> For the complete documentation index, see [llms.txt](https://tlcfem.gitbook.io/suanpan-manual/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://tlcfem.gitbook.io/suanpan-manual/section/sectionnm/nm2d2.md).

# NM2D2

$$N$$-$$M$$ Interaction Inelastic Section (Linear Hardening)

## References

1. [10.1061/JSENDH.STENG-12176](http://dx.doi.org/10.1061/JSENDH.STENG-12176)

## Syntax

### Option One

```
section NM2D2 (1) (2...9)
# (1) int, unique section tag
# (2) double, EA
# (3) double, EI
# (4) double, yielding axial force
# (5) double, yielding strong axis moment
# (6) double, c
# (7) double, isotropic hardening parameter H
# (8) double, kinematic hardening parameter K
# (9) double, linear density
```

If the `NM2D2` section is defined by using the above command, it is assumed that the $$N$$-$$M$$ interaction surface is defined as follows.

$$
f=1.15p^2+m\_s^2+3.67p^2m\_s^2-c
$$

where $$p$$ and $$m\_s$$ are normalised axial force and moment (about the strong axis). This surface is suitable for I-sections.

### Option Two

One may wish to customise the surface by assigning different weights and orders, it is possible by using the following syntax.

```
section NM2D2 (1) (2...9) [(10) (11) (12) ...]
# (1) int, unique section tag
# (2) double, EA
# (3) double, EI
# (4) double, yielding axial force
# (5) double, yielding strong axis moment
# (6) double, d
# (7) double, isotropic hardening parameter H
# (8) double, kinematic hardening parameter K
# (9) double, linear density
# (10) double, a_i
# (11) double, b_i
# (12) double, c_i
```

In the above command, parameters `(10)`, `(11)` and `(12)` form a triplet and can be appended as many groups as the analyst wishes. The surface is assumed to possess the following form,

$$
f=\sum\_{i=1}^na\_ip^{b\_i}m\_s^{c\_i}-d.
$$

For example, the previous surface $$f=1.15p^2+m\_s^2+3.67p^2m\_s^2-c$$ can be equivalently expressed with the second syntax as follows.

```
section NM2D2 (1) (2...9) 1.15 2. 0. 1. 0. 2. 3.67 2. 2.
```

The only validation implemented is the number of triplets. The command takes $$3n$$ parameters and interprets them accordingly. Please make sure the definition is correct.

## Remarks

The true hardening ratio is defined as

$$
\dfrac{H+K}{1+H+K},
$$

given that $$H$$ and $$K$$ are hardening ratios based on plastic strain.

## Output Type

This model supports the following additional history variables to be recorded.

| variable label | physical meaning                 |
| -------------- | -------------------------------- |
| YF             | yielding flag (vector of size 2) |


---

# Agent Instructions
This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com.

## Querying This Documentation
If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter, and the optional `goal` query parameter:

```
GET https://tlcfem.gitbook.io/suanpan-manual/section/sectionnm/nm2d2.md?ask=<question>&goal=<endgoal>
```

`ask` is the immediate question: it should be specific, self-contained, and written in natural language.
`goal` is optional and describes the broader end goal you are ultimately trying to accomplish on behalf of the user. GitBook uses it to tailor the answer towards what is most useful for that goal.

The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.