# RD-E: 4701 Concrete Validation with Kupfer Tests

Concrete validation with Kupfer tests.

## Options and Keywords Used

- Concrete material law (/MAT/LAW24 (CONC), /MAT/LAW81)
- Brick elements
- Solid property (/PROP/TYPE14 (SOLID))
- Boundary condition (/BCS)
- Imposed displacement (/IMPDISP)
- Imposed velocity (/IMPVEL)
- Pressure load (/PLOAD)

## Input Files

The following input file is used in this example:

<install_directory>/hwsolvers/demos/radioss/example/47_concrete_test/Kupfer_tests/

## Model Description

For stability reasons, 1 element models must use a time step scale factor of 0.1.

`q`_{a}= 1.1 and`q`_{b}= 0.05 (default values)`I`_{solid}= 24`I`_{frame}= 2 (co-rotational formulation)`I`_{strain}= 1 (to post-treat stains)

In this example, two material laws, /MAT/LAW24 and /MAT/LAW81, will be compared to the experiment data.

The following system is used: mm, ms, g, MPa

^{1}used:

**Concrete Material Law (/MAT/LAW24)**- Initial density
- 0.0022 $\left[\frac{g}{m{m}^{3}}\right]$
- Concrete elasticity Young's modulus
- ${E}_{c}=31700[\mathrm{MPa}]$
- Poisson's ratio
- $\nu =0.22$
- Concrete plasticity initial value of hardening parameter
- ${k}_{y}=0.35$
- Concrete plasticity dilatancy factor at yield
- $${\alpha}_{y}=-0.6$$
- Concrete plasticity dilatancy factor at failure
- $${\alpha}_{f}=0.2$$

**Data Read Kupfer Experimental Data**- Concrete uniaxial compression strength
- $${f}_{c}=32.22[\mathrm{MPa}]$$
- Concrete uniaxial tension strength
- 0.01 $${f}_{c}$$, then set $$\raisebox{1ex}{${f}_{t}$}\!\left/ \!\raisebox{-1ex}{${f}_{c}$}\right.=0.1$$ (Default=0.1 in LAW24)
- Concrete biaxial strength
- 1.15 $${f}_{c}$$, then set $$\raisebox{1ex}{${f}_{b}$}\!\left/ \!\raisebox{-1ex}{${f}_{c}$}\right.=1.15$$

**Concrete Material Law (/MAT/LAW81)**- Initial density
- 0.0022 $\left[\frac{g}{m{m}^{3}}\right]$
- Bulk modulus
- $K=\frac{{E}_{c}}{3\left(1-2\nu \right)}=\text{18869}\text{.048[MPa]}$
- Young's modulus
- ${E}_{c}=31700[\mathrm{MPa}]$
- Poisson's ratio
- $\nu =0.22$
- Shear modulus
- $G=\frac{{E}_{c}}{2\left(1+\nu \right)}=\text{12991}\text{.8[MPa]}$

- Friction angle
- $\varphi =\text{68}{\text{.35}}^{\circ}$
- Ratio
- $\alpha =\text{0}\text{.4186898}$
- Cap limit pressure set constant
- ${P}_{b}=0.838\text{,}\text{\hspace{0.17em}}{f}_{c}\text{=27[MPa]}$
- Cap beginning pressure
- ${P}_{a}=\alpha \cdot {P}_{b}\text{=0}\text{.351,}\text{\hspace{0.17em}}{f}_{c}=11.305\text{[MPa]}$
- Material cohesion set constant
- $c=0.169175\text{,}\text{\hspace{0.17em}}{f}_{c}=5.4508\text{[MPa]}$

### Simulation Iterations

^{2}tests.

