Senior product manager
MSC.Software
Santa Ana, Calif.
www.mscsoftware.com

 An automotive steering knuckle is one component that frequently carries loads from several directions, making it a candidate for biaxial fatigue analysis. The graph shows the history from first load case in the Y axis. The software identifies a few details to the right. Angle versus principal stress is a plot of wp versus the maximum absolute principal stress for all time points at the critical node, 7977. Higher stress levels tend to line up vertically at a particular angle, suggesting mobility is minimal and that uniaxial conditions exist. Mobility refers to a stress factor with a wide range of excursions -- it's not stationary along one plane. Stress cycles that show mobility should be eliminated with peak-and-valley-slicing because they have no influence on damage to the component under test. A plot of biaxial ratio versus maximum absolute principal stress is for all time points at the critical node 7977. The biaxial ratio, ae, tends to line up vertically close to zero for this node, indicating a uniaxial condition for higher stress values. The lower stress values should be gated out. The plot displays the number of times each angle, wp, appeared during loading. A spike indicates a predominate angle, in this case about -40°. Other angles appear occasionally but are due to lower stress cycles. Mohr's circles indicate three special biaxial conditions. When biaxiality analysis is negative (as indicated by the Mohr's circles of stress), the maximum shear plane (where cracks tend to initiate) is oriented as shown in the left diagram. In early initiation stages, cracks grow mainly along the surface in shear (Mode 2) before becoming normal to the maximum stress (Mode 1).

The assumption that loads tug in one direction is a simplification that works well, to a point. In the real world, however, loads are simultaneously applied in several directions, producing stresses with no bias to a particular direction. In 3D geometry, these stresses are called multiaxial. To accurately calculate fatigue damage, analyses must identify multiaxial stresses and use appropriate algorithms.

But multiaxial fatigue analysis has been costly and time consuming because it required investigating stresses from loads applied at multiple locations and how they combine at particular or critical locations. A recent feature found in more advanced analysis packages reduces the time for multiaxial fatigue simulations by considering multiple load cases simultaneously with time variations to identify regions of multiaxial stress. In addition, the features examine only those regions that warrant multiaxial analyses.

A steering knuckle on a passenger car provides an example of multiaxial stress analysis. The steering knuckle has a strut mount at the top, ball joint at the bottom, and a steering arm on the side. The wheel spindle fits through a hole in the center. Driving a vehicle over a cobblestone slalom applies loads to the steering knuckle through the strut mount, lower ball joint, steering tie rod, and wheel axis -- a multiaxial load condition.

### Modeling the real world

Fatigue calculations are postprocessing functions, so they must be preceded by a linear or nonlinear finite-element analysis. The fatigue solver imports analysis results involving multiaxial repeated loads, fluctuating loads, and rapidly applied loads.

Cracks, the major sign of fatigue, usually begin on a component's surface unless there is a flaw in the material. Therefore, the fatigue solver uses only the surface elements and nodes from the results. This eliminates internal elements and nodes, which reduces processing time without affecting result accuracy. The process next identifies multiaxial-stress regions to further limit the solver's work. The software must calculate surface-resolved stresses (plane stresses on a structure's surface) to correctly calculate multiaxial relationships and assessments. For example, the two principal stresses are in the plane of the surface. The third principal stress, normal to the surface, is zero, unless the part is subject to internal pressure.

Shell models produce surface stresses by default. However, many solid models produce stress results in coordinate systems that must be transformed into surface resolved stresses. And it takes surface stresses to correctly calculate biaxiality ratios and perform multiaxial assessments.

The software easily finds surface stresses. It generates a vector file for coordinate transformations. A function calculates normals (the Z axis) and defines a local coordinate system at each surface node. Stress results from the fatigue analysis are written in terms of this coordinate system, limiting multiaxial analysis to the X-Y planes. This puts all stresses into a consistent coordinate system.

Users assign fatigue properties in a material-information form for potentially hundreds of different groups and materials, including corrections for surface finish and treatments. Our steering-knuckle analysis is based on properties from as-cast specimens with the same surface and finish.

To transfer loads to components as realistically as possible, they are applied using rigid elements at precise locations. The steering knuckle model was constrained at the wheel center and 12 load cases were applied, including three forces (1,000 N in X, Y, and Z) at the lower ball joint, steering arm and strut mount, and three moments (1,000 N-mm) at the strut mount. Any real-world loading condition from the test track can be replicated using a combination of these 12 load cases.

Results from the fatigue analysis are stored in a database. They show that greatest damage or shortest life appears to be near the loading devices at the end of the steering arm. A list function shows node 7977 to be the critical one with a life of 330 loading cycles, while most remaining nodes fall below a critical damage cutoff. This is user defined but a frequent default value is 20% of ultimate tensile strength (UTS).

