Parker Hannifin Corp.
Although finite-element analysis has become an indispensable part of engineering, a few users knock it saying the stresses it finds are always too high or too conservative. But is that true?
Take a grooved bar, for example, made of 4340 alloy steel with ultimate tensile strength, Stu, of 160 ksi. Give it a fatigue load of 0 to 30,000 lb and an ultimate axial load of 40,000 lb. This is a simple axisymmetric case (rotationally symmetric about an axis) so a 2D FEA model should work. However, we'll use a 3D FEA model to make a point. After a linear-static stress analysis, results show the maximum (peak) stress is 43.88 ksi with a dummy axial load of 10,000 lb. Because it is a linear analysis, we can easily scale the stresses for different loads against the dummy load. Therefore, the maximum (peak) stress,
at the ultimate load of 40,000 lb. The maximum fatigue stress,
But what is the margin of safety, and how many cycles can the bar endure before failure?
Engineer A may conclude that the margin of safety at the ultimate load is based on the maximum stress value from FEA.
Engineer B, however, uses hand calculations to quickly verify engineer A's result and finds the stress in the groove
= 79.58 ksi nominal stress, where F = axial load, A = section area, and
Can the grooved bar (Stu = 160-ksi steel) hold 175-ksi peak stress at the notch without failure? The answer is yes. But when looking at the accompanying images, what fatigue stress should you capture from the gradient stress maps?
Stress concentrations in ductile materials
Handling FEA stress correctly requires a good understanding of the stress-concentration effect, quantified as a factor . (It's also called the notch factor.) The theoretical stress-concentration factor is based on a theoretical elastic, homogeneous, isotropic material, and can be expressed as
= maximum (peak) stress, and nom = nominal or average stress. For the grooved bar, = 2.2, and is found from Chart 2.19 of Peterson's Stress Concentration Factors (2nd Ed., 1997). Thus the maximum stress at 10,000 lb axial loading can be calculated by multiplying F/A by the appropriate stress-concentration factor, which is
= 43.76 ksi.
It matches FEA results. It tells us that the stress at the grooved area is 2.2 times higher than in the smooth area, and that an FEA program can detect this high stress at the notch.
You can see that is actually the ratio of the FEA peak stress, , to the calculated nominal stress, :
which matches Peterson's Kt value.
But what does maximum stress, max, mean to us? For most practical purposes, a grooved bar (with stress-concentration factor) can carry the same static load as an ungrooved bar. That means if you were using max to calculate the margin of safety the answer would be conservative. How could that happen?
It seems that the stress raiser (i.e., the stress concentration or notch effect) is suppressed by something in the ductile materials at static loading. The behavior is simple: stress at the grooved or notched area rises faster than in the core area. When it reaches the material yielding strength, the localized yielding ( nonlinear elongation) starts from the grooved surface. However, the core of the bar cannot significantly elongate without yielding the entire cross section. Therefore, peak stress is smoothed out through the entire cross section so, eventually, the grooved bar carries the same static load as the ungrooved bar. This is why Kt is customarily ignored for static loads in most engineering practices.
Therefore, the correct answer for margin of safety at the ultimate load in the above case is:
Recall that is the nominal stress:
Therefore, the peak stress from FEA can be converted into the nominal stress using a known for a reasonable margin of safety:
(This matches F/A above.)
We must point out that 99% of FEA used in machine-design practices ( unlike metalforming applications) are linear due to small material deformations. It's also less expensive than other methods. If the FEA model accurately includes geometric irregularities (small fillets and notches, for example) and the mesh is fine enough to detect them, the stress from FEA will be the peak stress. Peak stress could be higher than the nominal stress that does not include Kt. This is why some engineers complain that FEA stresses are too high to show any margin of safety for most designs. The high figures come from linear FEA's inability to reflect the nonlinear behavior of ductile materials. This could be misleading and may become a barrier to using FEA in static analysis. For most static analyses, nominal (average) stresses should be applied instead of peak stress because most engineering materials we deal with are ductile, where Kt is insignificant. Consequently, peak stress from FEA can be converted to nominal stress using an appropriate Kt. And if Kt is unavailable, nominal stress can be calculated by conservatively averaging stress layers cross the entire section for the optimal design.
A BRITTLE MATERIAL
Now consider a grooved bar of 440C CRES. It's a brittle material. (Defined in Margins of safety for ductile and brittle materials para. 2.1.2). Local yielding is not likely to happen at the notched area. The peak stress will initiate cracks in the notch and lead to a sudden rupture. In this case, must be taken into account even for static loading. Therefore, a peak stress from FEA should be applied directly in the margin of safety equation. For brittle materials, it becomes
Note that the peak stress, , is used instead of the average stress, .
Now consider a fatigue case, one with fluctuating or repeated loading. Extensive research over the last century has given us a basic understanding of fatigue-failure mechanisms. Stress concentrations play a critical role in fatigue damage. Fatigue life is quite different for the same material at fatigue loading with different notch factors. A stress riser is always a key factor in the fatigue life.
The conventional procedure for fatigue analysis first calculates maximum and minimum stresses under the fatigue loads, Smax and Smin. Nominal stresses do not take into account geometric irregularities or stress-concentration factors. The mean stress, Sm, alternating stress Salt, and stress ratio, R, are calculated using:
Secondly, find a stress-concentration factor from Kt charts such as Peterson's Stress Concentration Factors, or based on the experience data for typical geometries. Finally, determine the fatigue life using fatigue curves (S-N curves) such as those in MIL-HDBK-5, which originated from a large number of fatigue tests of numerous materials at different fatigue loads and Kts. In this way, the accuracy of a predicted fatigue life largely depends on the fidelity of the Kt estimation.
For real-world engineering, estimating Kt becomes difficult with irregular part geometries. FEA helps here because it can find stresses in complex parts and assemblies. Fatigue analysis using FEA will accurately define stress-concentration effects in that all geometric features, even small fillets and notches, can be included in models with a sufficiently small mesh.
Handling FEA fatigue stresses correctly also requires good understanding of fatigue stress-concentration factor, Kf. It's found from
where = fatigue stress at (notched), and
Sf = fatigue stress at
= 1 (unnotched)
The relation between the fatigue stress-concentration factor and the stress-concentration factor is
where q is the fatigue-notch sensitivity and 0q 1. Here, q = 0 for no notch and q = 1 for a full notch. Average fatigue-notch-sensitivity values for some typical materials can be found in Figure 1.31 in Peterson's Stress Concentration Factors.
The relationship between Kf and Kt shows that q plays the important roll in the fatigue-stress-concentration factor. It should be obvious that Kf Kt. When q is not available, conservative results come from using Kf = Kt or q = 1. S-N curves with Kt = 1 in MIL-HDBK-5 are typically applied to FEA results. By knowing the q effect, it can be shown that S-N curves with Kt = 1 still produce conservative fatigue calculations for FEA applications because it assumes q = 1 or Kf = Kt. That's why the surface factor is usually ignored in FEA for average or machined surfaces.
Now lets return to the example of finding fatigue life cycles for the grooved bar of 4340 steel. For fatigue loading of 0 to 30,000 lb, we can scale the maximum stress from the FEA result:
Fatigue life then comes from a best fit to S-N curves for 4340 alloy-steel bar with Kt = 1 (unnotched) from MIL-HDBK-5J, Figure 22.214.171.124.8(c): Number of fatigue cycles,
Nf, = 88,000 when Smax = 131.64 ksi and R = 0
The most accurate fatigue data comes from MIL-HDBK-5 and equivalents based on many fatigue tests. But be careful. It's possible to introduce significant error into calculations by interpolating and extrapolating from MILHDBK-5 to find new fatigue curves.
Margins of safety for ductile and brittle materials
Here's a programmed format for finding margins of safety and cycle life using linear FEA results.
1.0 Initial calculations.
1.1 Compute the peak stress, Smax, at the fatigue load using an FEA program. (Smin can be scaled from Smax.)
1.2 Calculate the mean stress, Sm, alternating stress, Salt, and stress ratio R from
1.3 Determine cycle life, Nf, from fatigue curves (S-N diagrams) in MIL-HDBK-5 or equivalents using a stress-concentration factor Kt = 1, provided FEA stress results detect geometric irregularities such as small fillet radii or notches. A surface or degradation factor is generally ignored in FEA practices unless the part has severe surface conditions, such as anodizing.
2.0 An analysis procedure for static loading. 2.1 Determine whether the material is ductile or brittle. It must be treated differently for the sake of stress-concentration behaviors.
2.1.1 For ductile materials, Kt is customarily ignored in static loads on common engineering materials with some ductility, and those that behave in a ductile manner. Consequently, peak stress, max, can be converted into a nominal or average stress, nom, assuming that a) The FEA model includes detailed features (geometric irregularities) such as small fillet radii and notches, and that it is b) Meshed with a resolution fine enough to detect those geometric irregularities. Most engineering materials are considered ductile. Examples include 15-5 PH CRES, 17-7 PH CRES, PH13-8MO CRES, 300 Series CRES, Inconel, and alloys of aluminum, titanium, and steel.
2.1.2 For brittle materials, and for those normally ductile but behave in a brittle manner under special conditions, Kt must be taken into account even for static loading. Brittle here means both brittle and relatively homogeneous. For materials permeated with internal discontinuities, such as gray cast iron, stress risers usually have little effect, regardless of the nature of loading. This is because a surface of geometric irregularities seldom causes more severe stress concentrations than those already associated with the internal irregularities. Therefore, the peak stress from an FE model that includes geometric irregularities can be applied directly.
2.2 Use FEA software to find peak stress, max, at the ultimate load. 2.3 Find Kt. For ductile materials (para. 2.1.1), find the stress-concentration factor from charts (such as Peterson's Stress Concentration Factors ). If Kt cannot be calculated, estimate the average stress, nom, by averaging the FEA stress layers conservatively across the entire section and go to para. 2.4.3;
For brittle materials (para. 2.1.2), skip the Kt calculation and go to para. 2.4.3;
2.4 Margins of safety.
2.4.1 For ductile materials (2.1.1), find nom from max/Kt.
2.4.2 For brittle materials (2.1.2) directly apply the FEA peak stress, max
2.4.3 Calculate a margin of safety at the ultimate load.
For ductile materials,
For brittle materials,