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Keywords: reinforced concrete, interaction curves, size effect, limit states, concrete in compression, ductility, stress-strain relationship.

Summary

In order to evaluate the progressive damage produced by crushing of compressed concrete, a numerical model has been proposed in [1]. With this model, it was possible to reproduce the post-peak branch in the moment curvature $(M-\mu)$ relationship of RC beams in bending. The $M-\mu$ diagrams obtained, also applied to elements in bending and compression, seem to be close to the curves obtained by testing RC beams with and without stirrups [2]. Moreover, the stress $\sigma_c$ of compressed concrete is a function of the strain $\varepsilon_c$ and of the extension $y_{c,max}$ of the compressed zone, according to the size effect theory proposed by Hillerborg [3]. However, in the case of beams in bending, the stress $\sigma_c$ obtained with the proposed model seems to depend also on the cross-sectional curvature $\mu$ .

All this aspects cannot be computed by means of Eurocode 2 [4], where the size effect on the structural response of compressed concrete is not taken into account. In the Eurocode 2 the concrete in compression is modelled by the parabola-rectangle $\sigma-\varepsilon$ diagram. Thus in a generic cross-section the maximum moment is reached only when one of the materials attains its limit strain.

The aim of this paper is to highlight how Eurocode 2 is not able to reproduce the structural effects produced by the post peak branch of $\sigma-\varepsilon$ . The influence of the mechanical behaviour of compressed concrete on the interaction curves of RC cross-sections is investigated by adopting different approaches. For a given value of the normal force , the maximum bending moments obtained by the proposed model are compared to the ones of Eurocode 2 [4]. Both the approaches are applied to similar cross-sections, whose dimensions are obtained by scaling the size factor $\chi$ . In each case, there is a deterministic evaluation of the maximum bending moment, so the partial factors for materials are not applied. The non-dimensional bending moment , normal force and mechanical reinforcement ratio $\omega$ are introduced in order to compare the numerical results. Varying the dimensions of the cross-section, the proposed model provides different stress distributions in the compressed concrete and, consequently, more than one interaction curves. These curves, obtained for symmetrically reinforced cross-sections and for asymmetrically reinforced ones (no reinforcement in the compressed zone), with size factors $\chi = 1$ and $\chi = 20$ , are compared with the curves computed by assuming the parabola rectangle stress-strain diagram for concrete in compression [4]. For a given value of , a reduction of the bending moment with the increase of $\chi$ is obtained. This phenomenon is particularly evident for $n \cong 0.4$ , in the section symmetrically reinforced and in the case of high mechanical reinforcement ratio.

From a practical point of view, it is also interesting to evaluate the rotation $\varphi$ of a beam portion having a length equal to the height of its cross-section. If the value of the curvature at maximum bending moment is considered in the evaluation of $\varphi$ , the effect of the inelastic behaviour of the beam are also included in the rotation. The $\varphi-n$ curves obtained in case of asymmetrically reinforced concrete sections and for different size factors $\chi$ , show an increase of $\varphi$ when decreases. Similarly, for a given value of , there is an increase of $\varphi$ with the decrease of $\chi$ . When and $\chi = 20$ , the rotation $\varphi$ can be times lower than $\chi = 1$ . This difference considerably decreases with the increase of and $\omega$ , and is particularly evident in beams with lower $\omega$ and . This scaling behaviour cannot be reproduced with the classical approach proposed in the Eurocode 2 [4].

In conclusion, with the proposed model for compressed concrete, the softening branch of the stress strain relationship, and its effects on moment curvature diagrams, can be defined. The post-peak branch of the $\sigma-\varepsilon$ diagram clearly shows a size effect, which remarkably affects the cross-sectional strength of a RC beam and its corresponding rotation $\varphi$ . The decrease of $\varphi$ , observed when $\chi$ increases, can be explained as a reduced capability of larger structures to bear plastic deformations. Since size-effect is not currently considered into Eurocode 2, it is desirable that future code requirements will be take into account these phenomena.

References

1: A.P. Fantilli, D. Ferretti, I. Iori, P. Vallini, "A Mechanical Model for the Fail-ure of Compressed Concrete in R/C Beams", Journal of Structural Engineering, ASCE, 128(5), 637-645, 2002. doi:10.1061/(ASCE)0733-9445(2002)128:5(637)
2: A.P. Fantilli, I. Iori, P. Vallini, "A Mechanical Model for the Confined Com-pressed Concrete of RC Beams", 1st fib Congress: Concrete Structures in the 21st century, October 13-19 2002, Osaka, Japan.
3: A. Hillerborg, "Fracture mechanics concepts applied to moment capacity and rotational capacity of reinforced concrete beams", Engineering Fracture Mechanics, 35(1/2/3), 233-240, 1990. doi:10.1016/0013-7944(90)90201-Q
4: European Committee for Standardization, "Eurocode 2: Design of concrete structures- Part 1: general rules and rules for buildings", Brussels, 2001.

