Comparison with the non-linear structural response Motivated by the need for load case superposition within

load case combinations, the ultimate limit state assess-ment approach described in the previous section was based on linear-elastic finite element analysis even though the material behaviour is highly non-linear (section 2.2). It must be said that the use of linear-elastic analysis when de-signing TRC composites is on the safe side due to the un-derlying strain-hardening material response in tension. In-deed, stress redistributions within the shell lead to a larger load-carrying capacity than predicted by the linear-elastic analysis.

In order to investigate the structural redundancies for the given TCR roof structure and to provide a realistic prediction of the shell deflections for the serviceability limit state, non-linear simulations were performed for se-lected horizontal and vertical load case combinations. For this purpose an anisotropic damage model of the micro-plane type [19, 20], reflecting the evolution of finely dis-tributed, oriented cracks as direction-dependent damage variables, has been formulated and implemented [21, 22].

The material-specific damage function reflecting the strain-hardening tensile response was calibrated based on the stress-strain curves obtained from the TRC tensile tests described in section 2.2 (Fig. 2b).

To evaluate the loadbearing reserves of the TRC roof structure assessed in section 4.2, let us consider a load case consisting of only permanent loads, including self-weight of the shell (material density ρ = 0.224 kN/m³), additional load on top of the shell (distributed load g_k = 0.20 kn/m²) and vertical load due to the attached façade (line load g_b= 0.35 kN/M). The utilization ratio ob-tained for this load case using the assessment procedure described based on a fine finite element discretization, which includes the stress peak values at the shell edges, equates to ηnmd= 0.27. In the subsequent non-linear analy-sis this load case has been considered as a reference load with the load factor λref= 1.0.

Fig. 7 shows the dependency between the increasing load factor λ and the corner deflection w obtained from

122

A. Scholzen/R. Chudoba/J. Hegger · Thin-walled shell structures made of textile-reinforced concrete

Structural Concrete (2015), No. 1

the non-linear simulation. As mentioned above, the para-meters of the anisotropic damage model were identified based on the tensile tests conducted and therefore reflect the material behaviour corresponding to the mean values of the cross-sectional strength characteristics in Eq. (14).

Ultimate failure was reached for the load factor λinel = 9.08.

For the comparison with the linear analysis, let us first determine the load factor corresponding to full uti-lization at the ultimate limit state for the design values of material strength (Eq. 16), which is λuls = l/ηnmd = 3.7.

Further, the load factor corresponding to the mean strength characteristics (see Eq. (15)) is obtained as λel= γtex/χ · λuls = 1.5/0.81 · 3.7 = 6.85. Comparing this load factor with the prediction obtained using the non-linear model (λinel = 9.08), the structural reserves due to stress-redistribution within the shell can be evaluated for the load case considered as λinel/λel= 1.32. Fig. 8 depicts the spatial distribution of damage calculated using the non-linear simulation for the load levels specified in Fig. 7. The distribution of the maximum value of the micro-plane damage, i.e. max(ω), on the top and bottom surfaces of the quarter of the shell is depicted for each load level. The analysis shows that the propagation of damage, i.e. matrix cracking, starts in the middle of the Fig. 7. Evaluation of structural reserves using the non-linear, anisotropic strain-hardening damage model for increasing vertical loading

Fig. 8. Distribution of the maximum damage within the TRC shell for the four loading levels given in Fig. 7

A. Scholzen/R. Chudoba/J. Hegger · Thin-walled shell structures made of textile-reinforced concrete shell edges and propagates along the ring of principal

tensile stresses obtained using the linear FE analysis (see companion paper [12], Fig. 7). At the ultimate limit state the damage spreads over a large area of the shell, indicating a high amount of stress redistribution during the loading process. These redistributions, not reflected in the linear-elastic design, are the source of the load-bearing reserves identified using the non-linear simula-tion.

