DESIGN OF COLUMNS

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DESIGN OF COLUMNS

Grouping of columns:
Group-1 (Axial columns C6,C7,C10,C11)
Group-2 (Bi-axial columns C1,C4,C13,C16)
Group-3 (Uni-axial columns C2,C3,C5,C8,C9,C12,C14,C15)

Design of group-1 axial columns (internal columns):
Size of column = 450×450mm
Moment at the interior support ‘M’ =
                 wd=14.53kN/m2  ;   wL= 6.75Kn/m2
                    l2= 6.6m              ; ln = 6.6-0.45 = 6.15m
stiffness of column Kc =  = 1.13×106
stiffness of slab Ksb      =  = 0.62×106
   MX=  = 52.88kNm
           Since the slab around the column have same dimensions in both directions,
   MY = 52.88kN/m.
Total load from slab and drop = 21.2805kN
Axial load on ground floor = w×L1×L2 = 21.2805×6.6×6.6 = 926.97 KN.
Axial load for 4storey = 926.97×5 = 4634089kN
Factored self weight of column = 1.5×0.45×0.45  = 7.59 KN
Total axial load = PU = 4642.48 KN
Slenderness ratio: λ =  =  = 4.33<12 (hence designed as short column)
                                Effective length (le) = 0.65×l = 1.95m
Eccentricity calculations: ey = ex = +  = 21>20.                                                                   
          From page 71 of IS-456: minimum eccentricity = 0.005D
                                                                           = 0.005×450 = 22.5>21
Longitudinal reinforcement: 
                          PU    = 0.4fck Ag + (0.67fy- 0.4fck)Asc
        4642.48×106   = 0.4× 20×( 450× 450) + (0.67×415 - 0.4× 20)Asc
                              Asc     = 1119.29 mm2
                    Provide 14bars of 32mm diameter.
Spacing in longitudinal direction:  450-40-40 = 370
                                                           = 92.55
                    Provide spacing of 92.55mm in longitudinal direction
Spacing in transverse direction:  450-40-40 = 370
                                                           = 123.33
                    Provide spacing of 123.33mm in transverse direction
          Lateral reinforcement:
Spacing as per code : pitch and diameter of lateral ties
1)    Least lateral dimension = 450mm
2)    16×ɸ                              = 16  = 512mm
3)    300mm
Size of lateral ties =  =  = 8mm
Hence provide lateral ties of 8mm diameter at 300mm c/c
Design of group-2 Bi-axail columns(corner columns):
Size of column = 450×450mm
Load on ground floor = w×L1×L2 = 21.2805×3×3 = 191.52 KN.
Load for 4-storey       = 191.52×5 = 957.21KN
Factored self weight of column = 1.5×0.45×0.45  = 7.59 KN
Total bi-axial load (Pu)     = 965.21KN
Moment in X-direction ‘MUX = 75% of the total moment on column from slab
                                                 = 0.75×189.972
                                                 = 142.479kNm
Moment in Y-direction ‘MUY’ = 142.479kNm
Slenderness ratio: λ =  =  = 4.33<12 (hence designed as short column)
                                Effective length(le) = 0.65×l = 1.95m

Minimum eccentricity as per code , ey min = ex min = +  = 21>20
Minimum eccentricities are less than the applied eccentricities no modification to MUX , MUY is called for:
Trial section: 
  Longitudinal reinforcement:
  MU= 1.15   = 1.15  = 231.705KNm
Assuming d1 (using 8mm ties &25mm bars) = 40+ +8 = 60.5mm
   =  = 0.134  0.15
S.F ratio:   =  = 0.238
B.M ratio:  = = 0.127

Referring to chart 45 of SP:16 :   = 0.105
                                            Pt = 0.105×20 = 2.11>0.8 and <4.0
 Ast =  = 4272.75mm2
Provide 12 bars of 25mm diameter.

