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Making metal forming math more meaningful

Speak in a language that the audience understands and that makes sense

We all know that informed decisions lead to better results. Too frequently, though, we make manufacturing decisions from abstractions without understanding the sources or the risks associated with the abstractions.

The graphs and curves we use in sheet metal forming are simply abstractions derived from estimates taken from test observations. When we use these estimates in simulation, we build in another layer of abstraction from the press floor. The best tool we have to reverse the uncertainty of abstraction is understanding.

When we understand how mathematical models are developed, we are able to make better decisions. Algebraic equations and graphs drive many decisions in forming sheet metal. We use forming limit diagrams and stress/strain curves to decide which materials to use and how to form those materials into desired geometries. Maintenance activities are often based on equipment life cycle curves and Weibull distributions of failure events. Management activities are measured by distributions of events, failure rates, and progress-against-historical results.

Too frequently, though, the employees who measure our results and the managers who are tasked to respond to those statistics fail to understand and properly communicate how results were determined and what they really mean. Sometimes misleading representations are intentional, or sometimes they are poorly prepared. In other cases, poor samples were taken or were measured improperly.

Math is an expressive language. We use variables to represent real-world observations. Equations express relationships between observations and events. Unfortunately, some of us understand the language better than most. To some, long strings of Greek symbols have no more meaning than listening to a “Star Trek” fan speaking Klingon. A speaker may be impressed with his own fluency, but if his audience doesn't understand him, the presentation failed.

If you present mathematical equations, you are responsible for ensuring that your audience understands what you are discussing. In other words, don't follow the time-honored tradition of flashing a string of Greek symbols on a screen, saying, “And here is the equation for _________.” This presentation method pretty much ensures a number of results:

  1. Your audience won't understand you.
  2. Nobody will be impressed by your superior intellect.
  3. Some will doubt your own understanding of your equation, assuming you want to move on quickly to avoid embarrassing questions.
  4. A significant percentage of your audience will find themselves fighting off a nap.

For any topic your only real goal is to communicate. Effective communication requires sharing your knowledge with your audience in a way that is both meaningful and concise. If the audience does not understand the principles and relevance of your knowledge, you failed in your mission.

Anyone who follows politics or advertising knows how graphs are misused to lead the viewer to false conclusions. Too frequently we fall into undisciplined habits that can lead the audience astray. Visual representations can easily lead to poor conclusions when we select improper values and scales for our axes or use an origin that misrepresents the variability of comparative numbers. Both presenters and audience members must consider the following points in developing understanding:

  1. What are the represented values of each axis, and are they appropriate or standard to the topic under discussion?
  2. If the origin (intersection of the X and Y axes) is not (0,0) or a standard value, does it amplify or minimize observation variability in a misleading way?
  3. How many samples were taken to arrive at the graph coordinates?
  4. What are the characteristics of outliers, and do you understand why they are inconsistent with other observations?
  5. If sample sizes are small, do we know if relationships are linear or nonlinear?
  6. Are observation variabilities understood and explainable in plain English?

Sheet metal forming relies on a number of graphical representations of material behavior. The two most commonly used graphs are the stress/strain curve and the forming limit diagram. The stress/strain curve is determined during a tensile test.

The stress/strain curve demonstrates the yield strength, ultimate tensile strength, plastic strain, and elastic strain of the material under a load until failure. Stress (Y axis) is measured in pascal units. The scale is KPa or MPa. Strain (X axis) is measured as a percentage of elongation of the sample material. These two simple statements demonstrate the opportunities for misrepresentation and flawed conclusions.

A chart showing a difference in sample results in KPa may appear to show a significant difference, while it is truly insignificant when shown in MPa. Likewise, a 10 percent elongation on a large sample is not the same as a 10 percent elongation on a smaller sample. It is important that the measures and scale of each axis be well-explained and that the audience knows the test standard used. We cannot simply assume our representation to be comparable without knowing the scales of representation and the conditions of the test.

The forming limit curve (FLC) demonstrates the formability characteristics of sheet metal. The curve shows the maximum amount of major strain the material can sustain, at a given minor strain, before necking. In a given presentation of an FLC, one of the most important considerations is the number of samples tested to define the values of the curve. If a small sample is used and a linear relationship is assumed, your results may be unexpected. (Those results, in fact, may be related to the real relationship being nonlinear.) A reliable sample will include enough observations to show the true shape of the curve, outliers, and their expected causes.

The FLC is properly defined through regression analysis. This means that some of the test samples failed below the curve. Defining a meaningful safety margin, therefore, requires you to understand the statistical dispersion of the sample observations and the confidence level of the sample size.

Die construction, tryout, and manufacturing are expensive activities. Reliance on unsubstantiated assumptions can prove costly during implementation and may hamper troubleshooting efforts.

Your test results and their values are critical in developing a reliable forming operation. Here are a few things to keep in mind:

  1. The curves and diagrams you use in your forming decisions are not absolutes, but estimations from test observations.
  2. Frequently the communication of mathematical information and its meaning is handled too casually by presenters.
  3. Sharing mathematical results is a communication effort. Presentations are successful only when the presentation includes all important variables and samples are truly representative. It is incumbent on the presenter to explain important relationships and significant testing variables to offer useful and meaningful information for the selection of materials, simulation, and testing.

About the Author
4M Partners LLC

Bill Frahm

President

P.O. Box 71191

Rochester Hills, MI 48307

248-506-5873