Centroids and Moment of Inertia Calculation.Pptx

14 January 2011 Mechanics of Materials CIVL 3322 / MECH 3322 Centroids and <a href="/tags/Moment_(physics)/" rel="tag">Moment</a> of Inertia Calculations Centroids n n n x A yA z A ∑ i i ∑ ii ∑ i i x i=1 i=1 i=1 = n y = n z = n A A A ∑ i ∑ i ∑ i i=1 i=1 i=12 Centroid and <a href="/tags/Moment_of_inertia/" rel="tag">Moment of Inertia</a> 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 i=1 x = n 1 in A ∑ i i=11 in1 in 1in3 in6 Centroid and Moment of Inertia Calculations 3 14 January 2011 An Example ¢ We want to locate both the x and y centroids n xA ∑ ii i=1 x = n 1 in A ∑ i i=11 in1 in 1in3 in7 Centroid and Moment of Inertia Calculations An Example ¢ There isn’t much of a chance of developing a function that is easy to integrate in this case n xA ∑ ii i=1 x = n 1 in A ∑ i i=11 in1 in 1in8 Centroid and Moment of Inertia3 Calculations in 4 14 January 2011 An Example ¢ We can break this shape up into a series of shapes that we can find the centroid of n xA ∑ ii i=1 x = n 1 in A ∑ i i=11 in1 in 1in9 Centroid and Moment of Inertia3 Calculations in An Example ¢ There are multiple ways to do this as long as you are consistent n xA ∑ ii i=1 x = n 1 in A ∑ i i=11 in1 in 1in3 in 10 Centroid and Moment of Inertia Calculations 5 14 January 2011 An Example ¢ First, we can develop a rectangle on the left side of the diagram, we will label that as area 1, A n 1 xA ∑ ii i=1 x = n 1 in A ∑ i i=1 A 1 in 11 in 1in11 Centroid and Moment of Inertia3 in Calculations An Example ¢ A second rectangle will be placed in the bottom of the figure, we will label it A2 n xA ∑ ii 1 in i=1 x = n A A ∑ i 1 in 1 i=1A 2 1 in 1in3 in 12 Centroid and Moment of Inertia Calculations 6 14 January 2011 An Example ¢ A right triangle will complete the upper right side of the figure, label it A3 n xA ∑ ii 1 in i=1 x = n A A ∑ i 1 in 1 i=1 A3 A 2 1 in 1in3 in 13 Centroid and Moment of Inertia Calculations An Example ¢ Finally, we will develop a negative area to remove the quarter circle in the lower left hand corner, label it A4 n xA ∑ ii 1 in i=1 x = n A A ∑ i 1 in 1 i=1 A3 A A 4 2 1 in 1in14 Centroid and Moment of Inertia3 Calculationsin 7 14 January 2011 An Example ¢ We will begin to build a table so that keeping up with things will be easier ¢ The first column will be the areas n xA 1 in ∑ ii i=1 ID Area x = n 2 (in ) A A 2 1 in ∑ i 1 A i 1 1 A3 = A2 3A3 1.5 A A 1 in A4 -0.7854 4 2 1in3 in 15 Centroid and Moment of Inertia Calculations An Example ¢ Now we will calculate the <a href="/tags/Distance/" rel="tag">distance</a> to the local centroids from the y-axis (we are calculating an x-centroid) n xAii ID Area xi 1 in ∑ 2 i 1 (in ) (in) = x = n A1 2 0.5 Ai A2 3 2.5 1 in ∑ A1 A i=1 A3 1.5 2 3A4 -0.7854 0.42441 A 1 in A4 2 1in3 in 16 Centroid and Moment of Inertia Calculations 8 14 January 2011 An Example ¢ To calculate the <a href="/tags/Top_(toy)/" rel="tag">top</a> term in the expression we need to multiply the entries in the last two columns by one another n xA ∑ ii i 1 ID Area xi