Mathematical Problems in Engineering

Research Article

Dynamic Analysis of Partially Embedded Structures Considering Soil-Structure Interaction in Time Domain

Table 1

The Newmark algorithm and iteration procedure used in the analysis.
(A) Initial conditions
 (1) Form stiffness, mass, and damping matrices of the structure 𝐾 , 𝑀 , 𝐶
 (2) Initial values 𝑢 0 , ̇ 𝑢 0 , ̈ 𝑢 0
 (3) Select time step and parameters 𝛼 = 0 .25, 𝛿 = 0 . 5 . Calculate integration constants 𝑎 0 = 1 𝛼 Δ 𝑡 2 , 𝑎 1 = 𝛿 𝛼 Δ 𝑡 , 𝑎 2 = 1 𝛼 Δ 𝑡 , 𝑎 3 = 1 2 𝛼 𝑎 − 1 4 = 𝛿 𝛼 − 1 , 𝑎 5 = Δ 𝑡 2 𝛿 𝛼 − 2 , 𝑎 6 = Δ 𝑡 ( 1 − 𝛿 ) , 𝑎 7 = 𝛿 Δ 𝑡
 (4) Form effective stiffness matrix 𝐾 𝐾 = 𝐾 + 𝑎 0 𝑀 + 𝑎 1 𝐶
(B) For each time step
  (1) Use 𝑡 + Δ 𝑡 ̈ 𝑢 𝑔 and calculate dynamic load due to ground motion (4.2) 𝑡 + Δ 𝑡 𝐹 𝑠 𝑟
  (2) Iterative procedure
  (2.1) Consider 𝑡 + Δ 𝑡 ̈ 𝑢 𝑘 𝑏 , 𝑘 = 1 , 2 , 3 , … 𝑡 + Δ 𝑡 ̈ 𝑢 1 𝑏 = 𝑡 ̈ 𝑢 𝑏
  (2.2) Calculate the interaction load induced on soil structure interface (4.6) using 𝑡 + Δ 𝑡 ̈ 𝑢 𝑘 𝑏 𝑡 + Δ 𝑡 𝐹 𝑏 𝑟
  (2.3) Calculate the dynamic load 𝑡 + Δ 𝑡 𝑅 𝑡 + Δ 𝑡 𝑅 = 𝑡 + Δ 𝑡 𝐹 𝑟 𝑠 𝑠 𝑡 + Δ 𝑡 𝐹 𝑟 𝑠 𝑏 + 𝑡 + Δ 𝑡 𝐹 𝑏 𝑟
  (2.4) Calculate effective load 𝑅 at time 𝑡 + Δ 𝑡 𝑡 + Δ 𝑡 𝑅 = 𝑡 + Δ 𝑡 𝑅 + 𝑀 ( 𝑎 0 𝑡 𝑈 + 𝑎 2 𝑡 ̇ 𝑈 + 𝑎 3 𝑡 ̈ 𝑈 )
+ 𝐶 ( 𝑎 1 𝑡 𝑈 + 𝑎 4 𝑡 ̇ 𝑈 + 𝑎 5 𝑡 ̈ 𝑈 )
  (2.5) Applying Gauss reduction scheme, displacements are calculated at time 𝑡 + Δ 𝑡 𝐾 𝑡 + Δ 𝑡 𝑈 = 𝑡 + Δ 𝑡 𝑅
  (2.6) Calculate acceleration and velocities at time 𝑡 + Δ 𝑡 𝑡 + Δ 𝑡 ̈ 𝑈 = 𝑘 + 1 ̈ 𝑢 𝑠 𝑘 + 1 ̈ 𝑢 𝑏 𝐾 + 1 = 𝑎 0 ( 𝑡 + Δ 𝑡 𝑈 − 𝑡 𝑈 ) − 𝑎 2 𝑡 ̇ 𝑈 − 𝑎 3 𝑡 ̈ 𝑈 𝑡 + Δ 𝑡 ̇ 𝑈 = 𝑡 ̇ 𝑈 + 𝑎 6 𝑡 ̈ 𝑈 + 𝑎 7 𝑡 + 4 Δ 𝑡 ̈ 𝑈
  (2.7) Calculate the tolerance between two successive iterations T L R = ‖ 𝑡 + Δ 𝑡 ̈ 𝑢 𝑏 𝐾 + 1 − 𝑡 + Δ 𝑡 ̈ 𝑢 𝐾 𝑏 ‖
  (2.8) Check TLR. If TLR < 0.001, the iterative procedure finishes otherwise steps 2.2 to 2.8 are repeated using 𝑡 + Δ 𝑡 ̈ 𝑢 𝑏 𝑘 + 1