Civil Engineering Reference
In-Depth Information
however, does not alter the equilibrium condition, as shown by the two force triangles in
the longitudinal and transverse directions. To take care of the nonuniform distribution of the
longitudinal steel we simply define
N
=
n
d
v
, which is the total force in the longitudinal steel
to resist shear.
Equilibrium of the main body in the longitudinal, transverse and diagonal directions gives
the following three equations:
N
cot
V
=
α
r
(2.38)
V
=
n
t
d
v
tan
α
r
(2.39)
V
=
(
σ
d
h
)
d
v
sin
α
r
cos
α
r
(2.40)
This set of three Equations (2.38)-(2.40) is identical to the set of three Equations (2.4)-(2.6)
for element shear, if the latter three equations are multiplied by the length
d
v
.
Assuming the yielding of the longitudinal and transverse steel, then
N
=
N
y
,
n
t
=
n
ty
,
and
V
=
V
y
. Multiplying Equations (2.38) and (2.39) to eliminate
α
r
gives
(
N
y
/
V
y
=
d
v
d
v
)
n
ty
(2.41)
Combining Equations (2.38) and (2.39) to eliminate
V
we have
N
y
/
d
v
n
ty
tan
α
r
=
(2.42)
In design, the total longitudinal steel force
N
y
is divided equally between the top and
bottom stringers. For the design of the bottom steel bars, this bottom tensile force of
N
y
/2 due
to shear is added to the longitudinal tensile force
M
d
v
due to bending. The design of the top
steel bars, however, is less certain. In theory, the top tensile force of
N
y
/2 due to shear could
be subtracted from the longitudinal compressive force
M
/
d
v
due to bending. Such measure
would be nonconservative. Perhaps a better approach is to select the larger of the two forces
M
/
d
v
or
N
y
/2, for design purposes.
In the design of beams, it is also cost effective to select an
/
α
r
value greater than 45
◦
, because
the transverse steel, in the form of stirrups, is more costly than the longitudinal steel bars.
2.1.4 Equilibrium in Torsion
2.1.4.1 Bredt's Formula Relating
T
and
q
A hollow prismatic member of arbitrary bulky cross section and variable thickness is subjected
to torsion, as shown in Figure 2.6(a). According to St. Venant's theory the twisting deformation
will have two characteristics. First, the cross-sectional shape will remain unchanged after the
twisting; and second, the warping deformation perpendicular to the cross-section will be
identical throughout the length of the member. Such deformations imply that the in-plane
normal stresses in the wall of the tube member should vanish. The only stress component
in the wall is the in-plane shear stress, which appears as a circulating shear flow
q
on the
cross-section. The shear flow
q
is the resultant of the shear stresses in the wall thickness and