xi*Area 1 in = x = n (in2) (in) (in3) A A1 2 0.5 1 ∑ i A2 3 2.5 7.5 1 in i=1 A1 A A3 1.5 2 3 3A4 -0.7854 0.42441 -0.33333 A 1 in A4 2 1in17 Centroid and Moment of Inertia Calculations 3 inAn Example ¢ If we sum the second column, we have the bottom term in the division, the total area nID Area xi xi*Area xA ∑ ii 2 3 (in ) (in) (in ) x = i=1 A1 2 0.5 1 1 in nA2 3 2.5 7.5 A ∑ i A3 1.5 2 3 i=1 A -0.7854 0.42441 -0.33333 1 in A 4 1 A3 5.714602 A 1 in A4 2 1in3 in 18 Centroid and Moment of Inertia Calculations 9 14 January 2011 An Example ¢ And if we sum the fourth column, we have the top term, the area moment ID Area xi xi*Area n (in2) (in) (in3) xAii A1 2 0.5 1 ∑ i=1 A2 3 2.5 7.5 x = 1 in n A3 1.5 2 3 A A -0.7854 0.42441 -0.33333 ∑ i 4 i=1 5.714602 11.16667 1 in A 1 A3A 1 in A4 2 1in3 in 19 Centroid and Moment of Inertia Calculations An Example ¢ Dividing the sum of the area moments by the total area we calculate the x-centroid n ID Area x x*Area i i xAii (in2) (in) (in3) ∑ x = i=1 A1 2 0.5 1 1 in n A2 3 2.5 7.5 A ∑ i A3 1.5 2 3 i=1 1 in A4 -0.7854 0.42441 -0.33333 A1 A 5.714602 11.16667 3 x bar 1.9541 A 1 in A4 2 1in3 in 20 Centroid and Moment of Inertia Calculations 10 14 January 2011 An Example ¢ You can always remember which to divide by if you look at the final units, remember that a centroid is a distance n ID Area x x*Area i i xAii (in2) (in) (in3) ∑ x = i=1 A1 2 0.5 1 1 in n A2 3 2.5 7.5 A ∑ i A3 1.5 2 3 i=1 1 in A4 -0.7854 0.42441 -0.33333 A1 A 5.714602 11.16667 3 x bar 1.9541 A 1 in A4 2 1in3 in 21 Centroid and Moment of Inertia Calculations An Example ¢ We can do the same process with the y centroid n ID Area x x*Area i i yAii (in2) (in) (in3) ∑ y = i=1 A1 2 0.5 1 1 in n A2 3 2.5 7.5 A ∑ i A3 1.5 2 3 i=1 1 in A4 -0.7854 0.42441 -0.33333 A1 A 5.714602 11.16667 3 x bar 1.9541 A 1 in A4 2 1in3 in 22 Centroid and Moment of Inertia Calculations 11 14 January 2011 An Example ¢ Notice that the bottom term doesn’t change, the area of the figure hasn’t changed n ID Area x x*Area i i yAii (in2) (in) (in3) ∑ y = i=1 A1 2 0.5 1 1 in n A2 3 2.5 7.5 A ∑ i A3 1.5 2 3 i=1 1 in A4 -0.7854 0.42441 -0.33333 A1 A 5.714602 11.16667 3 x bar 1.9541 A 1 in A4 2 1in3 in 23 Centroid and Moment of Inertia Calculations An Example ¢ We only need to add a column of yi’s ID Area xi xi*Area yi (in2) (in) (in3) (in) n A1 2 0.5 1 1 yA A2 3 2.5 7.5 0.5 ∑ ii i=1 A3 1.5 2 3 1.333333 y = n A4 -0.7854 0.42441 -0.33333 0.42441 A ∑ i 5.714602 11.16667 i=1 x bar 1.9541 1 in1 in A 1 A3A 1 in A4 2 1in24 Centroid and Moment of Inertia Calculations 3 in12 14 January 2011 An Example ¢ Calculate the area moments about the x- axis ID Area xi xi*Area yi yi*Area n (in2) (in) (in3) (in) (in3) yAii A1 2 0.5 1 1 2 ∑ A 3 2.5 7.5 0.5 1.5 i=1 2 y = n A3 1.5 2 3 1.333333 2 Ai A4 -0.7854 0.42441 -0.33333 0.42441 -0.33333 ∑ 5.714602 11.16667 i=1 x bar 1.9541 1 in1 in A 1 A3A 1 in A4 2 1in25 Centroid and Moment of Inertia Calculations 3 inAn Example ¢ Sum the area moments ID Area xi xi*Area yi yi*Area (in2) (in) (in3) (in) (in3) n A1 2 0.5 1 1 2 yA