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UNIVERSITY OF CALIFORNIA, IRVINE
Finite Element Modeling and Stress Analysis of Underground Rock Caverns
DISSERTATION
submitted in partial satisfaction of the requirements for the degree of
DOCTOR OF PHILOSOPHY
in Civil Engineering
by
Amber Jasmine Greer
Dissertation Committee: Professor Maria Feng, Chair
Professor Lizhi Sun Professor Mark Bachman
2012
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© 2012 Amber Jasmine Greer
ii
DEDICATION
To
my Lord and Savior, Jesus Christ, my parents, my best friend, my incredible friends and the remarkable girls I have been able to teach and see grow over the past two years
iii
TABLE OF CONTENTS
LIST OF FIGURES .................................................................................................................... VI
LIST OF TABLES ....................................................................................................................... X
ACKNOWLEDGEMENTS ....................................................................................................... XI
CURRICULAUM VITAE ........................................................................................................ XII
ABSTRACT OF THE DISSERTATION............................................................................... XIII
CHAPTER 1 INTRODUCTION............................................................................................. 1
1.1 Introduction to the Longyou Grottoes................................................................................. 2 1.1.1 Introduction to Cavern 2 ................................................................................................. 5
1.2 Scope of Dissertation .......................................................................................................... 9
CHAPTER 2 ON-SITE INVESTIGATIONS ...................................................................... 11
2.1 Purpose of Performing On-Site Investigations ................................................................. 11
2.2 Introduction to System Identification ............................................................................... 12
2.3 On-Site Investigations ....................................................................................................... 15 2.3.1 First On-Site Investigation ............................................................................................ 16 2.3.2 Second On-Site Investigation ........................................................................................ 20
2.4 Comparison of Results ...................................................................................................... 28
CHAPTER 3 JUSTIFICATION FOR A SIMPLIFIED MODEL OF CAVERN 2 ......... 30
3.1 Finite Element Modeling in Underground Rock Caverns ................................................ 30
3.2 Development of Four-Part Criterion for Model Simplification ........................................ 33 3.2.1 Element Sizing and Mesh Generation ........................................................................... 33 3.2.2 Stress Levels under Static Loading ............................................................................... 34 3.2.3 Dynamic Properties ...................................................................................................... 35 3.2.4 Sensitivity Analysis of Material Properties to Dynamic Characteristics ..................... 36
3.3 Implementation of Four-Part Criterion into Cavern 2 ...................................................... 37 3.3.1 Model Generation ......................................................................................................... 37 3.3.2 Implementation of Four-Part Criterion ........................................................................ 39
3.3.2.1 Selection of Appropriate Model ............................................................................ 47
iv
3.4 Calibration of Simplified Model ....................................................................................... 48 3.4.1 Determination of Boundary Conditions ........................................................................ 48 3.4.2 Determination of Water Saturation Levels and Material Properties ........................... 49
3.5 Conclusions ....................................................................................................................... 54
CHAPTER 4 COMPARATIVE ANALYSIS OF TWO DEVELOPED LOCAL MODELS WITH THE CALIBRATED GLOBAL MODEL ................................................. 56
4.1 Introduction ....................................................................................................................... 56
4.2 Optimization Techniques .................................................................................................. 58 4.2.1 Subproblem Approximation Method ............................................................................. 59 4.2.2 First-Order Optimization .............................................................................................. 61
4.3 Development of the Two Local Models ........................................................................... 63 4.3.1 Derivation of Updating Parameters ............................................................................. 63 4.3.2 Application of Optimization Algorithms to Determine Structural Parameters for the Two Local Models ..................................................................................................................... 66
4.4 Comparison of Global and Local Models ......................................................................... 74 4.4.1 Assumptions for Comparative Analysis ........................................................................ 74 4.4.2 Results from Comparative Analysis .............................................................................. 77
4.5 Conclusions ....................................................................................................................... 85
CHAPTER 5 STATE OF STRESS ANALYSIS ON CAVERN 2 ...................................... 87
5.1 Introduction ....................................................................................................................... 87
5.2 Discussion of possible failure criteria ............................................................................... 90 5.2.1 Mohr-Coulomb Failure Criterion ................................................................................. 90 5.2.2 Drucker-Prager Failure Criterion ................................................................................ 93 5.2.3 Hoek-Brown Failure Criterion ..................................................................................... 95
5.3 Determination of Applicable Failure Criterion and Material Model ................................ 98
5.4 Application of Hoek-Brown failure criterion on Chosen FE Model .............................. 115 5.4.1 Assumptions for Non-Linear State of Stress Analysis ................................................. 115 5.4.2 Results from Non-Linear State of Stress Analysis....................................................... 122
5.5 Conclusions ..................................................................................................................... 129
CHAPTER 6 CONCLUSIONS AND FUTURE WORK .................................................. 131
6.1 Conclusions ..................................................................................................................... 131
v
6.2 Original Contributions .................................................................................................... 135
6.3 Future Work .................................................................................................................... 136
REFERENCES .......................................................................................................................... 139
APPENDIX A: DERIVATION OF EQUIVALENT SPRING CONSTANTS .................... 146
vi
LIST OF FIGURES
Figure 1-1: Location of Longyou Grottoes in China ................................................................. 3
Figure 1-2: Topographic map of the Fenghuang Hill with all 24 caverns locations and separated into three categories. ............................................................................... 4
Figure 1-3: The interior and geographic position of Cavern 2 in the primary cavern cluster (Li, Yang et al. 2009): (a) Entrance, (b) Back Wall, and (c) Side Wall of Cavern 2 .... 6
Figure 1-4: Detailed comparison of the columns in Cavern 2: (a) Column 1, (b) Column 2, (c) Column 3, and (d) Column 4 .................................................................................. 8
Figure 2-1: Flow-Chart of the results obtained from each on-site investigation ..................... 16
Figure 2-2: Pictures of the (a) the fiber optic accelerometers used and (b) the data acquisition system used to collect the acceleration data ......................................................... 17
Figure 2-3: Pictures during the on-site investigation in August 2010 ..................................... 18
Figure 2-4: Sample time segment both pre and post processed for test set-up 1 ..................... 18
Figure 2-5: Sample PSDF for both test set-ups ....................................................................... 19
Figure 2-6: Test Set-Ups for Columns 1 and Columns 3 ......................................................... 21
Figure 2-7: Sample time histories for test-set ups 1, 2 and 3 ................................................... 22
Figure 2-8: Sample time histories for test-set ups 4, 5 and 6 ................................................... 22
Figure 2-9: Sample PSDF results for test set-up (a) 1, (b) 2, (c) 3, (d) 4, (e) 5 and (f) 6 ........ 24
Figure 2-10: PSDF Results for (a) Column 1 and (b) Column 3 for all test set-ups (Test SU) 25
Figure 2-11: Sample FDD results for test set-up (a) 1, (b) 2, (c) 3, (d) 4, (e) 5 and (f) 6, where SSV1 and SSV2 are the singular values .............................................................. 26
Figure 2-12: FDD results for all test set-ups ............................................................................ 27
Figure 2-13: A comparison of (a) traditional mode shapes and the extracted mode shapes for (b) Column 1 and (c) Column 3 .......................................................................... 28
Figure 3-1: Illustrative comparison of the models developed: (a) M1, (b) M2, and (c) M3 ... 38
Figure 3-2: Arrangement of material property assignment for the 4-sections for (a) M1, (b) M2 and (c) M3 ...................................................................................................... 43
Figure 3-3: Sensitivity Analysis on the Dynamic Properties by varying Poisson’s Ratio....... 44
vii
Figure 3-4: Sensitivity Analysis on the Dynamic Properties by varying the density .............. 45
Figure 3-5: Sensitivity Analysis on the Dynamic Properties by varying the elastic modulus . 46
Figure 3-6: Boundary Conditions for the FE model ................................................................ 48
Figure 3-7: Variation of Saturation Levels and Material Properties Results for Column 1 for the 1st and 2nd natural frequency and material definitions for each property set .. 52
Figure 3-8: Variation of Saturation Levels and Material Properties Results for Column 3 for the 1st and 2nd natural frequency and material definitions for each property set .. 53
Figure 4-1: Geometrical dimensions for (a) Column 1 and (b) Column 3 for the local models developed of each column .................................................................................... 64
Figure 4-2: Graphical representation of the conversion of the columns to beam-beam models for (a) a translational spring and (b) a rotational spring ...................................... 65
Figure 4-3: Results from optimization algorithms: (a) Comparison of objective function (F = frequency residual only, Eq. 11, and MF = frequency residual plus MAC values, Eq. 12) and (b) percent change with calibrated global model .............................. 73
Figure 4-4: Optimization results from force determination for (a) Column 1 and (b) Column 3............................................................................................................................... 76
Figure 4-5: Comparison of global model and local models for column 1 for the 50-year time history analysis for the displacement, Von Mises’ stress and vertical stress for (a) 1 percent, (b) 3 percent, and (c) 5 percent material degradation .......................... 81
Figure 4-6: Comparison of global model and local models for column 1 for the 50-year time history analysis for 1st, 2nd, 3rd principal stresses for (a) 1 percent, (b) 3 percent, and (c) 5 percent material degradation .................................................................. 82
Figure 4-7: Comparison of global model and local models of column 3 for the 50-year time history analysis for the displacement, Von Mises’ stress and vertical stress for (a) 1 percent, (b) 3 percent, and (c) 5 percent material degradation .......................... 83
Figure 4-8: Comparison of global model and local models for column 3 for the 50-year time history analysis for the 1st, 2nd, 3rd principal stresses for (a) 1 percent, (b) 3 percent, and (c) 5 percent material degradation .................................................................. 84
Figure 5-1: Graphical Representation of Mohr-Coulomb Failure Criterion (Zhao 2000) ........ 91
Figure 5-2: Stress-Strain curves from test data for saturated and unsaturated specimens ...... 100
Figure 5-3: Specimen used for the tri-axial tests .................................................................... 101
Figure 5-4: Mohr-Coulomb, Drucker-Prager, and Hoek-Brown failure criteria matched to Test Data ..................................................................................................................... 102
viii
Figure 5-5: Finite Element Model of Specimen with applied stresses and blue symbols for fixed boundary conditions................................................................................... 104
Figure 5-6: Comparison of progression of failure for stress-strain data obtained from rock samples and an assumed elastic stress-strain curve ............................................ 106
Figure 5-7: Stress-Strain Curves used in the Sensitivity Analysis ........................................ 108
Figure 5-8: Progression of failure based on the defined failure criterion and specified stress-strain curve for an assumed confining pressure of 0 MPa (2 = 3 = 0) ............ 111
Figure 5-9: Progression of failure based on the defined failure criterion and specified stress-strain curve for an assumed confining pressure of 5 MPa (2 = 3 = 5) ............ 112
Figure 5-10: Progression of failure based on the defined failure criterion and specified stress-strain curve for an assumed confining pressure of 10 MPa (2 = 3 = 10) ....... 113
Figure 5-11: Progression of failure based on the defined failure criterion and specified stress-strain curve for an assumed confining pressure of 15 MPa (2 = 3 = 15) ....... 114
Figure 5-12: Sample sets of the stress-strain curves assuming material degradation of 1% decrease per year ............................................................................................... 117
Figure 5-13: Sample sets of the stress-strain curves assuming material degradation of 3% decrease per year ............................................................................................... 118
Figure 5-14: Sample sets of the stress-strain curves assuming material degradation of 5% decrease per year ............................................................................................... 119
Figure 5-15: Sample sets of the stress-strain curves assuming material degradation of 10% decrease per year ............................................................................................... 120
Figure 5-16: Time when the first element fails in the Cavern 2 model due to the influence of material degradation and horizontal pressure, l ................................................. 123
Figure 5-17: Time of complete failure of Cavern 2 model due to the influence of material degradation and horizontal pressure, l ............................................................... 124
Figure 5-18: Comparison of Maximum Vertical Stress in Column 1 assuming (a) 3%, (b) 5%, and (c) 10% material degradation at failure ...................................................... 127
Figure 5-19: Comparison of Maximum Vertical Stress in Column 3 assuming (a) 3%, (b) 5%, and (c) 10% material degradation at failure ...................................................... 128
Figure 5-20: Comparison of Minimum Vertical Stress in Column 1 assuming (a) 3%, (b) 5%, and (c) 10% material degradation at failure ...................................................... 128
ix
Figure 5-21: Comparison of Minimum Vertical Stress in Column 3 assuming (a) 3%, (b) 5%, and (c) 10% material degradation at failure ...................................................... 129
Figure A-1: Showing equivalence of using stiffness method to determine spring stiffness for an axial spring and the Free Body Diagram used in the derivation ..................... 147
Figure A-2: Showing equivalence of using stiffness method to determine spring stiffness for a rotational spring and the free body diagram used for the derivation ................... 151
x
LIST OF TABLES
Table 2-1: Summary of PSDF results for test set-ups 1 and 2 ...................................................... 20
Table 2-2: Summary of directional placement of accelerometers, location of each, and the number of data sets for each test set-up ...................................................................... 21
Table 2-3: Average Results from PSDF for Column 1 and Column 3 ......................................... 25
Table 2-4: Average FDD results obtained for different test set-ups ............................................. 27
Table 3-1: Stress values and respective TRD values for M2 and M3 .......................................... 40
Table 3-2: Natural frequencies and respective TRD values for M2 and M3 ................................ 41
Table 3-3: Comparison between On-Site and FE Model Results for Column 1 and Column 3 ... 49
Table 3-4: Sample Results for 2 Meter Saturation Level and Modified Property Set 18 ............. 51
Table 4-1: Parameters used in optimization functions with initial values, and the upper and lower bounds ......................................................................................................................... 68
Table 4-2: Column 1 and 3’s optimization results ........................................................................ 70
Table 4-3: Column 1 and 3’s spring constant values based on optimization results .................... 71
Table 4-4: Summary of Assumptions used in Comparative Analysis .......................................... 76
Table 4-5: Results at the present state (Year 0) for the Local and Global Models ....................... 80
Table 5-1: Comparison of Test Data to Three Different Failure Criteria ................................... 101
Table 5-2: Parameter Assignment for each Failure Criterion ..................................................... 102
Table 5-3: Comparison of Test Data to Three Different Failure Criteria assuming a Stress-Strain Curve as the Material Model ..................................................................................... 105
Table 5-4: Comparison of Test Data to Three Different Failure Criteria assuming an Elastic Material Model .......................................................................................................... 105
Table 5-5: Comparison of Test Data to Three Different Failure Criteria (MC = Mohr-Coulomb, DP = Drucker-Prager, HB = Hoek-Brown) assuming different Stress-Strain Curves (Combined Error) ...................................................................................................... 108
xi
ACKNOWLEDGEMENTS
I would like to first and foremost convey my deepest appreciation to my committee chair, Dr. Maria Feng. Through her guidance and support I have been able to develop my skills not only in academics, but also in my daily life. She showed me the way to complete this dissertation in a manner that I never knew possible. If it wasn’t for her giving me not only structured guidance, but also the freedom needed to explore various possibilities, I am not sure the outcomes would have been as great as they developed into. I would also like to thank the other two members of my committee, Dr. Lizhi Sun and Dr. Mark Bachman. Dr. Sun was instrumental in helping me to expand my thoughts in regards to the finite element portion of this dissertation. Dr. Bachman guided me through the difficult process of not only my research, but also the NSF funded IGERT program, where I learned the importance of interdisciplinary research. He helped me to understand my duties and this allowed me to focus my research and keep myself on track to graduate in a timely manner. I would also like to a give special thanks to Dr. Chikoasa Tanimoto, Dr. Yoshinori Iwasaki, Dr. Encong Liu, Dr. Keigo Koizumi and Dr. Yoshi Fukuday for their support and continual advice on the experimental and analytical framework for this research. In addition, I would like to extend thanks to my fellow research mates, many whom have graduated. If it wasn’t for their kindness, generosity and just continual laughter, there would have been many a days of just pure frustration. A huge depth of appreciation goes to my parents for always supporting me and giving me a home that was always loving and caring. Without their encouragement I would have been lost early on. Furthermore, I would like to thank my best friend, Jamie Robertson, for never letting me give up on my dream and dealing with me during this tumultuous time in my life. Also, I want to give a warm node of appreciation to Lucas Wagner for not only rereading and correcting this dissertation, but also being a huge support to me and for always being a bright light of encouragement.
Throughout my career at UCI, I have been able to conduct research in China through the support of the Ministry of Education, Science, Sports and Culture, by a Grant-in-Aid for Scientific Research (A), No. 22254003. Also this research was funding by the National Science Foundation, Integrative Graduate Education and Research Traineeship (IGERT) program (NSF IGERT DGE 0549479).
Last, but certainly not least, I would like to thank my Lord and Savior, Jesus Christ for giving me the strength to keep moving forward when I thought all I wanted to do was quit. Thank you for loving me enough to save me.
xii
CURRICULAUM VITAE Amber Jasmine Greer
EDUCATION Ph.D. in Civil Engineering July 2012 Civil and Environmental Engineering Department
University of California, Irvine Ph.D. Dissertation: “Finite Element Modeling and Stress
Analysis of Underground Rock Caverns” Advisor: Professor Maria Feng
M.S. in Civil Engineering June 2010 Civil and Environmental Engineering Department
University of California, Irvine Advisor: Professor Maria Feng
B.S. in Civil Engineering June 2009 Civil and Environmental Engineering Department
University of California, Irvine
PUBLICATIONS Journal Papers Greer A., Feng M.Q., and Gomez H. “Modeling of Underground Rock Caverns”. Submitted to Computers and Geotechnics. Greer A., Feng M.Q. “Development and Comparison of global and local models of a 2000 year-old Rock Cavern” in preparation.
xiii
ABSTRACT OF THE DISSERTATION
Finite Element Modeling and Stress Analysis of Underground Rock Caverns
By
Amber Jasmine Greer
Doctor of Philosophy in Structural Engineering
University of California, Irvine, 2012
Professor Maria Feng, Chair
The 2000 year old Longyou Grottoes in China have become of significant interest to the
research community, but the amount of available information on the historical underground rock
caverns is limited. With the possibility of failure increasing with each passing year, there has
been augmented need to identify possible mechanisms of failure for each cavern. This ultimately
requires a state of stress analysis of the caverns to identify potential areas of failure.
However, in order to determine the state of stress in the present, as well as the future,
requires an advanced analysis of the cavern cluster. This dissertation was able to successfully
combine the technique of both rock engineering and structural engineering to develop and
analyze an advanced finite element model for an identified cavern at the Longyou site (Cavern 2).
The research progressed in four stages. In the first stage, ambient vibration measurements were
obtained over two on-site investigations on two specific columns in the cavern. From these
results, a simplified global finite element model was calibrated (second stage). In order to
adequately identify the proper level of simplification, the development of a four-part criterion
was used to aid in the simplification process. In the third stage, a comparative analysis was
conducted between the simplified global model and two local models created using the results
xiv
from the on-site investigation and two different optimization techniques. Consequently, it was
determined the global model is a more adequate fit for future analysis, which led directly into the
final stage where an advanced non-linear finite element analysis was performed on the global
model. In addition, it was determined that the Hoek-Brown failure criterion shows the most
appropriate representation of the characteristics of the rock material.
Through this research, it was determined the cavern would have increased probability of
failure surrounding one of the columns, which would fail due to an exceedance of the tensile
strength at the junction between the top of the column and the roof. Furthermore, it is advised
that retrofit techniques should be designed and applied at this location.
1
CHAPTER 1 Introduction
Traditionally when one investigates possible failure of a rock structure in the field of rock
engineering, there is usually a specific process to follow: (1) extensive testing is conducted on
the rock type, (2) then greater depth of research is conducted into the specifics of the rock, i.e.
the physical and chemical characteristics, (3) next an applicable failure criterion is assigned to
the rock type, (4) then all the characteristics are cumulatively collected together to be applied to
a defined finite element model, and (5) finally a case specific analysis is conducted in order to
investigate the possible ways of failure. This process has been corroborated through research
projects of different rock structures including: Bet Gurvin (Hatzor, Talesnick et al. 2002),
Xiaolangdi Powerhouse (Huang, Broch et al. 2002), Tel Beer Sheva (Hatzor and Benary 1998),
and Zedekiah Cave (Bakum-Mazor, Hatzor et al. 2009). Unfortunately in the examples above, if
displacement measurements were collected from on-site investigations, the results are used for
comparative purposes only and not for the actual development or calibration of the defined finite
element model.
Consequently, one of the primary goals of this dissertation is to bridge the gap between
structural engineering and rock engineering. Customarily, for any structural engineering problem
the on-site results are vital to develop a finite element model and are not used for comparison
purposes. By utilizing many of the techniques developed in the field of structural engineering,
this will help to better analyze the rock structures and give a possible failure mechanism with
higher accuracy. Nevertheless, there are a plethora of different rock structures, which could be
focused on-including, but not limited to underground rock caverns, tunnels, retaining walls,
underground railroad systems, and underground sport complexes. However, for the purposes of
the research conducted in this dissertation, the focus will be on underground rock caverns.
2
Underground rock caverns have an aesthetic appeal of being historical structures or
monuments, which receive a large volume of visitors each year. However, considering the nature
of their construction, i.e. being underground, this does raise questions of structural integrity such
as, how long can the cavern remain intact before progressive or sudden collapse occurs. This
consequently allows for high-profile research to be conducted with a significant amount of
support coming from both tourist agencies and governmental institutions. In order to answer the
question of structural integrity, the performance of advanced analysis should and needs to be
conducted. However, without an appropriate methodology that combines both structural
engineering and rock engineering approaches, the results could be misleading.
1.1 Introduction to the Longyou Grottoes
There is a specific set of underground rock caverns, which will be investigated and be the
focus hereinafter. The cavern cluster was discovered in June 1992, when local farmers were
searching for a water supply near their village in Longyou Country, Zhejiang Province, China
(Figure 1-1), when they came across and dewatered five large and complete rock grottoes (or
caverns), traditionally called the Longyou Grottoes. Through one artifact found on-site, it is
currently believed the Longyou Grottoes were made by man over 2000 years ago.
There are several unique features, which has drawn the attention of not only scientists but
also archeologists, historians, and engineers: their proximity to the surface, the characteristics
soft-medium hard surrounding rock (known as argillaceous siltstone), the size of the caverns
(spanning 15 to 40 m and heights of 10 to 20 m), the low number of slim supporting columns for
each cavern (ranging from 2 to 4), and the long-term integrity of the cavern cluster. Even with
this vast amount of attention from researchers, there is still no quantifiable timeline for when
they were built, why or how they were constructed, and the reason for their construction.
3
Figure 1-1: Location of Longyou Grottoes in China
However, not all of the 24 caverns identified in the cluster have been dewatered (five
remain filled with water). The reason for this is that of the 19 caverns dewatered, 14 have either
partially or completely collapsed due to shear or tensile failure. Even the five main caverns have
shown continual crack propagation due to physical and chemical weathering (Yue, Fan et al.
2010). This progression has lead to retrofit techniques to the columns for three of the five main
caverns (Caverns 1,3,4), but recently increased attention has been drawn to Cavern 2 to
determine if a need for an applied retrofit technique is appropriate. The topographical location
0 20 40 km
JiangxiProvince
QiandaoLake
LongyouGrottoes
Quzhou
Zhejiang Province
Jinhua
Ningbo
Eas
t Chi
na S
ea
Hangzhou
Hangzhou Harbour
Taizhou Harbour
Wenzhou
N
SOUTHKOREA
MONGOLIA
RUSSIA
INDIA
Beijing
Tibet NanjingShanghai
PACIFIC OCEAN
Guangzhou TAIWAN
4
for each of the 24 caverns is shown in Figure 1-2, with them separated into the three categories:
excavated, unexcavated, and partially or completely failed caverns.
Figure 1-2: Topographic map of the Fenghuang Hill with all 24 caverns locations and separated into three categories.
Over the last fifteen years, research has been progressing quickly to answer many of the
questions poised earlier in the section. Consequently, numerous papers have been published on
the Longyou Grottoes. All the papers can be broken down into five main categories: basic
information in regards to the Longyou Grottoes (Lu 2005; Yang, Yue et al. 2010); studies into
the characteristics of the rock material, i.e. argillaceous siltstone, (Guo, Li et al. 2005; Li and
Tanimoto 2005; Cui, Feng et al. 2008; Li, Wang et al. 2008; Li, Wang et al. 2008; Yue, Fan et al.
