# SteelBRB

Steel Model For Modelling BRB

The `SteelBRB` material defines a steel model. It can be used to model buckling restrained braces. It uses exponential type functions thus resembles the [`RambergOsgood`](/suanpan-manual/material/material1d/hysteresis/rambergosgood.md) material.

References are available:

1. <https://doi.org/10.1016/j.jcsr.2011.07.017>
2. <https://doi.org/10.1016/j.jcsr.2014.02.009>

## Syntax

If tension response is identical to compression response, users can use the following command to define the material.

```
material SteelBRB (1) (2) (3) (4) (5) (6) (7) [8]
# (1) int, unique material tag
# (2) double, elastic modulus
# (3) double, yield stress
# (4) double, plastic modulus
# (5) double, saturated stress, \sigma_s
# (6) double, \delta_r
# (7) double, \alpha
# [8] double, density, default: 0.0
```

If tension response is different from compression response, which is often the case in modelling BRB, users can use the following command.

```
material SteelBRB (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) [11]
# (1) int, unique material tag
# (2) double, elastic modulus
# (3) double, yield stress
# (4) double, plastic modulus
# (5) double, tension saturated stress
# (6) double, tension delta_r
# (7) double, tension alpha
# (8) double, compression saturated stress
# (9) double, compression delta_r
# (10) double, compression alpha
# [11] double, density, default: 0.0
```

## Remarks

1. In the original model, a linear response is defined before the initial yield stress, this feature is disabled. Thus, the monotonic loading response is similar to that of the `RambergOsgood` material.
2. In the original model, the elastic and plastic moduli could be different under tension and compression. The implemented model uses identical values for these two parameters.

## History Variable Layout

| location             | value                      |
| -------------------- | -------------------------- |
| `initial_history(0)` | accumulated plastic strain |
| `initial_history(1)` | plastic strain             |

## Theory

The model is described in incremental form. The total strain increment is decomposed into elastic and plastic part,

$$
\dot\sigma=E(\dot\varepsilon-\dot\varepsilon\_p).
$$

For loading, the increment of plastic strain is defined as an implicit function, that is

$$
\dot\varepsilon\_p=\Big|\dfrac{\sigma-\sigma\_p}{\sigma\_y}\Big|^\alpha\dot\varepsilon,
$$

in which $$\sigma\_p$$ is associated with $$\varepsilon\_p$$ via plastic modulus $$H$$,

$$
\sigma\_p=H\varepsilon\_p,
$$

and

$$
\sigma\_y=\sigma\_s-(\sigma\_s-\sigma\_{y,0})\exp(-\dfrac{\mu}{\delta\_r})
$$

where $$\sigma\_s$$ is the saturated stress, $$\sigma\_y$$ is the yield stress, $$\delta\_r$$ is a constant parameter that controls the speed of isotropic hardening, $$\mu$$ is the accumulated plastic strain which is defined to be

$$
\dot\mu=|\dot\varepsilon\_p|.
$$

In the above formulation, $$\sigma\_s$$, $$\alpha$$ and $$\delta\_r$$ could be different in tension and compression.

The development of plastic strain $$\varepsilon\_p$$ is activated when

$$
\dot\varepsilon\sigma<0
$$

## Example

```
material SteelBRB 1 2E5 400. 2E3 660. .2 .6 450. .15 .4
materialTest1D 1 1E-4 40 80 120 160 200 240 280 320 360 400 400
```

![example one](/files/-MQ5ry5FhhVGyeMuA6jZ)

## Accuracy

```py
from plugins import ErrorLine
# note: the dependency `ErrorLine` can be found in the following link
# https://github.com/TLCFEM/suanPan-manual/blob/dev/plugins/scripts/ErrorLine.py

young_modulus = 2e5
yield_stress = 4e2
hardening_ratio = 0.01

with ErrorLine(
    f"material SteelBRB 1 {young_modulus} {yield_stress} {hardening_ratio * young_modulus} {1.5 * yield_stress} .2 .6 {2 * yield_stress} .15 .4",
    ref_strain=yield_stress / young_modulus,
    ref_stress=yield_stress,
) as error_map:
    error_map.contour("steelbrb", center=-5, size=5, type={"abs"})
```

![accuracy analysis](/files/o8wDq0F0xEfhgLQzkoWx)


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