Civil Engineering Reference
In-Depth Information
cause the diagram is discontinuous. There the concrete is assumed to be cracked and un-
able to resist tension. The value shown opposite the steel is the fictitious stress in the con-
crete if it could carry tension. This value is shown as
f
s
/
n
because it must be multiplied by
n
to give the steel stress
f
s
.
Examples 2.2, 2.3, and 2.4 are transformed-area problems that illustrate the calcula-
tions necessary for determining the stresses and resisting moments for reinforced concrete
beams. The first step to be taken in each of these problems is to locate the neutral axis,
which is assumed to be located a distance
x
from the compression surface of the beam. The
first moment of the compression area of the beam cross section about the neutral axis must
equal the first moment of the tensile area about the neutral axis. The resulting quadratic
equation can be solved by completing the squares or by using the quadratic formula.
After the neutral axis is located, the moment of inertia of the transformed section is cal-
culated, and the stresses in the concrete and the steel are computed with the flexure formula.
EXAMPLE 2.2
Calculate the bending stresses in the beam shown in Figure 2.7 by using the transformed area
method:
n
9 and
M
70 ft-k.
SOLUTION
Taking Moments about Neutral Axis (Referring to Figure 2.8)
x
2
(12
x
)
(9)(3.00)(17
x
)
6
x
2
459
27.00
x
Solving by Completing the Square
6
x
2
27.00
x
459
x
2
4.50
x
76.5
(
x
2.25)(
x
2.25)
76
.5
(2.25)
2
x
2.25
76.5
(2.25)
2
9.03
x
6.78
Moment of Inertia
I
(
3
)(12)(6.78)
3
(9)(3.00)(10.22)
2
4067 in.
4
Figure 2.7
Figure 2.8
