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A mixed explicit–implicit time integration approach for nonlinear analysis of base-isolated structures

Abstract

The paper investigates the accuracy, the stability and the computational efficiency of a mixed explicit–implicit time integration approach proposed for predicting the nonlinear response of base-isolated structures subjected to earthquake excitation. Adopting the central difference method for evaluating the response of the nonlinear base isolation system and the Newmark’s constant average acceleration method for estimating the superstructure linear response, the proposed partitioned solution approach is used to analyze a 3D seismically isolated structure subjected to a bidirectional earthquake excitation. Both isolation systems adopting lead rubber and friction pendulum bearings are considered. Numerical results show that the computational time required by the proposed method, in spite of its conditional stability arising from the use of the central difference method in the explicit integration substep, is clearly reduced in comparison to the widely used implicit time integration method adopted in conjunction with the pseudo-force approach (i.e., pseudo-force method). As a matter of fact, the typical low stiffness of the isolation system leads to a critical time step larger than the one used to define the ground acceleration accurately and the proposed method preserves its computational efficiency even in the case of isolators with very high initial stiffness (i.e., friction pendulum bearings) for which the critical time step size could become smaller.

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Correspondence to Fabrizio Greco.

Appendix: solution algorithm

Appendix: solution algorithm

The proposed solution algorithm is given in the following:

  1. 1.

    Initial calculations:

    1. 1.1

      Form superstructure mass matrix \({{\varvec{M}}_s}\), damping matrix \({{\varvec{C}}_s}\) and stiffness matrix \({{\varvec{K}}_s}\) and base isolation system mass matrix \({{\varvec{m}}_b}\), damping matrix \({{\varvec{c}}_b}\) and stiffness matrix \({{\varvec{k}}_b}\).

    2. 1.2

      Initialize superstructure displacement, velocity and acceleration vectors \({{\varvec{u}}_s}\left( 0 \right)\), \({\dot {{\varvec{u}}}_s}\left( 0 \right)\), \({\ddot {{\varvec{u}}}_s}\left( 0 \right)\), and base isolation system displacement, velocity and acceleration vectors \({{\varvec{u}}_b}\left( 0 \right)\), \({\dot {{\varvec{u}}}_b}\left( 0 \right)\), \({\ddot {{\varvec{u}}}_b}\left( 0 \right)\); then calculate:

      $${{\varvec{u}}_b}\left( { - \Delta t} \right)={{\varvec{u}}_b}\left( 0 \right) - \Delta t~{\dot {{\varvec{u}}}_b}\left( 0 \right)+\frac{{{{\left( {\Delta t} \right)}^2}}}{2}{\ddot {{\varvec{u}}}_b}\left( 0 \right)~.$$
    3. 1.3

      Select time step \(\Delta t\), \(\Delta t \leqslant \Delta {t_{cr}}\), and calculate the integration constants:

      $${a_1}=\frac{2}{{{{\left( {\Delta t} \right)}^2}}}\quad {a_2}=\frac{1}{{{{\left( {\Delta t} \right)}^2}}}\quad {a_3}=\frac{1}{{2\Delta t}}\quad {a_4}=\frac{4}{{{{\left( {\Delta t} \right)}^2}}}\quad {a_5}=\frac{2}{{\Delta t}}\quad {a_6}=\frac{4}{{\Delta t}}~.$$
    4. 1.4

      Form effective mass matrix and effective stiffness matrix:

      $$\begin{aligned} {{\varvec{M}}^*}&= {a_2}~{{\varvec{m}}_b}+{a_3}~\left( {{{\varvec{c}}_b}+{{\varvec{c}}_1}} \right) \\ {{\varvec{K}}^*}&= {a_4}~{{\varvec{M}}_s}+{a_5}~{{\varvec{C}}_s}+{{\varvec{K}}_s}~. \\ \end{aligned}$$
    5. 1.5

      Triangularize \({{\varvec{M}}^*}\) and \({{\varvec{K}}^*}\):

      $$\begin{aligned} {{\varvec{M}}^*}&= {{\varvec{L}}_{\text{e}}}~{{\varvec{D}}_{\text{e}}}~{\varvec{L}}_{e}^{{\text{T}}} \\ {{\varvec{K}}^*}&= {{\varvec{L}}_{{\text{i}}}}~{{\varvec{D}}_{{\text{i}}}}~{\varvec{L}}_{{i}}^{{\text{T}}}~. \\ \end{aligned}$$
  1. 2.

    Calculations for each time step:

    1. 2.1

      Compute the state of motion at each seismic isolation bearing at time \(t\).

    2. 2.2

      Compute the resultant nonlinear forces vector \({{\varvec{f}}_n}\left( t \right)\) at the center of mass of the base isolation system.