A2 3 2.5 7.5 0.5 1.5 ∑ ii i 1 A3 1.5 2 3 1.333333 2 = y = n A4 -0.7854 0.42441 -0.33333 0.42441 -0.33333 A 5.714602 11.16667 5.166667 ∑ i i=1 x bar 1.9541 1 in1 in A 1 A3A 1 in A4 2 1in26 Centroid and Moment of Inertia Calculations 3 in13 14 January 2011 An Example ¢ And make the division of the area moments by the total area ID Area xi xi*Area yi yi*Area (in2) (in) (in3) (in) (in3) nA1 2 0.5 1 1 2 yA ∑ ii A2 3 2.5 7.5 0.5 1.5 y = i=1 A 1.5 2 3 1.333333 2 n 3 A A4 -0.7854 0.42441 -0.33333 0.42441 -0.33333 ∑ i 5.714602 11.16667 5.166667 i=1 x bar 1.9541 y bar 0.904117 1 in1 in A 1 A3A 1 in A4 2 1in27 Centroid and Moment of Inertia Calculations 3 inParallel 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=+ 28 Centroid and Moment of Inertia Calculations 14 14 January 2011 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=+ 29 Centroid and Moment of Inertia Calculations 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=+ 30 Centroid and Moment of Inertia Calculations 15 14 January 2011 Another Example ¢ We can use the parallel axis theorem to find the moment of inertia of a composite figure 31 Centroid and Moment of Inertia Calculations Another Example y 6" 3"6" x6"32 Centroid and Moment of Inertia Calculations 16 14 January 2011 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"33 Centroid and Moment of Inertia Calculations 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"34 Centroid and Moment of Inertia Calculations 17 14 January 2011 Another Example We can locate the centroid of each area with respect the y axis. y 6" 3" ID Area xbari (in2) (in) I 36 3 6" I II 9 7 II x III 27 6 III 6"35 Centroid and Moment of Inertia Calculations Another Example From the table in the back of the book we find that the moment of inertia of a rectangle about its y-centroid axis is 1 3 Ibh= y 6" 3" y 12ID Area xbar 6" I i II (in2) (in) x III I 36 3 6" II 9 7 III 27 6 36 Centroid and Moment of Inertia Calculations 18 14 January 2011 Another Example In this example, for Area I, b=6” and h=6” 1 3 Iininy = (66)( ) 12 4 y 6" 3" Iiny =108ID Area xbar 6" I i II (in2) (in) x III I 36 3 6" II 9 7 III 27 6 37 Centroid and Moment of Inertia Calculations Another Example For the first triangle, the moment of inertia calculation isn’t as obvious y 6" 3"6" I II x III 6"38 Centroid and Moment of Inertia Calculations 19 14 January 2011 Another Example The way it is presented in the text, we can only find the Ix about the centroid y 6" 3" h x 6" I II x III b 6"39 Centroid and Moment of Inertia Calculations Another Example The change may not seem obvious but it is just in how we orient our axis. Remember an axis is our decision. h x y 6" 3" b x 6" I II x b III 6" h40 Centroid and Moment of Inertia Calculations 20 14 January 2011 Another Example So the moment of inertia of the II triangle can be calculated using the formula with the correct orientation. 