Hot Spot Stress

HyperLife calculates hot spot stress based on grid point force of a toe/root element.

In the original approach suggested by Fermér, Andréasson, Frodin, grid point forces contributed to the element of interest were directly used without any "adjustment". Later research [P. Fransson and G. Pettersson, 2000] showed that averaging grid point forces made grid point forces less mesh sensitive.

The calculation method implemented in HyperLife includes weighted averaging of grid point forces, as:

HyperLife identifies potential damage locations at weld toe, weld root, weld throat and their evaluation surface from weld elements location and its normal direction (as shown in previous sections).
For the corresponding root and toe damage locations (in adjacent elements), and in the throat (in the weld element), local coordinate systems are constructed. The local X axis is constructed in the direction perpendicular to and away from the corresponding weld element face at the center of the adjacent element on the weld line (the X axis is in located in the plane of the adjacent element). The local Y axis is constructed perpendicular to this X axis in the plane of the adjacent element.

Figure 1. Seam Weld Stress Calculation Example
Grid point forces are calculated at grids (Q and R) of the adjacent element (2) along the weld line. Grid point force contributions are sourced from elements attached (1 and 3) to the adjacent element.
Grid point forces are calculated as:(1)
$f_{Q} = f_{Q 1} + f_{Q 2}$
(2)
$f_{R} = f_{R 2} + f_{R 3}$

Moments are written as:(3)
$m_{Q} = m_{Q 1} + m_{Q 2}$
(4)
$m_{R} = m_{R 2} + m_{R 3}$
At each node ( $Q$ and $R$ ) of the adjacent element that lies on the weld line, averaged grid point forces/moments weighted by length of the adjacent element and the attached element on the weld toe line are calculated.
Weighted forces are written as:(5)
$f_{Q}^{R I G H T} = \frac{L_{2}}{L_{1} + L_{2}} f_{Q}$
(6)
$f_{R}^{L E F T} = \frac{L_{2}}{L_{2} + L_{3}} f_{R}$

Weighted moments are written as:(7)
$m_{Q}^{R I G H T} = \frac{L_{2}}{L_{1} + L_{2}} m_{Q}$
(8)
$m_{R}^{L E F T} = \frac{L_{2}}{L_{2} + L_{3}} m_{R}$
Line Forces and Moments are calculated based on the weighted grid point forces and moments. These line forces and moments from both ends of the adjacent element on the weld line are averaged to generate the line force and moment at the midpoint ( $L$ ).
Line Forces are calculated as:(9)
$f_{L Q}^{R I G H T} = \frac{2}{L_{2}} (2 f_{Q}^{R I G H T} - f_{R}^{L E F T})$
(10)
$f_{L R}^{L E F T} = \frac{2}{L_{2}} (2 f_{R}^{L E F T} - f_{Q}^{R I G H T})$

Line Moments are calculated as:(11)
$m_{L Q}^{R I G H T} = \frac{2}{L_{2}} (2 m_{Q}^{R I G H T} - m_{R}^{L E F T})$
(12)
$m_{L R}^{L E F T} = \frac{2}{L_{2}} (2 m_{R}^{L E F T} - m_{Q}^{R I G H T})$

The averaged line forces and moments at the midpoint, $L$ , for element 2 are:(13)
$f_{2} = \frac{f_{L Q}^{R I G H T} + f_{L R}^{L E F T}}{2}$
(14)
$m_{2} = \frac{m_{L Q}^{R I G H T} + m_{L R}^{L E F T}}{2}$
The line force and moment at the midpoint are then resolved in the local coordinate system constructed in step 2 to generate $f_{2 X}$ and $m_{2 Y}$ respectively. This force and moment pair leads to tensile and bending stresses in the adjacent element with respect to the weld line. These are the forces that are used to calculate stresses.
Stresses are then calculated normal to the weld line for the adjacent element from this force moment pair. Stresses are calculated for both top and bottom of the shell element, and depending on the type of weld, either one or both are used for fatigue calculations. This is the final hot spot stress used in further S-N Fatigue Damage Evaluation. For addition information, refer to Stress-Life (S-N) Approach.(15)
$σ_{T O P} = \frac{f_{2 X}}{T} + 6 \frac{m_{2 Y}}{T^{2}}$
(16)
$σ_{B O T T O M} = \frac{f_{2 X}}{T} - 6 \frac{m_{2 Y}}{T^{2}}$