Duration-of-load and Creep Effects in Thick MPB Strand Based Wood Composite by Bing Wang B.Sc., Chongqing University, 2000 M.A.Sc., Tongji University, 2003 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Forestry) The University of British Columbia (Vancouver) April 2010 © Bing Wang, 2010 Abstract British Columbia (BC) is in the midst of the largest outbreak of the Mountain Pine Beetle (MPB) ever recorded in western Canada. Technologies capable of converting stained lumber into market acceptable products are urgently required to reduce the impact of the growing volume of MPB killed lumber on the profitability of forestry in BC. New, thick MPB strand-based structural composite products can be produced and help absorb a large volume of MPB wood. With appropriate mechanical properties, such products can be used as beams, headers, and columns in the low-rise commercial, multi-family residential and single family residential markets. This work was focused on the duration-load and creep behaviour of thick MPB strand-based wood composite. The beam specimens were made in the Timber Engineering and Applied Mechanics Laboratory at UBC. A series of tests were conducted on the matched groups to investigate the creep-rupture behaviour. These investigations comprised of ramp load tests at three loading rates, long-term constant load tests at two stress levels and cyclic bending tests at six stress levels. A damage accumulation model was developed to study the creep-rupture behaviour. This model stipulates that the rate of damage growth is given in terms of the current strain rate and the previously accumulated damage, and a 5-parameter rheological model is applied to describe the viscoelastic constitutive relationship to represent the time-dependent strain, while the damage accumulation law acts as the failure criterion. The results of the long-term constant load tests ii were then interpreted by means of the creep-rupture model which had been shown to be able to represent the time-dependent deflection and time-to-failure data for different stress levels. The predictions of the model were verified using results from ramp load tests at different loading rates and results from cyclic loading tests at different stress levels. The creep-rupture model incorporates the short term strength of the material, the load history and predicts the deflection history as well as the time-to-failure. As it is a probabilistic model, it allows its incorporation into a time-reliability study of wood composites’ applications. iii Table of Contents Abstract………………………………………………….……………………………………..ii Table of Contents………………………………………..………………………………………iv List of Tables.…………………………………………………………………………………..viii List of Figures……………………………………………………………………………….......ix Acknowledgments.………………….………………………………………………………......xii Dedication……………………………………………………………………………………...xiii Chapter 1 General Introduction and Research Objectives…………………………………..1 1.1 Introduction………………………………………………………………………………..1 1.2 Background and research significance…………………………………………………2 1.3 Objective and methods……………………………………………………………………6 Chapter 2 A Literature Review of Duration-of-load and Creep Effects………………….... 8 2.1 Summary…………………………………………………………………………………8 2.2 Introduction………………………………………………………………………………..8 2.3 Duration-of-load model…………………………………………………………………...9 2.3.1 Damage accumulation models……………………………………………………….11 2.3.2 Fracture mechanics approach……………………………………………………….16 2.3.3 Strain energy approach……………………………………………………………...19 2.4 Creep modeling………………………………………………………………………….20 2.4.1 Mathematical modeling of viscoelastic creep………………………………………20 iv 2.4.2 Modeling of mechano-sorptive behavior……………………………………………23 2.5 Conclusions………………………………………………………………………………28 Chapter 3 Modeling of Creep-rupture Behavior………………….…………………............29 3.1 Summary…………………………………………………………………………………29 3.2 Introduction………………………………………………………………………………30 3.3 Constitutive relationship………………………………………………………………30 3.3.1 3-element model……………………………………………………………………..31 3.3.2 4-element model……………………………………………………………………..35 3.3.3 5-parameter model…………………………………………………………………37 3.4 Damage accumulation model…………………………………………………………38 3.5 Application of creep-rupture model to different load cases…………………………..42 3.5.1 Constant load………………………………………………………………………..42 3.5.2 Ramp load…………………………………………………………………………..43 3.5.3 Experimental load history…………………………………………………………45 3.5.4 Fatigue load…………………………………………………………………………48 3.6 Conclusions and discussions……………………………………………………………52 Chapter 4 Testing of Duration-of-load and Creep Behavior of MPB Strand-based Wood Composite…………………………………………………………………….....54 4.1 Summary…………………………………………………………………………………54 4.2 Introduction………………………………………………………………………………55 4.3 Materials and methods…………………………………………………………………...55 4.3.1 The MOE-matching technique……………………………………………………57 v 4.3.2 Ramp load test……………………………………………………………………...59 4.3.2.1 Control group ………………………………………………………………...59 4.3.2.2 Different rates of loading ………………………...…………………………...60 4.3.3 Creep-rupture test…………………………………………………………………...61 4.3.4 Cyclic bending test………………………………………………………………….64 4.4 Experimental results…………………………………………………………………….69 4.4.1 MOE test results……………………………………………………………………69 4.4.2 Ramp load test results……………………………………………………………..71 4.4.2.1 Control group test results ……...……………………………………………71 4.4.2.2 Ramp load test results for different rates of loading ……...……….……….73 4.4.3 Creep-rupture test results………………………………………………………...…75 4.4.4 Cyclic bending test results…………………………………………………………78 4.5 Conclusions………………………………………………………………………………83 Chapter 5 Calibration and Verification of the Creep-rupture Model…………………….85 5.1 Summary…………………………………………………………………........................85 5.2 Introduction………………………………………………………………………………85 5.3 Calibration of the creep-rupture model…………………………………………………87 5.3.1 Calibration of the creep model………………………………………………………87 5.3.2 Calibration of the damage accumulation model …………………………………….94 5.4 Verification of the creep-rupture model…………………………………………………98 5.4.1 Verification with ramp load test results……………………………………………...98 5.4.2 Verification with fatigue test results……………………………………………...100 5.5 Conclusions……………………………………………………………………………..103 vi Chapter 6 Concluding Remarks and Future Work…………………………………….......105 6.1 Concluding remarks…………………………………………………………………….105 6.2 Future areas of research………………………………………………………………...107 References……………………………………...........................................................................109 Appendix I.1 Experimental Results and Model Simulation of Strain History under Constant Load - Level I (27 MPa)……………………………………………116 Appendix I.2 Experimental Results and Model Simulation of Strain History under Constant Load - Level П (33 MPa)………………………………………….132 Appendix II Experimental Results of Strain History under Cyclic Load………………...148 Appendix Ш Calibration Results of Creep Parameters………………………………162 Appendix IV Experimental Data of Short-term Strength and MOE for Control Group Test Sample……………………………….…......................................................164 vii List of Tables Table 4.1: Loading data of cyclic bending test...……………………………………………...67 Table 4.2: A summary of statistical data on MOE…………………………………………70 Table 4.3: A summary of statistical data on short-term strength………...…………………71 Table 4.4: Distribution characteristics of Three-parameter Weibull Distribution fitted to short-term strength data…………………………………………………….…...72 Table 4.5: Rates of loading and corresponding time-to-failure in ramp load tests…….74 Table 4.6: Time-to-failure and mechanical properties of groups tested at different rates of loading…………………………………………………………………………74 Table 4.7: Independent-samples t-test for bending strength…………..……………………74 Table 4.8: Time-to-failure and deflection at failure of broken specimens in creep-rupture test…………………………………………………………………………………77 Table 4.9: Cyclic bending test results.………………………………………………………...79 Table 5.1: A summary of statistical data on creep parameters………………………………90 Table 5.2: Distribution characteristics of two-parameter Weibull distribution fitted to creep data………………………………………………………………………………...93 Table 5.3: Correlation coefficients between creep parameters…………………………........93 Table 5.4: Calibration result of parameter B……………………………………………........97 Table 5.5: Experimental results and model prediction of time-to-failure under ramp load………………………………………………………………………………99 viii List of Figures Figure 3.1: 3-element viscoelastic model..……………………………………………………31 Figure 3.2: Creep compliance of 3-element model…………………………………………...34 Figure 3.3: Time-dependent strain of 3-element model……………………………………34 Figure 3.4: Strain rate of 3-element model …………………………………………………...34 Figure 3.5: 4-element viscoelastic model……………………………………………………35 Figure 3.6: Time-dependent strain of 4-element model……………………………………...36 Figure 3.7: Strain rate of 4-element model……………………………………………………36 Figure 3.8: Time-dependent strain of 5-parameter model…………………………………...37 Figure 3.9: Strain rate of 5-parameter model………………………………………………...38 Figure 3.10: Constant load……………………………………………………………………..42 Figure 3.11: Ramp load………………………………………………………………………...43 Figure 3.12: Plot of strain rate versus time under ramp load…………………………….....44 Figure 3.13: Experimental load history…………………………….........................................46 Figure 3.14: Triangular cyclic loading scheme………………………………………………..48 Figure 3.15: Strain-time relationship under triangular cyclic load ………………………51 Figure 4.1: A typical MPB log used in this work……………………………………………56 Figure 4.2: A typical hand formed mat………………………………………………………..56 Figure 4.3: Thick MPB strand-based wood composite specimens…………………………57 Figure 4.4: Test apparatus of MOE test……………………………………………………….58 Figure 4.5: Test apparatus of short-term bending test……………………………………….59 Figure 4.6: Setup of creep-rupture test………………………………………………………..62 Figure 4.7: Loading system of creep-rupture test…………………………………………….63 ix Figure 4.8: Data acquisition system…………………………………………………………...63 Figure 4.9: Deflection measuring device for creep-rupture test……………………………..64 Figure 4.10: Loading Scheme of cyclic bending test………………………………………….65 Figure 4.11: Setup of cyclic bending test……………………………………………………...68 Figure 4.12: Loading program of cyclic bending test………………………………………68 Figure 4.13: Deflection measuring device for cyclic bending test…………………………69 Figure 4.14: Relationship between MOE and density………………………………………..70 Figure 4.15: Cumulative distribution curve of experimental data and Three-parameter Weibull Distribution fitted to short-term strength………………………………………72 Figure 4.16: Relationship between MOR and MOE for short term test sample…………73 Figure 4.17: Cumulative distribution curves of time-to-failure under constant load….......78 Figure 4.18: The typical tension failure mode………………………………………………80 Figure 4.19: Comparison between creep and fatigue test results……………………………82 Figure 5.1: Experimental load history………………………………………………………...86 Figure 5.2a: Two-parameter Weibull Distribution fitting curve of ………………….......91 Figure 5.2b: Two-parameter Weibull Distribution fitting curve of ……………………...91 Figure 5.2c: Two-parameter Weibull Distribution fitting curve of ………………………92 Figure 5.2d: Two-parameter Weibull Distribution fitting curve of ……………………...92 Figure 5.2e: Two-parameter Weibull Distribution fitting curve of ………………………93 Figure 5.3: Comparison between experimental results and model simulation of time-tofailure under constant load………………………………………………………………..97 Figure 5.4: Comparison between experimental results and model prediction of time-tofailure under ramp load…………………………………………………………………100 x Figure 5.5: Relationship between median of number of cycles to failure and stress ratio..102 Figure 5.6: Experimental and model predicted ranges of number of cycles to failure under cyclic load…………………………………………………………………………………103 xi Acknowledgements I would like to express my deepest gratitude to my research supervisor, Dr. Frank Lam, for his guidance and support throughout this work. Great appreciation is also given to the supervisory committee members, Dr. Ricardo Foschi and Dr. Reza Vaziri, for their valuable suggestions, encouragement and support all the time. Thanks to the Timber Engineering and Applied Mechanics team at UBC for their years of support. Special thanks to Mr. George Lee and Dr. Azzeddine Oudjehane for their help and time. Thanks to the Forestry Innovation Investment Ltd. for their financial support. Finally I would like to take this opportunity to express my deepest gratitude to my parents for their unconditional love and selfless support throughout my life. I am also deeply grateful to all my friends for their caring, understanding and being on my side all the time. xii To my parents xiii Chapter 1 General Introduction and Research Objectives 1.1 Introduction A comprehensive program of the development of thick strand-based wood composites manufactured from Mountain-Pine-Beetle (MPB) killed lumber has been carried out in the Department of Wood Science at the University of British Columbia, Canada. The objective of this program is to develop a basic understanding of such products’ fundamental properties. This knowledge can be used to calibrate a mechanics based model for strength properties prediction of thick MPB strand-based products. As part of the program, the work conducted in this thesis is focused on the research of the duration-of- load (DOL) and creep behavior of such wood based composites. Chapter 1 gives a brief introduction of the research background, objectives and methods. A literature survey of the research work which has been done in the field of durationof-load and creep is carried out in Chapter 2. In Chapter 3, a creep-rupture model is developed where the accumulated damage is linked to creep deformation. This creep-rupture model incorporates the short term strength of the material, the load history and predicts the deflection history as well as the time-to-failure. As it is a probabilistic model, it allows its incorporation into a time-reliability study of wood composites’ applications. Chapter 4 investigates the DOL and creep behavior of thick strand-based wood composite in a series of tests which include ramp load tests, long-term constant load tests and cyclic bending tests. Then in Chapter 5 the creeprupture model developed in Chapter 3 is calibrated against experimental data of long-term constant load tests and verified using the data from ramp load and cyclic bending tests. Finally, the implications of the work conducted in this thesis as well as future research are discussed in Chapter 6. 1 1.2 Background and research significance British Columbia (BC) is in the midst of the largest outbreak of the Mountain Pine Beetle (MPB) ever recorded in western Canada. The epidemic is concentrated in central and Southern Interior BC where the MPB (Dendroctonus ponderosae) primarily infests Lodgepole pine (Pinus contorta) trees. Once a healthy tree is attacked the MPB bores egg galeries into the tree sap wood disrupting the water transportation in the tree. In addition, it also carries blue staining Ophiostomatoid fungi such as Ophiostoma clavigerum (Lim et al., 2004). The fungi and the beetle share a symbiotic relationship where the fungi ‘uses’ the beetle as a ‘vector’, a means to spreading to new trees while the beetle larvae use the fungi as a food source. The blue staining fungi spread throughout the tree sapwood decreasing the moisture content and weakening the tree defense mechanisms, eventually leading to tree death. The MPB life cycle lasts one year, with the healthy trees usually attacked in the summer time, this is known as the ‘green phase’ of infection where the tree foliage still retains its green color. The next generation of beetles tends to bore out of the tree in the summer of the following year, known as the ‘grey stage’ of infection when the tree is beginning to die and the foliage color begins to fade from green to red. Finally, death of the tree usually occurs 1 year later, the ‘red phase’ when the foliage turns red and the sap is stained blue throughout most of the tree. It is believed that a prolonged winter period of 40oC significantly reduces the MPB larvae populations in infected trees, and thus limits the mature MPB population the following summer. However, the current infestation has been exacerbated by progressively milder winters and longer, hotter summers allowing for a longer mating season and a greater population of flying adult beetles which increases the rate and geographic range of infection. The MPB epidemic coupled with limitation of annual allowable cut will result in a large volume of older dead lodgepole pine coming into the log supply for BC forest industry in the next decade and beyond. 