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Stochastic finite element analysis of the load-carrying capacity of laminated wood beam-columns Folz, Bryan Russell 1997

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S T O C H A S T I C FINITE E L E M E N T ANALYSIS OF T H E L O A D - C A R R Y I N G C A P A C I T Y OF L A M I N A T E D W O O D B E A M - C O L U M N S by BRYAN RUSSELL FOLZ B.Sc, Simon Fraser University, 1982 M.A.Sc., The University of British Columbia, 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Civil Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February 1997 ©Bryan Russell Folz, 1997 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. Department of The University of British Columbia Vancouver, Canada Date /AAXC// /Zj /95>? DE-6 (2/88) Abstract A relatively simple, yet comprehensive, stochastic finite element model is presented herein for predicting the response variability in the load-carrying capacity of glued-laminated wood columns, beams and beam-columns. In this problem spatial stochasticity occurs in the elastic modulus and the compressive and tensile strengths of the lamination material. Central to this investigation is the calibration of the stochastic material models to actual test data. The finite element model is based on a one-dimensional higher order shear deforma-tion beam theory. Material behaviour is, in general, nonlinear; being perfectly brittle in tension and yielding in compression. Material failure occurs in a lamina or at an end-joint wherever the tensile strength is exceeded. The beam-column model employs a full nonlinear solution strategy, tracing the load-displacement response up to collapse which may result from progressive material failure and/or an overall loss of structural stabil-ity. Column analysis is formulated as an eigenvalue problem and yields the critical elastic buckling load. Strength and stiffness properties of the lamination material are modeled as one-dimensional homogeneous stochastic fields using a spectral approach. Realizations of each material property are simulated using a series representation of the stochastic field as a summation of sinusoids, each with a random phase, and weighted according to the spectral density function. The collapse load response statistics of a glued-laminated member are determined through a Monte Carlo simulation, which employs the stochastic finite element model. ii The stochastic modeling of the elastic modulus of the lamination material was char-acterized as a one-dimensional nonergodic process and its defining parameters were cal-ibrated against available test data. As an application, the response variability in the elastic buckling load of glued-laminated columns was investigated. It was concluded that this problem did not warrant stochastic modeling of the elastic modulus; a random variable approach was deemed sufficiently accurate. This was followed by a sensitivity study on the response variability in the collapse load of glued-laminated beams. In this investigation both the elastic modulus and the tensile strength of each lamina were modeled as one-dimensional correlated stochastic fields. The material models were not calibrated to experimental data. Rather, the defining parameters of the stochastic material models were varied to determine their influence on the response of the beam. It was observed that the specification of the stochastic tensile strength model was of primary importance in influencing the beam response, whereas the elastic modulus and its cross-correlation with the strength had only a marginal influence. Barrier crossing analysis was used to calibrate the defining parameters of the tensile strength process so that simulated strength profiles reproduced the cumulative distribu-tion of minimum tensile strength obtained from test specimens. This tensile strength model was utilized in the stochastic finite element beam model ULAG: Ultimate Load Analysis of Glulam. The collapse load predictions of ULAG were found to be in good agreement with recent test results on full-scale glued-laminated beams. ULAG was then applied to demonstrating and quantifying the statistical size effect in beam strength. As a final application, ULAG was combined with a reliability assessment procedure to optimize, at the manufacturing stage, the load-carrying capacity of a glued-laminated beam. in Table of Contents Abstract ii Table of Contents iv List of Tables vii List of Figures ix Acknowledgements xiii 1 Introduction 1 1.1 Preliminary Remarks 1 1.2 Scope of Present Work 6 2 A Glued-Laminated Wood Beam-Column Model 11 2.1 Introduction 11 2.2 Formulation of the Problem 13 2.2.1 Beam-Column Kinematics 14 2.2.2 