Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

504
STRESS TRANSFORMATIONS
In words, the maximum in-plane shear stress τ max is equal in magnitude to one-half of the
difference between the two in-plane principal stresses.
Normal Stresses on maximum In-plane
Shear Stress Surfaces
Unlike principal planes, which are free of shear stress, planes subjected to τ max usually
have normal stresses. After substituting angle functions obtained from Equation (12.14)
into Equation (12.5) and simplifying, we find that the normal stress acting on a plane of
maximum in-plane shear stress is
σ avg
σx
=
+ σ
2
y
(12.17)
The normal stress σ avg is the same on both τ max planes.
Absolute maximum Shear Stress
In Equation (12.15), we derived an expression for the maximum shear stress magnitude
acting in the plane of a body subjected to plane stress. We also found that the maximum
in-plane shear stress τ max is equal in magnitude to one-half the difference between the two
in-plane principal stresses [Equation (12.16)]. Let us briefly consider a point in a stressed
body in which stresses act in three directions. We ask the question, “What is the maximum
shear stress for this more general state of stress?” We will denote the maximum shear stress
magnitude on any plane that could be passed through the point as τ abs max to differentiate it
from the maximum in-plane shear stress τ max . In the body at the point of interest, there will
be three orthogonal planes with no shear stress: the principal planes. (See Section 12.11.)
The normal stresses acting on these planes are termed principal stresses, and, in general,
they each have unique algebraic values (i.e., σ p1 ≠ σ p2 ≠ σ p3 ). Consequently, one principal
stress will be the maximum algebraically (σ max ), one principal stress will be the minimum
algebraically (σ min ), and the third principal stress will have a value in between these two
extremes. The magnitude of the absolute maximum shear stress τ abs max is equal to one-half
of the difference between the maximum and minimum principal stresses:
τ
absmax
σ
=
max
− σ
2
min
(12.18)
For example, if stresses act only
in the x–y plane, then the z face
of a stress element is a principal
plane.
Furthermore, τ abs max acts on planes that bisect the angles between the maximum and minimum
principal planes.
When a state of plane stress exists, normal and shear stresses on the out-of-plane face
of a stress element are zero. Since no shear stresses act on it, the out-of-plane face is a
principal plane and the principal stress acting on it is designated σ p3 . Therefore, two principal
stresses σ p1 and σ p2 act in the plane of the stress, and the third principal stress, which
acts in the out-of-plane direction, has a magnitude of σ p3 = 0. Thus, for plane stress, the
magnitude of the absolute maximum shear stress can be determined from one of the following
three conditions:
(a) If both σ p1 and σ p2 are positive, then
τ
absmax
σ p − σ p σ p − 0 σ p
=
= =
2 2 2
1 3 1 1