Structural Concrete - Hassoun

7.9 Moment–Resistance Diagram (Bar Cutoff Points) 281
c b + k tr 2.51 + 0.244
= = 2.16 < 2.5, use 2.16
d b 1.27
( ) 3 (60,000)(1.0)(1.0)(1.0)
l d =
40 (1) √ (1.28) =41.8in.
4000(2.16)
Class B splice = 1.3(32.83) =54.3in.
7.9 MOMENT–RESISTANCE DIAGRAM (BAR CUTOFF POINTS)
The moment capacity of a beam is a function of its effective depth, d, width, b, and the steel area for
given strengths of concrete and steel. For a given beam, with constant width and depth, the amount
of reinforcement can be varied according to the variation of the bending moment along the span. It
is a common practice to cut off the steel bars where they are no longer needed to resist the flexural
stresses. In some other cases, as in continuous beams, positive-moment steel bars may be bent up,
usually at 45 ∘ , to provide tensile reinforcement for the negative moments over the supports.
The factored moment capacity of an underreinforced concrete beam at any section is
M u = φA s f y
(
d − a 2
)
(7.18)
The lever arm (d–a/2) varies for sections along the span as the amount of reinforcement varies;
however, the variation in the lever arm along the beam length is small and is never less than the value
obtained at the section of maximum bending moment. Thus, it may be assumed that the moment
capacity of any section is proportional to the tensile force or the area of the steel reinforcement,
assuming proper anchorage lengths are provided.
To determine the position of the cutoff or bent points, the moment diagram due to external
loading is drawn first. A moment–resistance diagram is also drawn on the same graph, indicating
points where some of the steel bars are no longer required. The factored moment resistance of one
bar, M ub ,is
where
(
M ub = φA sb f y d − a )
2
(7.19)
a =
A sf y
0.85f cb
′
A sb = area of one bar
The intersection of the moment–resistance lines with the external bending moment diagram
indicates the theoretical points where each bar can be terminated. To illustrate this discussion,
Fig. 7.13 shows a uniformly loaded simple beam, its cross section, and the bending moment diagram.
The bending moment curve is a parabola with a maximum moment at midspan of 2400 K ⋅ in.
Because the beam is reinforced with four no. 8 bars, the factored moment resistance of one
bar is
(
M ub = φA sb f y d − a )
2
a =
A sf y 4 × 0.79 × 50
0.85f cb ′ =
0.85 × 3 × 12 = 5.2in.
(
M ub = 0.9 × 0.79 × 50 20 − 5.2 )
= 620K ⋅ in.
2