Civil Engineering Reference
In-Depth Information
S
y
δ
x
y
y
2
δ
y
y
b
0
A
G
y
1
z
F
IGURE
13.20
Determination of form
factor
β
in which
S
is the applied shear force,
A
is the cross-sectional area of the beam section
and
β
is a constant which depends upon the distribution of shear stress through the
beam section;
β
is known as the
form factor
.
To determine
β
we consider an element
b
0
δ
x
of a beam
subjected to a vertical shear load
S
y
(Fig. 13.20); we shall suppose that the beam
section has a vertical axis of symmetry. The shear stress
τ
is constant across the width,
b
0
, of the element (see Section 10.2). The strain energy,
y
in an elemental length
δ
δ
U
, of the element
b
0
δ
y
δ
x
,
from Eq. (10.20) is
τ
2
2
G
×
δ
U
=
b
0
δ
y
d
x
(13.17)
Therefore the total strain energy
U
in the elemental length of beam is given by
y
2
x
2
G
δ
τ
2
b
0
d
y
U
=
(13.18)
y
1
Alternatively
U
for the elemental length of beam is obtained using Eq. (13.16); thus
S
y
A
2
β
2
G
×
U
=
×
A
δ
x
(13.19)
Equating Eqs (13.19) and (13.18) we have
S
y
A
2
y
2
β
2
G
×
x
2
G
δ
τ
2
b
0
d
y
×
A
δ
x
=
y
1
whence
y
2
A
S
y
τ
2
b
0
d
y
β
=
(13.20)
y
1
The shear stress distribution in a beam having a singly or doubly symmetrical cross
section and subjected to a vertical shear force,
S
y
, is given by Eq. (10.4), i.e.
S
y
A
¯
y
=−
τ
b
0
I
z
