100 lbf
+y
+x"
+x'
+y'
45 degs +y" 60 deg
+x
F = 70.7 lbf (x'-direction) F = 86.6 lbf (x"-direction)
F = 100 lbf
+ +
(y-direction)
70.7 lbf (y'-direction) 50 lbf (y"-direction) 2
3. Prof. M. E. Tuttle
University of Washington
Normal and Shear Forces
• A "normal" force acts perpindicular to a surface
• A "shear" force acts tangent to a surface
P = Normal Force
V = Shear Force
3
4. Prof. M. E. Tuttle
University of Washington
Forces Inclined to a Plane
• Since forces are vectors, a force inclined to a plane can always
be described as a combination of normal and shear forces
Inclined Force
P = Normal Force
V = Shear Force
4
5. Prof. M. E. Tuttle
University of Washington
Moments
• A moment (also called a "torque" or a "couple") is a force
which tends to cause rotation of a rigid body
• A moment is also vectoral quantity (i.e., a 1st-order tensor)...
M
M
5
6. Prof. M. E. Tuttle
University of Washington
Static Equilibrium
• A rigid solid body is in "static equilibrium" if it is either
- at rest, or
- moves with a constant velocity
• Static equilibrium exists if: Σ F = 0 and Σ M = 0
50 lbf 50 lbf
+y
(40 lbf) (40 lbf)
(30 lbf) +x (30 lbf)
60 lbf
(30 lbf) (BALL ACCELERATES) (30 lbf) (NO ACCELERATION)
(40 lbf) (40 lbf)
50 lbf 50 lbf 6
7. Prof. M. E. Tuttle
University of Washington
Free Body Diagrams
and Internal Forces
• An imaginary "cut" is made at plane of interest
• Apply Σ F = 0 and Σ M = 0 to either half to determine
internal forces
F F
R (= F)
"cut"
(or)
R (= F)
F F 7
8. Prof. M. E. Tuttle
University of Washington
Free Body Diagrams
and Internal Forces
• The imaginary cut can be made along an arbitrary plane
• Internal force R can be decomposed to determine the normal
and shear forces acting on the arbitrary plane
F
R (= F) P
"cut" V
F F F 8
9. Prof. M. E. Tuttle
University of Washington
Stress:
Fundamental Definitions
• Two "types" of stress:
normal stress = σ = P/A
shear stress = τ = V/A
where P and V must be uniformly distributed over A
P = Normal Force σ = P/A
V = Shear Force
τ = V/A
9
A = Cross-Sectional Area
10. Prof. M. E. Tuttle
University of Washington
Distribution of Internal Forces
• Forces are distributed over the internal plane...they may
or may not be uniformly distributed
F M
σ = F/A σ= ?
"cut" "cut"
F F M M 10
11. Prof. M. E. Tuttle
University of Washington
Infinitesimal Elements
• A free-body diagram of an "infinitesimal element" is used to
define "stress at a point"
• Forces can be considered "uniform" over the infinitesimally
small elemental surfaces
+x
+z
+y
dx
dy
dz 11
12. Prof. M. E. Tuttle
University of Washington
Labeling Stress Components
• Two subscripts are used to identify a stress component, e.g.,
"σxx" or "τxy" (note: for convenience, we sometimes write
σx = σxx, or σxy = τxy)
• 1st subscript: identifies element face
• 2nd subscript: identifies "direction" of stress
τ
xy
+y
σ xx +x
σ
xx
12
13. Prof. M. E. Tuttle
University of Washington
Admissable Shear
Stress States
τyx
+y +y +y
+x
τxy τxy +x
τxy τxy +x
τxy
τyx
ΣF = 0 ΣF = 0 If: τyx = τxy
ΣM = 0 ΣM = 0 ΣF = 0
(inadmissable) (inadmissable) ΣM = 0
(admissable)
13
14. Stress Sign Conventions
• The "algebraic sign" of a cube face is positive if the outward
unit normal of the face "points" in a positive coordinate
direction
• A stress component is positive if:
• stress component acts on a positive face and "points" in
a positive coordinate direction, or
• stress component acts on a negative face and "points"
in a negative coordinate direction.