2 In addition to reducing the aesthetic qualities of wood, blue-stain also alters the morphology and chemistry of infested wood and these properties are related to the time-since-tree-death. Blue stain develops quickly in the sapwood of dying trees and carries over into products made from the stained logs as well. Infested trees also dry and develop splits and checks as the drying stresses are relieved. Such physical conditions lower its commercial value for use in lumber and pulp. Furthermore, an increased volume percentage of fines and residues is expected from the processing of this resource. The Ministry of Forests and Range analysis shows that, at the current rate of spread of MPB infestation, 80 percent of the merchantable pine could be killed by 2013. In a large portion of the Interior BC, pine makes up over 50 percent of the harvestable timber (British Columbia’s Mountain Pine Beetle Action Plan 2006-2011). According to this statistical information, the MPB killed timber would make up over 40% of the province’s lumber production by 2013. Therefore technologies capable of converting stained lumber into market acceptable products are urgently required to reduce the impact of the growing volume of MPB killed lumber on the profitability of forestry in BC. The government is encouraging alternate timber processing industries, including the production of oriented strand board (OSB), to utilize MPB killed timber. Currently 10% of Lodgepole pine supply is used in the manufacture of plywood and OSB (Harrison, 2006).Sawmills in MPB infested regions will increasingly be processing beetle-killed lodgepole pine timber. Currently some OSB (oriented strand board) mills are manufacturing OSB panels with a mixture of MPBkilled and aspen wood. These mills are addressing many technical issues associated with OSB manufacturing with MPB-killed wood including processing of the dry logs, excessive fine 3 content, etc. By the nature of the manufacturing process of such wood-based products, large defects such as knots and other strength-reducing characteristics are either eliminated or dispersed throughout the cross section to produce a more homogeneous product. Manufacturers can control the production process and thus the material’s short-term strength, stiffness, density and variability of these properties. Using the OSB production technology as a base, thicker products with additional strand alignment can be produced to be used in structural applications. New, thick MPB strand-based structural composite products can help absorb a large volume of MPB wood. With appropriate mechanical properties, such products can be used as beams, headers, and columns in the low-rise commercial, multi-family residential and single family residential markets. Given the improved product consistency as an engineered wood product, it is especially suitable for low rise commercial and multi-family residential applications where load demands are higher compared to single family applications. As an engineered wood product, one of the key properties is its structural response under long term loading. The phenomenon of creep, i.e. the time-dependent increase of deformation or deflection of structural members under a constant stress, is an important material characteristic. It can lead to structural failure as either excessive deformation or worse as a collapse. It is necessary to limit the deflection because excessive deformation will cause damage to associated building materials such as the cracking of plaster and splintering of windows and jamming of sliding doors. At the same time sagging beams are unsightly and convey an impression of impending collapse. Deflection limits are also required to avoid floor vibration problems. As creep can lead to collapse, duration-of-load (DOL) is another important material characteristic. The deterioration of strength under continued application of constant stress is called duration-of4 load effect which is manifested as the failure after a given period of loading. This failure or rupture under long-term loading is termed creep-rupture (Laufenberg, 1988) which is used to describe the combined phenomena of duration-of-load and creep. All wood-based products are susceptible to creep and duration-of-load effects. Thus, in-service stiffness must be known and controlled to limit the component’s creep deformation and also it is important to quantify the load duration effect to ensure reliability throughout its intended service life. The creep and duration-of-load effects are of great interest in timber engineering applications as well as to wood composite manufacturers concerned with the introduction of new building products and the implementation of new codes for engineering design in wood. Since the early 1970s, a significant amount of work has been conducted on measuring and empirically modeling the time-dependent strength behavior of structural size lumber. However, limited published information on creep-rupture behavior in wood-based composite lumber products hinders the development of a reliability-based assessment for these newer products. The performance of structural wood composites is not well understood and some consider this aspect as the ‘Achilles heel’ for the further development of structural wood composites. A new ASTM Standard Specification (ASTM D6815) has been developed to provide a procedure for testing and evaluating duration-of-load and creep effects of wood and wood-based materials. The intention of this procedure is to demonstrate the engineering equivalence of wood-based products used in dry service conditions in terms of duration-of-load and creep effects to that of solid sawn lumber. Equivalence demonstrated in this specification is dependent upon the evaluation of a product’s 90-day (minimum) creep-rupture performance when subjected to a constant load level of 55% of the 5-th percentile of the short-term strength. In this evaluation, three criteria must be satisfied: (1) adequate strength over a 90-day period, (2) decreasing creep rate, and (3) limited fractional 5 deflection (ratio of deflection after 3 months to deflection after 1 minute). This standard is considered to be a pass-fail procedure and does not attempt to develop a duration-of-load factor for a specific product. Further, there is no intent to establish the long-term creep-rupture response from this 3-month test at a relatively low load level. The work conducted in this thesis is of great research and practical significance in that it represents the first known attempt to model and evaluate the duration-of-load and creep effects and fatigue behavior of thick strand-based wood composites made from MPB-killed lumber. The distinctive advantage of this creep-rupture model is that it permits the prediction of damage produced by an arbitrary random load sequence and its convenient usage in reliability-based design formats. The damage accumulation model developed in this work takes into account the deformation history; herein the model provides a description of the time-dependent deflection to present a complete picture from creep deflection to a final rupture event. In this manner, the creep-rupture model incorporates the short term strength of the material, the load history and predicts the deflection history as well as the time-to-failure. As it is a probabilistic model, it allows its incorporation into a time-reliability study of wood composites’ applications. Meanwhile an experimental database on the time-dependent mechanical properties of the strandbased wood composite products has been generated for the purpose of model calibration and future research. 1.3 Objective and methods The objective of this thesis is to investigate and evaluate the creep-rupture behavior of strandbased wood composites under sustained loading and to provide a better understanding of the inservice stiffness and the possibility of rupture of such wood composites throughout their 6 intended service life. In order to model the duration-of-load and creep effects in strand-based wood composites, a damage accumulation model was proposed to study the creep-rupture behaviour. This model stipulates that the rate of damage growth is given in terms of the current strain rate and the previously accumulated damage, and a 5-parameter rheological model is applied to describe the viscoelastic constitutive relationship to represent the time-dependent strain, while the damage accumulation law acts as the failure criterion. Therefore how strength is affected by the passage of time, and how the deflection of structural members is influenced by time is studied with a unified approach. Meanwhile, MPB strand-based wood composite product was developed and prototype samples were processed in the Timber Engineering and Applied Mechanics Laboratory at the University of British Columbia. Then a series of tests were carried out to investigate the creep and duration-of-load effects of thick MPB strand-based wood composite beams. The testing comprised ramp load test at three different loading rates, longterm constant load test at two stress levels and cyclic bending test at six stress levels. Then the creep-rupture model was calibrated against the experimental data of long-term constant load test and verified using the results from ramp load test and cyclic bending test. This model allows one to gain deeper insight into the interdependent duration-of-load and creep effects in strand-based wood composites. 7 Chapter 2 A Literature Review of Duration-of-load and Creep Effects 2.1 Summary Presented here is a literature review in fields of duration-of-load and creep. This work will help to establish a unified approach to investigate the in-service stiffness and possibility of rupture of strand-based wood composite products under a range of loading conditions. 2.2 Introduction The phenomenon of creep, i.e. the time-dependent increase of deformation of material under a constant stress, is an important material characteristic because it can lead to structural failure as either excessive deformation or worse as collapse. There are several reasons why it is necessary to limit the deflection: 1) to avoid damage to associated building materials, e.g., cracking of plaster, splintering of windows; 2) to prevent functional failures such as jamming of sliding doors; 3) appearance – sagging beams reduce the aesthetic quality of products and convey an impression of impending collapse; 4) floor vibrations – deflection limits were supposed to avoid vibration problems; and 5) ponding of flat roofs (Madsen, 1992). Creep can lead to collapse. A typical creep curve can be divided into three phases including primary, secondary and tertiary creep. The tertiary creep phase can lead to material failure due to creep-rupture. In the timber engineering literature creep-rupture is commonly referred to as a “duration-of-load” effect on time-to-failure under constant load. Previous work has treated timeto-failure under constant load and creep as uncoupled problems. That is, the two basic behaviors, 8 deformation (creep) and time-to-failure (duration-of-load), have been examined independently. Models employed to characterize either the deformation or the time-to-failure are intended to provide a mathematical description of the behavior for a given environmental condition and a range of loading conditions so that the long-term behavior of a material may be predicted in a variety of structural applications (Laufenberg, 1988). Considering the continuous nature of creep-rupture behavior, development of new models reflecting the interdependent relationship between duration-of-load and creep will be an important topic of research. Therefore, a literature review in both fields of duration-of-load and creep is performed first with the aim of understanding the state of the art needed to establish a unified approach to address both issues. 2.3 Duration-of-load model The duration of load is a phenomenon that has puzzled wood scientists and timber engineers for a long time. It has been observed that wood can carry a higher load (or stress) for a short period of time than it can carry for a long period of time. The relationship between the applied stress and time-to-failure is called the duration-of-load (DOL) effect. The DOL effect should not be assumed to be caused by deterioration such as that associated rot or other biological attack but rather a phenomenon in itself (Madsen, 1992). It is one of the distinctive characteristics of wood that its apparent strength is influenced by the intensity and duration of the applied load. Thus it is important to quantify this load duration effect in order to be able to calculate the reliability throughout a structure’s service life. The application of damage-accumulation models and linear viscoelastic fracture mechanics are well-established tools for the prediction of duration of load response in lumber. 9 In the latter half of the 20th century the DOL factors used in Canadian and US allowable stress codes were based on Wood’s (1951) work on small defect-free specimens. The DOL factors were applied to adjust a range of design stresses for lumber and other wood products as well as connectors for different in-service loading conditions. Wood indicated that the relationship between the bending stress ratio (maximum apparent bending strength divided by short-term strength) and the logarithm of the time over which the stress was applied was slightly different for ramp and constant loading, and was slightly curvilinear. An empirical hyperbolic curve, known as the “Madison Curve”, was chosen to fit the trends of Wood’s (1951) long-term constant loading data, rapid-loading test results and the impact point established by Elmendorf. The horizontal asymptote of this hyperbola was 18.3 percent, a stress ratio level for which the duration of loading was presumed to be infinite. By the early 1970s there was abundant evidence to indicate that the creep rupture response of structural timber beams differed considerably from the classic case for small clear test pieces described by the Madison Curve. In Canada the first DOL research on dimension lumber was carried out by Madsen (1971,1973) who tested No.2 grade nominal 2- by 6-in. (standard 38- by 140-mm) hem-fir lumber under bending with several stepwise constant bending regimes. One of Madsen’s conclusions was that the time effect was strongly dependent upon strength level. Lowstrength structural timber not only had less DOL effect than high strength timber, but it also had a DOL effect that was significantly less severe than that predicted from the Madison Curve which is based on small clear test pieces. Later, Madsen and Barrett (1976) and Foschi and Barrett (1982) showed that the relationship between applied constant load and time-to-failure for small defect-free specimens differs from a similar relationship in full-size lumber. 10 The above test work clearly indicated that the DOL factors predicted by Madison curve were conservative and in order to obtain a more realistic prediction of time-to-failure, attention moved to the possible application of reliability-based design principles for the assessment of the reliability of timber members under in-service loading conditions. In particular, this approach led to the adoption of the concept of damage accumulation. In the Canadian Standard CSA O86, the DOL factor was based on the research work of Foschi in 1989 (Foschi, 1989), and the damage state variable α 30 was used to estimate the reliability of the design at the end of the service life of 30 years. 2.3.1 Damage accumulation models Following the procedures used in metal fatigue to predict number of cycles to failure, the process leading to failure in wood was modeled as an accumulation of damage (Wood Design Manual, 2001). The cumulative damage we are concerned with stems from material behavior at the microscopic level. Because knowledge of behavior at this level is generally incomplete, it is difficult to postulate corresponding models for damage accumulation based on physical laws. As an alternative, cumulative damage models have been proposed based on our understanding of the phenomena at the macroscopic level and upon experimental data (Yao, 1987). The distinctive advantage of a damage accumulation model is that it permits the prediction of damage produced by an arbitrary random load sequence and its convenient usage in reliability-based design formats. Several types of damage accumulation models have been recorded, of which the most important models are summarized in the following: 11 Barrett and Foschi model I (Barrett and Foschi, 1978) if τ > τ 0 dα dt = a (τ − τ 0 ) b α c if τ ≤ τ 0 =0 Barrett and Foschi model II (Barrett and Foschi, 1978) dα dt = a(τ − τ 0 ) b + λα if τ > τ 0 if τ ≤ τ 0 =0 Foschi and Yao model (Foschi and Yao, 1986) dα dt = a(σ (t ) − τ 0σ S ) + c(σ (t ) − τ 0σ S ) α b n =0 if σ (t ) > τ 0σ S if σ (t ) ≤ τ 0σ S Gerhards’ model (Gerhards and Link, 1987) dα dt = exp(−a + bτ ) = exp(− a + b σ (t ) σ S ) Schaffer model (Schaffer, 1973) dα dt = −α exp(− a + b σ (t ) (1 − α )) In all the models α is the damage state variable (“𝛼𝛼 = 0” in undamaged state and “𝛼𝛼 = 1” at failure), τ is the stress ratio defined as the applied stress σ (t ) divided by the short-term strength σ S (i.e. τ = σ (t ) σ S ); τ 0 is the stress threshold below which damage is assumed not to accumulate, and a, b, c, λ and n are model parameters. It should be noted that both the shortterm and the long-term strength cannot be known for the same structural test piece, because the test piece can be tested only once. This problem was usually resolved by side matched control to estimate the short-term strength for long-term test specimens. 12 The Madison Curve generally overestimated time-to-fracture for both shear and bending strength. This overestimate decreased as the assumed creep rupture threshold increased. Unfortunately, changes in the threshold level had little effect on the accuracy of the fit in the range of available data (Barrett and Foschi, 1978). As for the Barrett and Foschi models, in model I, the damage rate was expressed as the product of a stress-dependent factor and a damage-dependent factor. In model II, the damage rate was given as the sum of a stress-dependent term and a damage-dependent term. Both models considered a stress threshold below which the damage rate vanished. For each model, stress histories corresponding to three types of tests were considered: a step function load test and a ramp load test and a combination of an initial ramp load followed by a constant load. The predictions of the models for these two cases were used to obtain model parameters by fitting to constant and ramp load test data. The data used corresponded to small, clear specimens of Douglas-fir in bending. They also found that model II was a more flexible model and provided a better fit of constant load data for shear and bending of small clear specimens. The Foschi and Yao model also postulated that damage accumulated at a rate which was not only a function of the applied stress σ (t ) , but also a function of already accumulated damage α . It was a more realistic model because in some of the previous models if damage had accumulated to a significant level the slightest damage would continue to grow under the absence of stress and therefore material failure could occur under no stress. According to the expanded series of damage accumulation rate, dα dt = F0 (σ ) + F1 (σ )α + F2 (σ )α 2 + 13 Foschi and Yao model used up to the first order damage-dependent term. Stress histories corresponding to three types of tests were considered in the model formulation: a ramp load test, a constant load test and arbitrary load history test. The model was calibrated with the data from the experiment for Western hemlock lumber at two constant stress levels which was conducted by Foschi and Barrett since 1977 (Foschi and Barrett, 1982), and the model followed the experimental trends very well. In Gerhards’ model the exponential damage rate model (EDRM) was used to describe duration of load data on lumber in bending. The model was used to fit the ramp and constant load experimental data by an iterative reweighted nonlinear least squares procedure. Except for the rapid ramp results, the EDRM provided a reasonable fit to constant load and ramp load data for lumber containing an edge knot. However it was also concluded that the model seemed to overpredict residual strengths of specimens surviving constant load. It should be noted that in Gerhards’ model, the damage accumulation rate is only a function of applied stress. Yao (1987) calibrated Gerhards’ model to the same Western hemlock bending data and concluded that the EDRM was too stiff to represent the data trend and that this fact revealed the effect of the damage already accumulated in the process of damage accumulation. Gerhards’ model could not fit the long-term experimental results because it lacked a damagedependent factor. Fridley et al. (1992c) proposed a modified EDRM model where the damage rate was a function of stress, temperature and moisture content. The stress ratio was adjusted depending on the temperature and moisture conditions of the lumber. In this way, additional damage was 14 associated with mechano-sorptive effects, and thus the influence of changes in moisture content on creep-rupture behavior was studied. Work from 1950s to 1980s on load duration