Constitutive Model with Stochastic Material Properties 16 2.2.3 Nonlinear Equilibrium Equations 20 2.3 Finite Element Formulation 21 2.3.1 Solution of the Nonlinear Equilibrium Equations 23 2.3.2 Simplified Laminated Beam Model 28 2.3.3 Simplified Laminated Column Model 29 iv 2.4 Numerical Examples and Model Verification 30 2.4.1 Sinusoidal Loading of a Simply Supported Beam 30 2.4.2 Geometric Nonlinear Response of a Laminate 33 2.4.3 Ultimate Load Analysis 33 2.5 Concluding Remarks 35 3 E las t ic M o d u l u s of W o o d as a Stochastic Process 46 3.1 Introduction 46 3.2 Experimental Evaluation of the MOE of Wood 47 3.3 Simulating MOE Profiles 50 3.4 Accuracy of the Experimental Data 53 3.5 Response Variability in the Buckling Load of Columns 54 3.6 Concluding Remarks 57 4 S F E A — A Sensi t iv i ty S tudy 66 4.1 Introduction 66 4.2 Stochastic Material Modeling 67 4.3 Stochastic Finite Element Beam Model 70 4.4 Response Variability Analysis 71 4.5 Size Effect Study 74 4.6 Concluding Remarks 75 5 S F E A — Ca l ib ra t i on , Ver i f icat ion and App l i ca t ion 89 5.1 Introduction 89 5.2 Barrier Crossing Analysis of the Tensile Strength Process 90 5.3 U.B.C. Laminating Stock and Beam Test Program 93 5.4 Calibration ofthe Tensile Strength Process 96 v 5.5 ULAG Computer Program 98 5.6 Verification of the ULAG Computer Program 101 5.6.1 16 ft Beams 102 5.6.2 24 ft Beams 103 5.7 Size Effect Study using ULAG . . . 104 5.8 Reliability Analysis using ULAG and RELAN 106 5.9 Concluding Remarks 109 6 Conclus ions 131 6.1 Summary 131 6.2 Concluding Remarks 134 6.3 Further Areas of Research 135 Bib l iography 137 A Theory and Simulat ion of Stochast ic Fie lds 144 A.l Introduction 144 A.2 Spectral Representation of Stochastic Processes 145 A.2.1 One-dimensional univariate homogeneous stochastic processes . . 145 A.2.2 One-dimensional bivariate homogeneous stochastic processes . . . 148 A.3 Simulation of Stochastic Processes 151 A.4 Spectral Moments 154 A.5 Barrier Crossing Analysis 155 A.6 Probability Distribution of Minimum Values 157 A.6.1 Two-State Markov Crossing Assumption 158 B F in i te Element Shape Funct ions 162 vi List of Tables 1.1 Literature review - summary of problems analyzed and SFEM utilized. . 9 1.2 Literature review (continued) - description of stochastic material/geometry models 10 2.1 Finite element solutions for the maximum deflection of simply supported beams ( 0 - linear) 36 2.2 Finite element solutions for the maximum deflection of simply supported beams (ip ~ cubic) 36 2.3 Finite element solutions for the shear force and bending moments in simply supported beams 37 2.4 Analytical and finite element predictions for the ultimate load-carrying capacity of a simply supported beam 37 3.1 Spectral parameter values for G^K) 58 4.1 Material Model 3 tensile strength statistics 77 4.2 Load-carrying capacity statistics for the reference beam as a function of simulation size 77 4.3 Load-carrying capacity statistics obtained from the various material models. 78 4.4 Load-carrying capacity statistics for be ^ bs (Material Model 1, with 8$ = 0.20 and bs = 1.0 m) 79 4.5 Load-carrying capacity statistics using different spectral density functions (Material Model 2, with 6S = 0.20) 79 vii 4.6 MOR statistics for beams of varying length (Material Model 2) 80 4.7 MOR statistics for beams of varying depth (Material Model 2) 80 5.1 Summary of U.B.C. test program - lamination stock I l l 5.2 Calibration results for Grades B and C lamination stock I l l 5.3 Load-carrying capacity statistics from the 16 ft U.B.C. beam tests and the ULAG predictions 112 5.4 Load-carrying capacity statistics from the 24 ft U.B.C. beam tests and the ULAG predictions 112 5.5 Summary of a size effect study using ULAG 112 5.6 Dead and live load statistics 113 5.7 2-P Weibull load-carrying capacity statistics for 16 ft beams, varying the number of Grade B laminae, with no end-joints 113 5.8 2-P Weibull load-carrying capacity statistics for 16 ft beams, varying the number of Grade B laminae, with end-joints 113 5.9 Reliability analysis of 16 ft beams, varying the beam spacing and the number of Grade B laminae, with no end-joints 114 5.10 Reliability analysis of 16 ft beams, varying the beam spacing and the number of Grade B laminae, with end-joints 114 5.11 Allowable spacing for the 16 ft beams, ensuring a reliability level of (3 = 3.0.115 vm L i s t o f F i g u r e s 2.1 Typical glued-laminated beam-column layout 38 2.2 General state of deformation of a beam-column segment 38 2.3 Cumulative distribution of tensile and compressive strength of SPF test specimens (Buchanan 1984) 39 2.4 Experimental axial stress-strain curve from a defect-free wood specimen (Malhotra and Bazan 1980) 40 2.5 Idealized axial stress-strain behaviour of softwood 41 2.6 Normal stress distribution for beam under sinusoidal loading 42 2.7 Shear stress distribution for beam under sinusoidal loading 42 2.8 Asymmetric [9O4/O4] laminated beam configuration 43 2.9 Load-displacement response of an asymmetric [904/04] laminated beam. . 