# RD-E: 4701 Concrete Validation with Kupfer Tests

Concrete validation with Kupfer tests.

Radioss includes the material models LAW24 and LAW81 to model concrete failure modeling under compression and tension. In this example, the simulation results are compared to the experiment data.

## Input Files

The following input file is used in this example:

## Model Description

A 10 mm concrete cube is modeled using one brick element with the same boundary conditions as the experimental tests.

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

Solid properties are:
• qa = 1.1 and qb= 0.05 (default values)
• Isolid= 24
• Iframe= 2 (co-rotational formulation)
• Istrain= 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

The material data 1 used:
Concrete Material Law (/MAT/LAW24)
Initial density
0.0022 $\left[\frac{g}{m{m}^{3}}\right]$
Concrete elasticity Young's modulus
${E}_{c}=31700\left[\mathrm{MPa}\right]$
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
${\alpha }_{f}=0.2$
Concrete uniaxial compression strength
${f}_{c}=32.22\left[\mathrm{MPa}\right]$
Concrete uniaxial tension strength
0.01 ${f}_{c}$, then set ${f}_{t}}{{f}_{c}}=0.1$ (Default=0.1 in LAW24)
Concrete biaxial strength
1.15 ${f}_{c}$, then set ${f}_{b}}{{f}_{c}}=1.15$
All other parameters can be left as default in LAW24 because the default values are representative of generic concrete materials.
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
${E}_{c}=31700\left[\mathrm{MPa}\right]$
Poisson's ratio
$\nu =0.22$
Shear modulus
$G=\frac{{E}_{c}}{2\left(1+\nu \right)}=\text{12991}\text{.8[MPa]}$
The following data was calculated by curve fitting the experimental data using the solidThinking Compose script, included with the input files.
Friction angle
$\varphi =\text{68}{\text{.35}}^{\circ }$
Ratio
$\alpha =\text{0}\text{.4186898}$
Cap limit pressure set constant
Cap beginning pressure
Material cohesion set constant
Note: In this example, stresses are scaled by ${f}_{c}$. A common practice for concrete materials is to define pressures as positive in compression. Stresses are then negative in traction or tension.

### Simulation Iterations

The purpose of this example is to compare the simulation results to experimental data from the Kupfer 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 ${f}_{c}$
The triaxiality can be computed using the principal stresses:(1) ${\sigma }^{*}=\frac{{\text{σ}}_{m}}{{\sigma }_{VM}}$

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

Figure 3 shows concrete material experiment failure stress (scaled by ${f}_{c}$) in the stress space.

## Results

### Failure Results with LAW24 and LAW81

The failure curve in LAW24 is:(2) ${r}_{f}=\frac{1}{a}\left(b+\sqrt{{b}^{2}-a\left({\sigma }_{m}-c\right)}$

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

Radioss will curve fit Equation 2 using the different strength input ${f}_{c}$, ${f}_{t}}{{f}_{c}}$, to generate the ${r}_{f}$ failure curve (green).
The failure curve and yield curve in LAW81 are the same, and can be described in two parts:
1. $p\le {P}_{a}$

It is linear with $p\mathrm{tan}\varphi +c$

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

2. ${P}_{a} (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) $\sqrt{1-{\left(\frac{{\sigma }_{m}-27}{27-11.305}\right)}^{2}}\cdot {\sigma }_{m}\cdot \mathrm{tan}\left({68.35}^{\circ }\right)+5.4508$
These two parts of the failure curve for LAW81 are:
The failure results with LAW24 and LAW81 under different loading paths (from Kupfer tests) show as:
Comparing LAW24 and LAW81 failure results with experiment data, the LAW81 results are better than LAW24.
The failure results in LAW81 match the experiment data even in cap region. For the LAW24 results, most are well match the analytical results; except CC00 shows a little bit larger difference with analytical curve, but it is almost the same as the experiment data. The following CC00 stress-strain diagram shows almost the same failure stress.

### 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 ${H}_{t},\text{ }\text{\hspace{0.17em}}{D}_{\mathrm{sup}},{\epsilon }_{\mathrm{max}}$. 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 ${f}_{t}}{{f}_{c}}=0.1$ (default) in input.

In LAW81, the same bulk and shear modulus are used for tension and compression. With LAW24 it is possible to use $E=\left(1-{D}_{\mathrm{sup}}\right)\cdot {E}_{c}$ to represent a residual stiffness in the concrete after softening. This is not possible with LAW81.

### 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