A load p at the joint is applied at an angle ߠ ³ so

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A load P at the joint is applied at an angle ߠ ³ so that the horizontal and vertical components of the applied load are given, respectively, as: ܲ ൌ ܲ ߠ ǡ ܲ ൌ ܲ ߠ ² The design variables for the problem are selected as ܣ and ܣ ² The design objective is to minimize the total mass ൌ ߩ݈ሺʹξʹ ܣ ൅ ܣ ² The constraints in the problem are formulated as follows: a) The stresses in members 1, 2 and 3, computed as: ߪ ξଶ ሺ஺ ାξଶ ቃ Ǣ ߪ ξଶ ሺ஺ ାξଶ Ǣ ߪ ξଶ ቂെ ሺ஺ ାξଶ , are to be limited by the allowable stress for the material. b) The axial force in members under compression, given as: ܨ ൌ ߪ ܣ , is limited by the buckling load, L²H²³ െܨ ாூ ³ RU െߪ ாఉ஺ ൑ ߪ ǡ or where the moment of inertia is estimated as: ܫ ൌ ߚܣ ³ ߚ ൌ constant.
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Download free eBooks at Fundamental Engineering Optimization Methods 40 Graphical Optimization c) The horizontal and vertical deflections of the load point, given as: ݑ ൌ ξଶ ௟௉ ǡ ݒ ൌ ξଶ ௟௉ ሺ஺ ାξଶ ሻா ǡ are to be limited by ݑ ൑ ο ǡ ݒ ൑ ο ² d) To avoid possible resonance, the lowest eigenvalue of the structure, given as: ߞ ൌ ଷா஺ ఘ௟ ሺସ஺ ାξଶ ³ where ߩ is the mass density should be higher than a specified frequency, i.e., ߞ ൒ ሺʹߨ߱ ² e) The design variables are required to be greater than some minimum value, i.e., ܣ ǡ ܣ ൒ ܣ ௠௜௡ ² For a particular problem, let ݈ ൌ ͳǤͲ݉ǡ ܲ ൌ ͳͲͲ݇ܰǡ ߠ ൌ ͵Ͳιǡ ߩ ൌ ʹͺͲͲ ௞௚ ǡ ܧ ൌ ͹Ͳܩܲܽǡ ߪ ͳͶͲܯܲܽǡ ο ൌ ο ൌ ͲǤͷ ܿ݉ǡ ߱ ൌ ͷͲܪݖǡ ߚ ൌ ͳǤͲǡ ܣ ௠௜௡ ൌ ʹܿ݉ ² 7KHQ³ ܲ ξଷ ǡ ܲ Ǣ D and Then, and the resulting optimal design problem is formulated as: Minimize ݂ሺܣ ǡ ܣ ሻ ൌ ʹξʹ ܣ ൅ ܣ Subject to: ݃ͳǣ ʹǤͷ ൈ ͳͲ ିସ ξ͵ ܣ ͳ ൫ܣ ൅ ξʹ ܣ ቉ െ ͳ ൑ Ͳǡ ݃ʹǣ ʹǤͷ ൈ ͳͲ ିସ ቈെ ξ͵ ܣ ͳ ൫ܣ ൅ ξʹ ܣ ቉ െ ͳ ൑ Ͳǡ ݃͵ǣ ͷ ൈ ͳͲ ିସ ൫ܣ ൅ ξʹ ܣ െ ͳ ൑ Ͳǡ ݃Ͷǣ ͳǤͲʹ ൈ ͳͲ ି଻ ξ͵ ܣ ͳ ܣ ൫ܣ ൅ ξʹ ܣ ቉ െ ͳ ൑ Ͳǡ ݃ͷǣ ͵Ǥͷ ൈ ͳͲ ିସ ܣ െ ͳ ൑ Ͳǡ ݃͸ǣ ʹ ൈ ͳͲ ିସ ܣ ൅ ξʹ ܣ െ ͳ ൑ Ͳǡ ݃͹ǣ ʹ ൈ ͳͲ ିସ ܣ െ ͳ ൑ Ͳǡ ݃ͺǣ ʹ ൈ ͳͲ ିସ ܣ െ ͳ ൑ Ͳǡ ݃ͻǣ ͳǤ͵ͳ͸ ൈ ͳͲ ିହ ൫Ͷܣ ൅ ξʹ ܣ ൯ െ ͳ ൑ Ͳ ³ ݃ͳͲǣ ʹͶ͸͹ܣ െ ͳ ൑ Ͳ ² The problem was graphically solved in Matlab (see Figure 3.3). The optimum solution is given as:
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