Real-time hybrid test of a LNG storage tank with a variable curvature friction pendulum system

Abstract

To accurately reproduce the seismic response of the liquefied natural gas (LNG) storage tank equipped with the variable curvature friction pendulum system (VCFPS), a real-time hybrid (RTH) experiment, also known as a real-time substructure experiment, is conducted on it in this study. A typical LNG storage tank with a capacity of 160,000 m³ is employed as the numerical substructure simulated using the MATLAB/Simulink, while the variable curvature friction pendulum bearing (VCFPB) is utilized as the experimental substructure tested using the compression-shear equipment. Thereafter, the validity and feasibility of the RTH experiment are verified using the SAP2000 results. Finally, the working performance of the VCFPB is evaluated scientifically, comprehensively, reasonably, and efficiently. The results show that the VCFPB is very effective in avoiding the resonance phenomenon. It can be seen from the displacement of isolation layer that the VCFPB meets the design requirement. The maximum relative deviations between the RTH test results and the SAP2000 results are 3.45% for the displacement of isolation layer, 4.27% for the base shear, and 1.49% for the liquid sloshing height, respectively. The RTH test is stable and reliable and the predicted results are highly accurate and effective. The RTH test method proves to be accurate in the prediction of the seismic response of the LNG storage tank equipped with the VCFPBs.

Appendices

Appendix 1

This appendix contains the detailed information about the RTH experiment.

The schematic of the test setup consists of shaking table (ST), electro-hydraulic servo actuator (ATR), steel frame (Fm), four lateral supports (LS-1, LS-2, LS-3, and LS-4), five connectors (CTR-1, CTR-2, CTR-3, CTR-4, and CTR-5), and five hinges (HG-1, HG-2, HG-3, HG-4, and HG-5). The lateral supports (LS-2 and LS-4) are used to make the actuator move together with the shaking table during the experiment, the same as the hinges (HG-2, HG-4, and HG-5), and connectors (CTR-2, CTR-4, and CTR-5). The lateral supports (LS-1 and LS-3) fixed to the ground floor are used to provide the relative movement of the test specimen of the VCFPB.

To maintain translational motion of the test specimen during the test, this test specimen of the VCFPB (see Fig. 13) is divided into the upper and lower identical storeys, with four reduced scale submodels arranged on each storey. That is, this test specimen has eight reduced scale submodels in total. Please note that one reduced scale model is split into four reduced scale submodels.

Fig. 13

Test specimen of the VCFPB

The displacement ratio of the reduced scale model to the prototype is 1/8 because of the geometrical similarity principle. Thus, the displacement feedback from the numerical substructure is multiplied by 1/8 to determine the input of the experimental substructure.

Based on the equal pressure on the PTFE plate, and combined with the geometrical similarity principle, the horizontal force ratio of the prototype to the reduced scale model is 8². Plus, there are 360 prototypes of the base isolation bearings in this LNG storage tank. Thereby, the force feedback from the force sensor (L1 or L2) is multiplied by (1/2 × 8² × 360) as the input of the numerical substructure.

Appendix 2

In the Euler­Gauss method, the responses during each step can be calculated from the initial conditions (displacement and velocity) at the beginning of this step and from the loading history during this step. Based on the initial conditions: the velocity $\left\{ {\dot{x}_{i} } \right\}$ at i-th time step, the velocity $\left\{ {\dot{x}_{{i + {1}}} } \right\}$ at (i + 1)-th time step can be obtained using the constant average acceleration method, and can be expressed as follows:

$$ \left\{ {\dot{x}_{{i + {1}}} } \right\} = \left\{ {\dot{x}_{i} } \right\} + \left( {\left\{ {\ddot{x}_{i} } \right\} + \left\{ {\ddot{x}_{{i + {1}}} } \right\}} \right)\frac{{\left( {\Delta t} \right)}}{{2}}, $$

(20)

where ∆t is the time interval.

By integrating Eq. (20) with the initial conditions: the displacement $\left\{ {x_{i} } \right\}$ at i-th time step, the displacement $\left\{ {x_{i + 1} } \right\}$ at (i + 1)-th time step can be got and expressed as follows:

$$ \left\{ {x_{{i + {1}}} } \right\} = \left\{ {x_{i} } \right\} + \left\{ {\dot{x}_{i} } \right\}\left( {\Delta t} \right) + \left( {\left\{ {\dot{x}_{i} } \right\} + \left\{ {\ddot{x}_{{i + {1}}} } \right\}} \right)\frac{{\left( {\Delta t} \right)^{2} }}{{4}} $$

(21)

and solving Eqs. (20) and (21) for the acceleration and velocity at (i + 1)-th time step results in:

$$ \left\{ {\ddot{x}_{{i + {1}}} } \right\} = \frac{{4}}{{\left( {\Delta t} \right)^{2} }}\left( {\left\{ {x_{{i + {1}}} } \right\} - \left\{ {x_{i} } \right\}} \right) - \frac{{4}}{\Delta t}\left\{ {\dot{x}_{i} } \right\} - \left\{ {\ddot{x}_{i} } \right\} $$

(22)

$$ \left\{ {\dot{x}_{{i + {1}}} } \right\} = \frac{{2}}{\Delta t}\left( {\left\{ {x_{{i + {1}}} } \right\} - \left\{ {x_{i} } \right\}} \right) - \left\{ {\dot{x}_{i} } \right\} $$

(23)

The back-substitutions of Eqs. (22) and (23) into Eq. (18) and simplifying give [45]:

$$ \left( {\frac{{4}}{{\left( {\Delta t} \right)^{2} }}\left[ {m_{{\text{N}}} } \right] + \frac{{2}}{{\left( {\Delta t} \right)}}\left[ {c_{{\text{N}}} } \right] + \left[ {k_{{\text{N}}} } \right]} \right)\left\{ {x_{{i + {1}}} } \right\} + \left\{ {R_{{\text{E}}} \left( {\ddot{x}_{{i + {1}}} ,\dot{x}_{{i + {1}}} ,x_{{i + {1}}} } \right)} \right\} = \left\{ {\left( {F_{{{\text{eq}}}} } \right)_{{i + {1}}} } \right\}, $$

(24)

where $\left\{ {\left( {F_{{{\text{eq}}}} } \right)_{{i + {1}}} } \right\} = \left\{ {F_{{i + {1}}} } \right\} + \left[ {m_{{\text{N}}} } \right]\left\{ {\ddot{x}_{i} } \right\} + \left( {\frac{{4}}{\Delta t}\left[ {m_{{\text{N}}} } \right] + \left[ {c_{{\text{N}}} } \right]} \right)\left\{ {\dot{x}_{i} } \right\} + \left( {\frac{{4}}{{\left( {\Delta t} \right)^{2} }}\left[ {m_{{\text{N}}} } \right] + \frac{{2}}{{\left( {\Delta t} \right)}}\left[ {c_{{\text{N}}} } \right]} \right)\left\{ {x_{i} } \right\}.$

The absolute and relative tolerances of the solver are the system default value and 0.001, respectively. The time step is variable in accordance with them.