2010); studies focusing on the stabilization problems of the Longyou Grottoes (Li, Mu et al.
0 60 m59.4
47.64
47.12
57.4
5759.8
16
76
8
17
4 5
32 1
1415
1312
1110
9
1819
20
2122
23
24 N
30
ExcavatedCaverns
Unexcavated Caverns
Partly/CompletelyFailed Caverns
#
#
#
5
2005; Li and Tanimoto 2005); advanced failure analysis of specific caverns (Yang, Li et al.
2005; Guo, Yang et al. 2006; Guo, Yang et al. 2007); and studies into possible retrofit techniques
for specific caverns (Yang, Xu et al. 2000; Li, Yang et al. 2009; Zhu, Chang et al. 2009).
Unfortunately, none of the papers analyze the structural integrity of the caverns or
attempt to use displacement or acceleration measurements to calibrate or correctly define finite
element models. Without an accurately defined model, the results become arbitrary estimations
of possible mechanisms of failure. Within this dissertation, proven structural engineering
techniques will be combined with rock engineering to determine the possible failure mechanisms
of one cavern at the Longyou Grottoes: Cavern 2.
1.1.1 Introduction to Cavern 2
Much of the research conducted on the Longyou Grottoes has primarily focused on the
cluster as a whole or the behavior of the argillaceous siltstone. Considering Cavern 2 is the
largest of the cavern cluster, spanning almost 35 meters in both directions with a maximum
height of 15 meters and is the picture for advertisements used for the historical site, gives good
cause to be researched in depth. Figure 1-3 displays the location of Cavern 2 relative to the other
four main caverns on display and pictures inside the cavern, as a reference.
Recent publications by Guo et al (2005) and Yue et al (2010) have discussed the dire
need for some action to be taken to ensure proper integrity of the cavern. Both papers discuss
several pressing issues, which could be potential problems: the proximity to the Qu River, the
high levels of precipitation each year causing saturation of the surrounding rock, direct sunlight
coming in at the entrance of the cavern and the close proximity of Cavern 2 to both Cavern 1 and
Cavern 3 with connecting walls only 1 and 3 meters thick, respectively. Even though tests have
been conducted on the mechanical and physical properties of other caverns (Yue, Fan et al.
6
2010), no information is available about the specific properties, i.e. Elastic Modulus, density,
Poisson’s Ratio, compressive or tensile strength of the rock, and the present state of stress, inside
of Cavern 2. Based on the above mentioned concerns, it became vital to try and accurately
determine the possible failure mechanisms of Cavern 2, which will consequently be the focus for
the rest of this dissertation.
Figure 1-3: The interior and geographic position of Cavern 2 in the primary cavern cluster (Li, Yang et al. 2009): (a) Entrance, (b) Back Wall, and (c) Side Wall of Cavern 2
40m200
Cavern 1
(c)(a)
(b)
(c)
(a)
C1
(b)
C2C3
7
In order to better understand and visualize the various idiosyncrasies of Cavern 2, the
following list describes several pertinent characteristics of the cavern:
1) Is approximately 2 meters below ground level and does not have a uniform cross-section.
2) The maximum length, width and height are 34 meters, 35 meters and 15 meters,
respectively.
3) The roof of the cavern is not horizontal, but slopes from horizontal at inconsistent angles
ranging from 17° to 32°.
4) The interior cavity is supported by four columns ranging from 5 meters to 11 meters tall.
Each column’s cross-section is an isosceles triangle, with the areas ranging from 1.20 m2
to 2.38 m2. A detailed comparison of the columns is shown in Figure 1-4, where it is
clearly seen that column 1 is the tallest, while column 4 is the shortest.
5) The surrounding rock mass is argillaceous siltstone and is categorized as a soft rock,
which means its properties can be affected significantly when saturated in water. Through
material property tests, the dry to fully saturated conditions in unit weight vary from 21.8
to 23.5 kN/m3 and its Elastic Modulus can range from 4.5 to 3.03 GPa, respectively (Guo,
Li et al. 2005).
6) The connecting walls to Cavern 1 and Cavern 3 are approximately 1 meter and 3 meters
thick, respectively.
8
Figure 1-4: Detailed comparison of the columns in Cavern 2: (a) Column 1, (b) Column 2, (c) Column 3, and (d) Column 4
(a) Column 1 (b) Column 2
Plan View Side View
7.71
m
10.6
5 m
2.71
m
1.38 m
1.38 m
Plan View Side View
Geometrical PropertiesIx = 0.76 m4*
Iy = 0.12 m4**
J = 0.96 m4***
A = 1.87 m2****
Geometrical PropertiesIx = 0.48 m4
Iy = 0.16 m4
J = 0.63 m4
A = 1.57 m2
(c) Column 3
Plan View Side View
2.80
m
8.46
m
1.70 m
1.70 m
Geometrical PropertiesIx = 1.03 m4
Iy = 0.38 m4
J = 1.41 m4
A = 2.38 m2
(d) Column 4
Plan View Side View
2.09
m
4.85
m
1.15 m
1.15 mGeometrical Properties
Ix = 0.29 m4
Iy = 0.09 m4
J = 0.38 m4
A = 1.20 m2
2.34
m
1.34 m
1.34 m
* Ix = Moment of Inertia around the X-Axis** Iy = Moment of Inertia around the Y-Axis
*** J = Radius of Gyration**** A = Area
9
1.2 Scope of Dissertation
Due to recent and collapses, it is imperative that we understand the state of stress for
Cavern 2. This state of stress analysis is essential to determine the likelihood of failure for the
cavern. Ultimately, each chapter builds upon the previous to understand the state of stress of
Cavern 2 and a possible mechanism of failure is found, which could occur in the next 30 years.
Chapter 2 presents the results from the on-site investigations of Cavern 2. In total there
were two investigations performed over a two-year period. Ambient vibration measurements
were extracted for columns 1 and 3 inside Cavern 2. The main goal of these experiments was to
determine the dynamic characteristics, i.e. natural frequencies and mode shapes, of each column,
so they could be used for model development purposes in chapters 3 and 4.
Chapter 3 focuses on calibrating a finite element model of Cavern 2. In light of the
complex geometry and boundary conditions for the cavern, an investigation into the proper
simplified model was necessary. Consequently, three finite element models are created and
compared using a developed four-part criterion for model simplification. In the end, an
appropriate finite element model is selected and calibrated using the results from the on-site
investigations from chapter 2.
Chapter 4 uses the results extracted from the on-site investigations to analyze and
determine if the use of local models is an adequate representation of Cavern 2. Since the results
obtained from chapter 2 are only for columns 1 and 3, it is only appropriate to develop local
models of each column and then compare them with the global model developed from chapter 3.
A 50-year non-linear time history analysis is conducted on each of the finite element models.
This allows for direct comparisons to be made not only on stress levels for static situations, but
10
also to analyze the changes in the stress levels over time. This comparative study is vital to
determine the proper finite element model to use for the state of stress analysis in chapter 5.
Chapter 5 focuses on conducting a state of stress analysis on Cavern 2. From chapter 4 it
is determined that the global model should be used for this analysis. In order to determine the
state of stress and consequently the possible failure mechanisms of the cavern, an investigation
into the proper failure criterion is performed. Three criteria are considered: Mohr-Coulomb,
Drucker-Prager, and Hoek-Brown, but through a comparative analysis it was determined the
Hoek-Brown failure criterion is the most appropriate criterion to represent the behavior of the
argillaceous siltstone of Cavern 2. Ultimately, a non-linear time history analysis is performed on
the finite element model considering several different assumptions. Based on these assumptions,
there are important conclusions to draw about the state of stress of Cavern 2 and the possible
failure mechanisms.
Finally Chapter 6 discusses the major conclusions from each chapter, the original
contributions and suggestions for future work.
11
CHAPTER 2 On-Site Investigations
2.1 Purpose of Performing On-Site Investigations
One of the primary goals of this dissertation is to perform a stress analysis and determine
the possible failure mechanisms for Cavern 2. However, to reach this goal a finite element (FE)
model needs to be developed and calibrated. In order to calibrate the FE model, on-site
investigations are necessary to determine the appropriate parameters, which need to be emulated
by the FE model.
Through the years, several different approaches have become available for on-site
investigations and they can be broken down into two categories: destructive testing and non-
destructive testing. Determining the suitable approach is based on the required information
needed to create the FE model. If the ultimate strength of the structure is desired, then destructive
testing is the better approach, but as its name suggests will cause irreparable damage to the
structure. Normally destructive testing is only performed on replicas when complete failure of
the structure is needed. Ultimately, using a non-destructive technique is found more beneficial
for many structures, including underground rock caverns. There are several different types of
non-destructive testing, e.g. flat-jack testing (Binda, Lualdi et al. 2008), sonic tomography
(Binda, Saisi et al. 2003), strain gauges (Moyo, Brownjohn et al. 2005), acoustic emissions
(Carpinteri, Invernizzi et al. 2009), and ambient vibration testing (Feng and Kim 2006; Kim and
Feng 2007). The flat-jack test allows the current stress state and elastic modulus to be extracted.
However, it does require several small incisions to be made in the rock structure and only
specific sections can be tested at a time (Rossi 1987). Since the historical society in Longyou has
been clear on the damage allowed to each cavern, this approach may not be recommended for
Cavern 2. Sonic tomography uses x-ray technology to visually represent the inside of the
12
structure (Binda, Saisi et al. 2003), which would help to identify the depth of the fractures in the
cavern, but does have limitations on depth. Another option is to capture ambient vibration data
using accelerometers, which is a widely accepted non-destructive technique in the civil
engineering field for complex and large structures (Wu and Li 2004; Jaishi and Ren 2005; Wang,
Li et al. 2010). Considering the complex nature of Cavern 2, this approach will yield the results
desired and allow specific locations inside the cavern to be measured with ease, without
interfering with the historical ambiance of the cavern. Thus vibration analysis will be used
exclusively in this work.
Over the course of this research project, two different on-site investigations were
performed and valuable information was obtained. For each investigation, system identification
techniques (explained in the next section) are performed to determine the dynamic properties for
the locations measured. At the end of the chapter a comparison of the data from the two
investigations will be presented and specific dynamic properties will be identified.
2.2 Introduction to System Identification
Since the on-site ambient vibration measurements are in the form of acceleration data, the
application of system identification techniques is available to extract the dynamic characteristics.
The system identification problem using measured data is an inverse problem, due to the indirect
identification of structural parameters based on the measured dynamic properties of a full-scale
structure.
Normally when identifying a structure’s dynamic characteristics classical system
identification techniques require measured data for both input (or forces being applied to the
system) and output loads (or the response of the system) (Koh, Hong et al. 2003). Through the
years, numerous system identification techniques have been developed considering various cases
13
of available measured data such as Multiple-Input Multiple-Output (MIMO), Single-Input
Multiple-Output (SIMO), and Single-Input Single-Output (SISO). When working with broad-
banded excitation, i.e. ambient vibration, it becomes difficult to measure the input loads and the
output only problem is the only viable alternative.
The relationship between the unknown inputs, , and the measured responses, , is
shown in the following equation:
( 1 )
where is the (s x s) power spectral density (PSD) matrix of the input, s is the number of
inputs, is the (m x m) PSD matrix of the responses, m is the number of responses,
is the (m x s) frequency response function (FRF) matrix, the overbar signifies the
complex conjugate and superscript, T, denotes the transpose (Brincker, Zhang et al. 2001). There
are several techniques developed over the years to solve the output only problem: Ibrahim time
domain technique (Mohanty and Rixen 2003), eigensystem realization algorithm (Juang and
Pappa 1984), stochastic subspace identification algorithms (Overschee and Moor 1996) and
frequency domain decomposition (Brincker, Zhang et al. 2001). It is important to note the first
three identification techniques use mathematical representations of the system to identify the
system’s structural parameters.
The first three system identification techniques extract the modal parameters based on
time domain information. The Ibrahim time domain technique uses an algorithm based on time
responses of multiple outputs, which aid in determining the modal parameters. A downfall of this
algorithm is at least 2N response locations need to be measured to identify a model of order N
(Mohanty and Rixen 2003). While the eigensystem realization algorithm (ERA) consists of two
major parts: the formulation of the minimum-order realization and modal parameter
14
identification. In order to quantify the system and noise modes, two indicators, the modal
amplitude coherence and the modal phase colinearity, are calculated. Stochastic subspace
identification algorithms compute state space models from a given output data. The physical
system is represented by a mathematical model in the form of input, output, and state variables
related by first-order differential equations (Overschee and Moor 1996).
Frequency domain decomposition (FDD) is an extension of the basic frequency domain
technique (or peak picking technique) and is one of the only output-only techniques that extract
modal parameters in the frequency domain. The basic frequency domain technique is popular for
two main reasons. First, users can directly work with the spectral density function, which helps
them have a feel for the behavior of the structure just by looking at the peaks in the spectral
density functions. Second, the basic technique is effective at estimating the natural frequencies
and mode shapes of a structure if the modes are well separated. The technique uses the
assumption that well-separated modes can be estimated directly from the power spectral density
matrix at the peak. Unfortunately, many civil infrastructures, i.e. bridges and tall buildings, have
modes which are very close together, making the implementation of the frequency domain
technique difficult. FDD builds on the user friendliness and simplicity of the basic frequency
domain approach, while at the same time removing all the disadvantages (Brincker, Zhang et al.
2001). Ultimately, FDD will be the approach used for the purposes of system identification in
this chapter.
When using FDD for identification, the first step is to estimate the power spectral density
(PSD) matrix. Once the estimated output PSD, , is known at discrete frequencies
, it is then decomposed using the singular value decomposition of the matrix, with the
results shown in Equation (2).
15
( 2 )
where the matrix , , … , is a unitary matrix holding the singular vectors , and
is a diagonal matrix holding the scalar singular values . When there is a dominating mode,
there will be a peak at the kth mode or a close mode. If there is only one mode dominating then
there will only be one term in the outputted PSD. In this case the first singular vector from the
unitary matrix is an estimate of the mode shape, with an auto-PSD function, which has both a
corresponding singular value and a single degree of free system. In order to validate the PSD
function is the peak, one has to compare the mode shape with ones around the peak obtained. If
the singular vector is found to have a high modal assurance criterion (MAC) (Allemang 2003),
then the corresponding singular value is associated with the SDOF density function. Ultimately,
the natural frequencies and related damping can be obtained from the SDOF density function
(Brincker, Zhang et al. 2001).
2.3 On-Site Investigations
Two on-site investigations were completed on Cavern 2. However, due to sensing
capabilities only certain information is available for each investigation. In order to clarify the
available information from each investigation a flow chart is presented in Figure 2-1, with
detailed results explained for each investigation in the next sections.
16
Figure 2-1: Flow-Chart of the results obtained from each on-site investigation
2.3.1 First On-Site Investigation
The first on-site investigation was performed from August 7-13, 2010, under humid
conditions. The temperature averaged around 30°C for the week, with no apparent rainfall
occurring for the entire week. There seemed to be some level of saturation, but the actual amount
could not be determined.
Acceleration measurements were obtained using fiber optic accelerometers (FOAs)
developed by the Feng Research Group at University of California, Irvine, shown in Figure 2-
2(a) (Feng and Kim 2006; Kim and Feng 2007). The use of these FOA was chosen over
traditional accelerometers for two reasons. The first reason being the size of the FOA is
considerably smaller than the traditional accelerometers, which makes travel and mounting
inside the cavern easier. The second reason has to do with the capabilities of the FOA. Due to
their Moiré-Fringe design, this gives them the capability to measure higher frequencies (up to 50
Hz) and be sensitive to very small accelerations (~0.001 gals), which ultimately makes them
ideal for use in stiff structures. Each FOA was connected to a data acquisition system where
real-time measurements were captured for each test set-up, shown in Figure 2-2(b).
First On-Site Investigation Second On-Site Investigation
Frequency Domain DecompositionPower- Spectral Density Function
Acceleration Data Acceleration Data
Natural Frequencies Mode ShapesDominant Frequencies
17
Figure 2-2: Pictures of the (a) the fiber optic accelerometers used and (b) the data acquisition system used to collect the acceleration data
Two FOAs simultaneously collected ambient vibration measurements at two locations on
column 3, Figure 2-3. One was located 0.5 m from the bottom of the column (Location 1), while
the other was 3.5 meters from the bottom (Location 2). For test set-up 1 and 2, both FOAs were
placed in concurrent directions, the x-direction and y-direction, respectively. For each test set-up
only three measurements were performed, due to the lack of time available in the cavern. After
reviewing the collected data, Figure 2-4, it is evident some post processing techniques should be
applied to the data. Since the FOAs themselves are intended for use in a frequency range below
50 Hz, a band pass filter was applied outside the frequency range of 1 to 50 Hz. Also, to remove
the trend from the data caused by shifting of accelerometer during testing, a linear data
correction factor was utilized. Figure 2-4 shows the raw and post-processed time-history data and
now either the dominate or natural frequencies can be extracted from each data set. As expected,
the middle location has a higher response than the bottom location. This happens because of the
location of each FOA: Location 1 is surrounded with more rock material and would cause the
response at this location be lower than Location 2.
18
Figure 2-3: Pictures during the on-site investigation in August 2010
Figure 2-4: Sample time segment both pre and post processed for test set-up 1
It is customary when applying system identification techniques to have three or more
simultaneous measurements, but considering for this investigation only two locations were
measured; only the power spectral density function (PSDF) was extracted from the data. The
PSDF transforms the data from time domain to frequency domain, showing the frequency
content of the retrieved response. A sample PSDF for both test set-ups 1 and 2 are shown in
0 1 2 3 4 5
-1
-0.5
0
0.5
1
Acc
eler
atio
n (g
al)
Raw Data - Location 1
0 1 2 3 4 5-0.4
-0.2
0
0.2
0.4Post-Processed - Location 1
0 1 2 3 4 5
-0.5
0
0.5
Time (min)
Acc
eler
atio
n (g
al)
Raw Data - Location 2
0 1 2 3 4 5-0.4
-0.2
0
0.2
0.4
Time (min)
Post-Processed - Location 2
19
Figure 2-5. For each set-up, the dominant frequency in both locations correlates well. Also,
Location 2 shows a higher response amplitude in the frequency domain for both test set-ups. A
lower frequency was shown for test set-up 1 than test set-up 2, which can be attributed to the
cross-section of the column: the length in the X-direction is half of the Y-direction. This
adequately explains the differences in the frequencies. The results from both PSDFs showed the
dominant frequency for test set-up 1 was 22.22 Hz and test set-up 2 was 34.91 Hz, with a
summary of results in Table 2-1. For the various time segments, the extracted frequencies are
close together. Deviations could be related the movement of visitors inside Cavern 2, but are
relatively small.
Figure 2-5: Sample PSDF for both test set-ups
0
0.01
0.02
0.03
(a) Test Set-Up 1
Am
plit
ude
10 15 20 25 30 35 400
0.02
0.04
0.06
Frequency (Hz)
(b) Test Set-Up 2
Loc 1 Loc 2
20
Table 2-1: Summary of PSDF results for test set-ups 1 and 2
Test Set-Up 1 Test Set-Up 2
Time Segment Time Segment
1 2 3 4 5 6
Location 1 (Hz) 22.22 21.73 22.22 34.91 ---- 34.42
Location 2 (Hz) 22.22 22.71 22.22 34.91 ---- 36.62
2.3.2 Second On-Site Investigation
The second on-site investigation occurred during April 1-8, 2011, with similar humidity
conditions to August 2010, however, the main difference was the temperature averaged around
16° Celsius and there was rainfall during the week. For the entire week, water was constantly
dripping from the ceiling and it was apparent the rock at the surface was saturated, but the level
of saturation of the surrounding rock mass could not be determined.
Unlike the first on-site investigation, three FOAs simultaneously collected ambient
vibration measurements at three locations of columns 1 and 3, shown in Figure 2-6. For column
3, the same two locations from the first on-site investigation were tested again, with an additional
location added to the top of the column (Location 3); detailed information is available in Table
2-2. Three test set-ups and multiple time segments are recorded for both columns, with the
accelerometers mounted in purely x-, purely y- and a combination of the x- and y-directions. For
each of the test set-ups multiple time segments were recorded with the purpose of comparing the
measured data.
21
Figure 2-6: Test Set-Ups for Columns 1 and Columns 3
Table 2-2: Summary of directional placement of accelerometers, location of each, and the number of data sets for each test set-up
Column 1 Column 3
Location from
Ground (m)
Test Set-Up Location from
Ground (m)
Test Set-Up
1 2 3 4 5 6
Location 1 1.65 X X Y 0.47 X X Y
Location 2 4.87 X Y Y 3.53 X Y Y
Location 3 8.19 X X Y 7.52 X X Y
Data Sets ---- 22 12 11 ---- 18 21 18
The same post-processing techniques performed on the data from the first on-site
investigation were applied to this data as well, i.e. the application of a band pass filter outside the
1 to 50 Hz frequency range and a linear correction factor to remove the obvious trend in the data.
The results are shown in Figure 2-7 and Figure 2-8. For column 1, the highest response is
coming from Location 3, due in part to its proximity to the hole at the entrance of Cavern 2. This
22
fact though not surprising, is not self-evident. For column 3, the highest response corresponds
with location 2, which is consistent with the confinement of the column.
Figure 2-7: Sample time histories for test-set ups 1, 2 and 3
Figure 2-8: Sample time histories for test-set ups 4, 5 and 6
-0.1
0
0.1
Test Set-Up 1
Location 1
-0.1
0
0.1
Acc
eler
atio
n (g
al)
Location 2
0 1 2 3 4 5
-0.1
0
0.1 Location 3
-0.1
0
0.1
Test Set-Up 2
Location 1
-0.1
0
0.1 Location 2
0 1 2 3 4 5
-0.1
0
0.1
Time (min)
Location 3
-0.1
0
0.1
Test Set-Up 3
Location 1
-0.1
0
0.1 Location 2
0 1 2 3 4 5
-0.1
0
0.1 Location 3
-0.1
0
0.1
Test Set-Up 4
Location 1
-0.1
0
0.1
Acc
eler
atio
n (g
al)
Location 2
0 1 2 3 4 5
-0.1
0
0.1 Location 3
-0.1
0
0.1
Test Set-Up 5
Location 1
-0.1
0
0.1 Location 2
0 1 2 3 4 5
-0.1
0
0.1
Time (min)
Location 3
-0.1
0
0.1
Test Set-Up 6
Location 1
-0.1
0
0.1 Location 2
0 1 2 3 4 5
-0.1
0
0.1 Location 3
23
The PSDF was performed on all data sets, in order to directly compare the results with
the first on-site investigation. For some of the test data sets, a clear peak is not definitive and user
discretion is used to determine the peak. Normally, the frequency corresponding to the highest
peak was chosen. If there was no discernible peak, then either a value was not chosen for that
particular test segment or a frequency close to the ‘peak’ value was selected. Sample PSDFs for
each test set-up are shown in Figure 2-9, for column 1 (test set ups 1, 2, and 3) and column 3
(test set-ups 4, 5, and 6). It is interesting to note that depending on the test set-up, the location of
the dominant peak changes. This is seen to be true for all the data sets, not just the sample sets
shown below. Another observation is the correlation of the frequency peaks with the direction of
the accelerometer. If the accelerometers are facing the same direction, the dominant frequencies
in the PSDF correlate accordingly, e.g. test set-ups 1 and 3 have peaks around the same
frequencies, while test set-up 2 does not. This means the peak in the dominant frequency has a
direct correlation to the placement of the FOA.
For each of the time segments, the peak in the PSDF is collected and an average of all the
results is given in Table 2-3. Looking at Figure 2-10, the first dominant frequency in both
columns 1 and 3 show stability in the dominant frequency, but the second frequency shows some
instability in the dominant frequency. Because of this, the second dominant frequency should be
regarded as a rough estimate rather than a distinct value when used in further analysis.
