Structural Design of Footings by bfk20410

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```									Structural Design of Footings

Prof. Jie Han, Ph.D., PE
The University of Kansas
Outline of Presentation

• Introduction
• Design for Square Footings
• Design for Continuous Footings
• Design for Rectangular Footings
• Connections with Superstructures

Pu = 1.2 D + 1.6 L     (ACI)

Pu = 1.25 D + 1.75 L   (AASHTO)

Minimal Requirements

d
T
db
3in.
3in.
Flexural steel

Effective depth

d = T − 3in. − d b ≥ 6in.
Minimal thickness
T ≥ 12in.
Common Selection of Materials

Concrete

Compressive strength

fc' = 2,000 to 3,000psi (15 to 20MPa)

Steel

Grade 40 steel       fy = 40,000psi (300MPa)
Grade 60 steel       fy = 60,000psi (420MPa)
Design Criterion for Shear

Design Criterion

Vuc ≤ φVnc

Vuc = factored shear force on critical surface
φ = resistance factor for shear = 0.75
Vnc = nominal shear capacity on critical surface
Nominal Shear Capacity

Vnc = Vc + Vs

Vnc = nominal shear capacity on critical surface

Vc = nominal shear capacity of concrete

Vs = nominal shear capacity of steel (ignored)
Design for Square Footings
Design of Square Footings
for One-Way Shear
Pu
B                    c

d
T
d

Factored shear force on one critical surface
Pu B − (c + 2d )
Vuc   =
2      B
No need to check one-way shear if there is no applied moment
Design of Square Footings
for One-Way Shear
Pu
Mu
c
Vu

d

d

Most critical face

Factored shear force on one critical surface
2       2
⎛ B − c − 2d ⎞ ⎛ Pu 3Mu ⎞ ⎛ Vu ⎞
Vuc   =⎜            ⎟ ⎜ +      ⎟ +⎜ ⎟
⎝     B      ⎠ ⎝ 2   B ⎠ ⎝ 2⎠
Design of Square Footings
for Two-Way Shearing
B         c                                           Pu

d/2                                      Mu
d/2
T
d                           d

c+d
Most critical face

Factored shear force on one critical surface

⎛ Pu Mu ⎞⎛ Base area of outer block ⎞ ⎛ Pu Mu ⎞⎛ B 2 − (c + d )2 ⎞
Vuc   =⎜ +       ⎟⎜                        ⎟=⎜ +       ⎟⎜
⎜
⎟
⎟
⎝ 4 c + d ⎠⎝                        ⎠ ⎝ 4 c + d ⎠⎝
2
total base area                        B         ⎠
Design of Square Footings
for Two-Way Shearing

Pu
Vu
Most critical faces

Most critical faces
c+d

Factored shear force on one critical surface

⎛ Pu ⎞ ⎛ Vu ⎞ ⎛ B − (c + d ) ⎞
2      2                                     2      2              2
⎛ Pu ⎞ ⎛ Vu ⎞ ⎛ base area of outer block ⎞
2
Vuc   = ⎜ ⎟ +⎜ ⎟ ⎜                               ⎟ = ⎜ ⎟ +⎜ ⎟ ⎜     ⎜
⎟
⎟
⎝ 4 ⎠ ⎝ 2⎠ ⎝                             ⎠   ⎝ 4 ⎠ ⎝ 2⎠ ⎝
2
total base area                             B         ⎠
Nominal One-Way Shear Capacity

Vnc = Vc = 2bw d fc'         English Units

1
Vnc   = Vc = bw d fc'       SI Units
6
Vnc = nominal shear capacity on one critical surface (lb or N)

Vc = nominal shear capacity of concrete (lb or N)

bw = length of one critical surface = B (in or mm)

d = effective depth (in or mm)

fc’ = 28-day compressive strength of concrete (psi or MPa)
Nominal Two-Way Shear Capacity
– English Units
Smallest of      Vnc = Vc = 4b0d fc'
⎛     4⎞
⎜ 2 + ⎟b 0d f c'
Vnc = Vc = ⎜
⎝     βc ⎟
⎠
⎛     αsd ⎞
Vnc = Vc = ⎜ 2 +
⎜          ⎟b 0d f c'
⎝     4b 0 ⎟
⎠
Vnc = nominal shear capacity on one critical surface (lb)

