A DYNAMIC STRUCTURAL ANALYSIS OF TREES SUBJECT TO WIND LOADING BY KENNETH RONALD JAMES Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy October 2010 Melbourne School of Land and Environments The University of Melbourne ABSTRACT A dynamic structural analysis of trees under wind loading is described. This project examines open grown trees from a structural perspective under real wind conditions in urban areas. Open grown trees differ from forest grown plantation trees due to morphological differences, in particular the slenderness ratio of the tree and the size of branches. Branches play a dominant role in modifying the dynamic response of open grown trees under wind loading. Previous theoretical models of trees have not included branches as dynamic elements. A range of open grown trees were studied, based on their structural shapes and branching morphology, including Palm (Washingtonia robusta), Italian cypress (Cupressus sempervirens), Hoop pine (Araucaria cunninghamii), Red gum (Eucalyptus tereticornis), NZ Kauri pine (Agathis australis), Spotted gum (Corymbia maculata), She oak (Allocasuarina fraseriana). New instruments have been designed to measure dynamic wind loads on trees during storms. These instruments are attached to the trunk at the base of a tree and measure the strain of the outer fibres of the trunk as it bends in the wind. Two sensors are orthogonally oriented on a trunk, usually in a North/South and East/West direction to record the complex sway response of trees due to wind blowing from any direction. Each sensor is accurate to one micron and readings at 20Hz record the dynamic motion of the tree. The tree is calibrated using a static pull test so that strain readings can be converted into bending moment values which are used as the measure of wind load. Wind speed, wind direction, and temperature are also recorded via a data logger and computer. This method can be used on all trees and differs from most previous studies which have put instruments into the upper canopy of a tree and have therefore been limited to excurrent shaped trees with a central trunk form. Trees have been monitored under wind storm conditions for several years and the results indicate that trees sway in a complex manner, due to the dynamic contribution of branches. In order to explain this complex motion a new theoretical model is proposed which includes branches as dynamic masses attached to the trunk. This model represents the tree as a multi-degree-of-freedom system which is different from previous studies that used a single degree-of-freedom model. The new model uses dynamic masses attached to other dynamic masses to represent the branches of a tree i and introduces the concept of mass damping. Mass damping occurs when two or more masses dynamically interact in a complex manner to cancel out the overall motion with an out-of-phase response. This is a component of the overall damping in a tree which acts to dissipate energy and minimize damage in wind storms. The application of dynamic measurement of forces will lead to a better understanding of how trees withstand winds. Knowledge of the actual wind forces allow a better understanding of tree stability and has implications for pruning techniques, and safe work practices during tree felling and dismantling operations. ii DECLARATION This is to certify that i. the thesis comprises only my original work towards the PhD except where indicated in the Preface, ii. due acknowledgement has been made in the text to all other material used, iii. the thesis is less than 100,000 words in length, exclusive of tables, maps, bibliographies and appendices. Signed ____________________________ Date: _____________________ iii ACKNOWLEDGEMENTS I gratefully acknowledge the assistance of Associated Professor Dr Nick Haritos for his friendly guidance on the technical and mathematical treatment of dynamics and its application to the study of trees. It has been a pleasure to work on this project with him over a number of years. Dr Brian Kane, University of Massachusetts, has been invaluable for his guidance and discussion of details regarding trees, assistance in collecting data and discussions at numerous conferences, presentations and workshops in Australia and the United States. Dr Peter Ades and Dr Greg Moore are acknowledged for their academic guidance and support in reviewing the thesis. Phil Kenyon is thanked for his challenging ideas and understanding of real trees and was the first person to offer some different views that eventually grew into this study. Ross Payne has been invaluable for his technical assistance and many in depth discussion on the finer points of instrumentation, electronics