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Real-time substructure testing (RST) algorithm is a newly developed integration method for real-time hybrid simulation (RTHS) which has structure-dependent and explicit formulations for both displacement and velocity. The most favourable characteristics of the RST algorithm is unconditionally stable for linear and no iterations are needed. In order to fully evaluate the performance of the RST method in solving dynamic problems for nonlinear systems, stability, numerical dispersion, energy dissipation, and overshooting properties are discussed. Stability analysis shows that the RST method is only conditionally stable when applied to nonlinear systems. The upper stability limit increases for stiffness-softening systems with an increasing value of the instantaneous degree of nonlinearity while decreases for stiffness-hardening systems when the instantaneous degree of nonlinearity becomes larger. Meanwhile, the initial damping ratio of the system has a negative impact on the upper stability limit especially for instantaneous stiffness softening systems, and a larger value of the damping ratio will significantly decrease the upper stability limit of the RST method. It is shown in the accuracy analysis that the RST method has relatively smaller period errors and numerical damping ratios for nonlinear systems when compared with other two well-developed algorithms. Three simplified engineering cases are presented to investigate the dynamic performance of the RST method, and the numerical results indicate that this method has a more desirable accuracy than other methods in solving dynamic problems for both linear and nonliner systems.

Real-time hybrid simulation (RTHS) is an advanced technique for assessing the seismic behaviours of structures, especially large-scale and ultralimit complex systems [

In RTHS, integration algorithms are effective methods to solve discrete equations of motion for structural dynamics [

More and more researchers focus on developing new types of integration algorithms, which can achieve both explicit formulation and unconditional stability at the same time. Chang [

Though the performance of the RST method in solving dynamic problems for linear systems has been well studied, its application to nonlinear systems need to be further discussed and evaluated. In this work, the numerical properties of the RST method in RTHS are discussed for nonlinear structural dynamic problems and compared with other two structure-dependent integration algorithms (i.e., CEM and CRM) since they have quite similar formulations and numerical properties. Firstly, the stability and the upper stability limit of the RST method for nonlinear systems with softening and hardening stiffness are discussed using the spectra radius method. Then, the numerical dispersion and energy dissipation characteristics caused by the numerical damping are evaluated. In addition, the overshooting behaviour [

The differential equation of motion for a SDOF system under external excitations can be expressed in time domain as [

The general expressions of the displacement and velocity increments for the RST, CEM, and CRM methods is found to be_{1}, _{2}, _{3}, and _{4} for the three methods are_{N} _{E}, _{N} = _{N}/2_{n}, _{E} = _{E}/2_{n}, _{n} is the natural frequency of the system determined from the initial system.

Obviously, the coefficients in each method are not all constant, and some of them depend on the initial structural properties (_{n}) and time step ∆_{N} + _{E}) and (_{N} + _{E}), respectively. The velocity of the CEM method at (

For a MDOF, the coupled differential equation of motion should be written in a matrix form as follows:

For simplicity, _{j} is expressed as follows:

For a MDOF nonlinear system, equations (_{0} is the damping matrix generally determined from the initial structural properties; and _{0} is the initial stiffness matrix.

The unconditional stability of the RST method for linear elastic systems has been verified by authors [

In order to realistically reflect the change of stiffness during the test, a parameter, defined as the ratios of the stiffness at the end of each time step _{i} over the initial stiffness _{0}, is introduced to monitor this change. That is,_{i} _{+} _{1} is the instantaneous degree of nonlinearity. When _{i} _{+} _{1} = 1, it means that the system is linear, and no stiffness change occurs during the test. When _{i} _{+} _{1}>1, it represents that the instantaneous stiffness becomes harder at the end of the (_{i} _{+} _{1} < 1 denotes instantaneous stiffness soften.

From equations (_{i+1} = [_{i+1}],

Based on the algorithm stability analysis theory, a stable computation of an integration algorithm can be obtained when the spectral radius _{1}, _{2}, and _{3} are three coefficients indicating half of the trace, sum of principal minors, and determinant of

After substituting equations (

Variations of upper stability limits with

Two principal eigenvalues at the (

In general, numerical dispersion and energy dissipation characteristics are two indexes evaluating the accuracy of an integration algorithm. The former characteristic is usually expressed by the relative period error

The variations of PE are shown in Figure

Variations of relative period errors with (

Variations of relative period errors with (

The variations of _{1} and _{2} derived from equations (_{2} equals to 1 for all the three methods, which leads to _{∞} = 1 and the variations of

Variations of numerical damping ratios with (

In general, two primary factors are taken into account for choosing an appropriate time step Δ_{h} that is noteworthy in the dynamic analysis, and its value depends both on the higher modal frequency of the system and the effective high frequency of the dynamic load [

According to the basic theory of structural dynamics, reliable solutions can be achieved when Δ_{h}) [_{0} = 10^{2}, 10^{4}, 10^{6}) and different values of Δ

In this section, three numerical examples are applied to examine the numerical properties of the RST method when compared with the CRM and CEM methods for both linear elastic and nonlinear systems.

