Civil Engineering Reference
In-Depth Information
The four available equations and their unknowns are:
Type of equation
Equations
Unknowns
85
f
c
ba
A
s
f
s
f
s
Equilibrium of forces
A
s
f
y
=
0
.
+
a
(3
.
102)
0
85
f
c
ba
d
2
a
.
−
f
s
Equilibrium of moment about
T
=
ϕ
M
u
a
(3
.
103)
u
A
s
f
s
(
d
d
)
+
−
ε
s
ε
u
=
d
c
−
ε
s
a
Compatibility of compression steel
(
c
=
a
/β
1
)
(3
.
104)
c
f
s
=
E
s
ε
s
ε
s
≤
ε
y
f
s
,ε
s
.
Constitutive law of compression steel
for
(3
105a)
f
s
=
ε
s
>ε
y
f
s
,ε
s
.
f
y
for
(3
105b)
It should be pointed out that an approximation has been introduced in writing Equations
(3.102) and (3.103). The compression steel force
A
s
f
s
should have been written as
A
s
(
f
s
−
0
85
f
c
) to include the negative force attributed to the area of concrete displaced by the
compression steel. Since 0
.
85
f
c
is smaller than
f
s
.
by an order of magnitude, this small force
85
f
c
) has been neglected in the two equations.
Examination of the unknowns in the four equations points indicates two methods to solve
these equations:
A
s
(0
.
Method 1.
(Trial-and-error method)
=
/β
1
) and calculate the compression steel strain
ε
s
Step 1:
Assume a value of depth
a
(
c
a
from the compatibility condition of Equation (3.104).
Step 2:
Calculate the compression steel stress
f
s
from the stress-strain relationship of com-
pression steel (Equation 3.105a or b).
Step 3:
Insert
f
s
into the force equilibrium equation (3.102) to calculate a new value of the
depth
a
. If the new
a
is the same as the assumed
a
, a solution is obtained. If not, assume
another value of
a
and repeat cycle. The convergence is usually quite rapid.
Step 4:
Once the depth
a
and the compression steel stress
f
s
is solved, insert them into the
moment equilibrium equation (3.103) to calculate the moment
M
u
.
Method 2
. (solve quadratic equation)
ε
s
from Equation (3.105a) into Equation (3.104)
and express the compatibility of compression steel by a new equation in terms of the
compression steel stress
f
s
and the depth
a
.
Step 2:
Solve the new compatibility equation simultaneously with the force equilibrium equa-
tion (3.102) to obtain the stress
f
s
Step 1:
Insert the compression steel strain
and the depth
a
. This is the process of solving a quadratic
equation. If
f
s
≥
f
y
, then use
f
s
=
f
y
according to Equation (3.105b) and recalculate the
depth
a
from Equation (3.102).
Step 3:
Substitute the stress
f
s
and the depth
a
into Equation (3.103) to determine the moment
M
u
.
