1
5.1. A beam of square cross section
a
×
a
is bent about a diagonal axis by a
moment
M
, as shown in Figure P5.1. Find the relation between
M
and the radius
of curvature
R
, if the material obeys the constitutive law
σ
zz
=
E
parenleftbigg
e
zz
+
e
3
zz
e
2
0
parenrightbigg
.
M
a
a
y
O
x
Figure P5.1
Using equations (5.17), we have
σ
zz
=
E
parenleftbigg
y
R
+
y
3
R
3
e
2
0
parenrightbigg
.
To perform the integral in the second of (5.16), we first note that the width of the
section,
w
(
y
)
in Figure P5.1.1, varies linearly from
√
2
a
at
y
=
0 to zero at
y
=
a
/
√
2
and hence
w
(
y
) =
√
2
a
parenleftBigg
1

√
2
y
a
parenrightBigg
=
√
2
a

2
y
.
2
a
a
y
O
w(y)
Figure P5.1.1
It follows that
M
=
M
x
=
integraldisplay integraldisplay
A
σ
zz
ydA
=
2
integraldisplay
a
/
√
2
0
σ
zz
w
(
y
)
ydy
=
2
E
integraldisplay
a
/
√
2
0
parenleftbigg
y
R
+
y
3
R
3
e
2
0
parenrightbigg
(
√
2
a

2
y
)
ydy
,
where we have used the symmetry of the section about the
x
axis to perform the
integral only for the upper triangle and then multiply by 2.