effect and the rheological behavior of composite wood-based panel products was reviewed by Laufenberg (1987, 1988) and augmented with new experimental data (McNatt and Laufenberg, 1991; Laufenberg et al., 1999). Laufenberg (1999) summarized a cooperative research program between the USDA Forest Service, Forest Products Laboratory (FPL) in Madison, and Forintek Canada Corp. in Vancouver (now FP Innovations). This research program provided detailed creep-rupture and some creep information for composite panel products, such as plywood, OSB and waferboard. The study included creeprupture testing using three rates of ramp loading and three levels of constant loading under one environmental condition, and creep testing at two low constant load levels under three environmental conditions for a 6-month period. Predictions of creep-rupture behavior were obtained through Gerhards’ Exponential Damage Accumulation Model (EDRM) and exponential model, which for constant load test takes the following form, SL = A + B ln T EDRM yielded some parameter estimates with wide confidence limits. It was concluded that the exponential model provided parameters that appear appropriate; however the model had no mechanism for inclusion of specimens that survived or failed on uploading. Moreover, the model appeared to be unduly sensitive to the reference strength chosen if the sample had few failures in the first few days of testing. In order to analyze the creep testing data, a number of empirical and viscoelastic models were examined to characterize the time-dependent behavior of all the specimens tested. Each modeling method had its advantages and limitations. It was then concluded that the four-element creep model reported by Pierce et al. (1977) provided the best fit 15 for the measured medium-term deflections or strains. However the simplicity of the twoparameter power function creep model was highly appealing for practical purposes. In spite of its objective to use analytical methods for describing and predicting panel behavior during creep to rupture, the phenomena of creep and creep-rupture were actually studied separately in terms of modeling. Thus, the damage was not related to actual creep behavior. Brandt and Fridley (2003) studied the load-duration behavior of wood-plastic composites, and the experimental data was fitted to both Gerhards’ EDRM and Wood’s model by the least square regression method. It was found that the wood-plastic composites tested exhibited a more pronounced load-duration response than that of solid wood. However, the load-duration behavior trend of the selected WPC formulations was determined to be similar to that of solid sawn lumber, although a rotation of the EDRM curves was observed. When wood and wood-based products are subjected to cyclic load, the creep behavior is closely related to fatigue behavior. Lam (1982) calibrated the Foschi and Yao model against the fatigue test data to predict the relationship between load levels and number of cycles to failure for small specimen in bending. Dinwoodie et al. (1995) studied the influence of slow cyclic fatigue on the creep behavior of chipboard at a range of stress levels and moisture contents. 2.3.2 Fracture mechanics approach Another promising approach employed fracture mechanics to study the process of slow crack growth through timber. Pitched at the microscopic level, this approach provided some complementary information on cause and effect in the failure process. Fracture mechanics has 16 been successfully applied to time to failure studies in timber (Johns and Madsen, 1982; Nielsen, 1986; Sorensen and Svensson, 2004). DVM (Damaged Viscoelastic Material) Theory was proposed by Nielsen (Nielsen, 1986, 2000, 2004). In Nielsen’s work an integrated materials concept for the behavior of wood was established: wood is a cracked viscoelastic material the mechanical behavior of which can only be described in sufficient detail by coupling the theories of viscoelasticity and fracture mechanics. The ingredients of DVM-model were viscoelasticity, elastic-viscoelastic analogy and Dugdale crack model (including crack closure). The rate of crack growth was given as: dk π 2 = dt 8qτ σ cr σl k (σ σ cr ) 2 [(k (σ σ 2 cr ) ) −1 2 ] −1 1b Where b and τ are parameters from the creep function, J (t ) : [ J (t ) = 1 + (t τ ) b ]E With E = Young’s modulus σ l = theoretical (un-damaged) strength σ cr = reference strength (1 + b )(2 + b ) q= 2 1b k = l l 0 = damage ratio (dimensionless crack length ratio) However, parameters for these advanced models were difficult to generate and in some cases may have little or no means of assessing confidence in the estimates (Link et al., 1988). Thus, 17 their usage in reliability-based design formats may be limited. Yet to be developed, however, is the model which can link the deformation and failure process. In Nielsen’s latest paper (Nielsen, 2004), it was demonstrated that the influence of moisture and load variations on lifetime and residual strength (re-cycle strength) of wood can be considered by previously developed DVM theory. The common, controlling factor was creep, which can be modified by introducing a special moisture dependent relaxation time in the well-known PowerLaw creep expression. However it was recognized that the moisture distribution model described in this paper was rather crude and had to be refined in further research. Its purpose was just to demonstrate how moisture variations provoke increasing creep of wood while decreasing lifetime. In Johns and Madsen (1982), the duration of load effect was treated using a viscoelastic, limited ductility fracture mechanics model based on Nielsen’s theory. The model was explained and developed in a general way, then modified for use with commercial lumber. The problems of assigning correct creep function parameters and values of stress ratio for use in calculations involving the model were discussed. In Part II of this paper, the viscoelastic fracture mechanics model developed in Part I was calibrated using a 70-day duration of load test on commercial lumber. Sorensen and Svensson (2004) developed a model based on Griffith energy relation for a fracturing material to account for both stress and moisture dependency on the accumulation of damage in wood. In expressing strain needed in this theory, the power law creep model was 18 adopted. The creep law was modified with an accelerating factor, e C ∆m , to account for the moisture change dependency. The case with time varying stress was modeled by stepwise constant stress states and the accumulated damage, α , was determined by Miners rule for cumulative damage. Failure occurs when damage α equaled unity. Then the parameters in the model were fitted to data relevant for Nordic structural timber using the Maximum Likelihood method. The probability of failure as function of time was estimated by simulation. 2.3.3 Strain energy approach Another method to study the load-duration effects in structural lumber is the strain energy approach proposed by Fridley. In Fridley et al. (1992a), a critical strain-energy-density failure criterion was introduced to describe the load-duration behavior of structural lumber. A linear, viscoelastic constitutive model modified to account for hygrothermal effects developed by Fridley et al. (1992b) was used in conjunction with the critical strain energy failure criterion to predict the load-duration effects. For extensive ramp, constant, and step-constant load tests in various constant and cyclic environments, a critical strain-energy density was identified which forecasted impending failure. Impending failure was defined as the initiation of member failure, excluding partial failures that do not influence subsequent member behavior. The critical time, which was the time corresponding to the occurrence of the critical strain energy and impending member failure, was predicted using the strain energy model. It was concluded that the model can predict the critical time quite accurately when the creep response was predicted accurately. A loss of accuracy was observed when mechano-sorptive creep occurred owing to the reduced accuracy of the constitutive model for such behavior. A comparison of the strain energy model with damage accumulation models was presented in this paper. The damage models were 19 developed with a definition of failure as the total collapse of a member with a total loss of load carrying capacity. While the strain energy model defined failure as the exceeding of a critical strain energy density that corresponds to the initiation of final member failure. Therefore, strain energy approach was relatively conservative in comparison. 2.4 Creep modeling Creep occurs in most structural materials given the appropriate environmental conditions. Creep should be thought of as a time-dependent increase of deformations or deflections at constant loads as distinct from the deformations or deflections caused by changes in the stresses or loads. The creep deformations are most pronounced in bending members but it does occur in other types of structural members as well (Madsen, 1992). Creep includes three distinct types of behavior, which are difficult to separate because they can all operate simultaneously. These are: time-dependent (viscoelastic) creep, mechano-sorptive (moisture-change) creep, and the pseudocreep and recovery that have been ascribed to differential swelling and shrinkage. This literature survey is focused on the former two types of creep. 2.4.1 Mathematical modeling of viscoelastic creep A considerable amount of research has been conducted to define the creep behavior of wood mathematically. Among those work many empirical models have been proposed for describing such behavior. Their relative merits depend on how easily their constants can be determined and how well they fit the experimental results (Dinwoodie, 2000). One classical model which has been successfully applied to descriptions of creep in bending of timber over relatively long time spans is the so-called power-law creep function, of general form, 20 ε (t ) = ε 0 + at m Where ε (t ) is the time-dependent strain, ε 0 is the initial elastic strain, t is the elapsed time, and a and m are material-specific parameters to be determined experimentally (Schniewind, 1968). But more recent efforts (Gowda et al., 1996; Nielsen, 2000) have modeled relative creep normalized by the initial elastic strain. The prime advantage of using a power function to describe creep is its representation as a straight line on a log-log plot, thereby making curve fitting easier than using other models. Some other empirical models, such as exponential function, semilogarithmic function and second-degree polynomial are also available (Dinwoodie, 2000). Alternatively, models with more physical interpretation have been proposed. Creep behavior in timber can be interpreted with the aid of mechanical (rheological) models comprising different combinations of springs and dashpots. Some commonly used models are 1) the Kelvin model, which is a Hookean spring in parallel with a Newtonian dashpot; 2) the Maxwell model, which is a Hookean spring in series with a Newtonian dashpot; 3) a three-element model, which is comprised of a Hookean string added serially to a Kelvin model; and 4) the Burger model, which is the Kelvin model with a Maxwell model added in series. Pierce and Dinwoodie (1977, 1979) showed that the viscoelastic behaviour of chipboard under sustained load in bending could be represented by 3- or 4-element spring and dashpot models. A set of experiments was carried out on the creep behaviour of five commercially-available types of chipboard under 3-point sustained loading at a constant temperature and humidity. Creep curves based on 3- and 4-element rheological models were fitted to the data from each specimen 21 using an iterative least squares computer program developed by the authors. The models took the following form: a) for the 3-element model Y = β 1+ β 2 [1 − exp(− β 3t )] b) for the 4-element model Y = β 1+ β 2 [1 − exp(− β 3t )] + β 4 t These two equations gave the deformation Y at time t in terms of the material constants of the spring and dashpot components. It was concluded that although both models gave high multiple correlations, the 4-element model provided a consistently better overall fit. Also it should be noted that the 4-element model almost always overestimated the end deflection whereas the 3element model almost always erred on the low side. There was some evidence to show that as the stress level decreased the fit of the 3-element model improved at large time values. At low stress levels there may be little or no viscous flow and the deformation may tend towards a fixed horizontal asymptote. On the other hand the 4-element model predicted too high a deformation since it assumed a constant rather than a decreasing rate of creep which occurs in reality. It appeared that β 4 may be a function of time rather than a pure constant, and if this was true, neither the 3-element nor the 4-element model was entirely adequate. Then Pierce and Dinwoodie (1985) developed a modified 5-parameter model which added a power term that reduces the contribution of the viscous component over time as shown below: Y = β 1+ β 2 [1 − exp(− β 3t )] + β 4 t β 5 Where 0 < β 5 < 1 and the viscous component had a gradually reducing flow rate rather than a constant flow rate β 4 as in the 4-parameter model. The resulting 5-parameter model was shown to be superior to the 4-element model for long-term predictions of creep deflection particularly at 22 the lower stress levels. The efficacy of the models was tested on 7-10 years data, and the comparison with actual deflection confirmed the unsuitability of the 4-element model as a predictive tool (Dinwoodie et al. 1990). 2.4.2 Modeling of mechano-sorptive behavior Many attempts have been made since the late 1970s to develop a model for mechano-sorptive behavior, which is the deformation directly related to the interaction of moisture content changes and mechanical stressing. In particular, the researchers of Finland, Sweden and the UK have done a lot of work on this subject. A few of these models have been descriptive in nature. They sought relationships at the molecular, ultrastructural or microscopical levels. But most of them have been either purely mathematical, with the aim of producing a generalized constitutive equation; or partly mathematical, where the derived equation was linked to the physical phenomenon, or change in structure of the timber under stress. The descriptive models tried to relate the hydrogen bond breakage and subsequent reformation to mechano-sorptive behavior. Van der Put (1989) proposed a thermodynamic theory and thought it was possible to explain the mechano-sorptive effect by a general deformation kinetics model, as a bond breaking and remaking process, causing internal shifts of adjacent layers with respect to each other, due to sorption. This concept has been further developed by Hunt and Gril (1996) by linking it to the theory of physical aging, which is the time dependent approach of a polymer to thermodynamic equilibrium. Hoffmeyer (1989) introduced the hypothesis that the microfailures, known as slip planes, were the main cause of mechano-sorptive behavior of wood subjected to moderate or high compression or bending stresses parallel to grain. 23 Many of the mathematical models were developed aiming at establishing the constitutive relationship equations. They are based on the concept that the total strain comprises four components – an elastic strain, a creep strain covering normal time-dependent viscoelastic deformation, a swelling or shrinkage strain, and lastly a strain that covers mechano-sorptive behavior. Usually Maxwell, though sometimes Kelvin rheological units, were employed in model formation. Many of the quantitative models assumed for the simplicity that the terms for the various phenomena can be added. One of the first attempts to produce a quantitative description of mechano-sorptive behavior was done by Ranta-Maunus (1975) who developed a theory of hydroviscoelasticity which describes a functional relationship between deformation and stress, time, temperature and moisture content. A somewhat similar approach to constitutive modeling was adopted by Martensson (1988) to describe mechano-sorptive behavior of hardboard under a uniaxial tensile stress. It was assumed that the total strain rate was given as the sum of linear elastic strain rate, creep strain rate and stress-dependent moisture induced strain rate in which the free moisture induced strain rate was included. The tests showed that the effect of moisture variations on tensile creep was surprisingly small, while it was found to be much greater in relaxation tests. In a later paper, Martensson (1994) proposed a theoretical constitutive model for the case of uniaxial stress in the longitudinal direction. The constitutive relation was still given in rate form, and the total strain rate was assumed to consist of four main parts, one elastic part, one part describing creep under constant humidity conditions, one part describing the shrinkage-swelling behaviour and finally one mechano-sorptive part. Her model also incorporated the effect of strain and stress on shrinkage-swelling that had been found in experiments into the formulation of shrinkage- 24 swelling strain rate. In order to reflect the decreasing mechano-sorptive strain rate after the first sorption period, a mechano-sorptive limit had been introduced into the model. In more recent research, by analogy to corresponding rheological viscoelastic models, the mechano-sorptive Maxwell model or mechano-sorptive Kelvin model were adopted, where the effect of time was assumed to be replaced by the cumulative amount of moisture content change. The time-temperature-moisture equivalence principle was also applied in an attempt to cover a wide moisture content and temperature range so that the long-term performance of a material can be predicted from short-term data. This principle was based on time-temperature superposition (TTS) principle proposed by Williams et al. (1955) and has been expended and applied to the viscoelastic properties of wood and wood-based products (Hanhijärvi, 1999, 2000, 2001; Tajvidi et al., 2005). The time-temperature-moisture equivalence principle suggested that the effect of temperature/moisture-content on the viscoelastic creep amount can be modeled by defining a ‘material time’, which is the real time multiplied with a temperature/moisture-content dependent coefficient. Hanhijärvi (1999-2000) proposed a model which took into account all the needed strain terms: hygroexpansion (shrinkage, swelling), hygrothermal, elastic, viscoelastic and mechano-sorptive strains. Elastic and viscoelastic strains were modeled jointly as generalized Kelvin material, using the time-temperature-moisture-content superposition principle. Mechano-sorptive creep was modeled with a series of modified ‘mechano-sorptive Kelvin elements’, in which a part of the strain