Civil Engineering Reference
In-Depth Information
than
a
max
=
0
.
375
β
1
d
t
, and substituting it into Equation (3.89) gives the value of
A
s
1
for the
first cycle as
M
u
A
s
1
=
f
y
d
2
(3.92)
a
1
ϕ
−
Then inserting
A
s
1
into Equation (3.89) to calculate the depth
a
2
:
A
s
1
f
y
a
2
=
(3.93)
0
.
85
f
c
b
If
a
2
=
a
1
, a solution is found. If
a
2
=
a
1
, then assume another
a
2
value and repeat the cycle.
The convergence is usually very rapid, and two or three cycles of trial and error is usually
sufficient to give an accurate solution.
Second Type of Design (find cross sections of concrete and steel)
Given:
M
u
,
f
y
,
f
c
and
ε
u
Find:
b
,
d
,
A
s
and
a
Notice that four unknowns are shown. Obviously, only two of these four unknowns can be
determined by the two available equations. Therefore, two unknowns will have to be assumed
in the process of design. It is a characteristic of design problems that the number of unknowns
frequently exceeds the available number of equations.
The three equilibrium equations and their unknowns are:
Type of equation
Equations
Unknowns
Equilibrium of forces
A
s
f
y
=
0
.
85
f
c
ba
b
A
s
a
(3
.
94)
A
s
f
y
d
1
a
2
d
Equilibrium of moment about
CM
u
=
ϕ
−
.
dA
s
a
(3
95)
85
f
c
bda
1
a
2
d
Equilibrium of moment about
T
=
ϕ
0
.
−
bd
a
(3
.
96)
u
Two methods of design have been used:
First Method.
Assume a suitable percentage of steel
ρ
max
.This
means that we are specifying the ratio of the two quantities that are being designed.
From equilibrium of forces (Equation 3.94):
ρ
=
A
s
/
bd
,say
ρ
=
0
.
5
a
d
=
A
s
f
y
ρ
f
y
85
f
c
bd
=
(3.97)
0
.
0
.
85
f
c
/
d
from Equation (3.97) into the moment equilibrium equation about
C
(Equation 3.95), and noticing
A
s
=
ρ
Inserting
a
bd
, we can solve for a parameter
bd
2
for the concrete
cross-section:
M
u
bd
2
f
y
1
f
c
=
(3.98)
59
ρ
f
y
ϕρ
−
0
.
