Idealized stress-strain, characteristic strength

Idealised stress-strain diagram

The actual stress-strain curves of concrete and steel are irregular. For design, they are replaced by simplified (idealised) curves given in IS 456:2000.

Concrete (Fig. 21): a parabola from zero to a strain of 0.002, then a horizontal line up to 0.0035; the peak design stress is $0.67f_{ck}/1.5=0.446f_{ck}$.

Steel:

• HYSD bars (Fig. 23B): linear up to $0.8f_y$ (slope $E_s=2\times10^5$ N/mm$^2$), then a curve up to $f_y$ with a plastic strain of 0.002; the design curve is scaled by $1/1.15$ to peak at $0.87f_y$.
• Mild steel (Fig. 23A): linear up to $f_y$, then horizontal (elastic-perfectly plastic); the design curve is capped at $0.87f_y$.

 Concrete (IS 456 Fig. 21)        Steel (Fe415, Fig. 23B)
 stress                           stress
 0.67fck |   _______________      fy   |     ______  actual
         |  /               |     0.87fy|  __/ ______ design
 (design |/                 |           |_/ /
 0.446fck)                  |           |/ /
         +------------------>          +----------------> strain
        0.002          0.0035            Es=2x10^5   0.87fy/Es+0.002

Characteristic strength

• Concrete: the compressive strength $f_{ck}$ of 150 mm cubes at 28 days, below which not more than 5% of test results are expected to fall. M20 means $f_{ck}=20$ N/mm$^2$.
• Steel: the yield stress (or 0.2% proof stress) $f_y$, below which not more than 5% of test results are expected to fall. Fe415 means $f_y=415$ N/mm$^2$.

Both are obtained from $f_k=f_m-1.65s$, where $f_m$ is the mean strength and $s$ the standard deviation.