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Effect of proof loading on the strength of lumber Gyamfi, Charles Kumi 2002

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E F F E C T O F P R O O F L O A D I N G O N T H E S T R E N G T H O F L U M B E R by C H A R L E S K U M I G Y A M F I B.Sc, The University of Science and Technology, Kumasi, Ghana, 1991 M . S c , The University of Science and Technology, Kumasi, Ghana, 1995 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E STUDIES Faculty of Forestry (Department of Wood Science) We accept this thesis as conforming tojtye required standard T H E UNIVERSITY O F BRITISH C O L U M B I A January 2002 © Charles Kumi Gyamfi, 2002 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Faculty of Forestry Department of The University of British Columbia Vancouver, Canada 2 1 5*wy, 2 0 0 2 A B S T R A C T A Monte Carlo simulation program was used in this study to evaluate the performance of No.2 and better grade combination of 38 mm x 140 mm Western Hemlock lumber proof loaded in bending. Two proof load magnitudes of 19.54 MPa and 23.05 MPa representing the 2 n d and 5 t h percentile load levels of the short-term strength respectively were selected for the proof testing program to investigate the impact of proof loading. The rate of loading for all the testing programs was 2679 MPa/hr. The Canadian damage accumulation model was incorporated into the simulation program to calculate the probability and time-of-failure of every member as well as the strength of the survivors after testing at the two proof load magnitudes. Damage accumulated in a member was quantified based on the strength of the weak survivors. It was found that damage accumulated appears to be greater in the weaker survived members tested at the 2 n d percentile load level compare to the weaker survivors tested at the 5 t h percentile load level. Forty pieces of the 2000 members tested at the 2 n d percentile load level broke during proof loading whereas 101 pieces failed when the 2000 members were proof loaded at the 5 t h percentile load level. Plots of cumulative probability distributions indicated no significant difference between the original short-term strength and the strength of the survivors tested at both proof load levels. However, approximately 11% damage was accumulated in the weakest survived members proof loaded at the 2 n d percentile proof load level compared to a damage level of 1.2% experienced by the weakest survived member proof loaded at the 5 t h percentile load level. It appears damage did not accumulate in the survivors with bending strength values beyond 20.69 MPa and 24.48 MPa for the 2 n d and 5 t h percentile proof load levels respectively. Therefore, when a proof load magnitude of 23.05 MPa is selected to evaluate the performance of 38 mm x 140 mm Western Hemlock lumber in bending, more reliable survivors may be obtained compare to a proof load magnitude of 19.54 MPa. ii T A B L E O F C O N T E N T S Page Abstract ii Table of Contents iii List of Tables v List of Figures vi List of Notations vii Acknowledgements viii Dedication ix CHAPTER ONE INTRODUCTION 1 1.1 Background 1 1.2 Objective 2 CHAPTER T W O LITERATURE SURVEY 3 2.1 The Importance of Proof Testing/Loading 3 2.2 The Methods of Proof Testing 4 2.3 Reverse Proof loading 4 2.4 Time-to-Failure Modeling 5 2.5 Previous Damage Models 6 2.6 The Accumulation of Damage Model 6 2.6.1 Applications of The Canadian Damage Model . 8 2.6.2 The Damaged Viscoelastic Materials 8 2.7 Effect of Service Conditions 9 2.7.1 Temperature and Moisture Effects 9 2.7.2 Rate and Duration of Loading 10 2.8 Fatigue 11 iii CHAPTER T H R E E M E T H O D O L O G Y 13 3.1 Monte Carlo Simulation Program 13 3.2 Proof Load Levels 14 3.3 Proof Load History 15 3.4 Simulations of a Number of Replications 17 3.5 Probability of Failure 17 3.6 The Residual Strength of Survived Members 17 CHAPTER FOUR RESULTS AND DISCUSSION 18 4.1 The Residual Strength of the Survivors 18 4.2 The Incipient Damage of Weak Survived Members 22 4.3 The Reliability of Members 24 4.4 Optimum Number of Members for Better Simulation Results 25 CHAPTER FIVE CONCLUSIONS 26 5.0 Conclusions 26 LITERATURE CITED 27 APPENDIX A: Simulation (Proof Load) Program 30 APPENDIX B: Integration Procedure to Determine Time-to-failure 34 iv LIST O F T A B L E S Page 1. Damage Accumulation Models 6 2. Load Duration Factors to Check Design Values 11 3. Distribution Parameters for Simulation Program 13 v LIST O F FIGURES Page 1. Load and Resistance distribution during Proof Testing 3 2. Theoretical Effects of Proof loading on lumber strength 7 3. Loading Regime of Foschi et al. (1989) 15 4. Time to failure Cumulative Distribution from Experiment and Simulation 15 5. Proof load and time-to-failure History 16 6a. Cumulative probability distributions of bending strength for the original Specimens, broken pieces and the survivors after proof loading to the 2 n d percentile load magnitude 18 6b. Details of the lower 30% Cumulative probability distributions of bending Strength for the original specimens, broken pieces and the survivors after Proof loading to the 2 n d percentile load magnitude 19 7a. Cumulative probability distributions of bending strength for the original Specimens, broken pieces and the survivors after proof loading to the 5' percentile load magnitude 20 7b. Details of the lower 30% Cumulative probability distributions of Bending Strength for the original specimens, broken pieces and the survivors after Proof loading to the 5 t h percentile load magnitude 20 8. Damage accumulated based on 2 n d Percentile Proof Loading 22 9. Damage accumulated based on 5 t h Percentile Proof Loading 23 10. Residual strength distribution of the survivors after proof loading to the 2 n d and 5 t h percentile of the Short term Strength 24 11. Comparison of CDF's for four different replications 25 v i LIST O F NOTATIONS a Damage level 0"0 Threshold stress ratio xs Short-term strength T c Proof load magnitude T( t) Applied stress t Time t0 Time-to-failure during proof loading tc Time-to-failure at the end of proof loading /R Residual strength of survivors O Performance factor Pf Probability of failure T f Time-to-failure D n Nominal dead load Ln Nominal live load Kd Load duration factor K s Rate of loading R Resistance L Load PL1 Proof load level 1 or 2 n d percentile load level PL2 Proof load level 2 or 5 t h percentile load level vii A C K N O W L E D G E M E N T S I would like to express my profound gratitude to Dr. Frank Lam, my Supervisor for accepting me as his graduate student. His guidance, advice and criticisms as well as the financial support provided during my studies are highly appreciated. I would like to also record the invaluable assistance of Dr. Sarath Abayakoon, a Research Engineer in