Let us finally remark that the degree of structural re-dundancy depends on the smoothness of the stress field induced by a particular load case. When a concentrated local load with highly localized stresses is considered, the effect of the in-plane stress redistribution might also be less pronounced or even negligible. Furthermore, no re-dundancy can be expected for pure uniaxial bending of the shell without any membrane stresses and, thus, no possibility for the damage to propagate to other zones of the shell. Nevertheless, the study performed demon-strates that the linear-elastic design approach used pro-vides a lower bound estimate of the ultimate load level.

5 Further issues addressed in the structural assessment Due to limitations of space, the present paper could not cover all aspects included in the design and assessment of the TRC structure actually built. Special attention has been paid to the design and assessment of joints with the aim of assuring a smooth load transfer between structural elements. Further, the serviceability limit state assessment was performed based on the maximum deflections evalu-ated, also including the effect of creep. Another issue to be considered in a general assessment method are zones ex-posed to a significant biaxial tensile loading that might lead to a strength reduction in one direction due to longi-tudinal cracks induced by the tensile load in the perpen-dicular direction.

6 Conclusions

The present paper introduces a design approach for TRC shell structures based on cross-sectional strength charac-teristics determined experimentally. The description of the design focuses on the assessment at the ultimate limit state considering the interaction between normal forces and bending moments. In order to take into account all load case combinations and at the same time the orienta-tion of the stress resultants with respect to the orientaorienta-tion of the reinforcing fabrics, an automated assessment tool has been formulated and implemented. The design ap-proach described has been successfully applied and tested with respect to its practical feasibility for the TRC pavilion built at RWTH Aachen University.

Acknowledgements

The authors wish to thank the German Research Founda-tion (DFG) for financial support within the collaborative research centre SFB 532 “Textile reinforced concrete – de-velopment of a new technology” and DFG project CH 276/2-2.

References

1. Hegger, J., Voss, S.: Investigation of the loadbearing behav-iour and potential of Textile Reinforced Concrete. Engineer-ing Structures, 2008, 30, No. 7, pp. 2050–2056.

2. Hegger, J., Will, N., Bruckermann, O., Voss, S.: Loadbearing behaviour and simulation of textile reinforced concrete. Ma-terial and Structures, 2006, 39, No. 8, pp. 765–776.

3. Voss, S.: Ingenieurmodelle zum Tragverhalten von textilbe-wehrtem Beton (Design models for the loadbearing behav-iour of textile reinforced concrete). Dissertation, Institute of Structural Concrete (IMB), RWTH Aachen University, 2008, ISBN 3-939051-03-9, ISSN 0949-7331 (in German).

4. Schladitz, F., Lorenz, E., Curbach, M.: Biegetragfähigkeit von textilbetonverstärkten Stahlbetonplatten (Bending Capacity of Reinforced Concrete Slabs Strengthened with Textile Re-inforced Concrete). Beton- und Stahlbetonbau, 2011, 106, No. 6, pp. 377–384 (in German).

5. Hegger, J., Horstmann, M.: Sandwichfassaden aus Textilbe-ton – Numerik und Ingenieurmodelle (Sandwich facades made of Textile Reinforced Concrete – numerical investiga-tions and engineering models). Bautechnik, 2011, 88, No. 6, pp. 373–384 (in German).

6. Schnabel, A., Grieß, T.: Production of non-crimp fabrics for composites; In: Non-crimp fabric composites: manufactur-ing, properties and applications, Lomov, S. V. (ed.), Wood-head Publishing Series in Composites Science and Engineer-ing, No. 35, Woodhead, Oxford, 2011.

7. Hartig, J., Jesse, F., Schicktanz, K., Häußler-Combe, U.: Influ-ence of experimental setup on the apparent uniaxial tensile loadbearing capacity of Textile Reinforced Concrete speci-mens. Materials and Structures, 2012, 45, pp. 433–446.

8. Colombo, I. G.., Magri, A., Zani, G., Colombo, M., di Prisco, M.: Textile Reinforced Concrete: experimental investigation on design parameters. Materials and Structures, 2013, 46, pp.

1953–1971.

