# MULTIAXIAL

For multiaxial calculations, the winLIFE MULTIAXIAL module is required in addition to winLIFE BASIC.

## From Component Loading to Local Stress/Strain

The knowledge of the local stress (stress, strain) is an essential prerequisite for a fatigue life calculation. There are different problems that need to be solved using different theoretical approaches. On the one hand, the type of component to be analysed (rigid, flexible, multi-body system) plays a role, and on the other hand, the type of load.

Loads can be specified as load-time histories (time series), as load spectra (frequency of load steps) or as spectral density of the load as a function of frequency (power density spectrum). When to use each of these methods is briefly explained here.

### SUPERPOSITION OF FE UNIT LOAD CASES: RIGID BODIES UNDER THE INFLUENCE OF TIME-VARYING LOADS

If a rigid body is subjected to one or more load variables (force, moment), the locally occurring stresses and strains can be calculated by combining the (measured) load-time function with statically determined unit load cases.

The stress tensors obtained from the unit load cases are scaled with the measured load-time functions and superposed for each time step. The result is a stress tensor-time function which is used as the basis for the damage accumulation calculation. This method is applicable if the deformations of the body are small relative to its dimensions.

For this example (figure) we therefore need:

• The curve of the forces as a function of time (time series): F1(t),F2(t),F3(t)
• The results of the associated FE unit load cases.

In each case, a force FFE1, FFE2, FFE3 acts with the same line of action and point of application as the associated force. The results of the FE calculation are the stress tensors in each node of interest (the surface) for each load case.

### NON-LINEAR, TRANSIENT ANALYSIS: VARIABLE COMPONENT GEOMETRY AND TIME- AND/OR DIRECTION-VARYING LOADING

If a body changes its geometry significantly or if the directions of the acting forces change or if inertial forces occur, the superposition method described above is no longer suitable for the calculation. An example of this is an excavator (Fig.) whose bucket is moved in such a way that the three angles alfa, beta and gamma change over time. In addition, the external load changes due to the moving load. In this case, the behaviour of the excavator can be calculated using a MBS/FEM simulation. The forces and stresses at each point of interest can be calculated for each point in time. The stress tensor, which fully describes the stress state, can also be specified.

If you now export the stress tensors for the nodes of interest k for each time step t, a fatigue life calculation can be performed with winLIFE based on this. In this way, other geometrically non-linear variable components and vibration states can also be analysed.

## Components under the Influence of Rotating Principal Stresses (Multiaxial Stresses)

The calculation of components in which the principal stress directions rotate is considerably more complex than the calculation of components in which the principal stress direction does not change. This case, known as a multiaxial problem, usually has a larger number of external loads, but at least 2 external loads, e.g. a shaft under torsion and bending.

However, there are often dozens or even hundreds of independent loads, usually defined by measured time signals. Such problems can be found in various areas of mechanical engineering, such as car bodies, axle components, crankshafts, rotating hubs in wind turbines, etc.

The following figure shows an example of a dynamically loaded axle guide. It is loaded by a horizontal and a vertical force group F1 and F2. As the groups of forces are not proportional to each other, the direction of the principal stress can change (multiaxial problem).

The calculation time for multiaxial problems is considerably longer than for uniaxial or biaxial problems. Therefore, only the nodes on the surface are considered. Since damage usually originates from the surface, this restriction does not limit the solvability. As there is a planar stress state on the surface, the calculation is further simplified.

The principal stress as a function of time decides whether a problem should be treated as a multiaxial problem.

If the angle f or the ratio of the two principal stresses s2/s1 is variable over time, it means that we are dealing with a multiaxial case. Mohr’s stress circle can also be used to decide.

Because it is possible to calculate a multiaxial problem in a simple way without disadvantages if the change of stress direction is only small, the grade of multiaxiality must be determined at the start.  For this purpose WinLIFE shows the angle f and the principal stress ratio s2/s1 for characteristic time steps presented by a point ().The location of the points helps to identify whether a multiaxial problem really exists or if a simplified calculation can be done by assuming that the case is biaxial.