the Department of Wood Science who contributed to successfully run the simulation program of this study. The advice and suggestions of Dr. Dave Barrett and Dr. Helmut Prion, my supervisory committee members, who spent time to correct my draft thesis, are also appreciated. Thanks are due to George Lee, Winnie Louie and Bob Myronuk of the Wood Mechanics and Timber Engineering Laboratory, Department of Wood Science for their patience, sense of duty and work ethics during the laboratory testing exercises I did in the course of my program. To my colleagues Alex, Henry, Peggi, Alicia and Youhai Wang, and my friends Ross, Vinit and Aarti, I would like to say thanks for your encouragement. A number of awards from various sources were provided to support my graduate program at the University of British Columbia, for which I would like to record. The partial fellowship granted by the International Tropical Timber Organization (1TTO), Yokohama, Japan and the Ghana Timber Miller's Organization (GTMO) is greatly acknowledged. The International tuition fee scholarship provided by the Faculties of Forestry and Graduate Studies, the University of British Columbia is also highly appreciated. Lastly to my wife, Lily and daughter, Joycelyn, thank you for your prayers and love, and to my Lord Almighty God, Glory is yours. viii DEDICATION In memory of my Beloved Mother ix Chapter One 1.0 Introduction 1.1 Background Proof loading may be defined as a testing procedure to demonstrate the integrity of a manufactured product by subjecting the product to a specific load magnitude. This has been done in the past and very extensively in recent times to evaluate the strength properties of wood products. Although, the method can be destructive, it has been noted by Johnson (1980) as one of the most effective failure prediction techniques for structural wood members. Proof loading has been used in studies to obtain approximately the same ultimate strengths for duration of load experiments. For example, proof loading approximately to the ultimate strength of members reduced considerably the time required for duration of load studies. Researchers in wood engineering have designed structural systems using some of the newest products developed by industry. Due to these complex designs and construction, the social needs of North Americans and Europeans have improved, because of better housing facilities. The agencies responsible for developing and enforcing grading rules continue to work strictly requesting industries to produce quality products. In Canada for example, the National Lumber Grades Authority, N L G A (2001) prepared a standard recommending proof loading procedures to evaluate the tensile strength of every piece of structural finger-joined machine stress rated (MSR) lumber. It may be expensive for lumber mills to produce finger-joined MSR lumber because of the following two reasons; a target volume may take a longer time to produce, and secondly, weaker pieces may be destroyed or damaged during proof testing. The efficient use of wood products as structural members requires knowledge of their mechanical properties and the associated variability in these properties. Past and current studies directed toward understanding of basic failure criteria of wood members require this knowledge. 1 The application of the damage accumulation and fracture mechanics models involving numerical and experimental studies are critical requirements. For example, premature tension failure of wood joints is of great concern to the practicing structural engineering designers. In the furniture design and manufacturing industries, the design of members to be joined with connectors and/or adhesive is critical. Again, the need to develop in-plant quality control procedures for end-jointed structural members has been noted repeatedly in recent studies. Quality control standards can be improved in the areas of sampling techniques and the methods of product testing. The influence of proof loading on the strength of lumber was not done experimentally in this work because a large sample size is needed to estimate this relatively small effect. 1.2 Objective In the present study, the impact of proof loading on the bending strength of structural lumber was evaluated using the accumulation of damage model and a simulation program. 2 Chapter Two 2.0 Survey of Literature 2.1 The Importance of Proof Testing/Loading O'Connor and Shaw (2000) defined proof load as a load applied to a member or an existing system to determine or "prove" its load-bearing capacity. The load-bearing capacity of a bridge for example can be evaluated by proof loading. The results of this quality examination, enables a bridge to be rated and a weight limit imposed for vehicles usage. They reported that the most common proof loads are stationary and applied as a ramp load history. O'Connor and Shaw (2000) noted two important outcomes from any proof testing experiment: (i) The likelihood of failure during loading and (ii) Proof test may be used to reassess the probability of failure in a member or a system as indicated in Figure 1. For an assessment to be made after a program of proof testing, the initial selection of the magnitude of the proof load is critical. The estimates of resistance and strength of a member or a structural system enable the probability of failure to be determined since failure will occur for any value of the resistance, R, less than the load effect, L. Figure 1: Load and Resistance distribution during Proof Testing. (O'Connor and Shaw, 2000) 3 Figure 1 is the distribution of probability of failure (Pf) in a proof loaded member in service and the resistance offered by that member as described by O'Connor and Shaw (2000). The probability of failure (Pf) is estimated by Ang and Tang (1992) as in equation [1]. oo P f = P (R < L) = J FL(x)fR(x)dx [1] 0 2.2 The Methods of Proof Testing Two modes of loading have been widely used in proof testing experiments. These are bending and tensile proof load applications. The drawback of bending proof load application is that a survived member can develop compression damage. Strickler et al. (1970) studied proof loading structural end-jointed lumber and reported that a bending proof load of up to 90% of the expected strength do not significantly reduce the strength of end-joined lumber. End-joints can be loaded up to 70% of their expected ultimate strength in bending, (which is the maximum resistance that a member can offer to an applied load before breaking) without causing failure in a higher proportion of usable material. It was noted that a tension proof load offers the possibility to ensure higher tensile grades of lumber than will a bending proof load. Strickler et al. (1970) suggested that proof loading should be applied to a member in the same mode in which it will be used in service. 