Page 53 - Demo

CONCRETE TECHNOLOGYwww.cpi-worldwide.com CPI %u2013 Concrete Plant International %u2013 1 | 2026 51Simplified derivation of the tensile stress-strain relationship up to localizationThe matrix tensile strength fct is determined at the end of Phase 1 at the flexural linearity limit based on Equation (1). The evolution of tensile stresses beyond the linear-elastic tensile range is a priori unknown. A linear strain distribution over the cross-section is assumed. The stress-strain distribution in the compression zone is likewise assumed to be linear or bilinear. The different stress distribution approaches are illustrated in Figure 9.From the equilibrium conditions, the following equations are obtained:%u2022 Model 1 [15] (5)%u2022 Model 2 (6)%u2022 Model 5 (7)where(8)The elastic strain is determined according to Hooke%u2019s law, and the strains up to localization are calculated from the deformation at the bottom edge relative to the initial gauge length.Fig. 9: Assumed stress distributions.1+%ud835%udefd%ud835%udefd$100(%ud835%udf05%ud835%udf05 =%ud835%udf0b%ud835%udf0b2 %u2212 arctan (%ud835%udc4e%ud835%udc4e) %ud835%udf05%ud835%udf05 =%ud835%udf00%ud835%udf00%! %u2212 %ud835%udf00%ud835%udf00%\70 %ud835%udf09%ud835%udf09 = >%ud835%udf00%ud835%udf00!(>%ud835%udf00%ud835%udf00\ %ud835%udf09%ud835%udf09* + 3%ud835%udf09%ud835%udf09+ %u2212 12%ud835%udc40%ud835%udc40$,-%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%udefd%ud835%udefd $ %u210e100(%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\ 0,011%ud835%udc38%ud835%udc38%ud835%udc38%ud835%udc38%u210e %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\%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\ (2%u210e%ud835%udf05%ud835%udf05 + %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\bekaert.comDramix%u00aeSimplify. Save. Sustain.Build effortlessly with steel fibers.Steel fiber concrete reinforcementUsing Dramix%u00ae steel fibers to reinforce your concrete allows the creation of reliable and long-lasting precast elements of any design, no matter how complex. Contact our local expert: vinicio.casas@bekaert.com or visit our website: www.construction.bekaert.com/precast.