As was the case with first on-site investigation, there are some inconsistencies in the
dominant frequencies not only over the data sets, but also at locations along each of the columns.
As previously reasoned, these discrepancies could be directly related to the activity of visitors
inside Cavern 2, but further tests need to be conducted to definitively conclude this.
24
Figure 2-9: Sample PSDF results for test set-up (a) 1, (b) 2, (c) 3, (d) 4, (e) 5 and (f) 6
0
0.01
0.02(a) Test Set-Up 1
Loc 1 Loc 2 Loc 3
0
0.01
0.02 (b) Test Set-Up 2
0
0.005
0.01 (c) Test Set-Up 3
0
0.05(d) Test Set-Up 4
0
0.01
0.02
Am
plit
ude
(e) Test Set-Up 5
10 15 20 25 30 35 400
0.02
0.04(f) Test Set-Up 6
Frequency (Hz)
25
Figure 2-10: PSDF Results for (a) Column 1 and (b) Column 3 for all test set-ups (Test SU)
Table 2-3: Average Results from PSDF for Column 1 and Column 3
Column 1 Column 3
Test Set-Up Test Set-Up
1 2 3 4 5 6
Location 1 (Hz) 25.94 27.1 12.70 22.77 22.78 35.51
Location 2 (Hz) 25.91 12.98 12.90 22.88 35.63 35.90
Location 3 (Hz) 25.86 25.51 12.85 22.80 22.83 ----
The data was further transformed using FDD to extract the natural frequencies and mode
shapes. A sample of the FDD results for all test set-ups in Figure 2-11. Just as in the PSDF, the
peak or peaks for each data set are recognized as the natural frequencies. The results, Figure 2-
12, show stability for the first natural frequency of both columns and some instability for the
10
15
20
25
30Fr
eque
ncy
(Hz)
Test SU 1
(a) Column 1
Test SU 2 Test SU 3
0 5 10 15 2020
25
30
35
40 Test SU 4
0 5 10 15 20
Test SU 5
(b) Column 3
Time Segment0 5 10 15 20
Test SU 6
Loc 1 Loc 2 Loc 3
26
second natural frequency. Table 2-4 shows the average FDD results obtained for each of the test
setups. Just as in the case of the PSDF results, the first natural frequency will have a distinct
value (column 1 is 12.70 Hz and column 3 is 22.95 Hz), but the second natural frequency has
greater uncertainty and will be assumed to lie within some range. It is interesting to note, not all
the natural frequencies could be obtained from each test set-up. Figure 2-11, Figure 2-12 and
Table 2-4 all show the extraction of natural frequencies from the data sets is directly correlated
with the direction of the FOAs.
Figure 2-11: Sample FDD results for test set-up (a) 1, (b) 2, (c) 3, (d) 4, (e) 5 and (f) 6, where
SSV1 and SSV2 are the singular values
0
5 (a) Test Set-Up 1
0
2
4(b) Test Set-Up 2
0
1
2 (c) Test Set-Up 3
0
50(d) Test Set-Up 4
SSV1 SSV2
0
5
10
15
Am
plit
ude
(e) Test Set-Up 5
10 15 20 25 30 35 400
10
20(f) Test Set-Up 6
Frequency (Hz)
27
Figure 2-12: FDD results for all test set-ups
Table 2-4: Average FDD results obtained for different test set-ups
Column 1 Column 3
Test Set-Up Test Set-Up
1 2 3 4 5 6
f1 (Hz) --- 12.97 12.73 22.95 22.93 22.87
f2 (Hz) 26.12 25.43 --- --- 35.43 35.67
Furthermore, the corresponding mode shapes are determined for each data set, but they
do not correlate with the traditional mode shapes for the first and second natural frequencies. A
comparison between the traditional mode shapes and the extracted natural frequencies is shown
0 5 10 15 2024
25
26
27
28
Time Segment
2nd F
requ
ency
(H
z)
Set-Up 1 Set-Up 2 Set-Up 3
11
12
13
14
15Column 1
1st F
requ
ency
(H
z)
0 5 10 15 20
34
35
36
37
Time Segment
21
22
23
24
25Column 3
Set-Up 4 Set-Up 5 Set-Up 6
28
in Figure 2-13. The mode shapes are shown for only four of the test set-ups (for column 1 test
set-ups 1 and 3, and column 3 test set-ups 4 and 6) because they are the simplest mode shapes to
extract and should be the easiest to match with the traditional mode shapes. Considering the
rather large discrepancy between the two mode shapes, this brings doubt into the reliability of the
extracted mode shapes. This error could be attributed to the number of FOAs used to measure the
ambient vibration measurements. Based on the size of each column, an increased number of
FOAs would help to increase the accuracy of the extracted mode shapes.
Figure 2-13: A comparison of (a) traditional mode shapes and the extracted mode shapes for (b) Column 1 and (c) Column 3
2.4 Comparison of Results
Due to a lack of data for column 1 during the first on-site investigation, only a direct
comparison can be made from the results obtained for column 3 during both on-site
investigations. Even though there were few test results from the first investigation, there is still
enough to draw comparisons with the second investigation. First, the PSDFs show the same
(a) Traditional Mode Shapes (b) Column 1 (c) Column 3
f1 = 12.70 Hz f2 = 25.88 Hz f1 = 22.95 Hz f2 = 35.44 Hz
X
Z
Y
Z
X
Z
Y
Z
f1 f2
X
Z
Y
Z
29
dominant frequencies when compared with equivalent test set-ups. Second, there is not a
significant change between the natural frequencies between each of the investigations. The main
conclusion is the stiffness for column 3 didn’t change between investigations. This suggests the
column’s properties between each investigation are similar. There was some doubt regarding this
statement before performing the experiments, due to the extreme environmental conditions the
caverns are subjected to during the course of a year. However, this is proven differently by the
extracted data and shows the cavern does not experience extreme changes in the natural
frequency based on the time of year. Considering the lack of data for the first investigation, the
results from the second investigation can be used for further analysis.
When performing future analysis, the results of the FDD will be used. Considering the
lack of change between the two investigations, it is presumptive to say similar modeling
conditions can be applied when developing a FE model in the coming chapters.
30
CHAPTER 3 Justification for a Simplified Model of Cavern 2
3.1 Finite Element Modeling in Underground Rock Caverns
For years, the focus of research on underground rock structures has been on either
laboratory and field measurements or FE modeling. Nevertheless, current research has shifted
more towards FE modeling for several reasons. In spite of the plethora of data obtained from on-
site experiments, there still remain many parameters, e.g. material properties and geometrical
dimensions, which need to be determined and are always under review for which the data is
limited. While on the other hand, a FE model can consider the uncertainties in the structural
parameters. Considering the expense and destructive nature of on-site testing, in a developed
model many parameters can be assessed and modified if necessary. Since most underground rock
caverns are historical sites, this becomes an important advantage to consider. Even miniature or
replica models can be useful, but they have their own set of disadvantages, e.g. they are
expensive to make, scaling may not always be accurate, and determining in-situ conditions is
challenging. Often, modeling of underground rock structures focuses on investigating the
stability and failure mechanisms of the structure, while field measurements would need the
structure to fail for the mechanisms to be properly identified.
However, dealing with modeling issues of underground rock structures is different from
those in other engineering fields, e.g. aerospace or structural engineering. Many times the FE
modeling techniques and approaches will vary in both application and execution, which means
that the same technique is not usually applicable to all underground rock structures. The main
challenge then becomes to understand and recognize the distinctive features of the structures, and
ultimately to develop a good model. In most instances, the data available on geometry and
boundary conditions for most sites is limited. Considering most underground structures have
31
complex geometries, this makes generating a FE model, which represents the structure, rather
cumbersome.
Through the years, there have been numerous approaches to model different types of
underground rock structures. Due to computational restrictions, a majority of the models are,
unfortunately, only two-dimensional representations of the real structures. One of the first
continuum analysis was conducted on jointed blocky rock mass, where it was determined the
added structural affects of rock blocks helps to inhibit the formation of hinges, controls the stress
redistributions, and increases the stiffness of the rock mass (Chappell 1987). Next, two different
continuum analyses were performed on the failure of a fictional jointed rock roof. One analysis
specifically focused on the bending failure (Sofianos and Kapenis 1998), while the other focused
primarily on the analysis and design of the rock roof (Sofianos 1996). In another study, a
completely new computer method, deformation discontinuous analysis, was used on a water
storage system at Tel Beer Sheva, where it was determined there would be no internal block
crushing of the roof (Hatzor and Benary 1998). Deformation discontinuous analysis allows the
elements to be modeled as discontinuous rock media, but due to its high computational cost has
only been investigated in two-dimensional space (Shi and Goodman 1985; Shi 1992). Once this
new analysis technique was available, combinations of both continuous and discontinuous
analysis became popular. A two-dimensional study was conducted on the cavern openings at
Bet-Gurvin, where a combination of continuous and discontinuous analysis was implemented to
output the maximum stress levels for a single cavern opening and compute the increase in stress
levels when neighboring cavern openings are considered. It was concluded there is a 30 percent
increase in the stress levels when the surrounding cavern openings are considered in the analysis
(Hatzor, Talesnick et al. 2002). Another study on the Xiaolangdi Powerhouse, where the arching
32
theory was applied, it was determined the effect of adding fully grouted rockbolts and tensioned
cable anchors was instrumental to reinforce the cavern roof and walls. They were able to
conclude the use of the rockbolts and tensioned cable anchors in areas where the roof arch has
already formed may not be appropriate (Huang, Broch et al. 2002). Finally, while investigating
the roof stability of the Zedekiah Cave, a combination of the geological realistic fracture models
of mechanical layering and discontinuous deformation analysis concluded that as the length of
the roof increases the vertical deformation and settlement concurrently increases (Bakum-Mazor,
Hatzor et al. 2009).
Given the uncertainties of modeling underground rock caverns, many promising
directions have been pursued. Depending on the cavern type and the problem to be researched,
different methods have been used for model simplification. The use of the Voussoir beam theory
has become increasingly popular for modeling cavern roofs and roof failures (Sofianos 1996;
Hatzor and Benary 1998; Sofianos and Kapenis 1998; Huang, Broch et al. 2002) and cavern
openings (Bakum-Mazor, Hatzor et al. 2009). The derivation of new FE techniques has improved
the accuracy of rock modeling, from the infinite element, which aids in modeling underground
boundary conditions (Kumar 2000), to joint elements, which allows present joints to be modeled
in specific locations in the cavern with a high level of accuracy (Curran and Ofoegbu 1993).
Many of the advanced analyses are performed in two-dimensions, where a wide variety
of techniques are applicable. However, for the purposes of the analysis to be conducted on
Cavern 2, a two-dimensional model would not be appropriate. Considering the complexity and
non-symmetrical geometry of the cavern and the need to recognize possible failure mechanisms,
a three-dimensional model is necessary. Ultimately, the type of analysis will be continuous to
simplify the modeling issues, reduce the computational effort and reduce the possibility of
33
erroneous results. Given such a complex geometry, a simplified model could be more
appropriate and computationally feasible. Unfortunately, an investigation into a simplified model
has never been done and consequently will be the primary focus of this chapter. Since there is no
criterion available to aid in the selection of a proper simplified model, a four-part criterion is
developed, which includes several factors to aid in the selection. The four-part criterion will
include both static and dynamic comparison factors because Cavern 2 is a host to both static
loads, i.e. self-weight, and dynamic loads generated from the large volumes of visitors. The
criterion is then applied to three developed models, with different levels of simplification, and an
appropriate selection is made. In the final section this identified model is calibrated using the
dynamic properties extracted from the on-site investigations. This calibrated model will be the
ultimate FE model used for all other analysis in this dissertation.
3.2 Development of Four-Part Criterion for Model Simplification
The purpose of developing a four-part criterion system for model selection is to quantify
the differences, both quantitative and qualitative, between an ‘exact’ model, i.e. model based on
the exact geometry, and a simplified model. These quantifications will then be exploited to
determine the proper simplified model for use in a future analysis. The four-part criterion
examines four different areas of model generation: (1) mesh skewness, (2) stress levels under
static loading, (3) dynamic properties and (4) sensitivity of dynamic properties to material
properties. The added vibration caused by visitors, considering Cavern 2 is a historical site,
increases the importance of a dynamic investigation.
3.2.1 Element Sizing and Mesh Generation
When dealing with meshing for a FE model, it is important to understand the
characteristics of the structure in question. If the elements used in the mesh are too large then the
34
results may be erroneous because information is lost in the elements, but if meshed elements are
too dense this causes the computational time to increase substantially. Methods have been
developed to deal with these issues and include mapped meshing (Zsaki 2010) and automatic
mesh generation (Zienkiweicz, Taylor et al. 2005).
In order to determine whether the mesh will properly identify the intended behaviour
when further analysis is conducted, a comparative measure called skewness, a mesh factor, is
utilized. The skewness of a meshed element can be defined as, how close to ideal (i.e. equilateral
or equiangular) is the face of the element used. The value of the skewness ranges from 0 to 1,
where 0 indicates an equilateral face (desired value) and 1 indicates a completely degenerate face
(worst case). For the purposes of this measure, the element with the largest mesh skewness will
define the mesh skewness for the whole model. By considering the element geometry, this will
allow the user to identify weaknesses in the mesh generation and possible areas of improvement.
3.2.2 Stress Levels under Static Loading
For each given structure the important stress level calculations, i.e. vertical and horizontal
compressive or tensile stress, will vary. Engineering judgment is required when determining the
proper stress level calculations to use for the purposes of comparison. There are certain stress
levels, which have more significance than others, and should carry an increased weight in the
decision-making. For instance, when investigating the opening of an underground rock cavern,
the on-site conditions and geometry affect the behavior of the opening. If the opening is near
ground level, then the surrounding soil pressure may have a larger effect than the vertical load,
and the horizontal stress is the important stress measurement. However, if the opening is below
the ground surface, then a combination of the geometry and the added weight of the soil above
may cause the compressive vertical stress to be an appropriate measurement. The identification
35
of the appropriate comparative measure is essential in determining the selection of a proper
simplified model.
The error between a specified ‘exact’ or reference model should be equitable to the error
in the model itself. Error in a FE model comes from four main areas: assumptions in model
development, meshing, material properties, and assumed boundary conditions. The introduction
of an error factor, such as the total relative deviation, will provide a quantitative value between
the reference model and the simplified models. This error value should be assigned an
appropriate value to remain within the bounds of other incorporated errors and could range
anywhere from 0 to 10 percent, depending on the desired accuracy. Total relative deviation is
calculated by using the following equation:
Total Relative Deviation TRD2
, ( 3 )
where is the weight assigned to each stress value and is the stress in the simplified model
and is the stress in the reference model.
In addition to the quantitative error of Equation ( 3 ), there is a significant qualitative
error assessment of the model, to ensure reasonable results. This includes analyzing the stress
contours in important sections of the cavern, i.e. roof, walls, columns, and entrances. The
incorporation of both aspects into the static criteria will allow a properly represented simplified
model to be distinguishable amongst all the rest.
3.2.3 Dynamic Properties
Investigation of the modal frequencies and mode shapes are vital in understanding the
dynamic characteristics of any structure. Considering the size of most underground rock caverns,
a direct comparison of modal frequencies, using the total relative deviation from Equation ( 3 )
36
and a qualitative comparison of mode shapes is adequate. The inclusion of the weight factor can
either be included or ignored depending on the number of modes used for comparison. The
higher modes generally are less accurate and influential as the lower modes. Just as in the static
analysis, a qualitative assessment of the mode shapes is just as important as quantifying the error
in modal frequencies. The combination of both will give the user an idea of how the underground
cavern will behave under dynamic loading.
3.2.4 Sensitivity Analysis of Material Properties to Dynamic Characteristics
It is important to realize that a majority of underground rock caverns are equivalent in
size to half a football field, which causes difficulties in properly identifying parameters that have
a significant influence on the dynamic characteristics. A sensitivity analysis will show how
changing the material properties affect the dynamic properties throughout the entire structure.
Since the material properties can vary throughout the entirety of the cavern, having an analysis,
which quantifies this variance in terms of dynamic properties, will help to identify inappropriate
simplified models. Two different models of a cavern may behave similarly as a complete
structure, but there is no guarantee the structures or models will behave the same if broken down
into several segments. The use of a localized sensitivity analysis, usually implemented to
determine the most sensitive parameter for FE updating, will help to quantify the desired
behavior for the segmented sections.
In general, the sensitivity analysis computes the sensitivity coefficient, Sj, (Brownjohn,
Xia et al. 2001) which is defined as
(4 a)
(4 b)
37
where the numerator and denominator of Eq. (4b) shows the measured output, , and
perturbation of the parameter, , used. The sensitivity matrix can be computed by varying any
number of properties (material, geometrical, boundary, etc.) either directly or through
perturbation methods. For the purposes of this criterion, the material properties will be the focus
of variation, with a proper range defined by the user and the outputted natural frequencies will be
compared with each of the defined FE models.
Ultimately, the combination of all four parts will help in the final selection of a simplified
model.
3.3 Implementation of Four-Part Criterion into Cavern 2
3.3.1 Model Generation
In order to implement the four-part criterion formulated in Section 3.2, three models of
Cavern 2 are constructed using various techniques, shown in Figure 3-1. The first model (M1) is
developed using 3D scan data taken on-site from 2002, with no simplifications made to the
cavern geometry and will be used unless one of the other simplified models is deemed
appropriate. On one side of the model, shown in Figure 3-1(a), there is an extra section, which is
included to aid in the meshing and consequently reduces the mesh skewness. The second model
(M2) took advantage of the 3D scan data as reference values only, and simplified the geometry
by using completely vertical walls, a constant roof slope, isosceles triangles for the columns, and
the plan view was modified for ease of meshing. The third model (M3) is developed based on
simplified drawings given by Li et al (2009) for the plan view and Yue et al (2010) for the side
view. The third model is of interest considering most FE models are generally developed using
drawings as the only available reference and in some cases these drawings lack accuracy in
dimensions and details. One important difference between M3 and the other two models is the
38
exclusion of the hole at the entrance of Cavern 2. Considering neither published work describes
this area, it was intentionally left out to show how this model varies significantly due to differing
amounts of available information. An illustrative comparison of the models is shown in Figure 3-
1 and includes for each model a side view, top view and isometric view from the bottom.
Figure 3-1: Illustrative comparison of the models developed: (a) M1, (b) M2, and (c) M3
C1
C4
C3 C2 C1
C2C3
C4
C1C4
C2C3
C1
C2C3
C4
C1
C3C2C4
C1
C2
C3
C4
C1: Column 1 C2: Column 2 C3: Column 3 C4: Column 4
(c) Model 3: M3
Side View
Top View
X
Y
Y
Z
Y
Z
Y
Z
X
Y
X
Y
XY
Z
Isometric View
Isometric View
Isometric ViewTop ViewSide View
Top ViewSide View
(b) Model 2: M2
(a) Model 1: M1
XY
Z
XY
Z
39
3.3.2 Implementation of Four-Part Criterion
All three models are developed in a FE program called ANSYS© and are meshed using a
combination of user-defined and computer generated meshing with three solid element types: 10-
node tetrahedral, 20-node hexagonal and 13-node pyramidal being applied at specific locations
in the cavern. The number of elements used is 34,236, 15,593, and 14,135, with a mesh skewness
of 0.999, 0.947, and 0.870, for M1, M2, and M3 models, respectively. By observation, the M3
model has the least number of elements and reduced mesh skewness. Considering the simplicity
of the model, the fact that this result occurred is understandable.
When performing the calculations for the stress levels, natural frequencies, mode shapes,
and sensitivity analyses, the general assumption is as follows: the boundary conditions are
completely fixed on all sides of the model, and the material properties are 4.5 GPa, 2300 kg/m3,
and 0.25 for the elastic modulus, density, and Poisson’s ratio, respectively, based on published
values (Yue, Fan et al. 2010).
For comparative purposes of stress levels, there are five different measurements used in
the TRD equation (Eq. ( 3 )): Z-direction, Von Mises’, first principal, second principal, and third
principal stresses. Each stress level is taken as the largest absolute value in the model, with
lateral stress attributed by only one meter from the surrounding rock. The total weights used are
0.500, 0.125, 0.125, 0.125, and 0.125, for each of the stresses mentioned above. The stress in Z-
direction carries a higher weight in the TRD equation, since the self weight of the cavern could
cause stress concentrations to form in specific locations. These locations could be identified later
as possible areas of crack formation and will need to be investigated. All the other stress levels
carry the same significance, thus are equally weighted.
40
Ultimately, the values from M1 are used as the reference stress values and the M2, M3
values are the comparative stress measurements. The results for each stress level is shown in
Table 3-1, with the total relative deviation for M2 and M3 are 0.12 and 0.32, respectively. It is
interesting to note, for certain stress values the M3 model does have a closer approximation to
the M1 model, but this occurs for only two of the five cases where the difference with the M2
model is not large. In the instances where the M2 model has a closer stress value to the M1
model, there seems to be a larger difference between the M1 model and the M3 model, which
causes the TRD value to become higher for the M1 model. This gives reason to suspect that even
though the M2 model is simplified, there are stress levels similar to those in the M1 model and
may be an appropriate simplified model.
Table 3-1: Stress values and respective TRD values for M2 and M3
Stress Calculation Weighting Factor M1 (MPa) M2 (MPa) M3 (MPa)
Z-Direction, z 0.500 2.69 3.06 2.19
Von Mises’, M 0.125 -3.71 -3.25 -2.33
First Principal, 1 0.125 1.22 1.05 1.10
Second Principal, 2 0.125 -0.75 -0.74 -0.45
Third Principal, 3 0.125 -3.22 -3.56 -2.54
TRD -- 0.12 0.32
When comparing the stress contours, there are definite similarities between all three
models, however, there are inconsistencies between the M3 model and the other two models. The
columns in the M3 model have substantially different stress contours than those in the M1 model,
which raises serious concerns over the validity of the model. This could have occurred due to the
41
simplicity of the model, i.e. length and shape of columns varies drastically from the other two
models. While there are consistencies in the column contours between the M1 and M2 models,
considering M2 is overly simplified, it will likely not obtain the right qualitative behavior.
When analyzing the dynamic properties of the cavern models, only the global modes are
investigated and their corresponding mode shapes. Table 3-2 shows the first, second, and third
natural frequencies for each model. Only the first three natural frequencies are used in the
comparison because generally in civil infrastructure the most contributing modes are the first few
and the others have little to no effect on the dynamic characteristics of the structure under
excitation loads. Fortunately, the mode shapes for each model are the same: the first frequency
and second frequency involve a local mode around the column near the entrance and back right
corner, respectively, while the third mode contains movement in the roof only. This leaves the
TRD to be the significant comparative factor for the dynamic properties. The TRD value for the
M3 model is higher than the M2 model, which may be attributed to the geometry, specifically the
roof height and consequently the column heights.
Table 3-2: Natural frequencies and respective TRD values for M2 and M3
Natural Frequency Weighting Factor M1 (Hz) M2 (Hz) M3 (Hz)
f1 0.33 11.38 10.44 8.14
f2 0.33 11.52 10.96 9.46
f3 0.33 14.75 15.03 11.05
TRD -- 0.06 0.24
For the sensitivity analysis of the dynamic characteristics, the material properties, i.e.
elastic modulus, density and Poisson’s ratio, were varied locally. The sections of the cavern are
42
segmented as shown in Figure 3-2 with both 3D scan data and visual inspection playing a role in
the segmentation. Since the opening in the roof receives the highest environmental influence, it
was separated into its own segment (C4), with the other segments (C1, C2, C3) separated in a
somewhat arbitrary, but still using the visual inspection of the cavern as a reference. The elastic
modulus is used as the primary focus for comparative purposes (Figure 3-5), since the sensitivity
of the natural frequencies to the elastic modulus is higher ( 18.33%) than both the density (
7.25%) and Poisson’s ratio ( 0.52%) combined. Only a 33%, 20%, and 15% variation is
used for the elastic modulus, density and Poisson’s ratio, respectively, i.e. deviations from the
material properties obtained from published works is assumed small.