9. Larrinaga, P., Chastre, C., San-Jose, J. T., Garmendia, L.:

Non-linear analytical model of composites based on basalt textile reinforced mortar under uniaxial tension. Compos-ites: Part B, 2013, No. 55, pp. 518–527.

10. Tysmans, T., Adriaenssens, S., Cuypers, H., Wastiels, J.: Struc-tural analysis of small span textile reinforced concrete shells with double curvature. Composites Science and Technology, 2009, 69, pp. 1790–1796.

11. Scholzen, A., Chudoba, R., Hegger, J.: Dünnwandiges Scha-lentragwerk aus Textilbeton: Entwurf, Bemessung und baupraktische Umsetzung. Beton- und Stahlbetonbau, 2012, 107, No. 11, pp, 767–776.

12. Scholzen, A., Chudoba, R., Hegger, J. (2015): Thin-walled shell structures made of textile-reinforced concrete – Part I:

Structural design and construction. Structural Concrete, 16:

106–114. doi: 10.1002/suco.201300071.

13. Rypl, R., Chudoba, R., Mörschel, U., Stapleton, S. E., Gries, T., Sommer, G.: A novel tensile test device for effective testing of high-modulus multi-filament yarns. Journal of Industrial Textiles, 2014, published online, doi: 10.1177/

1528083714521069.

14. Hinzen, M., Brameshuber, W.: Loadbearing Behaviour of Textile Reinforced Concrete with Short Fibres. In: Fibre Re-inforced Concrete: Challenges and Opportunities, Proc. of 8th RILEM Int. Symposium, Barros, J. A. O., (ed.), Portugal, 19–21 Sept 2012.

15. Mobasher, B., Peled, A., Pahilajani, J.: Distributed cracking and stiffness degradation in fabric-cement composites. Mate-rials and Structures, 2006, 39, pp, 317–331.

16. Rypl, R., Chudoba, R., Scholzen, A., Voˇrechovsky´, M.: Brittle matrix composites with heterogeneous reinforcement:

Multi-scale model of a crack bridge with rigid matrix. Composites Science and Technology, 2013, 89, pp. 98–109.

17. DIN EN 1992-1-1, Eurocode 2: Design of concrete structures – Part 1-1: General rules and rules for buildings; German ver-sion EN 1992-1-1:2004 + AC:2010.

18. DIN EN1990, Eurocode: Basis of structural design; German version EN 1990:2002 + A1:2005 + A1:2005/AC:2010.

19. Jirásek, M.: Comments on microplane theory. Mechanics of Quasibrittle Materials and Structures, Hermes Science Pub-lications, 1999, pp. 55–77.

20. Carol, I., Jirásek, M., Bazˇant, Z.: A thermodynamically con-sistent approach to microplane theory. Part I. Free energy and consistent microplane stresses. International Journal of Solids and Structures, 2001, 38, No. 17, pp. 2921–2931.

21. Scholzen, A., Chudoba, R., Hegger, J.: Damage Based Model-ing of Planar Textile-Reinforced Concrete Structures. In:

Proc. of Int. RILEM Conference on Material Science, Brameshuber, W. (ed.), Aachen, 2010, pp. 362–370.

22. Scholzen, A., Chudoba, R., Hegger, J.: Calibration and Valida-tion of a Microplane Damage Model for cement-based Com-posites applied to Textile Reinforced Concrete, In: Proc. of Int. Conf. on Recent Advances in Nonlinear Models – Struc-tural Concrete Applications, Barros, H. Faria, R., Pina, C., Ferreira, C. (eds.), ECCOMAS thematic conf., 2011, Coim-bra, pp. 417–427.

