Create coupling lines

About coupling lines

Coupling lines are used in design problems where there are multiple limiting constraints. In other words, failure may occur by more than one mechanism. For example, a rod loaded in compression may fail by buckling, or by direct compressive failure. Coupling lines allow both failure modes to be considered in the selection process, and enable the dominant failure mode to be identified for a given material choice.

An example coupling line chart is shown above. This considers the case of a hollow piston rod loaded in compression. The length and outer radius of the rod are fixed. The objective is to minimize the mass, and the limiting constraints are that it must not fail by elastic buckling or direct compressive fracture.

The performance indices for both limiting constraints are plotted on the chart axes. The coupling line represents the point where the failure mode switches from one mode to the other. Materials above the coupling line will fail by buckling and those below will fail by compressive fracture. Materials on the line are equally likely to fail by buckling or yielding.

The position of the coupling line is determined by the geometric variables that are fixed by the design. In this case, the fixed variables are the rod length and the outer radius.

Material selection is performed by creating a selection box and moving it along the coupling line to reduce the number of selected materials. Suitable materials are found in the bottom left of the chart if the aim is to minimize the performance indices, or in the top right of the chart if the aim is to maximize the performance indices.

Create a coupling line chart

Before creating the chart, identify what the limiting constraints and material indices are:

1. Identify what the two limiting constraints are.
   • For the piston rod example above, the two constraints are the rod must not fail by yield in compression, and must not fail by elastic buckling.
2. Derive material indices for the constraints.
   1. Compressive fracture will occur if the stress, sigma, in the rod exceeds the compression strength, σ_c, of the material. To avoid failure requires .
   2. The objective is to minimize the mass. The mass of a thin-walled piston rod tube is: m = 2π RtLρ, where R is the tube outer radius, t is the wall thickness, L is the tube length, and ρ is the material density.
   3. Remove the free variable t (wall thickness) from the objective equation, by substituting for t using the equation for compressive failure. This gives the equation for minimizing rod mass:
      
      • The first performance index to minimize is:
   4. The constraint that the rod does not fail by buckling requires a mass of:
      
      where E is the Young’s modulus.
      • The second performance index to minimize is:
3. Derive a coupling equation that connects the two indices.
   • The selection that meets both criteria is found by equating the two expressions for mass to produce a coupling equation:

     m₂ = m₁

     

     The coupling equation is therefore:
     

     with a coupling constant of:
     

     Taking logs of both sides of the coupling equation, produces: log M₂ = log M₁ + log C_c. This is an equation in the form y = m x + c.

These equations can now be used to create a coupling line chart:

4. Create a log chart with the performance indices M₁ on the x-axis and M₂ on the y-axis.
   • Performance indices can be added to a chart axis by building an attribute expression. On the axis tab in the Stage Settings dialog, click Advanced and create an attribute expression from the list of attributes and constants.
5. Create a display line, and position it so that it goes through a certain point:
   1. Calculate the value of the coupling constant C_c based on the fixed variables. For example, for an L/R ratio of 30, C_c = 5.5 (NB. Young’s modulus has units of GPa and compression strength has units of MPa in GRANTA EduPack).
   2. Pick an arbitrary numeric value for M₁, and use the coupling equation to calculate what the corresponding M₂ value would be.
   3. Click Index line and create a display line of slope 1.
   4. Right-click on the display line and change the properties so that the line passes through the point of the specific values of M₁ and M₂ calculated above.
6. Create a selection box, with a corner of the box positioned on the coupling line.
   • If the objective is to minimize the indices, then the top right corner of the box will be on the line.
   • If the objective is to maximize the indices, then the bottom left corner of the box will be on the line.