# Calculate Linearized Stress

Decompose a through-thickness elastic stress field into equivalent membrane, bending, and peak stresses for comparison with appropriate allowable limits.

## Calculation Method

SimSolid method for calculating linearized stress.

Linearized stress decomposes a through-thickness elastic stress field into equivalent membrane, bending, and peak stresses for comparison with appropriate allowable limits.

- The stress is extracted by interpolation in a local coordinate system at all
the points along the line. The local coordinate system is based on the start
and end points of the stress linearization segment. 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:
Z

_{local}= X_{local}x Y_{global}Y

_{local}= Z_{local}x X_{local} - 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}x Y_{local}

- If the local x-axis is not parallel to the global
y-axis:
- From the extracted stress values above, the average membrane stress tensor +
bending stress tensors at the entry and exit points are calculated using
numerical integration.
$${\sigma}_{i}^{m}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{L}{\displaystyle {\int}_{-L/2}^{L/2}{\sigma}_{i}dx}$$

$${\sigma}_{iS}^{b}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\frac{6}{{L}^{2}}{\displaystyle {\int}_{-\frac{L}{2}}^{\frac{L}{2}}{\sigma}_{i}xdx}$$

$${\sigma}_{iE}^{b}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-{\sigma}_{iS}^{b}$$

$${\sigma}_{i}^{m}$$ =

*i*^{th}component of membrane stress$${\sigma}_{i}$$ =

*i*^{th}component of extracted stress value$${\sigma}_{iS}^{b}$$ =

*i*^{th}component of bending stress at the entry$${\sigma}_{iE}^{b}$$ =

*i*^{th}component of bending stress at the exitL = Length of the Stress linearization segment

x = position of a point along the segment

- Peak stress and membrane and bending stress are also calculated at the entry
and exit.
$${\sigma}_{iS}^{p}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\sigma}_{iS}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\left({\sigma}_{i}^{m}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\sigma}_{iS}^{b}\right)$$

$${\sigma}_{iE}^{p}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\sigma}_{iE}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\left({\sigma}_{i}^{m}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\sigma}_{iE}^{b}\right)$$

$${\sigma}_{iS}^{p}$$ =

*i*^{th}component of peak stress at the entry$${\sigma}_{iE}^{p}$$ =

*i*^{th}component of peak stress at the exit - Finally, invariants for the membrane, membrane + bending, and peak stresses are calculated.