2.4.3.2 Modified Characteristic Steel Curve

Modified Characteristic Steel Curve

The Tension Stiffening effect can also be taken into account by means of a modified characteristic steel curve. The approximate minor tangential stiffness (abrupt change in case of re-cracking) during crack development is determined by distinguishing between crack formation and final crack state.

Uncracked – state I (0 < σ_s ≤ σ_sr)

${\epsilon }_{sm} = {\epsilon }_{s1 }$

State of first crack formation (σ_sr < σ_s ≤ 1.3σ_sr)

${\epsilon }_{sm} = {\epsilon }_{s2} - \frac{{\beta }_t \left({\sigma }_s - {\sigma }_{sr}\right) + \left(1.3 {\sigma }_{sr} - {\sigma }_s\right)}{0.3 {\sigma }_{sr}} \left({\epsilon }_{sr2} - {\epsilon }_{sr1}\right)$

State of final crack state (1.3 σ_sr < σ_s ≤ f_y)

${\epsilon }_{sm} = {\epsilon }_{s2} - {\beta }_t \left({\epsilon }_{sr2} - {\epsilon }_{sr1}\right)$

Plastic steel yielding until failure (f_y < σ_s ≤ f_t)

${\epsilon }_{sm} = {\epsilon }_{sy} - {\beta }_t \left({\epsilon }_{sr2} - {\epsilon }_{sr1}\right) + {\delta }_d \left(1 - \frac{{\sigma }_{sr}}{f_y}\right) \left({\epsilon }_{s2} - {\epsilon }_{sy}\right)$

Descriptions:

• ε_sm : mean steel strain
• ε_su : ultimate strain at failure of reinforced steel
• ε_s1 : steel strain in uncracked state
• ε_s2 : steel strain in cracked state (in crack)
• ε_sr1 : steel strain in uncracked state with crack internal forces
• ε_sr2 : steel strain in crack with crack internal forces
• β_t : factor for considering loading period or load repetitions
  • 0.40 short-term loading
  • 0.25 permanent load or frequent load changes
• σ_sr : stress in tension reinforcement, calculated based on cracked cross-section for action combination resulting in first crack formation
• σ_s : steel stress in cracked state (in crack) in [N/mm²]
• σ_d : factor for considering ductility of reinforcement
  • 0.8 highly ductile steel
  • 0.6 normally ductile steel