2.3 Reverse Proof loading The amount of damage due to proof loading is difficult to quantify as observed by many studies including the work by Marin and Woeste (1982). Based on reliability analysis, a 5 t h percentile proof load level was selected from the load distribution of No. 2 Southern Pine lumber tested to failure in bending on the edge. A reverse proof loading, which is the application of load to members in sequence causing compression, and then tension in the extreme fibers was used to 4 investigate the amount of damage. Marin and Woeste (1982) found out that damage was not significant at the 5 t h percentile proof load magnitude. Therefore, proof loading at a level lower than the 5 t h percentile load level may not induce significant damage to members. However, a 5 t h percentile proof load magnitude seems to have resulted in 27% reduction in the bending strength of Southern Pine lumber. Eby (1981) on the other hand noted that proof testing is not always destructive. It is the best quality control technique available to the manufacturer of many wood products including glue-laminated members. It permits the evaluation of every joint in the critical area of members. 2.4 Time-to-Failure Modeling For the interpretation of experimental results from proof loading programs, the development of a proper time-to-failure model is required. Time-to-failure models have been derived based on the concepts of damage accumulation, fracture mechanics and chemical kinetics models. In particular, the reliability of wood members using the probability approach has been useful to predict times to failure under ramp, dynamic and fatigue loading situations (Barrett, 1996). Table 1 presents six damage accumulation models developed over the years. The model of Foschi et al. (1989) was judged to be the most appropriate model in determining the accumulation of damage in a member because the other models have some limitations. 5 2.5 Previous Damage Models Table 1: Damage Accumulation Models Model Year Expression Limitation Wood 1951 da ( Y Does not depend on previous damage, a. Barrett and Foschi 1978 , da / Y c 2- ^ = a (T ( ( )—GO J + ca Gerhards 1979 da _[„_£,„,] ~dt~e Does not also include a threshold below which damage will not occur Gerhards and Link 1987 da [-fl+i)T(0/Ts] ~dt~e The model fits some but not all data well Foschi et al. 1989 ^ = a{r(t)-GoTs] + c(ju)-<JoTsjOc a 0 is the threshold stress ratio and xs is the short term strength. 2.6 The Accumulation of Damage Model The accumulation of damage in a member in service relates to the time-of-failure of that member. Damage accumulation modeling has proven to be an accurate approach to evaluate the impact of simulated loads and time histories on time to failure of wood members. The model of Foschi et al. (1989) is presented as shown in equation [2]. Let a be a damage state variable, which is a function of time t, normalized to take a value a = 0 in the initial state and a = l at failure. A damage accumulation model, which effectively describes the growth rate in a can be expressed by the differential equation [2]. The four model parameters b, c, n and c0 are assumed to be constants for a given member but vary randomly between members. With the short-term strength determined by ramp loading, the 6 parameter & is a function of the other parameters as shown in equation [3]. xs is the short term strength of the member which can be determined in a ramp load test of short duration with a constant rate of loading. a 0 is the threshold stress ratio. Whenever xt > a0x s damage starts to accumulate. There is no damage when xt < a0xs. When equation [2] is integrated for any load history xt, the results yield the damage a at any time t. The four independent damage model parameters b, c, n and o0 can be modeled as independent lognormally distributed variables. The short-term strength xs is also assumed to be lognormally distributed. a =[K s (b+l ) ] / [x s -o 0 T s ] b + 1 [3] Gerhards (1979) demonstrated using damage accumulation theory, the effects of proof loading on the strength of lumber. Figure 2 shows distributions of the static strength of lumber determined in a ramp loading case, and its residual strength after a proof testing program. 100 i — : 1 1 1 — . I 1 :—I 1 STRENGTH OO3 PSI) Figure 2: Theoretical Effects of Proof loading on lumber strength (Gerhards, 1979). 7 A 5 t n percentile load magnitude of approximately 10.18 MPa (1470 psi) was obtained from the lumber strength distribution with a minimum strength value of 4.57 MPa (663 psi). The rate of loading for the testing program was 0.003 MPa/sec (4.2 psi/sec). Each member of the population was subjected to a proofload level of 110% of the 5 t h percentile load in 10 seconds. Then, based on the cumulative damage theory, Gerhards (1979) observed that the distribution of the residual strength after proof testing does not overlap the original static strength distribution of lumber as indicated in Figure 2. The results of the study showed about 4.72% of the lumber population failed after proof loading. Of the survivors, a smaller number representing the weaker members had residual strength, T: 0<T<1350 psi, and approximately 0.5% of the population had residual strength between 1350 psi and 1470 psi. The rest of the survivors, about 94.7%, which represents the stronger members of the population, however had their residual strength exactly the same as their original strength. 2.6.1 Applications of The Canadian Damage Model Yao (1987) developed a damage model called the Canadian model based on the theories of probability of failure and reliability analyses. This model is useful in lumber property studies. A number of studies including those of Lau and Barrett (1998) and Norlin et al. (1999) have used the Canadian model to determine the probability of failure in lumber under varying loading conditions. 