Body and Surface Forces
Body Forces: F(x)
Surface Forces: T(x)
(a) Cantilever Beam Under Self-Weight Loading
S
(b) Sectioned Axially Loaded Beam
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Traction Vector
P3
P2
F
n
A
p
P1
(Externally Loaded Body)
F
A0 A
T n ( x, n)  lim
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
(Sectioned Body)
T n ( x, n  e1 )   x e1   xy e 2   xz e3
T n ( x, n  e 2 )   yx e1   y e 2   yz e3
T n ( x, n  e3 )   zx e1   zy e 2   z e3
Stress Tensor
y
yx
yz
zy
y
xy
x
zx xz
z
 x

  [ ]   yx
  zx

 xy
y
 zy
 xz 

 yz 
 z 
x
z
y
n
Traction on an Oblique Plane
Tn
T n  (  x n x   yx n y   zx n z )e1
 (  xy n x   y n y   zy n z )e 2
x
z
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
 (  xz n x   yz n y   z n z )e3
Ti n   ji n j
Stress Transformation
ij  QipQ jq  pq
 l1
Qij  l2

l3
m1
m2
m3
n1 
n2 

n3 
x   x l12   y m12   z n12  2(  xy l1m1   yz m1n1   zx n1l1 )
y   x l22   y m22   z n22  2(  xy l2 m2   yz m2 n2   zx n2 l2 )
z   x l32   y m32   z n32  2(  xy l3m3   yz m3n3   zx n3l3 )
xy   x l1l2   y m1m2   z n1n2   xy (l1m2  m1l2 )   yz ( m1n2  n1m2 )   zx ( n1l2  l1n2 )
yz   x l2 l3   y m2 m3   z n2 n3   xy (l2 m3  m2 l3 )   yz ( m2 n3  n2 m3 )   zx ( n2 l3  l2 n3 )
zx   x l3l1   y m3m1   z n3n1   xy (l3m1  m3l1 )   yz ( m3n1  n3m1 )   zx ( n3l1  l3n1 )
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Two-Dimensional
Stress Transformation
y
y'
 cos  sin  0
Qij   sin  cos  0


0
1
 0
x'


x
x   x cos 2    y sin 2   2 xy sin  cos 
y   x sin 2    y cos 2   2 xy sin  cos 
xy   x sin  cos    y sin  cos    xy (cos 2   sin 2 )
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
x 
y 
xy 
x   y
2
x   y
2
 y  x
2


x   y
2
x   y
2
cos 2   xy sin 2
cos 2   xy sin 2
sin 2   xy cos 2
Principal Stresses & Directions
1 ,  2 , 3
det[ ij  ij ]  3  I12  I 2   I 3  0
y
2
2
yx
yz
zy
y
1
xy
1
x
zx xz
z
3
x
3
z
(General Coordinate System)
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
(Principal Coordinate System)
Traction Vector Components
N
A
N Tn n
n
Tn
S  (| T n |  N 2 )1/ 2
S
S
S 2  ( N   2 )( N   3 )  0
Admissible N and S values
lie in the shaded area
S 2  ( N   2 )( N   3 )  0
S  ( N   3 )( N  1 )  0
2
N
3
2
1
S 2  ( N  1 )( N   2 )  0
S 2  ( N  1 )( N   2 )  0
S 2  ( N   3 )( N  1 )  0
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Mohr’s Circles of Stress
Example 3-1 Stress Transformation
For the given state of stress below, determine the principal stresses and directions and find the
traction vector on a plane with unit normal n = (0,1,1)/2.
3
 ij  1

1
1
0
2
1
2

0
The principal stress problem is started by calculating the three invariants, giving the result
I1 = 3, I2 = -6, I3 = -8. This yields the following characteristic equation
  3  3 2  6  8  0
The roots of this equation are found to be  = 4, 1, -2. Back-substituting the first root into the
fundamental system (1.6.1) gives
 n1(1)  n 2(1)  n3(1)  0
n1(1)  4n 2(1)  2n3(1)  0
n1(1)  2n2(1)  4n3(1)  0
Solving this system, the normalized principal direction is found to be n(1) = (2, 1, 1)/6. In
similar fashion the other two principal directions are n(2) = (-1, 1, 1)/3, n(3) = (0, -1, 1)/2.
The traction vector on the specified plane is calculated using the relation
Ti n
Elasticity
3
 1
1
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
1
0
2
1   0  2 /

