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hw4_sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139 2.002 MECHANICS AND MATERIALS II SOLUTIONS FOR HOMEWORK NO. 4 Problem 1 (20 points) Part A: Because the beam is still in elastic region, the stress field can be expressed as: σ ( y ) = yM y I (1) Since the most highly stressed region is at the verge of yielding, we have | σ ( y ) max | = | − yM y I | y = ± h/ 2 = σ y M y = σ y I h/ 2 (2) For the diamond-orientation beam, I is calculated as: h h 4 h 2 2 dA = 2 ( y ) × 2 dy = I (3) = y 12 2 0 The area moment of inertia for this diamond-orientation is the same as the cross-section appears as a square. Substitution of I into Eq. 2, we get that: M 2 σ y h 3 y ( diamond ) = (4) 12 Part B: h h 3 h 2 σ 2 y M L ( diamond ) = σ y ( y ) dA = 2 σ y y × ( y ) × 2 dy = (5) 6 2 0 Part C: M L ( diamond ) 2 σ y h 3 6 = = 2 (6) 2 σ y h 3 M y ( diamond ) 12 σ When the cross-section appears as a square, M y is calculated as: y I σ y h 4 / 12 σ y h 3 M = = = y ( square ) y max h/ 2 6 1 (7)
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and M L is: h M h 3 L ( square ) = σ ( y ) dA = 2 σ y y × hdy = (8) 2 σ y 4 0 So the ratio of M L /M y for square-orientation is: M L ( square ) σ y h 3 3 4 M = = (9) y ( square ) σ y h 3 2 6 Part D: The ratio of the first-yield bending moments for the two orientations is: M y ( diamond ) 2 σ y h 3 12 M = = 1 / 2 (10) y ( square ) σ y h 3 6 The first-yield bending moment is calculated by: σ y I y M y = (11) max Since σ y is a constant, and I is the same for these two orientations, the above ratio is only determined by the ratio of y max . The ratio of limit moment for the two orientations is: M L ( diamond ) 2 σ y h 3 2 6 M = = 2 < 1 . 0 (12) L ( square ) σ y h 3 3 4 The limit moment for a symmetrical cross-section is calculated by: M L = 2 σ y ydA (13) A/ 2 where A/ 2 is the 1 / 2 area of the cross section in which y 0. For the diamond-orientation, though the maximum value of y is 2 times larger than the square-orientation, the major part of its cross-section is located at the region with small y coordinate. Thus the ratio of limit moment calculated above is smaller than one. Since M y ( diamond ) < M y ( square ) and M L ( diamond ) < M L ( square ) , we should use the beam in the square-orientation.
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