Civil Engineering Reference
In-Depth Information
(7.8b)
compressive stress of a non-softened standard cylinder, taken as positive
(
e
o
is defined as the
strain at the maximum compressive stress of non-softened concrete and can
be taken as -0.002. The factor
s
d,
e
d and
e
o are negative for compression). The strain
x
is a softening coefficient suggested to be
(7.8c)
The softening coefficient
x
, which is less than unity, is the reciprocal of the
coefficient
given in previous references (Vecchio and Collins, 1981; Hsu,
1984). The Poisson ratio µ in Eqn (7.8c) is taken as 0.3.
The stress-strain relationship in the r-direction can be expressed by
e
r
£e
cr
l
s
r
=
E
c
e
r
where
E
c
is the initial modulus of elasticity of concrete, taken to be -2
f
c
/
(7.9a)
e
o
with
e
cr
is the strain at cracking of concrete taken to be
f
cr
/
E
c
and
f
cr
is
the stress at cracking of concrete assumed to be
e
o
=-0.002,
where
f
'
c
and
f
cr
are
expressed in psi
(7.9b)
Eqns (7.9a and b) are plotted in Figure 7.5(b).
The stress-strain relationships for the longitudinal and transverse steel
bars are assumed to be elastic-perfectly plastic
e
l
‡e
l
y
f
l
=f
ly
e
l
<
e
l
y
f
l
=E
s
e
l
e
t
‡e
ty
f
t
=
f
ty
e
t
<
e
ty
f
t
=
E
s
e
t
where
E
s
is the modulus of elasticity of steel bars,
f
lr
,
f
ty
are yield stresses of
longitudinal and transverse steel bars, respectively and
e
ly
,
e
ty
are yield
strains of longitudinal and transverse steel bars, respectively.
The general equations of the softened truss model theory, Eqns (7.4), (7.7-
7.10) are described in a summary paper (Hsu, 1988). The equations for deep
beams, Eqns (7.5) and (7.6), are given in a separate paper (Mau and Hsu, 1987b).
(7.10a)
(7.10b)
(7.10c)
(7.10d)
7.3.5
Solution algorithm
Eqns (7.6) to (7.10) are to be solved for a pair of given
p
and
v
. However,
