عدم قطعیت و تجزیه و تحلیل حساسیت مدل های خزشی برای پارامترهای ورودی ناهمبسته و همبسته

The intention of this paper is to evaluate the uncertainties and sensitivities of creep prediction models of standard concrete. The development of creep prediction models has been a field of extensive research and many different models have already been proposed. The four major models are: model GL2000 by Gardner and Lockman, model MC90 according to CEB-FIP Model Code 1990, model ACI209 according to the American Concrete Institute and model B3 by Bažant and Bajewa. First, a sensitivity study is performed in order to determine the parameters which mostly contribute to the uncertainties of the model prediction. This is done for uncorrelated and correlated input parameters and the differences are pointed out. Due to high parameter correlation, most standard sensitivity methods are not applicable and, therefore, a new method developed by Xu and Gertner is applied. Second, the uncertainties of the creep prediction for all of the models are compared and reveal significant differences. Due to the consideration of parameter and model uncertainties, a measure for the total variation of the model response is achieved. A special FEM code is developed to include the existing creep models in structural analysis. Utilising the FEM application, arbitrarily distributed creep strains, non-linear creep and single reinforcement bars can be taken into account. Finally, this FEM code is used and the prescribed creep models and uncertainty method are applied to a pre-stressed concrete bridge. The uncertainties of the loss of pretensioning force and axial shortening are calculated and show a reduction in comparison to uncertainties of pure creep strain. Together, the presented method proves its ability to determine uncertainties and sensitivities of models of time-dependent behaviour for uncorrelated and correlated parameters in an efficient way.

The prediction of long-term deformation of concrete and reinforced concrete structures has been a field of extensive research for many decades and several different creep models have been developed so far. These models vary in terms of theory, complexity and described phenomena. Advanced creep formulations take into account non-linear creep, aging of concrete, additional damage due to creep, and secondary and tertiary creep phases. Moreover, they can simulate loading–unloading processes. A general approach to evaluate the quality of these different models for their specific design purposes does not yet exist and is the object of the authors’ work. One part of this evaluation method is the estimation of sensitivities of the model prediction towards the input parameters and the calculation of uncertainties of the model’s prognosis, presented in this paper. The investigation of uncertainties concerning time-dependent behaviour of concrete has been the subject of much research. Madsen and Bažant [1] studied the parameter- and model uncertainties of creep model BP by Bažant, Kim and Panula. Diamantidis et al. [2] extended it to determine the influence of varying humidity on the creep coefficient of model C78, according to the Model Code 1978. Further uncertainty surveys of creep models were proposed by Bažant and Liu [3], using the effective Latin Hypercube Sampling. In [4], Bažant and Bajewa propose a simple approach to consider the creep uncertainties in the design of structures. Besides the material uncertainties, further external uncertainties were considered in the context of time-dependent deformation of reinforced concrete structures in [5], [6] and [7]. Yang [8], [9] and [10] determined the uncertainties and sensitivities of creep and shrinkage models ACI209 and MC90 and their effect on pre-stressed elements. Howells et al. [11] did an intensive study of sensitivities of creep and shrinkage models using local sensitivity measures. All the mentioned research work on uncertainties and sensitivities of creep of concrete has assumed uncorrelated input parameters and, neglected appropriate parameter correlation. When assuming uncorrelated input parameters, there are many different techniques to determine global sensitivities. A good overview of global sensitivity analysis is given by Saltelli et al. [12] and [13]. Most common are the Response Surface Methods (RSM), variance-based methods considering first order effects (VBFO) and total effects (VBTE), Fourier Amplitude Sensitivity Test (FAST), and the Random Balance Design (RBD). The application of sensitivity methods to models including correlated input parameters is still studied today. The most strategies can be found in [14], [15], [16], [17] and [18]. In 2008 Xu and Gertner [18] published an approach using linear regression models (RSM) to determine uncorrelated and correlated sensitivities. This is used in the following. As mentioned before, all previous sensitivity and uncertainty analyses of creep models assumed uncorrelated input parameters. In the scope of this paper, the effect of the consideration of parameter correlation in the context of a global sensitivity and uncertainty analysis is presented, applying the method by Xu and Gertner. In the following, four different creep models are compared: model GL2000 by Gardner and Lockman [19], model according to ACI209 [20], model according to CEB-FIP Model Code 90 [21] and model B3 by Bažant and Bajewa [22]. All of the models are summarised in [23]. In Section 2 the creep models and their theory are briefly explained. Section 3 reveals the method of uncertainty modelling and the main parts of the sensitivity method by Xu and Gertner. The results of the sensitivities and uncertainties regarding the prediction of creep strains are presented in Section 4. A comparison of the models and the effect of parameter correlation are given. The effects of these uncertainties on the structural response of a pre-stressed concrete bridge is demonstrated in the last section, showing the uncertainties of the pretensioning force and the axial shortening caused by creep.

Sensitivity and uncertainty analyses have been conducted for four popular creep models of standard concrete: GL2000, MC90, ACI209 and B3. Analysis has been performed for both uncorrelated and correlated input parameters. In order to investigate the sensitivities for highly correlated parameters, a new method proposed by Xu and Gertner is used. In general, models GL2000 and ACI209 are most sensitive to Young’s modulus. The sensitivity of Young’s modulus for model MC90 is high for uncorrelated parameters, as well, but reduced assuming parameter correlation. In that case, the sensitivity of the strength increases. Furthermore it is revealed that the consideration of parameter correlation decreases the sensitivity of the strength for model B3 distinctively and humidity becomes most important. Evaluating the uncertainties of the creep prediction, model GL2000 and B3 perform best. Especially due to high model uncertainties, MC90 and ACI209 include uncertainties of up to View the MathML sourceCV=0.33. The uncertainties of the model’s prediction are generally high and are affected by parameter correlation in different ways, either increasing (MC90) or decreasing (B3) the uncertainty. This evaluation is also obtained in the study of a pre stressed concrete bridge, but the uncertainties are reduced due to the interaction of creep strains and loss of tension force. The effect of parameter correlation is more pronounced in the case of pure creep strains considering only parameter uncertainties. Taking into account parameter- and model uncertainties, e.g., for the calculation of the bridge, the influence of the parameter correlation is minor. The calculated creep uncertainties should be taken into account in the design of structures, which are sensitive to creep deformations, e.g., concrete arches or long span pre stressed girders and bridges. The presented coefficient of variation can be used to determine the 90%- or 95%- confidence interval of the creep strains [22]. These values can be considered as upper and lower bounds of the creep strain in an ordinary, deterministic, time-dependent analysis of concrete structures.