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
Factored Normal Load

Pu = 1.2 D + 1.6 L     (ACI)


Pu = 1.25 D + 1.75 L   (AASHTO)



  D = dead load
  L = live load
      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
or shear loads
        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

For grade 40 (metric grade 300) steel

                As > 0.0020 Ag


For grade 60 (metric grade 420) steel

                As > 0.0018 Ag


      Ag = gross cross-sectional area
 Design Data for Steel Reinforcing Bars
                                            Nominal dimensions
 Bar size           Available
                                                     Cross-sectional
designation          grades             Diameter, db
                                                        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
 Design for Compressive Loads
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
Design for Moment Loads

                                1200d b
Development length      ldh =
                                   f c'




                                    ldh



                 12db
        Design for Shear Loads


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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