2.6.2 The Damaged Viscoelastic Materials The Damaged Viscoelastic Material (DVM) theory has been shown by Nielsen (1992) to predict residual strength of wood fairly well. From the field of fracture mechanics theory, wood was considered as a cracked viscoelastic material. Failure occurs in a member when the rate of 8 crack growth reaches an infinite value. In a recent study by Nielson (2000), the lifetime and residual strength of wood subjected to static and variable loads was predicted based on fracture mechanics. It was observed that load cycling reduces the strength and lifetime of visco-elastic materials, which requires consideration in the design of structures. The rate of damage in an elastic material such as wood subjected to varying load was determined by Nielson (2000) as follows: Assuming a crack or split exists in a member and depending on a load level and duration, the crack propagates at its tip. The damage rate was modeled based on the energy dissipation involved in the crack opening process, the distance traveled by the crack and duration of loading the member. Nielson was thus able to derive equations to determine the residual strength of wood. 2.7 Effect of Service Conditions 2.7.1 Temperature and Moisture Effects Lau and Barrett (1998) performed a reliability analysis on wood tension members and used it as a direct function of time to failure. This work focused on the structural behavior of light-frame wood members subjected to tensile loading at elevated temperature history. A model based on linear kinetic theory for strength as a function of temperature and stress was developed using the Canadian damage accumulation model and the pyrolytic process expressed as a form of damage. Two grades of 38mm x 89mm structural MSR lumber were tested in tension at three different rates of loading, and at 20, 150, 200 and 250°C. It was observed that lumber grade appears not to significantly affect the reliability of lumber to survive fire damage. The behavior of lumber under constant loading at 250°C was also predicted fairly well by their model. Galimard and Lasserre (2000) also studied the stress/strain characteristics of laminated veneer lumber (LVL) beams subjected to varying loads and climatic histories. A failure criterion based on strain energy was implemented after ramp testing of 50 L V L beams to failure. It was 9 reported that the damage parameter appears to be a good predictor of the duration of load effect in different climatic conditions. They suggested however that the damage parameter must be tested simultaneously for a wider range of climatic conditions and load applications. In their work, Galimard and Lasserre (2000) determined a global damage parameter in tension and compression failure modes in Winter. The value of the damage parameter was 1.06. This value varies for simulated stress levels for Spring and Autumn (l<oc<1.05), and the cause, could be due to the influence of moisture effects. 2.7.2 Rate and Duration of Loading The rate of loading of lumber in service or in a laboratory study influences its mechanical properties. Barrett and Lau (1994) noted that, wood products property test standards specify the rate of loading in all testing programs. Past and current studies have followed documented test standards such as those of the American Society for Testing and Materials (ASTM) and the British Standards Institution (BSI). The rate of loading of a member should be different for different testing procedures in which the load is applied to the member in tension, bending or compression parallel or perpendicular to the grain in a short-term (Karacabeyli and Barrett, 1993). In this way, tests results from laboratory testing programs may be compared. The duration of load effects on the strength properties of lumber have been examined in many studies including that of Gerhards and Link (1987) and recently by Gerhards (2000). Barrett (1996) also presented a comprehensive review of the past, present and future load duration effects research programs and investigations presently underway in different institutions of the world. The final research report by Gerhards (2000) presents results of an investigation on graded Douglas-fir 38mm x 89mm lumber subjected to constant bending loads of various magnitudes and durations. Gerhards (2000) recommended, based on prediction equations of this study and previous ones, that a safety factor of 2.0 for a 10-year load duration is more 10 appropriate for Douglas-fir bending allowable properties than the 1.62 factor presently utilized in design calculations. In Canada, the expression for checking the reliability of members under various load cases is of the form as shown in equation [4], and has duration of load factor term; 0>K(iR0.o5 = 1.25Dn + 1.5Ln [4] Where, R o . o 5 = is the 5 t h percentile of the specified strength <D is the performance factor D n and L„ are the nominal dead and live loads, and ICj is the duration of load factor Table 2: Load Duration Factors for various Duration of Loads to Check Design Values Load Duration K d Explanations Short-term 1.15 Specified loads duration <7 days (continuously or cumulatively), e.g., wind loading, earthquake, impact and form or false work. Standard term 1.00 Specified loads with duration between that of short-term and permanent cases, e.g., loading from snow; live loads from occupancies, wheel loads on bridges; dead loads in combination with the above. Permanent 0.65 Specified loads with duration more or less continuous, e.g., dead loads or dead loads plus live loads where the live loads will be continuously applied. Source: C A N / C S A 086.1-M89 Clause 4.3.1 Table 4.3.1.2 The duration of load factor Kd, (Table 2) for the three categories of load durations have different values specified for use in the design expression in equation [4]. 2.8 Fatigue Fatigue may be defined as a process of damage accumulation under cyclic loading. Norlin et al. (1999) studied the shear behavior of laminated Douglas fir veneer under cyclic 11 loading and observed three modes of failure after testing and visual inspection. Rolling shear specimens and longitudinal shear specimens were constructed based on orientation of cross-ply veneers. After calibration of the damage model developed by Foschi et al. (1989) to experimental and simulated data, the following conclusions were drawn: (i) The damage model provided a good prediction of the fatigue behavior of the two failure modes, (ii) The rolling shear mode exhibited higher fatigue resistance than the longitudinal shear mode. The mechanisms of failure in specimens of clear black spruce blocks subjected to forces under compression parallel to grain was also studied by Gong and Smith (2000). They noted that the damage accumulated due to creep and fatigue in spruce was different. In their creep tests, damage develops due to existing kinks formed during initial loading. The accumulation of damage during the fatigue tests was due to existing kinks formed during the initial load cycle, and newly formed kinks during load cycling. Damage was quantified based on the number of kinks formed during relative cyclic creep. According to Gong and Smith (2000), deformations that accumulated in their creep tests specimens were larger than those in the fatigue tests specimens. The practical implication of their work is that higher stressed creep tests using constant loads should not be used to predict the behavior of members under cyclic load conditions. 