Question

Assuming that the bridge segments are free to pivot at each intersection point, what is the tension T in the horizontal segment directly above the point where the object is attached? If you find that the horizontal segment directly above the point where the object is attached is being stretched, indicate this with a positive value for T. If the segment is being compressed, indicate this with a negative value for T. The answer should be in terms of M (mass) and g (gravity).

Answer 1

Since the bridge and all segments of it are static, the sum of the torques acting on any portion of the bridge you choose is zero for any pivot point you may choose. See if you can find a rigid portion of the bridge and a wisely chosen pivot to which you can apply this powerful fact.
Consider the triangular portion shown in bold and let x be the pivot. (This choice eliminates the torques due to the tensions in the beams that attach at point x.) Find the torques on this left hand triangle (which can be considered a solid piece because of the connections). Remember that counterclockwise torque is positive. Assume that the horizontal segment above   is being stretched, so that the force that the tension in this segment exerts on the bold triangle is directed to the right. Express the torque in terms of  T, L , and Fp.

Answer in terms of T and L :
Tt = (TL.sqrt 3) / 2
Summation Tx =  -LFp - T sqrt[L^2 - (L/2)^2]
The negative value of the tension shows that the segment is actually under a compressible load.