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 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.

Young's modulus (psi)

Shearing modulus for the beam material (psi)

Overhang length of the member (inch)

Design stress of the weld (psi)

Design normal stress for the beam material (psi)

Maximum deflection (inch)

Load (lb)

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

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

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

Volume of weld material ()

Volume of bar ()

Total material cost to be minimized.

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

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

The member thickness is greater than the weld thickness

The weld thickness must be larger than a defined minimum

The deflection cannot exceed the maximum deflection

The buckling load is greater than the applied load

Size constraints

Collect all the constraints

Weld stress

Primary stress acting over the weld throat

Secondary torsional stress.

Moment of P about center of gravity of weld setup

Polar moment of inertia of weld

Bar bending stress

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

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)

Hence the minimum cost and optimized dimensions are