# Load History Overview

## Static Fatigue Analysis: Linear Superposition of Multiple FEA/Load Time History Load Cases

When there are several load cases at the same time, all of which vary independently of one another, the principle of linear superposition will be used to combine all load cases together to determine the stress variation at each calculation point due to the combination of all loads. The formula is:(1)
${\sigma }_{ij}\left(t\right)=\sum _{k=1}^{n}\left(\frac{{\sigma }_{ij,k}}{{P}_{FEA,k}}{P}_{k}\left(t\right)\right)$

Where $n$ is the total number of load cases, ${P}_{k}\left(t\right)$ and ${\sigma }_{ij}\left(t\right)$ are, respectively, the time variation of the kth load time history and the total stress tensor, and ${P}_{FEA,k}$ and ${\sigma }_{ij}\left(t\right)$ are, respectively, the kth load magnitude and stress tensor from FE analysis.

The following equation depicts how LDM, Scale, and Offset values work together to scale the FEA stress tensor at time t.(2)
${\sigma }_{ij}\left(t\right)=\frac{{\sigma }_{ij.FEA}}{LDM}\left(P\left(t\right)Scale+Offset\right)$
Where:
${\sigma }_{ij}\left(t\right)$
Results stress tensor at time t
${\sigma }_{ij.FEA}$
Stress tensor from static analysis
$P\left(t\right)$
The y point value of load-time history at time t

## Transient Fatigue Analysis

During Transient Fatigue Analysis, the load-time history input is not required, as it is calculated internally during Transient Analysis.

## Load Time History Compression

This option is used to save calculation time. It will remove small cycles (defined by a gate value) and intermediate points.
When removing small cycles, adjacent turning points, where the difference is less than or equal to the maximum range multiplied by relative gate value, will be removed from each channel. However, phase relationship will be maintained, when peaks and valleys occur on different channels at different times. This is shown by the sample above. In the first channel (top), the points at time 4 and 5 will be removed when the absolute gate equals one, while in the second channel (bottom), the points at time 1 and 2 will not be removed in order to keep the phase relationship between channels.

Removing intermediate points is another important mechanism to save computation time. If any point on the load-time history is neither a peak nor valley point, it will not contribute in determining any stress cycle. Such points could be screened out in the fatigue computation without losing the accuracy, but the computation time could be saved significantly. For example, the left column in Figure 2 shows three load-time histories of three super-positioned loadcases, respectively. After removing the intermediate points, the three load-time histories are obtained as in the right column, which can produce the same fatigue results as the left column, but use much less time. This mechanism is built in HyperLife and is effective automatically.