Structural Reliability Sensitivities under Nonstationary Random Vibrations

Algorithm 1

data input 𝐌,𝐂,𝐊,𝑆₀,𝜔_𝑓,𝜉_𝑓,𝜑(𝑡)

𝐏_𝐵=(1/2)(𝐈−Δ𝑡𝐀)

𝐏_𝐹=(1/2)(𝐈+Δ𝑡𝐀)

𝐑(1)=𝟎; (definition of initial condition for 𝑡=0 both on 𝐑 and on (𝐑_,𝑏)

for 𝑗=1 to 𝑛_𝑏

 𝐒(𝑗,1)=0;

end; (of initial condition definition)

for 𝑖=1 to ( 𝑛_𝑡−1) (beginning of time integration)

  𝐍(𝑖)=(Δ𝑡/2)(𝐁(𝑖+1)+𝐁(𝑖))

  𝐂(𝑖)=𝐏_𝐹𝐑(𝑖)+𝐑(𝑖)𝐏^𝑇_𝐹+𝐍(𝑖)

  𝐑(𝑖+1)= lyap (𝐏_𝐵,−𝐂(𝑖)) (matlab command for stationary Lyapunov eq.)

  𝐑(𝑖+1)𝐏_𝐵+𝐏_𝐵𝐑^𝑇(𝑖+1)−𝐂(𝑖)=0

  for 𝑗=1 to 𝑛_𝑏 (cycle for each design vector element)

   𝐌(𝑖,𝑗)=(Δ𝑡/2)𝐀_𝑏(𝑗)(𝐑(𝑖)+𝐑(𝑖+1))+(𝐑(𝑖)+𝐑(𝑖+1))𝐀^𝑇_𝑏(𝑗)

   𝐃(𝑖,𝑗)=𝐏_𝐹𝐒(𝑖,𝑗)+𝐒(𝑖,𝑗)𝐏^𝑇_𝐹+𝐌(𝑖,𝑗)

   𝐒(𝑖+1,𝑗)= lyap (𝐏_𝐵,𝐃(𝑖,𝑗))

  end (of cycle on design vector elements)

end (of time integration)