A visual representation of the relationships between the Poisson’s ratio or density and the
natural frequency is shown in Figure 3-3 and Figure 3-4, respectively. The largest variance in the
natural frequencies is observed when the models are assumed to be one solid (C), with similar
responses shown by each model. However, when the models are broken down into four segments
(C1, C2, C3, and C4) more significant comparisons can be made. Even though the variances
within each segment are not large, several conclusions can be drawn from each figure. The
correlations between the M1 model and the other models when varying Poisson’s ratio are worse
than any of the other material properties. Considering the maximum change in the natural
frequencies for any of the segments is only 0.20 Hz, this is not a huge cause for concern. Now
looking at the density sensitivity results, the correlation is better than the Poisson’s ratio, but not
by much. The behaviors are similar for most of the segments between the M1 model and the M2
model, but there is no identifiable similarity between the M1 model and the M3 model. Now
looking specifically at Figure 3-5 and the variation of elastic modulus, the comparison of the M1
model and the M3 model shows a similar response for one section for each of the natural
43
frequencies observed. For the first and third natural frequencies the section is C2, while for the
second natural frequency the section is C3. The story is completely different when comparing
the M1 model and the M2 model. By examining the graphs side by side, it is clear there is a
similarity in behavior for both models for the first and second natural frequencies. The only
evident variation is between their third natural frequencies. However, considering the same
section (C2) has the largest influence, this inconsistency is not a cause for major concern.
Figure 3-2: Arrangement of material property assignment for the 4-sections for (a) M1, (b) M2 and (c) M3
C2
C3
C4
C1 Y
X
C1 Y
X
C2
C3C4
Y
X
C4
C1
C3
C2
(a) M1
(b) M2
(c) M3
44
Figure 3-3: Sensitivity Analysis on the Dynamic Properties by varying Poisson’s Ratio
-20 -10 0 10 2011.37
11.38
11.39
11.4
11.41
11.42
11.43
1st N
atur
al F
req
(Hz)
11.5
11.52
11.54
11.56
11.58
11.6
2nd N
atur
al F
req
(Hz)
14.7
14.72
14.74
14.76
14.78
14.8
14.82
3rd N
atur
al F
req
(Hz)
M1
-20 0 2010.4
10.42
10.44
10.46
10.48
10.5
Percent Change of Poissons Ratio (%)
10.94
10.95
10.96
10.97
10.98
10.99
11
12.4
12.45
12.5
12.55M2
-20 -10 0 10 20
8.1
8.12
8.14
8.16
8.18
8.2
8.22
9.4
9.42
9.44
9.46
9.48
9.5
9.52
11
11.02
11.04
11.06
11.08
11.1
11.12M3
C C1 C2 C3 C4
45
Figure 3-4: Sensitivity Analysis on the Dynamic Properties by varying the density
-10 -5 0 5 1010.8
11
11.2
11.4
11.6
11.8
12
12.2
1st N
atur
al F
req
(Hz)
10.5
11
11.5
12
12.5
2nd N
atur
al F
req
(Hz)
14
14.5
15
15.5
16
3rd N
atur
al F
req
(Hz)
M1
-10 -5 0 5 109.8
10
10.2
10.4
10.6
10.8
11
11.2
Percent Change of Density (%)
10.4
10.6
10.8
11
11.2
11.4
11.6
11.8
12
12.2
12.4
12.6
12.8
13
13.2
M2
-10 -5 0 5 10
7.8
8
8.2
8.4
8.6
8.8
9
9.2
9.4
9.6
9.8
10
10.2
10.5
11
11.5
12M3
C C1 C2 C3 C4
46
Figure 3-5: Sensitivity Analysis on the Dynamic Properties by varying the elastic modulus
-20 0 209
10
11
12
13
1st N
atur
al F
req
(Hz)
9.5
10
10.5
11
11.5
12
12.5
13
13.5
2nd N
atur
al F
req
(Hz)
12
13
14
15
16
17
3rd N
atur
al F
req
(Hz)
M1
-20 0 208.5
9
9.5
10
10.5
11
11.5
12
Percent Change of Elastic Modulus (%)
9
9.5
10
10.5
11
11.5
12
12.5
12
13
14
15
16
17
M2
-20 0 206.5
7
7.5
8
8.5
9
7.5
8
8.5
9
9.5
10
10.5
11
9
10
11
12
13M3
C C1 C2 C3 C4
47
3.3.2.1 Selection of Appropriate Model
In this section the selection of an appropriate model is presented. First, looking at the
meshing criteria, the M3 model has a decreased mesh skewness and number of elements. This,
unfortunately, is not an adequate reason to use this simplified model. By looking next at the
static and dynamic criterion, then comparing the results with the M1 model shows clearly the M3
model is not appropriate for future analysis. One reason comes from the TRD values of 0.32 and
0.24 for the static and dynamic analysis, respectively. The next reason for exclusion comes from
the sensitivity analysis. Since there is only one section behaving similarly with the reference
model, the M1 model, this shows there is a low correlation between the two models and now
definitely should be disqualified as a possible simplified model based on the accuracy level
trying to be attained. Developing the model from only published sources and not having any on-
site investigative results prevented this model from being a suitable choice.
Looking now at the M2 model, a higher correlation to the M1 model is evident. Not only
looking at the TRD values, which are 0.12 and 0.06, for static and dynamic, respectively, but
also taking a look at the qualitative level, illustrates a closer resemblance between the two
models. The stress contours around the areas of high concentration, mainly the columns,
demonstrate there is hardly any variance between the two models. A final comparative measure
is the sensitivity analysis on the dynamic properties. By examining Figure 3-5 closely, even with
variation in magnitude of the change, the majority of the sections show the same behavior as the
material properties change for each section. In the end, taking into consideration both the
qualitative and quantitative measures, from above, it is concluded M2 is an appropriate model to
use for further analysis and will be the model calibrated in the next section.
48
3.4 Calibration of Simplified Model
Calibration of the FE model will be broken down into two parts. First, the appropriate
boundary conditions will be established for the model and extracted natural frequencies will be
compared with the second on-site investigation. Then, the material properties and saturation
levels will be examined for the model, based on current environmental conditions during each of
the on-site investigations.
3.4.1 Determination of Boundary Conditions
Since Cavern 2 is below the ground level, a fixed support for the surrounding rock mass
is an appropriate assumption. However, due to the proximity of Cavern 2 to both Cavern 1 and
Cavern 3, a completely confined rock mass may not be suitable. The resulting boundary
conditions take into consideration this relationship and through the use of both the 3D scan data,
plan view (Li, Yang et al. 2009), and side view (Yue, Fan et al. 2010), a proper set of boundary
conditions is established. Ultimately, it is determined the areas of close proximity to the two
surrounding caverns should be treated as free supports and the other surrounding rock mass as
fixed supports, shown in Figure 3-6.
Figure 3-6: Boundary Conditions for the FE model
C1
C2C3
C4
Front
Back
Lef
t Right Front
BackRight
Left
FIXED FIXED
FIXED FIXEDFREE FREECavern 1
Cavern 3
49
The natural frequencies identified from the ambient vibration measurements are used to
calibrate the FE model. White noise signals are inputted to the model through the supports and
the columns responses at the sensor locations are then computed, from which the natural
frequencies are extracted using FDD. The analytical natural frequencies are finally compared
with the experimental ones. The results gave the first and second natural frequencies from
column 1 as 9.96 Hz and 17.19 Hz, and column 3 as 17.97 Hz and 25.78 Hz, respectively.
Considering the difference between model and on-site frequencies is higher than 20% for both
columns, as tabulated in Table 3-3, the determination of proper material properties and saturation
levels still needs to be investigated.
Table 3-3: Comparison between On-Site and FE Model Results for Column 1 and Column 3
Natural Frequency On-Site (Hz) FE Model (Hz) Error (%)
Column 1 f1 12.70 9.96 -21.57
f2 25.43 17.19 -32.40
Column 3 f1 22.95 17.97 -21.70
f2 35.43 25.78 -27.24
3.4.2 Determination of Water Saturation Levels and Material Properties
Conditions during both on-site investigations, i.e. water dripping down from the ceiling
between the front column and the wall connecting Cavern 1 and Cavern 2, confirms the
argillaceous siltstone is saturated. However, there is uncertainty to the actual saturation levels in
the surrounding rock mass. Four saturation levels (2m, 4m, 6m, and 8m) below ground level are
investigated with the material properties from Yue et al (2010) being utilized. The inclusion of
various material properties is assumed due to the different conditions of the cavern on a day-to-
50
day basis. The cavern itself, for modeling purposes, is broken down into 3 sections: front (C3
and C4, Figure 3-2), middle (C2, Figure 3-2) and back sections (C1, Figure 3-2), with each
section’s properties changing during each analysis. From the sensitivity analysis conducted
during the four-part criterion, the dynamic properties have highest sensitivity to the elastic
modulus, which is now varied in each section. Each section’s elastic modulus ranges from 3.5
GPa to 5.5 GPa, based on previously published works, with property set 1 having the smallest
elastic modulus and property set 27 having the largest, with a constant density of 23 kN/m3 and a
Poisson’s ratio of 0.25.
From the results of the analysis the variation of the material properties over the separate
sections has a higher impact on the natural frequencies for the columns than the changing
saturation level, as seen in Figure 3-7 and Figure 3-8. Looking specifically at the first and second
natural frequencies for both columns, as the elastic modulus increases and the saturation level
decreases this causes an increase in the natural frequency to be observed. This can be identified
as an almost linear trend for the first natural frequencies, but there seems to be no consistent
trend shown for the second natural frequency. The only visible consistency is for a majority of
the property sets the eight meter saturation level outputs a lower natural frequency, while the two
meter saturation level outputs are higher. Nevertheless, the saturation level is determined to be 2
meters and the elastic modulus as 4.5 GPa for the front and 5.5 GPa for the middle and back
sections (property set 18), based on the results from the FE model. This corresponds to higher
elastic modulus assignments than those previously published by Li et al. (2009). The
corresponding first and second natural frequencies according to the FE model are 10.16 Hz and
19.53 Hz for column 1, and 18.95 Hz and 26.76 Hz for column 3. Table 3-4 shows a sample of
the results obtained for an assumed 2 meters of saturation and the percent difference between the
51
on-site measurements. The first (1) and last (27) property sets are included to show the upper and
lower limits of the results, while property set 14 shows the results obtained assuming the material
properties of published works (Li, Yang et al. 2009). It was determined that property set 18 has
the lowest percent difference among all the sets, with a maximum percent difference of 23 for
both columns. However, this percent difference is reduced to 21 percent when the densities for
the front, middle and back sections are reduced to 21.8 kN/m3 (18-modified), to correspond with
previously published works (Li, Yang et al. 2009), but the saturated section remains the same.
Considering the instability of the second natural frequency during the on-site investigation, the
close agreement for the first natural frequency is adequate for this calibrated FE model.
Table 3-4: Sample Results for 2 Meter Saturation Level and Modified Property Set 18
Property Set
Elastic Modulus Assignment (MPa)
Column 1 Column 3
Sat. Front Middle Back f1 (Hz) f2 (Hz) f1 (Hz) f2 (Hz)
1 3.03 3.50 3.50 3.50 8.79
(-31%) 16.02
(-36%) 16.02
(-29%) 23.44
(-31%)
14 3.03 4.50 4.50 4.50 9.57
(-25%) 18.55
(-26%) 17.19
(-23%) 25.2
(-26%)
18 3.03 4.50 5.50 5.50 10.16
(-20%) 19.53
(-23%) 18.95
(-16%) 26.76
(-21%)
18-Modified
3.03 4.50 5.50 5.50 10.16
(-20%) 19.92
(-21%) 19.34
(-13%) 27.15
(-20%)
27 3.03 5.50 5.50 5.50 10.16
(-20%) 18.95
(-25%) 19.14
(-15%) 25.78
(-24%)
52
Figure 3-7: Variation of Saturation Levels and Material Properties Results for Column 1 for the 1st and 2nd natural frequency and material definitions for each property set
15
16
17
18
19
20
2nd F
req
(Hz)
8.5
9
9.5
10
1st F
req
(Hz)
2m 4m 6m 8m
0 5 10 15 20 253
3.5
4
4.5
5
5.5
6
Property Set
Ela
stic
Mod
ulus
(M
Pa)
Front Middle Back
53
Figure 3-8: Variation of Saturation Levels and Material Properties Results for Column 3 for the 1st and 2nd natural frequency and material definitions for each property set
22
23
24
25
26
27
2nd F
req
(Hz)
15
16
17
18
19
1st F
req
(Hz)
2m 4m 6m 8m
0 5 10 15 20 253
3.5
4
4.5
5
5.5
6
Property Set
Ela
stic
Mod
ulus
(M
Pa)
Front Middle Back
54
3.5 Conclusions
In order to perform further analysis on Cavern 2, such as when and how the cavern will
fail, it became imperative to create a global FE model of the structure. However due to the
complex nature of Cavern 2, the question was posed: Is a simplified model a reasonable option
or should a more intricate model be used for further analysis? It is a commonly known fact that a
simplified model helps to reduce the number of elements, computational effort, and the
probability of erroneous results from irregular element definition, factors which greatly aids the
challenging analysis mentioned above.
Within this chapter the development and application of a four-part criterion enabled this
question to be answered, and the determination of an appropriate level of simplification for a
defined FE model was quantified. Understanding the historical context of the Longyou Grottoes,
the inclusion of both static and dynamic characteristics into the criterion resulted in a well-
rounded comparative analysis. The criterion helped to distinguish between the obvious pros and
cons of having a simplified or an exact model and choose, with confidence, a model for future
analysis. Even though the four-part criterion was developed only for Cavern 2, there is no reason
to assume it could not also be applied to other underground rock structures. The essential
characteristics are the same between Cavern 2 and other underground rock structures, i.e. high
volume of visitors, complex geometries, unknown material properties, and complicated boundary
conditions, implying the application of the four-part criterion would be useful for other structures.
Regarding Cavern 2, three models were developed and one was selected to be calibrated.
When comparing the models, it was assumed the M1 model, based on the exact geometry would
be selected for further use unless one of the simplified models, the M2 model or the M3 model,
showed a high correlation to the M1 model and was deemed appropriate. Based on the different
55
model definitions, it became abundantly clear that the use of both 3D scan data and published
works were crucial in the determination of modeling parameters and depicted obvious
advantages of having reliable data when determining geometry, material properties and boundary
conditions. Purely relying on published works, the M3 model, showed not to be sufficient in
determining the modeling parameters based on high TRD values for both the static and dynamic
criterion. The sensitivity analysis conducted on the dynamic properties revealed for all the
models treating Cavern 2 as only one solid has an increased impact on the resulting natural
frequencies than separating the cavern into different solids. Ultimately, when comparing the M1
model with the M2 model there was a close correlation of the TRD values for both the static and
dynamic parts of the criterion. In addition, the M2 model has a lower mesh skewness and
element count compared with the M1 model. Combining the two points, this allowed for a
confident selection of the M2 model.
The M2 model was then calibrated using on-site ambient vibration measurements.
Frequency domain decomposition was applied to the outputted measurements from the FE model
and the natural frequencies were extracted. Taking into consideration the proximity of Cavern 1
and Cavern 3, the use of published works and the on-site investigation aided in the assignment of
proper boundary conditions. Saturation levels and material properties were determined by
assuming a variation in both, based on on-site environmental conditions, and showed the elastic
modulus to be higher than previously published. Based on the location of the columns relative to
the assigned saturation levels, it was observed the change in material properties has a higher
influence on the natural frequencies for both columns. In the end, a calibrated model based on
the M2 model was developed and now can be used with the utmost confidence during the
execution of the future non-linear analysis performed in the coming chapters.
56
CHAPTER 4 Comparative Analysis of Two Developed Local Models with
the Calibrated Global Model
4.1 Introduction
Considering the size of underground rock caverns, using the information gathered from
on-site testing and the creation of a FE model to investigate the structural stability of the entire
structure becomes challenging. This has been clearly shown in research conducted on numerous
underground rock caverns including Bet-Gurvin in Israel (Hatzor, Talesnick et al. 2002),
Xiaolangdi Powerhouse in China (Huang, Broch et al. 2002), Tel Beer Sheva in Israel (Hatzor
and Benary 1998) and the Zedekiah Cave in Israel (Bakum-Mazor, Hatzor et al. 2009).
Nevertheless, if it is possible to recognize areas having increased damage or crack
propagation, on-site testing can commence on these sections only. This isolation gives rise to the
possibility of reducing the generated model of the entire structure or in this case Cavern 2. There
are two main advantages for creating these submodels or local FE models. First, when the local
FE model is developed this will ultimately cause the number of elements used to decrease and
will correspondingly increase the mesh consistency. Second, when extracting the dynamic
properties from the local FE model, a higher probability of correlation with the on-site
experimental results can be achieved. However, in order to obtain these geometrically and
dynamically realistic local FE models, the use of model updating techniques should be
incorporated (Zarate and Caicedo 2008). Model updating techniques do have some fairly crucial
limitations including increased computational effort and difficulties finding unique solutions, but
considering the size and limited number of structural parameters to be identified, they are prime
candidates for implementation into the local FE models, where these limitations become
irrelevant.
57
Several examples and implementations of model updating techniques, which determine
the structural parameters for the global model and local models, include: inverse eigen sensitivity
(Gentile and Saisi 2006; Aoki, Sabia et al. 2008), Douglas Reid (Gentile and Saisi 2006),
pseudo-inverse method (Wu and Li 2004), Bayesian Estimation technique (Wu and Li 2004),
subproblem approximation method (Jaishi and Ren 2005; ANSYS 2010) and first-order
optimization method (Jaishi and Ren 2005; Wang, Li et al. 2009; ANSYS 2010; Wang, Li et al.
2010).
Recently, there have been successful attempts in the field of local or sub- modeling. First,
local models of the Basilica of Pilar in Zaragoza, Spain were created and compared to a
developed global model. It was determined the local model could possibly be appropriate when
future experimentation is conducted and material models are further investigated (Romera,
Hernandez et al. 2008). Second, a sub-model of the central buckle of the Runyang Suspension
Bridge in China was created. The model was compared against field measurements and deemed
appropriate as a theoretical reference for analyzing and designing rigid central buckles in the
future (Wang, Li et al. 2009). Third, solid-to-solid and shell-to-solid sub-models were created for
solder joint reliability in electronic structures and were calibrated with experimental data. Based
on these models, multiple failure analysis and transient dynamic analysis were conducted
without interference of extra computational effort from global models (Lall, Gupte et al. 2007).
Next, the simplification of a beam-column joint was used to detect and locate damage in a
symmetric structure (Titurus, Friswell et al. 2003). Furthermore, the optimization of fuselage
panels for aircraft structures was instrumental to determine the effective parameters for a
globalized model of the aircraft itself (Price 2000). Finally, the creation of substructural system
identification methods has been beneficial to determine the structural parameters of a system by
58
dividing the system into multiple segments. The employment of genetic algorithms creates the
identification method, which is both easy to implement and has desirable characteristics because
it is a global search method (Koh, Hong et al. 2003).
For the purpose of this chapter, two local models will be developed for Cavern 2 based
on the two columns tested during the on-site investigations. Based on the geometric complexity,
size, and lack of significant research, on Cavern is a ripe candidate for a comparative analysis
between local and global models. The global model used in the analysis will be the one defined
in the previous chapter, but both local models will be developed in this chapter. The local models
will be constructed using two optimization algorithms: subproblem approximation and first-order
optimization methods, due to their ease of implementation into the ANSYS© program. Each
algorithm will be explained in more detail in the next section. Based on the results from the on-
site investigation, two different objective functions, frequency residual only and frequency
residual plus the mode shapes will be updated for each local model with the hope of
understanding the effects of both optimization algorithms and objective functions on the updated
structural parameters. However, considering the limited number of measurement locations, the
number of updating parameters will be simplified using beam theory and the complete derivation
is shown in Appendix A. Ultimately, the primary goal of this chapter is to directly compare the
local and global models using a time history analysis over a 50-year period, assuming the
material model is elastic and the material is degrading over time (Guo, Li et al. 2005).
4.2 Optimization Techniques
One of the many purposes of applying optimization algorithms is to reduce the error
between experimental and FE model outputted results by adjusting the assigned structural
parameters, i.e. elastic modulus, density and Poisson’s ratio. During the development of the two
59
local models, shown in the next section, two optimization algorithms: subproblem approximation
method (Jaishi and Ren 2005; ANSYS 2010) and first order optimization method (Wang, Li et al.
2009; ANSYS 2010; Wang, Li et al. 2010) are implemented.
4.2.1 Subproblem Approximation Method
The first optimization algorithm used in the determination of the structural parameters for
the local models is the subproblem approximation method. The subproblem approximation
method can be described concisely as an advanced-zero method, which ultimately means there is
only a requirement on the values of the dependent variables and not their derivatives. Two
concepts are important to consider for the subproblem approximation method: (1)
approximations are used for the objective function and state variables, or the dependent variables
which change as the design variables change, and (2) a conversion of the constrained
optimization problem to an unconstrained problem is necessary to apply the approximations
developed.
When executing this method into ANSYS© a relationship is established between the
objective function and the design variables or the independent variables which directly affects
the design objective by curve fitting. This relationship is achieved by calculating the objective
function for a specified number of design sets with designated design variables and through these
calculations a least squares fit is made between the data points. The resulting curve, i.e. surface,
is called an approximation of the design objective. After each optimization computation, a new
data point is generated and the objective function approximation is concurrently updated.
Consequently, it is this approximation which is minimized instead of the actual objective
function. In this same manner, the state variables are treated the same. For each of the
60
optimization approximations, a quadratic plus cross terms fit is used for the objective function,
but a quadratic fit is used for the state variables.
Since approximations are used for the both the objective function and state variables, this
means that the optimum design will only be as good as the assigned approximations. In order to
create approximations with higher accuracy, the implementation of random design sets is
conducted, before the optimization algorithm is executed, to establish accurate approximations of
the state variables and the objective function. Since a high number of random designs are
required to generate these approximations, this may cause an increase in the time to solve the
optimization problem. However, this time could be reduced if the infeasible design sets are
discarded as they appear in the random design generation process.
The second concept in the subproblem approximation method is to transform the
constrained optimization problem to an unconstrained one by introducing penalty functions.
Mathematically, the added penalty functions are formed through a truncated Taylor series
expansion. The creation of the unconstrained optimization problem for the subproblem
approximation method uses only the values from the dependent variables, i.e. the objective
function and state variables, and their derivatives are approximated by means of the least squares
fitting. The unconstrained optimization equation is shown below:
Minimize: , Π Π ,
( 4 )
where , represents the unconstrained objective function that varies with the design
variables, , and parameter, . is the penalty function used to enforce design variable
61
constraints; is the number of design variables and , , are the number of state
variables; and , , and are the penalty functions for state variable constraints. A sequential
unconstrained minimization technique is used to solve specific design iterations. Convergence is
assumed to occur when the present, previous or best design set is feasible and the difference
between two objective functions is below a specified tolerance. Only at this instance will
termination occur and the structure’s parameters are assumed optimized.
4.2.2 First-Order Optimization
The other optimization algorithm used to determine the structural parameters of the local
models is the first-order optimization method. Similar to the subproblem approximation, the
first-order optimization method converts the objective function from a constrained problem to an
unconstrained problem through the use of penalty functions. However, unlike the subproblem
approximation method, approximations are not created, but the actual FE representation is
minimized instead. If you were to directly compare the subproblem approximation method and
the first-order optimization method’s computational effort, then it is generally known that the
first-order optimization method has a higher computational effort, but consequently it is more
accurate. However, it is important to consider that high accuracy does not necessarily mean the
best solution will be obtained.
There are several situations to be aware of when running an optimization analysis using
the first-order optimization method. Based on the evolution of the design space, there is a
possibility this optimization method may not converge in a feasible design space. This means
the first-order optimization method has a higher likelihood of reaching a local minimum, which
may not be physically reasonable. This happens due in large part to the definition of the initial
start values and how the optimization algorithm goes from one existing point in a design space
62
and works its way to a minimum. If a starting point is too close to a local minimum, then it is
likely this point will be found instead of the global minimum. If this occurs, it is recommended
to generate random designs and use the feasible design sets as the initial starting points when
performing the optimization algorithm. Another situation to be aware of is the objection function
tolerance. If the tolerance is too tight this might cause an increased number of iterations to be
performed unnecessarily close to the optimal design. Since this method solves the actual FE
representation and not an approximation, it will attempt to find an exact solution based on the
defined tolerances.
