Object Split into Sections

After breaking the cross section into its composite parts, determine the area, the location of the centroid, and the centroidal moments of inertia for each part.

The centroid and the area of each part were found in the previous section, Centroid: Composite Parts. The moments of inertia for each part can be found from the tables in the Sections Appendix.

For the axis system as shown, the properties for part 1 are,

     x₁ = 1 cm    y₁ = 3.5 cm     A₁ = 6 cm²

     I_x'1 = 1/12 2(3)³ = 54/12 cm⁴

     I_y'1 = 1/12 3(2)³ = 24/12 cm⁴

The properties for part 2 are,

     x₂ = 8 cm    y₂ = 1 cm     A₂ = 32 cm²

     I_x'2 = 1/12 16(2)³ = 128/12 cm⁴

     I_y'2 = 1/12 2(16)³ = 8192/12 cm⁴

And the properties for part 3 are,

     x₃ = 15 cm    y₃ = 7 cm     A₃ = 20 cm²

     I_x'3 = 1/12 2(10)³ = 2000/12 cm⁴

     I_y'3 = 1/12 10(2)³ = 80/12 cm⁴

Part 4 is a triangle and its properties are,

     x₄ = 17.33 cm    y₄ = 10.67 cm     A₄ = 8cm²

     I_x'4 = 1/36 4(4)³ = 256/36 cm⁴

     I_y'4 = 1/36 4(4)³ = 256/36 cm⁴

These can be used with the following equations to find the moments of inertia of the entire cross section with respect to the centroid of the cross section.

Here d_xi and d_yi are the distances from the centroid of the cross section (global centroid) to the centroid of any part i. These distances are given by the equations:

Substituting x, y and the results of the above equations give,

     d_x1 = 10.616 - 1 = 9.616 cm
     d_y1 = 4.218 - 3.5 = 0.718 cm 

     d_x2 = 10.616 - 1 = 2.616 cm     
     d_y2 = 4.218 - 1 = 3.218 cm 

     d_x3 = 10.616 - 15 = -4.384 cm
     d_y3 = 4.218 - 7 = -2.782 cm 

     d_x4 = 10.616 - 17.33 = -6.714 cm
     d_y4 = 4.218 - 10.67 = -6.452 cm 

Substituting each d_xi, d_yi, I_x'i, and I_y'i into the basic summation equations give

     I_x = 1,011 cm⁴

     I_y = 2,217 cm⁴

In the theory page, the polar moment of inertia was show to be equal to

     J_z = I_x + I_y

Substituting for I_x and I_y gives the polar moment,

     J_z = 3,228 cm⁴

Since this is not a symmetrical cross section, the product of inertia, I_xy is not zero. The produce of inertia is used for un-symmetrical bending which is not covered in this statics eBook.