Question

solve a-c

Problem 1 (Scorelator) Consider the truss bridge diagrammed below. The bridge is composed of trusses (steel beams) that are connected at joints or nodes. Each truss exerts a force, F, on the nodes that t touches. When the bridge is at rest, these forces balance. A positive force means that the beam is in tension: it is being pulled apart by the nodes. A negative force means that the beam is being pushed inwards and is in compression 3 3/2 12 Fo Fu 13 10 8 6 0.5 2.5 3.5 Figure 1: A truss bridge Cars and trucksl pass over this bridge, and we would like to figure out what forces will act on the bridge as its load changes. Let W, W7, Ws be the weights, measured in Newtons, of vehicles on the bridge located at nodes 6,7,8, respectively We can find equations for the unknown forces with Newton's second and third laws of motion. z, and y-forces sum to zero on node (2). e r and y-forces sum to zero on node (3) .r and y-forces sum to zero on node (4) . r-forces sum to zero on node (5) r and y-forces sum to zero on node (6) e and y-forces sum to zero on node (7) (10) r and y-forces sum to zero on node (8) (12) (13) r and y-forces sum to zero on node (6) . r and y-forees sum to zero on node (7) (10) . z and y-forces sum to zero on node (8) (12) (13) Write this linear system as a matrix equatio A bwhere a is a vector of the forces: (a) Use the backslash command to solve for a, the vector of forces on the bridge, when 11500, and w, = 15000, Save z the weights of the vehicles are W,.-10000. I粁 in A1.dat (b) Find the largest force (in absolute value) of the vector a. Save its absolute value in A2.dat. You may find some of the commands nax, abs, or norm useful. (c) Now suppose that we add weight to the truck at position 6 in increments of 100 New- tons until the bridge collapses. Each bridge member can withstand 30000 Newtons of compression or tensioni, positive or negative boroes.) Therelore, the bridge will collapse when the absolute value of the largest force is larger than 30000. Find the smallest weight of the truck at position 6 for which the bridge collapses. Save your answer as A3.dat. Find which force is the one that exceeded 30000. If the force is F, save the value of j in A4.dat.

Answer 1

0 0 1234567890123 100000-1 0-01-0000 1 000 70-07 0000 -1-0000001-00 100 7 F00 70 70 0000000 0. FS 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0-1 0 f? 0 0 0 0 0 0 0 0 0-1 010 FG 0 0 0 0 0 0 0 F5 0 0 0 0 0 0-1-1 -00000 71 1 100000000 F2-001-0 21-0 10000000000 Fi 70 70 00000000000

Inverse of A found by using MINVERSE in Excel: -0.35355 0.25 0.75 1.06066 0.75 0.5 0.5 0.25 0.25 -0.35355 0.70711 0.5 0.5 0.70711 0.5 0.35355 -1.06066 -0.35355-0.70711 -0.35355-0.35355 1.06066 0.75 0.25 -0.35355 0.25 0.5 0.5 0.75 0.75 0.353553 0.25 0.25 0.5 0.5 0.25 0.75 0.25 0.75 0.75 0.75 0.25 0.25 0 0 0 0.353553-0.35355 | 0.353553 | -0.70711 | 0.353553 |-1.06066 0 0 0.25 0.5 0.5 0.75 0.25 0.25 0.5 0.5 0.75 0.25 0.25 0.5 0.5 0.75 0.25 0.25 0.5 0.5 0.75 0.25 0.5 0.75 0.5 0.5 0.25 0.25 0.5 0.5 0.5 0.5 0.70711 -0.35355 | 0.353553-0.35355-0.70711-0.35355-0.35355 -0.35355 0.353553-0.35355 | 0.353553 | -0.70711 | 0.353553 0.353553 0.353553 0.70711 0.5 0.25 0.25 0.25 0.5 0.25 0.25 0.75 0.25 Now multiplying this inverse matrix to the right hand side vector b, we get the solution as

99-3. 12211 2 9 123456 24-12-21-53 91 4 123456