# MULTIAXIAL

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

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.

The example (figure) therefore requires:

• The curve of the forces as a function of time (time series): F1(t),F2(t),F3(t)
• The results of the corresponding 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 (multi-axial 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 stresses as a function of time determine whether there is a multiaxial problem or not. If the angle f or the ratio of the two principal stresses s2/s1 is variable with time, this is a multiaxial case.  This can be visualised using Mohr's circle.

If the change in principal stress direction is not very large, multiaxial problems can also be treated as biaxial/uniaxial problems for simplification without any disadvantages. This is interesting for the calculation, as it saves computation time, but also important for test rig testing, as instead of many test cylinders a single one can be used with almost the same damage result.

It is therefore important to assess the extent of the multiaxial problem, which is done using the graph in the following figure. For this purpose, the angle f and the principal stress ratio s2/s1 are each represented by a point for characteristic time steps (if the set of points represents an almost vertical line, there is no multiaxial problem).

### Damage Parameter

Since the stress state in the cutting plane is composed of normal and shear stress, a damage equivalent parameter must be determined. 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

### DIRECTION-DEPENDENT Fatigue Life Calculation / Welded Joints

Structural stress concepts are particularly common in wind energy and shipbuilding, as the very large components are difficult to calculate otherwise.  Several variants of structural stress concepts have now been integrated into winLIFE. These require the stress tensors extrapolated to the weld seam and the unit normal vectors in an input file.

Certain rules have to be applied for FE modelling.  Different methods according to GL, IIW, Marquis Bäckström are implemented in winLIFE.

## Procedure of a Fatigue Life Calculation

### UNIT LOAD CASES FROM FINITE ELEMENTS, SCALED WITH LOAD TIME HISTORIES histories

The calculation - somewhat simplified - proceeds as follows:

• In the first step, one or more FEM unit load cases are calculated.
• A material S-N curve must exist or be generated. This can be a stress-Wöhler curve or a strain-Wöhler curve.
• The calculation time can be considerably reduced if the calculation is restricted to selected nodes. winLIFE offers the option of pre-selecting the nodes to be calculated. However, the user can also individually select the nodes he considers critical.
• The use of hysteresis can significantly reduce the calculation time.
• The stress tensor of the elastic stresses is calculated for each selected node and time step. This is done by scaling the unit load cases with the specified loads.
• A stress decomposition is performed for a number of - typically 20 - section planes, depending on the user specification, and the shear and normal stresses are determined. The damage is then calculated for each of these planes, for which various assumptions can be made. The section with the greatest damage is the critical section.

### USE OF STRAIN GAUGES

Strain gauge measurements can be used as a basis for fatigue life calculations. Data from commonly used rosettes can be imported directly. The import programme is so flexible that any rosette shape and even very different data structures can be imported. A fatigue life calculation can then be made for the point at the measurement location, or it is often possible to convert the measured data to another point. This is often necessary as it is often not possible to take measurements at critical points.

# Modal Superposition

If the excitation frequency of dynamically loaded components is greater than 1/3 of the lower natural frequency of the component, the superposition and scaling of static unit load cases is no longer permissible, as the resulting vibration amplitudes cannot simply be scaled linearly. Instead, the excitation signal must be decomposed (e.g. by Fourier transform) into the signal components in which the natural frequencies are present. The stress state of the structure is determined for each natural frequency.

In modal superposition, two characteristic variables are determined for this purpose:

• The natural frequencies and the stress tensor associated with each natural frequency
• The modal coordinates represent the weighting factors by which the signal component acting at the natural frequency must be multiplied

The method is formally identical to static superposition.

## Measures to reduce the Calculation Time

If the load-time function is long, the computation time for multiaxial problems can be considerable, and the following measures have been taken to reduce the computation time. For example, not every time step of the given load-time function is taken into account, but only those time steps in which at least one of the load-time functions has a reversal point. A hysteresis can also be specified to reduce the number of reversal points.

These measures lead to a significant reduction in calculation time, but at the expense of calculation accuracy. It is therefore generally recommended to perform a first calculation step with the above measures for all nodes on the surface and to generate a list of nodes on the surface in the order of their damage.

In a second step, the calculation is carried out without the measures to reduce the calculation time, with only a number of nodes - e.g. 100 - selected by the user and taken from the list of nodes. In this case, for example, the 100 nodes with the greatest damage are calculated as accurately as possible without simplifications.

This procedure ensures that all nodes are analysed and that the maximum possible calculation accuracy is used for the critical nodes.

## Analysing the Results

In a multiaxial case, analysis of the results is particularly important and is provided as follows: Mohr's circle can be displayed for each individual time step and as a sum image for all time steps and for each section plane for each calculated node. Similarly, the principal stress vectors can be displayed for each individual time step and as a sum image for all time steps.

In addition, the post-processor of the FE program can display the damage sums calculated by winLIFE as coloured areas.

Since the prediction accuracy for the multiaxial case is worse than for the uniaxial/biaxial case, it is important to know the degree of multiaxiality, which is possible using the analysis tools mentioned above.

If the change in principal stress direction is found to be small, uniaxial/biaxial calculations can be used for simplicity.

If several loads act on a component, it is often of interest to know what influence the individual load has on the damage sum. This can be analysed using the partial load analysis.

The following three variants are analysed (the symbols known from set theory are used to identify the variants):

• ∃!  (= only one (load) exists): only one of the existing loads is taken into account, the others are all set to zero.
• ∃! (= exactly one (load) does not exist): one of the acting loads is set to 0, the other loads remain unchanged.
• ∀∃ (any): any combination can be chosen by the user.

For each existing load-time function, a column L1, L2, ... is created in which the multiplier is specified. If this is =1, the load-time function is used unchanged; if it is =0, it is set to =0.

The index column corresponds to the row number in the matrix.
The symbols known from set theory mean:

The entries in the above screen have the following meaning:

Row 5: Load 1 does not act, the other loads are unchanged.

Row 6: Load 2 does not act, the other loads are unchanged.

Row 7: Load 3 has no effect, the other loads are unchanged.

Row 8: Load 4 has no effect, the other loads are unchanged.

Row 9: Loads 1 and 2 are acting, the other loads are not.

Row 10: Loads 1 and 3 are acting, the other loads are not.

Row 11: Loads 2 and 2 are acting, the other loads are not.

Row 12: Loads 1 and 4 are acting, the other loads are not..

Row 13: Loads 2 and 4 are acting, the other loads are not.

Figure 1: Loads 3 and 4 are acting, the other loads are not.