Civil Engineering Reference
In-Depth Information
8
Beams in Shear
8.1 Plasticity Truss Model for Beam Analysis
8.1.1 Beams Subjected to Midspan Concentrated Load
A plasticity truss model of a beam subjected to a concentrated load at midspan is shown in
Figure 8.1(a). The truss model is made up of two parallel top and bottom stringers, a series
of transverse steel bars spaced uniformly at a spacing of
s
, and a series of diagonal concrete
struts. The beam can be divided into two types of regions (main and local) depending on the
α
r
angle of the diagonal concrete struts. The main region is the one where the
α
r
angle is a
constant, so that the series of diagonal struts are parallel to each other. Regularity of the truss
in this region makes simple analysis possible based on the sectional actions of
M
,
V
,
T
and
P
. Analysis of elements from such regions has been made in Sections 2.1 and 2.2. The local
region is the one where the
α
r
angle varies. Such regions lie in the vicinity of the concentrated
loads, including the end reaction region and the midspan region under the load.
In the end reaction region, the local effect is shown in Figure 8.1(a) as the 'fanning' of the
compression struts from the application point of the concentrated load. In this case the angle
varies from 90
◦
to
α
r
and each concrete strut has a narrow triangular shape. Since the angle of
the principal compressive stress is assumed to coincide with the angle of a concrete strut, the
compressive forces will radiate upward from the load application point to form truss action
with the forces in the top stringer and in the transverse steel bars. Similarly, Figure 8.1(a) also
shows the 'fanning' of the compression struts under the concentrated load at midspan. Because
the load application point is on the top surface, the compressive forces radiate downward and
form truss action with the forces in the bottom stringer and in the transverse steel bars.
Now let us analyze how the forces vary in the transverse and longitudinal reinforcement in
the main regions of the beam. The depth of the beam is represented by
d
v
, the distance between
the centerlines of top and bottom stringers. To simplify the analysis we select the following
three geometric relationships: (1) angle of concrete struts tan
α
r
=
5
/
3; (2) spacing of stirrups
s
α
r
angle is
close to 60
◦
, the maximum allowed by the ACI Code and should be the most economical. The
spacing
s
and the half-span
=
d
v
/
3; and (3) half-span length
/
2
=
4(
d
v
tan
α
r
)
=
(20
/
3)
d
v
=
20
s
.The
/
2 are typical in practice.
