Developing the force-displacement relationship of a controlled rocking wall

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EXPERIMENT DETAILS

The Wight study subjected 7 full-scale single storey PCM walls to a range of in-plane base excitations using the shake table at North Carolina State University. This was followed by a shake table test of a single storey PCM structure. Figure 4-3 presents photographs of an example single in-plane wall test and the test on the PCM structure. The current study focuses on the tests conducted on the wall depicted in Figure 4-3a. The geometry of the PCM wall for the current study is summarised in Figure 4-4. The PCM wall was constructed from 6” concrete masonry units with alternate masonry cells grouted. The concrete masonry units had an approximate density of 1600 kg/m3 . Compression tests on prism specimens and mortar cylinders, 36 days after grouting, revealed an average masonry crushing strength of 20.8 MPa and a mortar compression strength of 12.1 MPa.

INSTRUMENTATION AND DATA ACQUISITION

PCM wall systems are designed to endure large base excitations with damage isolated to the prestressing tendon with minimum effects to the walls themselves. Failures are defined by either 1) yielding of the prestressing tendon, 2) crushing of the compression rocking toe or 3) lateral displacement exceeding that permitted by design code or that can be accommodated by adjoining structural and non-structural components. With the failure mechanisms in mind, the transducers were set up to focus on capturing the tendon stress variation, wall behaviour near the compression toe area and the overall drift of the system. This setup is also ideally suited for analysing the dynamic behaviour of the system.

SHAKE TABLE MOTION SELECTION

The input records were selected to cover a broad range of earthquake events. The records came from historic earthquake events which are familiar to the earthquake engineering community. Care was taken to ensure the input records covered a range of seismological signatures such as moment magnitudes, peak ground acceleration, frequency content, duration and the presence of forward directivity. The amplitudes of selected records were scaled to represent a variety of shaking levels. Furthermore, the time ordinate for a number of the records were scaled down to ensure they fitted within the stroke limits of the shake table. In these instances, the walls were effectively subjected to new earthquake records.

EXPERIMENT RESULTS

As expected, the PCM wall accommodated the shake table tests with negligible damage. The post-tensioning tendon remained elastic throughout the tests despite the wall reaching a maximum displacement of 35.7 mm (1.5 % lateral drift). In all the test runs, the wall re-centered to within 0.8 mm of the in-plane alignment at the beginning of each run. The wall however did twist slightly about a vertical axis and resulted in a maximum residual out of plane displacement of 14 mm at one end of the wall when viewed in plan. After each test run, the tendon returned to approximately the same tension force.

FREE VIBRATION RESPONSE (TEST RUN 5)

The free vibration response of the PCM wall is of particular interest to the current study. This free vibration test provides an opportunity to examine the dynamic characteristics of the PCM wall system without additional external excitations. It should be kept in mind that although the response of a particular PCM wall is investigated here, the intent is to catalogue knowledge on the behaviour of controlled rocking systems in general. The free vibration test, test run 5, was initiated by applying the Sylmar ground motion record to the shake table until the time when the largest wall displacement was expected. The shake table was then stopped abruptly and the wall vibration was allowed to decay freely.

DEVELOPING THE FORCE-DISPLACEMENT RELATIONSHIP OF A CONTROLLED ROCKING WALL

Encouraged by the ability to predict the tendon and uplift behaviour of the controlled rocking wall using mostly a theoretical approach, it is hypothesised that the dynamic characteristics of the wall can also be predicted by making use of the newly developed formulas. It is anticipated that there may be intelligent modifications to be made to the established formulas for a classical free rocking system. One of the key differences between the controlled rocking system and a free rocking system is the presence of a large initial vertical force, applied by posttensioning tendons.

