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

• Substitute the value of θ s into either Equation (12.4) or (12.6), and compute τ max .
• If τ max is positive, draw the shear stress arrow on the θ s face in the direction that tends
to rotate the stress element counterclockwise. If τ max is negative, the shear stress
arrow on the θ s face should tend to rotate the stress element clockwise. Label this
arrow with the absolute value of τ max .
• Once the shear stress arrow on the θ s face has been established, draw appropriate
shear stress arrows on the other three faces.
• Compute the average normal stress acting on the maximum in-plane shear stress
planes from Equation (12.17).
• Show the average normal stress with arrows acting on all four faces. Use the
direction of the arrow to indicate whether the average normal stress is tension or
compression. Label a pair of the arrows with the magnitude of this stress.
• In general, the maximum in-plane shear stress element will include both normal and
shear stress arrows on all four faces.
507
PRESENTATION OF STRESS
TRANSFORMATION RESuLTS
Wedge-Shaped Stress Element
A wedge-shaped stress element can be used to report both the principal stress and maximum
in-plane shear stress results on a single element, as shown in Figure 12.16.
• The two orthogonal faces of the wedge element are used to report the orientation and
magnitude of the principal stresses.
• Follow the procedures given previously for the principal stress element in order
to specify the principal stresses acting on the two orthogonal faces of the wedge
element. Since these two faces are principal planes, there should not be a shear
stress arrow on either face.
• The sloped face of the wedge is oriented 45° away from the two orthogonal faces, and it
is used to specify the maximum in-plane shear stress and the associated normal stress.
• Draw a shear stress arrow on the sloped face, and label the arrow with the magnitude
of the maximum in-plane shear stress computed from Equation (12.15).
• There are several ways to determine the proper direction of the maximum in-plane
shear stress arrow. One particularly easy way to construct a proper sketch is as
follows: Begin the tail of the shear stress arrow at the σ p1 side of the wedge, and
point the arrow toward the σ p2 side of the wedge.
• Compute the average normal stress acting on the maximum in-plane shear stress
planes from Equation (12.17).
• Show the average normal stress on the sloped face of the wedge. Use the direction
of the arrow to indicate whether the average normal stress is tension or compression.
Label this arrow with the average normal stress magnitude.
y
σ
y
τ
xy
σ
x
x
θ
p
σ
avg
τ
max
45°
σ
p2
σ
p1
There is never
shear stress on
principal planes.
FIGURE 12.16
Label arrows with
stress magnitudes
(i.e., absolute values).
Use direction of arrow
to indicate signs.