Civil Engineering Reference
In-Depth Information
Now that the position of the neutral axis, represented by
k
, is solved, the rebar stresses
f
s
can be obtained from the equilibrium of moments about the resultant
C
using Equation (3.8):
M
A
s
(1
f
s
=
(3.17)
k
−
3
)
d
and the concrete stress
f
c
can be obtained either from the equilibrium of forces (Equation
3.10), or more directly in this case (since
M
is given) from the equilibrium of moments about
the tensile force
T
:
M
f
c
=
(3.18)
1
k
2
k
(1
−
3
)
bd
2
Once the stresses,
f
s
and
f
c
, are found, the strains,
ε
s
and
ε
c
, can be calculated from Hooke's
law by Equations (3.10) and (3.11). Knowing
k
and
ε
c
, the curvature
φ
can be calculated from
the compatibility equation (3.6).
3.1.1.5 Solution for Second Type of Analysis Problem
The solution of the second type of analysis is similar to the first type described above. The
similarity can be observed by examining the five unknown variables in the five available
equations. The problem posed is:
Given four variables:
b
,
d
,
A
s
,
f
s
(or
f
c
)
Find five unknown variables:
M
,
f
c
(or
f
s
),
ε
s
,
ε
c
,
k
The variables
f
c
and
f
s
in the parentheses should be understood as follows. If the given
variable
f
s
is replaced by
f
c
in the parenthesis, then the unknown variables
f
c
must be replaced
by
f
s
in the parenthesis.
The five equations, (3.7)-(3.11), are still valid in this case, but the list of unknowns under
the column heading
Unknowns
should be revised. The stress
f
s
(or
f
c
) becomes a known value
and should be replaced by the new unknown
M
. The most efficient algorithm of solution still
consists of the following three steps:
Step 1:
Utilize the stress-strain relationships of rebar and concrete to express Bernoulli's
compatibility equation in terms of stresses, thus arriving at Equation (3.12). In this way a
set of five equations is reduced to a set of three equations, (3.7), (3.8) and (3.12), which
involve only three unknowns,
M
,
f
c
(or
f
s
) and
k
.
Step 2:
Of the three equations, both Equation (3.7) and (3.12) contain the same two unknowns,
f
c
(or
f
s
) and
k
. Simultaneous solution of these two equations results in Equation (3.16),
expressing the unknown variable
k
.
Step 3:
Once the position of the neutral axis,
k
, is determined, the unknowns
f
c
(or
f
s
) can be
calculated from the force equilibrium equation, and the unknown
M
from any one of the
two moment equilibrium equations. The strains,
ε
s
and
ε
c
, can easily be calculated from the
stresses
f
s
and
f
c
by Hooke's laws.
