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Selecting creep models using Bayesian methods

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Abstract

The development of creep prediction models has been a field of extensive research and many different models have already been proposed. This paper presents an evaluation method of the prediction quality of creep models for specific experimental data. Within the scope of this paper, the model according to Bockhold and the model according to Heidolf are examined. First, the parameters of the models are identified with respect to existing experimental data. This is done using a sampling based approach of Bayesian updating developed by Bažant and Chern. In extension to the method by Bažant and Chern, the uncertainty coming from inaccurate measurement data is taken into account in the definition of the likelihood function within the updating algorithm. The more inaccurate the measurements are, the more uncertain the estimated model parameters and model prognoses become. The identification is performed for different short- and long-term creep tests. The intension is not to validate these models intensively, but to evaluate their prognoses for the individually tested creep behavior. The results show that the identifiability of the models’ parameters is different for both models and consequently the models prognoses differ in their uncertainties. Second, the models are evaluated using two different strategies: the stochastic model selection according to MacKay, Beck and Yuen based on the Ockham factor, and a comparison of the uncertainties taking into account parameter and model uncertainties. The results of the evaluation of the creep models differ for various experimental tests. Model Heidolf is more flexible and gives a better fit to the data, however, it fails to predict reliable long-term creep deformations using only short-term measurements compared to model Bockhold. Comparing the evaluation methods, the analysis of uncertainties of the creep prognosis proofs to be more stable than the evaluation using the stochastic model selection.

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Abbreviations

\(\alpha_{\rm c,cr}^{i}\) :

History variable of maximum stress (MN/m2)

αc,v-el, αc,v-pl :

Creep non-linearity functions of stiffness, model Heidolf (–)

\(a_{\text{c,v-el}}^{i,l}, a_{\text{c,v-pl}}^{j,l}\) :

Polynomial factors of creep non-linearity functions, model Heidolf (–)

b :

Parameter to distinguish \(\varepsilon_{\rm c,cr,in}\) in \(\varepsilon_{\rm c,cr,pl}\) and \(\varepsilon_{\rm c,cr,da}\) (–)

c0, c1 :

Normalizing constants

C cr,meas, C cr,model :

Creep compliance of measurements and model (m2/MN)

C i,j , COV:

Covariance matrix

CV:

Coefficient of variation (–)

CVmodel :

Coefficient of variation of model prognosis to measurements (–)

d :

Index for damper

\(D_{\rm c,cr}^{i}\) :

Damage parameter (–)

\(E \left[\ldots\right]\) :

Expected value

E c, E c,0 :

Young’s modulus, Young’s modulus at t 0 (MN/m2)

\(E_{\text{c,v-el}}^{i}, E_{\text{c,v-pl}}^{j}\) :

Stiffness of spring of Kelvin/Bingham-chain, model Heidolf (MN/m2)

\(e_{\rm d}, \varepsilon_{{\rm c},0}, g_{\rm d}\) :

Parameters of time invariant concrete model of Häußler-Combe [1] (–)

\(\eta_{\rm c}\) :

Viscosity of damper (MNdays/m2)

\(\varepsilon_{\rm c,el}, \varepsilon_{\rm c,pl}, \varepsilon_{\rm c,da}\) :

Short-term strains of concrete: elastic, plastic, damage (–)

\(\varepsilon_{\rm c,cr}, \dot{\varepsilon}_{\rm c,cr}, \ddot{\varepsilon}_{\rm c,cr}\) :

Creep strains, creep rate, creep acceleration, (–), (1/days), (1/days2)

\(\varepsilon_{\text{c,cr,v-el}}, \varepsilon_{\text{c,cr,v-pl}}\) :

Long-term strains of concrete: visco-elastic and visco-plastic (–)

\(\varepsilon_{\rm c,cr,el}, \varepsilon_{\rm c,cr,in}, \varepsilon_{\rm c,cr,pl}, \varepsilon_{\rm c,cr,da}\) :

Creep strains of concrete: elastic, inelastic, plastic and damage (–)

\(\varepsilon_{\rm c1}\) :

Concrete strain at concrete strength (–)

f c, f c,0 :

Short-term concrete strength, short-term concrete strength at t 0 (MN/m2)

f c,T , f c,T,0 :

Long-term concrete strength, long-term concrete strength at t 0 (MN/m2)

H :

Hessian matrix

i :

ith creep chain of model Bockhold

i, j :

ith Kelvin- and j-th Bingham-chain of model Heidolf

k :

Actual sample

K :

Number of samples

L :

Order of creep non-linearity functions of stiffness, model Heidolf

\(L\left(A|B\right)\) :

Likelihood of A under condition of B

m :

Number of creep chains, model Bockhold

m, n :

Number of Kelvin/Bingham-chains, model Heidolf

m :

Measurement point

M :

Number of measurement points

M j :

Model class j

n :

Exponent of damper non-linearity, model Bockhold (–)

n :

Number of actual time increment

N M :

Number of model classes in model group U

\(p\left(A\right)\) :

Probability density function of A

\(p\left(A|B\right)\) :

Probability density function of A under condition of B

\(P\left(A\right)\) :

Probability of A (–)

\(P\left(A|B\right)\) :

Conditional probability of A under condition of B (–)

\(P^{\prime}\left(A\right)\) :

Prior probability of A (–)

\(P^{\prime\prime}\left(A\right)\) :

Posterior (updated) probability of A (–)

s :

Index for spring

τ:

Time at the beginning of actual time increment (days)

\(\tau_{\rm c}^{i}, \tau_{\text{c,v-el}}^{i}, \tau_{\text{c,v-pl}}^{j}\) :

Retardation time of dampers (days)

\(\Updelta t\) :

Time step (days)

t, t 0, t unl. :

Actual time, time at beginning of loading, time at unloading (days)

\({\varvec{\theta}}\) :

Parameter vector

\({\varvec{\theta}}^{k}\) :

Parameter vector of sample k

\(\sigma_{\rm c}, \sigma_{\rm c,s}^{i}, \sigma_{\rm c,d}^{i}\) :

Concrete stress, stress of spring, stress of damper (MN/m2)

\(\sigma_{\bullet}\) :

Standard deviation of •

\(\sigma_{X_{m}}\) :

Standard deviation of measurements

U :

Group of available models

X m :

Measurement data at measurement point m

\(\Uppsi\) :

Model uncertainty factor (–)

\(\overline{\bullet}\) :

Mean values of •

\(\hat{\bullet}\) :

Optimal value of •

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Acknowledgments

This author gratefully acknowledges the support for this research, provided by the German Research Foundation (DFG) through the Research Training Group 1462.

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Authors and Affiliations

  1. Research Training Group 1462, Bauhaus-Universität Weimar, Berkaer Str. 9, 99423, Weimar, Germany

    Holger Keitel

  2. Department of Civil Engineering - Chair of Polymer Materials, Bauhaus-Universität Weimar, Coudraystr. 11a, 99423, Weimar, Germany

    Andrea Dimmig-Osburg

  3. Department of Civil Engineering - Building Materials and Building Technology Section, Katholieke Universiteit Leuven, Kasteelpark Arenberg 40, 3001, Heverlee, Belgium

    Lucie Vandewalle & Luc Schueremans

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  1. Holger Keitel
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  2. Andrea Dimmig-Osburg
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  3. Lucie Vandewalle
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  4. Luc Schueremans
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Correspondence to Holger Keitel.

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Keitel, H., Dimmig-Osburg, A., Vandewalle, L. et al. Selecting creep models using Bayesian methods. Mater Struct 45, 1513–1533 (2012). https://doi.org/10.1617/s11527-012-9854-x

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