David Dearth
President
Applied Analysis & Technology
Huntington Beach, Calif.

Edited by Paul Dvorak

Test plate with a hole
The hole produces a discontinuity and makes it necessary to apply stress concentration factors (SCF). (Figure 86 in Peterson's text in For further reading, shows Kt = 3.24.)

The hole produces a discontinuity and makes it necessary to apply stress concentration factors (SCF). (Figure 86 in Peterson's text in For further reading, shows Kt = 3.24.)


A linear-stress analysis with SCF, but without cold work, shows a peak-maximum principal stress from the FEA model at 38,400 psi.

A linear-stress analysis with SCF, but without cold work, shows a peak-maximum principal stress from the FEA model at 38,400 psi.


The residual-compressive stresses at the inside surface of the hole are balanced by residual tension stresses just beyond the edge of the inner surface.

The residual-compressive stresses at the inside surface of the hole are balanced by residual tension stresses just beyond the edge of the inner surface.


Adding stresses from the previous plots provides the combined effect of 16,790 psi at the hole surface. Peak stresses are now away from the surface.

Adding stresses from the previous plots provides the combined effect of 16,790 psi at the hole surface. Peak stresses are now away from the surface.


An analysis with MSC.Nastran shows that without cold work (left), end loading with an SCF yields a maximum principal stress at the hole's edge of 38,400 psi and fatigue life, N = 347,000 cycles. Superimposing the residual compressive stress field (right) yields a maximum principal stress at the hole's edge of 16,790 psi, and N = 4.5 X 106.

An analysis with MSC.Nastran shows that without cold work (left), end loading with an SCF yields a maximum principal stress at the hole's edge of 38,400 psi and fatigue life, N = 347,000 cycles. Superimposing the residual compressive stress field (right) yields a maximum principal stress at the hole's edge of 16,790 psi, and N = 4.5 X 106.


The value of N is read from a fatigue S-N curve modified to account for adjustments of the factors.

The value of N is read from a fatigue S-N curve modified to account for adjustments of the factors.


Doing so is critical to judging a component's durability for both its operational life and that of the equipment in which it is used. That much is common knowledge. But did you know that FEA can indicate how much cold work might improve a part's fatigue strength? It can. But lets start by examining material properties.

Material suppliers often publish fatigue-strength data, the results of tightly controlled laboratory tests with polished specimens. This data is rarely usable. In most cases, it must be adjusted to account for manufacturing operations. Material fatigue-strength data from a laboratory is a starting point. More conservative and traditional estimates of cyclic-loading performance comes by applying several modifying factors for conditions such as surface finish, component size, needed reliability, temperature, and stress concentrations. These are usually applied to a stress calculation as K factors.

In a manufacturing setting, common ways to lengthen a part's fatigue life introduces compressive residual stresses by:

  • Shot peening, or bombarding a surface with pellets
  • Autofrettaging, usually the repeated over-pressurization of a cylinder or hydraulic tube, and
  • Cold-work expansion, usually by pulling a tapered mandrel through holes.

So, can FEA simulate these methods of adding compressive-residual stresses and improving fatigue strength? And, where might it be useful to cold work components to improve fatigue

strength? One way to answer these questions is to work through a sample problem.

Although this discussion is necessarily limited, it provides good insight. Consider a plate with a hole subjected to a uniform axial load that generates a region of concentrated stress near the hole's edge. The hole could be for a rivet or a simple bearing.

The problem can be broken into simple steps.

Step 1: Estimate the peak stress at the hole's edge due to uniform tension loading and by applying a stress concentration factor.

Step 2: Estimate compressive-residual stresses when the hole's surface is uniformly and radially expanded.

Step 3: Add the stresses calculated in the previous tasks.

To gain confidence in the solution, we will present two approaches: Hand calculations using conventional equations found in most engineering textbooks on stress-concentration factors (SCFs) and results from a finite-element idealization.

For the first step, find the peak stresses at the hole's edge due to a uniform distributed end loading of 12,000 psi. The theoretical solution for peak magnification of a uniform end-stress loading at the section is found with:

max = Ktnom

where max = peak stress from the SCF, Kt = value for the SCF and is found in the Peterson text (in For further reading), and nom = nominal stress.

