1
4.3. A cantilever beam of rectangular cross section is loaded by a force
F
di
rected along the diagonal
AC
of the section as shown in Figure P4.3. Show that
the neutral axis in this case coincides with the other diagonal
BD
for all values
of the dimensions
a
,
b
of the rectangle.
F
A
B
C
D
a
b
Figure P4.3
The force
F
can be resolved into the components
Fa
√
a
2
+
b
2
;
Fa
√
a
2
+
b
2
shown in Figure P4.3.1(b).
a
b
F
FaL
√
2
2
a
+ b
FbL
√
2
2
a
+ b
Fb
√
2
2
a
+ b
Fa
√
2
2
a
+ b
(a)
(b)
(c)
Figure P4.3.1
It follows that the bending moments
M
x
,
M
y
on a crosssection distance
L
from
the free end are
2
M
x
=
FbL
√
a
2
+
b
2
;
M
y
=
FaL
√
a
2
+
b
2
,
as shown in Figure P4.3.1(c).
We also have
I
x
=
b
3
a
12
;
I
y
=
a
3
b
12
,
so
E
R
x
=
M
x
I
x
=
12
FbL
b
3
a
√
a
2
+
b
2
=
12
FL
b
2
a
√
a
2
+
b
2
E
R
y
=
M
y
I
y
=
12
FaL
a
3
b
√
a
2
+
b
2
=
12
FL
a
2
b
√
a
2
+
b
2
,
from equations (4.16, 4.17).
Substitution into equation (4.7) then gives
σ
zz
=
Ey
R
x

Ex
R
y
=
12
FLE
ab
√
a
2
+
b
2
parenleftBig
y
b

x
a
parenrightBig
.
The neutral axis corresponds to the condition
σ
zz
=
0 and hence
y
b
=
x
a
,