43 2.10 Load-displacement beam response to failure for varying Rs 44 2.11 Load-displacement beam response to failure for varying mg 44 2.12 Load-displacement beam-column response to failure for varying Rs- . . . 45 3.1 Schematic of a stiffness-based grading machine 58 3.2 Experimental J3n(x)-profiles 59 3.3 Ensemble statistics for E(x) 60 3.4 Cumulative distribution of specimen means E* and 2-P Weibull fit to data. 60 3.5 Ensemble statistics for e(x) 61 3.6 Ensemble spectral density GS(K) 61 ix 3.7 Simulated en(x) profiles 62 3.8 Cumulative distribution of and ag 63 3.9 Comparison between £n(x)-profiles and E„(x)-profiles 64 3.10 Cumulative distribution of and ae 65 3.11 COV in PCT for the laminated columns 65 4.1 Normalized spectral density function: g(K) = G(K.)/<72 81 4.2 Normalized Autocorrelation function: r(£) = R(£)/o~2 81 4.3 Progressive failure analysis to determine qu 82 4.4 Realization of tensile strength process (6s = 0.20 and bs = 0.25 m). . . . 83 4.5 Realization of tensile strength process (6s — 0.20 and bs — 0.50 m). . . . 83 4.6 Realization of tensile strength process (6$ — 0.20 and bs — 1.00 m). . . . 84 4.7 Realization of tensile strength process (6s = 0.20 and bs = 2.00 m). . . . 84 4.8 Realization of tensile strength process (6$ = 0.20 and bs — 4.00 m). . . . 85 4.9 Convergence of the stochastic finite element beam model 85 4.10 Distribution of load-carrying capacity qu (6s = 0.20 and i s = 1.0 m), . . 86 4.11 Response variability in qu 87 4.12 Response variability in 6qu 87 4.13 Size effect - mean MOR versus beam span ratio (6s = 0.20) 88 4.14 Size effect - mean MOR versus beam depth ratio (6S — 0.20) 88 5.1 Realization of tensile strength process showing Smin and strength barrier a. 116 5.2 Cumulative distribution of tensile strength obtained from barrier crossing analysis 116 5.3 Cumulative distribution of tensile strength of Grade B lamination stock. 117 5.4 Cumulative distribution of tensile strength of Grade C lamination stock. 117 5.5 Scatter plot of Grade B tensile strength versus modulus of elasticity. . . . 118 x 5.6 Scatter plot of Grade C tensile strength versus modulus of elasticity. . . . 118 5.7 Cumulative distribution of end-joint strength of Grade B and C lamination stock 119 5.8 Full-scale glued-laminated beam testing configurations 120 5.9 Spectral density functions for Grades B and C lamination stock 121 5.10 Realizations of Grade B tensile strength process 122 5.11 Realizations of Grade C tensile strength process 122 5.12 Cumulative distribution of simulated tensile strength of Grade B lamina-tion stock (L = 3.658 m) 123 5.13 Cumulative distribution of simulated tensile strength of Grade C lamina-tion stock (L = 3.658 m) 123 5.14 Assembly procedure for a glued-laminated member 124 5.15 Cumulative distribution of load-carrying capacity for the 16 ft beams -comparison of test results with ULAG predictions 125 5.16 First quartile cumulative distribution of load-carrying capacity for the 16 ft beams 125 5.17 Cumulative distribution of load-carrying capacity for the 16 ft beams - comparison of test results with ULAG predictions, assuming tensile strength to be a random variable 126 5.18 Cumulative distribution of load-carrying capacity for the 24 ft beams -comparison of test results with ULAG predictions 127 5.19 Cumulative distribution of load-carrying capacity for the 24 ft beams -comparison of test results with ULAG predictions, with no end-joint failures. 127 5.20 First quartile cumulative distribution of load-carrying capacity for the 24 ft beams 128 xi 5.21 Cumulative distribution of MOR for beams of various lengths (L0 — 4.725 m) 129 5.22 Cumulative distribution of MOR for beams of various depths (ha = 304rnm).129 5.23 Reliability analysis of 16 ft beams, varying the beam spacing and the number of Grade B laminae, with no end-joints 130 5.24 Reliability analysis of 16 ft beams, varying the beam spacing and the number of Grade B laminae, with end-joints 130 A. l Realization of a strength process showing the down-crossing of barrier a. 161 xn A c k n o w l e d g e m e n t s It is with deepest gratitude and appreciation that I acknowledge the support, direction and insights offered by Dr. Ricardo Foschi in overseeing this thesis to its completion. I would also like to express my thanks to Felix Yao for all his assistance, that was so cheerfully offered, whenever I