4. (Energy Methods) The truss shown in Fig. 4, is made of equilateral triangles with side length L and axial stiffness EA. Movable nodes are numbered starting with j = 1 and the bars starting with i= 1. At a node j, assume there are applied horizontal and vertical external loads P; and Q; and the corresponding displacements u; and vj. 5 16 6 17 7 18 8 19 9 6 7 8 9 10 11 12 13 14 15 j 2 3 4 5 10 1 2 P Qj 4 Figure 4: A multi-bar truss for problem four. (a) Write the equilibrium equations at each node relating the member forces to the external forces. Arrange them in the matrix form, [B]{F} = {P}, where {F}: = A F P F2 Q1 {P} = F19 (b) Show that the complementary energy stored in the truss can be written as U* : L 2EA {F}{F} L [{P}'[C]{P},_where [C] = [DD, D=B 1-1 2EA (c) Using Castigliano's first theorem, show that 11 1 {0} = {P} where {u}= = U11 (d) Compute the matrix [C] and obtain 10 due to Q7 = 1; while all other loads are zero. (e) Finally, develop code in MATLAB to solve {u} for P; = 10 kN and Qj = 100 kN. Elements are 6061-Al with dimensions A = 0.001 m and L = 1 m.