Structural Concrete - Hassoun

454 Chapter 13 Footings
For the values of V c1
and V c2
it can be observed that V c1
controls (less than V c2
) whenever
β c ≤ 2, whereas V c2
controls (less than V c1
) whenever β c > 2. This indicates that the allowable
shear V c is reduced for relatively long footings. The actual soil pressure variation along the long
side increases with an increase in β. For shapes other than rectangular, β is taken to be the ratio of
the longest dimension of the effective loaded area in the long direction to the largest width in the
short direction (perpendicular to the long direction).
For Eq. 13.8, α s is assumed to be 40 for interior columns, 30 for edge columns, and 20 for
corner columns. The concrete shear strength V c3
represents the effect of an increase in b 0 relative
to d. For a high ratio of b 0 ∕d, V c3
may control.
Based on the preceding three values of V c , the effective depth, d, required for two-way shear
is the largest obtained from the following formulas (φ = 0.75):
or
d 1 =
V u2
φ4λ √ f ′ cb 0
(where β ≤ 2) (13.9)
d 1 =
V u2
φ(2 + 4∕β)λ √ f ′ cb 0
(where β>2) (13.10)
V u2
d 2 =
φ(α s d∕b 0 + 2)λ √ (13.11)
f cb ′ 0
The two-way shearing force, V u2
, and the effective depth, d, required (if shear reinforcement
is not provided) can be calculated as follows (refer to Fig. 13.12):
1. Assume d.
2. Determine b 0 :b 0 = 4(c + d) for square columns, where one side = c. b 0 = 2(c 1 + d) + 2(c 2 + d)
for rectangular columns of sides c 1 and c 2 .
3. The shearing force V u2
acts at a section that has a length b 0 = 4(c + d)or[2(c 1 + d) + 2(c 2 + d)]
and a depth d; the section is subjected to a vertical downward load, P u , and a vertical upward
pressure, q u (Eq. 13.2). Therefore,
{
Pu − q u (c + d) 2 for square columns (13.12a)
V u2
=
P u − q u (c 1 + d)(c 2 + d) for rectangular columns (13.12b)
4. Determine the largest d (of d 1 and d 2 ). If d is not close to the assumed d, revise your assumption
and repeat. ACI Code, Sections 13.3.1.2 and 13.4.2.1 specifies depth of footing above
bottom reinforcement shall not be less than 6 in. for footing on soil, nor less than 12 in. for
footing on piles.
13.4.4 Flexural Strength and Footing Reinforcement
The critical sections for moment occur at the face of the column (section n–n, Fig. 13.13). The
bending moment in each direction of the footing must be checked and the appropriate reinforcement
must be provided. In square footings and square columns, the bending moments in both directions
are equal. To determine the reinforcement required, the depth of the footing in each direction may
be used. Because the bars in one direction rest on top of the bars in the other direction, the effective
depth, d, varies with the diameter of the bars used. An average value of d may be adopted. A
practical value of d may be assumed to be h − 4.5 in.