and data collection. His constant good humour and help on field trips to pull trees and collect data is gratefully acknowledged. Martin Norris, Arborist, Shire of Wellington, Sale, Victoria has been very supportive with his time and resources during field trips to Gippsland, Victoria, to study trees on windy sites. I have been fortunate to meet very supportive people on trips to international conferences including Nelda Matheny and Dr Jim Clark and thank them for their constant positive support over many years. I also wish to thank Dr Tom Smiley and Sharon Lilly for their discussions on trees and friendly support. My wife Emma and daughter Stephanie have been a great support and inspirational at times when the end did not seem in sight, and I thank them for their encouragement and patience over many years which have helped to keep me going. iv TABLE OF CONTENTS Page Chapter 1. INTRODUCTION 1 1.1 Trees and Structural Analysis 1 1.2 Static and Dynamic Methods 4 1.3 Tree Diversity and Data Sets 9 1.4 The Philosophy of Structural Assessment 14 1.5 Outline and Aims of the Research Project 16 Chapter 2. LITERATURE REVIEW 19 2.1 The Structural Analyses of Trees 19 2.2 Structural Properties of Trees and Wood 21 2.3 Static Analysis of Trees 27 2.3.1 Static pull tests 27 2.3.2 Static loads and tree failure 33 2.4 Wind and Trees 37 2.4.1 Critical wind speeds and wind loads on trees 39 2.4.2 Drag estimation and wind tunnel tests 47 2.5 Dynamic Analysis of Tree Structures and Tree Models 55 2.5.1 Introduction 55 2.5.2 Estimation of natural frequency 57 2.5.3 Spectral analysis 62 2.5.4 Damping 69 2.6 Field Measurements of Tree Sway 78 2.7 Instrumentation 84 Chapter 3. STRUCTURAL MODELS OF TREES 96 3.1 Previous Models of Trees 96 3.2 Proposed New Model 109 3.3 Mass Dampers and Structures 116 Chapter 4. METHOD AND MATERIALS 113 4.1 Introduction 118 4.2 Measuring Tree Structural Properties and Dynamic Wind Loads 119 4.3 Instrumentation 123 4.3.1 Strain meter design 123 4.3.2 Wind instrumentation 126 4.3.3 Software and analysis programs 129 4.4 Laboratory Testing 130 4.5 Static Analysis 131 4.5.1 Tree pull test 131 4.5.2 Relationship between strain readings and tree deflection 135 4.5.3 Relationship between strain readings and spring constant (k) 136 4.5.4 Relationship between Young‟s Modulus and spring constant (k) 137 v TABLE OF CONTENTS (cont.) Page 4.6 Analysis of Field Data 138 4.7 Dynamic Analysis 139 4.7.1 Tree motion and structural magnification 140 4.8 Vibration Analysis, Estimating Dynamic Characteristics of Trees 143 4.8.1 Introduction 143 4.8.2 Free response oscillations (the pull and release test) 144 4.8.3 Spectral analysis of tree dynamic response 146 4.8.4 Mean moment response to estimate drag coefficients 151 4.9 Multi-degree of Freedom Systems and Tuned Mass Dampers 151 Chapter 5. RESULTS 157 5.1 Static Tests 159 5.2 Field Measurements on Wind Loads on Trees 162 5.2.1 Time series results 162 5.2.2 Along wind, across wind and resultant response 164 5.2.3 Branch results 169 5.2.4 Wind loads maximum values 172 5.2.5 Wind loads and wind speeds (averaging method) 174 5.3 Dynamic Analysis of Tree Response 180 5.3.1 Free vibration and the pull/release test 180 5.3.2 Spectra 186 5.3.3 Transfer functions 188 5.3.4 Drag and damping ratio 199 5.3.5 Dynamic response factor 207 Chapter 6. DISCUSSION 211 6.1 Tree Dynamics and Mass Damping 211 6.2 Wind and Trees 219 6.3 Spectral Analysis and Trees 223 6.4 Drag Force and Drag Coefficient 226 6.5 Energy Transfer and Damping 229 Chapter 7. CONCLUSION 235 Chapter 8. BIBLIOGRAPHY 247 vi List of Figures Page Chapter 1 Figure 1.1 Basic difference between static and dynamic loads: (a) static loading; (b) 5 dynamic loading (Clough and Penzien 1993). Figure 1.2 Lumped-mass idealization of a simple beam (Clough and Penzien 1993). 7 Figure 1.3 Uniformly distributed mass along a beam with deflection comprising 8 superposition of simple mode shapes to produce a complex solution (Clough and Penzien 1993). Figure 1.4 Branch joint of a tree, with finite element representation to visualise stress 9 distribution (Mattheck 1990). Figure 1.5 Comparison of slenderness ratios of trees, (a) Hoop pine (Araucaria 13 heterophylla) (b) Red gum (Eucalyptus tereticornis) (c) NZ Kauri pine (Agathis australis) (d) Palm (Washingtonia robusta) (e) Italian Cypress (Cupressus sempervirens) and trees from selected references, (f) Sequoia (Sequoiadendron giganteum),(Niklas 1992) (g) lodgepole pine (Pinus contorta) (Rudnicki et al. 2001). Figure 1.6 Structural compressive tests on green wood for two tree species (Brudi, 2002). 14 Chapter 2 Figure 2.1 Young‟s modulus (E) of wood (balsa) in three directions.