Calculating slope of line from performance index

Use a performance index to calculate the slope of the index line.

  1. Identify the performance index to maximize. (By convention, performance indices are written so that maximizing the index optimizes the performance.)

    For a panel in bending, the index to maximize is:

    Performance index: M = E1/3/ρ

  2. Rearrange this equation in terms of the numerator:

    E = ρ3* M3

  3. Take the logarithm of each side:

    log(E) = 3log(ρ) + 3log(M)

    This is now in the form of an equation for a straight line (y=mx+c), where m is the gradient (or slope) and c is the y-intercept. This means that on a logarithmic chart of Young’s modulus vs. density, an index line of slope 3 will allow you to compare and select materials for a panel in bending.