Creating a coupling line chart

Plot a coupling line on a chart to identify failure modes during material selection.

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, 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, σc, 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 Cc. This is an equation in the form y = m x + c.

These equations can now be used to create a coupling line chart:
  1. Create a log chart with the performance indices M1 on the x-axis and M2 on the y-axis.
    1. 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.
  2. Create a display line, and position it so that it goes through a certain point:
    1. Calculate the value of the coupling constant Cc based on the fixed variables. For example, for an L/R ratio of 30, Cc = 5.5 (NB. Young’s modulus has units of GPa and compression strength has units of MPa in Granta Selector).
    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 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.
  3. 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.