5.8 Bending of a Simply Supported Beam by a Distributed Loading …
149
5.8 Bending of a Simply Supported Beam by a Distributed
Loading (Udl)
Consider a beam of rectangular cross-section having unit width, supported at the ends
and subjected to a uniformly distributed load of intensity q as shown in Fig. 5.10.
It is to be noted that the bending moment is a maximum at position x = 0 and
decreases with a change in x in either the positive or the negative direction. This is
possible only if the stress function contains even functions of x. Also, it should be
noted that σ y varies from zero at y = −h to a maximum value of −q at y = +h.
Hence, the stress function must contain odd functions of y.
Now, consider a polynomial of second degree with
b 2 = c 2 = 0
∴ φ 2 =
a 2
2
x
2
a polynomial of third degree with a 3 = c 3 = 0
∴ φ 3 =
b 3
2
x
2 y +
d 3
6
y
3
and a polynomial of fifth degree with a 5 = b 5 = c 5 = e 5 = 0
∴ φ 5 =
d 5
6
x
2 y
3
−
d 5
30
y
5
∵ f 5 = −
2
3
d 5
∴ φ = φ 2 + φ 3 + φ 5
or
Fig. 5.10 Beam subjected to Uniform load
149
5.8 Bending of a Simply Supported Beam by a Distributed
Loading (Udl)
Consider a beam of rectangular cross-section having unit width, supported at the ends
and subjected to a uniformly distributed load of intensity q as shown in Fig. 5.10.
It is to be noted that the bending moment is a maximum at position x = 0 and
decreases with a change in x in either the positive or the negative direction. This is
possible only if the stress function contains even functions of x. Also, it should be
noted that σ y varies from zero at y = −h to a maximum value of −q at y = +h.
Hence, the stress function must contain odd functions of y.
Now, consider a polynomial of second degree with
b 2 = c 2 = 0
∴ φ 2 =
a 2
2
x
2
a polynomial of third degree with a 3 = c 3 = 0
∴ φ 3 =
b 3
2
x
2 y +
d 3
6
y
3
and a polynomial of fifth degree with a 5 = b 5 = c 5 = e 5 = 0
∴ φ 5 =
d 5
6
x
2 y
3
−
d 5
30
y
5
∵ f 5 = −
2
3
d 5
∴ φ = φ 2 + φ 3 + φ 5
or
Fig. 5.10 Beam subjected to Uniform load