was irrecoverable unless the direction of the acting stress changed. Based on this model, Hanhijärvi et al. (2001) introduced a modified model, the novel features of which were the two-dimensional extension, the description of partially irrecoverable mechanosorptive strain 25 by introducing a plastic element, and a hyperelastic formulation consistent with the laws of thermodynamics. Thus, it defined the mechanical behavior of drying wood as a viscoelasticmechanosorptive-plastic one. The model was applicable over a wide range of temperature as well as moisture content (20°C-120°C; nearly 0% moisture content till fibre saturation), which was achieved through applying the time-temperature-moisture equivalence principle to the evolution laws for viscoelastic strain. Fridley et al. (1992b) developed a general stochastic creep model. It was a modified Burger model: the four basic creep model parameters were found to be lognormally distributed, and no significant correlations between elastic and viscous parameters were observed. Interpolating functions were used to estimate the model parameters as functions of their hygrothermal state. A fifth element was added in series to account for mechano-sorptive creep effect. The fifth element was defined as a function of the current time rate of change in moisture content. The mathematical modeling of creep in materials can be approached from two main directions: pure fitting of creep curves to experimental results or derivation of governing equations for creep by making assumptions about the material structure and the mechanisms causing creep. The latter seemed more useful in the material science sense, but the problem with material structure based interpretations lied in the difficulty of giving the ideas an exact mathematical formulation, which would fit well all the different aspects of the experiment results. It follows that the best result is achieved by a synthesis of these two approaches (Hanhijarvi, 1995b). Hanhijarvi (1995b) proposed a new general rheological model for the calculation of the creep of wood. The flow equation derived in the theory of molecular deformation kinetics was adjusted to 26 account for creep flow, moisture content change induced swelling/shrinkage and their combined effect by making an assumption that both of these processes activate the same bond breaking and reforming process. The rheological model was built by making the dashpots in a generalized Maxwell material model to obey the adjusted flow equation and by placing an additional swelling/shrinkage component to each parallel Maxwell element. It seemed that an explanation for the mechano-sorptive effect can be found at the coupling of the creep deformation process and moisture swelling/shrinkage and the non-linearity of the phenomena. This formulation was restricted to one-dimensional stress states which were coaxial to the preferred material directions. In a later paper (Helnwein et al., 2000) this model has been reformulated for 3dimensional states of stress. The published findings by Hanhajarvi (1995b), Hanhajarvi and Hunt (1998) and Hunt and Gril (1996) indicate that mechano-sorptive deformation may be accounted for by a coupling between creep and hygroexpension. The more recent findings by Hunt (1999) stated that time-dependent creep and mechano-sorptive creep were different means of reaching the same creep result. In his opinion, if creep was to be considered as a ‘unified’ process (in other words, the strains resulting from time and moisture changes cannot be simply added), then the important variables in the creep process were strain, which was the dependent variable describing the present state of the material, and strain rate, which was the independent variable describing the potential for further creep. This finding led to a new way of characterizing wood creep, namely plotting data in the form of strain rate against strain. Those work presented the views that viscoelastic creep and mechano-sorptive behavior were but two different manifestations of one basic relationship. 27 2.5 Conclusions In order to evaluate the creep-rupture behavior of strand-based wood composites under sustained loading and to provide a better understanding of the in-service stiffness and the possibility of rupture of such wood composites a literature review in both fields of duration-of-load and creep was performed. The aim is to gain a greater understanding of the state of the art needed to establish a unified approach to address both issues for the industry. Based on the literature review it seems that although the phenomenon of creep to rupture is a continuous process, early work in this area used to treat the effects of duration-of-load and creep separately in terms of modeling. With regard to DOL modeling, the application of damageaccumulation model is a well-established tool for the prediction of duration of load response in lumber. The distinctive advantage of a damage accumulation model is that it permits the prediction of damage produced by an arbitrary random load sequence and its convenient usage in reliability-based design formats. In Canadian Standard CSA O86, the DOL factor was based on the research work of Foschi in 1989 (Foschi, 1989) and the damage accumulation model proposed by Foschi and Yao (Foschi and Yao, 1986). While in terms of creep modeling, the rheological creep models shows good balance between model efficiency and accuracy. To reflect the interdependent relationship between duration-of-load and creep effects in the modeling, the development of a new creep-rupture model which links damage accumulation to creep deformation will be discussed in the next chapter. 28 Chapter 3 Modeling of Creep-rupture Behavior 3.1 Summary In order to investigate the creep and duration-of-load (DOL) effects in thick strand-based wood composites, a damage accumulation model was proposed to study the creep-rupture behaviour. This model stipulates that the rate of damage growth is given in terms of the current strain rate and the previously accumulated damage, and a 5-parameter rheological model is applied to describe the viscoelastic constitutive relationship to represent the time-dependent strain, while the damage accumulation law acts as the failure criterion. This creep-rupture model allows one to gain deeper insight into the interdependent duration-of-load and creep effects of strand-based wood composites. The distinctive advantage of this creep-rupture model is that it permits the prediction of damage produced by an arbitrary random load sequence and its convenient usage in reliability-based design formats. In addition, the damage accumulation model developed in this work takes into account the deformation history; herein the model provides a description of the time-dependent deflection to present a complete picture from creep deflection to a final rupture event. In this manner the creep-rupture model incorporates the short term strength of the material, the load history and predicts the deflection history as well as the time-to-failure. As it is a probabilistic model, it allows its incorporation into a time-reliability study of wood composites’ applications. 29 3.2 Introduction Although the phenomena of creep and creep-rupture are inter-related time processes, most of the available literature treats them as independent. Only Nielsen’s (Nielsen, 1986) Damaged Viscoelastic Material model attempts to consider the dependence. Creep modeling has concentrated in formulating constitutive relationships that would permit the calculation of strain history when that for the applied load is given. For DOL modeling, most models put an emphasis on the “final point”, namely how long the structural component can sustain under a given load. Hence the main priority was to predict the time to failure rather than the specific process of deformation. In order to reflect the inter-related relationship between duration-of-load and creep in terms of modeling, a modified damage accumulation model based on Foschi and Yao’s model (Foschi and Yao, 1986; Foschi, 1989; Yao, 1987) is proposed here, taking into consideration the deformation history. This model stipulates that the rate of damage growth is given in terms of the current strain rate and the previously accumulated damage, and a 5-parameter rheological model is applied to describe the viscoelastic constitutive relationship to represent the time-dependent strain, while the damage accumulation law acts as the failure criterion. The creep-rupture model allows one to gain deeper insight into the interdependent duration-of-load and creep effects in strand-based wood composites. 3.3 Constitutive relationship The behavior of linear viscoelastic materials can be represented by models built from discrete elastic and viscous elements. Pierce and Dinwoodie in their series of papers (1977, 1979) used 3and 4- element models to fit the creep data of chipboard. The researchers concluded that 430 element curve always provided the better overall fit to the data, and that the 4-element model always overestimated the end deflection whereas the 3-element almost always erred on the low side. It was explained that the 4-element model predicted too high a deformation since it assumed the viscous behavior was linear with time. Then later they proposed a modified 4element model in which the viscous component was non-linear with respect to time (Pierce et al. 1985). The resulting 5-parameter model was shown to be superior to the 4-element model for long-term predictions of creep deflection particularly at the lower stress levels (Dinwoodie et al. 1990). The choice of creep models should be based on the balance between model complexity and accuracy. The advantages and limitations of these three models are presented in the following sections. 3.3.1 3-element model The 3-element model is comprised of a spring added to a Kelvin body in series, as shown in Figure 3.1. k0 η1 k1 σ (t ) = σ a H (t ) Figure 3.1 3-element viscoelastic model Where H (t ) is Heaviside unit step function, 1, t ≥ 0 H (t ) = 0, t < 0 31 And k 0 , k1 and η1 are spring constants and dashpot damping coefficient respectively. Considering equilibrium, σ (t ) = σ a H (t ) = k 0 ε k = k1ε 1 + η1ε1 (3.1) 0 From continuity conditions: ε = ε k + ε1 (3.2) 0 Taking Laplace transforms of Eq. (3.1), we get, σ= σa p = k 0 ε k0 = k1 ε 1 + pη1 ε 1 (3.3) Therefore we have, εk = 0 ε1 = σa pk 0 (3.4.1) σa (3.4.2) p(k1 + pη1 ) Taking Laplace transforms of Eq. (3.2), ε = ε k + ε1 (3.5) 0 Substituting in Eqns (3.4.1) and (3.4.2), 1 1 + ( ) pk p k p η + 1 1 0 ε = σ a (3.6) Taking inverse Laplace transforms of Eq. (3.6) yielded, ( ) 1 1 − e −t τ 1 ε (t ) = σ a + H (t ) = σ a J (t ) k1 k0 (3.7) Hence the creep compliance J (t ) is: ( ) 1 1 − e −t τ 1 J (t ) = + H (t ) k1 k0 (3.8) 32 The time-dependent creep compliance and strain-time curve of 3-element model are plotted in Figure 3.2 and 3.3 respectively. We define the parameters, β1 = σs k0 β2 = , σs k1 , β3 = 1 τ1 = k1 η1 After some algebraic operations, Eq. (3.7) can be written as: ε (t ) = σa β1 + β 2 (1 − e −tβ σs [ Where τ 1 = 3 )] (3.9) η1 is the retardation time; k 0 represents the instant modulus of elasticity, and the k1 (1 − e ) , represents the delayed elastic or recoverable creep −t τ 1 second term in the bracket, k1 component and is associated with the combined effects of the spring constant k1 and the dashpot damping coefficient η1 ; σ a is the applied external load, and σ s is the average short-term strength of the member, measured in a ramp test of short duration, and thus σ a σ s represents the stress ratio. The strain rate can then be calculated by differentiation of Eq. (3.9), ε = σa ⋅ β 2 β 3 ⋅ e − tβ σs (3.10) 3 And the strain rate for the 3-element model is plotted in Figure 3.4. 33 J (t ) 1 k1 1 k0 t Figure 3.2 Creep compliance of 3-element model ε 3 (t ) σa ε ve = k1 σa σa k1 (1 − e ) −t τ 1 ε e = σ a k0 k0 t Figure 3.3 Time-dependent strain of 3-element model ε3 σa ⋅ β2β3 σs t Figure 3.4 Strain rate of 3-element model The 3-element model is the simplest of the rheological models, however, it does not include a viscous term which represents irrecoverable creep. Therefore, if the load is removed, the 34 deformation will be completely recovered after a sufficiently long time for all stress levels. However in reality there would be some remaining deformation after the load has been removed for a long period, especially at high stress levels. 3.3.2 4-element model The 4-element model is comprised of a Maxwell body and a Kelvin body in series, as shown in Figure 3.5. k0 η1 k1 η0 σ (t ) = σ a H (t ) Figure 3.5 4-element viscoelastic model Similarly we can get the constitutive relationship determined by the 4-element viscoelastic model. ( ) 1 1 − e −t τ 1 t σ + + = a β 1 + β 2 1 − e − tβ 3 + β 4 t η0 σ s k1 k0 ε (t ) = σ a ε = [ ( ) σa ⋅ (β 2 β 3 ⋅ e −tβ + β 4 ) σs ] (3.11) (3.12) 3 Where, β1 = σs k0 , β2 = σs k1 , β3 = 1 τ1 = k1 η1 , β4 = σs , and σ s is the average short-term strength. η0 35 The strain-time curve of the 4-element model is shown in Figure 3.6. The 4-element model includes a linear viscous term β 4 t , and thus it assumes a constant strain rate after a long time, as shown in Figure 3.7. This asymptote implies that the deformation will keep going with time and is nevertheless impractical during the service life of real buildings. The 4-element model would overestimate the amount of deflection contributed by the viscous term, particularly at the low stress levels. ε 4 (t ) ε ve = σa σa k1 (1 − e ) −t τ 1 ε v = σ a t η0 k1 σa ε e = σ a k0 k0 t Figure 3.6 Time-dependent strain of 4-element model σa (β 2 β 3 + β 4 ) σs ε4 σa β4 σs t Figure 3.7 Strain rate of 4-element model 36 3.3.3 5-parameter model The 5-parameter model is actually a modified 4-element model, with the viscous flow term expressed as a non-linear function of time. ( ) 1 1 − e −t τ 1 t β5 + + k1 η0 k0 ε (t ) = σ a ε = σa ⋅ (β 2 β 3 ⋅ e −tβ + β 4 β 5 t β σs 3 σa β 1 + β 2 1 − e − tβ 3 + β 4 t β 5 = σs [ 5 −1 ( ) ] (3.13) ) (3.14) Where, β1 = σs k0 , β2 = σs k1 , β3 = 1 τ1 = k1 η1 , β4 = σs , 0 < β5 < 1 η0 The 5-parameter model is more complicated, and it is more difficult to calibrate the model parameters with experimental data because two nonlinear terms are included ( e − tβ3 and t β5 ). However, it suggests a non-linear viscous flow term β 4 t β5 . Hence the deflection contributed by the viscous term is smaller than what is predicted by the 4-element model. Meanwhile the strain rate decreases with time rather than approach a constant value. The strain-time curve and strain rate curve are shown in Figures 3.8 and 3.9 respectively. ε 5 (t ) ε ve = σa k1 σa k1 (1 − e ) −t τ 1 ε v = σ a t β η0 5 σa ε e = σ a k0 k0 t Figure 3.8 Time-dependent strain of 5-parameter model 37 ε5 σa ⋅ β 2 β3 σs t Figure 3.9 Strain rate of 5-parameter model It is noted that when β 4 = 0 , the 5-parameter model becomes the 3-element model; meanwhile when β 5 = 1 and β 4 ≠ 0 , the 5-parameter model becomes the 4-element model. Since either the 3- or the 4- element model can be expressed as a special case of the 5-parameter model, this 5parameter viscoelastic model is used in this work. 3.4 Damage accumulation model In this work a damage accumulation model is proposed to represent the creep-rupture phenomenon and it is calibrated to available experimental data. The damage state variable α conceptually defines the state of damage between the virgin state (𝛼𝛼 = 0) and the state at failure (𝛼𝛼 = 1). Foschi and Yao’s model (Foschi and Yao, 1986) prescribed the rate of damage growth to be a function of the applied stress, as well as a function of already accumulated damage α . A model for the damage rate should be more realistically based on strain history. During creep, when strain increases with time, damage is accumulated in spite of the constant stress. During relaxation, when the stress decreases under constant strain, no additional damage should be accumulated despite decreasing levels of stress. Thus, the model presented here stipulates that 38 the rate of damage growth is given in terms of the current strain rate and the previously accumulated damage. The general description of the rate of damage growth takes the following form, dα = F (ε, α ) dt (3.15) Where α is the measure of damage, defined such that α = 0 identifies an undamaged initial state and α = 1 corresponds to failure; ε is the strain rate under a given load. Eq. (3.15) can be expanded into a power series of α , dα = F0 (ε ) + F1 (ε ) ⋅ α + F2 (ε ) ⋅ α 2 + dt (3.16) A particular form was proposed including the first order of damage-dependent term and firstorder of strain rate term: dα = ( Aε + C ) + (Bε + D ) ⋅ α dt (3.17) As previously stated, in the case of relaxation, where the strain rate is zero, no additional damage is accumulated and thus the damage growth rate should be zero, i.e., ε = 0 , dα =0 dt Substituting in Eq. (3.17), we have, C + Dα = 0 Since it holds true for every value of α , we can conclude that, C=D=0 Therefore Eq. (3.17) becomes, 39 dα = Aε + Bε ⋅ α dt (3.18) During unloading, the crack would not go on propagating and would close, but the damage is already there and irrecoverable, therefore the damage growth rate is zero in this case. Hence it is reasonable for us to assume that when ε ≤ 0, dα = 0. dt Or we can write Eq.(3.18) in a more general form as, dα = Aε p + Bε q ⋅ α dt (3.19) When p = q = 1 , Eq.(3.19) simplifies to Eq.(3.18) and A, B, p, q are model parameters. These parameters are assumed to be constant for a given structural member but vary between members. ε is the strain rate under a given load, which can be derived from the constitutive relationship with the help of rheological model. We define, F1 (t ) = Aε p (3.20.1) F2 (t ) = Bε q (3.20.2) Substituting Eq. (3.14) in Eqs. (3.20.1) and (3.20.2), we have, σ F1 (t ) = Aε = A a ⋅ β 2 β 3 ⋅ e −tβ 3 + β 4 β 5 t β 5 −1 σ s p σ F2 (t ) = Bε = B a ⋅ β 2 β 3 ⋅ e −tβ 3 + β 4 β 5 t β 5 −1 σ s q p q ( ( ) ) (3.21.1) (3.21.2) Eq. (3.19) can be written as: dα = F1 (t ) + αF2 (t ) dt (3.22) Or 40 dα − αF2 (t ) = F1 (t ) dt ( ) Multiplied by exp − ∫ F2 (t )dt on both sides, ( ) ddtα − exp(− ∫ F (t )dt )⋅ α ⋅ F (t ) = exp(− ∫ F (t )dt )⋅ F (t ) exp − ∫ F2 (t )dt ⋅ ∴ [ 2 )] ( 2 ( 2 1 ) d α ⋅ exp − ∫ F2 (t )dt = exp − ∫ F2 (t )dt ⋅ F1 (t ) dt (3.23) Integrating Eq. (3.23), [ ( ∴ α ⋅ exp − ∫ F2 (t )dt )] = ∫ exp(− ∫ F (t )dt )⋅ F (t )dt T T 0 0 2 (3.24) 1 Then theoretically we can calculate the damage at a certain time, α (T ) . However it should be ( ) noted that usually it is difficult to get a close form evaluation of − ∫ F2 (t )dt , due to the complexity of strain rate. When p = q = 1 , we can get a close form expression of α (T ) , as shown in the following equations. dα = Aε + Bε ⋅ α = ε ( A + Bα ) dt ∴ 1 dα ⋅ = ε ( A + Bα ) dt ∴ 1 d ⋅ [ln ( A + Bα )] = ε B dt ∴ d [ln( A + Bα )] = Bε dt (3.25) ∴ [ln( A + Bα )]0 = ∫ Bεdt = B[ε (T ) − ε (0 )] T T (3.26) 0 41 Therefore the mathematical form is quite neat and elegant. Moreover, it can be seen from Eq. (3.14) that the strain rate itself can be expanded into a power series of time from the perspective of mathematics, so we set p = q = 1 at present and Eq. (3.25) was used in the following discussion. However Eq. (3.19) can be checked of its accuracy and efficiency in future research. 