Quality assessment of material models for reinforced concrete flexural members
5.3 Total uncertainty and model quality evaluation
The total uncertainty of the material models is shown in Fig. 7a. The model quality illustrated in Fig. 7b is another expression of the total uncertainty (inversion of total un-certainty) in order to quantify the prediction quality. If the uncertainty of a model output is low, it means the model prediction has a high reliability and, consequently, a high prognosis quality is quantified. When the scatter of the model output due to the model error (model uncertainty) and the uncertain input parameters (parameter uncertain-ty) is lower, then the model prediction is more reliable.
The scale of the model quality varies significantly be-tween the uncracked stage and the crack formation stage as well as between the stabilized cracking stage and the Fig. 6. Load level dependency of model uncertainty and parameter
uncer-tainty for material models (a)
(b)
(c)
Fig. 7. Load level dependency of total uncertainty and model quality for material models
(a)
(b)
steel yielding stage. For the same load level, some samples remain in the uncracked stage and others have already reached tensile strains above the strains corresponding to the tensile strength. This behaviour results in a significant bending stiffness difference and recognizable variance (uncertainty). A similar behaviour occurs in the loading level between the stabilized cracking stage and the steel yielding stage.
The model quality tends to drop in the range of the crack initiation bending moment and the yielding bending moment. The consequence of the simplified linear-elastic material model is a significant reduction in the prognosis quality initiated with the beginning of the first bending stiffness degradation (caused by high model uncertainty).
The lower model uncertainty of the “br-func” model com-pared with the “lin-el”“ model in the stabilized cracking stage is overlapped by the higher parameter uncertainty.
Therefore, the partial model quality of both models is sim-ilar:
The highest prediction qualities of the tension stiffening models “e-func”, “multi-lin” and “mod-steel” are found in the stage of stabilized cracking. The low parameter uncer-tainty and the comparable prediction of the bending stiff-ness (model uncertainty) lead to a similar partial model quality. The quality of the more complex model with the modified steel strains (“mod-steel”) show the overall best model prediction quality over the entire loading level.
6 Conclusions
A comparatively large number of different models exist for the non-linear modelling of reinforced concrete cross-sec-tions and structures. For each application, it is not obvi-ous which model is the most appropriate one for describ-ing the physical phenomena. Therefore, model evaluation with the aid of uncertainty analysis is a powerful method-ology for comparing various model predictions in a quan-titative manner. In the design process of engineering struc-tures there is often a lack of experimental data, particularly during the preliminary design phase. The fo-cus of this paper is the assessment of the material models without experimental data in order to assist model selec-tion in these project phases.
Uncertainty analysis [16] according to the models of a reinforced cross-section enables a clear quantitative dif-ferentiation between the different model prognoses for all loading levels up to the steel yield stage. The results show that the model quality of the purely linear-elastic material model is generally opposite to that of the non-linear ten-sion stiffening material models. Furthermore, exclusive consideration of the concrete compressive non-linear be-haviour does not improve the overall prediction quality.
Using such simplified models for the simulation of struc-tures with a potentially non-linear response will lead to unreliable prognoses and should not be used for the simu-lation of, for example, restraint-sensitive structures. Mater-ial models with a high quantitative model quality provide reliable predictions and should be used in global structur-al models.
Mlin el Mbr func
MQPM |MQPM
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B. Jung/G. Morgenthal/D. Xu/H. Schröter · Quality assessment of material models for reinforced concrete flexural members
Structural Concrete (2015), No. 1
Owing to the crucial influence of the loading level on the load-deformation behaviour of a flexural reinforced concrete member, a clear assignment between the com-plexity and quality of models does not exist in general.
Further research is necessary in order to investigate the ef-fect of several cross-section types, reinforcing steel grades, stirrups, compression steel, concrete strength class and loading conditions. In addition, different material models for reinforced, prestressed, normal-strength and high-strength concrete can be generally assessed by the evalua-tion method presented in future work.
Acknowledgements
The first two authors gratefully acknowledge the support for this research provided by the German Research Foun-dation (DFG) via the research training group “Assessment of Coupled Experimental and Numerical Partial Models in Structural Engineering (GRK 1462)”. The close collabo-ration between Bauhaus-Universität Weimar and Tongji University is also acknowledged.
