A rectangular cross section alumina ( E=415GPa , ν=0.23 ) bar, of initial lengths 100mm in the x direction, 10mm in the y direction and 20mm in the z direction, is loaded in compression by a load of 50 kN applied in the x-direction. What are the strains and dimensions in each direction under this load?
Answers
Answer:
The cross-sectional area perpendicular to the force must be solved to get the area where the force is acting and solve for the axial stress being experienced by the bar.
A
=
(
10
m
m
)
(
20
m
m
)
A
=
200
m
m
2
The stress experienced by the bar will be equal to
σ
=
F
A
σ
=
−
50
k
N
200
m
m
2
σ
=
−
250
M
P
a
Using Hooke's Law, the strain on the axial direction can be solved.
σ
=
ϵ
E
ϵ
=
σ
/
E
ϵ
=
−
0.0006024
Under the compressive load, the dimensions on the y and z-axis will experience deformation proportional to the deformation on the x-axis. The strain on both directions will be equal to
ν
=
−
ϵ
t
r
a
n
s
,
y
ϵ
ϵ
t
r
a
n
s
,
y
=
0.0001386
This corresponds to a dimension on the y-axis of
y
′
=
(
1
+
ϵ
t
r
a
n
s
,
y
)
y
y
′
=
10.001386
m
m
For the strain and dimension on the z-axis,
ν
=
−
ϵ
t
r
a
n
s
,
z
ϵ
ϵ
t
r
a
n
s
,
z
=
0.0001386
This corresponds to a dimension on the z-axis of
z
′
=
(
1
+
ϵ
t
r
a
n
s
,
z
)
z
z
′
=
20.0028
m
m