Again, in practice, we don’t often need to integrate because most areas we encounter are composite areas.  Therefore, we can simplify the equations to

 

             I_x = S I_xi                         I_y = S I_yi                         I_xy = S I_xyi                      J = S J_i

 

Where the subscript i represents that the moment of inertia is for the i^th simple shape.

 

Although we have tables that give information about moments of inertia for simple shapes, the information given is generally for specific axes through the simple shape.  Most often, the given equations are with respect to axes that pass through the centroid.  The parallel axis theorem allows us to determine the moments of inertia for any set of axes parallel to the axes that pass through the centroid.

 

I_x = I_cx + A d_y²               I_y = I_cy + A d_x²               I_xy = I_cxy + A d_x d_y                     J = J_c + A d²

 

Where ( d_x , d_y ) are the coordinates of the centroid, the bar above the moment of inertia designates that the moment of inertia is with respect to a centroidal axis and  d² = d_x² + d_y² .

 

Often, information about the product of inertia is not given in the tables provided.  If there is an axis of symmetry, I_xy = 0 .  Therefore, we only need equations for unsymmetric shapes.  For these shapes, the sign will depend on the orientation.  For example, consider the right triangle.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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