Lecture #15

Reading: A:1-3

 

Centroids and Area Moments of Inertia

 

You were introduced to these topics in statics so we will only review quickly.  However, centroids and moments of inertia are very important concepts in mechanics of materials.

 

 

In general, the centroid of an area is found by

 

           x_c =  ∫  x dA  / A

 

           y_c =  ∫  y dA  / A

 

where ( x_c , y_c ) are the coordinates of the centroid and  A =  ∫  dA .

 

 

 

In practice, we do not often need to integrate because most areas we encounter can be treated as a composite area - an area made up of simple shapes.  Given that we have tables that locate the centroid of simple shapes for us, we can simplify the equations to

 

             x_c = S A_i x_ci / A                                      y_c = S A_i y_ci / A

 

Where ( x_ci , y_ci ) are the coordinates of the centroid of the simple shape A_i and A = S A_i .

 

We know that if there is an axis of symmetry, the centroid must be located on that axis.

 

In general, area moments of inertia are found from

 

             I_x =  ∫  y² dA                  I_y =  ∫  x² dA                  I_xy =  ∫  x y dA  J =  ∫  r² dA

 

where I_xy is the product of inertia and J is the polar moment of inertia.

 

Since r² = x² + y² , we can show that  J = I_x + I_y .

 

We know that I_x , I_y , and J are always positive, but I_xy can be positive or negative.

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