Looking more specifically at the mathematical construct of the first-order optimization
method, the use of gradient representations for the dependent variables with respect to the design
variables is created. During each iteration, gradient calculations, which employ either the
steepest descent or conjugate direction method, are performed to determine a search direction
and a line search strategy is assumed in order to minimize the unconstrained problem. Ultimately,
each iteration is composed of a number of sub-iterations, which include both search direction and
gradient computations. Consequently, each optimization iteration performs several analysis loops
before moving to the next iteration.
The unconstrained optimization equation for the first-order optimization method is given
here:
Minimize: ,ΠΠ
,
( 5 )
where , is the dimensionless unconstrained objective function; , , , are the
penalties applied to the constrained design and state variables; , , , are the number of
63
design and state variables; and Π refers to the objective function value that is selected from the
current group of design sets. As in the subproblem approximation method, convergence and
ultimately termination occurs when the present, previous or best design set is feasible and the
difference between two objective functions is within a specified tolerance.
4.3 Development of the Two Local Models
4.3.1 Derivation of Updating Parameters
For the local models, two models (column 1 and column 3, shown in Figure 4-1) are
developed for comparison with the global model. Each local model has the same cross-sectional
area and a similar height with the global model. The goal is to define the local models to have the
same dynamic properties as the on-site investigations results presented in Chapter 2. Based on a
preliminary analysis, it was determined that the roof slope does not affect the resulting natural
frequency enough to be included as part of the simplified models. Ultimately, the height was
determined from the global model by measuring from the ground level to where the column’s
midpoint intersects the ceiling.
Unfortunately when optimizing or updating the models, the number of updating
parameters is limited by the amount of information extracted from the on-site measurements
(Brownjohn, Xia et al. 2001). Given there were only three locations for each column tested
during the on-site experiments, it is important to determine the appropriate number of updating
parameters. In all the updating situations presented in this section, the number of parameters is
reduced to the significant parameters in order to produce a genuine improvement in the modeling
of the columns and reduce the ill-conditioning of the updating problem.
64
Figure 4-1: Geometrical dimensions for (a) Column 1 and (b) Column 3 for the local models developed of each column
Traditionally the use of linear springs would be added to the top of both local models
based on confinement of the columns. However, the number of updating parameters for such an
arrangement is nine: Elastic Modulus ( ), density ( ), Poisson’s Ratio ( ), translational spring
constants in all directions ( , , ) and rotational spring constants in all directions
( , , ). Due to the limited information extracted from the on-site data, a reduction in
the above parameters is investigated. This reduction has a twofold purpose: to ensure the
updating procedure returns significant and reliable results, while also decreasing the
computational effort needed to solve the updating procedure.
This reduction is completed by using classical beam theory, which explains any beam-
spring model is equivalent to a beam-beam model (shown in Figure 4-2). The basics of the
concept are explained below but a complete derivation of the conversion used in this section is
shown in Appendix A. Ultimately, the conversion to a beam-beam model simplifies the number
(a) Column 1 (b) Column 3
Plan View Side View
2.71
m
2.78
m
8.5
0m
10.0
0 m
1.38 m 1.70 m
1.38 m1.70 m
Plan View Side View
Geometrical PropertiesIx = 0.76 m4
Iy = 0.20 m4
J = 0.96 m4
A = 1.87 m2
Geometrical PropertiesIx = 1.03 m4
Iy = 0.38 m4
J = 1.41 m4
A = 2.38 m2
X
Z
X
Z
X
Y
X
Y
65
of updating parameters to four. Since the columns can be assumed to be long and slender, the use
of the basic principle of force or moment is equal to stiffness multiplied by displacement or
rotation is applied. When the displacement or rotation is set to one unit, then the force or moment
is equal to the stiffness. This principle along with the beam-stiffness matrix is utilized in the
derivation of the equivalent stiffness equations. The free-body diagram for the derivation is
shown in Figure 4-2, and the resulting equations for the translational spring constants are shown
in Equations ( 6 ) and ( 7 ), respectively, and the rotational spring constants are shown in
Equations ( 8 ) and ( 9 ), respectively.
Figure 4-2: Graphical representation of the conversion of the columns to beam-beam models for (a) a translational spring and (b) a rotational spring
3 12 12 18 3
3,
( 6 )
F
k1, E1, I1, L1
ks
kT
E1, I1, L1 E2, I2, L2
(a)
M
krs kT
E1, I1, L1 E2, I2, L2
(b)
k1, E1, I1, L1
E1, I1, L1 E2, I2, L2
F
d1
d2
d3
d4 d6
d5
E1, I1, L1 E2, I2, L2
d1
d2
d3
d4 d6
d5
M
Beam-Spring Model Beam-Beam Model Free Body Diagram
66
( 7 )
4 4 6,
( 8 )
( 9 )
where is or depending on the direction of the spring, is the elastic modulus, is the
shear modulus, is the polar moment of inertia, is the moment of inertia in the direction of
the spring and is the length of the column being modeled, and is the elastic modulus, is
the moment of inertia in the direction of the spring and is the length of the added beam.
For the purposes of beam-beam model developed, the moment of inertia for both beams
are the same and the length of the added beam is the length of the column. Since the spring
constants are a function of only the elastic modulus and Poisson’s ratio, each will be indirectly
determined to reduce the number of updating parameters in the optimization algorithms.
4.3.2 Application of Optimization Algorithms to Determine Structural Parameters for the
Two Local Models
Based on the simplicity of the local models developed in this section, a direct comparison
of the optimization algorithms, i.e. subproblem approximation method and first-order
optimization method, using two different objective functions can be performed with relative ease.
Because all optimization algorithms have their benefits and draw-backs, having a model with a
relatively simple geometry and boundary conditions allows for observations and comparisons to
67
be made between the material properties assignments for both columns and each optimization
method. In addition, the results can be compared directly with the calibrated global model.
Now specifically looking at the definition of the two objective functions, the value
minimized by the optimization algorithm, will include both the natural frequencies and mode
shapes extracted from the on-site investigations. Considering that the mode shapes have less
accuracy than the natural frequencies, one objective function reflects only the natural frequency
residual (Eq. ( 10 )), while the other reflects both the natural frequency residual and the mode
shapes (Eq. ( 11 )). In order to correlate the mode shapes from the analytical and experimental
data, the use of the modal assurance criteria (MAC) (Allemang 2003) is applied. The MAC value
mathematically represents the consistency between the analytical and the experimental mode
shapes (Teughels, Maeck et al. 2002). The two objective functions are shown in Equations ( 10 )
and ( 11 ):
( 10 )
1 , ( 11 )
where is the number of identified modes, is the natural frequency identified from
experimental results, is the natural frequency from the model, and is the modal
assurance criteria between the experimental and the mode shape identified from the analytical
model.
In order to ensure the design sets are within a feasible range, constraints are applied to
each design parameter to keep the variances within reasonable limits. The initial value, upper
bound, and lower bound of each design parameter are shown in Table 4-1. Considering the added
68
beam is designed to act as a spring, which is weightless, the assigned values for both the density
and Poisson’s ratio is assumed to be a value close to zero, in this case 0.01.
Table 4-1: Parameters used in optimization functions with initial values, and the upper and lower bounds
Parameter (MPa) (kg/m3) (Pa) (kg/m3)
Initial 4.50 2300 0.25 6E3 0.01 0.01
Lower Bound 4.00 2000 0.10 0 --- ---
Upper Bound 6.75 2600 0.40 8E30 --- ---
It is important to note, when implementing the subproblem approximation algorithm into
ANSYS©, there will be set of random designs generated in order to better define the
approximations for both the objective function and state variables (the updated structural
parameters). The implementation of the first-order optimization algorithm does not include
random design generation, but will begin its first optimization from the initial values mentioned
in Table 4-1.
In total four different analyses were run for each column and a summary of the results is
shown in Table 4-2, while the corresponding spring constants are shown in Table 4-3. From both
tables several conclusions can be drawn. First, the first-order optimization method gives similar
results for the final values of the updated structural parameters for both objective functions,
while there is a significant difference for the elastic modulus and Poisson’s ratio for the two
objective functions using the subproblem approximation method. This shows that the mode
shapes do not have a significant impact on the results for the first-order optimization method, but
have some influence on the subproblem approximation method results. Unfortunately, the results
from the subproblem approximation method directly rely on the approximations generated from
69
the random designs. If there are not enough initial random designs or they are similar, then the
final result would be different. Considering these pitfalls, the results from the subproblem
approximation method are not the wisest choice as the defined structural parameters for the next
section. Now looking at the results presented in Table 4- 3, the spring constant values for column
3 are larger than for column 1, which makes sense considering the increased confinement for
column 3 in the cavern. It is interesting to note, the spring constants in the z-direction are almost
100 times smaller than the ones calculated for both the x- and y- directions. This difference could
be related to the rock mass surrounding Cavern 2. In the case of the x- and y- directions the
influence of the surrounding rock mass could be as high as 300 meters, while the z-direction has
only two and four meters worth of rock mass before the ground surface for column 1 and column
3, respectively. This large difference clearly rationalizes the large variance between the spring
constant values. There is a relatively small difference between the spring constants in the x- and
y- directions, which stems from the geometrical shape of the columns: one direction is almost
twice as long as the other.
70
Table 4-2: Column 1 and 3’s optimization results
Column Objective Function
Optimization Algorithm
Updated Parameters
(GPa) (kg/m3) (GPa)
1
First Order* 4.29 2241.70
0.21 1.89E11
Sub App** 4.18 2098.70
0.115 3.34E10
First Order 4.30 2241.70
0.21 1.89E11
Sub App 4.59 2220.70
0.17 1.95E10
3
First Order 5.28 2229.80
0.38 1.89E11
Sub App 5.25 2190.80
0.35 4.91E11
First Order 5.14 2170.30
0.40 1.89E11
Sub App 5.96 2545.50
0.39 2.04E11
* First Order = first-order optimization method ** Sub App = subproblem approximation method
71
Table 4-3: Column 1 and 3’s spring constant values based on optimization results C
olu
mn
Ob
ject
ive
Fu
nct
ion
Op
tim
izat
ion
A
lgor
ith
m
Spring Constants
1
First Order* 4.31E17 1.12E17 8.01E8 1.44E19 3.73E18 1.70E8
Sub App** 7.62E16 1.98E16 7.81E8 2.54E18 6.60E17 1.80E8
First Order 4.31E17 1.12E17 8.03E8 1.44E19 3.73E18 1.70E8
Sub App 4.45E16 1.16E16 8.57E8 1.48E18 3.85E17 1.88E8
3
First Order 1.39E18 5.13E17 1.48E9 2.60E19 9.61E18 3.18E8
Sub App 3.60E18 1.33E18 1.47E9 6.75E19 2.50E19 3.23E8
First Order 1.39E18 5.13E17 1.44E9 2.60E19 9.61E18 3.05E8
Sub App 1.50E18 5.53E17 1.67E9 2.80E19 1.04E19 3.56E8
* First Order = first-order optimization method ** Sub App = subproblem approximation method
Graphical representations of the convergence for all the different analyses and a direct
comparative analysis with the calibrated global model are shown in Figure 4-3. The inclusion of
the mode shapes increases the value of the objective function, proving the mode shapes affect the
updated structural parameters and may diminish the accuracy of the model. Based on Figure 4-3,
the number of iterations for termination is about half for the subproblem approximation method
compared to the first-order optimization method, while the overall objective function value is
similar for both. Since the first-order optimization method costs more each iteration, requiring
more iterations gives this method a major disadvantage. However, in terms of convergence, the
first-order optimization method reaches a valley of the objective function quicker than the
72
subproblem approximation method. The subproblem approximation method may have a
decreased number of iterations (~20 total), but lacks uniform convergence, which reduces the
validity of the results relative to the first-order optimization method. The early termination of the
subproblem approximation method relies on the defined approximations of the objective function
and each state variable or updated structural parameter, which is a major disadvantage for this
optimization method.
Relative to the global model, the values of the updated parameters, with the exception of
the Poisson’s ratio, are below a ten percent difference, which means the structural parameters in
the global model don’t need to be modified or re-evaluated. It is interesting to note that the
percentage difference between the local and global values for the Poisson’s ratios for column 1 is
-40% and column 3 is 60%. Due to this large inconsistency, this parameter value will not be
changed in the global model either, but will remain pertinent to the local model. As a last
consideration comparing the local and global models, the relatively high value of the elastic
modulus of the added auxiliary beam element corresponds to high spring constants and may be
assumed to be close to fixed support boundary conditions for the local models.
Even though there are an increased number of iterations, the reliability of convergence
and the consistency of the results for both the columns and the objective functions significantly
outweigh this disadvantage for the first-order optimization method and the results from this
method will be used for future analysis. Considering the lack of accuracy for the mode shapes,
the results from the frequency residual only objective function will be the defined values for the
structural parameters used for both the columns in the comparative analysis in the next section.
73
FOO = First-Order Optimization SPA = Subproblem Approximation
Figure 4-3: Results from optimization algorithms: (a) Comparison of objective function (F
= frequency residual only, Eq. 11, and MF = frequency residual plus MAC values, Eq. 12) and (b) percent change with calibrated global model
0 10 20 30 40 500
0.5
1
1.5
2
2.5
3O
bjec
tive
Fun
ctio
n
Iteration
Column 1
0 10 20 30 40 50 60 70 800
0.5
1
1.5
2
2.5
Iteration
Column 3
FOO-F FOO-MF SPA-F SPA-MF
E1 p1 v1-40
-35
-30
-25
-20
-15
-10
-5
0
5
% C
hang
e w
ith
Glo
bal M
odel
Updated ParametersE1 p1 v1
-10
0
10
20
30
40
50
60
Updated Parameters
FOO-F FOO-MF SPA-F SPA-MF
74
4.4 Comparison of Global and Local Models
4.4.1 Assumptions for Comparative Analysis
For the purposes of Cavern 2, a comparison of the behavior over time is vital. Each
model has already been designed for the present condition of the cavern; however, the
assumption cannot be made that each model will behave similarly over time. Therefore each
model will be simulated over the course of 50 years, using several assumptions: (1) the material
model, (2) specified material degradation, (3) incorporation of surrounding rock mass, and (4)
the application of a failure criterion. A summary of the assumptions for both the local and global
model is shown in Table 4-4, with an explanation of each assumption presented below.
(1) When determining the material model and to keep it simple, a linear elastic model is used
for both the local and global models primarily for ease of computational effort.
Unfortunately, if the model becomes too complicated, then the direct behavioral changes
and comparisons will become difficult to identify. The assigned material properties from
calibration analysis are used for the global model and the optimization results are used for
the local model.
(2) Considering that some of the caverns in the cluster have failed in the recent years, an
assumption can be made that there is some form of material degradation of the rock over
time. When the term material degradation is used from this time on it directly refers to
the strength of the argillaceous siltstone decreasing over time. This assumption is
justified by work completed by Guo et al. (2005), where they described the concept of
disintegrative durability. Based on the wet and dry cycles each cavern experiences within
a given year, disintegrative durability quantifies the amount of rock diminished between
each cycle. The results stated there was a one percent decrease in the mass per wet and
75
dry cycle, justifying the assumption of a decrease in the strength of the argillaceous
siltstone. However, the decrease in strength cannot be quantified by the disintegrative
durability quantity alone because it is only a function of material weight and not strength.
This means there is no definitive conclusion on how the wet and dry cycle will affect the
defined material model of the rock mass. In the end, three different assumptions will be
used for the degradation of the material properties: a one percent, three percent, or five
percent decrease in the strength, i.e. a percent reduction of the elastic modulus each year
for both the local and global models.
(3) For the global model, only one meter of surrounding rock mass was incorporated into the
model. This was applied as a triangular pressure distribution to each side of the
surrounding walls of the model. Based on these stress results, the first-order optimization
method is applied to determine the equivalent force to apply to both local column models.
The goal was to minimize the objective function, which included both displacement and
stress values, i.e. Von Mises’ and vertical, at several locations along the length of both
columns. Ultimately, it was determined a force of 8.26 MPa and 17.1 MPa should be
incorporated into the local models of column 1 and column 3, respectively, shown in
Figure 4-4.
(4) There is no failure criterion applied because one of the main purposes of the models is to
assess the response of each model, not necessarily when or how it will fail. This will
allow a computational friendly analysis to be conducted.
76
Figure 4-4: Optimization results from force determination for (a) Column 1 and (b) Column 3
Table 4-4: Summary of Assumptions used in Comparative Analysis
Assumption Local Model Global Model
Material Model
Simple linear elastic model with properties assigned from optimization
analysis conducted in Section 4.3
Simple linear elastic model with properties assigned from calibration
analysis conducted in Section 3.3
Material Degradation
Assuming three different scenarios: a one percent, three percent or five
percent decrease in material strength each year
Same as local model
Incorporation of Surrounding
Rock Mass
Force is added to the top of each local model
A triangular pressure distribution is applied to the surrounding walls
assuming only 1 meter of surrounding rock mass affects
Cavern 2
Failure Criteria
There is no failure criterion applied Same as local model
0
10
20O
bjec
tive
Fun
ctio
n (a) Column 1
0
10
20
30(b) Column 3
0 20 40 60 800
20
40
60
Forc
e (M
Pa)
Iteration0 5 10 15 20
5
10
15
20
Iteration
77
4.4.2 Results from Comparative Analysis
This section is specifically designed to look at the comparative results from the non-
linear time history analysis conducted on both the local models and the global model.
Considering Cavern 2 needs to be analyzed over a long period of time, which in this case is 50
years; it makes logical sense that the results will be analyzed two ways: by comparing the
outputted results of the local models and the global model, and the behavior of the results. It is
important not only to understand what the output values from the FE model are, but also how the
models are behaving over time. In order to see these trends more clearly, the figures of the 50-
year history analysis (Figure 4-5, Figure 4-6, Figure 4-7, and Figure 4-8) only show the percent
change for each outputted parameter, while Table 4-5 shows the initial values of each FE model
at time equal to zero for each outputted parameter. After each analysis is run, a comparison is
made over six specific output measurements and their locations: the maximum displacement, the
maximum Von Mises’ stress, the minimum vertical stress, the maximum first principal stress, the
maximum second principle stress, and the minimum third principal stress.
Since the local models were optimized for the sole purpose of being directly compared
with the global model, it is interesting to note that the results from all the FE models are quite
different, shown in Table 4-5. Looking specifically at the initial values of each model for column
1, there is a definite trend of local model having lower stress values than the global model, which
means the global model is the more conservative estimation of the two. One possible reason for
this difference is due to the location of column 1 near the entrance of the cavern. The hole in the
ceiling is likely affecting the response of the column in the global model, making the stresses and
displacements higher, while the updated boundary conditions in the local model may not
incorporate this phenomenon in the same manner. However, this is not the case for column 3; in
78
all cases, the global model is the one with smaller results, which shows very clearly that the local
model is a more conservative estimation of stress values. Considering in all cases the initial
values are almost twice for one model as they are for the other, this shows there is little similarity
between the two models. These differences could be attributed to the application of the force on
the top of the local model, which was applied uniformly to the top of the column. However, in
the global model each column has a slanted roof and this may cause the force distribution not
only to vary at the boundary of the column, but also throughout the column.
In order to display all the results from the six outputted parameters, the results are split
into two different figures for each column. The first set of figures, Figure 4-5 and Figure 4-7 for
column 1 and column 3, respectively, displays the results from the displacement, the Von Mises’
stress and the vertical stress, while the second set of figures, Figure 4-6 and Figure 4-8 for
column 1 and column 3, respectively, displays the extremal results for the first, second and third
principal stresses. Each one of the figures show the percent change of each output parameter
under the varying levels of material degradation and the corresponding location in both the plan
view and z-direction location, for the global model (black symbols) and the local models (blue
symbols).
Making direct comparisons between the local and global models, it is evident for both
columns that the behavior of the displacement over time is almost identical; however none of the
behaviors of the stress output parameters are similar with the exception of the second principal
stress. The local model results have a higher percentage increase over time, which could show
these models would fail sooner than the global model. Another observation is the location of the
measured output parameter never correlates between the local and global model. This could
mean the oversimplification of the column geometrical shape, i.e. changing the column roof to
79
be flat in the local models relative to the slanted roof in the global model, changed the location of
the maximum or minimum output measurement.
Next, a direct comparison between each column is presented. There is quite a difference
in the maximum absolute value of the percentage change for the vertical stress, first principal
stress, and the third principal stress, but not each parameter, where column 3 is higher than
column 1. One plausible explanation is the applied force to column 3 is over twice the value for
column 1 due to the larger overhead rock mass, and as the material model changes, the
equivalent stress value will increase accordingly. Thus, the outputted stress values will be larger
for column 3 than column 1.
Lastly, there is a comparison among the different levels of material degradation. For an
assumed one percent material degradation per year, the relationship from year to year is linear.
However, this is not the case for an assumed three percent and five percent material degradation
per year, where the behavior becomes exponential for three percent and is highly exponential for
five percent. Even the absolute maximum percent change between each level of material
degradation increases by almost 800%. This consequently shows the level of material
degradation will have a significant effect on behavior of both global and local models during a
time history analysis. For the analyses performed in the next chapter, varying levels of material
degradation will need to be incorporated in order to ensure all possible outcomes are investigated.
Considering all the above results, the inconsistency of the results of the local models to
the global model reveals that the two models are different. Just by the definition of the global
model it does have a higher confidence level and considering the local model does not mimic the
global model shows it does not have the same confidence level. Furthermore, the global model
has the ability to consider not only the column in isolation but also the contribution of the other
80
columns and the surrounding conditions, i.e. Cavern 1 and Cavern 3. Since the local models are
inconsistent in the results and the difference in location, value, and behavior of the maximum
displacements, maximum Von Mises’ stress, minimum vertical stress, maximum first principal
stress, maximum second principal stress, and minimum third principal stress, shows the global
model is appropriate model for future analysis.
Table 4-5: Results at the present state (Year 0) for the Local and Global Models
Output Parameter Column 1 Column 3
Local Model Global Model Local Model Global Model
Max Displacement 1.57 mm 2.47 mm 2.76 mm 2.71 mm
Max Von Mises’ 0.775 MPa 1.275 MPa 3.15 MPa 1.72 MPa
Min Vertical Stress -0.795 MPa -1.283 MPa -3.16 MPa -1.90 MPa
Max 1st Principle Stress 12.74 KPa 39.38 KPa 58.94 KPa 11.22 KPa
Max 2nd Principle Stress 8.87 KPa 14.44 KPa 13.64 KPa 6.65 KPa
Min 3rd Principle Stress -0.795 MPa -1.292 MPa -3.30 MPa -1.97 MPa
81
Figure 4-5: Comparison of global model and local models for column 1 for the 50-year time
history analysis for the displacement, Von Mises’ stress and vertical stress for (a) 1 percent, (b) 3 percent, and (c) 5 percent material degradation
Z Location from Bottom (m)
Local Global
Disp. 9.69 11
Von Mises 1.35 9.90
Vertical 0.46 9.90
Location inPlan View
0
20
40
60
% C
hang
eD
ispl
acem
ent
(a) One Percent
0
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200
300
400(b) Three Percent
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(c) Five Percent
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% C
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on M
ises
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ertic
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tres
s
Time (Years)
G1% L1%
0 20 400
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0.4
Time (Years)
G3% L3%
0 20 400
0.5
1
1.5
2
Time (Years)
G5% L5%
82
Figure 4-6: Comparison of global model and local models for column 1 for the 50-year time history analysis for 1st, 2nd, 3rd principal stresses for (a) 1 percent, (b) 3 percent, and (c) 5 percent material degradation
Z Location from Bottom (m)
Local Global
1st Princ Stress 9.10 9.62
2nd Princ Stress 9.61 8.55
3rd Princ Stress 0.46 9.90
Location inPlan View
0
0.05
0.1
% C
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e1st
Pri
nc S
tres
s(a) One Percent
0
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0.6
(b) Three Percent
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2.5(c) Five Percent
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Time (Years)
G1% L1%
0 20 400
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Time (Years)
G3% L3%
0 20 400
0.5
1
1.5
Time (Years)
G5% L5%
83
Figure 4-7: Comparison of global model and local models of column 3 for the 50-year time history analysis for the displacement, Von Mises’ stress and vertical stress for (a) 1 percent, (b) 3 percent, and (c) 5 percent material degradation
Z Location from Bottom (m)
Local Global
Disp. 7.78 9.20
Von Mises 7.78 7.0
Vertical 7.78 0
Location inPlan View
0
20
40
60
% C
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eD
ispl
acem
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(a) One Percent
0
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400(b) Three Percent
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ises
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Time (Years)
G1% L1%
0 20 400
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0.6
0.8
1
Time (Years)
G3% L3%
0 20 400
1
2
3
Time (Years)
G5% L5%
84
Figure 4-8: Comparison of global model and local models for column 3 for the 50-year time history analysis for the 1st, 2nd, 3rd principal stresses for (a) 1 percent, (b) 3 percent, and (c) 5 percent material degradation
Z Location from Bottom (m)
Local Global
1st Princ Stress 8.04 6.44
2nd Princ Stress 5.97 4.60
3rd Princ Stress 7.78 0
Location inPlan View
0
0.05
0.1
0.15
% C
hang
e1st
Pri
nc S
tres
s(a) One Percent
0
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1(b) Three Percent
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G5% L5%
85
4.5 Conclusions
After examining the possibility of developing local models and comparing these with
global models for Cavern 2, it was ultimately determined the global model is the appropriate
representation. This main conclusion is drawn from the results presented throughout the entirety
of this chapter and is based on the development of the local models and the comparative analysis
of the local and global models.