3.5 Application of creep-rupture model to different loading conditions 3.5.1 Constant load Consider the case of long-term constant load, as shown in Figure 3.10. σ (t ) σa t Figure 3.10 Constant load The strain and strain rate under constant load are given by, ( ) 1 1 − e −t τ 1 t β5 σ a + ε (t ) = σ a + β 1 + β 2 1 − e − tβ 3 + β 4 t β 5 = k1 η0 σ s k0 ε = σa ⋅ (β 2 β 3 ⋅ e −tβ + β 4 β 5 t β σs 3 [ 5 −1 ( ) ] ) (3.13) (3.14) Substituting Eq.(3.13) in Eq.(3.26) and noting that α = 0 at t = 0 , we have, ln[A + Bα (T )] − ln A = B[ε c (T ) − ε c (0)] ∴ α (T ) = A [exp(B(ε c (T ) − ε c (0))) − 1] B (3.27) 42 ∴ α (T ) = A σ a β 2 1 − e −Tβ3 + β 4T β5 exp B B σs [ ( ) ] − 1 (3.28) At time to failure T = Tc , α (Tc ) = 1 . Thus, ln ( A + B ) − ln A = B[ε c (Tc ) − ε c (0 )] ∴ 1 B σa β 2 1 − e −Tc β3 + β 4Tcβ5 ln1 + = B A σs [ ( ) ] (3.29) Therefore, we can solve for Tc numerically using Eq. (3.29). 3.5.2 Ramp load Consider the case of ramp loading, as shown in Figure 3.11. σ (t ) σa Ka 1 Tr t Figure 3.11 Ramp load Based on Boltzman’s superposition principle we can derive the time-dependent strain under ramp load with the creep compliance obtained in the case of constant loading. According to Eq. (3.13), the creep compliance is: J (t ) = ε (t ) 1 1 tβ = + (1 − e −t τ ) + σ a k 0 k1 η0 5 (3.30) 1 43 β t 1 ( dσ t − s) 5 1 − (t − s ) τ 1 ds = ∫ + 1− e ∴ ε (t ) = ∫ J (t − s ) + ⋅ K a ds 0 0 k ds η0 0 k1 1 t β 5 +1 1 τ = K a + t − 1 1 − e −t τ 1 + η 0 (β 5 + 1) k 0 k1 k1 ( t ( ) ) K a σ s σ s τ 1 σ s σ s t β 5 +1 −t τ 1 t e − + 1− = + k1 η 0 (β 5 + 1) σ s k 0 k1 ( ) Ka β2 β 4 t β 5 +1 − tβ 3 1− e = + (β 1 + β 2 )t − (β 5 + 1) β3 σs ( ) (3.31) where β1 = σs k0 β2 = , σs k1 , β3 = 1 τ1 = k1 η1 , β4 = σs , 0 < β5 < 1. η0 Then we can get the first derivative of strain with respect to time as: 1 1 t β5 K a 1 − e −t τ 1 + ⋅ β 1 + β 2 1 − e −tβ 3 + β 4 t β 5 + = η0 σ s k 0 k1 ε = K a ⋅ J (t ) = K a ( [ ) ( ) ] The plot of strain rate is shown in Figure 3.12. ε Ka σs β1 t Figure 3.12 Plot of strain rate versus time under ramp load Substituting Eq.(3.31) into Eq.(3.26) and noting that α = 0 at t = 0 , we have, ln[A + Bα (T )] − ln A = B[ε r (T ) − ε r (0 )] 44 (3.32) ∴α (T ) = A [exp(B(ε r (T ) − ε r (0))) − 1] B ∴α (T ) = β2 β 4 T β 5 +1 A K a − Tβ 3 exp B ( ) β β T e 1 + − − + 1 − 1 2 ( B σ s β3 β 5 + 1) (3.33) ( ) (3.34) At time to failure T = Tr , α (Tr ) = 1 . Thus, ln ( A + B ) − ln A = B[ε r (Tr ) − ε r (0 )] β2 β 4 Trβ 5 +1 1 B Ka −Tr β 3 1− e ∴ ln1 + = + (β 1 + β 2 )Tr − (β 5 + 1) β3 B A σs ( ) (3.35) Therefore, we can solve for Tr numerically using Eq. (3.35). Note that the short term strength is determined using the ramp load test, i.e., when T = Ts , α = 1 . Thus, 1= β 4Tsβ5 +1 A K a −Ts β 3 exp B ( ) + − − + 1 T e β β β β 1 − 1 2 2 3 s ( B σ s β 5 + 1) ( ) (3.36) Therefore, A is not independent of parameters B and Ts , and it can be determined using Eq. (3.36). 3.5.3 Experimental load history In the DOL test load history is the combination of ramp load and constant load, as shown in Figure 3.13. 45 σ (t ) σs Ka σa 1 t a Ts t Figure 3.13 Experimental load history Accordingly, the time-dependent strain is expressed in two steps. When 0 ≤ t ≤ t a , Ka β 4 t β5 +1 − tβ 3 ( ) 1 t e / β β β β + − − + 1 2 2 3 (β 5 + 1) σs β 4 t aβ5 +1 Ka −t a β 3 / β3 + ∴ ε (t = t a ) = (β1 + β 2 )t a − β 2 1 − e (β 5 + 1) σs ( ε (t ) = ) ( (3.37.1) ) σa β 4 t aβ β2 −t β (1 − e ) + (β + 1) = (β 1 + β 2 ) − β 3t a σs 5 5 a 3 (3.37.2) When t > t a , the strain should be the superposition of the strain at t = t a and the strain caused by the constant load σ (t ) = σ a H (t − t a ) , which is the strain we get using Eq. (3.13) minus the effect of the instantaneous increase of the stress from zero to σ a at t = t a , i.e., ε (t ) = ε t + a [ σa β1 + β 2 (1 − e −(t −t σs a )β 3 ) + β (t − t ) ]− σ β5 4 a σs a ⋅ β1 [ σa β 4 t aβ σ a β2 −t β ( ) = + 1− e β 2 (1 − e −(t −t (β 1 + β 2 ) − + (β 5 + 1) σ s ta β3 σs 5 a 3 σa β 4 t aβ β2 −t β ( ) ( ) = + + β 2 (1 − e −(t −t 1− e β1 + β 2 − (β 5 + 1) ta β3 σs 5 a 3 a )β 3 a )β 3 ) + β (t − t ) ] β5 4 a ) + β (t − t ) β5 4 a (3.37.3) Considering, 46 β2 β (1 − e −t β ) = β 2 lim(β ⋅ e −t β ) = β ( 1 − e −t β ) = 2 lim 3 2 →0 β t β 3 t →0 ta β 3 t →0 3 a a 3 lim ta a 3 a 3 a a Therefore, σa β 4 t aβ β2 β −t β ( ) ( ) β β e + − 1 − + + β 2 (1 − e −(t −t )β ) + β 4 (t − t a ) 1 2 t →0 t →0 σ (β 5 + 1) ta β3 s σ σ = a (β1 + β 2 ) − β 2 + β 2 (1 − e −tβ ) + β 4 t β = a β1 + β 2 (1 − e −tβ ) + β 4 t β = ε c σs σs lim ε (t ) = lim a 5 a 3 a 5 3 a [ 3 5 ] [ 3 5 ] (3.38) Therefore when t a → 0 , i.e. K a → ∞ , ε → ε c . It is reasonable for the strain under experimental load to approach that under constant load when the loading rate is quite big. Similarly, the time-to-failure should be calculated in two steps. 1) For the ramp load from t = 0 to t = t a = σa Ka , We can get damage at T = t a according to Eq. (3.34), α c = α (t a ) = β 4 t aβ5 +1 A K a −t a β 3 exp B ( ) β β β β 1 t e + − − + 1 − 1 2 a 2 3 ( β 5 + 1) B σ s ( ) If t a = Ts , i.e. σ a = σ s , then α c = 1 and T f = Ts = σs Ka (3.39) . 2) For the constant load from t = t a to time-to-failure T f , Integrating Eq. (3.25) during this time interval, also noting that at time-to-failure T = T f , α = 1 , we have, [ ] ln ( A + B ) − ln ( A + Bα c ) = B ε (T f ) − ε (t a ) (3.40) Substituting Eq. (3.37.3) in Eq. (3.40) and after some algebraic operations, we have, 47 1 A+ B ln B A + Bα c [ ( ) σa − (T −t )β = β 2 1 − e f a 3 + β 4 (T f − t a )β5 σs ] (3.41) Therefore, we can solve for T f numerically using Eq. (3.41). Note that A is not independent of parameters B and Ts , and it can be determined by other parameters using Eq. (3.36). β 4Tsβ5 +1 A K a −Ts β 3 1 = exp B β3 + (β1 + β 2 )Ts − β 2 1 − e − 1 ( B σ s β 5 + 1) ( ) (3.36) 3.5.4 Fatigue load Compared with the creep-rupture test, the fatigue test can accelerate the process and provide supplementary data for model verification. A loading scheme with triangular load pulses was used which represented the gradual increase in load to some maximum value at a constant rate followed by a gradual decrease in load at the same rate, as shown in Figure 3.14. This load pulse may represent a short-duration snow wherein snow may accumulate then melt off gradually. σ max σ (t ) tc tc tc t Figure 3.14 Triangular cyclic loading scheme The stress history is given in Eq. (3.42): 48 tc K a (t − t sI ), t sI ≤ t ≤ t sI + 2 σ (t ) = − K (t − t ), t + t c ≤ t ≤ t a eI sI eI 2 (3.42) Where, t sI and t eI are the starting and ending time of the I-th cycle respectively, and K a is the loading rate, and 𝑡𝑡𝑐𝑐 is the duration of each cycle. The creep compliance is, J (t ) = ε (t ) 1 1 tβ = + (1 − e −t τ ) + σ a k 0 k1 η0 5 (3.30) 1 When t sI ≤ t ≤ t sI + tc 2 ε (t ) = ε (t sI ) + ∫ J (t − s ) t tsI dσ ds ds t 1 (t − s )β5 ⋅ K ds 1 = ε (t sI ) + ∫ + 1 − e −(t −s ) τ1 + tsI k η0 a 0 k1 1 1 (t − tsI )β5 +1 τ = ε (t sI ) + K a + (t − t sI ) − 1 1 − e −(t −tsI ) τ1 + k1 η0 (β 5 + 1) k0 k1 ( ) ( ) K a σ s σ s τ 1σ s σ s (t − t sI )β5 +1 −(t −tsI ) τ1 (t − tsI ) − = ε (t sI ) + + + 1− e k1 η0 (β 5 + 1) σ s k0 k1 ( ) Ka β2 β 4 (t − t sI )β5 +1 −(t −tsI )β3 = ε (t sI ) + + 1− e (β1 + β 2 )(t − t sI ) − (β5 + 1) β3 σs ( ) (3.43) Therefore, ε t sI + tc K a t c σ s σ s τ 1σ s ( ) t ε = + + − sI 2 σ s 2 k 0 k1 k1 σ = ε (t sI ) + max σs 2β (β1 + β 2 ) − 2 β 3t c t − c 2 1 − e 2 t c τ1 σ (t 2 )β5 + s c η 0 (β 5 + 1) t β5 − c β 1 − e 2 3 + β 4 (tc 2 ) (β 5 + 1) (3.44) 49 Eq. (3.44) showed that the increase in strain during the loading part within each cycle remained constant for this type of cyclic loading. When t sI + tc ≤ t ≤ teI 2 ε (t ) = ε (t sI ) + ∫ tsI + tsI = ε (t sI ) + ∫ tsI + tsI tc 2 tc 2 J (t − s ) t dσ dσ ds + ∫ tc J (t − s ) ds + t sI ds ds 2 β β t 1 1 1 1 ( ( t − s) 5 t − s) 5 −( t − s ) τ 1 −( t − s ) τ 1 + + + 1− e ⋅ K ds − ∫tsI + tc + 1 − e ⋅ K ds k1 η0 a η0 a 2 k0 k0 k1 ( ) ( 1 1 t τ = ε (t sI ) + K a + c − 1 e −(t −s ) τ1 k0 k1 2 k1 [ ] tsI + tsI 1 1 t τ − K a + t − t sI − c − 1 e −(t −s ) τ1 2 k1 k0 k1 [ tc 2 − ] t t tsI + c 2 1 η0 (β 5 − [(t − s ) ] + 1) t c β5 +1 tsI + 2 tsI 1 η 0 (β 5 [(t − s ) ] + 1) β5 +1 t ) tsI + tc 2 (3.45) Therefore, 2 t β +1 − c τ 1 β 5 +1 ( t c 2) 5 τ 1 2 1− e ε (t eI ) = ε (t sI ) + K a 2 −2 + η 0 (β 5 + 1) k1 t 2 Ka c tc β 5 2 σ sτ 1 1 − e − 2 τ1 + σ s (t c 2 ) 2 β 5 +1 −2 = ε (t sI ) + η 0 (β 5 + 1) σ s t c k 1 2 ( ) ( − β σ 2β = ε (t sI ) + max 2 1 − e 2 σ s β 3t c tc 3 ) 2 β5 + β 4 (t c 2 ) 2 β 5 +1 −2 (β 5 + 1) ( ) (3.46) Thus the residual strain in the I-th cycle is: ∆𝜀𝜀𝐼𝐼 = 𝜀𝜀(𝑡𝑡𝑒𝑒𝑒𝑒 ) − 𝜀𝜀(𝑡𝑡𝑠𝑠𝑠𝑠 ) = 𝜎𝜎𝑚𝑚𝑚𝑚𝑚𝑚 ��� 𝜎𝜎𝑠𝑠 2𝛽𝛽 2 �𝛽𝛽 3 𝑡𝑡 𝑐𝑐 �1 − 𝑒𝑒 2 𝑡𝑡 −� 𝑐𝑐 �𝛽𝛽 3 2 � + 50 𝑡𝑡 𝛽𝛽 5 𝛽𝛽 4 � 𝑐𝑐 � 2 𝛽𝛽 5 +1 �2𝛽𝛽5 +1 − 2�� (3.47) Eq. (3.47) showed that the residual strain in each cycle remained constant for this type of cyclic loading. For the first cycle, because no additional damage was accumulated when unloading, Eq. (3.25) is integrated within 0 ≤ t ≤ tc , 2 tc tc t ∴ [ln ( A + Bα )]02 = ∫ 2 Bεdt = B ε c − ε (0) 0 2 t ∴ ln ( A + Bα1 ) − ln ( A) = B ε c 2 (3.48.1) From the second cycle on it was assumed that the damage would begin to accumulate only when the strain exceeded the maximum strain the previous cycle had reached, and would only accumulate within loading period. The principle is demonstrated in Figure 3.15. 𝜀𝜀 ∆𝜀𝜀1 ∆𝜀𝜀2 ∆𝜀𝜀1 ∆𝜀𝜀2 𝑡𝑡 Figure 3.15 Strain-time relationship under triangular cyclic load 51 Therefore, ln(𝐴𝐴 + 𝐵𝐵 ∝2 ) − ln(𝐴𝐴 + 𝐵𝐵 ∝1 ) = 𝐵𝐵∆𝜀𝜀1 (3.48.2) ln(𝐴𝐴 + 𝐵𝐵 ∝3 ) − ln(𝐴𝐴 + 𝐵𝐵 ∝2 ) = 𝐵𝐵∆𝜀𝜀2 (3.48.3) ln �𝐴𝐴 + 𝐵𝐵 ∝𝑁𝑁𝑓𝑓 � − ln �𝐴𝐴 + 𝐵𝐵 ∝𝑁𝑁𝑓𝑓 −1 � = 𝐵𝐵∆𝜀𝜀𝑁𝑁𝑓𝑓 −1 (3.48.4) … Adding up Eqs. (3.48) and noting that at cycle to failure 𝛼𝛼𝑁𝑁𝑓𝑓 = 1, we have 𝑡𝑡 ln(𝐴𝐴 + 𝐵𝐵) − ln(𝐴𝐴) = 𝐵𝐵 �𝜀𝜀 � 2𝑐𝑐 � + ∆𝜀𝜀 ∙ �𝑁𝑁𝑓𝑓 − 1�� (3.49) From Eq. (3.44) we have 𝑡𝑡 𝑐𝑐 𝜀𝜀 � 2 � = 𝜎𝜎𝑚𝑚𝑚𝑚𝑚𝑚 ��� 𝜎𝜎𝑠𝑠 2𝛽𝛽 2 �(𝛽𝛽1 + 𝛽𝛽2 ) − 𝛽𝛽 3 𝑡𝑡 𝑐𝑐 �1 − 𝑒𝑒 And ∆𝜀𝜀 is expressed in Eq. (3.47). 𝑡𝑡 −� 𝑐𝑐 �𝛽𝛽3 2 �+ 𝑡𝑡 𝛽𝛽 5 𝛽𝛽 4 � 𝑐𝑐 � 2 (𝛽𝛽 5 +1) � (3.50) We can solve for number of cycles to failure as following: 𝑁𝑁𝑓𝑓 = ln (𝐴𝐴+𝐵𝐵)−ln (𝐴𝐴) 𝐵𝐵∙∆𝜀𝜀 − 𝑡𝑡 𝜀𝜀� 𝑐𝑐 � 2 ∆𝜀𝜀 +1 (3.51) 3.6 Conclusions and discussions A large amount of research on creep and load duration behavior of solid sawn lumber has been carried out by many researchers. However, such time-dependent performance of thick strandbased wood composites under long-term loading is not well understood or documented, especially with products manufactured from MPB killed lumber. In order to investigate the creep and duration-of-load effects of such products, a damage accumulation model was proposed, which links the accumulated damage to creep strains. This model stipulates that the rate of damage growth is given in terms of the current strain rate and the previously accumulated 52 damage, and a 5-parameter rheological model is applied to describe the viscoelastic constitutive relationship to represent the time-dependent strain, while the damage accumulation law acts as the failure criterion. The creep-rupture model was calibrated to test data in a later chapter to help one gain insight into the inter-related duration-of-load and creep effects in strand-based wood composites. The damage accumulation model proposed here includes the first order of damage-dependent term and first-order of strain rate term; however a more generalized form proposed in Eq. (3.19) can be investigated in future research. Also the model can be refined further by taking into account the influence of the change in moisture content. These issues will be addressed in ongoing work. The distinctive advantage of this creep-rupture model is that it permits the prediction of damage produced by an arbitrary random load sequence and its convenient usage in reliability-based design formats. Also the damage accumulation model developed in this work takes into account the deformation history; herein the model provides a description of the time-dependent deflection to present a complete picture from creep deflection to a final rupture event. In this manner the creep-rupture model incorporates the short term strength of the material, the load history and predicts the deflection history as well as the time-to-failure. As it is a probabilistic model, it allows its incorporation into a time-reliability study of wood composites’ applications. Meanwhile an experimental database on the time-dependent mechanical properties of an engineered wood product was generated for the purpose of model calibration and verification and for future research. 53 Chapter 4 Testing of Duration-of-load and Creep Behavior of MPB Strandbased Wood Composite 4.1 Summary The results from a duration-of- load and creep testing program on MPB strand-based wood composite beams are presented. Thick strand-based MPB wood composite beams were made in the Timber Engineering and Applied Mechanics Laboratory at UBC. Specimens were then divided into matched groups using the MOE matching technique. Three matched groups were tested with a ramp load at three different loading rates. Among them, one group was tested using a simple ramp loading that caused failure at around 1 minute to establish the basic short-term bending strength distribution for the population; the remaining two groups were tested at different loading rates to investigate the influence of rate of loading on strength property. An additional two matched groups were tested under long term constant load at two stress levels using third-point edge-bending test. The constant loading of experimental beams lasted for one year. The long term deflection was monitored and recorded at a pre-set frequency. Time-tofailure data was also obtained for all the broken specimens. Finally the low-cycle fatigue behavior was investigated. Two matched groups were tested with a triangular cyclic load history and six stress levels were chosen. The relationship between the number of cycles to failure and stress ratio was studied. The time-dependent strain curves were obtained to elucidate the lowcycle fatigue performance of the composite. 54 4.2 Introduction In order to absorb the large volume of mountain pine beetle (MPB) killed timber MPB strandbased wood composite product was developed. Based on production technology of thick OSB, prototype samples were processed in the Timber Engineering and Applied Mechanics Laboratory at the University of British Columbia. Creep and duration-of-load (DOL) are important mechanical properties of structural wood or wood composite members under long-term loading. These characteristics are of great interest in timber engineering applications as well as to wood composite manufacturers concerned with the introduction of new building products and the implementation of new codes for engineering design in wood. Therefore, as part of the development of thick MPB strand-based wood composite creep-rupture behavior was investigated in a series of experiments. The experimental data were used to calibrate and verify the creep-rupture model and to develop a database on the structural properties of the composite. 4.3 Materials and methods Thick MPB strand-based wood composite boards were manufactured in the laboratory at UBC based on the production technology of thick OSB. MPB wood strands used in this work were cut from MPB lodgepole pine logs from the Prince George area and obtained courtesy of Ainsworth Lumber Company. A typical MPB log is shown in Figure 4.1. MPB strands were cut to the following target size: 175 mm (7”) length, 25mm (1”) width and 1mm (0.04”) thickness. The resin used for panel production was PF resin (Phenol Formaldehyde). 55 Figure 4.1 A typical MPB log used in this work After being mixed with resin in the blender, the strands were arranged along the longitudinal direction of the final product layer by layer by hand to form a loose mat on a steel press sheet. Figure 4.2 shows a typical hand formed mat. The mat was then pressed in a 762mm by 762mm Pathex press equipped in a PRESSMAN control system to form a thick strand-based board. The pressing cycle was as follows: pre-heat the press platens to 50°C then place mat into the press, and press the mat for 45 minutes at 150°C and then removed from the press (Lam et al., 2008). Figure 4.2 A typical hand formed mat 56 Finally the finished thick strand-based MPB boards (760mm x 760 mm x 38mm) were cut into beam specimens. All beams were 41mm in width, 38mm in depth and 660mm in span, as shown in Figure 4.3. Figure 4.3 Thick MPB strand-based wood composite specimens Specimens were conditioned in the constant climate chamber with the environment of 20°C in temperature and 65% in relative humidity (RH). The weights of the specimens were monitored and recorded for around 10 consecutive days until a constant weight was reached. At this point it was considered that the moisture content equilibrium had been achieved and the specimens were ready for the third-point edge bending MOE test. 4.3.1 The MOE-matching technique Matching is a technique that attempts to subdivide the initial sample population into two or more separate groups that possess nearly identical distribution for bending properties (ASTM 57 Standards D6815-02a). The Modulus of Elasticity (MOE) was determined using a third-point edge-bending test with a span of 660mm. These moduli were ranked and used to assign the specimens to matched groups such that the distributions of E-values within each group were as similar as possible when compared between groups. Since the stiffness and strength are known to be positively correlated, this E-matching of groups was assumed to provide an approximation to a matching by strength (Foschi and Barrett, 1982). This was required to ensure that the different groups tested under long-term loading had approximately the same distribution of short-term strength. The conditioned specimens were tested in bending on edge, under third-point loading, as shown in Figure 4.4. Figure 4.4 Test apparatus of MOE test MOE tests on all the specimens were conducted, and then these specimens were divided into seven matched groups with thirty specimens in each group using the E-matching technique. Firstly, the specimens were ranked according to edgewise MOE in ascending order. Specimens with the seven lowest edgewise MOE values (denoted as L1-1, L1-2 … L1-7) were selected and 58 randomly assigned to one of the seven matched groups. Likewise the specimens with the next seven lowest edgewise MOE values (denoted as L2-1, L2-2 … L2-7) were selected and assigned. This process was repeated until all the specimens were assigned to the seven matched groups with 30 specimens in each group. Among the seven matched groups, three groups were tested with a ramp load, two groups were tested under long-term constant load, and the last two groups were tested with a cyclic load history. 