Minimum Weight Design of a Variable Shape Three-Bar Truss.
Numerical solution using the MATLAB Optimization Toolbox function FMINCON is required. Use of the MATLAB code templates posted on Blackboard is acceptable. You must submit functioning Matlab code which calls FMINCON directly. Individual work required.
See the problem statement (p. 430) for a complete description of the nominal problem. The project involves Problem 12.11 as stated, plus extensions. A brief overview is given here.
This problem involves six (6) design variables:
The length L is a fixed constant, nominally 10 in., although we'll look at the effect of varying L a bit later.
The design objective will be to minimize weight (or, equivalently, volume of material).
The structure is subject to stress constraints which are given as follows:
where j = 1,2,3 represent the following load cases and σij represents the normal stress in the i-th member at the j-th loading condition.
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Load Case
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Load Pj (lb.)
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Angle qj (deg.)
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j = 1
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40.0 x 103
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45
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j = 2
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30.0 x 103
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90
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j = 3
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20.0 x 103
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135
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Formulas for stress, deflection, and other essential quantities are given on p. 431. Variables u and v denote the horizontal and vertical components of deflection at node D. Use the given value of Young's modulus E = 3.0 x 107 psi, which is an approximate value for carbon and alloy steels at ambient temperature.
(a) Nominal Case:
Formulate and solve the minimum volume problem as stated on p. 430-431.
(b) Buckling Constraints Added:
Review the symmetric three-bar truss in Sec. 2.10, p. 38-41, and see the buckling constraints used there. Then, using the nominal formulation in part (a), add buckling constraints to this problem for each member, at each load condition, similar to those described on p. 41, of the form:
Since the geometry of the cross-section is not specified, b is a dimensionless parameter derived from the relation I = βA2, where I is the moment of inertia of the cross section, and A its area. This assumption is reasonable if the shape of the cross section is fixed and its dimensions are varied proportionally. A value of β = 1 may be assumed.
Obtain a new solution, and determine whether any buckling constraints are active. If no buckling constraints are active for L = 10 in., re-run the analysis by increasing L, and estimate the value of L at which at least one buckling constraint becomes active at the solution. Or, if no buckling constraints are active for reasonable values of L, so state. Summarize and discuss.
HINT: As you vary L and re-run FMINCON, the initial value of the design vector may need to be changed in order to obtain convergence.
c. Open-Ended Analysis:
You propose an extension of the problem as stated which you believe is of practical interest. Possible subjects could include, but are not limited to:
- Analysis with other load cases.
- Addition of deflection constraints.
- Analysis of constraint sensitivity.
- Exploration of other aspects of the problem, including possible multiple local minima, and their implications.