CONTINUUM MECHANICS for ENGINEERS

FIGURE 3.18A
Representative rotation of axes for plane stress.
FIGURE 3.18B
Transformation table for general plane stress.
The pictorial description of this plane stress situation is portrayed by the
block element of a continuum body shown in Figure 3.17A, and is sometimes
represented by a single Mohr’s circle (Figure 3.17B), the locus of which
identifies stress points (having coordinates σ N and σ S) for unit normals lying
in the x 1x 2 plane only. The equation of the circle in Figure 3.17B is
2
⎛ σ + σ ⎞
2
11 22
⎛ σ11 − σ22
⎞
⎜σ
N − ⎟ + ( σ S)
= ⎜ ⎟ + ( σ 12)
⎝ 2 ⎠ ⎝ 2 ⎠
(3.9-2)
1
from which the center of the circle is noted to be at σN = ( σ11 + σ22)
, σS =
2
0, and the maximum shear stress in the x1x2 plane to be the radius of the
circle, that is, the square root of the right-hand side of Eq 3.9-2. Points A and
B on the circle represent the stress states for area elements having unit
normals and , respectively. For an element of area having a unit normal
in an arbitrary direction at point P, we must include the two dashed circles
shown in Figure 3.17C to completely specify the stress state.
ê1 ê2 With respect to axes Ox′ x′ x′
rotated by the angle θ about the x3 axis
1 2 3
relative to Ox 1x 2x 3 as shown in Figure 3.18A, the transformation equations
for plane stress in the x 1x 2 plane are given by the general tensor transformation
formula, Eq 2.5-13. Using the table of direction cosines for this situation
as listed in Figure 3.18B, we may express the primed stress components in
terms of the rotation angle θ and the unprimed components by
2
2