Welded Beam Design Optimization - Maple Programming 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

restart:

Young's modulus (psi)

E30106: 

Shearing modulus for the beam material (psi)

G12106: 

Overhang length of the member (inch)

L14: 

Design stress of the weld (psi)

τ__max13600: 

Design normal stress for the beam material (psi)

σ__max30000: 

Maximum deflection (inch)

δ__max0.25: 

Load (lb)

P6000: 

Cost per unit volume of the weld material ($ inch-3)

C__10.10471:

Cost per unit volume of the bar ($ inch-3)

C__2 0.04811 : 

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

C__31 :

Cost Function

Volume of weld material (inch3)

V__weldx__12x__2:

Volume of bar (inch3)

V__barx__3x__4L+x__2:

Total material cost to be minimized.

fx__1,x__2,x__3,x__4C__1+C__3V__weld+C__2V__bar:

Constraints

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

con1 τ__max  τx__1,x__2,x__3,x__40:

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

con2 σ__maxσx__1,x__2,x__3,x__4 0:

The member thickness is greater than the weld thickness.

con3 x__4x__10:

con4 C__1x__12+C__2x__3x__4L+x__250:

The weld thickness must be larger than a defined minimum.

con5x__10.1250:

The deflection cannot exceed the maximum deflection.

con6δ__maxδx__1,x__2,x__3,x__40:

The buckling load is greater than the applied load.

con7 Pcx__1,x__2,x__3,x__4P0:

Size constraints

con8x__10.1, x__42.0,x__20.1,x__310:

Collect all the constraints

conscon1,con2,con3,con4,con5,con6,con7,con8:

Engineering Relationships

Weld stress

τx__1,x__2,x__3,x__4τ__d2+2τ__dτ__ddx__22 R+τ__dd2:

Primary stress acting over the weld throat

τ__dP2x__1x__2:

Secondary torsional stress.

τ__ddMRJ:

Moment of P about center of gravity of weld setup.

MPL+x__22:

Rx__224+x__1+x__322:

Polar moment of inertia of weld

J2x__1x__22x__2212+x__1+x__322:

Bar bending stress

σx__1,x__2,x__3,x__46PLx__4x__32:

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

δx__1,x__2,x__3,x__44PL3E x__4x__32:

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)

Pcx__1,x__2,x__3,x__44.013Ex__32x__4636L21x__32LE4G:

Optimization

boundsx__1=0..10,x__2=0..10 ,x__3=0..10,x__4=0..10:

Hence the minimum cost and optimized dimensions are

solOptimization:-Minimizefx__1,x__2,x__3,x__4,cons,bounds

sol:=1.72485230854216631,x__1=0.205729639770726,x__2=3.47048866582864,x__3=9.03662391069483,x__4=0.205729639770726

(6.1)