Building Design and Construction Handbook - Merritt - Ventech!

9.50 SECTION NINE
9.46.1 Singly-Reinforced Rectangular Beams
For a rectangular beam, reinforced with only tension steel (Fig. 9.12), the total
tension force in the steel at nominal (ultimate) strength is
It is opposed by an equal compressive force
T � A ƒ � �ƒ bd (9.23)
s y y
C � 0.85ƒ�b� c (9.24)
c 1
where ƒ�c � specified compressive strength of the concrete, psi
c � distance from extreme compression surface to neutral axis
�1 � a constant (given in Art. 9.44)
Equating the compression and tension forces at the critical section gives:
�ƒ y
c � d (9.25)
0.85� ƒ�
1 c
The criterion for compression failure is that the maximum strain in the concrete
equals 0.003 in/in. In that case:
0.003
c � d (9.26)
ƒ/E � 0.003
s s
where ƒ s is the steel stress, ksi, and E s � 29,000,000 psi is the steel modulus of
elasticity.
Tension-Steel Limitations. Under balanced conditions, the concrete will reach its
maximum strain of 0.003 in/in when the tension steel reaches its yield strength ƒ y.
Then, c as given by Eq. (9.26) will equal c as given by Eq. (9.25). Also, the
reinforcement ratio for balanced conditions in a rectangular beam with tension steel
only becomes:
0.85� ƒ� 87,000
1 c
�b � (9.27)
ƒ 87,000 � ƒ
y y
All structures should be designed to avoid sudden collapse. Therefore, reinforcement
should yield before the concrete crushes. Gradual yielding will occur if the
quantity of tensile reinforcement is less than the balanced percentage determined
by strength design theory. To avoid compression failures, the ACI 318 Building
Code, therefore limits the reinforcement ratio � to a maximum of 0.75�b. The Code also requires that � for positive-moment and negative-moment rein-
forcement be at least 3�ƒ�/ƒ and not less than 200/ƒ y to prevent sudden collapse
c y
when the design moment strength is equal to or less than the cracking moment.
This requirement does not apply, however, if the reinforcement area at every section
of the member is at least one-third greater than that required by the factored moment.
The ratio, 200/ƒ y, will govern, except when � 4400 psi. For a statically
determinate T-section with the flange in tension, � should be at least 6�ƒ�/ƒ with
the flange width used for determining �.
For flexural members of any cross-sectional shape, without compression reinforcement,
the tension reinforcement is limited by the ACI 318 Building Code so
ƒ� c
c y