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HW4+Solutions - HOMEWORK PROBLEMS Week 4 Due Friday 7...

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HOMEWORK PROBLEMS Week 4: Due Friday 7 October 1. Determine the coordinates ¯ x, ¯ y of the centroid C of the area shaded in Figure 1. 1 1 0 x y 2 y = x y = x Figure 1 Solution We can express the area as A = integraldisplay 1 0 integraldisplay x x/ 2 dydx = integraldisplay 1 0 parenleftbigg x 1 / 2 - x 2 parenrightbigg dx = bracketleftBigg 2 x 3 / 2 3 - x 2 4 bracketrightBigg 1 0 = 2 3 - 1 4 = 5 12 . It then follows that A ¯ x = integraldisplay 1 0 integraldisplay x x/ 2 xdydx = integraldisplay 1 0 parenleftBigg x 3 / 2 - x 2 2 parenrightBigg dx = bracketleftBigg 2 x 5 / 2 5 - x 3 6 bracketrightBigg 1 0 = 2 5 - 1 6 = 7 30 and A ¯ y = integraldisplay 1 0 integraldisplay x x/ 2 ydydx = integraldisplay 1 0 bracketleftBigg y 2 2 bracketrightBigg y = x y = x/ 2 dx = integraldisplay 1 0 parenleftBigg x 2 - x 2 8 parenrightBigg dx = bracketleftBigg x 2 4 - x 3 24 bracketrightBigg 1 0 = 1 4 - 1 24 = 5 24 . We concude that ¯ x = A ¯ x A = 7 30 12 5 = 14 25 ; ¯ y = A ¯ y A = 5 24 12 5 = 1 2 . 1
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2. A distributed load w ( x ) = 1 + 3 x - 2 x 2 acts on the cantilever beam AB , as shown in Figure 2, where x is measured in meters and w is in kN/m. Find the reactions (force and moment) at A . 2 1 m (kN/m) x w(x) = 1 + 3x - 2x A B Figure 2 Solution The reaction force (positive upwards) is R A = integraldisplay 1 0 w ( x ) dx = integraldisplay 1 0 parenleftBig 1 + 3 x - 2 x 2 parenrightBig dx = bracketleftBigg x + 3 x 2 2 - 2 x 3 3 bracketrightBigg 1 0 = 1 + 3 2 - 2 3 = 11 6 kN . The reaction moment (counterclockwise positive) is M A = integraldisplay 1 0 w ( x ) xdx = integraldisplay 1 0 parenleftBig x + 3 x 2 - 2 x 3 parenrightBig dx = bracketleftBigg x 2 2 + x 3 - x 4 2 bracketrightBigg 1 0 = 1 2 + 1 - 1 2 = 1 kN m . 2
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3. The L-shaped bar ACE in Figure 3 is pinned at A and makes contact through a frictionless roller with the inclined surface at E . It is loaded by a vertical force F at D . Draw a free-body diagram of the bar, determine the reactions at A , and hence find the internal forces (axial force, shear force and bending moment) at the point B .
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