Simplified Design for Beams and One-Way Slabs - Simplified Design of Reinforced Concrete Buildings - page 50

For roof beams and slabs, the values are intended for roofs subjected to normal snow or construction live loads
only, and with minimal water ponding or drifting snow problems.

Prudent choice of steel percentage can also minimize deflection problems. Members will usually be of
sufficient size, so that deflections will be within acceptable limits, when the tension reinforcement ratio used

in the positive moment regions does not exceed approximately one-half of the maximum value permitted.
For › = 4000 psi and f y = 60,000 psi, one-half of

ρ

ρ max is approximately one percent ( 0.01).

Depth selection for control of deflections of two-way slabs is given in Chapter 4.

As a guide, the effective depth d can be calculated as follows:

For beams with one layer of bars d = h - ( 2.5 in.)

For joists and slabs d = h - ( 1.25 in.)

3.3 MEMBER SIZING FOR MOMENT STRENGTH

equal
to about one-half of the maximum permitted. The ACI 318-11 does not provide direct maximum reinforcement
ratio for beams and slabs. The code indirectly defines reinforcements in terms of the net tensile strain in
extreme layer of longitudinal tension steel at nominal strength

As noted above, deflection problems are rarely encountered with beams having a reinforcement ratio ρ

ε t . For beams, slabs and members with factored

axial compressive load less than 0.10 › A g , ε t at nominal strength should not be less than 0.004. Figure 3-2

summarizes the code limitations and related definitions for the strain ε t . For our selected materials

( › = 4000 psi and f y = 60,000 psi), the maximum reinforcement ratio corresponding to ρ max =

0.0206. It is important to point out that the increase in the nominal moment capacity M n when using larger area
of reinforcement beyond the area associated with the strain limit of tension controlled section (

ε t = 0.004, is

ε t = 0.005), is

offset by the required decrease in the in the φ factor when calculating the design moment capacity φ M n .

To calculate the required dimensions for a member subjected to a factored moment M u , nominal flexural
strength equation for the section needs to be developed. The section's nominal flexural strength can be
calculated based on equilibrium and strain compatibility assuming the strain profile and the concrete stress
block shown in Figure 3-2 (See design assumption of ACI 318-11 Section 10.2). A simplified sizing equation
can be derived using the strength design approach developed in Chapter 6 of Reference 3.8.

Set ρ = 0.5 ρ max = 0.0103

M =φ A s f y (d − a/2) =φρ bdf y (d − a/2)

u

a = A s f y / 0.85 f c b ʹ =ρ df y /0.85 f c ʹ

0.5 ρ f y

⎛
⎜

⎞
⎟ =

M u / φ bd 2 =ρ f y 1 − R n

0.85 f c ʹ

0.5 ρ f y

⎛
⎜

⎞
⎟

R n =ρ f y 1 −

0.85 f c ʹ