4.3.2 Ramp load test 4.3.2.1 Control group One of the matched groups was tested using a simple ramp loading that caused failure at around 1 minute to establish the basic short-term strength distribution for the population. This group was called the control group. This test was carried out using the third-point edge-bending test under load control, as shown in Figure 4.5. The loading rate was set at 51.91N/sec (0.58MPa/sec), which resulted in a mean time-to-failure of 1.2 minutes. Figure 4.5 Test apparatus of short-term bending test 59 4.3.2.2 Different rates of loading The rate of loading is of relevance for structural design because the real life loads are either significantly faster or slower than the traditional testing standards. It is important to understand whether the loading rate influences the strength properties in a significant manner. Preliminary work with the effect of the rate of loading had been performed by Dr. R. A. Spencer of UBC who conducted a comprehensive bending experiment on 8 groups of 140 Douglas Fir specimens (Spencer, 1979). Each group was subjected to a different rate of loading and the average time to failure covered the range from 0.06 seconds to 3 days. Dr. Spencer’s work indicated that the effect of rate of loading was strength dependent but small at the level of the characteristic strengths (Spencer, 1979). Thompson et al. (1996) conducted four-point bending test on chipboard samples at three loading rates. The rate of loading was increased by a factor of 10 from low to medium, and by a factor of 125 from low to high. It was found that the mean bending strength for the samples tested at the high rate was greater than for those at the low rate, but the medium rate produced greater strength than both. In this work two matched groups of specimens were tested at two loading rates different from the control group; one was set at one tenth of the standard loading rate (0.058 MPa/sec), and the other at 1.8 times faster (1.03MPa/sec). The tests were carried out using the third-point edgebending method under load control. The test setup was the same as that used in control group test, as shown in Figure 4.5. 60 4.3.3 Creep-rupture test Two stress levels were chosen for the long-term constant load tests: 27 MPa and 33 MPa, corresponding to the 5-th percentile and 20-th percentile of the short-term strength distribution respectively. One group of specimens was assigned to the 27 MPa level and denoted as group GL, and another group of specimens was allocated to the 33 MPa level and denoted as group GH. Two steel frames were designed and manufactured to test 60 specimens simultaneously, as shown in Figure 4.6. The specimens were tested in bending on edge under third-point loading with a span of 660 mm. The constant load was applied by means of a lever system which amplified the weight of a steel cylinder by 7 and 10 times for group GL and group GH respectively, as shown in Figure 4.7. The specimens were loaded using a special hydraulic apparatus that lowered the steel cylinder weight at a controlled rate which was approximately the same as that used for the control group ramp loading test. The constant load test of those specimens which survived the initial loading was continued until they either failed or a pre-determined time period has elapsed. Failure was defined as specimen broke and the steel weight dropped to the floor. The experiment was conducted for a one year period. A data acquisition system was designed and used to monitor and record long-term deflection data at a pre-set frequency, as shown in Figure 4.8. The mid-span deflection of every specimen was measured by a linear variable differential transducer (LVDT), as shown in Figure 4.9. The transducer was connected to the data acquisition system at the other end. A temporary power 61 supply system was used to make certain that the data acquisition system continued working even in the case of short-term power shut down. The environment at the testing site was not strictly controlled. The temperature varied from 20 to 25ºC degree in the winter, although in the summer it occasionally reached 30ºC degree. The relative humidity varied from 25% to 50%. It was believed that the environmental variations were not drastic and reflected the mild and constant climate in British Columbia and the Pacific Northwest near the ocean. Figure 4.6 Setup of creep-rupture test 62 Figure 4.7 Loading system of creep-rupture test Figure 4.8 Data acquisition system 63 Figure 4.9 Deflection measuring device for creep-rupture test 4.3.4 Cyclic bending test The damage accumulation model should be tested with load histories not used for its calibration. Considering the creep test is very expensive and time consuming, low-cycle fatigue test is a preferable choice to accelerate the test and to provide supplementary data for model verification. The researchers at the University of Bath, UK have performed research work on fatigue and creep behavior of chipboard. Bonfield et al. (1994) conducted four-point bending test on structural grade chipboard which was subjected to fatigue and creep loads. The ratio of the minimum to the maximum fatigue stress (the R ratio) was set at R=0.01. It was observed that creep samples never failed before fatigue samples at the same peak stress level. However until close to the point of failure, creep strains were nearly always greater than fatigue strains on 64 elapsed time. At BRE (Building Research Establishment), UK, the slow cyclic fatigue of chipboard (on/off loading) was evaluated under various environmental conditions (Dinwoodie et al. 1995). The effect of fatigue loading frequency on the performance of chipboard was investigated later by Thompson et al. (1996). It was concluded that increasing the frequency increased the number of cycles to failure as did reducing the stress level. Also it was observed that the cyclic component of the fatigue load seemed to damage the sample whilst the static component produced the deflection. Pritchard et al. (2001) and Thompson et al. (2002) investigated the strength variability and fatigue performance of MDF, OSB and Chipboard. In this project two matched groups of specimens were tested with a triangular cyclic load under load control, as shown in Figure 4.10. Each cycle comprised of a loading and an unloading part. In the first half the load was increased to the maximum level at a constant rate and then in the second half the load was decreased to zero at the same rate. This load pulse may represent a short-duration snow wherein snow may accumulate and then melt off gradually. σ max σ (t ) tc tc tc t Figure 4.10 Loading scheme of cyclic bending test In this experiment the duration of each cycle was set at 60 seconds to get a number of cycles to failure within a reasonable time frame. Due to the capacity limitation of the machine, the maximum number of cycles the machine can produce is 300. In terms of loading level, if too 65 low, it would take too long for one specimen to fail so that we might not be able to get data of number of cycles to failure within machine limit. Conversely, if too high, many specimens might fail within the first cycle and the body of specimens could not be fully utilized. Hence it is very important to determine the appropriate load level to get as much valid data as possible within reasonable time frame. With a new product, limited information existed on the fatigue behavior; therefore the choice of the load level for the first specimen was based on 60 to 70% of the estimated mean static load capacity of the control group. The load levels for subsequent tests were chosen based on the relationship between load level and the number of cycles to failure information from previous tests. The peak load was converted to peak stress and six stress levels were chosen such that the peak stress within each cycle was approximately 65%, 75%, 85%, 90%, 105% and 115% of the mean short term bending strength obtained from control group test. Each matched group was divided into three sub-groups with ten specimens, with sub-group A comprised of the specimens with the lowest MOE, sub-group C comprised of the specimens with the highest MOE, and sub-group B represented the mid-range stiffness specimens. This Esequencing of sub-groups was assumed to provide an approximation to a sequencing by strength from the weakest to the strongest specimens, considering that the stiffness and strength are known to be positively correlated. The stress levels for sub-groups A were set as 65 to 75% of mean short term bending strength; 85 to 90% for sub-groups B; and approximately 105% to 115% for sub-groups C. In this way, weaker specimens were tested at lower stress levels while stronger specimens were tested at higher stress levels to get as much valid data as possible within reasonable time frame. The loading information is given in Table 4.1, where the stress ratio is defined as the ratio of peak stress to mean short term bending strength. 66 Table 4.1 Loading data of cyclic bending test Peak load Number of specimens Group 10 Mean stress ratio Mean (KN) STDEV (KN) GF1-A 2.427 0.216 0.66 10 GF1-B 3.212 0.326 0.86 10 GF1-C 3.993 0.090 1.05 10 GF2-A 2.806 0.090 0.75 10 GF2-B 3.358 0.273 0.89 10 GF2-C 4.428 0.205 1.16 The specimens were tested in bending on edge, under third-point loading, with a span of 660 mm. The experiment set up is shown in Figure 4.11. The cyclic triangular load was applied using a hydraulic driven machine under load control, as shown in Figure 4.12. Two LVDT’s were amounted on both sides of the specimen, and the average reading of those two transducers was taken as the mid-span deflection, as shown in Figure 4.13. 67 Figure 4.11 Setup of cyclic bending test Figure 4.12 Loading program of cyclic bending test 68 Figure 4.13 Deflection measuring device for cyclic bending test 4.4 Experimental results 4.4.1 MOE test results The summary of statistic analysis for MOE for all specimens and seven matched groups is shown in Table 4.2 respectively. The MOE was found to be highly related to the density of specimens. Here the density was obtained as the average density of the specimen. The relationship between MOE and density is shown in Figure 4.14. The manufacturers can control the production process and thus the material’s density and stiffness. 69 Table 4.2 A summary of statistical data on MOE Group Mean all 10.851 GPa (1.574·106 psi) Standard deviation 1.416 GPa (2.054·105 psi) 1 10.833 GPa (1.571·106 psi) 1.419 GPa (2.058·105 psi) 13.1% 2 10.861 GPa (1.575·106 psi) 1.446 GPa (2.097·105 psi) 13.3% 3 10.835 GPa (1.571·106 psi) 1.427 GPa (2.069·105 psi) 13.2% 4 10.876 GPa (1.577·106 psi) 1.451 GPa (2.105·105 psi) 13.3% 5 10.888 GPa (1.579·106 psi) 1.522 GPa (2.207·105 psi) 14.0% 6 10.810 GPa (1.568·106 psi) 1.379 GPa (2.0·105 psi) 12.8% 7 10.850 GPa (1.574·106 psi) 1.410 GPa (2.045·105 psi) 13.0% MOE C.O.V 13.1% 17 16 y = 0.026x - 5.5163 R² = 0.7828 15 MOE (GPa) 14 13 Series1 Linear (Series1) 12 11 10 9 8 500 550 600 650 700 750 800 Density (kg/m^3) Figure 4.14 Relationship between MOE and density 70 4.4.2 Ramp load test results 4.4.2.1 Control group test results A summary of statistics of short-term strength determined by the control group test is shown in Table 4.3. According to Canadian Wood Council (CWC) data, the mean bending strength determined by 2-P Weibull Distribution for solid sawn lumber, such as Douglas fir (DF), HemFir (HF) and Spruce-Pine-Fir (SPF), ranges from 16.50 MPa to 67.88 MPa (Foschi, 1989). Therefore the thick MPB strand-based wood composite lumber showed good mechanical properties and is suitable for structural applications. Normal, Lognormal, and Two-parameter Weibull and Three-parameter Weibull distributions were fitted to raw experimental data using a University of British Columbia computer program RELAN (Foschi et al., 1993), and it was found that Three-parameter Weibull distribution can represent the data satisfactorily, as shown in Figure 4.15. The distribution characteristics are listed in Table 4.4. Hence, Three-parameter Weibull distribution was used to calculate 5-th percentile and 20-th percentile of short-term strength to determine the loading levels for long-term constant load test. Table 4.3 A summary of statistical data on short-term strength Short-term strength Mean Standard deviation C.O.V 41.83 MPa (6.607 ksi) 9.966 MPa (1.445 ksi) 0.24 71 Cumulative probability Short-term strength (MPa) Figure 4.15 Cumulative distribution curve of experimental data and Three-parameter Weibull Distribution fitted to short-term strength Table 4.4 Distribution characteristics of Three-parameter Weibull Distribution fitted to short-term strength data Distribution Location (MPa) Scale (MPa) Shape Three-parameter Weibull 21.1174 23.4073 2.1379 The relationship between short-term strength (MOR) and MOE for control group test sample was studied. The raw experimental data of MOR and MOE are listed in Appendix IV. The shortterm strength (MOR) was found to be positively correlated to MOE, as shown in Figure 4.16, which gave support to the use of the MOE-matching technique. 72 70 60 y = 6.5632x - 29.269 R² = 0.8731 MOR (MPa) 50 Series1 40 Linear (Series1) 30 20 10 0 0 2 4 6 8 10 12 14 16 MOE (GPa) Figure 4.16 Relationship between MOR and MOE for short term test sample 4.4.2.2 Ramp load test results for different rates of loading The experimental data for the first two specimens of the fast group were lost, and therefore data analysis was performed on the remaining 28 specimens. In order to make comparisons between groups, 28 specimens were also chosen from the control group and slow group respectively for statistical study so that they were still matched groups in terms of MOE. The range of times-tofailure for slow, control and fast groups is listed in Table 4.5. The average time-to-failure covered the range from 46 seconds to 12 minutes. The corresponding mechanical properties of three groups are listed in Table 4.6. Independent sample t-test was conducted for slow and control groups, control and fast groups, and slow and fast groups respectively using the commercial statistical analysis software SPSS. The t-test result is listed in Table 4.7. This was done to test the significance of difference between mean bending strength obtained at different rates of loading. And the analysis shows that there is small increase in mean bending strength with increased rate of loading, but there is no significant difference in the means. 73 Table 4.5 Rates of loading and corresponding time-to-failure in ramp load tests Group Rate of loading (N/sec) Rate of stressing (MPa/sec) Slow 5.2 Control Fast Time-to-failure 0.058 Minimum (sec) 443 Average (sec) 712 Maximum (sec) 1041 52 0.58 47 74 104 94.3 1.03 32.5 46 60.5 Table 4.6 Time-to-failure and mechanical properties of groups tested at different rates of loading Time-to-failure (sec) MOE (GPa) MOR (MPa) Rate of loading (N/sec) Rate of stressing (MPa/sec) Mean C.O.V Mean C.O.V Mean C.O.V Slow 5.2 0.058 712 21% 10.989 13% 41.349 20% Control 52 0.58 74 23% 10.962 13% 42.78 22% Fast 94.3 1.03 46 16% 10.964 13% 43.471 19% Table 4.7 Independent-samples t-test for bending strength t-test for Equality of Means Samples t 95% Confidence Interval of the Difference P(T≤t) (2-tail) Mean Difference Lower Upper Slow and control groups -.599 .551 -1.43109 -6.21854 3.35636 Control and fast groups -.290 .773 -.69107 -5.46357 4.08142 Slow and fast groups -.970 .336 -2.12216 -6.50793 2.26360 74 4.4.3 Creep-rupture test results Long-term deflection data were measured at every three seconds for the first minute, every minute for the first ten minutes, every ten minutes for the first hour, every hour for the first week and every day till the specimen failed or at the end of six months. The deflection history was converted to strain history, and the time-dependent strain was plotted for every specimen, as shown in Appendix I, where the curves with rectangular markers are experimental results. Depending on the stress level applied and the strength of individual specimen, different stages of creep were observed in different specimens. At the lower stress level most of the broken specimens showed primary, secondary and tertiary creep. While at higher stress level, for some weaker specimens, only tertiary creep i.e. accelerating deformation rate was observed. It can be seen from the time-dependent strain curves that the failure mode was brittle. Therefore, it is especially important to have a good understanding of the development of deflection and probability of failure for this particular wood composite product under long term loading. Time-to-failure and deflection at failure of broken specimens within a one-year period are listed in Table 4.8. The times-to-failure obtained for both constant stress levels were ranked, and the corresponding cumulative distribution curves were plotted in Figure 4.17, where the time is expressed in hours and in a natural logarithmic scale. At the 33 MPa level (20-th percentile of short term strength), approximately 20% of the specimens broke during the application of the load and it confirmed that the between-groups matching was quite satisfactory. For the specimens that survived initial loading, approximately 80% failed under constant load within a period of one year. Thus approximately 85% of the sample had failed within one year of loading (including the initial load application). At the lower 27 MPa level (5-th percentile of short term strength), around 5% of the specimens failed during the application of load. For the specimens 75 that survived the initial loading, approximately 53% failed under constant load within one year of loading. Therefore in total approximately 55% of the sample had failed within one year of loading (including the initial load application). It is shown in Figure 4.17 that there is an upward departure from the initial trend occurring around three months (ln�𝑇𝑇𝑓𝑓 � = 7.68). Similar phenomenon in the same tests for hemlock lumber was observed and reported in Foschi and Barrett (1982). It would be ideal to continue this type of experiment for a longer time, for example a three-year test would be a realistic compromise. 