Civil Engineering Reference
In-Depth Information
solved. Depending on the given variables, the problems are generally categorized into five
types as shown in the following table:
Type of problem
Given variables
Unknown variables
(1) First type of analysis:
bd A
s
M
f
s
f
c
ε
s
ε
c
k
(2) Second type of analysis:
bd A
s
f
s
(or
f
c
)
Mf
c
(or
f
s
)
ε
s
ε
c
k
(3) Balanced design:
bd f
s
f
c
MA
s
ε
s
ε
c
k
(4) First type of design:
bd M f
s
(or
f
c
)
A
s
f
c
(or
f
s
)
ε
s
ε
c
k
(5) Second type of design:
b
(or
d
)
Mf
s
f
c
d
(or
b
)
A
s
ε
s
ε
c
k
The first two types of problem are called analysis, because the cross-sectional properties of
concrete and rebars,
b
,
d
and
A
s
, are given. The last three types are called design, because at
least one of these three cross-sectional properties, primarily
A
s
, is an unknown.
The last three types of problem were known as the allowable stress design method, prevalent
prior to 1971. The allowable stress design method had been made obsolete by the ultimate
strength design method and, therefore, will not be treated in this topic. However, the two types
of analysis problems are still very relevant at present for checking the serviceability criteria,
i.e. the deflections and the crack widths.
The problems in the analysis and design of flexural members boils down to finding the most
efficient way to solve the five available equations for each type of problem. The methodology
of the solution process is demonstrated in the next section, where the five equations are applied
to the two types of analysis problems.
3.1.1.4 Solution for First Type of Analysis Problem
We will first look at the first type of analysis problem, which is posed as follows:
Given four variables:
b
,
d
,
A
s
,
M
Find five unknown variables:
f
s
,
f
c
,
ε
s
,
ε
c
,
k
ε
c
and
k
, are defined in Figure. 3.1 for a singly
reinforced flexural member. Because four variables are given, the remaining five unknown
variables can be solved by the five available equations. These equations and their unknown
variables are summarized as follows:
ε
s
,
The nine variables,
b
,
d
,
A
s
,
M
,
f
s
,
f
c
,
Type of equations
Equations
Unknowns
1
2
Equilibrium of forces
A
s
f
s
=
f
c
kbd
f
s
f
c
k
(3
.
7)
A
s
f
s
1
d
k
3
Equilibrium of moments
M
=
−
f
s
k
(3
.
8)
ε
s
ε
c
=
1
−
k
Bernoulli compatibility
ε
s
ε
c
k
(3
.
9)
k
Hooke's law for rebar
f
s
=
E
s
ε
s
f
s
ε
s
(3
.
10)
Hooke's law for concrete
f
c
=
E
c
ε
c
f
c
ε
c
(3
.
11)
