Civil Engineering Reference
In-Depth Information
2.4
ULTIMATE OR NOMINAL FLEXURAL MOMENTS
In this section a very brief introduction to the calculation of the ultimate or nominal
flexural strength of beams is presented. This topic is continued at considerable length
in the next chapter, where formulas, limitations, designs, and other matters are pre-
sented. For this discussion it is assumed that the tensile reinforcing bars are stressed to
their yield point before the concrete on the compressive side of the beam crushes. We
will learn in Chapter 3 that the ACI Code requires all our beam designs to fall into this
category.
After the concrete compression stresses exceed about they no longer vary di-
rectly as the distance from the neutral axis or as a straight line. Rather, they vary much as
shown in Figure 2.11(b). It is assumed for the purpose of this discussion that the curved
compression diagram is replaced with a rectangular one with an average stress of ,
as shown in part (c) of the figure. The rectangular diagram of depth
a
is assumed to have
the same c.g. (center of gravity) and total magnitude as the curved diagram. (In Section
3.4 of Chapter 3 of this text we will learn that this distance
a
is set equal to
c
,
0.50
f
c
0.85
f
1
is a value determined by testing.) These assumptions will enable us easily to calculate the
theoretical or nominal flexural strength of reinforced concrete beams. Experimental tests
show that with the assumptions used here, accurate flexural strengths are determined.
To obtain the nominal or theoretical moment strength of a beam, the simple steps to
follow are used as is illustrated in Example 2.6.
1
c
, where
1.
Compute total tensile force
T
A
s
f
y
.
2.
Equate total compression force
C
c
ab
to
A
s
f
y
and solve for
a
. In this ex-
pression
ab
is the assumed area stressed in compression at The compres-
sion force
C
and the tensile force
T
must be equal to maintain equilibrium at the
section.
3.
Calculate the distance between the centers of gravity of
T
and
C
. (For a rectangu-
lar section it equals
d
0.85
f
c
.
0.85
f
a
/2.)
4.
Determine
M
n
, which equals
T
or
C
times the distance between their centers of
gravity.
Figure 2.11
Compression and tension couple at nominal moment.
