Mechanics of Materials - Civil Engineering

1 14 January 2011 1 Mechanics of Materials CIVL 3322 / MECH 3322 centroids and Moment of inertia Calculations Centroid and Moment of inertia Calculations 2 centroids x=xiAii=1n Aii=1n 11niiiniiyAyA=== z=ziAii=1n Aii=1n 14 January 2011 2 Centroid and Moment of inertia Calculations 3 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 Iz=Iz+Ay2P07_045 Centroid and Moment of inertia Calculations 4 14 January 2011 3 P07_045 Centroid and Moment of inertia Calculations 5 An Example Lets start with an example problem and see how this develops 11niiiniixAxA=== 1in1 in1 in3 in1 in6 Centroid and Moment of inertia Calculations 14 January 2011 4 An Example We want to locate both the x and y centroids 1in1 in1 in3 in1 in11niiiniixAxA=== 7 Centroid and Moment of inertia Calculations An Example There isn t much of a chance of

2 Developing a function that is easy to integrate in this case 1in1 in1 in3 in1 in11niiiniixAxA=== 8 Centroid and Moment of inertia Calculations 14 January 2011 5 An Example We can break this shape up into a series of shapes that we can find the centroid of 1in1 in1 in3 in1 in11niiiniixAxA=== 9 Centroid and Moment of inertia Calculations An Example There are multiple ways to do this as long as you are consistent 1in1 in1 in3 in1 in11niiiniixAxA=== 10 Centroid and Moment of inertia Calculations 14 January 2011 6 An Example First, we can develop a rectangle on the left side of the diagram, we will label that as area 1, A1 1in1 in1 in3 in1 inA111niiiniixAxA=== 11 Centroid and Moment of inertia Calculations An Example A second rectangle will be placed in the bottom of the figure, we will label it A2 1in1 in1 in3 in1 inA1A211niiiniixAxA=== 12 Centroid and Moment of inertia Calculations 14 January 2011 7 An Example A right triangle will complete the upper right side of the figure, label it A3 1in1 in1 in3 in1 inA1A2A311niiiniixAxA=== 13 Centroid and Moment of inertia Calculations An Example Finally.

3 We will develop a negative area to remove the quarter circle in the lower left hand corner, label it A4 1in1 in1 in3 in1 inA1A2A3A411niiiniixAxA=== 14 Centroid and Moment of inertia Calculations 14 January 2011 8 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 1in1 in1 in3 in1 inA2A3A1A411niiiniixAxA=== IDArea(in2) Centroid and Moment of inertia Calculations An Example Now we will calculate the distance to the local centroids from the y-axis (we are calculating an x-centroid) 11niiiniixAxA=== IDAreaxi(in2)(in) in1 in3 in1 inA2A3A1A416 Centroid and Moment of inertia Calculations 14 January 2011 9 An Example To calculate the top term in the expression we need to multiply the entries in the last two columns by one another 11niiiniixAxA=== IDAreaxixi*Area(in2)(in)(in3) in1 in3 in1 inA2A3A1A417 Centroid and Moment of inertia Calculations An Example If we sum the second column, we have the bottom term in the division, the total area 11niiiniixAxA=== IDAreaxixi*Area(in2)(in)(in3)

4 In1 in3 in1 inA2A3A1A418 Centroid and Moment of inertia Calculations 14 January 2011 10 An Example And if we sum the fourth column, we have the top term, the area moment 11niiiniixAxA=== IDAreaxixi*Area(in2)(in)(in3) in1 in3 in1 inA2A3A1A419 Centroid and Moment of inertia Calculations An Example Dividing the sum of the area moments by the total area we calculate the x-centroid 11niiiniixAxA=== IDAreaxixi*Area(in2)(in)(in3) in1 in3 in1 inA2A3A1A420 Centroid and Moment of inertia Calculations 14 January 2011 11 An Example You can always remember which to divide by if you look at the final units, remember that a centroid is a distance 11niiiniixAxA=== IDAreaxixi*Area(in2)(in)(in3) in1 in3 in1 inA2A3A1A421 Centroid and Moment of inertia Calculations An Example We can do the same process with the y centroid 11niiiniiyAyA=== IDAreaxixi*Area(in2)(in)(in3)