called upon him. The financial support of Forestry Canada, the Natural Science and Engineering Re-search Council of Canada and the Canadian Wood Council under a cooperative research partnership is also gratefully acknowledged. The loving support and encouragement offered by my wife, Ruby, and my children, Matthew and Bryanne, have been constant and freely given. In closing, I dedicate this thesis in loving memory to my Mother who is ever present in my thoughts. X l l l Chapter 1 Introduction 1.1 Preliminary Remarks Uncertainty abounds in all problems of structural engineering analysis and design. The uncertainty may, on the one hand, be due to environmental influences which impose themselves on a structure; as for example the time-variant force resulting from sea waves buffeting an offshore drilling platform. Alternatively, or in combination with the latter, the uncertainty may be inherent in the structure itself; for example the spatial variation in the bending strength and stiffness of composite fiber-reinforced laminated panels which results from the manufacturing process. The necessity to quantify the uncertainty in an engineering analysis depends largely on the degree to which the randomness presents itself. In cases where the level of randomness is relatively small the analyst can be justified in modeling and evaluating the associated problems within a deterministic framework. For this class of problems the conventional finite element method has proven to be an almost indispensable analysis tool. State-of-the-art general purpose finite element com-puter codes have made the response evaluation of complex structures routine within the structural engineering profession. When the level of randomness manifesting itself in a problem is deemed significant, 1 Chapter 1. Introduction 2 a probabilistic approach provides a more realistic and rational framework for the analy-sis. Central to this approach is the quantitative treatment of all significant uncertainty inherent in a problem; this includes the identification of the sources of uncertainty, the construction of appropriate probabilistic models and their incorporation within a struc-tural analysis model. For proper application of this probabilistic approach to real-world problems supporting data is requisite for the development of the probabilistic models. Unfortunately this information is very often lacking, especially for the class of problems which involve spatially stochastic material properties. This remains an outstanding prob-lem which is widely acknowledged (Vanmarcke et al. 1986), but as yet, it seems, is not being addressed with any sense of urgency. This very issue is central to the presentation which is made in this study. Further to this, in times past, the added complexity required by a stochastic analysis often limited its application to overly simplified structural mod-els. It is only with the recent increase in the computational capabilities of computing facilities that the stochastic analysis of complex structural problems has become viable. Significant research activity is now focusing on integrating the finite element method with methods of stochastic mechanics to analyze structures which exhibit spatial stochas-ticity in their material properties and/or geometry. This synthesis of methodologies is appropriately termed the stochastic finite element method (SFEM). In a stochastic fi-nite element analysis (SFEA) excitation of a structure may be deterministic, random or stochastic in nature. The SFEM, as it has developed in its various forms, is utilized to achieve one, or both, of the following objectives: 1) to provide a probabilistic descrip-tion of a structure's response to a given excitation (this exercise is commonly termed response variability analysis); and/or 2) to assess the reliability of a structure against various performance criterion. Application of the SFEM can broadly be categorized as either simulation, perturbation or reliability based. A brief summary of these approaches follows, with due note made of the benefits and shortcomings involved with each method. Chapter 1. Introduction 3 In conjunction with this a literature review of pertinent research work is summarized in Table 1.1, with particular attention given to the type of structural problems that have been analyzed within a stochastic finite element framework. This literature review of the subject should be viewed as only representative, and not exhaustive, of the work undertaken to-date. Monte Carlo simulation is a numerical experiment which involves the repeated ap-plication of deterministic analyses. For each trial in the experiment the basic random variables or stochastic processes involved