    3. 2.3

      Calculate the explicit integration substep effective load vector at time \(t\):

      $$\begin{aligned} {\varvec{P}}_{e}^{*}\left( t \right)= & - {{\varvec{m}}_b}{{\varvec{R}}_b}{{\ddot {{\varvec{u}}}}_g}\left( t \right) - {{\varvec{c}}^T}{{\dot {{\varvec{u}}}}_s}\left( t \right) - {{\varvec{k}}^T}{{\varvec{u}}_s}\left( t \right) - {{\varvec{f}}_n}\left( t \right) \\ & ~+\left[ {{a_1}{{\varvec{m}}_b} - {{\varvec{k}}_b} - {{\varvec{k}}_1}} \right]{{\varvec{u}}_b}\left( t \right)+\left[ { - {a_2}{{\varvec{m}}_b}+{a_3}\left( {{{\varvec{c}}_b}+{{\varvec{c}}_1}} \right)} \right]{{\varvec{u}}_b}\left( {t - \Delta t} \right)~~. \\ \end{aligned}$$
    4. 2.4

      Solve for base isolation system displacement vector at time \(t+\Delta t\):

      $${{\varvec{L}}_e}{{\varvec{D}}_e}{\varvec{L}}_{e}^{T}~{{\varvec{u}}_b}\left( {t+\Delta t} \right)={\varvec{P}}_{e}^{*}\left( t \right)~.$$
    5. 2.5

      Evaluate base isolation system velocity and acceleration vectors at time \(t\):

      $$\begin{aligned} {{\dot {{\varvec{u}}}}_b}\left( t \right)= & {a_3}~\left[ {{{\varvec{u}}_b}\left( {t+\Delta t} \right) - {{\varvec{u}}_b}\left( {t - \Delta t} \right)} \right]~ \\ {{\ddot {{\varvec{u}}}}_b}\left( t \right)= & {a_2}~\left[ {{{\varvec{u}}_b}\left( {t+\Delta t} \right) - 2{{\varvec{u}}_b}\left( t \right)+{{\varvec{u}}_b}\left( {t - \Delta t} \right)} \right]~. \\ \end{aligned}$$
    6. 2.6

      Calculate the implicit integration substep effective load vector at time \(t+\Delta t\):

      $$\begin{aligned} {\varvec{P}}_{i}^{*}\left( {t+\Delta t} \right)= & - {{\varvec{M}}_s}{{\varvec{R}}_s}{{\ddot {{\varvec{u}}}}_g}\left( {t+\Delta t} \right) - {a_3}~{\varvec{c}}~\left[ { - 4~{{\varvec{u}}_b}\left( t \right)+3{{\varvec{u}}_b}\left( {t+\Delta t} \right)+{{\varvec{u}}_b}\left( {t - \Delta t} \right)} \right] - {\varvec{k}}~{{\varvec{u}}_b}\left( {t+\Delta t} \right) \\ & +\left[ {{a_4}{{\varvec{M}}_s}+{a_5}{{\varvec{C}}_s}} \right]{{\varvec{u}}_s}\left( t \right)+\left[ {{a_6}{{\varvec{M}}_s}+{{\varvec{C}}_s}} \right]{{\dot {{\varvec{u}}}}_s}\left( t \right)+{{\varvec{M}}_s}{{\ddot {{\varvec{u}}}}_s}\left( t \right)~. \\ \end{aligned}$$
    7. 2.7

      Solve for superstructure displacement vector at time \(t+\Delta t\):

      $${{\varvec{L}}_i}{{\varvec{D}}_i}{\varvec{L}}_{{~i}}^{T}~{{\varvec{u}}_s}\left( {t+\Delta t} \right)={\varvec{P}}_{i}^{*}\left( {t+\Delta t} \right)~.$$
    8. 2.8

      Evaluate superstructure velocity and acceleration vectors at time \(t+\Delta t\):

      $$\begin{aligned} {{\dot {{\varvec{u}}}}_s}\left( {t+\Delta t} \right)&= {a_5}\left[ {{{\varvec{u}}_s}\left( {t+\Delta t} \right) - {{\varvec{u}}_s}\left( t \right)} \right] - {{\dot {{\varvec{u}}}}_s}\left( t \right) \\ {{\ddot {{\varvec{u}}}}_s}\left( {t+\Delta t} \right)&= {a_4}\left[ {{{\varvec{u}}_s}\left( {t+\Delta t} \right) - {{\varvec{u}}_s}\left( t \right)} \right] - {a_6}~{{\dot {{\varvec{u}}}}_s}\left( t \right) - {{\ddot {{\varvec{u}}}}_s}\left( t \right)~. \\ \end{aligned}$$
  1. 3.

    Repetition for next time step: replace \(t\) by \(t+\Delta t\) and repeat steps 2.1–2.8 for the next time step.

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Greco, F., Luciano, R., Serino, G. et al. A mixed explicit–implicit time integration approach for nonlinear analysis of base-isolated structures. Ann. Solid Struct. Mech. 10, 17–29 (2018). https://doi.org/10.1007/s12356-017-0051-z

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  • DOI: https://doi.org/10.1007/s12356-017-0051-z

Keywords

  • Base-isolated structures
  • Mixed time integration
  • Nonlinear dynamic analysis
  • Earthquake engineering