TABLE OF CONTENTS :

• ABSTRACT
• ACKNOWLEDGEMENT
• TABLE OF CONTENTS
• LIST OF FIGURES
• LIST OF TABLES
• NOTATIONS
• Chapter 1 INTRODUCTION
  • 1.1 Rocking Motion and Aseismic Structural Design
  • 1.2 Research Objective and Motivation
  • 1.3 Thesis Outline / Organisation
  • 1.4 Reference Frames and Co-ordinate systems
• Chapter 2 LITERATURE REVIEW
  • 2.1 The Development of Rocking Science
  • 2.2 Housner’s Simple Rocking Model
  • 2.3 Other Rocking Block Models
  • 2.4 Rocking Structural Systems
    • 2.4.1 Rigid interface solutions
    • 2.4.2 Two-spring foundation solutions
    • 2.4.3 Tensionless Winkler foundation solution
  • 2.5 Rocking Systems as Nonlinear Dynamic Systems
    • 2.5.1 Free standing block subjected to harmonic ground motion
  • 2.6 Dirac- Representation of the Rocking Impact
  • 2.7 Controlled Rocking Motion
  • 2.8 Conclusion
• Chapter 3 FREE ROCKING OBJECTS
  • 3.1 Experimental Setup
  • 3.2 Results and Analysis
    • 3.2.1 Comparing experimental results against Housner’s SRM model
    • 3.2.2 Energy content during a rocking cycle
    • 3.2.3 Support and impact forces of a rocking rigid block
    • 3.2.4 Energy dissipation due to impacts
  • 3.3 Summary of Findings
• Chapter 4 CONTROLLED ROCKING SYSTEMS
  • 4.1 Introduction
  • 4.2 Experiment Details
  • 4.3 Instrumentation and Data Acquisition
  • 4.4 Shake Table Motion Selection
  • 4.5 Experiment Results
    • 4.5.1 Free vibration response (Test run 5)
  • 4.6 Modelling the tendon extension
  • 4.7 Developing the force-displacement relationship of a controlled rocking wall
    • 4.7.1 Summary of findings from the free vibration response
  • 4.8 Predicting the Free Vibration Time-history
    • 4.8.1 Nonlinearly elastic equivalent sdof approximation
    • 4.8.2 Modified Housner – Substitute Gravity approach
    • 4.8.3 Sensitivity of the MHSG method to input parameters
  • 4.9 PCM Wall Response Subjected to Base Excitation
    • 4.9.1 Shake table tests with 75 kN initial tendon force
    • 4.9.1.1 Wall uplifts and tendon force increases
    • 4.9.1.2 Location of rotation centre
    • 4.9.1.3 Predicting the time-history response to base excitation
    • 4.9.1.4 Attempts to improve the equivalent SDOF procedure
    • 4.9.1.5 Sensitivity of controlled rocking system
    • 4.9.2 Shake table tests with 110 kN initial tendon force
    • 4.9.2.1 Predicting the time-history response subjected to base excitation
  • 4.10 Conclusion
• Chapter 5 SIMPLIFIED ROCKING STRUCTURAL SYSTEMS
  • 5.1 Introduction
  • 5.2 Experimental Validation Data
    • 5.2.1 Free rocking decays (FRD) – Solid Feet
    • 5.2.2 Free rocking decays (FRD) – Soft Rubber Pad
    • 5.2.3 Shake table tests
  • 5.3 Theoretical models of rocking structural systems
    • 5.3.1 Rigid lumped-mass structure rocking about fixed pivots on a rigid foundation
    • 5.3.1.1 Validating the analytical model
    • 5.3.2 Rigid lumped-mass rocking structure resting on two compression-only springs
    • 5.3.2.1 Validating the analytical model
    • 5.3.3 Elastic 2 DOF structure rocking about fixed pivots on a rigid foundation
      • 5.3.3.1 Validating the analytical model
• 5.4 Conclusion

• Chapter 6 CONCLUSIONS AND FUTURE WORK
  • 6.1 Behaviour of Rigid, Rocking Blocks
  • 6.2 Dynamics of Controlled Rocking Systems
  • 6.3 Simplified Rocking Systems
  • 6.4 Future Work
  • REFERENCES

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THE MECHANICS OF ROCKING STRUCTURES SUBJECTED TO GROUND MOTION