in the description of a problem are generated in accordance with their probability distributions. The response statistics, such as the mean, variance and exceedance probabilities, are then determined from the generated sample. As the number of replications becomes large these response quantities approach those of the parent population. Thus, a high degree of accuracy can be achieved with this method if one is willing to expend the computational effort. An important feature offered by Monte Carlo simulation is that it is readily adaptable to all types of structural analysis problems for which a finite element model is available. In fact, little modification of a deterministic finite element code is required in order to apply this method. This, along with the aforementioned control on the level of accuracy of the results, makes this approach very appealing. The main drawback to this method is its high computational overhead since fairly large samples are required to produce stable statistical results. This often cited negative aspect, however, is continually being weakened by advances in the performance of computers. As shown in Table 1.1, Monte Carlo simulation predates other SFEM, with work originating in the early 1970's. Since the application of this method is relatively straightforward research work employing this approach has generally focused on analyzing problems which reflect a fairly high degree of practicality and realism. This fact is apparent from examination of Table 1.1. The perturbation method involves a first- or second-order Taylor series expansion Chapter 1. Introduction 4 of the terms in the governing equations for a structure about the mean values of the basic random variables. By equating and solving equal-order terms in the expansion the second-moment response statistics can be obtained. In this method the basic random variables are characterized only by their first and second statistical moments; specification of the underlying probability distributions is not required. A limitation of this method is that it only provides acceptable results when the random fluctuations in the material and geometric properties are small (the upper limit on the coefficient of variation being of the order of 20%). Furthermore, since the expansion is mean-centered it is not generally suitable for the safety assessment of structures, which is sensitive to the tails of the probability distributions of the basic variables. As seen from Table 1.1, this method has been applied to a variety of linear and nonlinear static and dynamic problems. Whereas the perturbation method is principally concerned with second-moment anal-ysis of the response, reliability methods focus on evaluating the failure probability of a structure. With this method structural performance is expressed in terms of limit-state functions, which define surfaces separating the safe and failure sets in the space of the basic random variables. Exact evaluation of the failure probability requires, in general, a multidimensional integration of the joint density function of the basic random variables over the failure domain; in all but the most simple cases this becomes an intractable prob-lem. Thus, approximate methods have been devised. In particular, first and second order reliability methods (FORM/SORM) have found wide acceptance. These methods replace the multidimensional integration over the failure domain with a constrained optimization problem of finding the most likely failure point(s) on the limit-state surface(s). A de-tailed exposition of the underlying theory of this approach is given by Thoft-Christenson and Baker (1982) and Madsen et al. (1986). As noted from Table 1.1, Der Kiureghian and his collaborators are strong proponents of incorporating FORM/SORM procedures within a stochastic finite element framework. Chapter 1. Introduction 5 This method too is not without its limitations. Depending on the nature of the limit-state functions this method may encounter numerical stability problems in search of the most likely failure points. For example, if a limit-state involves the ultimate capacity of the structure, at which point the finite element model of the structure has a nearly singular stiffness matrix, the evaluation of the response gradient, required by the FORM algorithm, may not be able to be computed with sufficient accuracy to achieve a meaningful numerical solution. From this review of the literature it is observed that a broad range of structural problems have been formulated and examined within a stochastic finite element context. These are a testament to the vigorous research activity that has been occurring in this area. However, a critical