Explanation Given: The height of point B from the top ( h B ) is 3 m. The width of gate ( b ) is 3 m. The height between point A and B ( h A B ) is 3 m. The height between point B and point E ( h B E ) is 3 m. The length between point B and point C ( L B C ) is 3 m. The radius of the quadrant ( r ) is 3 m. Calculation: Calculate the load acting at point B ( w B ) . w B = ρ w g h B b w B = 1 , 000 kg / m 3 × 9.81 m/s 2 × 3 m × 3 m = 88 , 290 N / m Calculate the height of point A from the top ( h A ) . h A = h B + h A B h A = 3 m + 3 m = 6 m Draw the cross section of gate and mark the dimensions as shown in Figure (1). Calculate the load acting at point A ( w A ) . w A = ρ w g h A b w A = 1 , 000 kg / m 3 × 9.81 m/s 2 × 6 m × 3 m = 176 , 580 N / m Calculate the horizontal force acting at the bent plate due to water above point B ( F h ) 1 . ( F h ) 1 = w B h B ( F h ) 1 = 88 , 290 N / m × 3 m = 264 , 870 N Calculate the point of action of ( F h ) 1 , ( y 1 ) . y 1 = 1 2 × h B y 1 = 1 2 × 3 m = 1.5 m Calculate the horizontal force acting at the bent plate from point B to point A ( F h ) 2 . ( F h ) 2 = 1 2 ( w A − w B ) h A B ( F h ) 2 = 1 2 ( 176 , 580 N / m − 88 , 290 N / m ) × 3 m = 132 , 435 N Calculate the point of action of ( F h ) 2 , ( y 2 ) . y 2 = 1 3 × h A B y 2 = 1 3 × 3 m = 1 m Calculate the volume of the rectangular portion BCDE ( V 1 ) . V 1 = h B E × L B C × b V 1 = 3 m × 3 m × 3 m = 27 m 3 Calculate the volume quadrant portion ABC ( V 2 ) . V 2 = ( π 4 r 2 ) × b V 2 = π 4 ( 3 m ) 2 × 3 m = 21.205 m 3 The weight of the imaginary column of water above the gate = the vertical component of the resultant force acting on the gate. Calculate the vertical component force acting on the rectangular portion BCDE ( F v ) 1 . ( F v ) 1 = ρ w g V 1 ( F v ) 1 = 1 , 000 kg / m 3 × 9

2nd Edition

ISBN: 9780134649290

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 2, Problem 97P

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The horizontal and vertical reaction at point A

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