Alternative stress blocks

General stress blocks

Parabola-rectangles are commonly uses for concrete stress-strain curves. The parabolic curve can be characterised as

$$\frac{f}{f_c}=a\left(\frac{\varepsilon}{\varepsilon_p}\right)^2+b\left(\frac{\varepsilon}{\varepsilon_p}\right)$$

Define

$$f^{\prime}=\frac{f}{f_c}$$

and

$$\eta=\frac{\varepsilon}{\varepsilon_p}$$

If the curve is taken to be tangent to the plateau then at $n=1, f^{\prime}=1$ and $\frac{d f^{\prime}}{d \eta}=0$.

Solving for the coefficients gives $a=-1$ and $b=2$ so

$$f^{\prime}=2 \eta-\eta^2$$

The area under the curve is given by

$$A_p=\int_0^1 f^{\prime} d \eta=\left[\eta^2-\frac{\eta^3}{3}\right]_0^1=\frac{2}{3}$$

For bi-linear curve with the strain transition at $\varepsilon_b$ the area under the curve to $\varepsilon_p$ is

$$A_b=\frac{\eta_b}{2}+\left(1-\eta_b\right)=1-\frac{\eta_b}{2}$$

Equating the areas

$$1-\frac{\eta_b}{2}=\frac{2}{3}$$

or

$$\eta_b=\frac{2}{3}$$

So

$$\varepsilon_{c, b}=\frac{2}{3} \varepsilon_{c, p}$$

For a rectangular stress block with the strain transition at $\varepsilon_r$ the area under the curve to $\varepsilon_p$ is

$$A_b=1-\eta_r$$

Equating the areas

$$1-\eta_r=\frac{2}{3}$$

or

$$\eta_r=\frac{1}{3}$$

so

$$\varepsilon_{c J}=\frac{1}{3} \varepsilon_{c, p}$$