Page 54 - Demo

CONCRETE TECHNOLOGY52 CPI %u2013 Concrete Plant International %u2013 1 | 2026 www.cpi-worldwide.comValidation The calculated values for %u03be show good agreement with the experimental data, which is plausible given the direct relationship between the compression zone height and the measured curvature (see Tables 8 and 9). Deviations are observed for the tensile stress %u03c3tf.The variation of %u03c3t0 results in negligible deviations in %u03be but leads to significantly steeper slopes of %u03c3tf.Summary and outlookThis study investigates the behaviour of highly ductile concrete beams under flexural tensile loading in order to abstract their tensile properties. Two mix designs with normal and high compressive strength are examined. Deformation and crack development are analysed using digital image correlation (DIC), enabling precise assessment of strain distributions and crack widths. Established models from the literature are adopted, leading - via equilibrium conditions - to polynomial equations of varying order, the solutions of which provide the compression zone depth and the tensile stress at the bottom fibre.The adopted model - linear elastic behaviour in the compression zone and rigid-plastic behaviour in the tensile zone - yields a particularly simple and unique solution due to the resulting quadratic equation. Overall, good agreement between experimental results and model predictions is observed up to damage localization. For the compression zone height, Equation (9) is proposed as a valid approximation up to the point of localization: (9)As an approximation for damage localization, %u03be = 0.1 and M = 0.9 Mmax may be assumed, from which the simplified Equations (10) and (11) are derived. This approximation provides an initial estimate, but in individual cases represents only a coarse approximation:(10)(11)If only a load-deflection curve is available, the curvature %u03ba can be determined using an iterative procedure in accordance with [22]. For the final assessment, the database will be expanded in further analytical evaluations, and all regions of the load-deflection curve will be analysed. nReferences[1] K. Holschemacher, F. Dehn, M. Torsten und F. Lobisch, %u201eGrundlagen des Faserbetons,%u201c in BetonKalender 2017: Spannbeton, Spezialbetone, Berlin, Wien, Darmstadt, Ernst Sohn FmbH Co. KG,, 2017, pp. 453-454.[2] V. Mechtcherine, %u201eDehnungsverfestigender Beton,%u201c in Betonbauwerke f%u00fcr die Zukunft, Z%u00fcrich, Beuth Verlag, 2015, pp. 77-92.[3] V. Mechtcherine, %u201eHochduktiler Beton mit Kurzfaserbewehrung-Baustoffliche Grundlagen und bautechnische Anwendungen,%u201c Beton-und Stahlbetonbau 110, Bd. 1, pp. 50-58, 2015. [4] F%u00e9d%u00e9ration internationale du b%u00e9ton, fib Model Code for Concrete Structures 2010, Lausanne: Ernst & Sohn, 2013. [5] AFNOR - NF P 18-710, %u201eCompl%u00e9ment national %u00e0 l%u2019Eurocode 2 - Calcul des structures en b%u00e9ton : r%u00e8gles sp%u00e9cifiques pour les B%u00e9tons Fibr%u00e9s %u00e0 Ultra-Hautes Performances (BFUP),%u201c Association Fran%u00e7aise de Normalisation (AFNOR), Saint-Denis, 2016.