CIVL 2131Moment - Statics of Inertia Composite Areas A math professor in an unheated room is cold and calculating. Radius of Gyration ¢ This actually sounds like some sort of rule for separation on a dance floor. ¢ It actually is just a property of a shape and is used in the analysis of how some shapes act in different conditions. 2 Moment of Inertia - Composite Area Monday, November 26, 2012 1 Radius of Gyration ¢ The radius of gyration, k, is the square root of the ratio of the moment of inertia to the area I k = x x A I k = y y A J II+ k ==O xy O AA 3 Moment of Inertia - Composite Area Monday, November 26, 2012 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 2 IIAxyy=+ 2 IIAyxx=+ 4 Moment of Inertia - Composite Area Monday, November 26, 2012 2 Parallel Axis Theorem ¢ Since we usually use the bar over the centroidal axis, the moment of inertia about a centroidal axis also uses the bar over the axis designation 2 IIAxyy=+ 2 IIAyxx=+ 5 Moment of Inertia - Composite Area Monday, November 26, 2012 Parallel Axis Theorem ¢ If you look carefully at the expression, you should notice that the moment of inertia about a centroidal axis will always be the minimum moment of inertia about any axis that is parallel to the centroidal axis. 2 IIAxyy=+ 2 IIAyxx=+ 6 Moment of Inertia - Composite Area Monday, November 26, 2012 3 Parallel Axis Theorem ¢ In a manner similar to that which we used to calculate the centroid of a figure by breaking it up into component areas, we can calculate the moment of inertia of a composite area 2 IIAxyy=+ 2 IIAyxx=+ 7 Moment of Inertia - Composite Area Monday, November 26, 2012 Parallel Axis Theorem ¢ Inside the back cover of the book, in the same figure that we used for the centroid calculations we can find calculations for moments of inertia 2 IIAxyy=+ 2 IIAyxx=+ 8 Moment of Inertia - Composite Area Monday, November 26, 2012 4 2 IIAxyy=+ Parallel Axis Theorem 2 IIAyxx=+ ¢ HERE IS A CRITICAL MOMENT OF CAUTION ¢ REMEMBER HOW THE PARALLEL AXIS IS WRITTEN ¢ IF THE AXIS SHOWN IN THE TABLE IS NOT THROUGH THE CENTROID, THEN THE FORMULA DOES NOT GIVE YOU THE MOMENT OF INERTIA THROUGH THE CENTROIDAL AXIS 9 Moment of Inertia - Composite Area Monday, November 26, 2012 2 IIAxyy=+ Parallel Axis Theorem 2 IIAyxx=+ ¢ By example ¢ The Iy given for the Semicircular area in the table is about the centroidal axis ¢ The Ix given for the same Semicircular area in the table is not about the centroidal axis 10 Moment of Inertia - Composite Area Monday, November 26, 2012 5 Using The Table ¢ We want to locate the moment of inertia in the position shown of a semicircular area as shown about the x and y axis, Ix and Iy y 10" x 11 Moment of Inertia - Composite Area Monday, November 26, 2012 Using the Table ¢ First, we can look at the table and find the Ix and Iy about the axis as shown y 10" x 12 Moment of Inertia - Composite Area Monday, November 26, 2012 6 Using the Table ¢ In this problem, the y axis is 8” from the y centroidal axis and x axis is 6” below the base of the semicircle, this would be usually evident from the problem description 5" y 8" 10" 6" x 13 Moment of Inertia - Composite Area Monday, November 26, 2012 Using the Table ¢ Calculating the Iy you should notice that the y axis in the table is the centroid axis so we won’t have to move it yet 1 Ir= π 4 y 5" 8 y 1 4 Iy = π (5 in) 10" 8 x 4 Iy = 245.44 in 14 Moment of Inertia - Composite Area Monday, November 26, 2012 7 Using the Table ¢ Next we can calculate the area 2 π (5in) A = 5" 2 y 2 10" Ain= 39.27 x 15 Moment of Inertia - Composite Area Monday, November 26, 2012 Using the Table ¢ If we know that distance between the y axis and the ybar axis, we can calculate the moment of inertia using the parallel axis theorem 5" 2 y Iyy=+ I Ad x 8 in 2 6in 10" Ixx=+ I Ad y 2.12" x 16 Moment of Inertia - Composite Area Monday, November 26, 2012 8 Using the Table ¢ I changed the notation for the distances moved to avoid confusion with the distance from the origin 5" 2 y Iyy=+ I Ad x 8 in 2 6in 10" Ixx=+ I Ad y 2.12" x 17 Moment of Inertia - Composite Area Monday, November 26, 2012 Using the Table ¢ The axis we are considering may not always be a the origin. 