14
16. Prof. M. E. Tuttle
University of Washington
3-Dimensional Stress States
• In the most general case, six independent components of
stress exist "at a point"
+x
F1 +z σxx
F2 +y
M1 τxz
τzy
τxy
σzz
F3 σyy
F4
M2 16
17. Prof. M. E. Tuttle
University of Washington
Plane Stress
• If all non-zero stress components exist in a single plane, the
(3-D) state of stress is called "plane stress"
+y
+y
σyy σyy
+x τxy
+z
τxy
σxx
σxx
σxx σxx
+x
σyy
σyy 17
18. Prof. M. E. Tuttle
University of Washington
Uniaxial Stress
• If only one normal stress exists, the (3-D) state of stress is
called a "uniaxial stress"
+y
+y
+x
σyy σyy
+z
+x
σyy
σyy
16
19. Prof. M. E. Tuttle
University of Washington
Free Body Diagram Defines
the Coordinate System
F
+y +y
F σyy
"cut"
+x +x
F F F
17
20. Prof. M. E. Tuttle
University of Washington
Free Body Diagram Defines
the Coordinate System
F
y' y'
σy'y'
P V x' τx'y' x'
"cut"
F F F
18
21. Prof. M. E. Tuttle
University of Washington
Free Body Diagram Defines
the Coordinate System
F
"cut" y" y"
(a
plane) Py"y" σy"y"
x" x"
Vy"x" τy"x"
Vy"z" τy"z"
z" z"
F F F
21
22. Prof. M. E. Tuttle
University of Washington
Stress Transformations
Within a Plane
• Given stress components in the x-y coordinate system (σxx,
σyy, τxy), what are the corresponding stress components in
the x'-y' coordinate system?
σyy
+y σy'y' τx'y'
τxy
+y' σx'x'
? +x'
σxx
θ
+x
20
23. Prof. M. E. Tuttle
University of Washington
Stress Transformations
• Stress components in the x'-y' coordinate system may be
related to stresses in the x-y coordinate system using a free
body diagram and enforcing Σ F = 0
Σ Fx' = 0
+y σyy +y' +x'
τxy τx'y' θ
σxx σxx σx'x'
+x τxy
"cut" σyy
23
24. Prof. M. E. Tuttle
University of Washington
Stress Transformation
Equations
• By enforcing ΣFx' = 0, ΣFy' = 0, it can be shown:
σ xx + σ yy σ xx − σ yy
σ x' x' = + cos 2θ + τ xy sin2θ
2 2
σ xx + σ yy σ xx − σ yy
σ y' y' = − cos2θ − τ xy sin2θ
2 2
σ xx − σ yy
τ x'y' = − sin 2θ + τ xy cos 2θ
2
22
25. Prof. M. E. Tuttle
University of Washington
Use of Mohr's Circle of
Stress
σyy
+y
τxy
σxx τ
(σyy,τ xy) (σx'x' ,−τx'y' )
+x
2θ 2θ
(Equivalent)
σ
σy'y' τx'y' (σxx,−τxy)
(σy'y' ,τ x'y' )
+y' σx'x'
+x'
θ
25
26. Prof. M. E. Tuttle
University of Washington
Principal Stresses
• In the principal stress coordinate system the shear stress is
zero...
σyy
+y
τxy
τ
σxx
(σyy,τ xy)
+x
2θ'
(Equivalent) σ22 2θ' σ11 σ
σ22
+2
σ11 (σxx,−τxy)
+1
θ'
26
27. Prof. M. E. Tuttle
University of Washington
"Stress":
Summary of Key Points
• Normal and shear stresses are both defined as a (force/area)
• Six components of stress must be known to specify the state
of stress at a point (stress is a "2nd-order tensor")
• Since stress is a tensoral quantity, numerical values of
individual stress components depend on the coordinate
system used to describe the state of stress
• Stress is defined strictly on the basis of static equilibrium;
definition is independent of:
material properties
27
strain
28. Prof. M. E. Tuttle
University of Washington
Strain:
Fundamental Definitions
• "Strain" is a measure of the deformation of a solid body
• There are two "types" of strain; normal strain (ε) and shear
strain (γ)
(change in length )
ε= ; units = in / in, m / m, etc
(original length)
γ = (change in angle ) ; units = radians
28
29. Prof. M. E. Tuttle
University of Washington
3-Dimensional Strain States
• In the most general case, six components of strain exist "at