In order to create the local models, the use of model updating techniques were
investigated and applied. Considering the simplicity of the models and the few experimental
reference data for the optimization techniques, the number of possible updating parameters was
excessive. The use of classical beam theory reduced the number of updating parameters in half
and made the optimization problem appropriate. Two optimization algorithms (first-order
optimization and subproblem approximation methods) and two objective functions (frequency
residual only and frequency residual plus MAC value) were explored and evaluated. The results
from the first-order optimization method and an objective function of the frequency residual only
were found to have the highest level of confidence and the corresponding optimized material
property and boundary condition assignments were used for analysis in the later parts of this
chapter. Nevertheless, the elastic modulus and density for the local models correlate very well
with the global model, with less than a ten percent change between the models.
Even though the dynamic characteristics were updated to be similar in both models, the
results from a simple time history analysis were not always comparable. The percent change for
the displacement may be similar, but the other stress values, i.e. Von Mises’ stress, vertical stress,
and the first, second and third principal stresses, are not. Even more importantly, the
86
corresponding maximum locations for each model are not close to each other, which were
attributed to the oversimplified geometrical representation of both columns in the local models.
When conducting investigations into the possible failure mechanisms of Cavern 2, the
global model allows for a holistic approach, while the local models allow for a possible
examination into isolated failures. If one was to look only at dynamic characteristics, this would
give a false perception of validity of the local models. Even though the exact locations of
extremal stresses cannot be identified, these locations make more sense for the global model than
the local models. While the local models might have saved computational resources, they do not
adequately replicate the global model, so these models will no longer be investigated.
87
CHAPTER 5 State of Stress Analysis on Cavern 2
5.1 Introduction
Strength and failure are two diametrically-opposed concepts often used in research to
address one fundamental question: under what conditions and after which period of time will the
system no longer perform its intended functions. When dealing with systems such as
underground rock caverns, failure and safety converge and the end goal is the same: to determine
when the structure is no longer stable. Due to brittleness of the material type and the geometric
configuration, there is a fine line between complete failure and safety for the visitors coming to
the site. When specifically looking at the Longyou grottoes, caverns in the cluster have been
failing with increased frequency over the past five years. The implication for Cavern 2 is the
probability of decreased stability and increased failure is increasing with each passing day.
Ultimately, the inherent goal of this chapter is to use the state of stress of the Cavern 2 FE model
to determine under which situations will there be decreased stability and when does failure
become eminent.
When trying to institute the proper approach for this type of state of stress analysis for
underground rock caverns, it is crucial to understand successes in previous work. Several
different approaches have been established to help understand this failure problem. The
introduction of a factor of safety has been a successful approach (Asadollahi and Tonon 2010).
By relating the forces applied to a rock block to the weight of the block, the maximum force that
can be applied before instability is reached can be identified. However, this approach can be
quite cumbersome when applied to three dimensional models. Other approaches have used a
combination of plasticity theory and failure criteria to aid in the determination of collapse
mechanisms (Fraldi and Gurarracino 2009), while other approaches factor in the compressive
88
and tensile behavior of the brittle material; one can input this information into a FE model to
replicate the failure of the structure using the on-site data (Fishman 2008). However, specifically
looking at underground excavations, there have been instability or failure analyses conducted
using several approaches. These include the influence of residual tensile strength (Diederichs and
Kaiser 1999), the use of various failure criteria and the linear strength behavior along planes of
weakness (Serrano and Olalla 1998), the incorporation of on-site investigation results for model
calibration and state of stress analysis to determine stability (Ferrero, Segalini et al. 2010), and
an innovative new procedure of using real-time laser holographic interferential fringes to visually
identify distribution, mode, moving path and dynamic evolution of deformation and cracking of
the rock (Cai and Liu 2009). For the purposes of this study a suitable approach combines the FE
model, state of stress analysis and implementation of a proper failure criterion to determine
mechanisms of failure and a timeline to failure.
In order to study the state of stress of Cavern 2, it is important to investigate two main
scenarios: the materials’ inherent response to a variety of loads and the identification of failure
conditions, i.e., the use of a failure criterion, based on different stress states. Over the last century,
numerous failure criteria have been developed to satisfy the results obtained from the above
scenarios. However, each failure criterion is generally suitable for only specific types of
materials and they are generally separated into three main categories: the single shear stress
series of failure criteria (Jaeger, Cook et al. 2007), the octahedral-shear stress series of failure
criteria (Zhou, Tu et al. 2007), or the twin-shear stress series of failure criteria (Yu 1983). The
single-shear stress series are the most primitive failure criteria available and are generally the
easiest to use and implement into commercial FE software. This has led to many engineers to
automatically implement these criteria into their analysis, without even understanding or
89
investigating other options. Even though the other two categories revolve around either three
dimensional or non-linear failure criteria, their greater accuracy outweighs the challenges of
integration into commercial FE software. This is why for the purposes of this chapter several
failure criteria, i.e. Mohr-Coulomb (Jaeger, Cook et al. 2007), Drucker-Prager (Zhou, Tu et al.
2007), and Hoek-Brown (Hoek and Brown 1980), will be investigated for possible
implementation into the FE model. Mohr-Coulomb and Drucker-Prager are typical failure
criteria available in commercial software, while Hoek-Brown is a specifically-developed failure
criterion for underground rock structures.
Throughout the years, all three criteria have been compared to determine the appropriate
one for a particular application. Unfortunately, there has not been a consensus for one failure
criterion over another. When investigating the failure of intact bedded rock salt, it was found that
the Hoek-Brown failure criterion is most suitable (Liu, Ma et al. 2011), while the Mohr-Coulomb
failure criterion is recommended for wellbore stability analysis (Zhang, Cao et al. 2010). When
fitting failure criterion to test data of sandstone, norite and limestone, it was found the Hoek-
Brown failure criterion is the best fit (Pariseau 2007). Consequently, it is not exactly clear which
criterion should be used for further analysis, which motivates a comparative analysis.
The state of stress analysis for this chapter is broken into three parts. First, there will be a
brief overview and introduction of three main failure criteria mentioned above. Then a
comparative analysis will be conducted to determine the proper failure criterion and material
stress-strain curve for the argillaceous siltstone. Finally, the identified failure criterion and stress-
strain curve will be implemented into the FE model along with several other assumptions to
identify the time to failure and a possible mechanism of failure for Cavern 2.
90
5.2 Discussion of possible failure criteria
For the purposes of the stress analysis to be conducted in the later sections, determination
of an appropriate failure criterion is crucial. Considering the vast number of options for the
selection of failure criterion, an investigation into several different options is warranted. For
reasons previously stated, three different criteria will be investigated: Mohr-Coulomb (Jaeger,
Cook et al. 2007), Drucker-Prager (Zhou, Tu et al. 2007), and Hoek-Brown (Hoek and Brown
1980). The following sections introduce and familiarize each criterion for use in the later
comparison.
5.2.1 Mohr-Coulomb Failure Criterion
The simplest and most well-known failure criterion is the Mohr-Coulomb failure criterion
developed in 1773. Through his extensive experimental investigations, Coulomb found the
failure of rock or soil happens along a plane due to the shear stress acting along this same plane
(Jaeger, Cook et al. 2007). His main assumption was the failure of a rock or soil takes place
along a defined plane, at angle , due to shear stress, , as shown in Figure 5-1 (a). Along this
failure plane, the normal and shear stress will be used to define the failure criterion. The
development of Mohr’s circle, Figure 5-1 (b), is useful in determining the angle of internal
friction, 2 2(Jaeger, Cook et al. 2007). In general, the failure criterion can be written
as
tan ( 12 )
where is cohesion, is the normal stress acting on the plane A – B, and is the angle of
internal friction (Zhao 2000). It is important to note that, in general, rock engineering assumes all
forces applied to a rock element will be compressive. The analysis of rock, just as in the analysis
91
of concrete, is assumed to be compression oriented, with tension not having a significant
influence.
(a) Failure of a Soil Block (b) Mohr’s Circle of Failure Plane (c) Principal Stress Plot
Figure 5-1: Graphical Representation of Mohr-Coulomb Failure Criterion (Zhao 2000)
This general failure criterion can also be transformed using stress transformation
equations to generate a failure criterion as functions of the principal stresses, and , as seen
in Equations ( 13 ) and ( 14 ) and the graphical representation is shown in Figure 5-1 (c):
12
12
cos 2 ( 13 )
12
sin 2 ( 14 )
Substituting Equations ( 13 ) and ( 14 ) into the general failure criterion, Eq ( 13 ), yields:
2 cos 1 sin1 sin
( 15 )
Thus the rock will fail along the angle 4 2 if exceeds the value . From
Equation ( 15 ) and Figure 5-1 (c), the uni-axial compressive and tensile strengths, and ,
respectively, can be determined:
1
3
n
A
B
n
1 3
1
3
c
92
2 ∙ cos1 sin
( 16 )
2 ∙ cos1 sin
( 17 )
Inputting Equations ( 16 ) and ( 17 ) into Equation ( 15 ) gives two equivalent expression that
indicate failure:
( 18 )
1 ( 19 )
where and depend in general on the angle of fracture, as in their definitions (Eqs ( 16 )
and ( 17 )). For practical purposes, since the vast majority of rock can hardly handle tension, the
tensile strength is usually considered to be zero (Zhao 2000).
Due to the simplicity of the Mohr-Coulomb linear relationship, its implementation into
FE code has been very successful for a variety of situations. The criterion was applied into a 3D
FE analysis to simulate the construction sequence of a Metro Tunnel in Sao Paulo, Brazil made
out of porous clay (Sousa, Negro et al. 2011). Further application of this criterion was used to
create a mechanical convergence forecast model to predict the failure of tunnels constructed
under a variety of geological configurations (Serrano, Olalla et al. 2011). Even a drained
analysis using the Mohr-Coulomb failure criterion was applied to soft clay, stones and sand of
stone columns based on experimental results to determine the proper spacing and design
procedure for said columns (Ambily and Gandhi 2007). Finally, the criterion has been applied to
a numerical study of reinforced soil segmental walls under typical operating conditions to predict
response of the soil walls during construction and surcharge loading (Huang, Bathurst et al.
2009).
93
However, since the Mohr-Coulomb criterion essentially is an empirical relationship, it is
hard to determine the conditions where it is valid. In deriving this failure criterion the rock
material is assumed to have linear elastic properties, which is not true for anisotropic materials.
These two factors have led to a rather significant gap when it comes to theory and practice of
rock engineering (Schweiger 1994). In addition, since “failure” occurs when Mohr’s circle first
touches the failure line, this generally assumes the intermediate or principal stress in the 2-
direction, where , has no effect on the failure envelope created by the stresses in the
1- and 3- directions. This had been widely accepted until extensive true-triaxial compressive tests
conducted by Mogi (1971) showed in fact the intermediate stress has a significant effect on the
failure envelope. His research showed the exclusion of the intermediate principal stress tends to
increase the amount of stress a soil or rock specimen can handle before failure. This causes the
Mohr-Coulomb failure surface to be stronger than the actual failure surface, both in the field and
in the lab. These results brought into question the validity of the Mohr-Coulomb relationship,
sparking a new revolution of failure criteria, all trying to define the true failure of a specific rock
material.
5.2.2 Drucker-Prager Failure Criterion
Another failure criterion, which has been widely used in rock engineering, is the
Drucker-Prager failure criterion. Drucker and Prager initially developed this pressure-dependent
model, in order to determine if soil material had failed or undergone plastic yielding. In complex
terms, the Drucker-Prager failure criterion is a cone in the principal stress space, which is
centered along the principal space diagonal or the hydrostatic axis (Pariseau 2007). Over the last
half century, the failure criterion has been modified for applications to rock, concrete, polymers
and even foams (Arslan 2007). Examples of the failure criterion being applied to many different
94
materials include implementation into FE models of earth pressure problems (Schweiger 1994),
stress and seepage analysis of earth dams (Li and Desai 1983), three-dimensional FE modeling
of laterally-loaded piles (Brown and Shie 1990), wave propagation induced by underground
explosions (Ma, Hao et al. 1998), investigations into wellbore stability analysis (Zhang, Cao et al.
2010), and even into the non-linearity of soil for soil-structure interaction problems (Fourie and
Beer 1989).
The general form of the Drucker-Prager yield criterion is:
( 20 )
where for isotropic materials, reduces to the second invariant of the deviatoric stress for an
isotropic material, reduces to the first invariant of the stress for an isotropic material, and the
constants and are defined by experimental studies on the rock material.
In order to determine and , a brief review of invariants will be presented. The three
invariants of the stress are just the coefficients defined by the characteristic polynomial of the
stress tensor. The deviatoric stress is the difference between the total stress and the isotropic
stress, , where and is the mean stress. The deviatoric invariants
become the coefficients defined by the characteristic polynomial when determining the three
deviatoric stresses. The deviatoric stress is valuable in the calculations of distortion while the
isotropic stresses are helpful for volumetric calculations (Jaeger, Cook et al. 2007). Both
and can be represented in terms of the principal stresses, , , and , which is shown in the
equations below:
16
( 21 )
( 22 )
95
Based on experimental data, and can be either functions of the uniaxial compressive
and uniaxial tensile strength or of the cohesion and the angle of internal friction; both shown
below:
2
√3
6 cos
√3 3 sin
6 cos
√3 3 sin ( 23 )
1
√3
2 sin
√3 3 sin
2 sin
√3 3 sin ( 24 )
As seen in the definitions above, there are two representations of and . This is due to how the
Drucker-Prager failure criterion surrounds the Mohr-Coulomb failure criterion. Based on
whether the failure surface either circumscribes or inscribes the Mohr-Coulomb failure criterion
on the pi-plane determines which equation is used (Arslan 2007). The Mohr-Coulomb criterion
on the pi-plane has a non-smooth failure surface, which has been known to cause computational
problems. Since Drucker-Prager can create a smooth surface either inside or outside the Mohr-
Coulomb failure surface, this allows for easier implementation into advanced FE codes. However,
several researchers (Schweiger 1994; Yu, Zan et al. 2002; Zhou, Tu et al. 2007) have
investigated whether the higher computing costs yield more accurate results, but difficulties
representing the failure surface may lead to extraneous results.
5.2.3 Hoek-Brown Failure Criterion
There is one other failure criterion developed that may help to represent a more accurate
failure surface for underground rock excavation, as compared to both the Mohr-Coulomb and the
Drucker-Prager failure criteria, the Hoek-Brown failure criterion (Hoek and Brown 1980). Hoek
and Brown wanted to develop a failure criterion that would meet three needs. First, it should
adequately represent the response of an intact rock when loaded with the stresses of underground
96
rock excavation. Second, the criterion needs to be able to predict the influence of discontinuities
upon the behavior of the rock sample, even if the rock is anisotropic. Finally, it should be able to
predict the behavior of the full rock mass with even more discontinuities than encountered during
the investigation of the rock sample (Hoek and Brown 1980).
There were three main assumptions used when Hoek and Brown were developing the
failure criteria. The first was the definition of failure stress. They defined failure stress as “the
maximum stress carried by the specimen” because they believed that the most prominent
application for their criterion would be for underground rock excavations, which means the rock
engineers would primarily be concerned with failure at the excavation site. Their second main
assumption was to ignore the influence of the intermediate principal stress, just as with the
Mohr-Coulomb failure criterion. While this is not strictly true, as discussed earlier, they
understood it was an oversimplification. Then the final assumption was that as the number of
discontinuities increased, the overall strength of the anisotropic rock mass would tend to behave
as an isotropic rock (Hoek and Brown 1997).
This led Hoek and Brown to develop a failure criterion purely based on experimental data.
The empirical criterion they developed is:
( 25 )
where is the Hoek-Brown constant and is the compressive strength of the rock, both of
which depend on rock properties and the extent of rock discontinuities before being subjected to
stresses, and where ranges from 1 for intact rock to 0 for rocks with major discontinuities.
Through extensive laboratory triaxial tests, and m can be defined as:
97
∑ ∑∑ ∑
∑∑
∑ ( 26 )
1 ∑∑ ∑
∑∑
( 27 )
where and which are summed over for each individual experiment, of
which there are total.
However, if laboratory tests are not possible then and , which are both functions of
the Geological Strength Index, can be estimated using the Tables and Charts provided in Hoek
and Brown (1997).
When implementing the Hoek-Brown failure criterion either into a FE model or for
practical design purposes, it is important to understand the properties estimated exhibit a
distribution around the mean values. Even under ideal conditions, the distribution can have an
impact on the results of the FE model and the on design calculations for underground
excavations. Through a stability analysis of a tunnel, the properties were shown to have
lognormal distributions when only a simply support system was applied (Hoek 1998). Even more
importantly, in order to effectively implement the Hoek-Brown criterion there are three
parameters that need to be determined to estimate the strength and deformability of any jointed
rock mass: the uniaxial compressive strength, , the value of the Hoek-Brown constant, , and
the value of the Geological Strength Index. When testing any rock specimen for its geotechnical
properties, cohesion and the internal failure angle are the most commonly determined properties.
While there have been attempts to determine the Hoek-Brown parameters as functions of the
cohesion and the internal failure angle and even vice versa (Hoek and Brown 1997), these
98
estimated parameters have validation issues when actually employed for design purposes. Since
its formation, the Hoek-Brown failure criterion has been used in variety of different applications
including the modeling of brittle failure of rock (Hajiabdolmajid, Kaiser et al. 2002), brittle rock
failure of the Lotschberg base tunnel (Rojat, Labiouse et al. 2009), predicting the depth of brittle
failure around tunnels (Martin, Kaiser et al. 1999), and to understand the collapse mechanisms in
cavities and tunnels (Fraldi and Gurarracino 2009). Due to its freshness in the geotechnical field,
the primary applications have thus far revolved around either brittle rock failure or underground
excavation, which is why it is a viable candidate as the failure criterion for the argillaceous
siltstone in Cavern 2.
In recent years, a modified Hoek-Brown failure criterion has been developed to
specifically solve issues related to the strength anisotropy of intact rock, since the Hoek-Brown
criterion parameters are greatly influenced by this property (Saroglou and Tsiambaos 2008).
There has been successful implementation of this modified criterion in the probabilistic analysis
and design of shallow strip footings resting on rock masses (Mao, Al-Bittar et al. 2012), in
developing a method for obtaining the ultimate bearing capacity (Serrano, Olalla et al. 2000),
and in the quantification of the upper bound solution for the ultimate bearing capacity (Yang and
Yin 2005). Ultimately, this modified criterion can predict the strength of intact rock, but can also
be applied to rock masses. For the purposes of the failure analysis the original Hoek-Brown
failure criterion, Equation ( 25 ), will be used.
5.3 Determination of Applicable Failure Criterion and Material Model
The determination of an applicable failure criterion involves investigating two important
aspects. The first aspect is to investigate the appropriate material property definition, where two
options will be explored. The first option assumes the material property is defined by a stress-
99
strain curve. Based on previous research, stress-strain curves were developed from axial tests on
both saturated and unsaturated specimens, shown in Figure 5-2 (Yang, Yue et al. 2010).
Considering there are only two tests for each specimen, the average of both tests will be used as
the primary stress-strain curve for future analysis. However, in order to determine the parameters
for the failure criteria, true-triaxial test data is necessary for curve fitting purposes, and this test
data is only available for saturated specimens. For the time being the unsaturated stress-strain
curve will be ignored. The second option assumes the material property is a purely elastic model
and has an elastic modulus of 4.5 GPa, as defined by the calibration analysis done in Chapter 3
for the saturated section of the cavern. A purely elastic model is the simplest material model
possible to incorporate into a non-linear FE analysis. In this section comparisons will be made
based on different material models and how the quantified failure surfaces respond. Ultimately, a
proper failure criterion and material model will be identified and applied for the state of stress
analysis on the entire cavern.
The first step of comparison will be to develop the failure surfaces. Based on previous
research (Yang, Yue et al. 2010), Table 5-1 gives the results from the tri-axial tests conducted on
the saturated specimens. Each of the specimens tested were cylindrical with a diameter of 40 mm
and a height of 76.5 mm, shown in Figure 5-3. All the tri-axial tests were conducted with a
constant pressure on the surface of the cylinder (σ σ ) and a linearly increasing pressure
being applied at the top (σ ). In order for the failure surfaces to be identified, each of the
respective criteria was fitted to the test data, shown in Figure 5-4, and each fitted parameter is
listed in Table 5-2. Future tri-axial tests can be conducted to determine the argillaceous
siltstone’s true behavior and increase the accuracy of the fitted parameters. Looking specifically
at the relative error in Table 5-1, this suggests the Hoek-Brown failure criterion should be used
100
for future analysis, which is lowest for all the different tri-axial tests except when 10
MPa. Although this is not enough to determine if it is an appropriate criterion to use, so further
analysis will be performed to either corroborate or contradict this conclusion.
It is interesting to note the results from the fourth tri-axial test, where σ σ 15MPa,
because none of the failure criterion can adequately reflect the results. There could be two main
reasons for this discrepancy: the test could have resulted in erroneous results or the specimen
could have been too small to handle the excessive load being applied on all sides. Either way this
test needs to be reexamined in future tri-axial tests.
Figure 5-2: Stress-Strain curves from test data for saturated and unsaturated specimens
0 1000 2000 3000 4000 5000 60000
5
10
15
20
25
30
Stre
ss (
MPa
)
Strain ()
Unsat Test 1Unsat Test 2Unsat AvgSat Test 1Sat Test 2Sat Avg
101
Figure 5-3: Specimen used for the tri-axial tests
Table 5-1: Comparison of Test Data to Three Different Failure Criteria
Triaxial Set-Up,
Test Data,
Mohr-Coulomb
Error (%)
Drucker-Prager
Error (%)
Hoek-Brown
Error (%)
0 18.13 18.68 3.03 19.34 6.67 18.13 0
5 31.27 31.62 1.12 31.69 1.34 30.59 -2.17
10 44 44.55 1.25 44.05 0.11 41.33 -5.61
15 31.04 57.49 85.21 56.41 81.73 51.16 64.82
D = 40 mm
H =
76.
