FEA (Finite Element Analysis) methods often underestimate the impact of stress concentration on the strength of mechanical components. The example below walks through how stress concentration methodology found on Knovel can be used in the design phase to increase the strength of a mechanical part without FEA.
This common engineering problem comes in the form of a story from a bicycle component manufacturer. The manufacturer recently received a number of claims from customers related to the failure of pedal axles. In order to correct their design, an engineer was brought in to assess the problem and offer a solution.
An engineer was assigned the task of redesigning the axle to reduce stress by 10-15%, which was deemed to be sufficient in preventing failure of the axle.
(Configuration of axle at the point of failure, dimensions in millimeters)
What We Know:
- The part failure was due to stress concentration at the shoulder
- There were no problems with materials or dimensions
- Research best design of bicycle axle
- Design and test the design to ensure it will increase stress resistance
Researching Designs That Reduce Stress
The engineer starts with a search for ‘stress concentration’ (Click to run search)
Open the Peterson’s Stress Concentration Factors (3rd Edition) and review the table of contents. Chapter 3 of this book is dedicated to concentration factors at shoulder fillets:
(Click to enlarge)
And Section 3.4 Bending contains equations for calculations of stress concentration factors and recommended methods for reduction of stress concentrations.
Method 1: Streamline Fillet
One of these methods involves using a streamline fillet.
Looking through Chapter 3 for more information on streamline fillets, the engineer finds a table with proportions for these fillets (Table 3.1 below).
Unfortunately, the fillet designed according to this table would have increased the diameter of the axle shank to at least (1.475 x 12 mm) + 2 mm = 19.7 mm, where 1.475 is the fillet to axle diameter ratio (see highlighted value in the Table 3.1), 12 mm is the axle diameter, and 2 mm are added to account for a shoulder required to mount the bearing.
Such an increase in the axle diameter is not acceptable because of extra weight and design considerations.
Method 2: Variable Radius Fillet
Another method, suitable for any type of loading (including torsion), is recommended on page 140. It involves a compound radius fillet (also called double radius fillet):
(Click to enlarge)
Design & Testing of Variable Radius Fillet
At this point, the engineer decides to determine if he can reduce stress by using a double radius fillet (see Figure 3.11). He turns to Section 3.5.3 Compound Fillet.
Specifically, he wants to find out if using the radius r2=6 mm and r1=1.4 mm would reduce maximum stresses in the axle.
Two distinct maximum stress concentrations have to be calculated: one for the circumferential line I and another for the circumferential line II.
The drawing of the resulting axle with the diameter at circumferential line I 12 mm and diameter at circumferential line II 13.6 mm is shown in Fig. 2 below:
The chart (click to enlarge) for stress concentration factor for this case is provided on the page 165 of the book:
Since this chart is interactive on Knovel, the engineer uses it to quickly determine stress concentration factors for the axle with modified design:
Using Knovel Graph Digitizer for Chart 3.10 (see picture below), let’s determine stress concentration factors for original condition and for circumferential line I and calculate stress reduction:
(Click to enlarge)
Now let’s determine stress concentration factor for circumferential line II:
The stress concentration factor at this section is obviously higher. However, since the diameter is significantly larger, the nominal stresses are smaller and, as a result, the concentrated stresses are smaller than those for the circumferential line I:
These preliminary calculations have shown that the nominal stresses at the circumferential line II are 68.7% the nominal stresses at the circumferential line I. Thus, the maximum stresses are 88.4% of the maximum stresses at the circumferential line I.
The engineer would like now to verify these calculations because one point was extrapolated outside the range of the chart and therefore could be inaccurate. To calculate, he uses the equations provided at the top of the chart. For the original design with single radius r=2 mm (see Fig. 1):
And then for the double fillet design with radius r2 increased to 6 mm as shown in Fig. 2:
The calculations (click here to see all calculations) demonstrate that replacing a fillet with single radius r = 2 mm with a double radius fillet having r2 = 6 mm decreases the stresses by about 16% which is sufficient to solve the problem.
A few more methods of reducing stress concentration are proposed in Section 3.6 Methods of Reducing Stress Concentration at a Shoulder. The most interesting method for the pedal axle is a relief groove.
The advantage of this method is that the configuration of the axle remains the same.
The calculations demonstrate that replacing a fillet with single radius (r = 2 mm) with a double radius fillet (with r2 = 6 mm) decreases the stresses by about 16%, which is sufficient to reduce the stress of the axle to acceptable levels.