Civil Engineering Reference
In-Depth Information
, which is less than one. These values
generally vary from 0.90 for bending down to 0.65 for some columns.
In summary, the strength design approach to safety is to select a member whose com-
puted ultimate load capacity multiplied by its strength reduction factor will at least equal
the sum of the service loads multiplied by their respective load factors.
Member capacities obtained with the strength method are appreciably more accurate
than member capacities predicted with the working-stress method.
of each member by the
strength reduction factor
3.4
DERIVATION OF BEAM EXPRESSIONS
Tests of reinforced concrete beams confirm that strains vary in proportion to distances
from the neutral axis even on the tension sides and even near ultimate loads. Compression
stresses vary approximately in a straight line until the maximum stress equals about
This is not the case, however, after stresses go higher. When the ultimate load is
reached, the strain and stress variations are approximately as shown in Figure 3.1.
The compressive stresses vary from zero at the neutral axis to a maximum value at or
near the extreme fiber. The actual stress variation and the actual location of the neutral
axis vary somewhat from beam to beam depending on such variables as the magnitude
and history of past loadings, shrinkage and creep of the concrete, size and spacing of ten-
sion cracks, speed of loading, and so on.
If the shape of the stress diagram were the same for every beam, it would easily be
possible to derive a single rational set of expressions for flexural behavior. Because of
these stress variations, however, it is necessary to base the strength design on a combina-
tion of theory and test results.
Although the actual stress distribution given in Figure 3.2(b) may seem to be impor-
tant, any assumed shape (rectangular, parabolic, trapezoidal, etc.) can be used practically
if the resulting equations compare favorably with test results. The most common shapes
proposed are the rectangle, parabola, and trapezoid, with the rectangular shape used in
this text as shown in Figure 3.2(c) being the most common one.
If the concrete is assumed to crush at a strain of about 0.003 (which is a little conser-
vative for most concretes) and the steel to yield at
f
y
, it is possible to make a reasonable
derivation of beam formulas without knowing the exact stress distribution. However, it is
necessary to know the value of the total compression force and its centroid.
c
.
0.50
f
Figure 3.1
Ultimate load.