76 Table 4.8 Time-to-failure and deflection at failure of broken specimens in creep-rupture test Group GL (27 MPa) GH (33 MPa) Specimen GL-1 GL-2 GL-3 GL-4 GL-5 GL-6 GL-7 GL-8 GL-9 GL-10 GL-11 GL-12 GL-13 GL-14 GL-15 GL-16 GL-19 GH-1 GH-2 GH-3 GH-4 GH-5 GH-6 GH-7 GH-8 GH-9 GH-10 GH-11 GH-12 GH-13 GH-14 GH-15 GH-16 GH-17 GH-18 GH-19 GH-20 GH-21 GH-22 GH-23 GH-25 GH-27 GH-28 Time-to-failure (hr.) 7.53 0.11 99.03 0.03 1637.10 796.10 12.98 1373.10 1.40 1350.10 1.42 2654.12 127.83 1163.10 80.23 7841.25 5246.23 0.019 6.90 0.023 0.047 0.038 0.021 0.041 0.044 1.27 0.544 6.13 0.215 0.218 1.35 1.72 1062.97 197.30 54.70 949.97 23.49 373.30 191.50 2273.92 1838.92 2889.99 3294.99 77 Deflection at failure (mm) 11.60 10.46 11.82 8.38 15.50 14.69 11.91 13.94 9.82 15.54 9.38 14.88 10.96 10.36 11.71 18.22 15.12 18.17 13.22 15.87 12.36 14.46 15.36 11.98 17.43 11.71 10.90 12.29 12.49 8.47 10.48 10.29 14.69 11.43 14.31 14.32 10.26 12.16 12.27 14.37 16.43 12.04 10.54 0.9 0.8 0.7 Cumulative Probability 0.6 0.5 33MPa (N=30) 0.4 27MPa (N=30) 0.3 0.2 0.1 0 -6 -4 -2 0 2 4 6 ln�𝑇𝑇𝑓𝑓 � = 7.68 8 10 LN (Time-to-failure) (Hour) Figure 4.17 Cumulative distribution curves of time-to-failure under constant load 4.4.4 Cyclic bending test results Results from the cyclic test program are summarized in Table 4.9. Each of the two matched groups was divided into three sub-groups with sub-group A comprised of specimens with the lowest MOE, sub-group C comprised of specimens with the highest MOE, and sub-group B represented the middle stiffness specimens. Since the stiffness and strength are known to be positively correlated, this E-sequencing of sub-groups was assumed to provide an approximation to a sequencing by strength from the weakest to the strongest specimens. GF1-A and GF2-A were still matched groups in terms of MOE, so were GF1-B and GF2-B, and GF1-C and GF2-C. The peak loads in sub-groups A to C corresponded to bending stress ratio of approximately 65%, 78 75%, 85%, 90%, 105%, and 115% of mean bending strength obtained from the short term bending test with the control group. Within each group, the variability in the number of cycles to failure was found to be large. The large variability was consistent with fatigue performance of plywood (Kommers, 1993 and 1994), and fatigue behavior of veneer wood plates (Lam, 1992). By comparing the matched sup-groups, i.e. GF1-A and GF2-A, GF1-B and GF2-B, GF1-C and GF2-C, it can be seen that the number of cycles to failure decreased with increased stress ratio. Figure 4.18 shows a typical tension failure mode observed during the cyclic bending test. Table 4.9 Cyclic bending test results Peak load Number of specimens Group 10 Number of cycles to failure Mean stress ratio Nonfailed (>300) 2 Mean (KN) STDEV (KN) GF1-A 2.427 0.216 0.66 77 12.7 10 GF1-B 3.212 0.326 0.86 83 137.4 3 10 GF1-C 3.993 0.090 1.05 75 91.6 1 10 GF2-A 2.806 0.090 0.75 70 113.1 0 10 GF2-B 3.358 0.273 0.89 6 5.4 2 10 GF2-C 4.429 0.207 1.16 42 50.9 2 79 Mean STDEV Figure 4.18 The typical tension failure mode The strain-time curves are shown in Appendix II. During the loading part within each cycle the deflection increased with time; while upon unloading, deflection decreased with time but did not reach zero. The elastic deformation was recovered completely when the load was decreased to zero, while there was irrecoverable viscous deformation and the viscoelastic deformation had no time to recover completely - a residual deformation remained at the end of each cycle. Total recovery cannot take place during cyclic loading of this type because the next cycle began before the recovery was complete. It can be seen from those strain-time curves that the residual deformation in each cycle almost remained constant till when it was close to the breaking point, and at that time the residual deformation was increased at an accelerated rate leading to failure. The last stage exhibited characteristics associated with tertiary creep. 80 Results from the cyclic bending test were compared with the creep test results. Some specimens in the fatigue test were loaded to the same stress level as those specimens tested under constant load (33MPa) and a comparison was made between corresponding specimens which possessed approximately the same strength properties, as shown in Figure 4.19. During the loading period within the first cycle, the specimens deflected to the same level as the corresponding specimens reached under constant load, and then upon unloading the deflection decreased but did not reach zero i.e. there was residual deformation remained within each cycle. It was the residual deformation accumulated with cycles that caused faster deformation, leading to failure. Considering there were fines in the strand-based wood composite beam specimens, after the fines’ failure and unable to contribute to the strength and stiffness of the specimen, there was less chance for the longer strands to take on the additional stress in this type of cyclic loading. And therefore there might be less opportunity of the stress redistribution than in constant loading. This is an interesting phenomenon which needs more investigation in future research. Based on the strain-time curves obtained from creep and fatigue tests, it can be seen that most of the specimens failed when the strain reached around between 4.5⋅10-3 and 5.0⋅10-3. It also demonstrated that during the process of creep-rupture damage was more related to deflection. Therefore it is reasonable for the proposed damage accumulation model to be expressed in terms of strain rate, other than stress. 81 Figure 4.19 Comparison between creep and fatigue test results 82 4.5 Conclusions The results from the duration-of- load and creep testing program on MPB strand-based wood composite beams were presented. The laboratory-made thick, strand-based MPB wood composite specimens were divided into matched groups using MOE matching technique. Three matched groups were tested with a ramp load at three different loading rates. Among them, one group was tested using a simple ramp loading that caused failure at around 1 minute to establish the basic short-term bending strength distribution for the population. The thick MPB strandbased wood composite lumber showed good mechanical properties in terms of bending strength, compared to that of solid sawn lumber. It gave support to its use in structural applications. And the remaining two groups were tested at different loading rates to investigate the influence of rate of loading on strength property. It was conclude that there was a small, not very significant increase of strength with increased rate of loading. Another two matched groups were tested under long term constant load at two stress levels using the third-point edge-bending test. The constant load experiment was continued for a period of one year. The long term deflection was monitored and recorded at a pre-set frequency. Additionally, time-to-failure data was obtained for all broken specimens. The brittle failure mode was observed during the test, and therefore it is especially important to have a good understanding of the development of deflection and probability of failure for this particular wood composite product. The third part of the testing program was cyclic bending test. Two matched groups were tested with a triangular cyclic load history. Six stress levels were chosen such that the peak bending stress within each cycle was approximately 65%, 75%, 85%, 90%, 105%, and 115% of the mean 83 bending strength obtained from the short term bending test with the control group. Stronger specimens were tested at higher stress levels and weaker specimens at lower levels to get as much valid data as possible within the reasonable time frame. The results of specimens under cyclic loading were compared with corresponding specimens under constant load. It was concluded that the number of cycles to failure decreased with increased stress ratio, and that cyclic load was more damaging than constant load. The experimental research conducted in this work contributes to the study of the duration-ofload and creep behavior of MPB strand-based wood composite product under long term loading. The results provided an experimental database for the calibration of the creep-rupture model developed in this work. 84 Chapter 5 Calibration and Verification of the Creep-rupture Model 5.1 Summary The results of the long-term constant load tests have been interpreted by means of a creep rupture model. This model is capable of representing the time-dependent deflection and time-tofailure data for different stress levels. The predictions of the model have been verified using results from ramp load tests at different loading rates as well as results from cyclic loading tests at different stress levels. The creep-rupture model incorporates the short term strength of the material, the load history and predicts the deflection history as well as the time-to-failure. As it is a probabilistic model, it allows its incorporation into a time-reliability study of wood composites’ applications. 5.2 Introduction The creep-rupture model developed in Chapter 3 was calibrated to the experimental results of long-term constant load tests at two stress levels using the third-point edge bending method. One group was loaded at the 5-th percentile of short-term strength, which was denoted as GL. The other matched group was loaded to the 20-th percentile of short-term strength, which was denoted as GH. The loading scheme is depicted in Figure 5.1. 85 σ (t ) σs Ka σa 1 t a Ts t Figure 5.1 Experimental load history According to the correspondence principle: If a viscoelastic beam is subjected to loads which are all applied simultaneously at 𝑡𝑡 = 0 and then held constant, its stresses are the same as those in elastic beam under the same load, and its strains and displacements depend on time and are derived from those of the elastic problem by replacing 𝐸𝐸 by 1⁄𝐽𝐽(𝑡𝑡), where 𝐽𝐽(𝑡𝑡) is the creep compliance (Flügge, 1967). Based on the corresponding elastic solution derived from Euler- Bernoulli beam theory, we have the following set of equations for the third-point edge-bending test: 𝜎𝜎𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑀𝑀 𝐼𝐼 𝑃𝑃𝑃𝑃 ∙ 𝑦𝑦𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑏𝑏ℎ 2 (5.1) 𝑃𝑃𝑃𝑃 (5.2) 23𝑃𝑃𝐿𝐿3 (5.3) 𝜀𝜀𝑚𝑚𝑚𝑚𝑚𝑚 = 𝜎𝜎𝑚𝑚𝑚𝑚𝑚𝑚 𝐽𝐽(𝑡𝑡) = 𝑏𝑏ℎ 2 𝐽𝐽(𝑡𝑡) 𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚 = 𝜀𝜀 𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚 = 23𝑃𝑃𝐿𝐿3 𝐽𝐽(𝑡𝑡) = 108𝑏𝑏ℎ 3 𝐽𝐽(𝑡𝑡) 1296𝐼𝐼 108ℎ (5.4) 23𝐿𝐿2 Thus we can convert load to stress and deflection to strain based on Eqs. (5.1) and (5.4). 86 5.3 Calibration of the creep-rupture model 5.3.1 Calibration of the creep model The creep model (Eq. (3.37)) which has been developed in Chapter 3 contains five parameters (𝛽𝛽1, 𝛽𝛽2 , 𝛽𝛽3, 𝛽𝛽4 and 𝛽𝛽5 ), as shown in the following: When 0 ≤ t ≤ t a , Ka β 4 t β5 +1 − tβ 3 ε (t ) = / β3 + (β1 + β 2 )t − β 2 1 − e (β 5 + 1) σs K β t β5 +1 ∴ ε (t = t a ) = a (β1 + β 2 )t a − β 2 1 − e −ta β 3 / β 3 + 4 a (β 5 + 1) σs ( ) ( = (3.37.1) ) σa β 4 t aβ β2 −t β ( ) ( ) β β e + − − + 1 1 2 (β 5 + 1) β 3t a σs 5 a 3 (3.37.2) When t > t a , ε (t ) = ε t + a [ σa β 1 + β 2 (1 − e −(t −t σs a )β 3 ) + β (t − t ) ]− σ β5 4 a σs a ⋅ β1 [ = σa β 4 t aβ σ a β2 −t β ( ) ( ) − − + 1 + e β β β 2 (1 − e −(t −t 1 + 2 ( ) + 1 t β β σ σs 5 a 3 s = σa β 4 t aβ β2 −t β ( ) ( ) + − − + + β 2 (1 − e −(t −t 1 β β e 1 2 ( ) + 1 β β t σs 5 a 3 5 a 3 5 a 3 a )β 3 a )β 3 ) + β (t − t ) ] β5 4 ) + β (t − t ) 4 a a β5 (3.37.3) Where ��� σs is the average short-term strength, and β1 = σs k0 , β2 = σs k1 , β3 = 1 τ1 = k1 η1 , β4 = σs , 0 < β5 < 1. η0 And k 0 , k1 , η0 , η1 are properties of springs and dashpots in the viscoelastic model as described in Chapter 3. 87 The creep model was calibrated to the 6 months’ deflection data of 60 specimens one by one to determine the creep parameters, 𝛽𝛽1, 𝛽𝛽2 , 𝛽𝛽3, 𝛽𝛽4 and 𝛽𝛽5 , of individual specimen. A non-linear minimization procedure which employs a quasi-Newton method was used to determine the values of unknown parameters when the creep model was fitted to the experimental data of every specimen. After discarding the unreasonable data points, the calibration results of creep parameters are listed in Appendix Ш. A statistical analysis was then performed to determine the mean, standard deviation and probability distribution of every parameter, as well as the correlation relationship between parameters. The procedure was as follows: 1. The initial values for each of the five model parameters were selected. 2. The simulated time-dependent strain was compared to the experimental data and the residual function ∅ was computed as following: 2 ∅ = ∑𝑁𝑁 𝑖𝑖=1(𝜀𝜀𝑐𝑐𝑐𝑐 − 𝜀𝜀𝑚𝑚𝑚𝑚 ) (5.9) Where 𝑁𝑁 is the number of the data points on the experiment curve; 𝜀𝜀𝑐𝑐𝑐𝑐 is the simulated strain, and 𝜀𝜀𝑚𝑚𝑚𝑚 is the measured strain in the experiment. 3. The gradient of Eq. (5.9) required by the minimization routine was computed with respect to each of the distribution parameters. These partial derivatives were computed based on the following equations: 𝜕𝜕∅ 𝜕𝜕𝑥𝑥 𝑗𝑗 𝜕𝜕∅ 𝜕𝜕∅ = ∑𝑁𝑁 𝑖𝑖=1(𝜕𝜕𝜀𝜀 ∙ 𝜕𝜕𝜀𝜀 𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐 𝜕𝜕𝜀𝜀 𝑐𝑐𝑐𝑐 𝜕𝜕𝑥𝑥 𝑗𝑗 = 2(𝜀𝜀𝑐𝑐𝑐𝑐 − 𝜀𝜀𝑚𝑚𝑚𝑚 ) ) (5.10) (5.11) When 0 ≤ t ≤ t a , 𝜕𝜕𝜀𝜀 𝑐𝑐𝑐𝑐 𝜕𝜕𝛽𝛽 1 𝜕𝜕𝜀𝜀 𝑐𝑐𝑐𝑐 𝜕𝜕𝛽𝛽 2 𝑘𝑘 𝑠𝑠 = ��� ∙ 𝑡𝑡 𝜎𝜎 𝑘𝑘 𝑠𝑠 (5.12.1) 1 𝑠𝑠 = ��� �𝑡𝑡 − 𝛽𝛽 �1 − 𝑒𝑒 −𝑡𝑡𝛽𝛽3 �� 𝜎𝜎 𝑠𝑠 (5.12.2) 3 88 𝜕𝜕𝜀𝜀 𝑐𝑐𝑐𝑐 𝜕𝜕𝛽𝛽 3 𝜕𝜕𝜀𝜀 𝑐𝑐𝑐𝑐 𝜕𝜕𝛽𝛽 4 ∂ε ci ∂β 5 𝑘𝑘 𝛽𝛽 𝛽𝛽 𝑠𝑠 = ��� � 2 �1 − 𝑒𝑒 −𝑡𝑡𝛽𝛽3 � − 𝛽𝛽2 �𝑡𝑡𝑒𝑒 −𝑡𝑡𝛽𝛽3 �� 𝜎𝜎 𝛽𝛽 2 𝑘𝑘 𝑠𝑠 3 𝑡𝑡 𝛽𝛽 5 +1 𝑠𝑠 = ��� � 𝛽𝛽 𝜎𝜎 k 𝑠𝑠 𝜕𝜕𝛽𝛽 1 𝜕𝜕𝜀𝜀 𝑐𝑐𝑐𝑐 𝜕𝜕𝛽𝛽 2 𝜕𝜕𝜀𝜀 𝑐𝑐𝑐𝑐 = = 𝜕𝜕𝛽𝛽 3 = 𝜕𝜕𝛽𝛽 4 = 𝜕𝜕𝛽𝛽 5 = 𝜕𝜕𝜀𝜀 𝑐𝑐𝑐𝑐 𝜕𝜕𝜀𝜀 𝑐𝑐𝑐𝑐 (5.12.4) β t β 5 +1 = ��σ��s �− (β4 s When t > t a , 𝜕𝜕𝜀𝜀 𝑐𝑐𝑐𝑐 � 5 +1 5 (5.12.3) 3 +1)2 + β 4 t β 5 +1 β 5 +1 ∙ ln (t)� (5.12.5) 𝜎𝜎𝑎𝑎 (5.12.6) ��� 𝜎𝜎𝑠𝑠 𝜎𝜎𝑎𝑎 ��� 𝜎𝜎𝑠𝑠 𝜎𝜎𝑎𝑎 �1 − 𝛽𝛽 𝛽𝛽 1 3 𝑡𝑡 𝑎𝑎 �1 − 𝑒𝑒 −𝑡𝑡𝑎𝑎 𝛽𝛽3 � + �1 − 𝑒𝑒 −(𝑡𝑡−𝑡𝑡 𝑎𝑎 )𝛽𝛽 3 �� 𝛽𝛽 (5.12.7) � 2 �1 − 𝑒𝑒 −𝑡𝑡𝑎𝑎 𝛽𝛽3 � − 𝛽𝛽2 �𝑒𝑒 −𝑡𝑡𝑎𝑎 𝛽𝛽3 � + 𝛽𝛽2 (𝑡𝑡 − 𝑡𝑡𝑎𝑎 )𝑒𝑒 −(𝑡𝑡−𝑡𝑡 𝑎𝑎 )𝛽𝛽3 � ��� 𝜎𝜎 𝛽𝛽 2 𝑡𝑡 𝑠𝑠 𝜎𝜎𝑎𝑎 3 𝑎𝑎 � 𝛽𝛽 𝑡𝑡 𝑎𝑎 5 ��� 𝜎𝜎𝑠𝑠 𝛽𝛽 5 +1 𝜎𝜎𝑎𝑎 + (𝑡𝑡 − 𝑡𝑡𝑎𝑎 )𝛽𝛽5 � 𝛽𝛽 𝛽𝛽 𝑡𝑡 5 𝑎𝑎 �− (𝛽𝛽 4+1) 2 + ��� 𝜎𝜎 𝑠𝑠 (5.12.8) 3 5 𝛽𝛽 𝛽𝛽 4 𝑡𝑡 𝑎𝑎 5 𝛽𝛽 5 +1 (5.12.9) (𝑡𝑡𝑎𝑎 ) + 𝛽𝛽4 ∙ ln (𝑡𝑡 − 𝑡𝑡𝑎𝑎 ) ∙ (𝑡𝑡 − 𝑡𝑡𝑎𝑎 )𝛽𝛽5 � ∙ ln (5.12.10) 4. The minimization routine automatically modified the initial choices for the variables in a search for the minimum of Eq. (5.9). 5. Different choices of the probability distributions were fitted to the obtained creep parameters data using a UBC software RELAN (Foschi et al., 1993) to determine the probability distribution for each parameter. 6. The correlation relationship of creep parameters was studied using the commercial statistical analysis software SPSS. Statistical analysis was performed to determine the mean, standard deviation and probability distribution of every parameter as well as the correlation relationship between parameters. A summary of statistics of creep parameters is listed in Table 5.1. It was found that Two-Parameter Weibull distribution represented the data well, as shown in Figure 5.2a~5.2e. The calibration 89 results of 𝛽𝛽2 and 𝛽𝛽3 were found to be sensitive to the choice of initial values especially for the weak specimens which failed within a short period of time. This can be explained from the point of view of the physical meaning of those parameters. 𝛽𝛽2 represents the delayed-elastic strain and 𝛽𝛽3 represents how fast/slow this delayed-elastic strain can be fully developed. It is therefore important to set reasonable initial values. For 𝛽𝛽2 the initial value was set as one third of the instantaneous elastic strain. In terms of 𝛽𝛽3 we can assume that it would take a year for the delayed-elastic strain to be fully developed. The variability of 𝛽𝛽3 was found to be large which was also reflected in other researchers’ work (Pierce et al. 1985). With the help of RELAN, it was concluded that Two-Parameter Weibull distribution can represent the data best, fitting to most of the reliable data points. The distribution characteristics of two-parameter Weibull distribution are listed in Table 5.2, and the correlation coefficients between creep parameters are listed in Table 5.3. The experimental results and model simulation of time-dependent strain were compared for all 60 specimens as shown in Appendix I. The curves with rectangular markers are experimental results and the curves with diamond markers are model calculations. It can be observed from the strain-time curves that the model simulation agreed with the experimental results well. However it should be noted that the viscoelastic creep model applied here can not reflect tertiary creep where the rate of deformation is increasing. The brittle failure mode observed in the experiment was also difficult to be reflected in terms of modeling. Table 5.1 A summary of statistical data on creep parameters Mean β1 0.0034 β2 6.619⋅10-4 β3 2.78⋅10-6 β4 2.154⋅10-4 β5 STDEV 0.0008 3.612⋅10-4 4.17⋅10-6 2.042⋅10-4 0.1458 90 0.2832 Cumulative probability 𝛽𝛽1 × 10−3 Cumulative probability Figure 5.2a Two-Parameter Weibull Distribution fitting curve of 𝛃𝛃𝟏𝟏 𝛽𝛽2 × 10−3 Figure 5.2b Two-Parameter Weibull distribution fitting curve of 𝛃𝛃𝟐𝟐 91 Cumulative probability 𝛽𝛽3 × 10−5 Cumulative probability Figure 5.2c Two-Parameter Weibull distribution fitting curve of 𝛃𝛃𝟑𝟑 𝛽𝛽4 × 10−4 Figure 5.2d Two-Parameter Weibull distribution fitting curve of 𝛃𝛃𝟒𝟒 92 Cumulative probability 𝛽𝛽5 Figure 5.2e Two-Parameter Weibull distribution fitting curve of 𝛃𝛃𝟓𝟓 Table 5.2 Distribution characteristics of two-parameter Weibull distribution fitted to creep data Scale (m) β1 0.0037 β2 0.000702 β3 2.26⋅10-6 β4 0.0002 β5 Shape (k) 4.2302 1.4609 1.1911 1.0436 2.4634 0.3071 𝑥𝑥 𝑘𝑘 Note: Two-parameter Weibull cumulative distribution function 𝐹𝐹(𝑥𝑥) = 1 − 𝑒𝑒 −�𝑚𝑚 � Table 5.3 Correlation coefficients between creep parameters β1 1 β2 0.213 β3 -0.031 β4 -0.469 0.309 β2 0.213 1 0.091 0.477 0.114 β3 -0.031 0.091 1 0.173 0.011 β4 -0.469 0.477 0.173 1 -0.081 β5 0.309 0.114 0.011 -0.081 1 β1 93 β5 5.3.2 Calibration of the damage accumulation model The damage accumulation model has been developed in Chapter 3, as shown in Eq. (3.41). 1 A+ B ln B A + Bα c [ ( ) σa − (T −t )β = β 2 1 − e f a 3 + β 4 (T f − t a )β5 σs Where αc is the damage at T = t a α c = α (t a ) = ] (3.41) β 4 t aβ5 +1 A K a −t a β 3 exp B ( ) t e + β + β − β 1 − β 1 − 1 2 a 2 3 ( B σ s β 5 + 1) ( ) (3.39) And ��� σs is the average short-term strength. Note that A is not independent of other parameters B and Ts , and it can be determined by other parameters using Eq. (3.36). 1= β 4Tsβ5 +1 A K a −Ts β 3 exp B ( ) T e 1 + − − + β β β β 1 − 1 2 2 3 s ( B σ s β 5 + 1) ( ) (3.36) σ Where Ts is the short-term time-to-failure for the individual specimen, determined by Ts = K s . a The model calibration procedure used here was based on previous studies on duration of load effects in lumber (Foschi, 1989). Because the creep parameters have been determined during the creep model calibration the only unknown parameter in duration-of-load model to be calibrated is 𝐵𝐵, since 𝐴𝐴 is determined by the short-term strength constraint of Eq. (3.36). The DOL model was fitted to time-to-failure data of long-term constant load tests. In the model simulation 𝐵𝐵 was modeled as a two-parameter Weibull variable and therefore the distribution characteristics, scale (m) and shape (k), are the variables to be determined. The short-term strength σs was represented by three-parameter Weibull distribution, the distribution characteristics of which have been obtained from standard short-term bending test with control group. The creep parameters, β1, β2 , 94 β3, β4 and β5, were modeled as correlated two-parameter Weibull variables which obey the probability distributions and correlation relationship determined in creep model calibration. A non-linear minimization procedure which employs a quasi-Newton method was used to determine the distribution characteristics (scale and shape) of parameter 𝐵𝐵 . The procedure was as follows: 1. The initial values for scale and shape of 𝐵𝐵 were chosen. 