# Seam Weld Fatigue

The seam weld fatigue implemented in SimSolid is based on linearization of stress in the conjunction with Volvo method and following that it predicts fatigue damage and life. Linearized stress decomposes a through-thickness elastic stress field into equivalent membrane and bending.

This method determines the contribution of bending to the total stress, and from this determines whether the weld is essentially stiff or flexible. The method typically requires two S-N curves. One is a bending S-N curve which is dominated by bending stress, and the other is a membrane S-N curve which dominated by membrane stress. Interpolation is made between the bending and membrane SN curves based on the degree of bending.

Predicted or likely failure (fatigue crack) locations at the weld toe are marked. These are the locations where the fatigue damage will be evaluated.

## Weld Stress Calculations

The seam weld fatigue damage calculation in SimSolid uses a structural stress at the location of interest using the stress linearization method (link to SimSolid linearized stress documentation) and the bending ratio associated with that stress. Linearized stresses are obtained in a local coordinate system along the stress linearization segment. The local coordinate system is based on the start and end points of the segment as shown in Figure 2. The X-axis of the system is along the segment from entry to exit points. The other two axes are calculated as follows:
• If the local X-axis is not parallel to the global Y-axis:
$\begin{array}{l}{Z}_{local}={X}_{local}*{Y}_{global}\\ {Y}_{local}={Z}_{local}*{X}_{local}\end{array}$
• If the local X-axis is parallel to the global Y-axis:

Local Y-axis ( ${Y}_{local}$ ) is negative of global-X if local-X is along positive global-Y, and vice versa.

${Z}_{local}={X}_{local}*{Y}_{local}$

From the extracted stress values above, the average membrane stress tensor plus the ending bending stress tensors at the entry and exit points are calculated using numerical integration.
${\sigma }_{i}^{m}=\frac{1}{T}\int {}_{-T/2}^{T/2}{\sigma }_{i}dx$
${\sigma }_{iS}^{b}=-\frac{6}{{T}^{2}}\int {}_{-T/2}^{T/2}{\sigma }_{i}xdx$
${\sigma }_{iE}^{b}=-{\sigma }_{iS}^{b}$
Where,
• ${\sigma }_{i}^{m}$ is equal to the ${i}^{th}$ component of membrane stress.
• ${\sigma }_{i}$ is equal to the ${i}^{th}$ component of extracted stress value.
• ${\sigma }_{iS}^{b}$ is equal to the ${i}^{th}$ component of bending stress at the entry.
• ${\sigma }_{iE}^{b}$ is equal to the ${i}^{th}$ component of bending stress at the exit.
• $L$ is equal to the ${i}^{th}$ Length of the stress linearization segment.
• $x$ is equal to the position of a point along the segment.
The stress quantity used for the seam weld damage parameter is the sum of the membrane and bending stresses.
Hence, the stress at the top and the bottom surface is derived from the below equations:
${\sigma }_{top}={\sigma }_{i}^{m}+{\sigma }_{iS}^{b}$
Note: Therefore, the calculation method used in the seam weld analysis engine is equally applicable to stresses calculated from either solid or thin-shell models. This generates top and bottom surface stresses in the same form as would be generated from a thin-shell model.

## Bending Ratio (r)

Experiments show that two types of SN curves are required to perform seam weld fatigue analysis based on a method suggested by M. Fermér, M Andréasson, and B Frodin. Based on lab tests, two SN curves were plotted (Figure 5). The upper curve is obtained in tests where the maximum stress is dominated by bending moment and the lower curve is obtained from tests where membrane force dominates the maximum stress.
The upper and lower curves are referred to as bending S-N curve and membrane S-N curve, respectively. It is recommended that membrane S-N curve should be used when membrane stress dominates in an element and a bending S-N curve should be used when bending stress dominates. Interpolation between the two curves may be carried out depending on the degree of bending dominance.
$r=\frac{\left|{\sigma }_{i}^{b}\right|}{\left|{\sigma }_{i}^{b}\right|+\left|{\sigma }_{i}^{m}\right|}$
Where,
• ${\sigma }_{i}^{b}$ is the maximum bending stress equal to $\frac{6}{{T}^{2}}\int {}_{-T/2}^{T/2}{\sigma }_{i}xdx$ .
• ${\sigma }_{i}^{m}$ is the maximum membrane stress equal to $\frac{1}{T}\int {}_{-T/2}^{T/2}{\sigma }_{i}dx$
The average bending ratio, $\left({r}_{b}^{AVG}\right)$ , is defined as:
$\left({r}_{b}^{AVG}\right)=\frac{{\sum }_{i=1}^{n}\left(r\left[{\sigma }_{TOP}^{2}\right]}{{\sum }_{i=1}^{n}\left[{\sigma }_{TOP}^{2}\right]}$
Where,
• ${\sigma }_{TOP}^{2}$ is the square of the maximum stress at the top surface when the damage is calculated, that is, at weld toe.
• $r$ is the bending ratio.

