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## Literature Review

Introduction

Structural analysis software has improved significantly over time. It is now possible to accurately simulate the seismic response of many specialist features in very high details. Yet, physical experiments remain important to help understanding the cause of structural failure. Experimental testing is crucial for verifying design assumptions and rules adopted in design codes and structural analysis software. Prior to the invention of the hybrid simulation method, the shake table and the quasi-static methods are the two most commonly used experimental techniques to simulate the seismic responses of structures.

The shake table testing method

A shake table test is often regarded as the most accurate seismic simulation method. In these tests, a specimen is placed on top of a platform that is accurately controlled to move in the same manner as the moving ground during an earthquake. Most shake tables for structural engineering purpose are powered by hydraulic actuators. For illustration, Figure 2-1 shows the shake table in the University of Auckland Structural Testing Laboratory (STL). Accurate control of the shake table accounting for stiffness of the table, table-specimen interaction and oil column characteristics are important to accurately replicate the ground displacement, velocity and acceleration. This is important to ensure the correct velocity and inertia dependent effects are replicated.

### The quasi-static testing method

The name of the method suggests that the load is applied slowly to the structure. Depending on the loading protocols, the quasi-static testing method has three main sub-categories:

Monotonic testing to failure, where the structure is loaded in one direction only. The load to the structure is gradually increased up to a certain magnitude or failure of the structure. Definition of structural failure varies among practices for example when the lateral load carrying capacity drops by more than 20% from the maximum load sustained during the test [3].

Cyclic testing to a predefined loading cycle that neither simulates specific displacement history nor the strain rate of an earthquake. Instead the structure is subjected to a number of loading cycles up to a predetermined ductility levels [3]. Researchers worldwide have used a range of loading patterns, as well as various definitions of yield and ultimate deformations, making comparison of results from different investigations difficult. Figure 2-2 shows a cyclic loading pattern which has been used in USA and Japan [3]. In the pattern, the yield displacement (i.e. *μ* = ±1) is taken as the displacement at first yield of the outer longitudinal reinforcing bars. Ductility level is increased one step at a time and the number of symmetrical loading cycles per ductility level varies between 2 in the US and 10 in Japan.

Cyclic testing to a predefined loading history, where the loading protocol is generated from a time history analysis. This type of test is more suitable for validations of the the numerical model in the time history itself.

Numerical errors

In a hybrid simulation, errors are inevitably introduced during numerical and experimental processes. The reliability of the result is therefore dependant to the magnitude and propagation characteristics of these errors. Numerical errors are the first source of inaccuracy in a hybrid simulation, arising from computation and modelling processes. For example, discrete parameters idealisation is only an approximate representation of a continuous specimen. Shing and Mahin [8] investigated the effect of structural idealisations in a hybrid simulation environment and concluded that the method is more suitable for structures whose significant portion of their masses are concentrated at limited number of degree-of-freedom (DOF). Much like a computer simulation, the method is less reliable if the mass is distributed evenly in the prototype structure or the discretisation assumption is not valid. Another source of numerical error is the numerical integration method used to solve the equation of motion. The selected parameters for the integration as well as the time step size must be able to accurately capture the dynamic response of the structure. However, it should be noted that the numerical errors are inherent in any numerical analysis, and arguably more important in a hybrid simulation setting. Well recognised strategies exist to minimise these through appropriate modelling assumptions and analysis parameters. There are also experimental errors, where previous research has shown that these can have significant effects on the accuracy of hybrid simulation results. Based on observations from their propagation characteristics, Shing and Mahin [1] classified experimental errors into random and systematic errors. Sections 2.4 and 2.5 present numerically simulated hybrid simulations of a single-degree-of-freedom (SDOF) structure under free vibration to demonstrate the effect of these errors. The SDOF structure has a natural period *T _{n}* of 0.16 s and an equivalent viscous damping

*ζ*of 5%. The structure is subjected to an impulse excitation of 1 g. The resulting displacement amplitude of the structure is close to 10 mm.