a point":
εxx, εyy, εzz, γxy, γxz, γyz
• Strain is a 2nd-order tensor; the numerical values of the
individual strain components which define the "state of
strain" depend on the coordinate system used
• This presentation will primarily involve strains which exist
within a single plane
29
30. Prof. M. E. Tuttle
University of Washington
Strain Within a Plane
• We are often interested in the strains induced within a
single plane; specifically, we are interested in two distinct
conditions:
• "Plane stress", in which all non-zero stresses lie within
a plane. The plane stress condition induces four
strain components: εxx, εyy, εzz, and γxy
• "Plane strain", in which all non-zero strain
components lies within a plane. By definition then,
the plane strain condition involves three strain
components (only): εxx, εyy, and γxy 30
31. Prof. M. E. Tuttle
University of Washington
Strain Within a Plane
∠ abc = π/2 radians ∠ abc < π/2 radians
+y +y
a
a ly + ∆ly
ly c
c
b b
+x +x
lx lx + ∆lx
Original Shape Deformed Shape
εxx = ∆lx/lx
εyy = ∆ly/ly
γxy = ∆(∠ abc)
29
32. Strain Sign Convention
• A positive (tensile) normal strain is associated with an
increase in length
• A negative (compressive) normal strain is associated with a
decrease in length
• A shear strain is positive if the angle between two positive
faces (or two negative faces) decreases
+y +y
+x +x
All Strains Positive εxx Positive
εyy and γ xy Negative 32
33. Prof. M. E. Tuttle
University of Washington
Visualization of Strain
F
+y (Loading)
εyy > 0
+x εxx < 0
γxy = 0
F 33
34. Prof. M. E. Tuttle
University of Washington
Visualization of Strain
F
+y' +x' (Loading)
εx'x' = εy'y' > 0
γx'y' > 0
45 deg
F 34
35. Prof. M. E. Tuttle
University of Washington
Strain Transformations
• Given strain components in the x-y coordinate system (εxx, εyy,
γxy), what are the corresponding strain components in the x'-y'
coordinate system?
+y +y'
+x'
?
ly + ∆ly θ
+x
lx + ∆lx εx'x' = ?
εxx = ∆lx/lx εy'y' = ?
εyy = ∆ly/ly γx'y' = ?
γxy = ∆(∠ abc) 35
36. Prof. M. E. Tuttle
University of Washington
Strain Transformation
Equations
• Based strictly on geometry, it can be shown:
ε xx + ε yy ε xx − ε yy γ xy
ε x' x' = + cos 2θ + sin2θ
2 2 2
ε xx + ε yy ε xx − ε yy γ xy
ε y' y' = − cos2θ − sin2θ
2 2 2
γ x'y' ε xx − ε yy γ xy
=− sin 2θ + cos 2θ
2 2 2
36
37. Prof. M. E. Tuttle
University of Washington
Well, I'll Be Darned!!!
• The stress transformation equations are based strictly on
the equations of static equilibrium
• The strain transformation equations are based strictly on
geometry
• Nevertheless, the stress and strain transformation
equations are nearly identical!!
37
38. Prof. M. E. Tuttle
University of Washington
"Strain":
Summary of Key Points
• ε = (∆ length)/(original length) γ = (∆ angle)
• Six components of strain specify the "state of strain"
• Strain is a 2nd-order tensor; numerical values of individual
strain components depend on the coordinate system used
• Strain is defined strictly on the basis of a change in shape;
definition is independent of:
material properties
stress
• "Suprisingly," the stress and strain transformation equations
are nearly identical
38
39. Prof. M. E. Tuttle
University of Washington
"Constitutive Models"
• The structural engineer is typically interested in the state of
stress induced in a structure during service
• The state of stress cannot be measured directly...
• The state of strain can be measured directly...