2 1 / 2   2 /
0 1 / 2  2 /
2

2
2 
Spherical, Deviatoric, Octahedral
and von Mises Stresses
~  ˆ
 ij  
ij
ij

~  1 

. . . Spherical Stress Tensor
ij
kk ij
3
1
ˆ ij   ij   kk  ij . . . Deviatoric Stress Tensor
3
1
1
1
 oct  (1   2   3 )   kk  I1
. . . Octahedral Normal
3
3
3
and Shear Stresses
1/ 2
1
1 2
2
2
2 1/ 2
 oct  [(1   2 )  ( 2   3 )  ( 3  1 ) ]  2 I1  6 I 2
3
3

 e   vonMises 

Elasticity

3
1
ˆ ij ˆ ij 
[( x   y ) 2  ( y   z ) 2  ( z   x ) 2  6( 2xy   2yz   2zx )]1 / 2
2
2
1
[(1   2 ) 2  ( 2   3 ) 2  ( 3  1 ) 2 ]1 / 2
2
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
. . . von Mises Stress
Stress Distribution Visualization Using
2-D or 3-D Plots of Particular Contour Lines
•
•
•
•
•
Particular Stress Components
Principal Stress Components
Maximum Shear Stress
von Mises Stress
Isochromatics (lines of principal stress difference =
constant; same as max shear stress)
• Isoclinics (lines along which principal stresses have
constant orientation)
• Isopachic lines (sum of principal stresses = constant)
• Isostatic lines (tangent oriented along a particular
principal stress; sometimes called stress trajectories)
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Example Stress Contour Distribution Plots
Disk Under Diametrical Compression
P
P
(a) Disk Problem
Maximum Shearing Stress Contours - Isochromatics
(b) Max Shear Stress Contours
(Isochromatic Lines)
von Mises Stress Contours
Sum of Principal Stress Contours - Isopachics
(d) Sum of Principal Stress Contours
(Isopachic Lines)
Elasticity
(e) von Mises Stress
Contours
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Maximum Principal Stress Contours
(c) Max Principal Stress
Contours
Stress Trajectories for Disk
(f) Stress Trajectories (Isostatic Lines)
Equilibrium Equations
Tn
S
F
 F  0   T
S
i
n
V
dS   Fi dV  0 
V
 xy   yx
 r  F  0  ij   ji   yz   zy
 zx   xz
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
 x  yx  zx


 Fx  0
x
y
z
 xy  y  zy


 Fy  0
x
y
z
 xz  yz  z


 Fz  0
x
y
z
Stress & Traction Components
in Cylindrical Coordinates
x3
z
z
z
rz
 r
σ   r

  rz
r
x2
r
x1

d
T   r e r    e    z e z
Tz   rz e r   z e    z e z
dr
Equilibrium Equations
 r 1  r  rz 1


 (  r    )  Fr  0
r
r 
z
r
 r 1    z 2


  r  F  0
r
r 
z
r
 rz 1  z  z 1


  rz  Fz  0
r
r 
z
r
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island

 z
 rz 
 z 

 z 
Tr   r e r   r e    rz e z
r

 r
Stress & Traction Components in
Spherical Coordinates
x3
R
R

σ    R
 R
R
R


R

 R 

  
  
TR   R e R   R e   R e

x2

 R

 
T   R e R    e   e
T   R e R   e    e
x1
Equilibrium Equations
 R 1  R
1  R 1


 (2 R         R cot )  FR  0
R R 
R sin  
R
 r 1  
1   1


 [(      ) cot   3 R ]  F  0
R R  R sin   R
 r 1  
1   1


 ( 2  cot   3 R )  F  0
R R  R sin   R
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island