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 77 PROCEEDINGS OF THE NINTH INTERNATIONAL CONFERENCE ON CIVIL AND STRUCTURAL ENGINEERING COMPUTING Edited by: B.H.V. Topping Paper 57 Size Effect of Compressed Concrete in the Ultimate Limit States of RC Elements A.P. Fantilli+, I. Iori* and P. Vallini+ +Department of Structural and Geotechnical Engineering, Politecnico di Torino, Italy Department of Civil Engineering, University of Parma, Italy doi:10.4203/ccp.77.57 purchase the full-text of this paper Full Bibliographic Reference for this paper A.P. Fantilli, I. Iori, P. Vallini, "Size Effect of Compressed Concrete in the Ultimate Limit States of RC Elements", in B.H.V. Topping, (Editor), "Proceedings of the Ninth International Conference on Civil and Structural Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 57, 2003. doi:10.4203/ccp.77.57 Keywords:* reinforced concrete, interaction curves, size effect, limit states, concrete in compression, ductility, stress-strain relationship. Summary In order to evaluate the progressive damage produced by crushing of compressed concrete, a numerical model has been proposed in [1]. With this model, it was possible to reproduce the post-peak branch in the moment curvature $(M-\mu)$ relationship of RC beams in bending. The $M-\mu$ diagrams obtained, also applied to elements in bending and compression, seem to be close to the curves obtained by testing RC beams with and without stirrups [2]. Moreover, the stress $\sigma_c$ of compressed concrete is a function of the strain $\varepsilon_c$ and of the extension $y_{c,max}$ of the compressed zone, according to the size effect theory proposed by Hillerborg [3]. However, in the case of beams in bending, the stress $\sigma_c$ obtained with the proposed model seems to depend also on the cross-sectional curvature $\mu$ . All this aspects cannot be computed by means of Eurocode 2 [4], where the size effect on the structural response of compressed concrete is not taken into account. In the Eurocode 2 the concrete in compression is modelled by the parabola-rectangle $\sigma-\varepsilon$ diagram. Thus in a generic cross-section the maximum moment is reached only when one of the materials attains its limit strain. The aim of this paper is to highlight how Eurocode 2 is not able to reproduce the structural effects produced by the post peak branch of $\sigma-\varepsilon$ . The influence of the mechanical behaviour of compressed concrete on the interaction curves of RC cross-sections is investigated by adopting different approaches. For a given value of the normal force , the maximum bending moments obtained by the proposed model are compared to the ones of Eurocode 2 [4]. Both the approaches are applied to similar cross-sections, whose dimensions are obtained by scaling the size factor $\chi$ . In each case, there is a deterministic evaluation of the maximum bending moment, so the partial factors for materials are not applied. The non-dimensional bending moment , normal force and mechanical reinforcement ratio $\omega$ are introduced in order to compare the numerical results. Varying the dimensions of the cross-section, the proposed model provides different stress distributions in the compressed concrete and, consequently, more than one interaction curves. These curves, obtained for symmetrically reinforced cross-sections and for asymmetrically reinforced ones (no reinforcement in the compressed zone), with size factors $\chi = 1$ and $\chi = 20$ , are compared with the curves computed by assuming the parabola rectangle stress-strain diagram for concrete in compression [4]. For a given value of , a reduction of the bending moment with the increase of $\chi$ is obtained. This phenomenon is particularly evident for $n \cong 0.4$ , in the section symmetrically reinforced and in the case of high mechanical reinforcement ratio. From a practical point of view, it is also interesting to evaluate the rotation $\varphi$ of a beam portion having a length equal to the height of its cross-section. If the value of the curvature at maximum bending moment is considered in the evaluation of $\varphi$ , the effect of the inelastic behaviour of the beam are also included in the rotation. The $\varphi-n$ curves obtained in case of asymmetrically reinforced concrete sections and for different size factors $\chi$ , show an increase of $\varphi$ when decreases. Similarly, for a given value of , there is an increase of $\varphi$ with the decrease of $\chi$ . When and $\chi = 20$ , the rotation $\varphi$ can be times lower than $\chi = 1$ . This difference considerably decreases with the increase of and $\omega$ , and is particularly evident in beams with lower $\omega$ and . This scaling behaviour cannot be reproduced with the classical approach proposed in the Eurocode 2 [4]. In conclusion, with the proposed model for compressed concrete, the softening branch of the stress strain relationship, and its effects on moment curvature diagrams, can be defined. The post-peak branch of the $\sigma-\varepsilon$ diagram clearly shows a size effect, which remarkably affects the cross-sectional strength of a RC beam and its corresponding rotation $\varphi$ . The decrease of $\varphi$ , observed when $\chi$ increases, can be explained as a reduced capability of larger structures to bear plastic deformations. Since size-effect is not currently considered into Eurocode 2, it is desirable that future code requirements will be take into account these phenomena. References 1 A.P. Fantilli, D. Ferretti, I. Iori, P. Vallini, "A Mechanical Model for the Fail-ure of Compressed Concrete in R/C Beams", Journal of Structural Engineering, ASCE, 128(5), 637-645, 2002. doi:10.1061/(ASCE)0733-9445(2002)128:5(637) 2 A.P. Fantilli, I. Iori, P. Vallini, "A Mechanical Model for the Confined Com-pressed Concrete of RC Beams", 1st fib Congress: Concrete Structures in the 21st century, October 13-19 2002, Osaka, Japan. 3 A. Hillerborg, "Fracture mechanics concepts applied to moment capacity and rotational capacity of reinforced concrete beams", Engineering Fracture Mechanics, 35(1/2/3), 233-240, 1990. doi:10.1016/0013-7944(90)90201-Q 4 European Committee for Standardization, "Eurocode 2: Design of concrete structures- Part 1: general rules and rules for buildings", Brussels, 2001. purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £123 +P&P)
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