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Parabola-rectangle

Parabola-rectangles are commonly uses for concrete stress-strain curves.

parabola rectangle consrete stress-strain curve diagram

The parabolic curve can be characterised as

ffcd=a(εεc)2+b(εεc)\frac{f}{f_{c d}}=a\left(\frac{\varepsilon}{\varepsilon_c}\right)^2+b\left(\frac{\varepsilon}{\varepsilon_c}\right)

At strains above εc\varepsilon_c the stress remains constant. For most design codes the parabola is taken as having zero slope where it meets the horizontal portion of the stress-strain curve.

ffcd=[1−(1−εεc)2]\frac{f}{f_{c d}}=\left[1-\left(1-\frac{\varepsilon}{\varepsilon_c}\right)^2\right]

The Hong Kong Code of Practice (supported by the Hong Kong Institution of Engineers) interpret the curve so that the initial slope is the elastic modulus (meaning that the parabola is not tangent to the horizontal portion of the curve).

ffcd=[1−(EEs)](εεc)2+(EEs)(εεc)\frac{f}{f_{c d}}=\left[1-\left(\frac{E}{E_s}\right)\right]\left(\frac{\varepsilon}{\varepsilon_c}\right)^2+\left(\frac{E}{E_s}\right)\left(\frac{\varepsilon}{\varepsilon_c}\right)

where the secant modulus is

Es=fcdεcE_s=\frac{f_{c d}}{\varepsilon_c}

In Eurocode the parabola is modified

ffcd=[1−(1−εεc)n]\frac{f}{f_{cd}}= \left[1-\left(1-\frac{\varepsilon}{\varepsilon_c}\right)^n\right]

and

n=2fc≤50MPan=1.4+23.4[(90−fc)/100]4fc>50MPa\begin{array}{ll} n=2 & f_c \leq 50 \mathrm{MPa} \\ \\ n=1.4+23.4\left[\left(90-f_c\right) / 100\right]^4 & f_c>50 \mathrm{MPa} \end{array}