124

A. Scholzen/R. Chudoba/J. Hegger · Thin-walled shell structures made of textile-reinforced concrete

Structural Concrete (2015), No. 1

Prof. Dr.-Ing. Josef Hegger RWTH Aachen University

Institute of Structural Concrete (IMB) Mies-van-der-Rohe-Str. 1

52074 Aachen Germany

[email protected] Dr.-Ing. Rostislav Chudoba RWTH Aachen University

Institute of Structural Concrete (IMB) Mies-van-der-Rohe-Str. 1

52074 Aachen Germany

[email protected] Dipl.-Ing. Alexander Scholzen RWTH Aachen University

Institute of Structural Concrete (IMB) Mies-van-der-Rohe-Str. 1

52074 Aachen Germany

[email protected]

Non-linear constitutive models for concrete in compression are frequently defined in design codes. The engineer generally uses either the linear (in SLS) or non-linear (in ULS) compression mod-el. However, a large variety of different approaches exists for de-scribing the behaviour of the cracked concrete tension zone, and the selection of a corresponding model is usually based on quali-tative engineering judgement. The aim of this paper is to assess the prediction quality of several concrete material models in or-der to provide a quantitative model selection. Therefore, uncer-tainty analysis is applied in order to investigate the model and pa-rameter uncertainty in the bending stiffness prognosis for flexural members. The total uncertainty is converted into a prognosis model quality that allows a quantitative comparison between the material models considered. The consideration of the reinforced concrete in tension is based on the characterization of the ten-sion stiffening effect, which describes the cracking in an average sense. In the interest of the practical applicability of the models considered, even for large structures, no discrete crack simula-tions based on fracture mechanics are considered. Finally, the assessment identifies that the prediction quality depends on the loading level and, furthermore, the quality across the models can be quantitatively similar as well as diverse.

Keywords: model evaluation, model quality, model uncertainty, parameter uncertainty, tension stiffening

1 Introduction

Numerical simulations are commonly used for analysing structural load-deformation behaviour, particularly in the case of complex systems with a high number of unknowns, geometrical and material non-linear behaviour, an irregu-lar geometry of the structure and a irregu-large number of load cases and combinations. In the field of structural design, a structural engineer has to decide which phenomena – rep-resented by partial models (PM), e.g. material models, soil models, interaction models, load models – should be con-sidered in the numerical simulation of the global structur-al model (GM) under different conditions and load cases.

In general, a high number of analytical and numeri-cal models for each of these partial models are applied in

construction projects and research studies. In addition, new models are being developed with additional knowl-edge according to the “real” phenomena. Therefore, the selecting adequate partial models in the global model is not a simple and trivial task. The importance of these par-tial models according to the global response of the struc-ture can be quantified using variance-based sensitivity analysis. In the case that a PM influences the structural behaviour, it is necessary to quantify its prediction quality MQ_PMand then subsequently combine both sets of infor-mation into the global model quality MQ_GMfor the entire structure [1].

When analysing the uncertainty in the model output, the quantitative model evaluation poses the question as to which model should be chosen in comparison to the oth-er models considoth-ered. This study considoth-ers the uncoth-ertain- uncertain-ties due to the non-deterministic input parameters and al-so uncertainties due to the model prediction error. This probabilistic model evaluation helps to achieve a definite model selection in a quantitative manner. Hence, the question as to which model is the most adequate of the ones considered can be answered by the use of uncertain-ty analysis. This most adequate model with highest predic-tion quality should be used in structural engineering prob-lems in order to achieve a greater confidence in the simulation results and to ensure a reliable structural de-sign.

Material modelling is a partial model with a poten-tially strong influence on the computational results and reliable prognosis models [2, 3, 4]. For instance, the analy-sis of internal section forces for restraint-sensitive struc-tures, e.g. pavements, bridge decks, walls, industrial floors, constrained slabs or integral and semi-integral bridges, is crucially dependent on the material model prediction [5].

Therefore, the evaluation of the partial model’s quality for reinforced concrete is the focus of this paper. Either pure-ly linear models, linear compressive models or non-linear models considering the tension stiffening effect are considered in the evaluation.

The uncertainty assessment on a structural level (continuous beams, frames) for the material non-linear simulation is not exclusively influenced by the partial model’s prediction of a certain structural element or cross-section. Owing to concrete cracking and internal force redistribution, the uncertainty in the structural sim-ulation is mainly influenced by several coupled elements Technical Paper

Quality assessment of material models