Civil Engineering Reference
In-Depth Information
ζ
f'
c
HOOP BAR
a
σ
d
LONGITUDINAL BAR
ε
ds
b
k
2
t
d
CENTERLINE OF
SHEAR FLOW ZONE
C
ε
d
t
d
h
σ
(x)
ε
t
d
dx
2
ψ
x
N.A.
(b) STRAIN DIAGRAM
(c) STRESS DIAGRAM
c
d
1
ε
ds
(a) CROSS-SECTION OF
CONCRETE STRUT
IN
S
-DIRECTION OF
FIG. 7.3
σ
ζ
f'
c
ε
d
s
=
ε
d
2
σ
(
ε
)
ε
N.A.
ε
ds
ε
d
ε
ASSUMED
(d) STRESS-STRAIN CURVE
OF SOFTENED CONCRETE
ε
d
Figure 7.4
Strains and stresses in concrete struts
The derivation of Equation [9] has been illustrated by a rectangular box section in Figure
7.3, because the imposed curvature is easy to visualize in such a section. In actuality, this
equation is applicable to any arbitrary bulky sections with multiple walls.
7.1.2.4 Strain Distribution in Concrete Struts
The curvature
derived in Equation [9] produces a nonuniform strain distribution in the
concrete struts. Figure 7.4(a) shows a unit width of a concrete strut in a hollow section with
a wall thickness
h
. The tension area in the inner portion of the cross-section is neglected.
The area in the outer portion that is in compression is considered to be effective to resist the
shear flow. The depth of the compression zone from the neutral axis (N.A.) to the extreme
compression fiber is defined as the thickness of the shear flow zone
t
d
. Within this thickness
t
d
, the strain distribution is assumed to be linear, as shown in Figure 7.4(b). This assumption
is identical to Bernoulli's plane section hypothesis used in the bending theory of Chapter 3.
The thickness
t
d
can, therefore, be related to the curvature
ψ
ψ
and the maximum strain at the
surface
ε
ds
by the simple relationship:
t
d
=
ε
ds
ψ
(7.23) or [10]