It is the weighted average of the bending ratio over all points in the loading time history.

An interpolation factor (IF) is now defined as:

The ${r}_{b}^{THR}$ value is defined by the bending ratio threshold in Fatigue solution settings. It is set to 0.5 by default. If average bending ratio (${r}_{b}^{AVG}$ ) is less than or equal to the bending ratio threshold (${r}_{b}^{THR}$ ), then the membrane S-N curve is used to assess damage. If average bending ratio is greater than the bending ratio threshold, then an S-N curve that is interpolated between membrane S-N curve and the bending S-N curve is used.

## Interpolation between Membrane and Bending S-N Curves

The linear interpolation method, as shown in Figure 6, uses the value of the interpolation factor (IF). For the interpolated curve, the calculation is done as follows, for the Fatigue Strength coefficient value (SRI1):
SRI1interpolated = SRI1membrane + (SRI1bending – SRI1membrane) ∙ IF
This defines the stress level at 1 cycle.
Nc1interpolated = 10(log10Nc1membrane + (log10 Nc1bending – log10(Nc1membrane) ∙IF))
S1interpolated = S1membrane + (S1bending – S1membrane) ∙ IF

This defines the stress level at Nc1interpolated cycles. These two points define the first section of the curve up to Nc1interpolated cycles. The last section is defined by finding a third point as follows. A life value is defined being 10 times greater of the Nc1 values for the stiff and flexible curves. From these, we can calculate S2bending and S2membrane. From these, we can interpolate to get S2interpolated which defines the high cycle part of the curve.

S2interpolated = S2membrane + (S2bending – S2membrane). IF

## Thickness

Optionally, a thickness (size effect correction) may be applied, based on thickness t of the part. It operates as follows:

There is no effect if $t\le {T}_{ref}$ (the reference thickness or threshold can be specified in the fatigue solution settings).

The fatigue strength is reduced by a factor of ${\left(\frac{{T}_{ref}}{t}\right)}^{n}$ , where n is the thickness exponent, if $t>{T}_{ref}$ , at all lifetimes (used as a factor to up the stress).

## Mean Stress Correction

FKM mean stress correction is supported for seam weld fatigue. Stress sensitivity can be defined in the fatigue solution settings dialog via the mean stress correction field. Mean stress correction for seam weld fatigue can be enabled through the seam weld fatigue solution settings dialog.

Based on FKM-Guidelines, the Haigh diagram is divided into four regimes based on the Stress ratio (R=Smin/Smax) values. The corrected value is then used to choose the SN curve for the damage and life calculation stage.

The FKM equations below illustrate the calculation of Corrected Stress Amplitude ( ${S}_{e}^{A}$ ). The actual value of stress used in the Damage calculations is the Corrected stress Amplitude (which is $2\cdot {S}_{e}^{A}$ ). These equations apply for SN curves that you input.
• Regime 1 (R>1.0): ${S}_{e}^{A}={S}_{a}\left(1-M\right)$
• Regime 2 ( $-\infty \le R\le 0.0$ ): ${S}_{e}^{A}={S}_{a}+M*{S}_{m}$
• Regime 3 (0.0<R<0.5): ${S}_{e}^{A}=\left(1+M\right)\frac{{S}_{a}+\left(M/3\right)*{S}_{m}}{1+M/3}$
• Regime 4 ( $R\ge 0.5$ ): ${S}_{e}^{A}=\frac{3{S}_{a}{\left(1+M\right)}^{2}}{3+M}$
Where,
• ${S}_{e}^{A}$ is the stress amplitude after mean stress correction (Endurance stress).
• ${S}_{m}$ is the stress amplitude, and M is the mean stress sensitivity.