• Hence, we must develop a relation between stress and
strain...this relationship is called a "constitutive model," and
the most common is "Hooke's Law"
39
40. Prof. M. E. Tuttle
University of Washington
Material Properties:
The Uniaxial Tensile Test
• Specimen is subjected to an axial tensile force, inducing a
uniform tensile stress in the "gage" region of the specimen
• Stress is increased until fracture occurs; strains are measured
throughout the test
• The ASTM has developed "standard" specimen
styles/procedures for most materials
F F
"Gage
Region" 40
41. Prof. M. E. Tuttle
University of Washington
The Tensile
Stress-Strain Curve
σ
Tensile Strength
Fracture Stress
x
0.2% Offset
Yield Strength
ε 41
0.2% Strain to Fracture
42. Prof. M. E. Tuttle
University of Washington
Load-Unload Cycles
σ σ σ
Tensile Strength Tensile Strength Tensile Strength
0.2% Offset 0.2% Offset 0.2% Offset
Yield Strength Yield Strength Yield Strength
ε ε ε
Elastic Strain Elastic Strain Inelastic Strain Elastic Strain
Inelastic
Strain
42
43. Prof. M. E. Tuttle
University of Washington
Material Property:
Young's Modulus
σ
• At stress levels below the
yield stress the response is
called "linear elastic"
0.2% Offset
Yield Strength
• The slope of the linear
In Linear Region:
region is called "Young's
E σ = Eε
modulus" or the "modulus of
elasticity", E
1
• In the linear region (only):
ε
σ = Eε (or) ε = σ/E
44. Prof. M. E. Tuttle
University of Washington
Material Property:
Poisson's Ratio
• A uniaxial tensile stress causes both a tensile axial strain (εa)
as well as a compressive transverse strain (εt)
• Poisson's ratio is defined as: ν = -(εt/εa)
F F
εa
εt 44
45. Prof. M. E. Tuttle
University of Washington
Material Properties:
The Torsion Test
• Cylindrical specimen is subjected to a torque, inducing a
uniform shear stress τxy in the gage region of the
specimen
• Shear stress increased until fracture occurs; shear strain
measured throughout test
+y
τ xy
+x
T
T
45
46. Prof. M. E. Tuttle
University of Washington
Material Property:
Shear Modulus
• At linear levels, the
slope of the shear τ
stress vs shear strain
curve is called the
"shear modulus," G
• In the linear region G
(only):
τ = Gγ (or) γ = τ/G
γ
46
47. Prof. M. E. Tuttle
University of Washington
"Isotropic" vs
"Anisotropic" Materials
• A material which exhibits the same properties in all
directions is called "isotropic"
2 2
1 1 2 1 1 2
E1 ≅ E2 E1 > E2
Isotropic Materials Anisotropic Materials
47
48. Prof. M. E. Tuttle
University of Washington
Material Properties
Summary
• Three material properties are commonly used to
describe isotropic materials in the linear-elastic region:
E, ν, and G
• It can be shown that (for isotropic materials) only two
of these three properties are independent. For
example:
E
G=
2(1 + ν)
48
49. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
3-Dimensional Stress States
• Assumptions:
• Linear-elastic behavior; i.e., yielding does not occur, so
the principle of superposition applies
• Isotropic material behavior
• Constant environmental conditions (e.g., constant
temperature)
49
50. Strains Induced by Individual Stress Components
(That is, Strain Caused During Six Different Tests)
=
εx xσ σxx /E
xx γx y τ= τx y /G = 2(1+ν)τx y /E
= -νεx x
εyy Induces:= -νσx x /E xy
εx x = εyy = εzz = γxz = γyz = 0
Induces:
εzz = -νεx x = -νσx x /E
γx y = γxz = γyz = 0
εyyσ= σyy /E
yy
= -νεyy
εx x Induces:= -νσyy /E γxz τ= τxz /G = 2(1+ν)τxz /E
xz
εzz = -νεyy = -νσyy /E εx x = εyy = εzz = γx y = γyz = 0
Induces:
γx y = γxz = γyz = 0
=
εzzσ σzz/E
zz
εx x Induces:= -νσzz/E
= -νεzz γyz τ= τyz /G = 2(1+ν)τyz /E
yz
εyy = -νεzz = -νσzz/E εx x = εyy = εzz = γx y = γxz = 0
Induces:
γx y = γxz = γyz = 0
51. Prof. M. E. Tuttle
University of Washington
3-D Form of Hooke's Law:
Apply Principle of Superposition
• The total strain εxx caused by all stresses applied
