Civil Engineering Reference
In-Depth Information
4.7
P
u
c
2
A
f
d
reqd
2
=
4.7q
u
c
2
=
P
u
A
f
d
reqd
=
2.2 q
u
c
2
=
2.2c
The above equation is in mixed units: P
u
is in kips, c is in feet, A
f
is in square feet, and d is in inches.
The one-step thickness equation derived above is applicable for both square and rectangular footings (using largest
value of c) and wall footings. Since f
y
has a larger influence on d than does › , the simplified equation can be used
for other concrete strengths without a substantial loss in accuracy. As shown in Fig. 7-1, this derivation assumes
uniform soil pressure at the bottom of the footing; for footings subject to axial load plus moment, an equivalent
uniform soil pressure can be used.
According to ACI 11.11, the shear strength of footings in the vicinity of the column must be checked for both
one-way (wide-beam) shear and two-way shear. Fig. 7-2 illustrates the tributary areas and critical sections for
a square column supported by a square footing.
For wide-beam shear:
V
u
≤
2
φ ʹ
f
c
b
w
d
for normalweight concrete
where b
w
is the width of the footing, and V
u
is the factored shear on the critical section (at a distance d from
the face of the column). In general,
(
)
V
u
=
q
u
b
w
c
−
d
The minimum depth d can be obtained from the following equation:
d
c
=
q
u
where q
u
in psi
q
u
+
2
φ ʹ
f
c
This equation is shown graphically in Fig. 7-3 for › = 3000, 4000, 5000 and 6000 psi.
For a footing supporting a square column, the two-way shear strength will be the lesser of the values of V
c
obtained from ACI Eqs. (11-32) and (11-33). Eq. (11-32) will rarely govern since t
he a
spect ratio b
o
/d will usually be
considerably less than the limiting value to reduce the shear strength below
4
f
c
b
o
d
ʹ
.
*
Therefore, for two-way
shear,
V
u
≤
4
φ ʹ
f
c
b
o
d
where, for a square column, the perimeter of the critical section is b
o
= 4(c
1
+ d). The factored shear V
u
on the
critical section (at d/2 from the face of the column) can be expressed as:
*
For square interior columns, Eq. (11-32) will govern where d/c
1
≤
0.25.
.
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