5 m
m 2
3
1
102
Figure 5-4: Mohr-Coulomb, Drucker-Prager, and Hoek-Brown failure criteria matched to Test
Data
Table 5-2: Parameter Assignment for each Failure Criterion
Mohr-Coulomb Drucker-Prager Hoek-Brown
Parameter Value Parameter Value Parameter Value
5.657 MPa 0.19 3.6
26.26° 7.49 1
Presently, the proper material model is still uncertain. To determine a suitable model, a
stress analysis is conducted comparing two material model assumptions already mentioned: a
stress-strain curve and a purely elastic model. This is achieved by creating a FE model in
ANSYS© of the same cylinder specimens tested for the tri-axial test (Figure 5-5), in order to
-5 0 5 10 150
10
20
30
40
50
60
1 (
MPa
)
3 (MPa)
Mohr-CoulombDrucker-PragerHoek-BrownTest Data
103
replicate the tri-axial tests. The FE model is relatively simple, with 1113 nodes and only the
bottom of the specimen has a boundary condition (fixed) and everywhere else is free.
Confinement is applied as constant pressure on the side walls of the cylinder, assumed to be
radially symmetric, i.e. , and a time-changing pressure applied on the top, assumed to be
. Each of the analyses conducted is a time-history analysis, and each failure criterion is
investigated in turn. This allows not only for the material models to be analyzed, but also the
failure criteria. At each time step, the stress state for every element is extracted and compared
with the failure surface of each criterion. If the stress in the FE model exceeds the allowed stress
(equivalent failure surface), then the corresponding element has also failed and is recorded as
such. This process continues until either all the elements of the model have failed or the material
model is no longer stable and consequently fails.
Each of the tri-axial tests, with confinement pressures of 0 MPa, 5 MPa, 10 MPa, and 15
MPa, are calculated using both material models. The results are shown in Table 5-3 and Table 5-
4 for the stress-strain curve and the elastic material model, respectively. As can easily be seen the
stress-strain material model has a closer correlation with the test data for only = 0 and 15,
while for = 5 and 10, the elastic model does much better. A visual comparison (Figure 5-6) of
the progression of failure further supports the previous conclusion. For each confined pressure, a
sudden failure is clearly seen for the stress-strain curve. However, for the elastic model, a more
progressive failure takes place for = 15 MPa. Even more importantly, when the failure starts
to occur it only takes an additional applied pressure of 2 MPa to cause 75 percent of the elements
to fail, but then it takes an application of 20 MPa to 30 MPa more of pressure to see complete
failure. This indicates that failure is sudden to a point, but then turns into a progressive failure.
But this is completely contradictory to the expected failure: rock specimens are brittle and should
104
fail suddenly. Considering this failure progression is seen for all the confining pressures except
15 MPa, this could cause erroneous results to be found. Therefore, the stress-strain curves from
the uniaxial compressive tests will be used as the material model. For the (un)saturated sections
of the simplified global cavern model the (un)saturated stress-strain curve will be implemented.
Other material properties identified in the previous chapters will remain unchanged.
Look specifically at the failure criteria, the Hoek-Brown is the most conservative of the
three, causing the model to fail sooner. However, since it was just determined that the material
model should be the stress-strain curve, further analysis should be conducted to make sure the
Hoek-Brown failure criterion is most suitable.
Figure 5-5: Finite Element Model of Specimen with applied stresses and blue symbols for fixed boundary conditions
2
3
1
105
Table 5-3: Comparison of Test Data to Three Different Failure Criteria assuming a Stress-Strain Curve as the Material Model
Triaxial Set-Up,
Test Data,
Mohr-Coulomb
Error (%)
Drucker-Prager
Error (%)
Hoek-Brown
Error (%)
0 18.13 18.4 1.49 18.4 1.49 18.6 2.59
5 31.27 23.4 -25.17 23.4 -25.17 23.4 -25.17
10 44 28 -36.36 28 -36.36 28 -36.36
15 31.04 29.4 -5.28 29.4 -5.28 29.4 -5.28
Table 5-4: Comparison of Test Data to Three Different Failure Criteria assuming an Elastic Material Model
Triaxial Set-Up,
Test Data,
Mohr-Coulomb
Error (%)
Drucker-Prager
Error (%)
Hoek-Brown
Error (%)
0 18.13 20.8 14.73 20 10.31 19 4.80
5 31.27 34 8.73 32.2 2.97 32 2.33
10 44 49 11.36 45 2.27 42 -4.55
15 31.04 55.4 78.48 55.4 78.48 55.4 78.48
106
Figure 5-6: Comparison of progression of failure for stress-strain data obtained from rock
samples and an assumed elastic stress-strain curve
15 16 17 180
50
100Test Data Stress-Strain Curve
2 = 3 = 0 MPa
Mohr-Coulomb Drucker-Prager Hoek-Brown
15 20 25 30 35 40
Elastic Material Model
2 = 3 = 0 MPa
20 21 22 230
50
100
2 = 3 = 5 MPa
20 30 40 50 60
2 = 3 = 5 MPa
23 24 25 26 27 280
50
100
Perc
ent o
f E
lem
ents
Fai
ling
2 = 3 = 10 MPa
40 50 60
2 = 3 = 10 MPa
25 26 27 28 290
50
100
2 = 3 = 15 MPa
1 (MPa)50 51 52 53 54 55
2 = 3 = 15 MPa
1 (MPa)
107
It is important to keep in mind the lack of tri-axial test data brings into question the
accuracy of the fitted stress-strain curve, shown in Figure 5-2. Even if this was not an issue,
researchers have commented on possible inaccuracies going from a confined lab test to a field
test, with higher strength results reported from laboratory tests than what is actually found in the
field (Benz and Schwab 2008). In order to investigate this issue, a sensitivity analysis is
conducted to determine the effect of inaccuracies in the tri-axial result data. This sensitivity
analysis assumes the true stress-strain curves differ in strength by ±5%, ±10%, ±50%, +75%,
+100%, and +200%, shown in Figure 5-7. Ultimately, the stress-strain curves will be compared
with the tri-axial data, but this sensitivity analysis will help to investigate how the different
failure criteria function under different material assumptions. The same finite element model will
be used to perform the sensitivity analysis, as was used in the previous analysis.
To directly compare each of the material assumptions and the failure criteria, the use of
total relative deviation (TRD), is used:
Total Relative Deviation TRD1
4
2
( 28 )
where is the minimum stress causing failure by the given failure criterion and is the test
data from the tri-axial tests. This TRD value will give a direct comparison between each of the
failure criteria without just looking at the visual representations shown in Figure 5-8 through
Figure 5-11.
108
Figure 5-7: Stress-Strain Curves used in the Sensitivity Analysis
Table 5-5: Comparison of Test Data to Three Different Failure Criteria (MC = Mohr-Coulomb, DP = Drucker-Prager, HB = Hoek-Brown) assuming different Stress-Strain Curves
(Combined Error)
TRD Percent Change (%)
-50 -10 -5 0 5 10 50 75 100 200
MC 0.40 0.15 0.13 0.11 0.11 0.10 0.09 0.13 0.16 0.23
DP 0.40 0.15 0.13 0.11 0.11 0.10 0.09 0.12 0.16 0.21
HB 0.40 0.15 0.13 0.11 0.11 0.10 0.09 0.12 0.16 0.17
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 55000
5
10
15
20
25
30
35
40
45
50
55St
ress
(M
Pa)
Strain ()
-50%-10%-5%05%10%50%75%100%200%
109
Based on Table 5-5, the lowest TRD value is based on the Hoek-Brown failure criterion
with a 50% change from the true stress-strain curve, which is a significant increase from the
assumed stress-strain curve and great care needs to be used before it is implemented in FE model.
The visual representation of failure is shown in Figure 5-8, Figure 5-9, Figure 5-10, and
Figure 5-11 for the results assuming a confining pressure of 0 MPa, 5 MPa, 10 MPa, and 15 MPa,
respectively. It is important to mention, an assumed stress-strain curve decreased by 50% from
the original could not be analyzed for two confining pressures: 10 MPa and 15 MPa. Due to the
high decrease in strength, the argillaceous siltstone could not withstand any pressure from the σ
direction when such high pressures were being applied in both the σ and σ directions, i.e. the
rock failed prematurely. This concludes that this stress-strain curve is not appropriate to use for
further analysis.
A noticeable trend seen in all the figures is that, as the percentage of increase of the
stress-strain curves gets larger, the progression of failure is seen more clearly. This relates
directly to the assumption that there will be signs of failure before the cylinder completely fails.
However, looking at recent failures of the Longyou grottoes, there were not clear signs of failure,
just sudden failure of the columns and entrances (Li, Yang et al. 2009), which implies assuming
such a large increase in the stress-strain curves is unrealistic. Since the progression of failure is
similar to the elastic material model results from Figure 5-6, objections must be raised. Taking
into consideration both the low TRD value and the lack of sudden failure for the 50% increase
stress-strain curves, these this options now become unrealistic choices for the FE model.
Removing these stress strain curves leaves only a few options: no change, ±5%, and ±10%.
While there are only signs of progressive failure for the +5% and +10% curves at 0 MPa
confinement, just as with 0% change, any increased stress-strain curve should be eliminated from
110
further consideration in light of published results (Benz & Schwab 2008). This leaves three
options left: no change, -5% and -10% change, but it is not clear by the figures alone which is the
best to use. Referring specifically back to the TRD values to assist with the decision, it is clearly
seen 0% change stress-strain curve should and will be used for future analysis.
Switching gears to focus on the failure criterion results, there are several conclusions that
can be drawn. The first is, like the other analysis, the Hoek-Brown failure criterion gives the
most conservative results, the Mohr-Coulomb failure criterion gives the least conservative and
the Drucker-Prager failure criterion is in between. However, it is interesting to note, the
progression of failure is similar for all three failure criteria. This shows the inclusion of the
intermediate principal stress is not a factor in the progression of failure, but it does have an effect
on when failure initiates, which makes sense based on the applied configuration of the confining
pressure. Finally, if there is sudden failure of the model, this occurs at exactly the same confining
pressure state for the failure criteria. This signifies that the failure of the model is not attributed
directly to the failure criterion, but to the strength of the material model. Ultimately, based on the
error calculations from the analysis conducted in this section, the best failure criterion to use for
future analysis is the Hoek-Brown failure criterion. Even though for sensitivity analysis there are
signs of progressive failure, the fact that it is the most conservative, helps to outweigh this minor
disadvantage.
In summary, when conducting the stress analysis for the next section, the material model
that will be used is the original stress-strain curve and the failure criterion will be Hoek-Brown
failure criterion.
111
Figure 5-8: Progression of failure based on the defined failure criterion and specified stress-
strain curve for an assumed confining pressure of 0 MPa (2 = 3 = 0)
20 30 40 500
50
100
+200% Change
2 = 3 = 0 MPa
Mohr-Coulomb Drucker-Prager Hoek-Brown
15 20 25 30 35
+100% Change
15 20 25 300
50
100
+75% Change
16 18 20 22 24 26
+50% Change
15 16 17 18 19 200
50
100
Perc
ent o
f E
lem
ents
Fai
ling
+10% Change
15 16 17 18 19
+5% Change
15 16 17 18 190
50
100No Change
15 15.5 16 16.5 17
-5% Change
14 14.5 15 15.5 16 16.50
50
100-10% Change
1 (MPa)
7 7.5 8 8.5 9 9.5
-50% Change
1 (MPa)
112
Figure 5-9: Progression of failure based on the defined failure criterion and specified stress-
strain curve for an assumed confining pressure of 5 MPa (2 = 3 = 5)
25 30 35 40 45 50 550
50
100
+200% Change
2 = 3 = 5 MPa
Mohr-Coulomb Drucker-Prager Hoek-Brown
25 30 35 40
+100% Change
26 28 30 32 34 360
50
100+75% Change
26 28 30 32
+50% Change
20 21 22 23 24 250
50
100
Perc
ent o
f E
lem
ents
Fai
ling
+10% Change
20 21 22 23 24
+5% Change
20 21 22 230
50
100No Change
18 19 20 21 22
-5% Change
17 18 19 20 210
50
100-10% Change
1 (MPa)10 11 12 13 14
-50% Change
1 (MPa)
113
Figure 5-10: Progression of failure based on the defined failure criterion and specified stress-strain curve for an assumed confining pressure of 10 MPa (2 = 3 = 10)
35 40 45 50 55 600
50
100
+200% Change
2 = 3 = 10 MPa
Mohr-CoulombDrucker-PragerHoek-Brown
35 40 45
+100% Change
36 38 400
50
100+75% Change
30 32 34 36
+50% Change
25 26 27 28 29 300
50
100
Perc
ent o
f E
lem
ents
Fai
ling
+10% Change
22 23 24 25 26 27 28
+5% Change
23 24 25 26 27 280
50
100No Change
20 21 22 23 24 25 26
-5% Change
1 (MPa)
20 21 22 23 24 250
50
100-10% Change
1 (MPa)
114
Figure 5-11: Progression of failure based on the defined failure criterion and specified stress-
strain curve for an assumed confining pressure of 15 MPa (2 = 3 = 15)
45 50 55 60 650
50
100
+200% Change
2 = 3 = 15 MPa
Mohr-CoulombDrucker-PragerHoek-Brown
45 46 47 48 49 50
+100% Change
40 42 44 460
50
100+75% Change
35 36 37 38 39 40 41
+50% Change
26 28 300
50
100
Perc
ent o
f E
lem
ents
Fai
ling
+10% Change
25 26 27 28 29 30
+5% Change
25 26 27 28 290
50
100No Change
22 23 24 25 26 27
-5% Change
1 (MPa)
20 22 24 260
50
100-10% Change
1 (MPa)
115
5.4 Application of Hoek-Brown failure criterion on Chosen FE Model
When studying or determining the state of stress of a cavern set, the determination of
proper assumptions is crucial. The conclusions from the previous chapters and sections are vital
to define the assumptions for the state of stress analysis. This analysis will quantify a possible
period of time when the cavern will experience high levels of stress and become prone to failure.
The analysis will look at the cavern as a global structure and also as a local structure, analyzing
results from both column 1 and 3, with the hopes of understanding possible mechanisms of
failure.
5.4.1 Assumptions for Non-Linear State of Stress Analysis
The assumptions for the analysis are as follows:
(1) The finite element model used will be continuous. Considering the size of the cavern and
the lack of data concerning discontinuities, it is appropriate to assume the surrounding
rock is primarily defined as a continuous intact rock mass. Also, if one was to include
the discontinuities into the finite element analysis, then the analysis becomes quite costly
due to the added complexity. So to avoid any erroneous results, a basic continuous model
is used, but the hope is to add the discontinuities in future work using such methods as
deformation discontinuous analysis (Shi and Goodman 1985).
(2) Based on the results from chapter 3 and 4, a simplified global finite element model of
cavern 2 (M2 from Figure 3-1) and not a local model of columns 1 or 3 will be used for
the state of stress analysis.
(3) Consequently, the material properties, i.e. Elastic Modulus, Poisson’s ratio, and density,
are determined based on the FDD results from the on-site measurements shown in
116
chapter 2. These results used to calibrate the finite element model shown as a result of
chapter 3. These will be the initial material properties used for the state of stress analysis.
(4) The material stress-strain curves will be used based on material tests conducted on the
argillaceous siltstone, for both the saturated and unsaturated conditions, shown in Figure
5-2.
(5) The concept of disintegrative durability mentioned in the previous chapter will be
included in this analysis as well. Considering there is no definitive conclusion on the
effect of the wet and dry cycles the argillaceous siltstone experience, four different
assumptions will be used for the degradation of the material properties, a one percent,
three percent, five percent and ten percent decrease in the strength, i.e. the slope of the
stress-strain curve for both the unsaturated condition and the saturated condition, each
year (shown in Figure 5-12 through Figure 5-15).
117
Figure 5-12: Sample sets of the stress-strain curves assuming material degradation of 1% decrease per year
0 1000 2000 3000 4000 5000 60000
2
4
6
8
10
12
14
16
18
Str
ess
(MP
a)
Strain ()
(b) Saturated Condition
0
5
10
15
20
25
30
Str
ess
(MP
a)
(a) Unsaturated Condition
0 yr 10 yr 20 yr 30 yr 40 yr 50 yr
118
Figure 5-13: Sample sets of the stress-strain curves assuming material degradation of 3% decrease per year
0 1000 2000 3000 4000 5000 60000
2
4
6
8
10
12
14
16
18
Str
ess
(MP
a)
Strain ()
(b) Saturated Condition
0
5
10
15
20
25
30
Str
ess
(MP
a)
(a) Unsaturated Condition
0 yr 10 yr 20 yr 30 yr 40 yr 50 yr
119
Figure 5-14: Sample sets of the stress-strain curves assuming material degradation of 5% decrease per year
0 1000 2000 3000 4000 5000 60000
2
4
6
8
10
12
14
16
18
Str
ess
(MP
a)
Strain ()
(b) Saturated Condition
0
5
10
15
20
25
30
Str
ess
(MP
a)
(a) Unsaturated Condition
0 yr 10 yr 20 yr 30 yr 40 yr 50 yr
120
Figure 5-15: Sample sets of the stress-strain curves assuming material degradation of 10% decrease per year
(6) Based on the analysis conducted at the beginning of the chapter, the Hoek-Brown failure
criterion will be applied to the calibrated simplified global FE model. Since data was only
collected for the saturated argillaceous siltstone there will be only one failure surface
0 1000 2000 3000 4000 5000 60000
2
4
6
8
10
12
14
16
18
Str
ess
(MP
a)
Strain ()
(b) Saturated Condition
0
5
10
15
20
25
30
Str
ess
(MP
a)
(a) Unsaturated Condition
0 yr 10 yr 20 yr 30 yr 40 yr 50 yr
121
used for the entire FE model with the parameters listed in Table 5-2. This will cause the
results to be of a more conservative nature.
(7) Considering the generated model only looks at a portion of the surrounding rock mass, an
appropriate confining pressure has to be applied to the sides of the FE model.
Unfortunately, there is almost an infinite amount of rock surrounding the cavern, so the
amount of the rock which influences the applied pressure is questionable. To incorporate
this assumption into the state of stress analysis, several different levels of pressure will be
used (ranging from 1 to 100 meters) determined by:
( 29 )
where is the pressure (Pa), is the density , is the height from the surface
level , is the gravitational constant , is the depth of the surrounding
rock mass to take into effect or the horizontal pressure effect , is the area where the
pressure is being applied , and is the soil coefficient to convert vertical soil
pressure to horizontal soil pressure can be found by:
1 ( 30 )
where is the Poisson’s Ratio of the argillaceous siltstone, which was determined in
chapter 2.
(8) Depending on the time of year, the level of saturation changes based on levels of
precipitation. In order to understand the influence of saturation on the outputted stress
levels inside the cavern, different levels of saturation will be investigated: no saturation,
half saturation, and full saturation. Based on the on-site investigations, it is determined
122
not all of the cavern becomes saturated during the rainy seasons. Consequently, only the
sections in the cavern determined to be saturated will have an additional weight
component. When the rock is saturated, there is additional weight from the water and this
will be added to the model.
5.4.2 Results from Non-Linear State of Stress Analysis
The non-linear analysis performed in this section is ultimately a time history analysis
over a period of at most 50 years. Each analysis assumes one saturation level, material property
or rock pressure to output a distinct set of results, which makes the analysis determinative in
nature with a total of 120 analyses performed.
The results first presented are in Figure 5-16, which shows the time when the first
element in the model fails due to various levels of material degradation and horizontal pressure.
Based on these results, when the effective rock depth, , reaches 50 meters for all the material
degradation, there are already elements which are failing within the 50 year time frame. This
shows there is a pressure that can be applied and from the first moment there are elements which
have reached and surpassed the Hoek-Brown failure surface. Ultimately, this is a scary situation
if this amount of pressure is actually being applied to the cavern walls, since failure could
already be progressing in the cavern.
Figure 5-17 shows when complete failure of the cavern occurs. Failure is quantified as
either when 75% of the elements fail based on the failure criterion or when the reduced strength
of the argillaceous siltstone causes the FE model to no longer be able to converge. It is
interesting to note, there are instances where the time of initial failure are the same as time of
complete failure, e.g. = 30 m and 10% material degradation. This failure is due to the reduced
strength of the material properties having reached their fundamental limit for stability. Then
123
there are instances where there are signs of failure by the elements (i.e. failure based on the
Hoek-Brown failure criterion), then at some later point in time the FE model completely fails.
The cases where there are no signs of failure are considered critical because this failure will be
instantaneous and without any warning.
Figure 5-16: Time when the first element fails in the Cavern 2 model due to the influence of material degradation and horizontal pressure, l
Assumed Horizontal Pressure Effect (m)
Per
cent
Cha
nge
of M
ater
ial
Deg
rada
tion
Per
Yea
r
1 10 15 20 25 30 35 50 75 100
10%
5%
3%
1%
0
10
20
30
40
50
42 28 20 20 15 15 10 0 0 0
50+ 50+ 50+ 44 40 36 15 0 0 0
50+ 50+ 50+ 50+ 50+ 50+ 25 0 0 0
50+ 50+ 50+ 50+ 50+ 50+ 50+ 0 0 0
yr
yr
124
Figure 5-17: Time of complete failure of Cavern 2 model due to the influence of material degradation and horizontal pressure, l
Looking specifically at each of the different material degradations, there are definite
conclusions to be drawn. In general, as the material degradation increases the time to failure
decreases, which is to be expected. Nevertheless, when the material strength decreases by one
percent each year, the results clearly show there is no amount of pressure which could be applied
where complete failure occurs within the 50 year period of analysis. Understanding the nature of
the caverns, there is a preconceived notion there is no cavern in the cluster, which will last
another 50 years or beyond without some sort of retrofit applied. Based on previous experience
with the other caverns, any result where the cavern lasts longer than 50 years can be deemed
unrealistic – thus a one percent material degradation is not a valid assumption to make. However,
this is not the case for both the three percent and five percent decrease in material strength. There
are instances when the complete failure is above 50 years, but they are at low ranges of pressure
only. Looking specifically at the results from the three percent decrease, even for the higher
pressures the time to failure will exceed 50 years and should be taken under advisement if used
Assumed Horizontal Pressure Effect (m)
Per
cent
Cha
nge
of M
ater
ial
Deg
rada
tion
Per
Yea
r
1 10 15 20 25 30 35 50 75 100
10%
5%
3%
1% 10
20
30
40
50 yr
yr
42 28 20 20 15 15 15 10 5 5
50+ 50+ 50+ 44 40 36 32 26 15 10
50+ 50+ 50+ 50+ 50+ 50+ 50+ 44 32 20
50+ 50+ 50+ 50+ 50+ 50+ 50+ 50+ 50+ 50+
125
for any future work. It might seem, that assuming a ten percent degradation of material strength
is too drastic for only a one percent disintegrative durability constant (Guo, Li et al. 2005). But,
the results show there may be some validity to this assumption. The complete failure of the
cavern is seen for all the pressure assumptions. If the horizontal pressure is large, the cavern
would fail within the next 5 years, which is possible, but highly unlikely, due to the condition of
the cavern in its present state. Alternatively, looking at the pressure effect, there is one major
conclusion to present, that as the pressure increases the time to failure drastically decreases.
When the pressure effect, , is 25 meters and above, the time to failure starts to decrease
dramatically, which signifies the pressure does carry significant influence in determining time to
failure and needs to be included when determining parameters for future work. Ultimately, a
three percent, five percent or ten percent material degradation could be a reasonable postulation
for future work, but the amount of horizontal pressure is vital as well and should be assumed be
at least 25 meters. Furthermore, caution needs to be used when any combination is used for
future work.
One distinction, not mentioned for either Figure 5-16 or Figure 5-17, was the saturation
level applied to the FE model for these specific initiations of failure or failure results.
Interestingly enough, the analysis showed there was no difference in results given the saturation
level. However, by digging deeper into the elemental results, some conclusions can be made
about saturation levels. Considering such emphasis has already been put on column 1 and
column 3, the outputted stress levels for these columns will be presented. The vertical stress for
both columns was the greatest of all stress levels and will be comparative measure. Figure 5-18
and Figure 5-19 shows the maximum vertical stress for column 1 and column 3, respectively,
while Figure 5-20 and Figure 5-21 show the minimum vertical stress for column 1 and column 3,
126
respectively. For all the figures, only a three percent, five percent and ten percent material
degradation is presented because a one percent change has already been deemed inappropriate.
Each figure shows all three saturation assumptions with the different levels of assumed
horizontal pressure effect.