[6] J. Yu, C. Leung und C. Lu, %u201eTension and shear behavior of hybrid-fiber pseudo-ductile cementitious composites (PDCC),%u201c Brittle Matrix Composites 11 - Proceedings of the 11th International Symposium on Brittle Matrix Composites BMC , pp. 161 -171, 2015. Table 8: Comparison between model and measurements V1-28-%u03c3t0 = 0,5%u03c3fl, elTable 9: Comparison between model and measurements V2-28-%u03c3t0 = 0,5%u03c3fl, el%ud835%udc53%ud835%udc53!\%ud835%udefd%ud835%udefd $ %u210e100(&,'1 + %ud835%udefd%ud835%udefd $ %u210e100(&,' %ud835%udc53%ud835%udc53(%,$% [%ud835%udc40%ud835%udc40%ud835%udc40%ud835%udc40%ud835%udc40%ud835%udc40] (1)%ud835%udf05%ud835%udf05 =%ud835%udf0b%ud835%udf0b2 %u2212 arctan (%ud835%udc4e%ud835%udc4e) (2)%ud835%udf05%ud835%udf05 =%ud835%udf00%ud835%udf00%! %u2212 %ud835%udf00%ud835%udf00%\70 (3)%ud835%udf09%ud835%udf09 = >%ud835%udf00%ud835%udf00!(>%ud835%udf00%ud835%udf00\(4) %ud835%udf09%ud835%udf09* + 3%ud835%udf09%ud835%udf09+ %u2212 12%ud835%udc40%ud835%udc40$,-%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%u210e* = 0 (5)(2%ud835%udc38%ud835%udc38%ud835%udc38%ud835%udc38%u210e %u2212 %ud835%udf0e%ud835%udf0e\ + 2%ud835%udf0e%ud835%udf0e\(6)13 %ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udf07%ud835%udf07*%u210e*%ud835%udf09%ud835%udf09* +%ud835%udc4f%ud835%udc4f%ud835%udc4f+6 G((1 %u2212 %ud835%udf07%ud835%udf07) $(%ud835%udf07%ud835%udf07 + 2)%ud835%udf0e%ud835%udf0e! + (2%ud835%udf07%ud835%udf07 + 1)%ud835%udf0e%ud835%udf0e!& + C2%ud835%udf0e%ud835%udf0e\+%u2212%ud835%udc4f%ud835%udc4f%ud835%udc4f6%ud835%udf05%ud835%udf05 I(4%u210e%ud835%udf05%ud835%udf05 %u2212 %ud835%udf00%ud835%udf00\+%ud835%udc4f%ud835%udc4f6%ud835%udf05%ud835%udf05+ I(2%u210e+%ud835%udf05%ud835%udf05+ %u2212 %u210e%ud835%udf05%ud835%udf05%ud835%udf05%ud835%udf05\+)%ud835%udf0e%ud835%udf0e\(7)%u0009%u0009%u0009%u0009%u0009%ud835%udf07%ud835%udf07 = N%ud835%udf00%ud835%udf00!&%ud835%udf00%ud835%udf00!%u2264 1%u0009%u0009|%ud835%udf00%ud835%udf00!&| %u2264 |%ud835%udf00%ud835%udf00!|1%u0009%u0009%u0009%u0009%u0009%u0009%u0009%u0009%u0009%u0009%u0009%u0009|%ud835%udf00%ud835%udf00!&| %u2265 |%ud835%udf00%ud835%udf00!|(8)%ud835%udf09%ud835%udf09 =%u2212%ud835%udf0e%ud835%udf0e\2%ud835%udc38%ud835%udc38%ud835%udc38%ud835%udc38%u210e %u2212 %ud835%udf0e%ud835%udf0e\%ud835%udc5a%ud835%udc5a%ud835%udc5a%ud835%udc5a%ud835%udc5a%ud835%udc5a %ud835%udf0e%ud835%udf0e\(9)%ud835%udf0e%ud835%udf0e\+(1 %u2212 %ud835%udf09%ud835%udf09) %u2212 %ud835%udf0e%ud835%udf0e\%ud835%udc38%ud835%udc38%ud835%udc38%ud835%udc38%u210e90 %u2212 %ud835%udf0e%ud835%udf0e\ (10)%ud835%udf00%ud835%udf00\(11)%ud835%udc53%ud835%udc53!\%ud835%udefd%ud835%udefd $ %u210e100(&,'1 + %ud835%udefd%ud835%udefd $ %u210e100(&,' %ud835%udc53%ud835%udc53(%,$% [%ud835%udc40%ud835%udc40%ud835%udc40%ud835%udc40%ud835%udc40%ud835%udc40] (1)%ud835%udf05%ud835%udf05 =%ud835%udf0b%ud835%udf0b2 %u2212 arctan (%ud835%udc4e%ud835%udc4e) (2)%ud835%udf05%ud835%udf05 =%ud835%udf00%ud835%udf00%! %u2212 %ud835%udf00%ud835%udf00%\70 (3)%ud835%udf09%ud835%udf09 = >%ud835%udf00%ud835%udf00!