Consider the truss shown in the figure. The bars have lengths as shown in the figure...
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Consider the truss shown in the figure. The bars have lengths as shown in the figure and are made of a material with Young's modulus E and density p. The force F > 0 and the angle a = 30. We want to find the cross-sectional areas A and A such that the weight is minimized, the stresses in the bars do not exceed the allowable stress o, and the tip displacement & does not exceed S = L/E. A1, E, L/cos a A2, E, L F a. Obtain the axial forces in all members using the method of joints (nodal equilibrium). b. Obtain an expression for the tip displacement & using the principle of virtual work. c. Formulate the constrained optimization problem using the information above. d. Solve the problem graphically. Here, you can use Matlab to plot the contours of the objective function and the feasible domain. In addition, verify the solution obtained graphically using Matlab's fmincon function. Provide snippets of the source code and the solution obtained with fmincon. By looking at the graphical solution, which constraints are active/inactive and what can you say about their Lagrange multipliers? Note 1: For parts (a)-(c) consider all variables (A1, A2, L, E, F, and p) as symbolic variables. For parts (d)- (e), assume F = 1, = 1, and L = 1. Note 2: The principle of virtual work states that the deflection of a truss, &, at a specific point, can be computed as n 8 = fiFili AE where fi is a virtual axial force in the i-th member obtained by applying a unit virtual force at the node where you want to compute & in the direction of the desired displacement. For the case of our truss, this corresponds to a unit force applied downward at the free end of the truss. Consider the truss shown in the figure. The bars have lengths as shown in the figure and are made of a material with Young's modulus E and density p. The force F > 0 and the angle a = 30. We want to find the cross-sectional areas A and A such that the weight is minimized, the stresses in the bars do not exceed the allowable stress o, and the tip displacement & does not exceed S = L/E. A1, E, L/cos a A2, E, L F a. Obtain the axial forces in all members using the method of joints (nodal equilibrium). b. Obtain an expression for the tip displacement & using the principle of virtual work. c. Formulate the constrained optimization problem using the information above. d. Solve the problem graphically. Here, you can use Matlab to plot the contours of the objective function and the feasible domain. In addition, verify the solution obtained graphically using Matlab's fmincon function. Provide snippets of the source code and the solution obtained with fmincon. By looking at the graphical solution, which constraints are active/inactive and what can you say about their Lagrange multipliers? Note 1: For parts (a)-(c) consider all variables (A1, A2, L, E, F, and p) as symbolic variables. For parts (d)- (e), assume F = 1, = 1, and L = 1. Note 2: The principle of virtual work states that the deflection of a truss, &, at a specific point, can be computed as n 8 = fiFili AE where fi is a virtual axial force in the i-th member obtained by applying a unit virtual force at the node where you want to compute & in the direction of the desired displacement. For the case of our truss, this corresponds to a unit force applied downward at the free end of the truss.
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