12 Chapter Three 3.0 Methodology 3.1 Monte Carlo Simulation Program A Monte Carlo simulation program (Appendix A) was modified to simulate the probability of failure and the corresponding time-to-failure of a random number of lumber specimens. The input data for the damage model for this study are given in Table 3. Since the parameters must be positive, lognormal distributions were selected to represent them. Barrett and Foschi (1982) determined these parameters for the short-term strength of No. 2 and better grade of 38mm x 140mm Western Hemlock dimension lumber in bending. These parameters represent the variables for the accumulation of damage model in equation [1]. Table 3: Distribution Parameters for Simulation Program Parameter Mean Standard Deviation 36.6566 3.78164 0.203312D-06 0.203823D-07 1.21218 0.11586 0.43639 0.05012 47.84 19.54 n o-o xs (MPa) K s (MPa/hr) DOL (hrs) 2679 43800 An expression for CI in terms of the other model parameters was obtained from the integration carried out in Appendix B and represented by equation [3] above. Similarly, integration was also used to estimate the time-to-failure (equation [5]) of every member in the calculation part of the program. Random numbers were then generated to compute the original 13 short-term strength of the members. The probability of failure (Pj) and the time-of-failure (Tf) were also determined using a pair of random seeds. -0 The residual strength, / R of survivors after proof loading was determined from equation [6] below. for T s > K s tc 3.2 Proof Load Levels Deciding on a load level to be used in any proof loading program is critical. A higher load level may lead to a greater amount of material failure during testing. On the other hand, a lower proof load magnitude may result in too few failures, thus preventing the establishment of a 5 t h percentile strength level. These were observed by Bechtel (1983) during the optimum load magnitude selection for proof testing studies. Two proof load levels were established based on the short-term strength distribution of the 2000 replicates. These load levels represent the 2 n d and 5 t h percentiles of the short-term strength distribution of Western Hemlock lumber. The figures were selected because design criteria base the allowable strength on the fifth percentile lower tail strength value due to safety reasons. Therefore, a 2 n d percentile load level also appears to be an appropriate value to evaluate the impact of proof loading on lumber strength. -[6] 14 3.3 Proof Load History Foschi et al. (1989) evaluated the duration of load effects of wood structures based on reliability analyses and incorporated the accumulation of damage model developed by Yao (1987). The model considered a particular load history as shown in Figure 3. CONSTANT LOAD J FAILURE 3 f TIME Figure 3: Loading Regime of Foschi et al. (1989) The cumulative probability distribution in Figure 4 illustrates the time-to-failure experimental and simulated results using the Canadian Damage Model derived by Foschi et al. (1989). The curves A and B are plots of experimental and simulated data from the load history shown in Figure 3. The model can be used to fit simulated data and to predict the time-to-failure of a member subjected to ramp and constant load. 3 ZD - D a m a g e M o d e l • S i m u l a t e d R a m p a n d C o n s t a n t l o a d s x S i m u l a t e d C o n s t a n t l o a d 3.00 -1.00 LOS T ( h o u r s ) 3.00 ' 4.00 8.00 Figure 4: Time-to-failure cumulative distribution from experiment and simulation (Foschi etal., 1989) 15 A stress magnitude of 19.54 MPa was determined as the proof load level, which corresponds to the 2 n d percentile lower tail of the short-term strength for the first ramp loading case. For the second simulation, a stress magnitude of 23.05 MPa was determined as the 5 t h percentile proof load level for the first ramp load. In both cases, the rate of proof loading was the same as the rate of ramp loading used to determine the short-term strength of the members. • T f t Figure 5: Proof load and time-to-failure history The load history for the present study is indicated in Figure 5. This is made up of two ramp load cases. For both proof load levels, the second ramp load was applied to the proof load survivors until complete failure of a member occurred. The rate of loading for the two ramp load cases were assumed to be the same and was K s= 2679 MPa/hr. For the first ramp load case, the time-of-failure, tc was calculated using equation [7]: t c =PL!/K s [7] Where PL) = 2 n d percentile proof load stress. When Tf < tc, the stress resulting in failure is determined by equation [8]: T s = t c.Ks [8] 16 and, whenever Tf > tc, the failure stress is determined by equation [9] T s = ( T f - t c ) K s [9] These procedures were repeated for the second proof load level, PL2. 3.4 Simulations of a Number of Replications To determine an optimum number of specimens to make good experimental deductions during proof loading, four different replications were simulated comprising 50, 100, 500 and 1000 members. The simulations were done at the 2 n d percentile proof load level to gather information on the number of pieces of Western Hemlock needed to make valid conclusions. 3.5 Probability of Failure The probability of failure of a member during proof load application was also simulated to gather data on the weaker members of the sample, which accumulated some damage. The probability of failure of each member is an indication of the reliability of the members, which is critical to select a particular grade of lumber for design. 3.6 The Residual Strength of Survived Members There is very little literature on how to quantify damage in a wood member. Many approaches have been used and evaluations carried out as cited in the literature survey. In this study, the residual strength of members who survived the proof load stress was evaluated. Damage was quantified based on the reduction of the short-term strength (STS) of the survivors as shown in equation [10]. Damage (%) = (Original STS - Residual Strength of Survivors) x 100 [10] (Original STS) 17 Chapter Four 4.0 Results and Discussion 4.1 The Residual Strength of Members that Survived Proof load Stress The residual strength of the survivors after proof testing is critical. The bending strength of the weaker members of the survivors was reduced. This is an indication that some damage accumulated in members during proof loading. Gerhards (1979) perceived damage in a wood member as the direct or linear accumulation of the loading duration. This relates to the time-to-failure of a member since a weaker piece will accumulate a higher magnitude of damage in a relatively shorter period during loading before failure. 