Table 6 Summary of previous models for strength–maturity relationship.

Kim et al. (2002a)

\( \frac{S}{{S_{u} }} = \left\{ {1 - \frac{1}{{\sqrt {1 + \sum\limits_{i = 1}^{n} {A\left[ {e^{{ - \frac{{E_{a} (i)}}{{R\left[ {T_{c} (i) + 273} \right]}}e^{{ - \alpha t_{i} }} }} +\, e^{{ - \frac{{E_{a} (i)}}{{R\left[ {T_{c} (i) + 273} \right]}}e^{{ - \alpha t_{i - 1} }} }} } \right](t_{i} - t_{i - 1} } )} }}} \right\} \)

where \( A = 1 \times 10^{7} \) (experimental constant); \( E_{a} (i) = 42830 - 43T_{c} (i) \) (in J/mol);

\( \alpha = 0.00017T_{c} (i) \); and \( t_{o} = 0.66 - 0.011T_{c} (i) \ge 0 \)

Pinto and Schindler (2010)

\( \frac{S}{{S_{u} }} = \frac{{k_{r} (t_{e}^{*} - t_{sr} )}}{{1 + k_{r} (t_{e}^{*} - t_{sr} )}} \)

where \( t_{sr} = t_{s} \exp \left[ { - \frac{{E_{s} }}{R}\left( {\frac{1}{{T_{c} (i) + 273}} - \frac{1}{{T_{r} + 273}}} \right)} \right] \); and

\( t_{e}^{*} = \sum\limits_{0}^{{t_{sr} }} {\exp \left[ { - \frac{{E_{s} }}{R}\left( {\frac{1}{{T_{c} (i) + 273}} - \frac{1}{{T_{r} + 273}}} \right)} \right]} \Updelta t_{i} + \sum\limits_{0}^{t} {\exp \left[ { - \frac{{E_{i} }}{R}\left( {\frac{1}{{T_{c} (i) + 273}} - \frac{1}{{T_{r} + 273}}} \right)} \right]} \Updelta t_{i} \)

Yang (2014)

\( \frac{S}{{S_{28} }} = \beta_{1} \frac{{k_{r} (t_{e} - t_{or} )}}{{1 + k_{r} (t_{e} - t_{or} )}} \)

where \( \beta_{1} = \frac{{S_{u} }}{{S_{28} }} = 1 + \frac{1}{{k_{r} \cdot 28}} \); \( k_{T} = k_{r} \exp \left[ { - \frac{{E_{a} (i)}}{R}\left( {\frac{1}{{T_{c} (i) + 273}} - \frac{1}{{T_{r} + 273}}} \right)} \right] \);

\( t_{or} = t_{sr} \exp \left[ { - \frac{{E_{s} }}{R}\left( {\frac{1}{{T_{c} (i) + 273}} - \frac{1}{{T_{r} + 273}}} \right)} \right] \);

\( t_{e} = \sum\limits_{0}^{{t_{sr} }} {\exp \left[ { - \frac{{E_{s} }}{R}\left( {\frac{1}{{T_{c} (i) + 273}} - \frac{1}{{T_{r} + 273}}} \right)} \right]} \Updelta t_{i} \)

\( + \sum\limits_{{t_{sr} }}^{3} {\exp \left[ { - \frac{{E_{a} (i)}}{R}\left( {\frac{1}{{T_{c} (i) + 273}} - \frac{1}{{T_{r} + 273}}} \right)} \right]} \Updelta t_{i} + \sum\limits_{3}^{t} {\exp \left[ { - \frac{{E_{a} (i)}}{R}\left( {\frac{1}{{T_{A3} + 273}} - \frac{1}{{T_{r} + 273}}} \right)} \right]} \Updelta t_{i} \) \( S_{28} = \left[ {\left( {\frac{{T_{A3} }}{{T_{r} }}} \right)^{2} (w/cm)^{4} + 0.97} \right](S_{28} )_{{T_{r} }} \); and

\( E_{a} (i) = E_{i} \cdot \exp ( - 0.00017T_{c} (i) \cdot t) \)