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Calculate the normal and shear stresses at point
O
o
n a su
r
face
whose outward normal is the bisector of the angle between the lines
Ox
and
Oz
O
and axes
x, y, z
.
.
1.44 Determine the extreme values of the shear stresses for the stress tensor
151
−
30
−
27
T
=
−
30
190
10
−
27
10
79
1.45 At a given point in a body all the normal stresses have the same value
σ
=
σ
=
σ
=−
pp
>
0
x
y
z
Show that any three mutually orthogonal directions are principal directions for this
state of stress (hydrostatic compression). Hydrostatic (fluid) pressure
p
is often shown
as
p
Show that hydrostatic pressure is an invariant.
1.46 An octahedral plane is one which intersects the principal axes at equal angles, that is,
the normal to this plane is
=−
(σ
+
σ
+
σ
)/
3
.
1
2
3
1
√
3
(
a
=
e
1
+
e
2
+
e
3
)
where
e
1
,
e
2
,
e
3
are unit vectors parallel to the principal axes. Show that the shear
stress on this plane, the
octahedral shear stress,
is given by
3
1
τ
oct
=
(σ
1
−
σ
2
)
2
+
(σ
2
−
σ
3
)
2
+
(σ
3
−
σ
1
)
2
1.47 The
principal directions
for strain are those directions for which the shear strain becomes
an extremum. If the stress and strain tensors are related by Hooke's law, prove that
the principal axes of stress coincide with the principal axes of strain.
1.48 A circle of radius 100 mm is inscribed on the surface of a thin sheet, which lies in
the
xy
plane and is stress-free. The plate is then subjected to the uniform stress field
σ
x
=
200 MPa,
σ
y
=
25 MPa,
σ
z
=
0,
τ
xy
=
125 MPa,
τ
yz
=
τ
xz
=
0
.
The material
In the stress state, the circle deforms into an
ellipse. Determine the lengths and orientations of the minor and major axes of this
ellipse.
1.49 A displacement field has been determined for an isotropic, linearly elastic solid as
properties are
E
=
210 GPa and
ν
=
0
.
3
.
Ky
where
K
is a positive constant. Determine the principal stress directions.
1.50 For the pressure vessel shown in Fig. P1.50 the circumferential normal stress (hoop
stress) is
u
=
K
(
x
−
y
)v
=
K
(
x
+
y
)w
=
σ
y
=
pr
/
t,
the longitudinal normal stress (axial stress) is
σ
x
=
pr
/(
2
t
)
,
and assume the radial direction stress varies linearly such that
σ
z
=−
p
(
1
−
z
/
t
).
Determine the location, orientation, and magnitude of the maximum shear stress.
1.51 A closed thin-walled cylinder is subject to an internal pressure of 1000 psi. The diam-
eter of the cylinder is 40 in. and the wall thickness is 1
2 in. What maximum shear
stress will occur? See Problem 1.50 for the formula for the cylinder stress.
/
Hint:
The radial stress is much smaller than the axial and hoop stresses, and can
be neglected.
20
,
000 psi
1.52 The thin rectangular element of Fig. P1.52 is in the state of plane stress. If the stress
components have magnitudes in the proportions
Answer:
τ
=
max
1
2
1
4
σ
/σ
/τ
=
1
/
/
identify the
x
y
xy
planes of maximum shear stress.
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