For the problem:
max = 3.24 (12,000 psi)
= 38,800 psi.

Linear stress without cold work shows results with the peak-maximum-principal stress plotted from the edge of the hole. For comparison, the FEA model estimates peak stresses at the edge of the hole at 38,400 psi or within ±1% of the value just calculated.

The second step approximates residual stresses that come from a forced and uniform displacement on the hole's surface. This is what happens when a tapered mandrel is pulled through the hole. (This cold working is sometimes called stress coining.) A feature in nonlinear FE software simulates such a uniform, radial displacement. In the software, let the radius increase by 0.00075 in. This FEA idealization simulates elastic-plastic analysis using an isotropic and kinematic-hardening rule.

The "isotropic + kinematic hardening" selection simulates the nonlinear material after yielding. One could choose "isotropic" or "kinematic" or "isotropic + kinematic." I choose both. Use Von Mises for the yield condition. After a material yields, it contains some residual strain.

There always seems to be some controversy as to what workhardening rule predicts a stress increment, which is consistent with the yield condition. I've found that selecting the "isotropic + kinematic" hardening rule works best to simulate ductile failure. The isotropic hardening rule alone tends to overestimate the hardening component. The kinematic-hardening rule alone tends to underestimate the hardening. The combination of isotropic + kinematic seems to work the best.

Residual stresses after cold expansion plots the stresses remaining after removing the enforced displacements.

The third step superimposes results from previous steps to estimate the stress distribution. The sum of linear SCF stress and cold work adds the previous two plots.

The combined results from the FEA models estimate a 16,790-psi peak stress at the hole's edge. Peak tension stresses are now away from the hole's edge where they are potentially less damaging. The magnitude of this peak stress can be adjusted by altering the cold-expansion distance — the mandrel.

Finally, we can compare estimated fatigue life (N as a number of cycles) due to the SCF alone and after the cold work with the simulated mandrel.

A few additional thoughts
Ever since early research on the failure of railway axles in the mid-nineteenth century, it has been known that compressive residual stress enhances fatigue life and that tensile residual stress reduces it. And subjecting components to full-reversing-cyclic fatigue loading (for example, 100 lb in tension and then compression) tends to reduce the benefits of cold expansion.

Certain environments let residual stresses relax or fade away. These include high temperatures and material overloading.

In fabrication, inducing compressive residual stresses by cold expansion (the mandrel method) often uses a split sleeve between material and mandrel to protect the hole's surface material. Excessive plastic deformation from too much cold working can cause cracks. Follow-up machining reams holes to slightly larger diameters and cleans up surface defects due to expansion.

For nonlinear material models, plastic deformations are assumed to follow isotropic and kinematic plastic-hardening rule with von Mises yielding criterion. This combination of nonlinear-material modeling produces slightly higher residual stress than using an isotopic-plastic-hardening rule alone. The rule is a way of estimating the effective mean or average stress.


FOR FURTHER READING
"Stress Concentrations Pose Sticky Meshing Problems," FE Update Column, MACHINE DESIGN magazine, July 8, 1999.
"When To Go Nonlinear With FEA," MACHINE DESIGN magazine, FE Update column, June 7, 2001.
R. E. Peterson, Stress Concentration Factors, Chapter 4, Holes – Fig. 86, John Wiley & Sons, 1974.


A little optional homework
Interested readers who would like to learn more on the topic of fatigue life can download several files from Author David Dearth. They are available in the online version of this article at machinedesign.comand in this box. For instance, RunNotes_SCFColdWork.pdf provides guidelines for running this type of simulation in FEA software.
SCF_HandCalcs.pdf
are Dearth's hand calculations.
SCFColdWork_v2006.MOD (WinZip file)is a Nastran model of the plate.
Static_SCF_wNoCW.bdf (WinZip file)and CW_Load-Unload_dR=0.00075inch.bdf (WinZip file)are other Nastran files.

Readers can also get a free NX Nastran and Femap demo on CD by registering at ugs.com/forms/femap_demo.shtml. At the bottom of the form select either a full version for 30 days, or a 300-node version. The latter does not expire but does have a model limit of 1,800-DOF (6 DOF/node). Author Dearth has formulated all sample problems for the FE Updates to run within the 300-node limit.

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