Problem 4-a P The semicircular rod ABCD is a

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Problem 4-a P The semicircular rod ABCD is a maintained in equilibrium by the small wheel at D and the rollers at B and C. Knowing that a = 45o, determine the reactions at B , C , and D.

Solving Problems on Your Own The semicircular rod ABCD is maintained in equilibrium by the small wheel at D and the rollers at B and C. Knowing that a = 45o, determine the reactions at B , C , and D. Problem 4-a P a A O D 45o 45o B C 1. Draw a free-body diagram of the body. This diagram shows the body and all the forces acting on it.

Solving Problems on Your Own The semicircular rod ABCD is maintained in equilibrium by the small wheel at D and the rollers at B and C. Knowing that a = 45o, determine the reactions at B , C , and D. Problem 4-a P 45o O D C A B a 2. Write equilibrium equations and solve for the unknowns. For two-dimensional structure the three equations might be: SFx = 0 SFy = 0 SMO = 0 where O is an arbitrary point in the plane of the structure or SFx = 0 SMA = 0 SMB = 0 where point B is such that line AB is not parallel to the y axis or SMA = 0 SMB = 0 SMC = 0 where the points A, B , and C do not lie in a straight line.

Draw a free-body diagram of the body. Problem 4-a Solution P 45o O D C A B a Draw a free-body diagram of the body. 45o O D C A B P sina P cosa R C/ 2 B/ 2 P

+ S Fx = 0: P cosa + B/ 2 _ C / 2 = 0 (2) Problem 4-a Solution 45o O D C A B P sina P cosa R C/ 2 B/ 2 P Write three equilibrium equations and solve for the unknowns. + S MO = 0: (P sina) R _ D (R) = 0 D = P sina (1) + S Fx = 0: P cosa + B/ 2 _ C / 2 = 0 (2) + S Fy = 0: _P sina + B/ 2 + C / 2 _ P sina = 0 _2P sina + B/ 2 + C / 2 = 0 (3)

(2) + (3) P(cosa _ 2sina) + 2 B/ 2 = 0 B = (2sina _ cosa) P (4) D C A B P sina P cosa R C/ 2 B/ 2 Problem 4-a Solution P (2) + (3) P(cosa _ 2sina) + 2 B/ 2 = 0 B = (2sina _ cosa) P (4) (2) _ (3) P(cosa + 2sina) _ 2 C/ 2 = 0 C = (2sina + cosa) P (5) 2 2

45o O D C A B P sina P cosa R C/ 2 B/ 2 Problem 4-a Solution P For a = 45o sina = cosa = 1/ 2 2 2 2 1 1 2 1 2 EQ. (4) : B = ( _ ) P = P ; B = P 45o EQ. (5) : C = ( _ ) P = P ; C = P 45o EQ. (1) : D = P/ 2 D = P/ 2 2 2 2 1 3 2 3 2