Structural Concrete - Hassoun

3.15 Analysis of T- and I-Sections 133
A general equation for calculating (Max A s ) in a T-section when a < t can be developed as
follows:
C = 0.85f ′ c[(b e − b w )t + ab w ]
For ε c = 0.003 and ε t = 0.005, then c/d = 0.003/(0.003 +0.005) = 0.375 (for the web). Hence,
a = β 1 c = 0.375 β 1 d
The maximum steel area is equal to C/f y and, therefore,
( 0.85f
′
)
Max A s = c
[(b
f e − b w )t + 0.375β 1 b w d] (3.71)
y
where Max A s is the maximum tension steel area that can be used in a T-section when a > t. For
example, for f ′ c = 3ksiandf y = 60 ksi, the preceding equation is reduced to:
For f ′ c = 4ksiandf y = 60 ksi,
Max A s = 0.0425[(b e − b w )t + 0.319b w d] (3.72)
Max A s = 0.0567[(b e − b w )t + 0.319b w d] (3.73)
In summary, the procedure to analyze a T-section, which can also be utilized for inverted
L-section, described later in Section 3-17, is as follows:
1. Determine the effective width of the flange b e (refer to Section 3.15.3). Calculate ρ max and
ρ min (or take from tables).
2. Check if a ≤ t as follows: a = A s f y ∕(0.85f ′ cb e ).
3. If a ≤ t, it is a rectangular section analysis.
a. Calculate φM n = φA s f y (d − a/2). Note that c = a/β 1 and ε t = 0.003(d t − c)/c ≥ 0.005 for
tension-controlled section and φ = 0.9.
b. Check that ρ w = A s /b w d ≥ ρ min .
c. Max A s can be calculated from Eq. 3.68 and should be ≥ A s used. When a < t, normally
this condition is met.
4. If a ′ > t, it is a T-section analysis:
a. Calculate A sf = 0.85f ′ ct(b e − b w )∕f y .
b. Check that (ρ w − ρ f ) ≤ ρ max (relative to the web area), where
ρ w = A s
b w d
and
ρ f = A sf
b w d
Or check that Max A s ≥A s used in the section, for φ = 0.9, (Eq. 3.71).
c. Check that ρ w = A s /b w d ≥ ρ min . This condition is normally met when a > t.
d. Calculate a =(A s − A sf )f y ∕0.85f ′ cb w (for the web section).
e. Calculate φ M n from Eq. 3.65.
Example 3.11
A series of reinforced concrete beams spaced at 7 ft, 10 in. on centers have a simply supported span of
15 ft. The beams support a reinforced concrete floor slab 4 in. thick. The dimensions and reinforcement
of the beams are shown in Fig. 3.33. Using f c ′ =3ksi and f y = 60 ksi, determine the design moment
strength of a typical interior beam.