5" 2 y Iyy=+ I Ad x 8 in 2 6in 10" Ixx=+ I Ad y 2.12" x 18 Moment of Inertia - Composite Area Monday, November 26, 2012 9 Using the Table ¢ If the y axis is 8 inches to the left of the centroidal axis, then the moment of inertia about the y axis would be IIAd2 5" yy=+ x y 422 8 in Iy =+245.44 in( 39.27 in)( 8 in) 4 6in 10" 2.12" Iy = 2758.72 in x 19 Moment of Inertia - Composite Area Monday, November 26, 2012 Using the Table ¢ The moment of inertia about the x axis is a slightly different case since the formula presented in the table is the moment of inertia about the base of the semicircle, not the centroid 5" y 8 in 6in 10" 2.12" x 20 Moment of Inertia - Composite Area Monday, November 26, 2012 10 Using the Table ¢ To move it to the moment of inertia about the x-axis, we have to make two steps 5" 2 y IIx= base − Ad( base to centroid ) 8 in 2 6in 10" 2.12" IIAdxx=+( centroid to x-axis ) x 21 Moment of Inertia - Composite Area Monday, November 26, 2012 Using the Table ¢ We can combine the two steps 2 IIx= base − Ad( base to centroid ) 2 IIAdxx=+( centroid to x-axis ) 22 IIx= base − Ad( base to centroid) + Ad( centroid to x-axis ) 5" y 8 in 6in 10" 2.12" x 22 Moment of Inertia - Composite Area Monday, November 26, 2012 11 Using the Table ¢ Don’t try and cut corners here ¢ You have to move to the centroid first 2 IIx= base − Ad( base to centroid ) 2 IIAdxx=+( centroid to x-axis ) 22 IIx= base − Ad( base to centroid) + Ad( centroid to x-axis ) 5" y 8 in 6in 10" 2.12" x 23 Moment of Inertia - Composite Area Monday, November 26, 2012 Using the Table ¢ In this problem, we have to locate the y centroid of the figure with respect to the base ¢ We can use the table to determine this This ybar is with respect the base of 4r 45( in) the object, not the x-axis. y == 33ππ 5" y yin= 2.12 8 in 6in 10" 2.12" x 24 Moment of Inertia - Composite Area Monday, November 26, 2012 12 Using the Table ¢ Now the Ix in the table is given about the bottom of the semicircle, not the centroidal axis ¢ That is where the x axis is shown in the table 5" y 8 in 6in 10" 2.12" x 25 Moment of Inertia - Composite Area Monday, November 26, 2012 Using the Table ¢ So you can use the formula to calculate the Ix (Ibase) about the bottom of the semicircle 1 Ir= π 4 5" base 8 y 8 in 1 4 Ibase = π (5 in) 6in 10" 2.12" 8 x 4 Ibase = 245.44 in 26 Moment of Inertia - Composite Area Monday, November 26, 2012 13 Using the Table ¢ Now we can calculate the moment of inertia about the x centroidal axis 2 IIAdbase=+ x base to centroid 2 IIxbase=− Adbase to centroid I245.44 in42 39.27 in 2.12 in 2 x =−( )( ) 5" y 4 Ix = 68.60 in 8 in 6in 10" 2.12" x 27 Moment of Inertia - Composite Area Monday, November 26, 2012 Using the Table ¢ And we can move that moment of inertia the the x-axis 2 Ixx=+ I Adcentroid to x-axis 42 2 Iininininx =+68.60( 39.27)( 6 + 2.12 ) 4 5" y Iinx = 2657.84 8 in 6in 10" 2.12" x 28 Moment of Inertia - Composite Area Monday, November 26, 2012 14 Using the Table ¢ The polar moment of inertia about the origin would be JIIOxy=+ 44 JO =+2657.84 in 2758.72 in 4 5" JO = 5416.56 in y 8 in 6in 10" 2.12" x 29 Moment of Inertia - Composite Area Monday, November 26, 2012 Another Example ¢ We can use the parallel axis theorem to find the moment of inertia of a composite figure 30 Moment of Inertia - Composite Area Monday, November 26, 2012 15 Another Example y 6" 3" 6" x 6" 31 Moment of Inertia - Composite Area Monday, November 26, 2012 Another Example ¢ We can divide up the area into smaller areas with shapes from the table y 6" 3" 6" I II x III 6" 32 Moment of Inertia - Composite Area Monday, November 26, 2012 16 Another Example Since the parallel axis theorem will require the area for each section, that is a reasonable place to start y 6" 3" ID Area (in2) I 36 6" I II 9 II III 27 x III 6" 33 Moment of Inertia - Composite Area Monday, November 26, 2012 Another Example We can locate the centroid of each area with respect the y axis.