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MEGR2144 Mechanics of Solids Extra Credit Problem 2 October 26, 2012 Due Date: November 19, 2012 The problem stated below is the second of the 3...

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MEGR2144 Mechanics of Solids Extra Credit Problem 2 October 26, 2012 Due Date: November 19, 2012 The problem stated below is the second of the 3 problems that will be counted as extra credit problems. A maximum of 4% bonus credit can be earned by solving this problem correctly. Your solution to the problem is due at 9:30am on September November 19, 2012. Late submissions will not be accepted. Also, please keep in mind that this problem is completely optional. Guidelines to submitting your solution: Use a US letter size paper and staple your solution sheets. Show all the work and arrange your work neatly. Graphs must look professional with all the axes and curves labeled correctly. A composite beam is made by bonding together two materials with elastic moduli E 1 and E 2 respectively. The cross-section of the composite beam is shown in the Fgure below. It is assumed that the cross-section is symmetric about a vertical axis y . Given that the beam is subjected to a pure bending moment M , 1. Show that the neutral surface is located at a distance y N from the bottom of the cross-section where y N is given by y N = E 1 ¯ y 1 A 1 + E 2 ¯ y 2 A 2 E 1 A 1 + E 2 A 2 , where A 1 and A 2 are the cross-sectional areas of materials 1 and 2 respectively. ¯ y 1 and ¯ y 2 are the distances of the centroids of areas 1 and 2 respectively from the base of the cross-section. 2. Show that the curvature is given by κ = 1 ρ = M b E 1 I z 1 + E 2 I z 2 where I z 1 and I z 2 are the second moments of areas 1 and 2 respectively about the z axis (neutral axis).
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MEGR2144 Mechanics of Solids Extra Credit Problem 2 October 26, 2012 Due Date: November 19, 2012 3. Show that the bending stress in the beam is given by σ i x = E i M b y E 1 I z 1 + E 2 I z 2 , where i takes on the value of 1 or 2 depending on the material. 4. Now, suppose that the beam is rectangular, material 1 is steel, and material 2 is wood. Let the cross- sectional dimensions be 2 in × 4 in for wood and 2 in × 0.25 in for steel. If the compoiste beam is subjected to a bending moment of 24000 lb-in, determine the maximum and minimum stresses in each of the two materials. Furthermore, sketch the distribution of the normal stress on a cross-section.
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