Stress-strain curves, design values of stresses

Concrete

Actual curve: stress rises almost linearly for the first part, then curves, reaches a peak at strain about 0.002, then falls (descending branch) until crushing at strain about 0.0035. It has no clear yield point.

Idealised curve (IS 456, Fig. 21): a parabola from zero to strain 0.002, then a horizontal line (constant stress) up to the ultimate strain 0.0035 in bending. The peak stress is $0.67f_{ck}$ (to allow for the difference between cube and in-situ strength). The ultimate strain is 0.0035 in flexure and 0.002 in pure axial compression.

$$\sigma_c=0.446f_{ck}\left[2\frac{\varepsilon}{0.002}-\left(\frac{\varepsilon}{0.002}\right)^2\right]\quad(\varepsilon\le0.002),\qquad \sigma_c=0.446f_{ck}\quad(0.002\le\varepsilon\le0.0035)$$

Steel

Actual curve:

• Mild steel (Fe250): linear to a definite yield point, a yield plateau, then strain hardening.
• HYSD / TMT bars (Fe415, Fe500): no definite yield point; the yield stress is taken as the 0.2% proof stress.

Idealised curve (IS 456, Fig. 23): Hooke's law up to $0.8f_y$ (linear, $E_s=2\times10^5$ N/mm$^2$), then non-linear up to $f_y$ with inelastic strain of 0.002. For design, stresses are divided by $\gamma_{ms}=1.15$, giving $0.87f_y$. Yield strain for design: $\varepsilon_y=\dfrac{0.87f_y}{E_s}+0.002$ (0.00380 for Fe415).

 Concrete (IS 456 Fig. 21)        Steel (Fe415, Fig. 23)
 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

Design values of stresses

The 0.446 factor (= 0.67/1.5) is applied to the stress block. The equivalent rectangular block used in design has an average stress $0.36f_{ck}$ acting over depth $x_u$, with its centroid at $0.42x_u$ from the compression edge.