Machinery's Handbook, 31st Edition
276 Curved Beams The accompanying diagram shows the dimensions of a clamp frame of rectangular cross section. Determine the maximum stress at points A and B due to a clamping force of 1000 pounds.
1,000 lbs
4
2
R
A B
6
24
c
1,000 lbs
The cross-sectional area = 2 3 4 = 8 square inches; the bending moment at section AB is 1000 (24 + 6 + 2) = 32,000 inch pounds; the distance from the center of gravity of the section at AB to point B is c = 2 inches; and using the formula on page 242 , the moment of inertia of the section is 2 3 (4) 3 ÷ 12 = 10.667 inches 4 . Using the straight-beam formula, page 274 , the stress at points A and B due to the bend ing moment is: . , S I Mc 10667 32 000 2 6000 psi # = = = The stress at A is a compressive stress of 6000 psi and that at B is a tensile stress of 6000 psi. These values must be corrected to account for the curvature effect. In Table 4 on page 275 for R / c = (6 + 2)/2 = 4, the value of K is found to be 1.20 and 0.85 for points B and A respectively. Thus, the actual stress due to bending at point B is 1.20 3 6000 = 7200 psi in tension, and the stress at point A is 0.85 3 6000 = 5100 psi in compression. To these stresses at A and B must be added, algebraically, the direct stress at section AB due to the 1000-pound clamping force. The direct stress on section AB will be a tensile stress equal to the clamping force divided by the section area. Thus 1000 ÷ 8 = 125 psi in tension. The maximum unit stress at A is, therefore, 5100 − 125 = 4975 psi in compression, and the maximum unit stress at B is 7200 + 125 = 7325 psi in tension. The following is a similar calculation using metric SI units, assuming that it is required to determine the maximum stress at points A and B due to clamping force of 4 kN acting on the frame. The frame cross section is 50 by 100 mm, the radius R = 200 mm, and the length of the straight portions is 600 mm. Thus, the cross-sectional area = 50 3 100 = 5000 mm 2 ; the bending moment at AB is 4000 (600 + 200) = 3,200,000 newton-millimeters; the distance from the center of gravity of the section at AB to point B is c = 50 mm; and the moment of inertia of the section is (from the formula on page 242), 50(100 3 )⁄12 = 4,170,000 mm 4 . Using the straight-beam formula, page 274, the stress at points A and B due to the bending moment is: , , , , . . s I Mc 4 170 000 3 200 000 50 384 384 newtons per millimeter megapascals 2 # = = = = The stress at A is a compressive stress of 38.4 N/mm 2 , while that at B is a tensile stress of 38.4 N/mm 2 . These values must be corrected to account for the curvature
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