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

planes whose normal is perpendicular to the z axis) can be determined from the three
principal stresses: σ p1 = 13.21 ksi, σ p2 = −11.21 ksi, and σ p3 = 0. The maximum
principal stress (in an algebraic sense) is σ max = 13.21 ksi, and the minimum principal
stress is σ min = −11.21 ksi. The absolute maximum shear stress can be computed
from Equation (12.18):
τ
abs max
σ
=
max
− σ
2
min
13.21 ksi − ( −11.21 ksi)
=
= 12.21 ksi
2
In this instance, the absolute maximum shear stress is equal to the maximum in-plane
shear stress. That will always be the case whenever σ p1 is a positive value and σ p2 is a
negative value. The absolute maximum shear stress will be greater than the maximum
in-plane shear stress whenever σ p1 and σ p2 are either both positive or both negative.
ExAmpLE 12.6
y
150 MPa
55 MPa
x
70 MPa
Consider a point in a structural member that is subjected to plane stress. Normal and shear
stresses acting on horizontal and vertical planes at the point are shown.
(a) Determine the principal stresses and the maximum in-plane shear stress acting at the
point.
(b) Show these stresses in an appropriate sketch.
(c) Determine the absolute maximum shear stress at the point.
Plan the Solution
The stress transformation equations derived in the preceding section will be used to compute
the principal stresses and the maximum shear stress acting at the point.
SolutioN
(a) From the given stresses, the values to be used in the stress transformation equations
are σ x = 70 MPa, σ y = 150 MPa, and τ xy = −55 MPa. The in-plane principal stresses
can be calculated from Equation (12.12):
σ
p1, p2
σx + σ y ⎛ σx − σ y⎞
= ±
τ
2 ⎝
⎜
2 ⎠
⎟ +
=
=
70 MPa + 150 MPa
±
2
178.0 MPa, 42.0 MPa
2
⎛
⎝
⎜
2
xy
70 MPa − 150 MPa
2
⎞
⎠
⎟
2
+ ( −55MPa)
The maximum in-plane shear stress can be computed from Equation (12.15):
2
τ
max
=±
⎛ σx
− σ
⎝
⎜
2 ⎠
=± 68.0 MPa
2
y⎞
2
⎟ + τ xy = ±
⎛ 70 MPa − 150 MPa ⎞
⎝
⎜
2 ⎠
⎟
2
+ ( −55MPa)
2
On the planes of maximum in-plane shear stress, the normal stress is simply the average
normal stress, as given by Equation (12.17):
σ
avg
σx
=
+ σ
2
y
=
70 MPa + 150 MPa
2
= 110 MPa = 110 MPa (T)
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