Moment capacities: MUX1, MUX2 [due to symmetry MUX1=MUX2]
           = 0.238 (as calculated earlier)
          P(provided) =  = 2.909
           =  = 0.145
         Assuming d1 (using 8mm ties &25mm bars) = 40+ +8 = 60.5mm
          =  = 0.134  0.15

Referring to chart 45 of SP-16:
 = 0.165
MUX1 = MUY1 = 0.165×20×4503 = 300.71kNm
Which is greater than MUX, MUY = 142.479kNm


Values of PUZ & :
  PUZ = 0.45fck Ag + (0.75fy – 0.45fck) ASC
        = 3603.35 KN             
   =  = 0.267 (which is lies between 0.2 &0.8)
   = 1.0 + (2.0 – 1.0)  = 1.11
                                                    

Check safety under bi-axial loading:
   +  =  +
                                   = 0.8<1.0
Hence the trail section is safe under the applied loading.
Lateral reinforcement:
The minimum diameter and the maximum spacing of the lateral ties are specified by the code [eq 13.9, 13.10]
   = greater of  & 6mm
 Spacing as per code: pitch and diameter of lateral ties
1)      Least lateral dimension = 450mm
2)      16×ɸ                              = 16  = 400mm
3)      300mm
Hence provide 8mm ties at 300 c/c
Arrangement of bars:  spacing = 450 – 40 – 40 =370
                                                  =  = 123.33mm
  Spacing is same in both directions.

Design of group-3 Uni-axial columns (exterior columns):
Size of column = 450×450mm
Load on ground floor = w×L1×L2 = 21.2805×3×6.6 = 421.344 KN.
Load for 4-storey       = 421.344×5 = 2106.72KN
Factored self weight of column = 1.5×0.45×0.45  = 7.59 KN
Total bi-axial load (Pu)     = 2114.31KN
Moment in X-direction     = 212.543Kn/m
The unbalanced moment in Y-direction = 441.346 – 315.135
                                                                = 126.211kN/m
Because of unbalanced moment, uni- axial columns acts as bi-axial columns.
Slenderness ratio: λ =  =  = 4.33<12 (hence designed as short column)
                                Effective length(le) = 0.65×l = 1.95m
Minimum eccentricity as per code , ey min = ex min = +  = 21>20
Minimum eccentricities are less than the applied eccentricities no modification to MUX , MUY is called for:
Trial section: 
  Longitudinal reinforcement:
  MU = 1.15   = 1.15  = 284.26KNm
Assuming d1(using 8mm ties &25mm bars) = 40+ +8 = 60.5mm
   =  = 0.134  0.15
S.F ratio :   =  = 0.522
B.M ratio:  = = 0.155

Referring to chart 45 of SP:16 :   = 0.135
                                                Pt = 0.135×20 = 2.7 >0.8 and <4.0
 Ast =  = 5467.5mm2
Provide 4 bars of 32mm diameter and 8bars of 25mm diameter.

Moment capacities : MUX1 , MUX2 [due to symmetry MUX1=MUX2]
            = 0.5220 ( as calculated earlier)
          P(provided) =  = 3.17
           =  = 0.158
         Assuming d1(using 8mm ties &25mm bars) = 40+ +8 = 60.5mm
          =  = 0.134  0.15
Referring to chart 45 of SP-16:
 = 0.178
MUX1 = MUY1 = 0.178×20×4503 = 324.405kNm
Values of PUZ & :
  PUZ = 0.45fck Ag + (0.75fy – 0.45fck) ASC
        = 3767.15KN              
  = 0.56 (which is lies between 0.2 &0.8)
   = 1.0 + (2.0 – 1.0)  = 1.6
Check safety under bi-axial loading:
   +  =  +
                                   = 0.72<1.0
Hence the trail section is safe under the applied loading.

Lateral reinforcement:
The minimum diameter and the maximum spacing of the lateral ties are specified by the code [eq 13.9, 13.10]
   = greater of  & 6mm
Spacing as per code: pitch and diameter of lateral ties
1)      Least lateral dimension = 450mm
2)      16×ɸ                              = 16  = 512mm
3)      300mm
Hence provide 8mm ties at 300 c/c
Arrangement of bars:  spacing = 450 – 40 – 40 =370
                                                  =  = 123.33mm
  Spacing is same in both directions
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