Welded Beam Design Optimization - Maple Help

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Welded Beam Design Optimization

 Introduction The diagram illustrates a rigid member welded onto a beam. A load is applied to the end of the member.     The total cost of production is equal to the labor costs (a function of the weld dimensions) plus the cost of the weld and beam material.   The beam is to be optimized for minimum cost by varying the weld and member dimensions x1, x2, x3 and x4. The constraints include limits on the shear stress, bending stress, buckling load, and end deflection. The variables x1 and x2 are usually integer multiples of 0.0625 inch, but for this application they are assumed continuous.   Reference: Ragsdell, K and Phillips, D. Optimal Design of a Class of Welded Structures using Geometric Programming. J. Eng. Ind., 98(3):1021–1025, 1976.

Parameters

 > $\mathrm{restart}:$

Young's modulus (psi)

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Shearing modulus for the beam material (psi)

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Overhang length of the member (inch)

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Design stress of the weld (psi)

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Design normal stress for the beam material (psi)

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Maximum deflection (inch)

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Cost per unit volume of the weld material ($inch-3)  > $\mathrm{C__1}≔0.10471:$ Cost per unit volume of the bar ($ inch-3)

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Labor cost per unit weld volume (\$ inch-3)

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Cost Function

Volume of weld material (${\mathrm{inch}}^{3}$)

 > $\mathrm{V__weld}≔{\mathrm{x__1}}^{2}\cdot \mathrm{x__2}:$

Volume of bar (${\mathrm{inch}}^{3}$)

 > $\mathrm{V__bar}≔\mathrm{x__3}\cdot \mathrm{x__4}\cdot \left(\mathrm{L}+\mathrm{x__2}\right):$

Total material cost to be minimized.

 > $\mathrm{f}≔\left(\mathrm{x__1},\mathrm{x__2},\mathrm{x__3},\mathrm{x__4}\right)\to \left(\mathrm{C__1}+\mathrm{C__3}\right)\cdot \mathrm{V__weld}+\mathrm{C__2}\cdot \mathrm{V__bar}:$

Constraints

The shear stress at the beam support location cannot exceed the maximum allowable for the material.

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The normal bending stress at the beam support location cannot exceed the maximum yield strength for the material.

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The member thickness is greater than the weld thickness.

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The weld thickness must be larger than a defined minimum.

 > $\mathrm{con5}≔\mathrm{x__1}-0.125\ge 0:$

The deflection cannot exceed the maximum deflection.

 > $\mathrm{con6}≔\mathrm{δ__max}-\mathrm{δ}\left(\mathrm{x__1},\mathrm{x__2},\mathrm{x__3},\mathrm{x__4}\right)\ge 0:$

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Size constraints

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Collect all the constraints

 > $\mathrm{cons}≔\left\{\mathrm{con1},\mathrm{con2},\mathrm{con3},\mathrm{con4},\mathrm{con5},\mathrm{con6},\mathrm{con7},\mathrm{con8}\right\}:$

Engineering Relationships

Weld stress

 >

Primary stress acting over the weld throat

 > $\mathrm{τ__d}≔\frac{\mathrm{P}}{\sqrt{2}\cdot \mathrm{x__1}\cdot \mathrm{x__2}}:$

Secondary torsional stress.

 > $\mathrm{τ__dd}≔\frac{\mathrm{M}\cdot \mathrm{R}}{\mathrm{J}}:$

Moment of P about center of gravity of weld setup.

 > $\mathrm{M}≔\mathrm{P}\cdot \left(\mathrm{L}+\frac{\mathrm{x__2}}{2}\right):$
 > $\mathrm{R}≔\sqrt{\frac{{\mathrm{x__2}}^{2}}{4}+{\left(\frac{\mathrm{x__1}+\mathrm{x__3}}{2}\right)}^{2}}:$

Polar moment of inertia of weld

 > $\mathrm{J}≔2\left(\mathrm{x__1}\cdot \mathrm{x__2}\cdot \sqrt{2}\cdot \left(\frac{{\mathrm{x__2}}^{2}}{12}+{\left(\frac{\mathrm{x__1}+\mathrm{x__3}}{2}\right)}^{2}\right)\right):$

Bar bending stress

 > $\mathrm{σ}≔\left(\mathrm{x__1},\mathrm{x__2},\mathrm{x__3},\mathrm{x__4}\right)\to \frac{6\cdot \mathrm{P}\cdot \mathrm{L}}{\mathrm{x__4}\cdot {\mathrm{x__3}}^{2}}:$

Bar Deflection: to calculate the deflection, assume the bar to be a cantilever of length L.

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For narrow rectangular bars, the bar buckling load is approximated by (Timoshenko, S and Gere, J. Theory of Elastic Stability. New York: McGraw-Hill. 1961. p.257)

 > $\mathrm{Pc}≔\left(\mathrm{x__1},\mathrm{x__2},\mathrm{x__3},\mathrm{x__4}\right)\to \frac{4.013\cdot \mathrm{E}\cdot \sqrt{\frac{{\mathrm{x__3}}^{2}\cdot {\mathrm{x__4}}^{6}}{36}}}{{\mathrm{L}}^{2}}\cdot \left(1-\frac{\mathrm{x__3}}{2\cdot \mathrm{L}}\cdot \sqrt{\frac{\mathrm{E}}{4\cdot \mathrm{G}}}\right):$

Optimization

 >

Hence the minimum cost and optimized dimensions are

 > $\mathrm{sol}≔\mathrm{Optimization}:-\mathrm{Minimize}\left(\mathrm{f}\left(\mathrm{x__1},\mathrm{x__2},\mathrm{x__3},\mathrm{x__4}\right),\mathrm{cons},\mathrm{bounds}\right)$
 ${\mathrm{sol}}{:=}\left[{1.72485230854216631}{,}\left[\mathrm{x__1}{=}{0.205729639770726}{,}\mathrm{x__2}{=}{3.47048866582864}{,}\mathrm{x__3}{=}{9.03662391069483}{,}\mathrm{x__4}{=}{0.205729639770726}\right]\right]$ (6.1)