First, the co-ordinates of joints A, B, C and D need to be found.
From the figure it can be seen that ΔABC is an equilateral triangle with each side of 5 feet. The height of the equilateral triangle will be equal to 5 cos30o, or 4.330 feet. Thus, the coordinates each joint will be,
A = [0, 0, 0] B = [-5, 0, 0]
C = [-2.5, 0, 4.33] D = [-2.5, 5, 4.33]
Since, there are only three unknown forces at joint D, the force analysis of the truss will begin at joint D. Recall, with 3D trusses, three equations can be used at each joint.
Let the force in member AD, BD and CD be labeled as FAD, FBD and FCD respectively.
Expressing each force acting at joint D in the vector notation gives,
W = -250 k
FAD = FAD (-2.5i + 5 j + 4.33k) / 7.071
FBD = FBD (2.5i + 5j + 4.33k) / 7.071
FCD = FCD 5j / 5
All forces have to be in equilibrium, giving
ΣF = 0
W + FAD + FBD + FCD = 0
Equating the i, j and k components gives,
ΣFx = 0
0.3536 FBD - 0.3536 FAD = 0
FBD = FAD
ΣFz = 0
0.6124 FAD + 0.6124 FBD - 250 = 0
By substituting FAD = FBD
0.6124 FAD + 0.6124 FAD - 250 = 0
FAD = 204.1 lb = FBD
ΣFy = 0
0.7071 FAD + 0.7071 FBD + FCD = 0
FCD = -288.6 lb
Members AD and BD are in tension and member CD is in compression.