1 3 Ibh= y 6" 3" y 361 3 6" I Iy = (63 in)( in) 36 II x 4 III Iy = 4.5 in 6"41 Centroid and Moment of Inertia Calculations Another Example The same is true for the III triangle 1 3 Ibh= y 6" 3" y 361 3 6" I Iy = (69 in)( in) 36 II x 4 III Iy =121.5 in 6"42 Centroid and Moment of Inertia Calculations 21 14 January 2011 Another Example Now we can enter the Iybar for each sub-area into the table Sub- y 6" 3" Area Area xbari Iybar (in2) (in) (in4) 6" I I 36 3 108 II x II 9 7 4.5 III III 27 6 121.5 6"43 Centroid and Moment of Inertia Calculations Another Example 2 We can then sum the Iy and the A(dx) to get the moment of inertia for each sub-area y 6" 3"6" I II x III 6" Sub- Iybar + 2 2 Area Area xbari Iybar A(dx) A(dx) (in2) (in) (in4) (in4) (in4) I 36 3 108 324 432 II 9 7 4.5 441 445.5 III 27 6 121.5 972 1093.5 44 Centroid and Moment of Inertia Calculations 22 14 January 2011 2 2 Sub-Area Area xbari Iybar A(dx) Iybar + A(dx) (in2) (in) (in4) (in4) (in4) I 36 3 108 324 432 II 9 7 4.5 441 445.5 III 27 6 121.5 972 1093.5 1971 Another Example And if we sum that last column, we have the Iy for the composite figure y 6" 3"6" I II x III 6" Sub- Iybar + 2 2 Area Area xbari Iybar A(dx) A(dx) (in2) (in) (in4) (in4) (in4) I 36 3 108 324 432 II 9 7 4.5 441 445.5 III 27 6 121.5 972 1093.5 1971 45 Centroid and Moment of Inertia Calculations Another Example We perform the same type analysis for the Ix y 6" 3"6" I II x ID Area III (in2) 6" I 36 II 9 III 27 46 Centroid and Moment of Inertia Calculations 23 14 January 2011 Another Example Locating the y-centroids from the x-axis y 6" 3"6" I Sub-Area Area ybari II x 2 III (in ) (in) 6" I 36 3 II 9 2 III 27 -2 47 Centroid and Moment of Inertia Calculations Another Example Determining the Ix for each sub-area y 6" 3"6" I II x Sub-Area Area ybari Ixbar III 6" (in2) (in) (in4) I 36 3 108 II 9 2 18 III 27 -2 54 48 Centroid and Moment of Inertia Calculations 24 14 January 2011 Another Example 2 Making the A(dy) multiplications y 6" 3"6" I II Sub- x 2 III Area Area ybari Ixbar A(dy) 6" (in2) (in) (in4) (in4) I 36 3 108 324 II 9 2 18 36 III 27 -2 54 108 49 Centroid and Moment of Inertia Calculations Another Example Summing and calculating Ix y 6" 3" Sub- I + xbar 6" I 2 2 Area Area ybari Ixbar A(dy) A(dy) II x 2 4 4 4 III (in ) (in) (in ) (in ) (in ) 6" I 36 3 108 324 432 II 9 2 18 36 54 III 27 -2 54 108 162 648 50 Centroid and Moment of Inertia Calculations 25 14 January 2011 Homework Problem 1 Calculate the centroid of the composite shape in reference to the bottom of the shape and calculate the moment of inertia about the centroid z-axis. 51 Centroid and Moment of Inertia Calculations Homework Problem 2 Calculate the centroid of the composite shape in reference to the bottom of the shape and calculate the moment of inertia about the centroid z-axis. 52 Centroid and Moment of Inertia Calculations 26 14 January 2011 Homework Problem 3 Calculate the centroid of the composite shape in reference to the bottom of the shape and calculate the moment of inertia about the centroid z-axis. 53 Centroid and Moment of Inertia Calculations 27

Centroids and Moment of Inertia Calculation.Pptx

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