1 -| 0.9 -0.8 -0.7 -abi) 0.6 -Pro 0.5 -01 > 0.4 -••a J3 0.3 -3 uin 0.2 -U 0.1 -0 -• 1 • I 1 / I / Short-term strength I / Residual strength 1 / 1 / Failed PL1 ' / - - - 19.54 MPa 20 40 60 80 100 120 140 160 Bending Strength (MPa) Figure 6a: Cumulative probability distributions of bending strength for the original specimens, the broken pieces and the survivors after proof loading to the 2 n d percentile load magnitude Figures 6a, 6b, 7a and 7b are the results of the proof loading program. Two proof load stress levels of 19.54 MPa and 23.05 MPa representing the second and fifth percentiles respectively of the short-term bending strength of 2000 pieces of 38mm x 140mm Western Hemlock were established. The cumulative probability distribution (CDF) graphs represent the original short-term bending strength values, the residual strength values of the survivors after 18 proof loading, and the failure stress of the weaker pieces of the 2000 members which were broken during proof loading. Bending Strength (MPa) Figure 6b: Details of the lower 30% cumulative probability distributions of bending strength for the original specimens, broken pieces and the survivors after proof loading to 2 n d percentile load magnitude The results indicate that a larger percentage of material failed in the members proof loaded to the 5 l h percentile load level compared to the members proof loaded to the 2 n d percentile load level. As many as 101 pieces representing 5.1% of those proof loaded to the 5 t h percentile level have no residual strength. This value is slightly higher than that determined by Gerhards (1979), who reported about 4.72% lumber failure in his work. Of the members proof loaded to the 2 n d percentile load level, 40 pieces representing 2% failed during proof loading. It appears the percentage of failure in a population and the amount of damage accumulated in a member have a direct relationship with the magnitude of proof load established as observed also by Bechtel (1983). The distributions indicated no detectable differences between the bending strength of the survivors and their original short-term bending strength values for both proof loads, except for 19 the weaker members of the survivors with bending strength below 20.50 MPa, which accumulated some magnitude of damage. Bending Strength (MPa) Figure 7a: Cumulative probability distributions of bending strength for the original Specimens, broken pieces and the survivors after proof loading to the 5 th percentile load magnitude Figure 7b: Details of the lower 30% cumulative probability distributions of bending strength for the original specimens, broken pieces and the survivors after proof loading to the 5 t h percentile load magnitude 20 The lower tail of the short-term strength distribution represents the members, which either were broken or weakened during proof loading. The survivors represent a greater percentage of members that did not break, although some had their short-term strength reduced due to the weakening effect of proof loading. The bending strength distribution of the survivors appears to be higher than that of their short-term strength. Figures 6b and 7b show details of the lower 30% cumulative probability distributions of the bending strength for the 2 n d and 5 t h percentile proof load magnitudes. For the same cumulative probability value, the bending strength of a weaker member of the survivors appears to have had an increased strength as shown by the lower tail of the curves. These differences in strength values may be considered significant because the 5 t h percentile strength value of the survivors, which is approximately 26.90 MPa indicates an increase of about 17% in strength over the original 5 t h percentile value which is 23.05 MPa as indicated in Figure 7b. The weaker survived members after proof loading to the 2 n d percentile load magnitude however also indicated a significant increase in strength of about 27% over the original short-term strength. These findings are important because the criteria may enable the wood industry select an appropriate load magnitude for any proof testing program with the aim of supplying good quality material for the design and construction of structural systems. 21 4.2 The Incipient Damage of Weak Survived Members Some damage was accumulated in the weaker pieces of the 2000 members tested at the simulated proof load levels, shown by Figures 8 and 9. The magnitude of damage appears to be insignificant to cause an adverse effect on wood structural systems that will be designed and installed using No. 2 and better grade of Western Hemlock. Generally, the damage accumulated in the members surviving the levels of proof loading reduced as the bending strength increased. 1 2 Residual Strength of Survivors (MPa) Figure 8: Damage accumulated based on the 2nd percentile proof loading Based on equation [10], the weakest survivor, which accumulated the largest damage at the 2 n d percentile proof load level had approximately 11% of its short term bending strength reduced. The 5 t h percentile proof load level on the other hand induced approximately 1.2% damage to the weakest survivor, also based on its short term bending strength. Beyond a strength of 20.50 MPa, it appears there was no damage accumulated in the members tested at the 2 n d percentile proof load level, whereas damage did not accumulate in members tested at the 5 th percentile proof load level with bending strength greater than 24.40 MPa. These values appear to be insignificant in specifying Western Hemlock lumber for design procedures where the 22 interest is in its bending strength. This result differs from the experimental findings by Marin and Woeste (1982), who reported damage levels up to 27% of the short-term bending strength of Southern Pine lumber. The use of a pure experimental approach to assess damage is difficult because of the need of large sample size. These observations confirm those made by Bechtel (1983) who stated that, an optimum load level is required to make the technique of proof testing economically viable. This could be based on the short-term strength distribution of members determined in a ramp load history. Therefore, proof loading can be a reliable quality control technique to eliminate weaker pieces of lumber from a population. 