Component strength Curve optimization for designers who like to think around corners

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Wuxi InfiMotion Technology Co., Ltd.

Why engineers should be well-versed in curve optimization is demonstrated by the authors here. They also provide tips on how designers and notches can work together more effectively, thus improving the lifespan of components.

In almost every technical construction, there are structural notches. Notches are practically unavoidable and arise, for example, from changes in cross-sections, drill holes, grooves, recesses, or threads. The problem is that notches are the biggest "enemies" of load-bearing structures. 
Why is that the case? There are three reasons for it: 
1. Notches are component killers.
2. Notches consume time.
3. Notches drive up costs.

Why notches are component killers

Because notches typically disrupt the flow of force, causing local stress concentrations that ultimately lead to fatigue fractures. For example, in an aluminum die-cast construction, if a notch increases nominal stresses by only 30 percent, this results in an 84 percent reduction in component lifespan with a Wöhler curve exponent k = 7. 
This means that while the notch-free component survives 1 million load cycles, the notched component will fail after only 160,000 load cycles. In common design practice, unfortunately, notches are often the major contributors to stress, leading to stress increases of 50 to 400 percent compared to nominal stresses. These notch-induced stress peaks frequently result in the catastrophic failure of the component. To prevent components from breaking, designers should master the topic of notch optimization thoroughly.

Why notches are time-consuming

Notches lead to time-consuming experimental trials – either in the laboratory or, undoubtedly more unpleasantly, during operation at the customer's site. An example: If you have to evaluate the lifespan of five components in a plastic injection molding construction, and each component must survive 1.5 million load cycles according to the specifications, the experiments with just one parameter set (e.g., f = 4 Hz; T = 23 °C; conditioned) take a whopping 520 hours or around 22 days. And this is still the best-case scenario. 
If the experiments reveal that the component does not meet customer requirements and fails prematurely, the game starts over: meaning redesigning the component again, changing the prototype tool again, producing prototypes again, and conducting all tests again. How much time is needed when multiple optimization loops with physical prototypes are required? Designers usually don't have that much time in component development. To shorten the development time for components, designers should master the topic of notch optimization thoroughly.

Why notches are cost drivers

For several reasons. In the scenario described earlier, with 520 hours of testing at an hourly rate of €45/hour (approx. USD49/hour), the testing costs amount to over €23,000 (approx.  USD 25,000). And that's just for one set of parameters. If one attempts to solve the issue of notch-induced reduction in lifespan by using more or higher-quality material, the problem takes a different route: additional material costs arise.

Suppose the designer needs only 5 grams more material – equivalent to a DIN A4 sheet – to meet the lifespan requirements. With a quantity of 2 million components and a material price of €3.1/kg (USD3.39/kg), this results in additional material costs of €31,000 (approx. USD34,000). To avoid notch-related additional costs in your constructions, it's crucial for designers to master the topic of notch optimization.

How radii lead to stress peaks

And now a practical consideration: How do you design cross-sectional transitions? How do you round sharp, right-angled, or blunt corners? Most designers achieve this with radii. And yes, a radius is better than a sharp corner. However, a radius, even a large one, causes stress concentrations that can trigger fractures. But why?

Explanation for this (Figure 1a):

• The maximum stresses calculated with FEM are the sum of nominal stresses and notch stresses.

• The magnitude of nominal stresses depends on the cross-sectional area

• As the cross-sectional area increases, nominal stresses decrease from the load introduction (top) to the load acceptance (bottom)

• However, notch stresses depend on the curvature of the shape. The greater the curvature, the more abrupt the force flow redirection, and the higher the notch stresses

And here's the crucial point: When you round the cross-sectional transition with a radius, the curvature-induced increase in notch stresses is significantly higher than the cross-sectional-induced decrease in nominal stresses. This leads to local stress peaks in the area of the radius transition.
  What does the FEM calculation say about the radii?

• Despite a large radius of R = 10 mm, local stress peaks of 143 MPa occur in the radius runoff.

• The cross-sectional nominal stresses, however, are only 107 MPa.

• This means: A radius (R = 10 mm) results in an increase in stress from 107 to 143 MPa. That's a 34 percent increase in stress due to a large radius.

You can avoid these radius-induced notch stresses – but only if you are willing to think a little around the corners.

What an optimal notch for homogeneous stresses looks like

Now, the thought process begins (Figure 1b): Through notch optimization, you aim to achieve a homogeneous stress distribution, a low stress level, and ultimately, a durable, lightweight, and cost-effective component. As you already know, the total stresses, which you can calculate using FEM, consist of cross-sectional nominal stresses and curvature-induced notch stresses. However, since the nominal stresses decrease from top (= smaller cross-section) to bottom (= larger cross-section), you must design the notch contour in a way that the notch stresses increase from top (= smaller curvature) to bottom (= larger curvature).

In this case – and this is the central idea – the decrease in nominal stresses is compensated by the increase in notch stresses, and you achieve a homogeneous stress distribution with all the resulting benefits. Simply put as a rule of thumb: Smaller cross-section, smaller curvature; larger cross-section, larger curvature.

Rule of thumb: Smaller cross-section, smaller curvature; larger cross-section, larger curvature.

With this insight, you can design the cross-sectional transitions with curvature-optimized splines for the same tension plate (Figure 1b). Pay particular attention to ensuring that the curvature profile increases as evenly as possible from top (smaller cross-section = smaller curvature) to bottom (larger cross-section = larger curvature). The result: the FEM stress plot corresponding to the sketch shows a homogeneous stress distribution; the nominal stresses are only 107 MPa, meaning there is no shape-induced increase in nominal stresses. It doesn't get any better than this!

Significance for the work of the designer

What can you take away from this example for your design practice?

• Firstly: The interplay between cross-sectional nominal stresses and curvature-induced notch stresses. If you internalize this somewhat abstract principle, you have an extremely useful thinking tool at your disposal.

• Secondly: In CAD, you shape the notch contours using the curvature function, which shows you how "smooth" the contour is. That's the first step. Using FEM, you check the form quality. The more homogeneous the stress distribution, the higher the form quality. That's the second step. After a few optimization loops – typically 3 to 7 iterations – you're done. Your component is in top form and ready for the upcoming load competition.

Example: High notch stresses due to bending loads

Another example (Figure 2): Bending stress in combination with a sharp corner. This is an extremely dangerous situation. Why? Firstly, under bending stress (unlike tensile stress), the nominal stresses are unevenly distributed, leading to local stress concentrations. Secondly, the force flow around the sharp corner must be abruptly redirected, which can result in high notch stresses. To solve this design problem, a) the principle of the interaction between nominal and notch stresses, and b) the method of curvature-based CAD design with subsequent FEM stress analysis (see above) can be helpful.

The FEM results (Figure 2) are self-explanatory: For a bending bracket with radii, the maximum tensile stresses are 155 MPa. With curvature-optimized spline contours, the tensile stresses are reduced to 84 MPa, which means 46 percent less 'stress' for the component. By achieving stress reduction, you can take advantage of one of the two benefits: 
1. You increase the lifespan of the construction with the same material usage, creating a technical advantage.
 2. You reduce material costs with the same lifespan, creating an economic advantage.

Engineers who think around the corner are clearly at an advantage.

In short: Engineers who think around the corner clearly have an advantage!

Do you want to learn more about the topic 'Notch Optimization in Construction Practice'? Or do you want to optimize established construction processes? If so, feel free to contact us. We look forward to hearing from you!