Stress Analysis Problem Solution: Mechanics of Materials

_______________________________________________________________________
PROBLEM (8.41) The shearing stress at a point in a loaded member is
x
y
τ
= 4 ksi (See Fig.
P8.41). The principal stresses at this point are 1
σ
= 5 ksi and 2
σ
= -8 ksi. Determine
x
σ
and y
σ
and
indicate the principal and maximum shear stresses on an appropriate sketch.
SOLUTION
22
15()
22
xy xy
σσ σσ
σ
+−
== + +
4
(1)
2
28()
22
xy xy
σσ σσ
σ
+−
=− = − + 2
4
3
(2)
Add Eqs. (1) and (2):
−=+ yx
σ
σ
(3)
Equation (1) becomes
22
51.5( )4
2
xy
σσ −
=− + +
or
246.10=− yx
σ
σ
(4)
Solving Eqs. (3) and (4):
ksiksi yx 62.662.3
−
=
=
σ
σ
We have
max 58 6.5
2ksi
τ
+
==
12(4)
2t an 38
3.62 6.42 o
p
θ
−
==
+
'3.62 6.62 3.62 6.62 cos38 4sin38
22
oo
x
σ
−+
=+ +
=
ksi546.204.45.1 =++−
Thus,
'19
o
p
θ =
6.5 ksi
x
’
19o
5 ksi
x
8 ksi
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