> For the complete documentation index, see [llms.txt](https://docs.enercalc.com/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.enercalc.com/e3d-user-manual/model-entities/toc161394014/toc161394018.md).

# Four-Node Shell Local Coordinate System

Understand how the default local coordinate system of a four-node shell element is defined in ENERCALC 3D.

The local coordinate system of a shell is determined by its four nodes, and an element local angle. The default (local angle equals 0 degrees) local coordinate system of a four-node shell is defined based on the shape of the shell element.

| Rectangular Shell                                                                                                                                                                                             | Non-Rectangular Shell                                                                                                                                                                                         |
| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| ![](https://560001759-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FWNe8U6EBBBdUJHdb2X9G%2Fuploads%2Fgit-blob-2e126d2a61f7310b46104e9ee5a005684044afde%2FFASTFR~1_img224.png?alt=media) | ![](https://560001759-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FWNe8U6EBBBdUJHdb2X9G%2Fuploads%2Fgit-blob-0e97081bffb8994b16d757107c6100b1e8eec588%2FFASTFR~1_img225.png?alt=media) |

For rectangular shells, the default local coordinate system is easily defined by the following: local x points from N1 to N2, local y points from N1 to N4 and local z is perpendicular to the shell surface.

For non-rectangular shells, the default local coordinate system is defined using the following procedures:

| Steps | Description                                                                                                       | Mathematical Notations                                                                                                      |
| ----- | ----------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------- |
| A     | Local z is perpendicular to the shell surface                                                                     | Let **V\*\*\*\*1** **=** N2 – N1\*\*,\*\* Let **V\*\*\*\*2** **=** N4 – N1 **V\*\*\*\*z** **= V\*\*\*\*1** x **V\*\*\*\*2** |
| B1    | For horizontal shells that are parallel to global XZ plane, local x is parallel to global X                       | For horizontal shells **V\*\*\*\*x** = **V\*\*\*\*X**                                                                       |
| B2    | For non-horizontal shells, **V\*\*\*\*x** is perpendicular to a plane formed by **V\*\*\*\*Y** and **V\*\*\*\*z** | For non-horizontal shells **V\*\*\*\*x** **= V\*\*\*\*Y** x **V\*\*\*\*z**                                                  |
| C     | **V\*\*\*\*y** is determined based on **V\*\*\*\*x** and **V\*\*\*\*z** and the right-hand rule                   | **V\*\*\*\*y** **= V\*\*\*\*z** x **V\*\*\*\*x**                                                                            |

For a shell with a non-zero local angle (γ), first follow the procedures above that determine the default local coordinate system. Then rotate the default system a γ angle about is its local z vector **V\*\*\*\*z**. The rotated **Vx, V\*\*\*\*y** and **V\*\*\*\*z** define the local coordinate system. The figure below shows the local coordinate systems of some shell elements with different local angles (γ)

![](https://560001759-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FWNe8U6EBBBdUJHdb2X9G%2Fuploads%2Fgit-blob-0ce3c341f85a9f64bf455354d5bbad858d9bedae%2FFASTFR~1_img226.png?alt=media)

Local coordinate systems for shells with different local angles


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