βc = long side length cl/short side length cs of columns
αs = 40, 30, and 20 for interior, edge, and corner columns
b0 = length of one critical surface = c + d (in)

c = column width (in or mm)           d = effective depth (in)
Nominal Two-Way Shear Capacity
– SI Units
1
Smallest of     Vnc = Vc = b 0 d f c'
3
1⎛     2⎞
Vnc = Vc = ⎜1 + ⎟b 0d f c'
6 ⎜ βc ⎟
⎝         ⎠
1⎛     αsd ⎞
Vnc = Vc = ⎜ 2 +        ⎟b 0d f c'
12 ⎜⎝   4b 0 ⎟⎠
Vnc = nominal shear capacity on one critical surface (N)

βc = long side length cl/short side length cs of columns
αs = 40, 30, and 20 for interior, edge, and corner columns
b0 = length of one critical surface = c + d (mm)

c = column width (mm)          d = effective depth (mm)
Critical Locations for Flexure

Critical sections
Flexural Design Principles

b        Neutral axis
0.85fc’

c                 C   a/2
a

d                                             d-a/2

fy               T=Asfy
Nominal Moment Capacity

⎛    a⎞
M n = As fy ⎜ d − ⎟
⎝    2⎠

ρdfy              As
a=                   ρ=
0.85fc'            bd

⎛     0.59 As f y ⎞
M n = As f y ⎜ d −
⎜            '
⎟
⎟
⎝        bf c     ⎠

As = cross-sectional area of reinforcing steel
ρ= steel ratio
b = width of flexural member
Required Cross-Sectional Area of
Reinforcing Steel
Design Criterion

Muc = φM n
Required Area of Reinforcing Steel

⎛ f c'b ⎞⎛                          ⎞
As = ⎜          ⎟⎜ d − d 2 − 2.355M uc   ⎟
⎜ 1.176 f ⎟⎜               φf c'b   ⎟
⎝        y ⎠⎝                       ⎠

Muc = factored bending moment
φ = 0.9 for flexure in reinforced concrete
b = width of flexural member
Factored Bending Moment
at Critical Section
Pu

Mu
c
Pu l 2 2Mu l
Muc   =       +
2B     B

b=?
l
B
Location of Critical Section for
Bending

c                    c                         c
c/4                                     cp

l                     l                         l
B                    B                        B

l = (B − c ) / 2   l = (B − c / 2) / 2      [                ]
l = 2B − (c + c p ) / 4

Concrete              Masonry                   Steel
Minimum Steel Area Requirement

As > 0.0020 Ag

As > 0.0018 Ag

Ag = gross cross-sectional area
Design Data for Steel Reinforcing Bars
Nominal dimensions
Bar size           Available
Cross-sectional
area, As
English       SI   English      SI     (in)   (mm) (in2)      (mm2)
#3      #10       40,60    300,420   0.375   9.5    0.11    71
#4      #13       40,60    300,420   0.500   12.7   0.20    129
#5      #16       40,60    300,420   0.625   15.9   0.31    199
#6      #19       40,60    300,420   0.750   19.1   0.44    284
#7      #22       60       420       0.875   22.2   0.60    387
#8      #25       60       420       1.000   25.4   0.79    510
#9      #29       60       420       1.128   28.7   1.00    645
#10     #32       60       420       1.270   32.3   1.27    819
#11     #36       60       420       1.410   35.8   1.56    1006
#14     #43       60       420       1.693   43.0   2.25    1452
#18     #57       60       420       2.257   57.3   4.00    2581
Development Length

- Develop proper anchorage

c
ld

l
B

ld = l − 3in. (or 70mm)
Development of Bars in Tension
ld 15 f y      αβγλ                       Atr fyt
=                            K tr =
d b 16 f c' ⎛ cd + K tr ⎞                260s ⋅ n
SI
⎜
⎜ d         ⎟
⎟
⎝     b     ⎠          Use Ktr=0 for conservative
α = reinforcement location factor
β = coating factor
γ = reinforcement factor
λ = lightweight concrete factor
ld = minimum required development length (mm)
db = nominal bar diameter (mm)
cd = half-spacing of bars or concrete cover dimension (mm)
s = max. c-c spacing of transverse reinf. within ld (mm)
n = number of bars
Atr = total cross-sectional area of all transverse reinf. (mm2)
fyt = yield strength of transverse reinforcement (MPa)
Development of Bars in Tension
ld   3 fy      αβγλ                        Atr fyt
=                            K tr =
d b 40 f c' ⎛ cd + K tr ⎞                1500s ⋅ n
⎜
⎜ d         ⎟
⎟
English
⎝     b     ⎠          Use Ktr=0 for conservative
α = reinforcement location factor
β = coating factor
γ = reinforcement factor
λ = lightweight concrete factor
ld = minimum required development length (in)
db = nominal bar diameter (in)
cd = half-spacing of bars or concrete cover dimension (in)
s = max. c-c spacing of transverse reinf. within ld (in)
n = number of bars
Atr = total cross-sectional area of all transverse reinf. (in2)
fyt = yield strength of transverse reinforcement (psi)
Definition of Transverse
Reinforcement Atr