Notation
EIM energy method with integral description of the material behaviour
GM global model
LHS Latin hypercube sampling PM partial model
b one-sided quantile value CoV (…) coefficient of variation
εΔMi model error with respect to reference model εMref error of reference model
Mi one specific model Mref reference model
μMi mean value of model response MQMPMi partial model quality
σMi standard deviation of model response V(…) variance of model response
Y model response
Y–
mean model response βt completeness factor
δ ductility factor of reinforcing steel Ecm secant modulus of elasticity of concrete Es modulus of elasticity of steel
fcm mean compressive strength of concrete fctm mean tensile strength of concrete fy yield strength of steel
ft tensile strength of steel
hc,eff effective depth of reinforced concrete sub -section
κ curvature
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Guido Morgenthal, Prof. Dr.
Institute of Modelling & Simulation of Structures
Bauhaus-Universität Weimar
Marinenstr. 13, 99421 Weimar, Germany [email protected] Bastian Jung, PhD student Research Training Group 1462 Bauhaus-Universität Weimar Berkaer Str. 9, 99425 Weimar, Germany Tel.: +49 (0) 3643 584107
Fax: +49 (0) 3643 584101 [email protected]
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B. Jung/G. Morgenthal/D. Xu/H. Schröter · Quality assessment of material models for reinforced concrete flexural members
Structural Concrete (2015), No. 1
Dong Xu, Prof. PhD
Department of Bridge Engineering Tongji University
Shanghai 200092, 1239 Siping Road, PR China [email protected]
Hendrik Schröter, PhD student Institute of Modelling & Simulation of Structures
Bauhaus-Universität Weimar
Marinenstr. 13, 99421 Weimar, Germany [email protected]
The loadbearing capacity of steel-concrete composite slabs us-ing thin-walled steel sheetus-ing with prepressed embossments is in most cases determined by their resistance in longitudinal shear.
The design of composite slabs still requires full-scale laboratory bending tests to be performed. Small-scale shear tests cannot in-clude all of the influences affecting the bent slab. However, by using an appropriate procedure, the shear characteristics ob-tained from such tests can be used to determine the bending capacity of the slab. Two such procedures are compared in this paper.
End restraints effectively increase the loadbearing capacity of the composite slabs. Two different types of easily assembled additional end constraints are also tested and compared in this paper. Small-scale tests are used to obtain their shear bearing characteristics and to predict the loadbearing capacity of bent slabs using these restraints.
Keywords: composite slab, prepressed embossment, thin-walled, longitudinal shear, small-scale test, loadbearing capacity, end restraints
1 Introduction
There are three failure modes affecting composite slabs in general: vertical shear failure mode, bending failure mode and longitudinal shear failure mode. The latter is typical for composite slabs, and means that the longitudinal shear resistance at the interface between steel and concrete has been reached. At first, the adhesive bond fixes the inter-face. The adhesive bond is brittle and heavily dependent on casting conditions; it is therefore not included in shear bearing resistance calculations. When the adhesive bond fails, frictional resistance and mechanical interlock be-come active. As the load increases, so slip occurs between the sheeting and the concrete. The contribution of friction is dependent on the magnitude of the support reaction and acts mainly above the support [1].
The failure mode in longitudinal shear is usually in-dicated by the sheeting separating from the concrete sur-face around the embossments (Fig. 1). Concrete cracking may occur simultaneously with slip in the form of macro -cracks due to bending and micro-cracks around
emboss-ments – in a similar way to around a reinforcing bar in re-inforced concrete [2]. Microcracks can result in abrasion around embossment edges or local concrete peeling when the embossments are close to each other [3]. The number of factors influencing the bearing capacity makes it diffi-cult to describe slab behaviour analytically.
The use of traditional shear studs as end anchorages is limited to steel beams and requires special welding equipment. The small thickness of the sheeting enables the use of common screws, which can be easily drilled in-to the sheeting before casting. Another possibility is in-to re-strain the sheeting at the webs so that it cannot easily sep-arate from the concrete surface. The effect of these two types of restraint is compared in this paper.
2 Bending tests