Test | Principle Stress | Triaxiality | Failure Stress |
---|---|---|---|

T000 Uniaxial tension |
$${\sigma}_{1}=0$$; $${\sigma}_{2}=0$$; $${\sigma}_{3}=1$$ | 1/3 | 0.1 $${f}_{c}$$ |

C000 Uniaxial compression |
$${\sigma}_{1}=-1$$; $${\sigma}_{2}=0$$; $${\sigma}_{3}=0$$ | -1/3 | $${f}_{c}$$ |

CC00 Biaxial compression |
$${\sigma}_{1}=-1$$; $${\sigma}_{2}=-1$$; $${\sigma}_{3}=0$$ | -2/3 | 1.15 $${f}_{c}$$ |

CC01 Compression/Compression |
$${\sigma}_{1}=-0.052$$; $${\sigma}_{2}=0$$; $${\sigma}_{3}=-1$$ | -0.5849 | 1.22 $${f}_{c}$$ |

TC01 Compression/Tension |
$${\sigma}_{1}=0.052$$; $${\sigma}_{2}=0$$; $${\sigma}_{3}=-1$$ | -0.3077 | 0.8 $${f}_{c}$$ |

TC02 Compression/Tension |
$${\sigma}_{1}=0.102$$; $${\sigma}_{2}=0$$; $${\sigma}_{3}=-1$$ | -0.2838 | 0.6 $${f}_{c}$$ |

TC03 Compression/Tension |
$${\sigma}_{1}=0.204$$; $${\sigma}_{2}=0$$; $${\sigma}_{3}=-1$$ | -0.2377 | 0.35 $${f}_{c}$$ |

with $${\text{\sigma}}_{m}=p=\frac{1}{3}\left({\text{\sigma}}_{1}+{\text{\sigma}}_{2}+{\text{\sigma}}_{3}\right)$$ and ${\text{\sigma}}_{VM}=\sqrt{\frac{1}{2}\left[{\left({\text{\sigma}}_{1}-{\text{\sigma}}_{2}\right)}^{2}+{\left({\text{\sigma}}_{2}-{\text{\sigma}}_{3}\right)}^{2}+{\left({\text{\sigma}}_{3}-{\text{\sigma}}_{1}\right)}^{2}\right]}$.

## Results

### Failure Results with LAW24 and LAW81

With $$b=\frac{1}{2}({b}_{c}+{b}_{t})$$

- $p\le {P}_{a}$
It is linear with $p\mathrm{tan}\varphi +c$

The failure is ${\sigma}_{m}\mathrm{tan}({68.35}^{\circ})+5.4508$

- ${P}_{a}<p\le {P}_{b}$ (cap)The cap curve is:
(3) $$\sqrt{1-{\left(\frac{p-{p}_{a}}{{p}_{b}-{p}_{a}}\right)}^{2}}\cdot \left(p\mathrm{tan}\varphi +c\right)$$ The failure is:(4) $\sqrt{1-{\left(\frac{{\sigma}_{m}-27}{27-11.305}\right)}^{2}}\cdot {\sigma}_{m}\cdot \mathrm{tan}({68.35}^{\circ})+5.4508$

### Results for Concrete Tension Tests

Concrete does not support very much load in tension. In LAW24 the uniaxial tensile failure (modeled by stress) and elastic modulus softening behavior is defined by $${H}_{t},\text{\hspace{0.05em}}\text{\hspace{0.17em}}{D}_{\mathrm{sup}},{\epsilon}_{\mathrm{max}}$$. The softening modulus $${H}_{t}=-{E}_{c}$$ (default) for tension is set. The peak for the above curve is at 0.1 where it is defined by $$\raisebox{1ex}{${f}_{t}$}\!\left/ \!\raisebox{-1ex}{${f}_{c}$}\right.=0.1$$ (default) in input.

### Conclusion

Under complex loading, the concrete mechanic failure behavior is shown using two Radioss material models LAW24 and LAW81 and results compared to experiments. For LAW24 the default values are a good choice, if no experimental data is available. For LAW81, the material parameters $\varphi ,c,\alpha ,{P}_{b}$ need to be calculated with curve fitting using at least four experimental tests.

### References

^{1}Han, D. J., and Wai-Fah Chen. "A nonuniform hardening plasticity model for concrete materials." Mechanics of materials 4, no. 3-4 (1985): 283-302

^{2}Kupfer, Helmut B., and Kurt H. Gerstle. "Behavior of concrete under biaxial stresses." Journal of the Engineering Mechanics Division 99, no. 4 (1973): 853-866