Atr = 2Ab               Atr = 4Ab
Modifiers for Development Length
Factor                   Condition                   Modifier

α       Top reinf. (bars w/ 12in. concrete below)     1.3
Bottom bars                                   1.0

β       Epoxy-coated bars w/ cover < 3db or           1.5
Clear spacing < 6db
All other epoxy-coated bars                   1.2
Uncoated reinforcement                        1.0
γ       No. 6 and smaller bars & deformed wire        0.8
No. 7 and larger bars                         1.0

λ       Lightweight-aggregate concrete                1.3
Normal weight concrete                        1.0
Development of Bars in Compression

ld = λ s lab
English
db f y
lab = 0.02               ≥ 0.0003d b f y
f c'
ld = minimum required development length (in)
lab = basic development length (in)
db = nominal bar diameter (in)
λs = modifying multiplier
Excess reinforcement: λs = required As/provided As
Spirally enclosed reinforcement: λs = 0.75

fc’ = compressive strength of concrete (psi)
Design for Continuous Footings
Continuous Footings

c

d         T              d

B
d
Transverse Reinforcement -
One-Way Shear

c                           ⎛ B − c − 2d ⎞
Vuc / b = Pu / b⎜            ⎟
⎝     B      ⎠

d       d          Vuc / b = φVnc / b   φ = 0.75
d               Pu / b(B − c )
d=                              English
B
48φB f + 2 Pu / b
c
'

1500 Pu / b(B − c )
d=                                SI
500φB f c' + 3Pu / b
Transverse Reinforcement -
Flexure

450

Zone of compression

No transverse steel is needed if the entire base is within
a 450 frustum
Transverse Reinforcement -
Flexure

c                     Pu / bl 2 2M u / bl
M uc / b =          +
2B        B

l                    M uc / b = φM n / b

d   Required Area of Reinforcing Steel

B
⎛ f c'b ⎞⎛                          ⎞
As = ⎜          ⎟⎜ d − d 2 − 2.355M uc   ⎟
⎜ 1.176 f ⎟⎜               φf c'b   ⎟
⎝        y ⎠⎝                       ⎠
Design for Rectangular Footings
Design Philosophy

• Similar to those for square footings
• Check one-way and two-way to determine minimal d & T
• Design long steel bars and evenly distribute them
• Design short steel bars & distribute the portion (E) of the
total short steel area within the inner zone

2
E=
L / B +1
Critical Shear Surfaces

One-way shear surface

Two-way shear surface
Long Steel and Short Steel

Long steel

Short steel
Distribution of Short Steel

B

Outer zone    Inner zone   Outer zone   B

2
E=
L / B +1

L
Connections with Superstructures
Use of Dowels for Connection

Wall steel
Concrete or
masonry
Lap joint
column
Dowel
Nominal column bearing capacity

Pnb = 0.85 f c' A1     on column (use column     f c' )
Pnb = 0.85 f c' A1s    on footing (use footing   f c' )

Design criterion
A1
Pu ≤ φPnb
Pu = factored column load                            A2
A1 = cross-sectional area of the column = c2
s = (A2/A1)0.5 < 2 if c + 4d < B; otherwise, s = 1
A2 = (c + 4d)2
φ = 0.7
Concrete or Masonry Columns

Use at least four dowels with a total area of steel, As, at least
equal to that of the column steel or 0.005 times the cross-
sectional area of the column, whichever is greater

Dowels may not be larger than #11 bars and must have a 90o
hook at the bottom

Normally, the number of dowels is equal to the number of vertical
bars in the column

1200d b
Development length      ldh =
f c'

ldh

12db

The minimum required dowel steel area

Vu
As =            Vu ≤ 0.2φf c' Ac
φf y μ

μ = 0.6 if the cold joint not intentionally roughened or
1.0 if the cold joint is roughened by heavy raking
or grooving

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