examination of this literature, as summarized in Table 1.2, reveals that, in general, little or no attempt has been made to correlate the stochastic material models used in these studies with experimental data. This in part falls back on the paucity of appropriate data with which to formulate the stochastic material models. Undoubtedly, this has influenced the direction of the research work. Principally, the cited research has focused on the computational performance of SFEA procedures. As can be seen from Table 1.2 the structural problems studied, have in the greater majority of cases, merely been examples to demonstrate the efficiency, robustness, accuracy, etc. of the proposed solution strategies. As a result the stochastic material models were typically assumed, for computational convenience, to be homogeneous, isotropic and Gaussian. Correlated material properties were generally avoided. Often, the chosen structural analysis problem only involved one stochastic material property. With the studies that employed the Monte Carlo simulation method the stochastic properties were numerically generated using a spectral approach as developed and refined by Shinozuka (Shinozuka and Jan 1972; Shinozuka 1987; Yamazaki and Shinozuka 1988; Shinozuka and Deodatis 1991). This involves a series representation of the stochastic Chapter 1. Introduction 6 field by a summation of sinusoidal terms, each with a random phase, and weighted by the spectral density function. On the other hand, the studies which utilized perturbation and reliability based methods generally specified the stochastic material properties in terms of the auto-correlation function. For simplicity, and lacking motivation for an alternate model, the auto-correlation function was assumed, in all cases, to be an exponential function with a single parameter (since isotropy was invoked in all of the 2-dimensional analyses) to measure the rate of fluctuation of the random field. It is interesting to note that the majority of these studies did not undertake a sen-sitivity study to determine the influence of the stochastic parameters on the structure's response and/or reliability. Nor was the question addressed as to whether or not the cho-sen structural problem merited a stochastic analysis. Finally, it is observed from Table 1.2 that only one of the cited studies calibrated their stochastic material model to actual data (Wang and Foschi 1992). A wider acceptance of the SFEM by the structural engineering community requires evidence of its ability to solve problems of practical importance, which hitherto may not have been evaluated satisfactorily by other methodologies. To achieve this there is an obvious need to provide realistic applications of this methodology to structural problems that, based on supporting evidence, warrant a stochastic analysis. 1.2 Scope of Present Work In this study a comprehensive stochastic finite element model is developed to evaluate the response variability in, and reliability assessment of, the load-carrying capacity of glued-laminated wood beam-columns, when the material properties which govern the response are modeled as stochastic fields. Collapse of the beam-columns may be reached Chapter 1. Introduction 7 either under progressive material failure and/or an overall loss of structural stability. A theoretical formulation of this problem is presented in detail in Chapter 2. This includes the development of a general structural analysis model for the ultimate load analysis of laminated beams, columns and beam-columns. A simple one-dimensional finite element formulation is adopted. A series of examples, of a statistically deterministic nature, are then examined to evaluate the overall performance of this finite element model. In Chapter 3, using available data collected from a standard test procedure which grades lamination stock, the lengthwise variability in the modulus of elasticity (MOE) of wood is examined and characterized as a one-dimensional nonergodic stochastic field. A stochastic model for the MOE, employing a spectral approach, is developed and calibrated to the test data. As an application, Monte-Carlo simulation is used to evaluate the response variability in the elastic buckling load of laminated wood columns. In Chapter 4 a general SFEA of the load-carrying capacity of laminated beams, under progressive tensile fracture is presented. In this analysis both the elastic moduli and the tensile strengths of the laminae are modeled as one-dimensional cross-correlated stochastic fields. The material models are not calibrated to experimental data. Rather, in this initial investigation, the defining parameters of the stochastic material models are varied