14 January 2011 
                                             Mechanics of Materials 
                                             CIVL 3322 / MECH 3322 
                                                             Centroids  
                                                                  and  
                                                      Moment of Inertia  
                                                           Calculations 
                                                       Centroids 
                                       n                   n                       n
                                     ∑xA                  ∑yA                    ∑zA
                                           i   i                 ii                     i  i
                               x = i=1              y = i=1                z = i=1
                                        n                    n                      n
                                      ∑A                   ∑A                      ∑A
                                             i                     i                      i
                                       i=1                  i=1                    i=1
                             2                  Centroid and Moment of Inertia Calculations 
                                                                                                                              1 
                                                                                                                                                                                                14 January 2011 
                                                                        Parallel Axis Theorem 
                                                         ¢  If you know the moment of inertia about a 
                                                              centroidal axis of a figure, you can 
                                                              calculate the moment of inertia about any 
                                                              parallel axis to the centroidal axis using a 
                                                              simple formula
                                                                                                    
                                                                                             I = I + Ay2
                                                                                                 z             z
                                                    3                                 Centroid and Moment of Inertia Calculations 
                                                                                                              P07_045 
                                                    4                                 Centroid and Moment of Inertia Calculations 
                                                                                                                                                                                                                                   2 
                                                                                                                                    14 January 2011 
                                                                            P07_045 
                                    5                      Centroid and Moment of Inertia Calculations 
                                                                 An Example 
                                        ¢  Lets start with an example problem and 
                                           see how this develops                                                 n
                                                                                                                ∑xA
                                                                                                                      ii
                                                                                                          x = i=1
                                                                                                                  n
                                                              1 in                                               ∑A
                                                                                                                       i
                                                                                                                 i=1
                                                   1 in
                                                                                                 1 in
                                                     1in
                                                                                3 in
                                    6                      Centroid and Moment of Inertia Calculations 
                                                                                                                                                            3 
                                                                                                       14 January 2011 
                                                   An Example 
                               ¢  We want to locate both the x and y 
                                                                                        n
                                  centroids                                            ∑xA
                                                                                            ii
                                                                                   x = i=1
                                                                                         n
                                                 1 in                                   ∑A
                                                                                             i
                                                                                         i=1
                                        1 in
                                                                            1 in
                                          1in
                                                               3 in
                            7                  Centroid and Moment of Inertia Calculations 
                                                   An Example 
                               ¢  There isn’t much of a chance of 
                                  developing a function that is easy to 
                                                                                        n
                                  integrate in this case                               ∑xA
                                                                                            ii
                                                                                   x = i=1
                                                                                         n
                                                 1 in                                   ∑A
                                                                                             i
                                                                                         i=1
                                        1 in
                                                                            1 in
                                          1in
                            8                  Centroid and Moment of Inertia Calculations 
                                                               3 in
                                                                                                                          4