Systematic experimental errors

Systematic errors in a hybrid simulation are those that have recurrence patterns and significantly affect the test result [1]. The most cases of systematic errors arise from position errors, i.e. the difference between the actual position of the structure and the value specified in the numerical model. Position errors may result from miscalibration of measurement transducers, digitisation error in signal conversions, poor actuator tuning, and actuator support deformation.

Transducer calibration error

Specimen displacements and forces are monitored using displacement transducer and load cell in a hybrid simulation, respectively. Converting the electrical voltage into physical measurement of force or displacement requires calibration factors. Although there are different possible combinations between miscalibrations in the force and displacement transducers, the following example simulates the case when both transducers are miscalibrated to a 5% increase in each calibration factor from the correct value. The result is presented in Figure 2-4 including the ideal solution of the simulation without simulated miscalibrations. The specific examples of the miscalibrations result in a higher apparent stiffness of the structure to the hybrid simulation coordinator as indicated in Figure 2-4a. The effect is also illustrated in the displacement time history in Figure 2-4b that shows faster amplitude decay in the simulation with miscalibrations.

#### Actuator displacement calibration error

Another opportunity for miscalibration arises when converting the computed displacement from the integration algorithm to the actuator command displacement. The actual position of the structure can either exceed (overshoot) or fall behind (undershoot) the desired displacement. The simulation results including the ideal response and the case where the calibration factor for the actuator displacement transducer is 10% above the true value. The simulated miscalibration causes the actual position of the structure to constantly overshoot the desired value, resulting in additional energy being dissipated from the structure.

Friction error

Frictions occur in various locations during a hybrid simulation, for example at the connections between the test equipment and the specimen. These frictions are considered contaminations to the load cell readings since they do not exist in real seismic event. The result of simulations considering constant friction force representing the total friction during each cycle of a hybrid simulation as well as an ideal simulation result. The friction force magnitude is 2% of the maximum force developed in an ideal simulation. Figure 2-6a indicates hysteretic behaviour from an otherwise linear elastic structure as a result of this friction which dissipates energy from the structure

1. Introduction

1.1 General

1.2 Research needs

1.3 Research objectives

1.4 Organisation of thesis

2. Literature Review

2.1 Introduction

2.2 The hybrid simulation method

2.3 Numerical errors

2.4 Systematic experimental errors

2.5 Random experimental errors

2.6 Implementation of the hybrid simulation method

2.7 Numerical integration method

2.8 Hybrid simulation with substructuring

2.9 Notable development in the hybrid simulation method

3. Delay Compensation for Fast hybrid simulation

3.1 Introduction

3.2 Prior research

3.3 Systematic displacement control errors and their effect on energy content

3.4 Computation of energy error

3.5 Numerical verification

3.6 Experimental validation

3.7 Stability

3.8 Summary

4. Improving hybrid simulation through Kalman filter

4.1 Introduction

4.2 Kalman filter

4.3 Influence of DOF coupling

4.4 Increasing simulation stability through Kalman filter

4.5 Effectiveness of Kalman filter algorithm for inelastic systems

4.6 Summary

5. Multi-axial Actuator Control

5.1 Introduction

5.2 Prior research

5.3 Quasi-static tests on RC walls

5.4 Hybrid simulations on RC column

5.5 Study on error accumulation

5.6 Summary

6. Displacement paths effect in hybrid simulations considering nonlinear responses

6.1 Introduction

6.2 Test setup and specimen

6.3 Displacement tracking strategies

6.4 Loading regime

6.5 Experiment results and discussions

6.6 Summary

7. Hybrid testing on concrete wall

7.1 Introduction

7.2 Methodology

7.3 Test setup

7.4 Instrumentations

7.5 Implementation problems during tests

7.6 Results and discussions on global responses

7.7 Results and discussions on local responses

7.8 Numerical validations

7.9 Summary

8. Conclusions

8.1 Future work

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