simultaneously is determined by "adding up" the strain εxx
caused by each individual stress component
σ xx −νσ yy −νσ zz
ε xx =
E + E + E + [0 ] + [0 ] + [0 ]
or
ε xx
1
[
= σ xx − νσ yy − νσ zz
E
]
51
52. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
3-Dimensional Stress States
• Following this procedure for all six strain components:
2(1+ ν)τ yz
1
[
ε xx = σ xx − νσ yy − νσ zz
E
] γ yz =
E
2(1+ ν)τ zx
1
[
ε yy = −νσ xx + σ yy − νσ zz
E
] γ zx =
E
2(1+ ν)τ xy
1
[
ε zz = −νσ xx − νσ yy + σ zz
E
] γ xy =
E
52
57. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Plane Stress
• For thin components (e.g., web or flange of an I-beam,
automobile door panel, airplane "skin", etc) the "in-plane"
stresses are much higher than "out-of-plane" stresses:
(σxx,σyy,τxy) >> (σzz,τxz,τyz)
• It is convenient to assume the out-of-plane stresses equal
zero...this is called a state of "plane stress"
+y
+x
+z
57
58. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Plane Stress
1
ε xx = (σ xx − νσ yy )
E
1
σ yy ε yy = (σ yy − νσ xx )
τ xy E
−ν
σ xx ε zz = (σ xx + σ yy )
E
τ xy 2(1+ ν)τ xy
γ xy = =
G E
γ xz = γ yz = 0 58
59. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Plane Stress
E
σ xx = 2 (ε xx + νε yy )
(1- ν )
σ yy E
τ xy σ yy = 2 (ε yy + νε xx )
(1- ν )
σ xx E
τ xy = Gγ xy = γ xy
2(1+ ν)
σ zz = τ xz = τ yz = 0 59
60. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Plane Stress
σ xx
1 ν 0 ε xx
E
σ yy = 2 ν 1 0 ε yy
(1− ν ) 1 − ν
τ xy 0
0 γ xy
2
60
61. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Plane Stress
• "Shorthand" notation:
{σ } = [ D]{ε }
where
1 ν 0
E
[ D] = 2 ν 1 0
(1 − ν ) 1− ν
0 0
2
61
62. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Plane Strain
• For thick or very long components (e.g., thick-walled pressure
vessels, buried pipe, etc) the "in-plane" strains are much
higher than "out-of-plane" strains: (εxx,εyy,γxy) >> (εzz,γxz,γyz)
• It is convenient to assume the out-of-plane strains equal
zero...this is called a state of "plane strain"
+y
+z +x
62
63. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Plane Strain
σ xx =
E
(1 + ν)(1− 2ν)
[ (1− ν)ε xx + νε yy ]
σ yy
τ xy σ yy =
E
(1+ ν)(1− 2ν)
[ νε xx + (1− ν)ε yy ]
νE
σ xx σ zz =
(1+ ν)(1− 2ν)
[ ] [
ε xx + ε yy = ν σ xx + σ yy ]
Eγ xy
τ xy = Gγ xy =
2(1+ ν)
τ xz = τ yz = 0 63
64. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Plane Strain
σ xx (1− ν) ν 0 ε xx
E ν ε
σ yy = (1− ν) 0 yy
(1+ ν)(1− 2ν) (1 − 2v) γ
τ xy
0 0
xy
2
64
65. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Plane Strain
• "Shorthand" notation:
{σ } = [ D]{ε }
where
(1− ν) ν 0
[ D] =
E ν (1− ν) 0
(1 + ν)(1− 2ν)
(1− 2v)
0 0
2
65
66. Hooke's Law
Plane Strain
1 − ν2 ν
ε xx = σ xx − σ yy
E 1− ν
σ yy 1− ν 2 ν
τ xy ε yy = σ yy − σ xx
E 1− ν
2(1+ ν)
σ xx γ xy = τ xy
E
ε zz = γ xz = γ yz = 0
[
....and: σ zz = ν σ xx + σ yy ] 66
67. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Uniaxial Stress
• Truss members are designed to carry axial loads only (i.e.,
uniaxial stress)
• In this case we are interested in the axial strain only (even
though transverse strains are also induced)
67
68. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Uniaxial Stress
• For: σyy = σzz = τyz = τxz = τzy = 0,
"Hooke's Law" becomes:
σ xx = Eε xx
• "Shorthand" notation:
{σ } = [ D]{ε } where [ D] = [ E ]
68
69. Prof. M. E. Tuttle
University of Washington
Hooke's Law
Summary of Key Points
• Hooke's Law is valid under linear-elastic conditions only
• The mathematical form of Hooke's Law depends on whether
the problem involves:
• Structures for which 3-D stresses are induced
• Structures for which (due to loading and/or geometry)
plane stress conditions can be assumed
• Structures for which (due to loading and/or geometry)
plane strain conditions can be assumed
• Structures for which (due to loading and/or geometry)
uniaxial stress conditions can be assumed
69