Modelling and quasistatic simulation of the stiffness degradation of concrete based on physical measurements

AIM: T he description of the mechanical properties of concrete is usually done with two pairs of values "stress-strain" within the "elastic area" which are defining the slope (Young's modulus) and the peak value (strength). T he post-peak-behaviour (softening) is usually defined only by a theory giving the decreasing curve e.g. exponential softening. T herefore the aim of this project is the metrological recording of the complete stress-strain-curve for the integration into the material model, reproducing the "linear" and the nonlinear area (softening) within the nonlinear simulation exactly. T he quality of simulation results should improve when using physical measured values as input data of the material model. PROCEDURE: According to the statistical evaluation of the experimentally determined material parameters (compressive-, bending -, tensile-, splitting tensile strength), the bending strength was identified as the material parameter with the least deviation from the mean value (relative scattering coefficient, the coefficient of variation). T his leads to the objective of implementing the electronically recorded measurement data of the carried out deformation controlled 4-point-bending-tests (according to test guideline German committee for reinforced concrete DAfStB) into the material model. T he experimentally recorded forces and displacements were linearly converted into normalized stresses and strains according to the rules of statics. T he input for the elastic-degrading material model of the steel-fibre-concrete are 20 pairs of values, ​​taken from the measured data of the 4-point bending test. T he material model (smeared cracking method) of the unreinforced fibre concrete and it’s softening is thus based exclusively on experimentally measured data, taken from the 4-point-bending-tests. T he theoretical material models available in "Ansys Mechanical", e.g. “Mohr-Coulomb” or “Drucker-Prager”, have not been applied here. Qeios, CC-BY 4.0 · Article, May 8, 2019

T he usual input data of the uniaxial tensile-and compressive strength were also not included in the simulation.
When evaluating the 4-point tests, the lowest force-displacement curve is decisive (minimum work performed); this corresponds to the main crack in midspan. In the area of the damage between the load applications, the constant (bending) normalstress corresponds to the equivalent stress because of the lack of (bending) shearstress. T he (bending) normal-stress together with the plastic strain are the experimental input values of the material model.

CONCLUSION:
Element size and deformation rate have to be minimal for a good quality simulation result. T here are deviations between simulation and experiment in the "elastic zone" and also concerning the peak value (bending strength) because within the simulation all cracks are smeared homogeneously.
Sufficient local and temporal discretization / sufficient small mesh size and deformation speed bring the nonlinear simulation close to the physical reality. T he optimum quality of results is achieved with a specific mesh density and a specific deformation speed -too small values, on the other hand, will worsen the quality of results (optimization process). T he elastic-degrading/multilinear-elastic simulation using a physically based material model showed no convergence problems. T he statistic analysis of the material parameters, determined by experiments with the small specimen, delivered the following "coefficient of variation" results for steel-fibreconcrete C30/37 containing 25kg/m³ steel fibres and 0,5kg/m³ plastic fibres. T he bending tensile strength can be identified as the material parameter with the lowest coefficient of variation. T herefore, according to the project's aim, the electronic Qeios, CC-BY 4.0 · Article, May 8, 2019 Qeios ID: 649497 · https://doi.org/10.32388/649497 2/19 measured data of the 4-point-bending-tests (test guideline "Steel fibre concrete" of the "DAFStB german committee for reinforced concrete") is implemented in the material model.     Element size and deformation rate have to be minimal for a good quality simulation result. T here are deviations between simulation and experiment in the "elastic zone" and    T ill about 300KN (15KN/m²) the massive and the hollow construction can be seen as equivalent, the curves are parallel to each other. T he massive construction will hold a higher load level for >400KN. T his load level won't be reached in reality, for normal use the level is scheduled with 5KN/m² (100KN). T he degree of restraint of the simulated supports has to be calibrated with the experimental strain measurements along the supports.
With smaller mesh size and smaller deformation rate, the simulation will get closer to physical reality. T he boundary conditions of the simulation have to be calibrated with the experimental strain measurements along the supports. T he elastic degrading/multilinear elastic simulation using a physically based material model showed no convergence problems. T he physically measured strains at the upper side of the ceiling (photogrammetry) will be compared with the simulated values; the differences will show good and bad areas. Finally, optimization potentials can be identified by specifying the origin of the differences.
T he element size converged at LESIZE=3cm, the deformation step size converged at 0,1mm. T he increase of local discretization or time discretization / the reduction of mesh size or deformation step size (time step size) will minimize the strain energy -the physical process will be reproduced.
Up to the target value (element size 3cm or deformation rate 0,1mm), the simulation results showed the behaviour of a monotonic convergence. When the target value was exceeded, the behaviour of the results changed to an oscillating convergence, which in turn severely degraded the quality of the results.
Subsequent some pictures of the last simulation / prediction for the experiment (LESIZE=0.03m deformation step=0.1mm). T he print of displacement is 10x inflated. 3. Qu a l i t y of r e su l t s 3. Qu a l i t y of r e su l t s T houg hts and remarks T houg hts and remarks Figure 26: Explicit methods calculate the lower sum, implicit methods deliver the upper sum [1] Above figure shows the reduction of the global error in connection with reduced time step ∆t; smaller stripes will reproduce the curve more exactly. But with a higher amount of stripes, the numerical mistake will also increase -see next figure.
Figure 27: Total error as the sum of rounding error and global error / method error [2] With the reduction of the global error for smaller time steps the rounding error will increase. Adding both errors will show an optimal time step size, where the total error will be minimized. From this follows that with a time step size chosen too small the result quality will get worse because of the increasing rounding error. [2] Qeios, CC-BY 4.0 · Article, May 8, 2019 T his relationship should also be valid for the element size -with smaller element size the rounding error will increase while the global error/method error will decrease. T herefore too small element sizes will worsen the result quality because of the increasing total error.
4 . S u m m a r y a n d ou t l ook 4 . S u m m a r y a n d ou t l ook With decreasing element size or deformation speed, the simulation gets closer to physical reality. Reducing these parameters over a certain level the force-displacementcurve did not get lower anymore and took place above the last curve -this behaviour has to be reviewed more exactly in future research projects. T he optimal result quality with a minimized total error can be reached with certain values for element size and deformation speed; these parameters have to be found. T he exceeding of these optimal parameters has to be avoided; choosing an element size or deformation speed too small will cause a significant worsening of the total result quality. T he origin of the abrupt worsening result quality should have it's source within the increasing numerical error of the calculation; with decreasing element size or time step (deformation step/load step) the numerical error increases.
Choosing the discretization of geometry or time will depend on the system stiffness. For a short beam, the parameters have to be much smaller compared to a ceiling with larger span length; e.g. applicable evaluation criteria could be the slenderness ratio. Even within the implicit quasistatic analysis, the discretization of time by deformation step or load step has a big influence on the simulation result. T herefore the quasistatic simulation has to be seen as a time-dependent simulation.
H ypothesis H ypothesis : : T he author formulates the hypothesis, that the convergence concerning the deformation speed (here: 0,1mm) results from the deformation speed of the 4-point- More detailed information concerning the project "CC-technology" can be found here: