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 andbeam 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 areusually integer multiples of 0.0625 inch, but for this application are assumed continuous.
Reference: K. Ragsdell and D. Phillips. 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)6> E d 30$10 :Shearing modulus for the beam material (psi)6> G d 12$10 :Overhang length of the member (inch)> L d 14 :Design stress of the weld (psi)> tmax d 13600 :Design normal stress for the beam material (psi)> smax d 30000 :Maximum deflection (inch)> dmax d 0.25 :Load (lb)> P d 6000 :Cost per unit volume of the weld material ($ inch-3)> C1 d 0.10471 :Cost per unit volume of the bar ($ inch-3)> C2 d 0.04811 :Labor cost per unit weld volume ($ inch-3)> C3 d 1 :Cost Function
3

Volume of weld material (inch )2> Vweld d x1 $x2 :3Volume of bar (inch )> Vbar d x3$x4$ L Cx2 :Total material cost to be minimized.
> f d x1, x2, x3, x4 / C1 CC3 $Vweld CC2$Vbar :Constraints
The shear stress at the beam support location cannot exceed the maximum allowable for the material> con1 d tmax Kt x1, x2, x3, x4 R 0 :The normal bending stress at the beam support location cannot exceed the maximum yield strength for the material> con2 d smaxKs x1, x2, x3, x4 R 0 :The member thickness is greater than the weld thickness> con3 d x4Kx1 R 0 :2> con4 d C1$x1 CC2$x3$x4$ L Cx2 K5 % 0 :The weld thickness must be larger than a defined minimum

> con5 d x1K0.125 R 0 :The deflection cannot exceed the maximum deflection> con6 d dmax Kd x1, x2, x3, x4 R 0 :The buckling load is greater than the applied load
> con7 d Pc x1, x2, x3, x4 KP R 0 :Size constraints> con8 d x1 R 0.1, x4 % 2.0, x2 R 0.1, x3 % 10 :Collect all the constraints> cons d con1, con2, con3, con4, con5, con6, con7, con8 :Engineering Relationships
Weld stressx22> t d x1, x2, x3, x4 /td C2$td$tdd$

22R

Ctdd :

Primary stress acting over the weld throatP> td d:2 $x1$x2
Secondary torsional stress.
M$R> tdd d:J
Moment of P about center of gravity of weld setupx2> M d P$ L C:
22

> Rdx2x1 Cx32

C:
4
2
Polar moment of inertia of weld2

> J d 2 x1$x2$ 2 $x2

12x1 Cx3

C

2

:

2

Bar bending stress> s d x1, x2, x3, x4 /

6$P$L

2 :

x4$x3

Bar Deflection. To calculate the deflection, assume the bar to be a cantilever of length L
3> d d x1, x2, x3, x4 /

4$P$L2 :E$ x4$x3

For narrow rectangular bars, the bar buckling load is approximated by (Timoshenko, S., and J. Gere,Theory of Elastic
Stability,McGraw-Hill, New York, 1961, p. 257)
2

4.013$E$> Pc d x1, x2, x3, x4 /Optimzation

6x3 $x4
362

L$ 1Kx3
2$L

$

E
4$G:

> bounds d x1 = 0 ..10, x2 = 0 ..10 , x3 = 0 ..10, x4 = 0 ..10 :Hence the minimum cost and optimized dimensions are> sol d Optimization:-Minimize f x1, x2, x3, x4 , cons, bounds
sol := 1.72485230854216631, x1 = 0.205729639770726, x2 = 3.47048866582864, x3 = 9.03662391069483, x4= 0.205729639770726(6.1