(>%ud835%udf00%ud835%udf00\(4) %ud835%udf09%ud835%udf09* + 3%ud835%udf09%ud835%udf09+ %u2212 12%ud835%udc40%ud835%udc40$,-%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%u210e* = 0 (5)(2%ud835%udc38%ud835%udc38%ud835%udc38%ud835%udc38%u210e %u2212 %ud835%udf0e%ud835%udf0e\(6)13 %ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udf07%ud835%udf07*%u210e*%ud835%udf09%ud835%udf09* +%ud835%udc4f%ud835%udc4f%ud835%udc4f+6 G((1 %u2212 %ud835%udf07%ud835%udf07) $(%ud835%udf07%ud835%udf07 + 2)%ud835%udf0e%ud835%udf0e! + (2%ud835%udf07%ud835%udf07 + 1)%ud835%udf0e%ud835%udf0e!& + C2%ud835%udf0e%ud835%udf0e\+%u2212%ud835%udc4f%ud835%udc4f%ud835%udc4f6%ud835%udf05%ud835%udf05 I(4%u210e%ud835%udf05%ud835%udf05 %u2212 %ud835%udf00%ud835%udf00\+%ud835%udc4f%ud835%udc4f6%ud835%udf05%ud835%udf05+ I(2%u210e+%ud835%udf05%ud835%udf05+ %u2212 %u210e%ud835%udf05%ud835%udf05%ud835%udf05%ud835%udf05\+)%ud835%udf0e%ud835%udf0e\(7)%u0009%u0009%u0009%u0009%u0009%ud835%udf07%ud835%udf07 = N%ud835%udf00%ud835%udf00!&%ud835%udf00%ud835%udf00!%u2264 1%u0009%u0009|%ud835%udf00%ud835%udf00!&| %u2264 |%ud835%udf00%ud835%udf00!|1%u0009%u0009%u0009%u0009%u0009%u0009%u0009%u0009%u0009%u0009%u0009%u0009|%ud835%udf00%ud835%udf00!&| %u2265 |%ud835%udf00%ud835%udf00!|(8)%ud835%udf09%ud835%udf09 =%u2212%ud835%udf0e%ud835%udf0e\2%ud835%udc38%ud835%udc38%ud835%udc38%ud835%udc38%u210e %u2212 %ud835%udf0e%ud835%udf0e\%ud835%udc5a%ud835%udc5a%ud835%udc5a%ud835%udc5a%ud835%udc5a%ud835%udc5a %ud835%udf0e%ud835%udf0e\(9)%ud835%udf0e%ud835%udf0e\+(1 %u2212 %ud835%udf09%ud835%udf09) %u2212 %ud835%udf0e%ud835%udf0e\%ud835%udc38%ud835%udc38%ud835%udc38%ud835%udc38%u210e90 %u2212 %ud835%udf0e%ud835%udf0e\ (10)%ud835%udf00%ud835%udf00\(11)%ud835%udc53%ud835%udc53!\%ud835%udefd%ud835%udefd $ %u210e100(&,'1 + %ud835%udefd%ud835%udefd $ %u210e100(&,' %ud835%udc53%ud835%udc53(%,$% [%ud835%udc40%ud835%udc40%ud835%udc40%ud835%udc40%ud835%udc40%ud835%udc40] %ud835%udf05%ud835%udf05 =%ud835%udf0b%ud835%udf0b2 %u2212 arctan (%ud835%udc4e%ud835%udc4e) %ud835%udf05%ud835%udf05 =%ud835%udf00%ud835%udf00%! %u2212 %ud835%udf00%ud835%udf00%\70 %ud835%udf09%ud835%udf09 = >%ud835%udf00%ud835%udf00!(>%ud835%udf00%ud835%udf00\ %ud835%udf09%ud835%udf09* + 3%ud835%udf09%ud835%udf09+ %u2212 12%ud835%udc40%ud835%udc40$,-%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%u210e* = 0 (2%ud835%udc38%ud835%udc38%ud835%udc38%ud835%udc38%u210e %u2212 %ud835%udf0e%ud835%udf0e\ + 2%ud835%udf0e%ud835%udf0e\13 %ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udf07%ud835%udf07*%u210e*%ud835%udf09%ud835%udf09* +%ud835%udc4f%ud835%udc4f%ud835%udc4f+6 G((1 %u2212 %ud835%udf07%ud835%udf07) $(%ud835%udf07%ud835%udf07 + 2)%ud835%udf0e%ud835%udf0e! + (2%ud835%udf07%ud835%udf07 + 1)%ud835%udf0e%ud835%udf0e!