22 23 24 25 26 Residual Strength of Survivors (MPa) Figure 9: Damage accumulated based on the 5th percentile proof loading There was relatively, a consistent trend of magnitude of damage accumulated in members proof loaded at the 2 n d percentile level as shown in Figure 8. The trend of magnitude of damage accumulated in the weaker survived members proof loaded at the 5 t h percentile level was inconsistent as shown in Figure 9. Some weaker members accumulated less damage than other members of those tested which have relatively higher bending strength, but accumulated a greater magnitude of damage. This can be attributed to the relatively low percentage of damage found in the 5 t h percentile proof load case. 23 4.3 The Reliability of Members Statistical and computer applications to determine the likelihood of failure of wood materials and structures under given conditions have become the norm to design for all of these materials in recent times. Residual Strength (MPa) Figure 10: Residual strength distribution of the survivors after proof loading to the 2 n d and 5 t h percentile of the short-term strength Figure 10 provides the results of a direct comparison of the bending strength distributions for the simulation programs at the 2 n d and 5 t h percentiles proof load levels. The results indicated that a 5 t h percentile proof load survivor can be a more reliable member than a 2 n d percentile proof load survivor. For the same cumulative probability, a member that survived a 5 t h percentile load level was stronger compared to a 2 n d percentile proof loaded member. Again, a 5 t h percentile proof load survivor may have better bending strength, than a 2 n d percentile proof load survivor because from earlier results, less damage was accumulated in the weaker survivors of the former. Although as many as 101 members, representing 5.1% of the sample population tested were broken during proof loading at the 5 percentile load level, compared to the 40 which also represents 2% of the population broke at the 2 n d percentile level, the higher loss of material may, however, provide better and more reliable survivors for construction. 24 4.4 The Optimum Number of Members for Better Simulation Results To enable a better comparison of results of the proof loading program, 2000 members of 38mm x 140mm, No. 2 and better grade of Western Hemlock lumber were tested by the simulation program. Figure 11 also shows the results of a comparison for four sets of replicates represented by 50, 100, 500 and 1000 members tested at the 2 n d percentile proof load magnitude. 0 50 100 150 200 Bending Strength (MPa) Figure 11: Comparison of CDF's for four different replications The results indicate that more than 500 pieces of lumber are needed to produce valid conclusions. The cumulative distribution frequency of the bending strength for the 500 and 1000 pieces overlapped from the lower tail to the upper tail of the distribution curve. The data points for the 50 and 100 pieces did not overlap with those of the 500 and 1000 pieces, but showed differences in strength values particularly at the upper tail of the curve from the 80-100% mark. The results also indicate that to do this study experimentally for relatively better information requires a larger sample size. 25 Chapter five 5.0 Conclusions This work has revealed the significance of proof testing programs as a quality control technique to sort and specify bending members of lumber for the design of wood structures. The Canadian damage model was successfully used to compute the time-to-failure of every member in a sample, proof loaded to a predetermined magnitude. Damage was quantified based on the original short-term bending strength of 2000 members and the residual strength of the weaker survivors after proof loading to failure. A fifth percentile proof load magnitude resulted in more material breakage but lesser damage accumulation in the weaker survivors. Thus, the members which survived the 5 t h percentile proof load magnitude appears to be more reliable members than the survivors obtained after proof loading to the 2 n d percentile load magnitude. The following are the main conclusions of the study: 1. Proof loading lumber members can provide reliable and good quality material. 2. The magnitude of damage accumulated in survivors is not significant. The residual strength of proof loaded material compares favorably with non-proof loaded material particularly the stronger members. 3. This work can be carried out experimentally but will require a large sample size. This step should be performed to evaluate the current approach. 4. For further work, the method should be repeated using lumber members of different dimensions loaded in tension. 5. Further studies can also be done using a different lumber species and/or different load cycles up to four ramp load cases and/or a constant load history. 26 L I T E R A T U R E C I T E D 1. American Society for Testing and Materials, 2000. Annual Book of A S T M Standards; Section four. Construction, 4 (10): 25-55. 2. Ang, A. H-S and W.H. Tang. 1992. Probability Concepts in Engineering Planning and Design. Vol. U, Decision, Risk and Reliability. Pp 333-337. 3. Barrett, J.D. 1996. Duration of Load - The Past, Present and Future. International COST 508 Wood Mechanics Conference, Stuttgart, Germany. 4. , and R.O. Foschi. 1978. Duration of Load and Probability of Failure in Wood. Canadian Journal of Civil Engineering, 5(4):505-532. 5. , and W. Lau. 1994. Canadian Lumber Properties, Canadian Wood Council document, Ottawa, Canada. 6. Bechtel, F.K. 1983. Proof test Load Value Determination for Maximum Economic Return. Forest Products Journal, 33(10): 30-38. 7. Eby, R.E. 1981. Proof Loading Finger-joints for Glulam Timber. Forest Products Journal, 31(1):37-41. 8. Foschi, R.O. and J.D. Barrett. 1982. Load Duration Effects in Western Hemlock Lumber. Journal of Structural Engineering Division, ASCE. 108(7): 1494-1510. 9. , B.R. Folz and F.Z. Yao. 1989. Reliability-Based Design of Wood Structures. Structural Research Series, Report No. 34, Department of Civil Engineering., UBC, Vancouver, Canada. 10. Galimard, P.J and B. Lasserre. 2000. Failure Criteria for Duration of Modeling of Structural Beams. Proceedings of The World Conference on Timber Engineering, Whistler, Canada. 11. Gerhards, C C . 1979. Time-Related Effects of Loading on Wood Strength: A Linear Cumulative Damage Theory. Wood Science 11(3): 139-144. 12. , 2000. Bending Creep and Load Duration of Douglas-fir 2 By 4's Under Constant Load for Up to 12-Plus Years. Wood and Fiber Science, 32(4):489-501. 13. , and C.L. Link. 1987. A Cumulative Damage Model to Predict Load Duration Characteristics of Lumber. Wood and Fiber Science, 19(2): 147-164. 14. Gong, M . and I. Smith. 2000. Failure Mechanism of Softwood under Low-cycle Fatigue Load in Compression Parallel to Grain. Proceedings of the World Conference on Timber Engineering, Whistler, Canada. 15. Johnson, R.A. 1980. Current Statistical Methods for Estimating Lumber Properties by Proofloading. Forest Products Journal, 30(1): 14-22. 16. Karacabeyli, E. and J.D. Barrett. 1993. Rate of Loading Effects on Strength of Lumber. Forest Products Journal, 43(5):28-36. 17. Lau, P.W. and J.D. Barrett. 1998. Modeling the Reliability of Wood Tension Members Exposed to Elevated Temperatures. Wood and Fiber Science, 30(3):223-237. 18. Marin, L A . and F.E. Woeste. 1982. Reverse Proofloading of Lumber. Forest Products Journal, 32(10):53-55. 19. National Lumber Grades Authority. 2001. Special Products Standard for Finger-joined Machine Graded Lumber. N L G A Standard document SPS 4-00. 