The structure ACD is constrained by a fixed support at D and

The structure ACD is constrained by a fixed support at D and a cable AB that is attached to the ground at B. Shaft CD lies along the z-axis, and arm AC is parallel to the y-axis. Coordinates for the points are: A = (0,2,2), B = (1,3,0), C = (0,0,2), and D = (0,0,0). The tension in AB is known to be 24 kN. (a) Express the force at A, due to the tension in AB, as a Cartesian vector (i.e. with x, y, and z components). (b) Calculate ALL components of the reaction at D. CLEARLY show the directions of all components on a FBD of ACD.

Solution

>> At point A, force will be in direction from A to B

As, A = (0,2,2)

B = (1,3,0)

C = (0,0,2)

and, D = (0,0,0)

>> So, AB = i + j - 2 k

>> Unit Vector along AB = (i + j - 2k)/[1² + 1² + 2²]^1/2 = 0.408 i + 0.408 j - 0.817 k

As, Magnitude of tension in wire AB = 24 kN

>> So, Force at A = 24*(0.408 i + 0.408 j - 0.817 k) = 9.80 i + 9.80 j - 19.60 k

>> Now, Considering member ACD,

Force at C will be equal and in opposite in direction to that of force at A

So, Fc = - Fa

=> Fc = - 9.80 i - 9.80 j + 19.60 k .........ANSWER.........