> 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/material/material1d/concrete/concretecm.md).

# ConcreteCM

Chang-Mander Concrete Model

## Syntax

```
material ConcreteCM (1) (2) (3) (4) (5) (6) [7] [8] [9] [10]
# (1) int, unique material tag
# (2) double, elastic modulus
# (3) double, compression strength, should be negative but sign insensitive
# (4) double, tension strength, should be positive but sign insensitive
# (5) double, NC
# (6) double, NT
# [7] double, strain at compression strength, default: -2E-3
# [8] double, strain at tension strength, default: 1E-4
# [9] bool string, linear transition switch, default: false
# [10] double, density, default: 0.0
```

## Remarks

1. The Chang-Mander concrete model uses Tsai's equation as backbone curves for both tension and compression.
2. Parameters `NC` and `NT` control the shapes of backbone curves. A detailed explanation is presented later.
3. In the original model, the transition between compression and tension could sometimes have larger stiffness than initial stiffness. This is unlikely to be true in reality.
4. A linear behaviour of part of hysteresis behaviour can be applied by turning on the linear transition switch. ***It shall be noted that a linear hysteresis rule is much more stable than the original version.***
5. The original CM model has some undefined behaviour which may cause stability issues. This is a simplified model.

## History Variable Layout

| location              | value                  |
| --------------------- | ---------------------- |
| `initial_history(0)`  | unload\_c\_strain      |
| `initial_history(1)`  | unload\_c\_stress      |
| `initial_history(2)`  | reverse\_c\_strain     |
| `initial_history(3)`  | reverse\_c\_stress     |
| `initial_history(4)`  | residual\_c\_strain    |
| `initial_history(5)`  | residual\_c\_stiffness |
| `initial_history(6)`  | unload\_t\_strain      |
| `initial_history(7)`  | unload\_t\_stress      |
| `initial_history(8)`  | reverse\_t\_strain     |
| `initial_history(9)`  | reverse\_t\_stress     |
| `initial_history(10)` | residual\_t\_strain    |
| `initial_history(11)` | residual\_t\_stiffness |
| `initial_history(12)` | connect\_c\_stress     |
| `initial_history(13)` | connect\_c\_stiffness  |
| `initial_history(14)` | connect\_t\_stress     |
| `initial_history(15)` | connect\_t\_stiffness  |
| `initial_history(16)` | inter\_strain          |
| `initial_history(17)` | inter\_stress          |
| `initial_history(18)` | reload\_c\_stiffness   |
| `initial_history(19)` | reload\_t\_stiffness   |

## General Description of the Model

### Unload from backbone

![example one](https://1124082496-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FEquxr3H9WpPg4iPvT2mB%2Fuploads%2Fgit-blob-3358f7df02c1cdf675047ac1c52d69c28deff908%2FCM1.svg?alt=media)

### Reload from unload branch

#### Reload before residual

![example two](https://1124082496-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FEquxr3H9WpPg4iPvT2mB%2Fuploads%2Fgit-blob-63e5f754d1f17c72f8cf26413cd788095991e775%2FCM2.svg?alt=media)

#### Reload between two residuals

![example three](https://1124082496-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FEquxr3H9WpPg4iPvT2mB%2Fuploads%2Fgit-blob-814e816e9be0e676ff5e467a1a7de2c7d47961cf%2FCM3.svg?alt=media)

#### Reload after the opposite residual

![example four](https://1124082496-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FEquxr3H9WpPg4iPvT2mB%2Fuploads%2Fgit-blob-52dd4c7738ff329eea60993d2f46e1788be22851%2FCM4.svg?alt=media)

#### Small Cycle

![example five](https://1124082496-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FEquxr3H9WpPg4iPvT2mB%2Fuploads%2Fgit-blob-5efbbdb74f784894820bf24fc35ade4d06096e99%2FCM5.svg?alt=media)

![example six](https://1124082496-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FEquxr3H9WpPg4iPvT2mB%2Fuploads%2Fgit-blob-dc2061475ca1ae00ac953514161aa8a82222c80a%2FCM6.svg?alt=media)

## Determination of Parameters

The Tsai's equation (Tsai, 1988) can be expressed as

$$
y=\dfrac{mx}{1+(m-\dfrac{n}{n-1})x+\dfrac{x^n}{n-1}}.
$$

where $$x=\varepsilon/\varepsilon\_c$$ or $$x=\varepsilon/\varepsilon\_t$$ is normalized strain and $$y=f/f\_c$$ and $$y=f/f\_t$$ are normalized stress. The stress decreases to zero if $$m(n-1)>n$$ and $$n>1$$. The initial stiffness is related to $$m\_t$$ and $$m\_c$$ by

$$
E\_0=mE\_s=m\_t\dfrac{f\_t}{\varepsilon\_t}=m\_c\dfrac{f\_c}{\varepsilon\_c}.
$$

Once $$E\_0$$ is given, $$m\_c$$ and $$m\_t$$ are determined automatically.

The parameter $$n$$ controls the slope of descending branch, normally $$n\_t>n\_c$$. Some empirical expressions are available to determine both $$m$$ and $$n$$. But most are unit dependent. Users shall do manual conversion. For example,

$$
m\_c=1+\dfrac{17.9}{f\_c}\quad{}f\_c\text{ in MPa},
$$

$$
m\_c=1+\dfrac{2600}{f\_c}\quad{}f\_c\text{ in psi},
$$

$$
n\_c=\dfrac{f\_c}{6.68}-1.85>1\quad{}f\_c\text{ in MPa},
$$

$$
n\_c=\dfrac{f\_c}{970}-1.85>1\quad{}f\_c\text{ in psi}.
$$


---

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