sm8 50

_______________________________________________________________________
PROBLEM (*8.50) At a particular point in a loaded member (see Fig. P8.50) there exists on
the horizontal plane a normal stress y
σ
= 2 ksi and a negative shear stress. One of the principal
stresses at the point is 1 ksi (tensile), and the maximum shearing stress has a magnitude of 5 ksi.
Calculate:
(a) The unknown stresses on the horizontal and vertical planes.
(b) The unknown principal stress.
Show the principal stresses and maximum shear stresses on a sketch of a properly oriented
element.
*SOLUTION
(a) 1,2 5
2
xy
σ
σ
σ
+
=± (1)
22
15
2
x
σ
σ
+
== −, ksi
x10
=
σ
22
max 10 2
5( )()
2xy
ττ
−
== +−
,
22
(5) 16 ( )
xy
τ =+− ksi
xy 3
−
=
τ
(b) From Eq. (1): ksi11
1=
σ
We have
12( 3)
2 tan 36.9
10 2 o
p
θ
−−
==−
−
'10 2 10 2 cos( 36.9 ) 3sin( 36.9 )
22 oo
x
σ
+−
=+ −−−
ksi118.12.36 =++=
Therefore,
' 18.5o
p
θ =−
10 2
'6
2ksi
σ
+
==
6 ksi 11 ksi
18.5o
5 ksi
x
1 ksi
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