Civil Engineering Reference
In-Depth Information
relationships of these three equations are described by the second type of expression for Mohr
circle in Figure 2.3(c).
The two equations of each set in the right-hand column are derived from the basic equations
in the left-hand column, assuming that the steels in both the longitudinal and transverse
directions have yielded. In the typical case of element shear the angle
α
r
at yield is obtained by
eliminating
q
from the first two basic equations, while the shear flow at yield,
q
y
, is obtained
by eliminating
α
r
.
The equations shown in Table 2.1 for pure bending, pure shear and pure torsion are all
derived in a consistent and logical manner. Such clarity in concept makes the interaction of
these actions relatively simple. These interaction relationships will be enunciated in the next
Section 2.2.
2.2 Interaction Relationships
2.2.1 Shear-Bending Interaction
A model of beam subjected to shear and bending is shown in Figure 2.7(a). The moment
M
creates a tensile force
M
d
v
in the bottom stringer and an equal compressive force in the top
stringer. The shear force
V
, however, is acting on a shear element as shown in Figure 2.7(d).
It induces a total tensile force of
V
tan
/
α
r
in the longitudinal steel, Figure 2.7(c). Due to
symmetry, the top and bottom stringers should each resist one-half of the tensile force,
(
V
tan
α
r
)
/
2. In the transverse direction, the shear force V will produce a transverse force
n
t
d
v
, in the transverse steel, Figure 2.7(b).
Figure 2.7
Equilibrium in shear-bending interaction