From the results in Figure 5-18 through Figure 5-21, many general observations can be
made. The first is that the added saturation levels decreases the z-direction stress for both the
maximum and minimum stress values. By adding the extra weight to the saturated elements, this
seemed to moderately help the maximum vertical stress to reach their respective critical values
slower. Specifically looking at the minimum z-direction stress, the added saturation causes the
stress to double for column 3, but not necessarily for all analysis cases in column 1. Interestingly
enough, the assumed material degradation or assigned pressure does not affect the outputted
stress results as much as varying levels of saturation. This could become crucial as the strength
of argillaceous siltstone continues to decrease each year, but the saturation level remains high.
Another observation is the stress at failure is similar for each of the material degradations,
but the only difference is the time of failure. For instance, if the horizontal pressure is influenced
by 100 meters of argillaceous siltstone, the maximum z-direction stress for column 1 and column
3 are approximately 2 MPa and 0 MPa, respectively. However, the time of failure is completely
different for each material degradation, i.e. 20 years, 10 years, and 5 years for 3 percent, 5
percent and 10 percent degradation, respectively. This shows the consistency of the model at
failure; however, the time of failure for each assigned material degradation is different, as is
expected.
Furthermore, the effect of incorporating different levels of pressure into the FE model
gives interesting results. Basically, as the amount of influencing argillaceous siltstone increases,
127
there is a higher likelihood of tensile stresses being outputted by the model, and the possibility of
tensile failure becomes more and more critical. This effect becomes increasingly grave for
column 1 (and only shows up for l = 100 m for Column 3). Between the maximum and minimum
z-direction stress values, column 1 shows a greater difference between the two, typically around
4 MPa, whereas column 3 differs only by about 0.5 MPa. This ultimately results in a higher
likelihood of tensile failure and possible instability for column 1 than column 3, which makes
sense due to the location, i.e. the whole near the entrance of the cavern, and the lower natural
frequencies, i.e. causes the column to be more flexible, for column 1 extracted from the on-site
investigation.
Figure 5-18: Comparison of Maximum Vertical Stress in Column 1 assuming (a) 3%, (b) 5%,
and (c) 10% material degradation at failure
0
1
2
(a) 3 Percent
0
1
2
Max
Z-D
ir S
tres
s,
z (M
Pa)
(b) 5 Percent
1 10 15 20 25 30 35 50 75 100
0
1
2
Assumed Horizontal Pressure Effect (m)
(c) 10 Percent
No Saturation Half Saturation Full Saturation
128
Figure 5-19: Comparison of Maximum Vertical Stress in Column 3 assuming (a) 3%, (b) 5%,
and (c) 10% material degradation at failure
Figure 5-20: Comparison of Minimum Vertical Stress in Column 1 assuming (a) 3%, (b) 5%,
and (c) 10% material degradation at failure
-1.5
-1
-0.5
0
0.5(a) 3 Percent
-2
-1
0
(b) 5 Percent
Max
Z-D
ir S
tres
s,
z (M
Pa)
1 10 15 20 25 30 35 50 75 100
-1.5
-1
-0.5
00.5
(c) 10 Percent
Assumed Horizontal Pressure Effect (m)
No Saturation Half Saturation Full Saturation
-2
-1
0
(a) 3 Percent
-2
-1
0
(b) 5 Percent
Min
Z-D
ir S
tres
s,
z (M
Pa)
1 10 15 20 25 30 35 50 75 100-2
-1
0
Assumed Horizontal Pressure Effect (m)
(c) 10 Percent
No Saturation Half Saturation Full Saturation
129
Figure 5-21: Comparison of Minimum Vertical Stress in Column 3 assuming (a) 3%, (b) 5%,
and (c) 10% material degradation at failure
5.5 Conclusions
Cavern 2 will fail. The only question is when. The many analyses done in this chapter
have sought to give a likely timeline to failure, given varying assumptions to several unknowns.
If one was able to identify some information on the rate of material degradation or the amount of
influencing argillaceous siltstone on the assumed horizontal pressure, then the time to failure
could be pinpointed more accurately.
Unfortunately, not a lot of information is available and the determination of failure
cannot be as accurate as one would like. Nevertheless, the FE non-linear analysis was able to put
limits on some of the major assumptions in the analysis. Through the recreation of the triaxial
tests, it was determined the developed-for-underground-rock-excavations failure criterion, Hoek-
-2
-1
0
(a) 3 Percent
-2
-1
0
(b) 5 Percent
Min
Z-D
ir S
tres
s,
z (M
Pa)
1 10 15 20 25 30 35 50 75 100
-2
-1
0
(c) 10 Percent
Assumed Horizontal Pressure Effect (m)
No Saturation Half Saturation Full Saturation
130
Brown failure criterion, is the appropriate criterion to use for the analysis, due to the minimal
error for all the tests compared with the other criterions and its performance with the sensitivity
analysis on the stress-strain data.
By breaking down the non-linear analysis, there are several conclusions that can be
drawn. The first is the material degradation is probably between 3 percent and 10 percent per
year. Based on previous experience with the other caverns, failure is expected to happen
sometime in the near future and is assumed within the next 50 years. When determining the
influence of the argillaceous siltstone on the horizontal pressure, this has a broad connection to
the material degradation assumed for the analysis. If one is to assume the degradation is 3
percent then the influence should 50 meters and above, while for 5 percent and 10 percent it is 20
meters and 10 meters, respectively. For the most conservative estimate of failure, due to the
simplicity of the FE model, full saturation at all times can be safely assumed. Considering the
high levels of tensile stress for column 1 as the horizontal pressure increases, gives insight into
how the cavern might initially fail due to tensile failure of the argillaceous siltstone in column 1.
Incorporating all these factors into the FE model, the timeline to failure will be between 5 and 30
years. This may seem to be a large uncertainty, but this at least gives a starting point for future
analysis. Summing up, the timeline to failure and a possible mechanism of failure for Cavern 2
were thoroughly investigated.
If a more clear and distinct timeline of possible failure is needed, there are other factors
not included in the analysis, which neglecting the some of the true behavior of the cavern. These
factors are discussed in more detail in the next section and are vital to pinpoint the failure
timeline accurately.
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CHAPTER 6 Conclusions and Future Work
6.1 Conclusions
Based on the on-site investigations several conclusions can be drawn:
1. The power spectral density functions for column 3 during both investigations showed similar
dominant frequencies when compared directly with equivalent test set-ups.
2. There is no significant change between the dominant frequencies between each investigation.
This means that the stiffness of column 3 did not change between investigations. This
strongly suggests the environmental condition of the surrounding rock mass was similar for
each investigation.
3. Based on FDD analysis, column 1 is more flexible than column 3 and is proven by lower
natural frequencies. This occurs due to the hole at the entrance of the cavern, which directly
affects the confinement at the top of column 1.
4. Based on the limited instrumentation used on each column, it is determined that while the
extracted natural frequencies are accurate, the mode shapes cannot be determined from the
data to high accuracy.
Based on the global model calibration several conclusions can be drawn:
1. The use of a four-part criterion has the ability to determine adequate levels of simplification
for a global FE model.
2. Three separate models were developed using varying techniques. It was concluded the use of
both 3D scan data, on-site investigations, and published works was necessary in order to
determine the geometry, material properties, and boundary conditions of the FE models. If
132
one was to purely rely on published works, this would result in insufficient modeling
parameters.
3. A sensitivity analysis conducted on the dynamic properties showed the treatment of the rock
structure as one solid has a higher impact on outputted natural frequencies, than if the FE
model were separated into several solids.
4. The close correlation of the TRD, stress contours, and mode shapes for both the static and
dynamic criteria justifies the selection of the simplified M2 model instead of the less accurate
M3 model or the more complicated M1 model.
5. The M2 model was calibrated using the frequency results from chapter 2. Boundary
conditions were assigned based on published works and visual inspection from the on-site
investigations with considerations based on the proximity of Cavern 2 to Cavern 1 and
Cavern 3.
6. Saturation levels and material properties were determined concurrently by varying both
parameters. The analysis showed the elastic modulus to be higher in certain sections than
previously published (Li, Yang et al. 2009; Yue, Fan et al. 2010).
Based on the comparison of global and local models several conclusions can be drawn:
1. When optimizing the simplified or local models, the number of updating parameters was
deemed excessive. They were reduced through the use of classical beam theory, derivation
shown in Appendix A, which cut the number of updating parameters in half.
2. Different combinations of optimization algorithms and objective functions were used to
update the material properties in the local models. The frequency residual only and first-order
133
optimization method was deemed the most appropriate optimization combination because of
the increased accuracy of the results obtained from the analysis.
3. The optimized high values of the spring constants were found to be similar to a fixed support,
a reassuring result.
4. When the global and local models were compared in a simple time history analysis, the
results were not very comparable. The percent change for the displacement may have been
similar, but stresses were not. Even more importantly, the corresponding maximum stress
locations of each model were not the same.
5. If one were to look at the dynamic characteristics only, this would give a false perception of
the validity of the local models. However, taking a deeper look into the location of the
maximum stress locations, the global model makes more sense than the local model.
6. Considering the inconsistencies in the results and the differences in location, value and
behavior of the maximum displacements, maximum Von Mises’ stress, minimum vertical
stress, maximum first principal stress, maximum second principal stress, and the minimum
third principal stress, the calibrated global FE model is the appropriate representation for use
in further analysis.
Based on the stress analysis, several conclusions can be drawn:
1. Using published tri-axial test data, parameters for three different failure criteria (Mohr-
Coulomb, Drucker-Prager, and Hoek-Brown) were determined. Hoek-Brown shows the
closest correlation to the tri-axial data.
2. An appropriate material model was evaluated by simulating the tri-axial test results
comparing two assumptions: a stress-strain material model and an elastic material model.
134
Based primarily on the error values, it was concluded the stress-strain material model will be
used for future analysis.
3. A sensitivity analysis was conducted on the stress-strain curve. Based on TRD values and
the resulting progression of failure, the material model assuming no change of the published
stress-strain curve should be used for further analysis.
4. Looking specifically at the failure criteria, the Hoek-Brown failure criterion gives the most
conservative results, the Mohr-Coulomb gives the least conservative and the Drucker-Prager
criterion is in-between the two. Based on the low TRD values and conservative nature of the
Hoek-Brown failure criterion, it was decided this is an appropriate representation for the
argillaceous siltstone.
5. The inclusion of the intermediate stress did not affect the progression of failure, but the only
noticeable change was when the applied pressure caused failure to be initiated.
6. Specifically looking at the results from the state of stress analysis, as the rate of material
degradation increased, the time to failure was reduced. It was also determined that the 1%
material degradation per year is an inappropriate based on its timeline to failure results, i.e.
50 years and above, which contradicts the assumed failure timeline.
7. Interestingly enough, as the horizontal pressure increases, due to a higher influence of the
surrounding argillaceous siltstone, the vertical stress switches to tensile stress in column 1.
This could result in column 1 failing due to tensile failure.
8. The saturation level does not increase or decrease the time to failure for a given set of
assumptions, i.e. material degradation and horizontal pressure, but there was a moderate
effect on the z-direction stress in column 3. Ultimately, it is a conservative assumption that
the cavern experiences full saturation at all times.
135
9. Incorporating all these factors into the FE model, the timeline to failure will be between 5
and 30 years. Though the uncertainty might seem quite larger, this is a more definitive
timeline then has ever been established before.
6.2 Original Contributions
The research presented in this dissertation has several unique contributions:
1. Ambient vibration measurements were collected over a three-year period in Cavern 2 for the
first time since its discovery. These measurements were vital to determine the natural
frequencies and mode shapes of column 1 and column 3 in Cavern 2.
2. Due to geometric complexity of Cavern 2, a four-part criterion for model simplification was
developed. This became instrumental to determine an adequate global simplified FE model to
represent Cavern 2. Such a criterion had never been developed for underground rock caverns.
An important aspect of the four-part criterion was including both static and dynamic
characteristics into the analysis. Since many underground rock structures experience both
types of loading conditions throughout the entirety of a day, it became crucial to investigate
the effects of both situations. The results confirmed the importance of including both
characteristics.
3. A comparative analysis was conducted between two local models of column 1 and column 3,
respectively, and a global model of Cavern 2. Studies generating local and global models are
typically seen in the structural engineering field, but never in the rock engineering field. This
study was ground-breaking to show how local models can be developed for isolated systems
in underground rock caverns.
4. Generally, the application of failure criteria is chosen based on the experience of the
researcher. There have been instances of direct comparison of various criteria for numerous
136
rock samples. However, there had never been this direct comparison of different failure
criteria for the argillaceous siltstone of the Longyou Grottoes. Based on the comparative
analysis conducted in this dissertation it was determined that the Hoek-Brown failure
criterion is the best representation for the argillaceous siltstone.
5. Through the non-linear analysis conducted in chapter 5, estimations of the possible timeline
to failure and failure mechanisms could be determined for Cavern 2. This estimation had
never been quantified before this analysis was executed. It was concluded the cavern is
primarily susceptible to failure in column 1 because of the increase in tensile stress, which
will surpass the argillaceous siltstone’s allowable tensile stress strength. Based on these
results, future work can be focused on fine-tuning the estimation to failure, but also
investigate appropriate retrofit techniques, which have the highest likelihood of increasing
the life of Cavern 2.
6. Ultimately, all of the research conducted in this dissertation on Cavern 2 was new and
innovative for the cavern cluster. Never before had there been in-depth and advanced FE
modeling completed on any of the Longyou Grottoes. The results of this dissertation show
the necessity of looking at several different aspects when creating a FE model.
6.3 Future Work
Based on the work accomplished in this dissertation there are three areas, which could be
of potential interest in the future: increased testing of the caverns, implementation of advanced
modeling methods, and an investigation into possible retrofit techniques.
When performing the additional testing to the caverns, it is important to understand the
results desired to be obtained from the data. For example, if the mode shapes are of interest,
denser instrumentation needs to be included in the field experiments, along with continuous
137
monitoring to understand the change in natural frequencies over time. By understanding this
change, then possible failure mechanisms could be identified easier. However, if more accurate
stress values in specific areas of the cavern are desired, then in-situ stress tests are more
appropriate tests to conduct. This could include such tests as the flat-jack test, sonography, or
any other tests deemed appropriate for use in the cavern. Alternatively, the strength of the
argillaceous siltstone could be investigated by performing more tri-axial tests on rock samples
from the Grottoes, allowing the Hoek-Brown parameters to be determined more accurately.
There are ways to increase the accuracy even more, if one was able to directly take the samples
from Cavern 2. Furthermore, there has been interest in using the results from this dissertation to
develop an early detection system for Cavern 2. The system could include both displacement and
visual monitoring sensors to identify when stabilities are present. Having the data streamed real-
time is crucial as well not only for safety purposes but also to increase the amount of data present
for review. Ultimately, this detection system will help to monitor the stability of Cavern 2,
allowing local researchers identify areas of instability and come up with a proper course of
action quickly.
Another area of future work would be to incorporate advanced modeling methods to
narrow the timeline to failure or to reduce the number of assumptions when performing the
failure analysis. From the on-site investigations, there are signs of discontinuities inside the
cavern. If the use of X-ray technology could pin-point the exact dimensions of the discontinuities,
then several different types of advanced analyses could be executed, including fracture analysis
and seepage analysis. The application of the deformation discontinuous analysis would be
beneficial to model and understand the impact of the discontinuities.
138
The last area of future work to be mentioned here has to deal with the investigation of
possible retrofit techniques. Considering that of the five main caverns, three have already been
retrofitted and considered the results from this dissertation, a definite assumption can be made
that at some point Cavern 2 will also need to be retrofitted. The question becomes how should
the cavern be retrofitted and which techniques should be implemented? There are several options,
which could be investigated including, shotcrete, rockbolts, and steel bar encasing (for the
columns only). There has been discussion into the possible retrofit techniques for Cavern 2 (Li,
Yang et al. 2009), but to date nothing has been put into action.
139
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Appendix A: Derivation of Equivalent Spring Constants
The main assumption for the derivation is there is one column with an attached rotational
spring or linear spring, which can be simplified to a slender column broken down into two
segments. Ultimately, the Stiffness Method is used to derive the spring values in terms of the
simplified slender column material and geometrical properties.
The equations below are used to determine the relationship between a force or moment
applied to the system and the displacement or rotation felt by the system, which is defined by a
stiffness factor, shown in Equations ( A 1 ) and ( A 2 ). This stiffness factor, is the stiffness for
the entire slender column, however, the stiffness interested in for the purposes of further analysis
is just the one portion. The stiffness of just the needed section will be calculated by Equation ( A
3 ) for the axial direction and ( A 4 ) for the rotational direction.
( A 1 )
( A 2 )
where is the equivalent force applied to the column, is the total vertical stiffness, is the
response displacement of the column, is the equivalent moment applied to the column, is
the total rotational stiffness, and is the response rotation of the column.
( A 3 )
( A 4 )
where is the vertical spring stiffness, is the vertical stiffness of the column in Longyou,
is the rotational spring stiffness, and is the rotational stiffness of the column in Longyou.
First the equivalent spring stiffness equations will be derived in the vertical direction.
Figure A-1 shows the transformation of the column-spring set-up to the simplified column set-up
and the free body diagram, which will be used for the derivation. For the purposes of applying
147
the Stiffness Method, each simplified slender column will be broken down into two elements and
three locations used for the combined stiffness matrix. At each location, there will be an applied
force and moment and a corresponding response displacement and rotation. Depending on the
type of spring to be considered, directly affects whether a force or moment will be applied at
location two. Each side of the slender column will be considered fixed, based on on-site
observation and the confinement of all the columns in Cavern 2.
Figure A-1: Showing equivalence of using stiffness method to determine spring stiffness for an axial spring and the Free Body Diagram used in the derivation
The generalized stiffness matrix and force vector used for the all the column segments are
shown in Equations ( A 5 ) and ( A 6 ).
12 6 12 66 4 6 212 6 12 66 2 6 4
( A 5 )
where is the elastic modulus of the section, is the moment of inertia of the section and is
the length of the section.
0 ( A 6 )
F
k1, E1, I1, L1
ks
kT
E1, I1, L1 E2, I2, L2 E1, I1, L1 E2, I2, L2
F
d1
d2
d3
d4 d6
d5
Beam-Spring Model Beam-Beam Model Free Body Diagram
148
where and are the force and moment applied at location 1, respectively, is the force
applied at location 2, and and are the force and moment applied at location 3, respectively.
The stiffness matrices for element 1 and element 2 are shown in Equations ( A 7 ) and
( A 8 ).
12 6 12 66 4 6 212 6 12 66 2 6 4
( A 7 )
where is the elastic modulus of element 1, is the moment of inertia for element 1, and is
the length for element 1.
12 6 12 66 4 6 212 6 12 66 2 6 4
( A 8 )
where is the elastic modulus of element 2, is the moment of inertia for element 2, and is
the length for element 2.
The combined stiffness matrix is shown in Equation ( A 9 ).
0
12 6 12 60 0
6 4 6 20 0
12 6 12 12 6 6 12 6
6 4 6 6 4 4 6 2
0 012 6 12 6
0 06 2 6 4
00
00
,
( A 9 )
where is the response displacement at location 2 and is the response rotation at location 2.
149
Looking at the matrix, there are only two equations, which needs to be solved with two
unknowns, shown in Equations ( A 10 ) and ( A 11 ).
12 6 ( A 10 )
0 6 4 ( A 11 )
Referring back to Equation ( A 1 ), in order to determine the stiffness in the axial
direction, force needs to be a function of displacement. For the purposes of the equations above,
solving in terms of is necessary and is shown in Equation ( A 12 ).
32
( A 12 )
Substituting Equation ( A 12 ) into Equation ( A 10 ) yields ( A 13 )
3 12 12 18 3,
( A 13 )
By substituting a unit displacement, 1, in Equation ( A 13 ) yields the axial stiffness,
shown in Equation ( A 14 ).
3 12 12 18 3,
( A 14 )
Finally, the equivalent spring stiffness can be determined by substituting Equation ( A
14 ) and Equation ( A 15 ), which is , into Equation ( A 3 ) yields Equation ( A 16 ). The
stiffness for is quantified by assuming the simplified column is only one segment and not two,
150
with one end fixed and the other end free. The force is applied to the free end and the appropriate
stiffness can be easily found in any structural analysis textbook.
3 ( A 15 )
3 12 12 18 3 3,
( A 16 )
Now Equation ( A 16 ) will be further transformed to be applied to both the x and y
directions, Equations ( A 17 ) and ( A 18 ), respectively. The only difference between the two
equations will be the value inputted for the moment of inertia, while every other factor would be
the same in both.
3 12 12 18 3
3,
( A 17 )
3 12 12 18 3
3,
( A 18 )
Second the equivalent rotational stiffness equations will be derived for the rotational
direction. Figure A-2 shows the transformation of the column-spring set-up to the simplified
column set-up and the free body diagram, which will be used for the derivation. For the purposes
151
of applying the Stiffness Method, each simplified slender column will be broken down into two
elements and three locations used for the combined stiffness matrix. At each location, there will
be an applied force and moment and a corresponding response displacement and rotation.
Depending on the type of spring to be considered, directly affects whether a force or moment
will be applied at location two. Each side of the slender column will be considered fixed, based
on on-site observation and the confinement of all the columns in Cavern 2.
Figure A-2: Showing equivalence of using stiffness method to determine spring stiffness for a rotational spring and the free body diagram used for the derivation
Considering the simplification is exactly the same for this set-up, as with the last, this
implies the Stiffness Matrices for elements 1 and 2, from Equations ( A 7 ) and ( A 8 ),
respectively, can be applied for this derivation. However, the force vector is not the same and is
shown in Equation ( A 19 )
0 ( A 19 )
where is the moment applied at location 2.
The combined stiffness matrix is shown in Equation ( A 20 ).
M
krs kT
E1, I1, L1 E2, I2, L2
k1, E1, I1, L1
E1, I1, L1 E2, I2, L2
d1
d2
d3
d4 d6
d5
M
Beam-Spring Model Beam-Beam Model Free Body Diagram
152
0
12 6 12 60 0
6 4 6 20 0
12 6 12 12 6 6 12 6
6 4 6 6 4 4 6 2
0 012 6 12 6
0 06 2 6 4
00
00
,
( A 20 )
Looking at the matrix, there are only two equations, which needs to be solved with two
unknowns, shown in Equations ( A 21 ) and ( A 22 ).
0 12 6 ( A 21 )
6 4 ( A 22 )
Referring back to equation ( A 2 ), in order to determine the stiffness in the rotational
direction, moment needs to be a function of rotation. For the purposes of the equations above,
solving in terms of is necessary and is shown in Equation ( A 23 ).
12
( A 23 )
Substituting Equation ( A 21 ) into Equation ( A 22 ) yields ( A 24 )
4 4 6,
( A 24 )
153
By substituting a unit rotation, 1, in Equation ( A 24 ) yields the axial stiffness,
shown in Equation ( A 25 ).
4 4 6,
( A 25 )
Finally, the equivalent spring stiffness can be determined by substituting Equation
( A 25 ) and Equation A26, which is , into Equation ( A 4 ) yields Equation A27. The
stiffness for is quantified by assuming the simplified column is only one segment and not
two, with one end fixed and the other end free. The moment is applied to the free end and the
appropriate stiffness can be easily found in any structural analysis textbook.
( A 26 )
4 4 6,
( A 27 )
Now Equation ( A 27 ) will be further transformed to be applied to both the x and y
directions, Equations ( A 28 ) and ( A 29 ), respectively. The only difference between the two
equations will be the value inputted for the moment of inertia, while every other factor would be
the same in both.
4 4 6,
( A 28 )
154
4 4 6,
( A 29 )
For the axial spring stiffness, the problem becomes relatively simple. For the
consideration of an axial force, there is assumed to be no rotation or vertical movement. This
implies the axial spring stiffness for the Z-direction reduces to just the stiffness coefficient,
shown in the Equation ( A 30 ) for a force assumption and Equation ( A 31 ) for a moment
assumption.
( A 30 )
where is the area of the column in plan view.
( A 31 )
where is the shear modulus and is the radius of gyration of segment 1.