### Damage Parameter

Since the stress situation in the cutting plane consists of normal and shear stresses, these must be used to ascertain a damage equivalent size. The following equivalent stress hypotheses or damage parameters are possible:

• Normal stress - , shear stress and modified von Mises criterion,
• Findley
• Smith Watson Topper, P. Bergmann, Socie and Fatemi Socie,

### Fatigue Life Calculation Depending on the Direction / Welded Joints

Particularly in the field of wind energy and ship building, structure stress concepts are common since very large components can hardly be calculated in any other way. In winLIFE several variations of structure stress concepts have now been included. You will need an entry file with the stress tensors extrapolated on the weld and the normal unit vectors.

## How a fatigue life calculation is carried out

### Using static FEA and superimpose according to (measured) load time histories

The calculation is carried out in the following steps as can also be seen in a simplified manner.

• Firstly, a FE loading condition must be calculated for each effective load.  This must be done with a “unit load”.
• A material S-N curve must be defined in the same way as a stress S-N curve for a uniaxial case.  In the case of Local Strain Approach an e-N-curve must be created.
• The time needed for the calculation can be considerably reduced if critical nodes are pre-selected.  This selection can either be made by the user entering node numbers, or winLIFE can perform an automatic analysis to find the nodes that are most likely to be the critical ones.
• If a hysteresis is carried out and if you only take into consideration the reversals in common, then the load-time function can be reduced to the events relevant to the damage. This considerably reduces the time needed for the calculation.
• The stress tensor for each selected node and each time step is calculated based on the unit load cases and the load-time functions.
• Then, according to the critical cutting plane method, the shear stress and the normal stress is calculated for each node and time step for every plane. With this data, an equivalent stress or a damage parameter can be calculated. There are several hypothesis and damage parameters available, which the user has to select.
• The equivalent stress available for each node, time step and cutting plane is classed according to the rainflow method and a damage calculation is carried out. The plane with the greatest damage is the critical one.  This result is taken as the damage for the node.

## Modale Superposition

You can analyse properly dynamically loaded components by static superposition as described before only if the frequency of the excitation is less than 1/3 ot the first natural frequency of the system. If the condition is not met you need to decompose the signal in single shares for each natural frequency (modal coordinates). Furthermore you need to calculate the stress tensor for each natural fequency.

To performe the modal superposition you have to calculate two charateristic quantities:

• The natural frequencies and the related stress tensor
• The modal coordinates. These represent the share of the signal which excites the structure in the related frequency

This procedure is formally identical to the static superposition.

### Using Strain gauges

When strain rosettes are used and the strain is measured, a fatigue calculation based on this data can be carried out. The data can be read directly and a flexible read-in tool is available (see next figure).

A fatigue prediction can be done for that point, where the measurement has been done.

## How to reduce the calculation time

If the load-time function is long, then the time needed for the calculation can be considerable. In order to reduce the load time function, the user should limit himself to the time steps where at least one load time function has a reversal. If a hysteresis is selected for each individual load-time function, the reversals can be reduced leading to a reduction in the number of time steps to be calculated.

The extensive possibilities for interactively processing the load-time function are also available in the multiaxial module. It is therefore possible to process the load-time function interactively.

## Analysis of the results

In a multiaxial case it is possible to analyse the results in the following ways:

• Mohr’s circle showing the critical cutting planes for each node and all considered time steps. The arising stress conditions can then be seen (diagram 6).
• largest principal stress vector for each node and all considered time steps

In addition there are numerous possibilities of showing the sum of damage with the FEA program post-processor.

The accuracy of the results for multiaxial problems will generally not be as good as those for uniaxial or biaxial problems. For this reason a conventional calculation should be carried out whenever possible in addition to the multiaxial calculation.

Reliable information can be ascertained regarding the critical places where a tear can be expected.  Combining test results and the Relative Miner’s Law, it is also possible to make helpful forecasts regarding the quantity.

## Partial load analysis

If several loads are acting on a component it is often interesting to know what influence the individual loads have on the damage sum. This can be ascertained with the partial load analysis.

We will now examine the following three alternatives. (To distinguish the alternatives, we use symbols recognisable from the set theory.)

• ∃ (= it only exists once) Only one of the existing loads is taken into account. The others are all set at Zero.
• - ∃ (=it does not exist exactly once) one of the acting loads is set at  = 0 while all other loads remain unchanged.
• ∀∃ (as required) the user can select combinations as required.

For each existing load-time-function a column L1, L2, .. is created for the multiplicator. If this =1, then the load-time-function is used unchanged. If it is =0 the corresponding load-time-function will be set at =0.

The index column relates to the matrix line number.