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Centroid & Moment of Inertia of
     Composite Figures


 Composite Figures – Made of more
     than one simple figures
Composite Figures Type - I




Centroidal Axes of the elements coincide
 with each other and also coincide with
   the centroid of the resultant figure
Composite Figures Type - II




Centroidal axes of elements are different and do
not coincide with centroid of composite figure
Composite Figures Type - I
  Y
            External Dia = D1;
            Internal Dia = D2
              IOX = IGX = IOY = IGY

        X     =
                   (
                π D −D 4
                       1
                               4
                               2   )
                   64

       IOz = IGz   =
                       (
                     π D −D4
                           1
                                       4
                                       2   )
                        32
Composite Figures Type - I
        Y
                          BH3 πD 4
                  IGX   =     −
                           12   64

H                             HB3 πD 4
    D                 IGY   =     −
            X                  12   64

                        BH(H + B ) πD
                               2   2     4

    B           IGz   =           −
                            12      32
Un Symmetric I Section –
        Composite Figure – Type - II
    3        Y
         20 cm
                      2 cm


             2 cm
30 cm
                 2


                     4 cm


         10 cm       X
    1
Un Symmetric I Section –
          Composite Figure – Type - II

Sl. Ai       Yi     Aiyi Aiyi2     I Self, OX
No (cm2)     (cm)   (cm3) (cm3)    (cm4)
1   40       2      80    160      53.33

2   48       16     768    12288 2304

3   40       29     1160 33640 13.33

∑
    128             2008 46088 2370.66
Un Symmetric I Section –
           Composite Figure – Type - II

Yb    =
        ∑ A y = 2008 = 15.6875cm
                i       i

        ∑ A 128     i

IOX   = ∑ I + ∑ A y = 48.459 × 10 cm
           self                 i
                                    2
                                    i
                                                3       4



 IGX = IOX − AY = 16.958 × 10 cm
                            2
                            b
                                            3       4


            3               3           3
        4(10 ) 24( 2 ) 2( 20 )
IGY   =       +       +        = 1682.667cm 4

          12     12      12
Un Symmetric I Section –
        Composite Figure – Type - II
    3           Y
            20 cm             Yb = 15.6875 cm
                        2 cm

        2
                2 cm
                              IGX = 16.958 × 10 cm
                                                3       4

30 cm

                              IGY = 1.682 × 10 cm
                                            3       4

                       4 cm


            10 cm      X
    1
Angle Section - Composite Figure – Type - II


         Y
4 mm         Y
                         Un Symmetric w.r.t both
 2                      the axes, hence Xb & Yb
                              is to be found
40 mm        G   X

                 6 mm
     1

     O   20 mm   X
Angle Section - Composite Figure – Type – II
           Calculating Yb, IOX, IGX
Sl. Ai   Yi   Aiyi  Aiyi2         I Self, OX
No (cm2) (cm) (cm3) (cm3)         (cm4)
1   120     3    360    1080      360


2   160     26   4160 108160 21333.33


∑   280          4520 109240 21693.33
Angle Section - Composite Figure – Type – II
                 Calculating Yb, IOX, IGX


      Y =
          ∑ A y = 4520 = 16.143 mm
                  i       i

           ∑ A 280
       b
                      i


IOX   = ∑I    + ∑ A y = 130.933 × 10 mm
           self ,OX               i
                                      2
                                      i
                                              3       4




  IGX = IOX − AY = 57.966 × 10 mm
                              2
                              b
                                          3       4
Angle Section - Composite Figure – Type – II
           Calculating Xb, IOY, IGY



Sl. Ai   Xi   Aixi  Aixi2        I Self, OY
No (cm2) (cm) (cm3) (cm3)        (cm4)
1   120    10    1200 12000      4000

2   160    2     320    640      213.33
∑   280          1520 12640      4213.33
Angle Section - Composite Figure – Type – II
                 Calculating Xb, IOY, IGY


      X =
           ∑ A x = 1520 = 5.43 mm
                       i       i

           ∑ A 280
        b
                           i


IOY   = ∑I    + ∑ A x = 16.853 × 10 mm
            self ,OY                   i
                                           2
                                           i
                                                   3       4




  IGY = IOY − AX = 8.597 × 10 mm   2
                                   b
                                               3       4
Angle Section - Composite Figure – Type - II
     Y
4 mm    Y               Y = 16.143 mm
                           b

 2                        Xb = 5.43 mm
40 mm     G      X
                        IGX = 57.966 × 10 mm
                                          3       4

                 6 mm
     1

     O   20 mm   X      IGY = 8.597 × 10 mm
                                          3       4
Composite Figure – Type - II


            2
10 cm                           Diameter of the circular
                                   opening = 6 cm
                           3
 20 cm


                        12 cm
        1




                10 cm
Composite Figure – Type - II


Sl. Ai        Yi     Aiyi     Aiyi2         I Self, OX
No (cm2)      (cm)   (cm3)    (cm3)         (cm4)
1   200       10     2000     20000         6666.67
2   50        23.33 1166.67 27218.33 277.78

3   -28.27 12        -339.29 -4071.5        -63.62
    221.73           2827.38 43146.83 6880.83
Composite Figure – Type – II
               Calculating Yb, IOX, IGX


 Y =
     ∑ A y = 2827.38 = 12.75 cm
                 i       i

     ∑ A 221.73
      b
                     i

I = ∑I
OX        + ∑ A y = 50.028 × 10 cm
            self ,OX             i
                                     2
                                     i
                                             3       4



