Algorithm 1 The pseudocode of cumulative damage

1:  Begin 2:  $n$ bridges with uncertain properties are generated 3:  Realizations of seismic occurrences associated with each bridge model are generated 4:  Time thresholds of interest, $m$, associated with corrosion are estimated 5:  Different time stages, $T$, are selected 6:  Initialize counters $i=1$, $k=1$ and $t_0=0$ 7:  while $k\le n$ 8:   while $i\le m$ 9:    $t=t_0+\mathsf{\Delta }{t}_{i+1}$ 10:   while $t\le T$ 11:      if $i=1$ 12:      The $i-\mathrm{th}$ and $\left(i+1\right)-\mathrm{th}$ intensities are associated with the $k-\mathrm{th}$ structural model 13:       Two seismic records are associated with the $i-\mathrm{th}$ and $\left(i+1\right)-\mathrm{th}$ intensities 14:       Each record is modified by a factor ${\psi }_e=i_{sim}/i_T$ that relates the intensity and the value of spectral acceleration at the fundamental period of the $k-\mathrm{th}$ system 15:       $D{i}_{corr}{}_{\left|y,t\right.}$ of the $k-\mathrm{th}$ system is calculated 16:       A random ground motion, $S_{ki}$, is modified by a factor, $\beta m$, that matches $D{i}_{corr}{}_{\left|y,t\right.}$ 17:       else 18:       A random seismic record, $r_i$, is associated with the $\left(i+2\right)-\mathrm{th}$ simulated intensity and is scaled by the factor ${\psi }_e$ 19:       The system is subjected to a seismic signal composed of the seismic record, $S_{ki}$, and the seismic record, $ri$ 20:       $D{i}_{corr}{}_{\left|y,t\right.}$ of the $k-\mathrm{th}$ system is calculated 21:       A ground motion, $S_{k\left(i+1\right)}$, is selected randomly, and it is modified by a factor, $\beta m$, that matches $D{i}_{corr}{}_{\left|y,t\right.}$ 22:       A reduction of the cross-sectional area of the reinforcement steel is performed 23:       The ground motion $S_{k\left(i+1\right)}$ at the stage $t$ is scaled up until the structure fails 24:    add one to the intensities counter 25:   add one to the simulated bridges counter 26:  end