Civil Engineering Reference
In-Depth Information
It can be seen that the strains
ε
and
ε
t
are constants from Equations (5.88) and (5.89). The
other variables
ε
d
from Equations (5.85)-(5.87) and (5.91), are all expressed in
terms of a single unknown variable
ρ
ρ
t
σ
d
and
α
r
. Substituting the strains
ε
,
ε
t
and
ε
d
from Equations
(5.88), (5.89) and (5.91) into Equation (5.70) results in
f
a
sin
α
r
cos
α
r
+
n
τ
t
cot
2
α
r
=
(5.92)
f
ta
sin
α
r
cos
α
r
+
n
τ
t
α
r
can be solved by Equation (5.92) using a trial-and-error method.
Examination of Equation (5.92) reveals two observations. First, when the allowable stresses
in the two directions are equal (
f
a
=
The angle
f
ta
), the Mohr compatibility truss model recommends
45
◦
. Second, the second terms in both the numerator and denominator
are smaller than the first terms by an order of magnitude. If the two
n
the use of an angle
α
r
=
τ
t
terms are neglected
in Equation (5.92), then
f
a
f
ta
cot
2
α
r
=
(5.93)
α
r
obtained from Equation (5.93) should be a close approximation to that obtained
from Equation (5.92) and becomes exact when
f
a
=
The angle
f
ta
. Once the angle
α
r
is determined, all
the nine unknown variables
ρ
,
ρ
t
,
σ
d
,
ε
,
ε
t
,
γ
t
,
ε
r
,
ε
d
,
α
r
, can be calculated.
5.4 Rotating Angle Softened Truss Model (RA-STM)
5.4.1 Basic Principles of RA-STM
In Section 5.3 we have studied the Mohr compatibility truss model which utilizes nine equations
to analyze the behavior of RC 2-D elements subjected to external membrane stresses. The nine
equations include three for equilibrium, three for compatibility and three for the constitutive
laws of materials. Since the constitutive equations for both concrete and steel are based on
Hooke's law, the predicted behavior of the 2-D element is linear. The Mohr compatibility truss
model, therefore, provides a method of linear analysis.
In this section we will introduce a method of nonlinear analysis for RC 2-D elements, which
also satisfies Navier's three principles of mechanics of materials. In this model the constitutive
equations are based on the actual, observed stress-strain relationships of concrete and steel.
The stress-strain curve of concrete must reflect two characteristics. The first is the nonlinear
relationship between stress and strain. The second, and perhaps more important, is the softening
of concrete in compression, caused by cracking due to tension in the perpendicular direction.
Consequently, a softening coefficient will be incorporated in the equation for the compressive
stress-strain relationship of concrete.
In view of the crucial importance of the softening effect on the biaxial constitutive laws
of reinforced concrete, this model has been named the 'softened truss model'. The word
'softened' implies two characteristics: first, the analysis must be nonlinear and, second, the
softening of concrete must be taken into account.
In Section 5.4 we will extend the applicability of the rotating angle softened truss
model to prestressed concrete (PC). The three equilibrium equations will include the terms
ρ
p
f
p
and
ρ
tp
f
tp
that represent the smeared stresses of the prestressing steel, as shown in