5 In1 in3 in1 inA2A3A1A422 Centroid and Moment of inertia Calculations 14 January 2011 12 An Example Notice that the bottom term doesn t change, the area of the figure hasn t changed 11niiiniiyAyA=== IDAreaxixi*Area(in2)(in)(in3) in1 in3 in1 inA2A3A1A423 Centroid and Moment of inertia Calculations An Example We only need to add a column of yi s 11niiiniiyAyA=== IDAreaxixi*Areayi(in2)(in)(in3)(in) in1 in3 in1 inA2A3A1A424 Centroid and Moment of inertia Calculations 14 January 2011 13 An Example Calculate the area moments about the x-axis 11niiiniiyAyA=== IDAreaxixi*Areayiyi*Area(in2)(in)(in3)(i n)(in3) in1 in3 in1 inA2A3A1A425 Centroid and Moment of inertia Calculations An Example Sum the area moments 11niiiniiyAyA=== IDAreaxixi*Areayiyi*Area(in2)(in)(in3)(i n)(in3) in1 in3 in1 inA2A3A1A426 Centroid and Moment of inertia Calculations 14 January 2011 14 An Example And make the division of the area moments by the total area 11niiiniiyAyA=== IDAreaxixi*Areayiyi*Area(in2)(in)(in3)(i n)(in3)

6 In1 in3 in1 inA2A3A1A427 Centroid and Moment of inertia Calculations Centroid and Moment of inertia Calculations 28 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 22=+=+yyxxIIAxIIAy14 January 2011 15 Centroid and Moment of inertia Calculations 29 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 22=+=+yyxxIIAxIIAyCentroid and Moment of inertia Calculations 30 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.

7 22=+=+yyxxIIAxIIAy14 January 2011 16 Centroid and Moment of inertia Calculations 31 Another Example We can use the parallel axis theorem to find the moment of inertia of a composite figure Centroid and Moment of inertia Calculations 32 Another Example yx6"3"6"6"14 January 2011 17 Centroid and Moment of inertia Calculations 33 Another Example yx6"3"6"6"IIIIII We can divide up the area into smaller areas with shapes from the table Centroid and Moment of inertia Calculations 34 Another Example Since the parallel axis theorem will require the area for each section, that is a reasonable place to start yx6"3"6"6"IIIIIIID Area (in2) I 36 II 9 III 27 14 January 2011 18 Centroid and Moment of inertia Calculations 35 Another Example We can locate the centroid of each area with respect the y axis.

8 Yx6"3"6"6"IIIIIIID Area xbari (in2) (in) I 36 3 II 9 7 III 27 6 Centroid and Moment of inertia Calculations 36 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 3112=yIbhyx6"3"6"6"IIIIIIID Area xbari (in2) (in) I 36 3 II 9 7 III 27 6 14 January 2011 19 Centroid and Moment of inertia Calculations 37 Another Example In this example, for Area I, b=6 and h=6 ()()3416612108==yyIininIinyx6"3"6"6"IIII IIID Area xbari (in2) (in) I 36 3 II 9 7 III 27 6 Centroid and Moment of inertia Calculations 38 Another Example For the first triangle, the moment of inertia calculation isn t as obvious yx6"3"6"6"IIIIII14 January 2011 20 Centroid and Moment of inertia Calculations 39 Another Example The way it is presented in the text, we can only find the Ix about the centroid xbhyx6"3"6"6"IIIIIIC entroid and Moment of inertia Calculations 40 Another Example The change may not seem obvious but it is just in how we orient our axis.

9 Remember an axis is our decision. xbhxhbyx6"3"6"6"IIIIII14 January 2011 21 Centroid and Moment of inertia Calculations 41 Another Example So the moment of inertia of the II triangle can be calculated using the formula with the correct orientation. ()() "3"6"6"IIIIIIC entroid and Moment of inertia Calculations 42 Another Example The same is true for the III triangle ()() "3"6"6"IIIIII14 January 2011 22 Centroid and Moment of inertia Calculations 43 Another Example Now we can enter the Iybar for each sub-area into the table yx6"3"6"6"IIIIIISub-Area Area xbari Iybar (in2) (in) (in4) I 36 3 108 II 9 7 III 27 6 Centroid and Moment of inertia Calculations 44 Another Example We can then sum the Iy and the A(dx)2 to get the moment of inertia for each sub-area yx6"3"6"6"IIIIIISub-Area Area xbari Iybar A(dx)2 Iybar + A(dx)2 (in2) (in) (in4) (in4)

10 (in4) I 36 3 108 324 432 II 9 7 441 III 27 6 972 14 January 2011 23 Centroid and Moment of inertia Calculations 45 Another Example And if we sum that last column, we have the Iy for the composite figure Sub-AreaAreaxbariIybarA(dx)2 Iybar + A(dx)2(in2)(in)(in4)(in4)(in4) "3"6"6"IIIIIISub-Area Area xbari Iybar A(dx)2 Iybar + A(dx)2 (in2) (in) (in4) (in4) (in4) I 36 3 108 324 432 II 9 7 441 III 27 6 972 1971 Centroid and Moment of inertia Calculations 46 Another Example We perform the same type analysi