Page 247 - Handbook of Civil Engineering Calculations, Second Edition

2.32 REINFORCED AND PRESTRESSED CONCRETE ENGINEERING AND DESIGN

5. Calculate the moment of inertia of the transformed cracked
section at the center
Referring to Fig. 19e and assuming tentatively that the neutral axis falls within the flange,
2
we see nA s 10(3.16) 31.6 sq.in. (203.88 cm ). The static moment with respect to the
2
1
neutral axis is Q /2(68y ) 31.6(20.5 y) 0; y 3.92 in. (99.568 mm). The neu-
tral axis therefore falls within the flange, as assumed. The moment of inertia with respect
4
4
2
1
to the neutral axis is I 2 ( /3)68(3.92)3 31.6(20.5 3.92) 10,052 in (41.840 dm ).
6. Calculate the deflection at midspan
Use the equation
L
2 5M 1 M 3
(37)
EI
48 8
4
4
1
where I average moment of inertia, in (dm ). Thus, I /2(9737 10,052) 9895 in 4
4
(41.186 dm ); E 145 1.5 33f c
) 0.5 57,600(2500) 0.5 2,880,000 lb/sq.in. (19,857.6
2
MPa). Then [22 1728/(2880 9895)](5 200/48 100/8) 0.244 in.
(6.198 mm).
Where the deflection under sustained loading is to be evaluated, it is necessary to
apply the factors recorded in the ACI Code.
Design of Compression Members
by Ultimate-Strength Method
The notational system is P u ultimate axial compressive load on member, lb (N); P b
ultimate axial compressive load at balanced design, lb (N); P 0 allowable ultimate axial
compressive load in absence of bending moment, lb (N); M u ultimate bending moment
in member, lb.·in (N·m); M b ultimate bending moment at balanced design; d
dis-
tance from exterior surface to centroidal axis of adjacent row of steel bars, in. (mm); t
overall depth of rectangular section or diameter of circular section, in. (mm).
A compression member is said to be spirally reinforced if the longitudinal reinforce-
ment is held in position by spiral hooping and tied if this reinforcement is held by means
of intermittent lateral ties.
The presence of a bending moment in a compression member reduces the ultimate ax-
ial load that the member may carry. In compliance with the ACI Code, it is necessary to
design for a minimum bending moment equal to that caused by an eccentricity of 0.05t
for spirally reinforced members and 0.10t for tied members. Thus, every compression
member that is designed by the ultimate-strength method must be treated as a beam col-
umn. This type of member is considered to be in balanced design if failure would be char-
acterized by the simultaneous crushing of the concrete, which is assumed to occur when

c 0.003, and incipient yielding of the tension steel, which occurs when f s f y . The
ACI Code set 0.75 for spirally reinforced members and 0.70 for tied members.

ANALYSIS OF A RECTANGULAR MEMBER
BY INTERACTION DIAGRAM
A short tied member having the cross section shown in Fig. 20a is to resist an axial load and
a bending moment that induces compression at A and tension at B. The member is made of
3000-lb/sq.in. (20,685-kPa) concrete, and the steel has a yield point of 40,000 lb/sq.in.