A biaxiality analysis tells whether the fatigue analysis is appropriate to the stress states in the component, among other things. It also:

• Calculates surface-resolved stresses by transforming stress results to local coordinate systems at each location where the x-y plane is the plane of the surface.
• Reorders principal stresses from the conventional order. sz is the surface normal stress and should be 0. s1 and s2 are ordered in magnitude. s1 is the largest in-plane principal in absolute value, which makes s2 the other in-plane stress.
• Determines the multi-axial ratio for every location at every time point: ae = s1/s2. The angle wp that s1 makes with the local X axis is also retained for each location at every time point.
• Describes the surface stress state completely by s1, ae and wp.

In addition, ae and the wp become a little unstable when stresses are small. A gate filter removes small stresses when calculating statistics with these parameters.

When biaxiality analysis is positive, cracks tend to be down through the thickness. These are more damaging for the same levels of shear strain. Uniaxial loading, a special case, refers to local stress-state variations, not the overall or global loading environment. Although the global loading imposes complex out-of-phase loads, variations to local stress are less complex at critical (most likely for cracks to start) locations because geometry imposes simplicity. For instance, a stress state at the edge of a thin metal sheet will always be uniaxial.

In addition, biaxiality analyses calculate and plot three main indicators: a mean-biaxiality ratio, biaxiality-ratio standard deviation, and angle spread. The mean biaxiality ratio is the biaxiality ratio average over the entire combined time signal for every location.

The average is used in calculations throughout the loading history. Values are ignored, however, when stress does not exceed a minimum value, by default 20% of UTS. Certain biaxiality ratios also indicate special loading conditions. For example, a mean ratio of zero indicates uniaxial or below minimum loading, -1 tells of pure shear or torsion, and +1 means equimultiaxial stress with 0.3 plane strain.

The biaxiality-ratio-standard deviation tells of the ratio's variability, such as, whether or not loading is proportional. Small ratio values indicate proportional loading, meaning that magnitudes of s1 and s2 vary proportionally to one another.

Large standard deviations in the multiaxial ratio indicate nonproportionality between these two stresses. Nonproportional loading is more difficult to handle and results may be misleading.

Angle spread indicates the mobility of the absolute maximum principal stress (wp ranges from 0 to 180°). For example, 45° or so does not indicate a problem, but movement of 90° or more indicates nonproportional loading, or the angle may occur when there is pure shear -- when stress flips through 90°.

Further postprocessing with biaxiality cross plots includes plotting all outputs, biaxiality versus principal stress, angle versus principal stress, and angle distribution.

In addition, the outputs display time variations of all parameters, such as biaxial ratio, ae, and angle wp, for critical locations. Time variations of these parameters are not as useful as cross-plots against principal stress for all time points.

### Setting up a fatigue analysis

CategoryPossible settingExplanation
1. Analysis MethodSTWSmith-Topper-Watson is a variant on the standard strain-life method that considers the mean stress of each cycle.
2. Plasticity CorrectionNeuberNeuber is the default elastic-plastic correction method.
3. Run Biaxiality AnalysisOn
4. Biaxiality CorrectionNone
5. Strain CombinationMaximum absolute principalThis is a default choice and is the principal strain having the largest magnitude. In a uniaxial test the selection would be Axial Strain.
6. Design Criterion50.0Design Criterion defaults to 50%, giving the component a 50% chance of surviving the calculated life. The probability is base on the scatter defined in the material parameters.
7. Factor of Safety AnalysisOffComponents intended for infinite life are best analyzed with Factor of Safety Analysis. This is not relevant to the steering knuckle.
Setting up fatigue analysis in MSC.Fatigue is done with simple inputs. First, enter General Setup Parameters. Then fill out the Solution Parameters form. Solution parameters for the steering knuckle appear in the table. Notice, biaxiality correction is set to none because we must identify the stress state (uniaxial, biaxial or triaxial) in various regions in the model before a multiaxial analysis.

### A summary in depth

The steering-knuckle example demonstrated a procedure for multiaxial analysis. If the stress state at a particular location is other than uniaxial, advanced solvers in MSC.Fatigue can account for proportional or nonproportional loading.

The proportional-loading approach is based on ae being nonzero but constant, and with minimal stress-tensor mobility. Material Parameter and Hoffman-Seeger are two methods for modifying uniaxial material properties. The material-parameter method makes a new set of parameters for each stress state. But it is only valid for use with a maximum strain based combination that indicates the maximum absolute principal.

The Hoffman-Seeger method makes the same assumptions, but it also makes what is called a Neuber correction in equivalent stress-strain space. Its advantage is that it predicts all principal stresses and strains, so it can be used with any equivalent stress or strain combination parameter.

Accounting for nonproportional loading is still a major research topic. Generally, predicting fatigue life from a nonproportional loading system can only be done properly by doing a critical-plane analysis using one of several critical plane algorithms. Critical-plane algorithms require multiple analyses at representative angles of wp, as well as adoption of a new counting procedure, taking into consideration that a cycle may begin on one plane and close in another. Notch correction for plasticity also becomes complicated and uses a kinematic hardening model (the equivalent of using Neuber and Masing's hypothesis for a uniaxial state). However, one should not assume a nonproportional loading situation just because of complex external loading and geometry.