20. Nielson, L.F. 1992. The Theory of Wood as a Cracked Visco-elastic Material. In: Madsen, B. Structural Behavior of Timber. Timber Engineering Limited North Vancouver, Canada. 21. . 2000. Lifetime and Residual Strength of Wood Subjected to Static and Variable Load. Part I: Introduction and Analysis. Holz als Roh- und Werkstoff, 58(1-2): 81-90. 22. Norlin, L.P., C M . Norlin, and F. Lam. 1999. Shear Behaviour of Laminated Douglas fir Veneer. Wood Science and Technology, 33(3): 199-208. 28 23. O'Connor, C. and P.A. Shaw. 2000. Bridge Loads: An International Perspective. Spon Press, NY, USA. Pp. 158-163. 24. Strickler, M.D., R.F. Pellerin and J.W. Talbott. 1970. Experiments in Proof Loading Structural End-Jointed Lumber. Forest Products Journal, 20(2):29-35. 25. Yao, F.Z. 1987. Reliability of Structures with Load History-Dependent Strength and an Application to Wood Members. M.Sc. Thesis, Department of Civil Engineering, The University of British Columbia, Vancouver, Canada. 26. Wood, L.W. 1951. Relation of Strength of Wood to Duration of Load. Forest Products Laboratory, U.S. Dept. of Agric, Rep. #1916. 29 APPENDIX A PROGRAM PROOFLOAD implicit real*8 (a-h,o-z) implicit real*8 (m) C O M M O N /C3/ RATE, PI2, Y(4), ICODE DIMENSION MEAN(4),SD(4),IS1(10),IS2(10),TEX( 10,2000), * FF1(4),FF2(4) OPEN (UNIT= 1 ,FILE='DOL.DAT.TXT') OPEN (UNIT=2,FILE='DOL.OUT',STATUS='UNKNOWN') OPEN (UNIT=3,FJLE='D0L1 .OUT',STATUS='UNKNOWN') ICOUNT1=0 ICOUNT2=0 DO 901= 1,4 WRITE (*,70) 70 F O R M A T (' M E A N =') READ (1,*) MEAN(I) WRITE (*,80) 80 F O R M A T (' S.D =') READ(1,*)SD(I) 90 CONTINUE WRITE (*, 100) 100 F O R M A T (/' ENTER T H E SHORT-TERM STRENGTH :7) WRITE (*,70) READ (1,*) S T M WRITE (*,80) READ (1,*) STSD WRITE (*, 110) 110 F O R M A T (/' ENTER T H E SHORT-TERM RAMP R A T E : ') READ (1,*) R A T E WRITE (*, 120) 120 FORMAT (/' ENTER T H E STRESS L E V E L :') READ (1,*) T A 130 WRITE (*,140) 140 FORMAT (/' ENTER T H E NUMBER OF REPLICATIONS :') READ (1,*) NREPL WRITE (*, 150) 150 F O R M A T (/' ENTER T H E RUN TIMES :') READ (1,*) NTLME WRITE (*,160) NTLME 160 F O R M A T (/' ENTER ', 12,' PAIRS OF R A N D O M SEEDS :') DO 1801= 1, NTLME WRITE (*,170)I 170 F O R M A T (IX,' T H E PALR NO.', 12,': ') READ (1 *) IS 1(1), IS2(I) 180 CONTINUE WRITE (*,190) 190 F O R M A T (/' ENTER T H E DURATION OF L O A D (IN HOURS) 30 READ (1,*) D T M E PI2 = (4.0DO*DATAN(1.0DO)) ** 2 PI2 = 8.0D0 / PI2 STCOV = STSD / S T M F l = DSQRT(DLOG(1.0D0+STCOV**2)) F2 = S T M / (DSQRT(1.0D0+STCOV**2)) DO 200 I =1,4 T T C O V = SD(I) / MEAN(I) FF1(I) = DSQRT(DLOG(1.0D0+TTCOV**2)) FF2(I) = MEAN(I) / (DSQRT(1.0D0+TTCOV**2)) 200 CONTINUE 210 DO 240 K = l .NTTME DO 2301= 1, NREPL ZR = RANDN(IS 1(K),IS2(K)) ST = F2 * DEXP(F1*ZR) WRITE(2,320)ZR, ST 320 FORMAT( lX , F10.5, 2X.F12.5) DO 220 J= 1,4 ZZ = RANDN(IS 1(K),IS2(K)) Y(J) = FF2(J) * DEXP(FF1(J)*ZZ) 220 CONTINUE C A L L TFAIL(ST, TA, TFF) IF (TFF .EQ. - 1.0D0) TFF = DTHME TEX(K,I) = TFF 230 CONTINUE 240 CONTINUE C ** STARTS RANKING ** NREPL2 = NREPL - 1 DO 270 K = 1, NTBVIE DO 2601= 1, NREPL2 DO 250 J = I, NREPL IF (TEX(K,J) .GT. TEX(K,I)) GO TO 250 T E M P A = TEX(K,J) TEX(K,J) = TEX(K,I) TEX(K,I) = T E M P A 250 CONTINUE 260 CONTINUE 270 CONTINUE C ** STARTS CALCULATION OF CUMULATIVE PROBABILITY OF FAILURE DO 3001= 1, NREPL TFT = 0.0D0 DO 280 K = 1, NTLME TFT = TFT + TEX(K,I) 280 CONTINUE TFT = TFT / NTIME IF (TFT .GE. DTJJVIE) GO TO 310 PI = (I - 0.3) / (NREPL + 0.4) TF = DLOGIO(TFT) WRITE (3,290) TF, PI, ICODE 31 290 F O R M A T (IX, F10.5, 2X, F8.5,2X, 13) 300 CONTINUE C 310 CLOSE ( U N T T ^ S T A T U S ^ K E E P ) 310 STOP END C C ** UNIFORM R A N D O M NUMBER GENERATOR ** R E A L * 8 FUNCTION RAND(I1,I2) LF (12 .EQ. 0) THEN 11 =MOD(Il + 1,32768) 12 = 3 ELSE II =MOD(Il*3 +12,32768) IF (12 .LT. 0) II = II + 2 11 =11 + (12- 1)/21845 LF (II .LT. 0) II = II - 32768 12 = MOD(I2*3,65536) END IF RAND = 65539.0 * II / 2147484800.0 LF (RAND .LT. 0) RAND = RAND + 2 RETURN END C C ** STANDARD N O R M A L R A N D O M NUMBER GENERATOR ** FUNCTION RANDN(IZ1, IZ2) IMPLICIT REAL*8(A - H,0 - Y) R M E A N = 0.0 RDT = 1.0 LP = 0 10Z1 =RAND(IZ1,IZ2) LF (Zl .GT. 0.5) GO TO 20 GO TO 30 20 LP = 1 Z l = 1.0-Zl 30 LF (Zl .LE. 0.0) GO TO 10 T E M = SQRT(-2.0*ALOG(Z1)) A l = 2.51557 + T E M * (0.802853 + TEM*0.010328) A2 = 1.0 + T E M * (1.432788 + TEM*(0.189269 + TEM*1.308E-3)) V V = T E M - A l / A 2 LF (LP .EQ. 1) GO TO 40 GO TO 50 40 V V = - V V 50 RANDN = V V * RDT + R M E A N RETURN END C SUBROUTINE TFALL(ST, TA, TF) IMPLICIT REAL*8(A - H,0 - Z) C O M M O N /C3/ RATE, PI2, Y(4), ICODE 32 C ** D A M A G E A C C U M U L A T I O N M O D E L ** C C * FOUR VARIABLES (NV = 4): C * Y(l) = EXPONENT BEN FIRST T E R M OF D A M A G E L A W C * Y(2) = COEFFICIENT C FOR SECOND T E R M OF D A M A G E L A W C * Y(3) = EXPONENT N IN SECOND T E R M OF D A M A G E L A W C * Y(4) = THRESHOLD STRESS RATIO 10 IF (TA .LT. ST) GO TO 20 TF = ST / R A T E ICOUNT1 =ICOUNT 1+1 ICODE=l RETURN C 20 FA = T A - Y(4) * ST C IF (FA .GT. 0.0) GO TO 30 C TF = -1.0 C RETURN c 30 TF = T A / R A T E C FS = ST - Y(4) * ST C AO = (Y( 1) + 1.0D0) * DLOG(FA/FS) C A1 = DLOG(RATE) + DLOG(Y( 1) + 1.0D0) - DLOG(Y(2)) - (Y(3) + 1.0D0) C 1 * DLOG(FA) C A2 = AO + A l C A2 = DEXP(A2) C A l = DEXP(Al) C A2 = DLOG((1.0DO+A2)/(1.0DO+A1)) - AO C A2 = DLOG(A2) - DLOG(Y(2)) - Y(3) * DLOG(FA) C TF = TF + DEXP(A2) 20B1 = Y(l)+ 1.0D0 B2 = Y(4) * ST F A = T A - B2 FS = ST - B2 IF (FA.LT.0.0) F A = 0.0 TF = (FS **B1 -FA**B1)**(1.0D0/B1) + B2 + T A TF = T F / R A T E ICOUNT2=ICOUNT2+1 ICODE=2 RETURN END 33 APPENDIX B Ramp load case to determine short-term strength. r{t)=Kst a is accumulated only when i{t) = a0xs ^ = a Ht)-<J0TS f + c [ T (0 -<X0T,]> a dt = afl+cfza where / , = \v(t)-<T0r sf and / 2 = [ T ( 0 - < T O T , J* T, i.e. t>tn a e •jcf2dt 7} 'f -\chdt dt jcf2 dt = (n + l)K! -{Kst-<JoTs)n+l=0 Qc«Ks a(Tf)-a (ta )= j a /, dt = « t - c0 T J + 1 (KsTf-C70tsri=l a • {b + \)Ks _ {b + \)Ks a is not independent of b! Damage Law: = a f, +c /, a dt 1 2 where / , = [r(t)-a0rs J and f2 = [r(f)- a0rs]" and t(t)= Kst So integrating [1] [1] for t0<t<tc [2] 34 Now jcf2dt = jc [ T ( 0 - <T 0T, J" = \c {Kst - o0xs)" dt = j—^—(Kst - a0xs) Eq.2 becomes a e {K,,-a0xsr* ,(n+l)tf ; dt Now a (r c )- a (f0 )= ja (AT, t-<j0xsf dt «(0 •(K,t-O0Ttf [(Ks tc - cB % s r - (Ks ta - o0 x s r ] a be )= 7 — ^ — iK< K ~ <*o T < f V c (fr+i)*, Now what about the second load cycle; Again no damage accumulation until t>tc+t0 with residual damage of a (tc )=a (tc +10) So we have T 1 f -\cf2dt 1 f . -[cf2dt , ae 1 = \afxe ' dt Again \cf2dt = C (K, t — o~0 xs )n+l =0 0 c « Ks => a{Tf)-a{tc-t0)= \a{Kst-a0xsfdt {{Kg 7} -a0 xs -[Ks (tc -t0)]-a0 xs } \y-ri;ixs {KsTf-a0xs} (b + l)Ks 1 = (b + \)K a [(KsTf-a0xsri+(Kstc-o0xsr] 35 36 

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