2. Values of β1, β2 , β3, β4 , β5, and σs were selected from the corresponding distributions generating a random sample of NR = 400 time-to-failure using Eq. (3.41). The correlation relationship between creep parameters was taken into account using the method described in Kiureghian et al. (1986). The sample of NR replications was ranked to obtain the cumulative distribution function of time-to-failure Tf . The following are the steps of how to generate correlated two-parameter Weibull random variables based on known marginal distributions and correlation relationship: 1) Generate independent standard normal random variables 𝑌𝑌1 , 𝑌𝑌2 , …, 𝑌𝑌𝑛𝑛 as IID N(0,1) 2) Generate correlated standard normal random variables 𝑍𝑍1 , 𝑍𝑍2 , …, 𝑍𝑍𝑛𝑛 𝑍𝑍 = 𝑀𝑀 + 𝐷𝐷𝐷𝐷𝐷𝐷 = 𝐿𝐿𝐿𝐿 (5.13) Where 𝑀𝑀 is mean vector and 𝐷𝐷 = 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑�𝜎𝜎𝑖𝑖𝑖𝑖 � = the diagonal matrix of the standard deviations, 𝜎𝜎𝑖𝑖 , and 𝐿𝐿 = a lower-triangular matrix obtained from Cholesky decomposition of the correlation matrix 𝑅𝑅 = �𝜌𝜌0𝑖𝑖𝑖𝑖 � such that 𝑅𝑅 = 𝐿𝐿𝐿𝐿𝑇𝑇 . And the correlation coefficients 𝜌𝜌0𝑖𝑖𝑖𝑖 between 𝑍𝑍𝑖𝑖 were obtained using the method by Kiureghian et. al. (1986) based on pre-described information of the target mean and standard deviation and correlation coefficients 𝜌𝜌𝑖𝑖𝑖𝑖 . 3) Generate correlated two-parameter Weibull random variables 95 ∅(𝑍𝑍𝑖𝑖 ) = 𝐹𝐹(𝑋𝑋𝑖𝑖 ) 𝑋𝑋𝑖𝑖 = 𝐹𝐹 −1 �∅(𝑍𝑍𝑖𝑖 )� = 𝑚𝑚𝑖𝑖 ∙ �−𝑙𝑙𝑙𝑙�1 − ∅(𝑍𝑍𝑖𝑖 )�� 1⁄𝑘𝑘 𝑖𝑖 (5.14) Return 𝑋𝑋𝑖𝑖 . 3. The simulated distribution was compared to the experimental data and the residual function ∅ was computed as following: ∅ = ∑Ni=1(ln (Tfi ) − ln (Tdi ))2 (5.15) Where 𝑁𝑁 is the number of the probability levels used for the comparison; Tfi is the simulated time-to-failure corresponding to the same probability level as the test result Tdi . 4. The gradient of Eq. (5.15) required by the minimization routine was computed with respect to each of the distribution parameters. These partial derivatives were computed numerically based on the following equations: ∂∅ ∂x j ∂∅ ∂T fi ∂T fi ∂x j ∂∅ ∂T fi = ∑Ni=1( ∙ ∂T fi ) ∂x j = 2 ∙ (ln(Tfi ) − ln(Tdi )) ∙ = − T+ fi −T fi 2∆x j (5.16) 1 T fi (5.17) (5.18) 5. The minimization routine automatically modified the initial choices for the variables in a search for the minimum of Eq. (5.15). The calibration results for the long-term constant load tests at two stress levels are shown in Figure 5.3. The distribution characteristics for B (scale and shape) are listed in Table 5.3. 96 0.9 0.8 0.7 Cumulative Probability 0.6 0.5 GH-experiment GL-experiment 0.4 GH-model GL-model 0.3 0.2 0.1 0 -6 -4 -2 0 2 4 6 8 10 12 14 LN (Time-to-failure) (Hour) Figure 5.3 Comparison between experiment results and model simulation of time-to-failure under constant load Table 5.4 Calibration result of parameter 𝐁𝐁 Scale (m) Shape (k) 1.8538⋅10-13 1.0149 As discussed in Chapter 3, the model used here is the simplified form of the damage accumulation model proposed in Eq, (3.19), but using p=q=1.0. This simplifies the math but, on the other hand, fixes some parameters and makes the model more rigid. The agreement of the experimental results with the simplified model is sufficiently good, but in future research, the 97 values of p and q could be relaxed to also be random parameters. In allowing for more flexibility, one would be able to obtain a closer fit. 5.4 Verification of the creep-rupture model 5.4.1 Verification with ramp load test results In a verification process, the creep-rupture model should be tested with load histories not used for its calibration. In this section the predictions of the model were verified using results from ramp loading tests with rates of loading producing times-to-failure from 33 seconds to 17 minutes. Creep-rupture model for ramp loading has been developed in Chapter 3: β 4Trβ5 +1 1 B Ka −Tr ⋅β 3 ln1 + = / β3 + (β1 + β 2 )Tr − β 2 1 − e (β 5 + 1) B A σs ( ) (3.35) As previously stated, A is not independent of other parameters B and Ts , and it can be determined by other parameters using Eq. (3.36). β 4Tsβ5 +1 A K a −Ts β 3 1 = exp B β3 + (β1 + β 2 )Ts − β 2 1 − e − 1 ( B σ s β 5 + 1) ( ) (3.36) σ Where Ts is the short-term time-to-failure for individual specimen, determined by Ts = K s . a As described in Chapter 4, three matched groups were tested with ramp load, i.e., control group, slow group and fast group. Considering that the information of the control group (i.e. standard short-term bending test) has been used when determining 𝐴𝐴 (see Eq. (3.36)) this group was satisfied automatically and the model predictions were compared with slow and fast group only. 98 The average times-to-failure are listed in Table 5.5. The comparison results are shown in Figure 5.4. It can be seen that the model prediction agreed with the experimental results satisfactorily. Table 5.5 Experimental results and model prediction of time- to-failure under ramp load Group Average time-to-failure (sec) Rate of loading (MPa/sec) Experiment Model Error Slow 0.058 712 735 3% Fast 1.03 46.1 41.4 10% 99 Figure 5.4 Comparison between experiment results and model prediction of time-to-failure under ramp load 5.4.2 Verification with fatigue test results In this section, the predictions of the model were verified using the results from cyclic bending test. The triangular cyclic load was applied where in each cycle the load was increased to the maximum stress level at constant rate and then decreased to zero gradually at the same rate. Creep –rupture model for this type of cyclic loading has been developed in Chapter 3. Number of cycles to failure was computed as following: 𝑁𝑁𝑓𝑓 = 𝑙𝑙𝑛𝑛(𝐴𝐴+𝐵𝐵)−𝑙𝑙𝑙𝑙 (𝐴𝐴) 𝐵𝐵∙∆𝜀𝜀 − 𝑡𝑡 𝜀𝜀� 𝑐𝑐 � 2 ∆𝜀𝜀 +1 (3.51) 100 Where ∆𝜀𝜀 = 𝜀𝜀(𝑡𝑡𝑒𝑒𝑒𝑒 ) − 𝜀𝜀(𝑡𝑡𝑠𝑠𝑠𝑠 ) = 𝑡𝑡 𝑐𝑐 𝜀𝜀 � 2 � = 𝜎𝜎𝑚𝑚𝑚𝑚𝑚𝑚 ��� 𝜎𝜎𝑠𝑠 𝜎𝜎𝑚𝑚𝑚𝑚𝑚𝑚 2𝛽𝛽 2 �𝛽𝛽 ��� 𝜎𝜎𝑠𝑠 2𝛽𝛽 2 �(𝛽𝛽1 + 𝛽𝛽2 ) − 𝛽𝛽 3 𝑡𝑡 𝑐𝑐 3 𝑡𝑡 𝑐𝑐 �1 − 𝑒𝑒 �1 − 𝑒𝑒 2 𝑡𝑡 −� 𝑐𝑐 �𝛽𝛽 3 � + 2 𝑡𝑡 −� 𝑐𝑐 �𝛽𝛽3 2 �+ 𝑡𝑡 𝛽𝛽 5 𝛽𝛽 4 � 𝑐𝑐 � 2 𝛽𝛽 5 +1 𝑡𝑡 𝛽𝛽 5 𝛽𝛽 4 � 𝑐𝑐 � 2 (𝛽𝛽 5 +1) �2𝛽𝛽5 +1 − 2�� (3.47) � (3.50) As previously stated, A is not independent of other parameters B and Ts , and it can be determined by other parameters using Eq. (3.36). 1= β 4Tsβ5 +1 A K a −Ts β 3 exp B ( ) + − − + T e 1 β β β β 1 − 1 2 2 3 s ( B σ s β 5 + 1) ( ) (3.36) σ Where Ts is the short-term time-to-failure for individual specimen, determined by Ts = K s . a Figure 5.5 shows the relationship between model predicted median of number of cycles to failure and stress ratio. The stress level was expressed as the ratio of peak stress to mean short term bending strength. It can be concluded from Figure 5.5 that the median of number of cycles to failure increased with decreased stress ratio. As described in Chapter 4, there were two matched groups of specimens, i.e. GF1 and GF2, tested with a cyclic bending load. Because we did not get sufficient valid data of number of cycles to failure from group GF1, the model prediction was compared with the experimental result of group GF2 only. Group GF2 was divided into three subgroups with subgroup A comprised of the weakest specimens, subgroup C comprised of the strongest specimens, and subgroup B consisted of the middle strength specimens. In order to make comparison, the model results were divided into three corresponding subgroups as well. Figure 5.6 shows a comparison between the model predicted ranges of number of cycles to failure and experimentally obtained data from three subgroups, and in this figure the number of 101 cycles to failure was plotted in logarithmic scale. There were two samples in subgroup A showing failure at a high number of cycles which was not predicted by the model. It should be noted that the division of subgroups from A to C was based on MOE, and it was assumed to provide an approximation to the sequencing by strength, considering MOE and strength are positively correlated. However there is possibility that the specimens with lower MOE are actually stronger. Most of the experimental data points falls in the model predicted range. Therefore it can be concluded that there is a good agreement between experimental results and model prediction overall, providing verification of the model predictions for cyclic loading test. 1.2 1 STRESS RATIO 0.8 0.6 median 0.4 0.2 0 0 50 100 150 200 250 300 350 400 NUMBER OF CYCLES-TO-FAILURE Figure 5.5 Relationship between median of number of cycles to failure and stress ratio 102 1.40 1.20 Stress ratio 1.00 0.80 experiment-A experiment-B experiment-C 0.60 model-A model-B model-C 0.40 0.20 0.00 1 10 100 1000 Number of cycles to failure Figure 5.6 Experimental and model predicted ranges of number of cycles to failure under cyclic load 5.5 Conclusions The results of the long-term constant load tests have been interpreted by means of a creep rupture model which has been shown to be able to represent the time-dependent strain and timeto-failure data at different stress levels. The predictions of the model have been verified using results from ramp load tests at different loading rates and results from cyclic loading tests at different stress levels. The importance of such a model lies in that it permits the extrapolation of test results for constant loads to predictions about the effect of random, cyclic loadings such as snow, traffic, or occupancy. The model also provides a description of the time-dependent 103 deflection to present a complete picture from creep deflection to a final rupture event. In addition the viscoelastic parameters in creep-rupture model have been treated as correlated random variables for the first time. In this manner, the creep-rupture model incorporates the short term strength of the material, the load history and predicts the deflection history as well as the timeto-failure. As it is a probabilistic model, it allows its incorporation into a time-reliability study of wood composites’ applications. In future research the effect of environmental changes more drastic than those observed during the present experiment should be considered. As a further study the model could be used to simulate the three-month testing as specified in ASTM standard to examine whether this testing procedure is adequate to establish the long-term creep-rupture response. 104 Chapter 6 Concluding Remarks and Future Work 6.1 Concluding remarks The largest outbreak of Mountain Pine Beetle (MPB) epidemic in Canada calls for technologies capable of converting stained lumber into market-acceptable products. This is neccessary to reduce the impact of the growing volume of MPB-killed lumber on BC forest products industry. New, thick MPB strand-based structural composite product can help absorb a large volume of MPB wood. The aim of this work was to investigate and evaluate the duration-of-load and creep effects of this wood based composite. Based on production technology of thick OSB, thick strand-based MPB wood composite beams were made in the Timber Engineering and Applied Mechanics Laboratory at UBC. All beams were 41mm in width, 38mm in depth and 660mm in span. Specimens were then divided into matched groups using the MOE matching technique. A series of tests were conducted on the matched groups to investigate the creep-rupture behavior of thick MPB strand-based wood composite beams. These investigations comprised of ramp load tests at three loading rates, longterm constant load tests at two stress levels and cyclic bending tests at six stress levels. The results from ramp load tests showed that there was a slight increase in strength with increased rate of loading, however not in a significant manner. Also the thick MPB strand-based wood composite lumber showed good mechanical properties in terms of bending strength, compared to that of solid sawn lumber. The long-term constant load tests were conducted for a period of one year and at two stress levels (27 MPa and 33 MPa) which were 5-th percentile and 20-th 105 percentile of short-term strength respectively. Two steel frames were designed and manufactured to facilitate 60 specimens being tested simultaneously. The specimens were tested in bending on edge under third-point loading with a span of 660 mm. The time-dependent deflection was monitored and recorded with a data acquisition system at a pre-set frequency, and the times-tofailure were obtained for all broken specimens. At the 33 MPa level approximately 85% of the sample had failed within one year of loading (including the initial load application). At the lower 27 MPa level around 55% of the sample had failed within one year of loading (including the initial load application). It was concluded from the cumulative probability curve of time-tofailure that there was an upward departure from the initial trend occurring around three months. The brittle failure mode was observed during the test, and therefore it is especially important to have a good understanding of the development of deflection and probability of failure for this particular wood composite product. Finally low-cycle fatigue tests were carried out under triangular cyclic bending load. Six stress levels were chosen such that the peak bending stress within each cycle was approximately 65%, 75%, 85%, 90%, 105%, and 115% of the mean bending strength obtained from the short term bending test. Stronger specimens were tested at higher stress levels and weaker specimens at lower levels to get as much valid data as possible within the reasonable time frame. It was concluded that the number of cycles to failure decreased with increased stress ratio, and that cyclic load was more damaging than a constant load. A damage accumulation model was developed to study the creep-rupture behavior. This model stipulates that the rate of damage growth is given in terms of the current strain rate and the previously accumulated damage, and a 5-parameter rheological model (Pierce et al. 1985) is applied to describe the viscoelastic constitutive relationship to represent the time-dependent strain, while the damage accumulation law acts as the failure criterion. Therefore how strength is 106 affected by the passage of time, and how the deflection of structural members is influenced by time, is studied with a unified approach. The results of the long-term constant load tests were then interpreted by means of the creep-rupture model which had been shown to be able to represent the time-dependent deflection and time-to-failure data for different stress levels. The predictions of the model were verified using results from ramp load tests at different loading rates and results from cyclic loading tests at different stress levels. The importance of such a model lies in that it permits the extrapolation of test results for constant loads to predictions about the effect of random, cyclic loadings such as snow, traffic, or occupancy. The model also provides a description of the time-dependent deflection to present a complete picture from creep deflection to a final rupture event. In addition the viscoelastic parameters in creep-rupture model have been treated as correlated random variables for the first time. For the prototype product evaluated, this model can be extrapolated to predict the long term performance beyond the testing period. The development of a duration-of-load factor needs comprehensive data and extensive testing. This model can be used to study this aspect from a probabilistic approach, thus reducing the time needed for an extensive test program. This creep-rupture model incorporates the short term strength of the material, the load history and predicts the deflection history as well as the time-to-failure. As it is a probabilistic model, it allows its incorporation into a timereliability study of wood composites’ applications. 6.2 Further areas of research The damage accumulation model developed in this work includes the first order of damagedependent term and first-order of strain rate term; however a more generalized form proposed in Eq. (3.19) could be investigated in future research. 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Dept. of Civil Engineering, University of British Columbia. 115 Appendix I.1 Experimental Results and Model Simulation of Strain History under Constant Load - Level I (27 MPa) 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 Appendix I.2 Experimental Results and Model Simulation of Strain History under Constant Load - Level П (33 MPa) 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 Appendix II Experimental Results of Strain History under Cyclic Load 148 149 150 151 152 153 154 155 156 157 158 159 160 161 Appendix Ш Calibration Results of Creep Parameters 𝛽𝛽1 2.99·10-3 2.97·10-3 3.26·10-3 4.74·10-3 4.17·10-3 4.13·10-3 4.48·10-3 4.05·10-3 4.36·10-3 3.90·10-3 2.85·10-3 3.97·10-3 3.90·10-3 3.60·10-3 3.79·10-3 3.57·10-3 3.56·10-3 3.84·10-3 3.73·10-3 3.72·10-3 3.44·10-3 3.17·10-3 3.44·10-3 2.79·10-3 2.65·10-3 2.85·10-3 2.62·10-3 4.40·10-3 1.21·10-3 3.51·10-3 1.47·10-3 2.23·10-3 2.91·10-3 2.93·10-3 4.03·10-3 4.38·10-3 3.33·10-3 3.92·10-3 4.05·10-3 4.38·10-3 3.70·10-3 𝛽𝛽2 1.00·10-3 9.99·10-4 1.00·10-3 1.06·10-3 1.12·10-3 9.99·10-4 6.23·10-4 9.99·10-4 4.72·10-4 1.12·10-4 1.31·10-3 9.91·10-4 3.06·10-4 5.64·10-4 3.57·10-4 1.68·10-4 5.18·10-4 3.41·10-4 4.24·10-4 4.90·10-4 4.91·10-4 2.05·10-4 3.31·10-4 4.04·10-4 3.95·10-4 3.01·10-4 2.53·10-4 9.98·10-4 1.02·10-4 9.99·10-4 1.00·10-4 9.72·10-4 9.99·10-4 9.87·10-4 9.96·10-4 1.00·10-4 9.86·10-4 6.29·10-4 1.14·10-3 1.03·10-3 9.77·10-4 𝛽𝛽3 3.00·10-6 3.00·10-6 3.00·10-6 3.16·10-6 2.03·10-7 3.00·10-6 2.78·10-6 3.00·10-6 2.78·10-6 3.05·10-6 9.99·10-7 2.99·10-6 4.94·10-6 1.66·10-6 2.22·10-6 3.07·10-6 1.47·10-6 1.88·10-6 1.80·10-6 1.95·10-6 1.43·10-6 1.50·10-6 1.20·10-6 1.76·10-6 1.29·10-6 1.90·10-6 1.68·10-6 3.00·10-6 3.07·10-6 3.00·10-6 3.00·10-6 2.78·10-6 3.00·10-6 2.96·10-6 2.99·10-6 3.00·10-6 2.96·10-6 2.78·10-6 1.97·10-7 8.29·10-7 1.37·10-7 162 𝛽𝛽4 7.61·10-4 6.55·10-4 3.87·10-4 2.16·10-4 4.49·10-4 7.41·10-4 2.16·10-4 7.10·10-5 2.16·10-4 5.23·10-5 4.91·10-4 2.16·10-4 1.89·10-5 8.03·10-5 2.90·10-5 3.67·10-5 6.98·10-5 2.17·10-5 2.27·10-5 2.19·10-5 3.53·10-5 8.27·10-5 5.48·10-5 9.93·10-5 1.42·10-4 1.18·10-4 1.32·10-4 8.80·10-5 5.68·10-4 2.76·10-4 3.24·10-4 8.56·10-4 3.99·10-4 5.17·10-4 4.34·10-4 2.16·10-4 2.52·10-4 1.93·10-4 2.16·10-4 1.93·10-4 1.28·10-4 𝛽𝛽5 0.101 0.194 0.273 0.521 0.273 0.323 0.543 0.328 0.398 0.275 0.0922 0.405 0.314 0.199 0.261 0.252 0.2 0.266 0.267 0.263 0.231 0.187 0.206 0.197 0.132 0.143 0.134 0.301 0.61 0.273 0.652 0.174 0.188 0.169 0.224 0.538 0.218 0.531 0.408 0.633 0.344 𝛽𝛽1 1.46·10-3 2.88·10-3 2.56·10-3 2.75·10-3 3.23·10-3 2.88·10-3 2.63·10-3 𝛽𝛽2 5.93·10-4 7.07·10-4 2.88·10-4 7.93·10-4 4.75·10-4 8.14·10-5 5.61·10-4 𝛽𝛽3 8.28·10-8 5.22·10-7 1.20·10-7 2.92·10-7 7.01·10-6 2.68·10-6 1.27·10-6 163 𝛽𝛽4 5.46·10-4 1.70·10-4 1.67·10-4 1.24·10-4 2.16·10-4 2.85·10-4 1.79·10-4 𝛽𝛽5 0.102 0.156 0.168 0.158 0.412 0.246 0.109 Appendix IV Experimental Data of Short-term Strength and MOE for Control Group Test Sample MOE (GPa) 9.01 9.06 9.19 9.41 9.43 9.54 9.67 9.72 9.85 9.92 10.00 10.11 10.21 10.33 10.42 10.57 10.73 10.83 11.09 11.14 11.33 11.58 11.67 11.81 12.17 12.22 12.35 13.37 13.87 14.40 Short-term strength (MPa) 25.82 31.32 29.18 27.92 30.60 34.49 38.32 36.70 29.78 33.28 33.60 40.64 38.05 36.30 41.07 39.63 38.47 43.82 42.97 44.81 48.05 48.88 52.20 55.67 55.99 49.33 54.72 59.15 56.75 57.50 164