& + C2%ud835%udf0e%ud835%udf0e\+%u2212%ud835%udc4f%ud835%udc4f%ud835%udc4f6%ud835%udf05%ud835%udf05 I(4%u210e%ud835%udf05%ud835%udf05 %u2212 %ud835%udf00%ud835%udf00\+%ud835%udc4f%ud835%udc4f6%ud835%udf05%ud835%udf05+ I(2%u210e+%ud835%udf05%ud835%udf05+ %u2212 %u210e%ud835%udf05%ud835%udf05%ud835%udf05%ud835%udf05\+)%ud835%udf0e%ud835%udf0e\%u0009%u0009%u0009%u0009%u0009%ud835%udf07%ud835%udf07 = N%ud835%udf00%ud835%udf00!&%ud835%udf00%ud835%udf00!%u2264 1%u0009%u0009|%ud835%udf00%ud835%udf00!&| %u2264 |%ud835%udf00%ud835%udf00!|1%u0009%u0009%u0009%u0009%u0009%u0009%u0009%u0009%u0009%u0009%u0009%u0009|%ud835%udf00%ud835%udf00!&| %u2265 |%ud835%udf00%ud835%udf00!|%ud835%udf09%ud835%udf09 =%u2212%ud835%udf0e%ud835%udf0e\2%ud835%udc38%ud835%udc38%ud835%udc38%ud835%udc38%u210e %u2212 %ud835%udf0e%ud835%udf0e\%ud835%udc5a%ud835%udc5a%ud835%udc5a%ud835%udc5a%ud835%udc5a%ud835%udc5a %ud835%udf0e%ud835%udf0e\%ud835%udf0e%ud835%udf0e\+(1 %u2212 %ud835%udf09%ud835%udf09) %u2212 %ud835%udf0e%ud835%udf0e\%ud835%udc38%ud835%udc38%ud835%udc38%ud835%udc38%u210e90 %u2212 %ud835%udf0e%ud835%udf0e\ %ud835%udf00%ud835%udf00\%ud835%udc53%ud835%udc53!\%ud835%udefd%ud835%udefd $ %u210e100(&,'1 + %ud835%udefd%ud835%udefd $ %u210e100(&,' %ud835%udc53%ud835%udc53(%,$% [%ud835%udc40%ud835%udc40%ud835%udc40%ud835%udc40%ud835%udc40%ud835%udc40] %ud835%udf05%ud835%udf05 =%ud835%udf0b%ud835%udf0b2 %u2212 arctan (%ud835%udc4e%ud835%udc4e) %ud835%udf05%ud835%udf05 =%ud835%udf00%ud835%udf00%! %u2212 %ud835%udf00%ud835%udf00%\70 %ud835%udf09%ud835%udf09 = >%ud835%udf00%ud835%udf00!(>%ud835%udf00%ud835%udf00\ %ud835%udf09%ud835%udf09* + 3%ud835%udf09%ud835%udf09+ %u2212 12%ud835%udc40%ud835%udc40$,-%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%u210e* = 0 (2%ud835%udc38%ud835%udc38%ud835%udc38%ud835%udc38%u210e %u2212 %ud835%udf0e%ud835%udf0e\ + 2%ud835%udf0e%ud835%udf0e\13 %ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udc4f%ud835%udf07%ud835%udf07*%u210e*%ud835%udf09%ud835%udf09* +%ud835%udc4f%ud835%udc4f%ud835%udc4f+6 G((1 %u2212 %ud835%udf07%ud835%udf07) $(%ud835%udf07%ud835%udf07 + 2)%ud835%udf0e%ud835%udf0e! + (2%ud835%udf07%ud835%udf07 + 1)%ud835%udf0e%ud835%udf0e!& + C2%ud835%udf0e%ud835%udf0e\+%u2212%ud835%udc4f%ud835%udc4f%ud835%udc4f6%ud835%udf05%ud835%udf05 I(4%u210e%ud835%udf05%ud835%udf05 %u2212 %ud835%udf00%ud835%udf00\+%ud835%udc4f%ud835%udc4f6%ud835%udf05%ud835%udf05+ I(2%u210e+%ud835%udf05%ud835%udf05+ %u2212 %u210e%ud835%udf05%ud835%udf05%ud835%udf05%ud835%udf05\+)%ud835%udf0e%ud835%udf0e\%u0009%u0009%u0009%u0009%u0009%ud835%udf07%ud835%udf07 = N%ud835%udf00%ud835%udf00!&%ud835%udf00%ud835%udf00!%u2264 1%u0009%u0009|%ud835%udf00%ud835%udf00!&| %u2264 |%ud835%udf00%ud835%udf00!|1%u0009%u0009%u0009%u0009%u0009%u0009%u0009%u0009%u0009%u0009%u0009%u0009|%ud835%udf00%ud835%udf00!&| %u2265 |%ud835%udf00%ud835%udf00!|%ud835%udf09%ud835%udf09 =%u2212%ud835%udf0e%ud835%udf0e\2%ud835%udc38%ud835%udc38%ud835%udc38%ud835%udc38%u210e %u2212 %ud835%udf0e%ud835%udf0e\%ud835%udc5a%ud835%udc5a%ud835%udc5a%ud835%udc5a%ud835%udc5a%ud835%udc5a %ud835%udf0e%ud835%udf0e\%ud835%udf0e%ud835%udf0e\+(1 %u2212 %ud835%udf09%ud835%udf09)%u2212 %ud835%udf0e%ud835%udf0e\90 %u2212 %ud835%udf0e%ud835%udf0e\ %ud835%udf00%ud835%udf00\where