     IGX = IOX − AY = 13.98 × 10 cm
                             2
                             b
                                         3       4



        20(103 ) 10(53 )     π(6 4 )
IGY   =         +        ×2−         = 1.81 × 103 cm 4
          12       12         64

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Cgmi3

  • 1. Centroid & Moment of Inertia of Composite Figures Composite Figures – Made of more than one simple figures
  • 2. Composite Figures Type - I Centroidal Axes of the elements coincide with each other and also coincide with the centroid of the resultant figure
  • 3. Composite Figures Type - II Centroidal axes of elements are different and do not coincide with centroid of composite figure
  • 4. Composite Figures Type - I Y External Dia = D1; Internal Dia = D2 IOX = IGX = IOY = IGY X = ( π D −D 4 1 4 2 ) 64 IOz = IGz = ( π D −D4 1 4 2 ) 32
  • 5. Composite Figures Type - I Y BH3 πD 4 IGX = − 12 64 H HB3 πD 4 D IGY = − X 12 64 BH(H + B ) πD 2 2 4 B IGz = − 12 32
  • 6. Un Symmetric I Section – Composite Figure – Type - II 3 Y 20 cm 2 cm 2 cm 30 cm 2 4 cm 10 cm X 1
  • 7. Un Symmetric I Section – Composite Figure – Type - II Sl. Ai Yi Aiyi Aiyi2 I Self, OX No (cm2) (cm) (cm3) (cm3) (cm4) 1 40 2 80 160 53.33 2 48 16 768 12288 2304 3 40 29 1160 33640 13.33 ∑ 128 2008 46088 2370.66
  • 8. Un Symmetric I Section – Composite Figure – Type - II Yb = ∑ A y = 2008 = 15.6875cm i i ∑ A 128 i IOX = ∑ I + ∑ A y = 48.459 × 10 cm self i 2 i 3 4 IGX = IOX − AY = 16.958 × 10 cm 2 b 3 4 3 3 3 4(10 ) 24( 2 ) 2( 20 ) IGY = + + = 1682.667cm 4 12 12 12
  • 9. Un Symmetric I Section – Composite Figure – Type - II 3 Y 20 cm Yb = 15.6875 cm 2 cm 2 2 cm IGX = 16.958 × 10 cm 3 4 30 cm IGY = 1.682 × 10 cm 3 4 4 cm 10 cm X 1
  • 10. Angle Section - Composite Figure – Type - II Y 4 mm Y Un Symmetric w.r.t both 2 the axes, hence Xb & Yb is to be found 40 mm G X 6 mm 1 O 20 mm X
  • 11. Angle Section - Composite Figure – Type – II Calculating Yb, IOX, IGX Sl. Ai Yi Aiyi Aiyi2 I Self, OX No (cm2) (cm) (cm3) (cm3) (cm4) 1 120 3 360 1080 360 2 160 26 4160 108160 21333.33 ∑ 280 4520 109240 21693.33
  • 12. Angle Section - Composite Figure – Type – II Calculating Yb, IOX, IGX Y = ∑ A y = 4520 = 16.143 mm i i ∑ A 280 b i IOX = ∑I + ∑ A y = 130.933 × 10 mm self ,OX i 2 i 3 4 IGX = IOX − AY = 57.966 × 10 mm 2 b 3 4
  • 13. Angle Section - Composite Figure – Type – II Calculating Xb, IOY, IGY Sl. Ai Xi Aixi Aixi2 I Self, OY No (cm2) (cm) (cm3) (cm3) (cm4) 1 120 10 1200 12000 4000 2 160 2 320 640 213.33 ∑ 280 1520 12640 4213.33
  • 14. Angle Section - Composite Figure – Type – II Calculating Xb, IOY, IGY X = ∑ A x = 1520 = 5.43 mm i i ∑ A 280 b i IOY = ∑I + ∑ A x = 16.853 × 10 mm self ,OY i 2 i 3 4 IGY = IOY − AX = 8.597 × 10 mm 2 b 3 4
  • 15. Angle Section - Composite Figure – Type - II Y 4 mm Y Y = 16.143 mm b 2 Xb = 5.43 mm 40 mm G X IGX = 57.966 × 10 mm 3 4 6 mm 1 O 20 mm X IGY = 8.597 × 10 mm 3 4
  • 16. Composite Figure – Type - II 2 10 cm Diameter of the circular opening = 6 cm 3 20 cm 12 cm 1 10 cm
  • 17. Composite Figure – Type - II Sl. Ai Yi Aiyi Aiyi2 I Self, OX No (cm2) (cm) (cm3) (cm3) (cm4) 1 200 10 2000 20000 6666.67 2 50 23.33 1166.67 27218.33 277.78 3 -28.27 12 -339.29 -4071.5 -63.62 221.73 2827.38 43146.83 6880.83
  • 18. Composite Figure – Type – II Calculating Yb, IOX, IGX Y = ∑ A y = 2827.38 = 12.75 cm i i ∑ A 221.73 b i I = ∑I OX + ∑ A y = 50.028 × 10 cm self ,OX i 2 i 3 4 IGX = IOX − AY = 13.98 × 10 cm 2 b 3 4 20(103 ) 10(53 ) π(6 4 ) IGY = + ×2− = 1.81 × 103 cm 4 12 12 64