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

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FIGURE 12.15
506
There is never
shear stress on
principal planes.
σ
y
τ
xy
σ
x
x
θ
θ
Label arrows with
stress magnitudes
(i.e., absolute values).
Use direction of arrow
to indicate signs.
p
s
45°
12.9 presentation of Stress Transformation Results
Principal stress and maximum in-plane shear stress results should be presented with a
sketch that depicts the orientation of all stresses. Two sketch formats are generally used:
(a) two square stress elements or
(b) a single wedge-shaped element.
σ
p2
P
S
σ
τ
σ
avg
max
σ
p1
avg
Two Square Stress Elements
Two square stress elements are sketched in Figure 12.15. One stress
element shows the orientation and magnitude of the principal
stresses, and a second element shows the orientation and magnitude
of the maximum in-plane shear stress along with the associated normal
stresses.
Principal Stress Element
• The principal stress element is shown rotated at the angle θ p
calculated from Equation (12.11):
tan2θ
p
=
σ
2τ
x
xy
− σ
This equation yields a value between +45° and −45° (inclusive).
• When θ p is positive, the stress element is rotated in a counterclockwise
sense from the reference x axis. When θ p is negative,
the rotation is clockwise.
• Note that the angle calculated from Equation (12.11) does not necessarily give the
orientation of the σ p1 plane. Either σ p1 or σ p2 may act on the θ p plane. The principal
stress oriented at θ p can be determined from the following rule:
• If σ x − σ y is positive, θ p indicates the orientation of σ p1 .
• If σ x − σ y is negative, θ p indicates the orientation of σ p2 .
• The other principal stress is shown on the perpendicular faces of the stress element.
• In the sketch, use the arrow direction to indicate whether the principal stress is tension
or compression. Label the arrow with the absolute value of either σ p1 or σ p2 .
• There is never a shear stress on the principal planes; therefore, show no shear stress
arrows on the principal stress element.
Maximum in-Plane Shear Stress Element
• Draw the maximum shear stress element rotated 45° from the principal stress element.
• If the principal stress element is rotated in a counterclockwise sense (i.e., positive θ p )
from the reference x axis, then the maximum shear stress element should be shown
rotated 45° clockwise from the principal stress element. Therefore, the maximum
shear stress element will be oriented an angle θ s = θ p − 45° relative to the x axis.
• If the principal stress element is rotated in a clockwise sense (i.e., negative θ p ) from
the reference x axis, then the maximum shear stress element should be shown
rotated 45° counterclockwise from the principal stress element. Therefore, the
maximum shear stress element will be oriented an angle θ s = θ p + 45° relative to
the x axis.
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