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

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STRESS TRANSFORMATIONS
If the normal of a surface lies in
the x–y plane, then the stresses
that act on that surface are
termed in-plane stresses.
In many situations, a stress element (which represents the state of stress at a specific point)
will have only normal stresses acting on its x and y faces. In these instances, one can conclude
that the x and y faces must be principal planes because there is no shear stress acting on them.
Another important application of the preceding statement concerns the state of plane
stress. As discussed in Section 12.4, a state of plane stress in the x–y plane means that there
are no stresses acting on the z face of the stress element. Therefore,
σz = τzx = τ zy =
If the shear stress on the z face is zero, one can conclude that the z face must be a principal
plane. Consequently, the normal stress acting on the z face must be a principal stress—the
third principal stress.
The Third principal Stress
In the previous discussion, the principal planes and principal stresses were determined for a
state of plane stress. The two principal planes found from Equation (12.11) were oriented at
angles of θ p and θ p ± 90° with respect to the reference x axis, and they were oriented so that their
outward normal was perpendicular to the z axis (i.e., the out-of-plane axis). The corresponding
principal stresses determined from Equation (12.12) are called the in-plane principal stresses.
Although it is convenient to represent the stress element as a two-dimensional square,
it is actually a three-dimensional cube with x, y, and z faces. For a state of plane stress, the
stresses acting on the z face—σ z , τ zx , and τ zy —are zero. So, since the shear stresses on the
z face are zero, the normal stress acting on the z face must be a principal stress, even though
its magnitude is zero. A point subjected to plane stress has three principal stresses: the
two in-plane principal stresses σ p1 and σ p2 , plus a third principal stress σ p3 , which
acts in the out-of-plane direction and has a magnitude of zero.
0
orientation of maximum In-plane Shear Stress
To determine the planes where the maximum in-plane shear stress τ max occurs, Equation (12.6)
is differentiated with respect to θ and set equal to zero, yielding
dτ
nt
dθ
=−( σ − σ )cos2θ − 2τ sin 2θ
= 0
(12.13)
x y xy
The solution of this equation gives the orientation θ = θ s of a plane where the shear stress
is either a maximum or a minimum:
tan2θ
s
σx
− σ
=−
2τ
xy
y
(12.14)
This equation defines two angles 2θ s that are 180° apart. Thus, the two values of θ s are 90°
apart. A comparison of Equations (12.14) and (12.11) reveals that the two tangent functions
are negative reciprocals. For that reason, the values of 2θ p that satisfy Equation (12.11) are
90° away from the corresponding solutions 2θ s of Equation (12.14). Consequently, θ p and
θ s are 45° apart. This property means that the planes on which the maximum in-plane shear
stresses occur are rotated 45° from the principal planes.
maximum In-plane Shear Stress magnitude
Similar to the way the principal stresses are computed, there are two methods for computing
the magnitude of the maximum in-plane shear stress τ max .