Page 266 - Mechanical Behavior of Materials

268 Chapter 6 Review of Complex and Principal States of Stress and Strain

One method of evaluating Eq. 6.45(a) and obtaining the corresponding axis rotation is to use Mohr’s
circle.
The principal shear stresses occur on planes inclined 45 with respect to the principal normal
◦
stress axes. These are given by

|σ 2 − σ 3 | |σ 1 − σ 3 | |σ 1 − σ 2 |
τ 1 = , τ 2 = , τ 3 = (6.46)
2 2 2
One of the values τ 1 , τ 2 , τ 3 is the maximum shear stress that occurs for all possible choices of
coordinate axes. For x-y plane stress, special care is needed that all three of Eq. 6.46 are considered,
because the principal shear stress in the x-y plane may not be the largest. It is useful to envision
three different Mohr’s circles, one for each plane perpendicular to a principal normal stress axis.
The radii of these are the principal shear stresses.
The octahedral normal and shear stresses occur on planes that intercept the principal normal
stress axes at equal distances from the origin. Their values are given by

σ 1 + σ 2 + σ 3
σ h = (6.47)
3
1 2 2 2
τ h = (σ 1 − σ 2 ) + (σ 2 − σ 3 ) + (σ 3 − σ 1 ) (6.48)
3
where σ h is also called the hydrostatic stress.
Principal normal strains and principal shear strains occur in a manner analogous to principal
stresses. The same equations apply by replacing stresses with strains as follows:
γ xy γ yz γ zx
σ x ,σ y ,σ z → ε x ,ε y ,ε z , τ xy ,τ yz ,τ zx → , , (6.49)
2 2 2
Even plane stress causes a three-dimensional state of strain. However, for isotropic materials, and
also for orthotropic materials stressed in a plane of material symmetry, the out-of-plane shear strains
γ yz and γ zx are zero, permitting two-dimensional analysis to be performed in the x-y plane, despite
the presence of a nonzero ε z .

NEW TERMS AND SYMBOLS

axes rotation angles: θ n and θ s principal axes (1, 2, 3)
direction cosines: l, m, n principal normal strains: ε 1 , ε 2 , ε 3
generalized plane stress principal normal stresses: σ 1 , σ 2 ,σ 3
Mohr’s circle principal shear strains: γ 1 , γ 2 , γ 3
octahedral normal (hydrostatic) principal shear stresses: τ 1 , τ 2 , τ 3
stress, σ h strain gage rosette
octahedral planes stress invariants: I 1 , I 2 , I 3
octahedral shear stress, τ h transformation equations
plane strain transformation of axes
plane stress