Civil Engineering Reference
In-Depth Information
From Figure 2.6 (a) it can be seen that
r
d
t
in the integral is equal to twice the area of the
shaded triangle formed by
r
and d
t
. Summing these areas around the whole cross-section
results in:
r
d
t
=
2
A
o
(2.45)
where
A
o
is the cross-sectional area bounded by the center line of the shear flow. This parameter
A
o
is a measure of the lever arm of the circulating shear flow and will be called the lever arm
area. Substituting 2
A
o
from Equation (2.45) into Equation (2.44) gives:
T
2
A
o
=
q
(2.46)
Equation (2.46) was first derived by Bredt (1896).
2.1.4.2 Torsion Equations
A shear element isolated from the wall of a tube of bulky cross-section, Figure 2.6(b), may
be subjected to a warping action in addition to the pure shear action discussed above. This
warping action will be taken into account in Chapter 7. If the warping action is neglected, then
this shear element becomes identical to the shear element in Figure 2.2 which is subjected
to pure shear only. As a result, the three equilibrium Equations (2.4)-(2.6) derived for the
element shear in Figure 2.2 become valid. Substituting
q
from Equation (2.46) into Equations
(2.4)-(2-6), we obtain the three equilibrium equations for torsion:
N
p
o
(2
A
o
) cot
T
=
α
r
(2.47)
T
=
n
t
(2
A
o
)tan
α
r
(2.48)
T
=
(
σ
d
h
)(2
A
o
)sin
α
r
cos
α
r
(2.49)
N
=
n
p
o
. This is because
n
, which is the longitudinal
force per unit length, must be multiplied by the whole perimeter of the shear flow
p
o
to arrive
at the total longitudinal force due to torsion
N
.
Assuming the yielding of the longitudinal and transverse steel, then
Notice in Equation (2.47) that
N
=
N
y
,
n
t
=
n
ty
,
and
T
=
T
y
. Multiplying Equations (2.47) and (2.48) to eliminate
α
r
gives
(2
A
o
)
(
N
y
/
T
y
=
p
o
)
n
ty
(2.50)
Combining Equations (2.47) and (2.48) to eliminate
T
we have
(
N
y
/
p
o
)
tan
α
r
=
(2.51)
n
ty
2.1.5 Summary of Basic Equilibrium Equations
A summary of the basic equilibrium equations for bending, element shear, beam shear and
torsion is given in Table 2.1. The table includes 18 equations, three for bending and five each
