Civil Engineering Reference
In-Depth Information
It is interesting to point out that the equilibrium of a parallel coplanar force system will
furnish two independent equations. The three simplest forms of equilibrium equations are:
(1) equilibrium of forces in the longitudinal direction; (2) equilibrium of moments about the
resultant compression force of concrete; and (3) equilibrium of moments about the tensile
force of rebars. The selection of two of these three equilibrium equations is strictly a matter
of convenience. It must be emphasized that the equilibrium condition in bending can only be
used to solve two unknowns, even if all three equilibrium equations are used.
The solution of the five unknowns (
f
s
,
f
c
,
ε
c
and
k
) by the five equations, (3.7)-(3.11), is
facilitated by identifying the unknown variables in each equation. These variables are shown
after each equation under the column heading
Unknowns
.
In the set of five equations, (3.7)-(3.11), the first two equilibrium equations deal with
unknown stresses, while the third equation expresses Bernoulli's compatibility condition in
terms of unknown strains. Substituting the stress-strain relationships of Equations (3.10) and
(3.11) into Equation (3.9) we obtain Bernoulli's compatibility equation in terms of stresses:
ε
s
,
f
s
nf
c
=
1
−
k
(3.12)
k
where
n
modulus ratio. With this simple maneuvering, we have now reduced a
set of five equations to a set of three equations, (3.7), (3.8) and (3.12), which involve only
three unknowns,
f
s
,
f
c
and
k
.
Of the three equations (3.7), (3.8) and (3.12), the first and the last have the unknown
k
and
the unknown stress ratio
f
s
/
=
E
s
/
E
c
=
f
c
. Substituting the ratio
f
s
/
f
c
from Equation (3.7) into Equation
(3.12) gives an equation with only one unknown
k
:
kbd
2
nA
s
=
1
−
k
(3.13)
k
Defining
ρ
=
A
s
/
bd
=
percentage of rebars, Equation (3.13) becomes
k
2
n
1
−
k
=
(3.14)
ρ
k
The unknown
k
can be solved by Equation (3.14), using whatever method that is convenient.
Two methods are generally used. The first is the trial-and-error method. A value of
k
is assumed
and inserted into Equation (3.14). If the equation is satisfied, the assumed
k
is the solution.
If the equation is not satisfied, another
k
value is assumed and the process is repeated. This
trial-and-error method could be quite efficient if the
k
value could be closely estimated in the
first trial. It is, therefore, convenient for engineers with experience.
The second method is to rewrite Equation (3.14) in the form of a second-order equation:
k
2
+
(2
n
ρ
)
k
−
2
n
ρ
=
0
(3.15)
From Equation (3.15),
k
can be determined by the formula for quadratic equation:
(
n
k
=
ρ
)
2
+
2
n
ρ
−
n
ρ
(3.16)