An undamped 4-story building

A linear 4-story shear building with no structural damping is shown in Figure _{1} = 10^{8} kg, _{2} = 10^{5} kg, _{3} = 10^{5} kg, _{4} = 10^{3} kg, and ^{7} N/m. The natural frequencies of the system are 5.9940, 116.95, 306.59, and 1906.9 rad/s for the four modes, respectively; the normalized modal matrix is given by Φ = [_{1}_{2}_{3}_{4}], where _{1} = [0.9970 0.9990 1.0000 1.0000]^{T}, _{2} = [−0.0002 0.6293 1.0000 1.0000]^{T}, _{3} = [0.0006–1.5882 0.9412 1.0000]^{T}, and _{4} = [−0.0000 0.0001–0.0101 1.0000]^{T}. It is clear that the system is intentionally designed to have a relatively high frequency for the forth mode. The first two stories are referred as the experimental substructures while the upper two stories are taken as the numerical substructures.

The free vibration responses of the system subjected to zero initial velocity and displacement _{0} = (_{1} + _{2} + 0.5 _{4}) are calculated. The time step of Δ_{4} = 3.036. The displacement response obtained from the undamped displacement equation based on the structural dynamics theory [

It is shown in Figure

Numerical modelling of a 4−story shear building.

Free vibration responses for the 4th story of the system. (a) Displacement. (b) Velocity. (c) Acceleration.

A 3-story building with a vibration isolator

A linear 3-story shear building with a vibration isolator is shown in Figure ^{3} kg, _{N} = 360 × 10^{7} N/m, _{E} = 9 × 10^{6} N/m, _{N} = 0, and _{E} = 115 × 10^{5} N s/m. The natural frequencies of the whole system are 3.7459, 114.10, 212.17, and 277.17 rad/s for the four modes, respectively; the normalized modal matrix is given by Φ = [_{1}_{2}_{3}_{4}], where _{1} = [0.9231 0.9231 1.0000 1.0000]^{T}, _{2} = [−1.0000–0.5225 0.4245 1.0000]^{T}, _{3} = [1.0833–1.0000 −1.0833 1.0000]^{T}, and _{4} = [−1.0000 2.3657–2.3657 1.0000]^{T}.

The seismic performance of the system is studied by exciting the ground acceleration record of EL Centro (1940 NS) with peak ground acceleration (PGA) scaled to 0.8 g. The system is subjected to zero initial velocity and initial displacement. The time step of Δ_{4} = 0.883. The responses obtained from the Newmark method with constant average acceleration (N4 algorithm) [

Numerical modelling of a 3-story shear building with a vibration isolator.

Seismic responses for the 3rd story of the system. (a) Displacement. (b) Velocity. (c) Acceleration.

A 5-story building with softening stiffness

A 5-story shear building (as shown in Figure ^{5} kg, _{0} = 10^{8} N/m, and _{i} is the interstory drift for the _{1}_{2}_{3}_{4}_{5}], where _{1} = [0.2828 0.5263 0.7368 0.8947 1.0000]^{T}, _{2} = [−0.8235–1.1176 −0.5882 0.3160 1.0000]^{T}, _{3} = [1.3571 0.3837–1.2143 −0.7143 1.0000]^{T}, _{4} = [1.7000–1.4000 −0.5372 1.9000–1.0000]^{T}, and _{5} = [−1.8614 3.1643–3.5366 2.6059–1.0000]^{T}. Assuming Rayleigh damping [

The seismic responses of the system with zero initial conditions is studied by exciting the ground acceleration record of EL Centro (1940 NS) with peak ground acceleration (PGA) scaled to 0.9 g. The results obtained from N4 algorithm with a time step Δ_{5} = 0.1933. Seismic responses for the 5th story and the shear force-displacement curves for the 1st story are plotted in Figures

Numerical modelling of a 5-story shear building with softening stiffness.

Seismic responses for the 5th story of the system. (a) Displacement. (b) Velocity. (c) acceleration.

Hysteretic loops for the 1st story.

Applications of the RST method to nonlinear systems in RTHS have been evaluated and compared with other two structure-dependent explicit integration algorithms, i.e., the CRM and CEM methods, since they have quite similar formulations and numerical properties. Stability analysis indicates that the RST method is only conditionally stable when applied to nonlinear systems, and the instantaneous degree of nonlinearity will exert great influence on the upper stability limit of the method. For stiffness-softening systems, the upper stability limit increases with an increasing value of the instantaneous degree of nonlinearity, while for stiffness-hardening systems, it decreases when the instantaneous degree of nonlinearity becomes larger. Meanwhile, the initial damping ratio of the system has a negative impact on the upper stability limit especially for instantaneous stiffness softening systems, and a larger value of the damping ratio will significantly decrease the upper stability limit of the RST method.

Three simplified engineering structural systems are presented as numerical examples to demonstrate the computational stability, accuracy, and overshooting behaviours of the three methods. Numerical results demonstrate that overshooting behaviours will not appear if the time step is reasonably chosen for integration. It is also illustrated that the RST and CEM method have a better accuracy than the CRM method in solving dynamic problems for both linear and nonlinear systems. However, as the velocity of the CEM method is in an implicit form, the RST method provides more benefits in computational efficiency as both the displacement and velocity are in an explicit form.

The data used to support the findings of this study are included within the article. The results of this paper can be verified according to the data and methods given in the paper.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This paper was based upon work supported by the National Natural Foundation of China (Grant nos. 41904095, 51979027, and 51908048), Natural Science Foundation of Hebei Province (Grant no. E2019210350), Natural Science Foundation of Shaanxi Province (Grant nos. 2019JQ-021), State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures at Shijiazhuang Tiedao University (Grant nos. ZZ2020-04), and Fundamental Research Funds for the Central Universities (Grant nos. DUT19JC23 and DUT19RC(4)020).