Civil Engineering Reference
In-Depth Information
The solution for the analysis type of problems turns out to be quite simple when the
unknowns in each equation are examined. It can be seen from Equation (3.84) that the depth
a
can be determined directly from the equilibrium of forces,
C
and
T
. Then the moment
M
u
could be calculated either from the equilibrium of moments about the compressive force
C
(Equation 3.85), or from the equilibrium of moments about the tensile force
T
(Equation 3.86).
From the equilibrium of forces:
A
s
f
y
a
=
(3.87)
0
.
85
f
c
b
and from the equilibrium of moments about the compression force
C
:
A
s
f
y
d
a
2
M
u
=
ϕ
−
(3.88)
It should again be emphasized that the depth of the neutral axis
c
, which is equal to
/β
1
, can be determined directly from the equilibrium condition (Equation 3.87), without
using the compatibility condition and the stress-strain relationship of steel. This is a special
characteristic of mild steel, under-reinforced, concrete beams.
a
3.2.2.6 Design of Ductile Sections
First Type of Design (find area of steel)
Given:
b
,
d
,
M
u
,
f
y
,
f
c
and
ε
u
Find:
A
s
and
a
The three equilibrium equations and their unknowns are:
Type of equation
Equations
Unknowns
85
f
c
ba
Equilibrium of forces
A
s
f
y
=
0
.
A
s
a
(3
.
89)
A
s
f
y
d
a
2
Equilibrium of moment about
C
=
ϕ
−
A
s
a
(3
.
90)
u
85
f
c
ba
d
a
2
Equilibrium of moment about
T
=
ϕ
0
.
−
a
(3
.
91)
u
Examination of the two unknowns
A
s
and
a
in the three equations reveals two ways to arrive
at the solution:
First way.
The depth
a
is solved from the equilibrium of moment about
T
(Equation 3.91).
Then the depth a is inserted into either Equation (3.89) or Equation (3.90) to solve for the steel
area
A
s
. This way of solution is quite straightforward, but not often used because Equation
(3.91) is a quadratic equation for the depth
a
. The solution of a quadratic equation by hand is
somewhat tedious.
Second way
. The depth
a
is obtained by solving Equations (3.89) and (3.90) simultaneously
using trial-and-error method. This solution is actually quite simple for an experienced engineer
because the depth
a
can be closely estimated in the first trial. Assuming a depth
a
1
to be less