over a certain range in order to determine their influence on the response variability in the load-carrying capacity of the beam. It is observed from this sensitivity study that the modeling of the tensile strength as a stochastic field is of paramount importance in this problem, whereas the influence of the elastic modulus and its cross-correlation with the strength is of only secondary importance. In Chapter 5 barrier crossing analysis of stochastic field theory is used to calibrate the defining parameters of the tensile strength process of the lamination stock so that Chapter 1. Introduction 8 simulated strength profiles reproduce the distribution of minimum tensile strength ob-tained from test specimens. This tensile strength model is then incorporated within the stochastic finite element beam model ULAG (Ultimate Load Analysis of Glulam). The accuracy of ULAG's predictions are verified against recent test results on the load-carrying capacity of glued-laminated beams. ULAG is then applied to quantifying the statistical size effect in beam strength. Finally, the capability of the ULAG program is further demonstrated by optimizing the load-carrying capacity of a glued-laminated beam under a target reliability constraint. In Appendix A an overview of the basic theory of one-dimensional homogeneous stochastic fields is presented. The topics which are covered include: spectral repre-sentation of stochastic fields and their numerical simulation, spectral moments and the distribution of minimum values of a stochastic process over a given length using barrier-crossing analysis. The techniques which are presented in this appendix are used through-out this study to develop, calibrate and simulate the stochastic material models. The presentation made in this study is, for the most part, specific to glued-laminated wood beam-columns through the calibration of the stochastic material models to actual test data. However, it is believed that the methodology presented is general and can be applied equally to other structural composites, which display similar material behaviour, such as advanced fiber-reinforced composites. Table 1.1: Literature review - summary of problems analyzed and SFEM utilized. Reference Problem Analyzed SFEM Astill et al. 1972 Impact loading of concrete cylinders. S Shinozuka and Astill 1972 Vibration and buckling of a beam-column. S and P Shinozuka 1972 Statistical size effect and first-failure analysis of concrete beams. S Shinozuka and Lenoe 1976 Size effect phenomenon in ceramic plates. S Yamazaki et al. 1988 Tensile loading of a square plate. S and P Deodatis 1989 Dynamic response of a plate on a nonlinear foundation. S Deodatis et al. 1989 First-failure analysis of laminated plates in uniaxial tension. S Handa and Andersson 1981 Stress analysis of a wooden roof truss. P Hisada and Nakagiri 1985 Stress analysis of a plate under uniaxial tension. P Liu et al. 1986 Wave propagation in an elastic-plastic bar. P Liu et al. 1987 Transient analysis of a continuum with a circular hole. P Teigen et al. 1991 Failure analysis of a concrete portal frame. P Der Kiureghian 1985 Displacement and stress limit-states of a clamped beam. R Der Kiureghian and Ke 1988 Stress limit-state of a composite plate. R Liu and Der Kiureghian 1991 Displacement and stress limit states of a plate with a circular hole. R Wang and Foschi 1992 Displacement limit-state of a laminated wood beam. R Note: S = simulation, P = pei turbation and R = reliability based method. Table 1.2: Literature review (continued) - description of stochastic material/geometry models. Reference Stochastic Properties Stochastic Model Sensitivity Study Calibration to Data Astill et al. 1972 concrete strength and mass density 2D, H, C, G, SR no no Shinozuka and Astill 1972 elastic modulus and mass density area and moment of inertia ID, H, C, G, SR no no Shinozuka 1972 concrete strength 2D, H, G, SR no no Shinozuka and Lenoe 1976 uniaxial strength 2D, H, G, SR no no Yamazaki et al. 1988 elastic modulus 2D, H, G, SR no no Deodatis 1989 elastic modulus and foundation rigidity 2D, H, I, G, SR yes no Deodatis et al. 1989 strength (5 components) 2D, H, I, G, SR no no Handa and Andersson 1981 elastic modulus, area and moment of inertia ID, H, I, AC(e) no no Hisada and Nakagiri 1985 plate boundary ID, H, AC(e) no no Liu et al. 1986 yield stress ID, H, AC(e) no no Liu et al. 1987 elastic modulus 2D, H, AC(e) no no Teigen et al. 1991 moment capacity and geometric imperfections ID, H, I, AC(e) no no Der Kiureghian 1985 flexural rigidity ID, H, G, AC(e) yes no Der Kiureghian and Ke 1988 bulk and shear moduli 2D, H, G, AC(e) yes no Liu and Der Kiureghian 1991 elastic modulus and Possion's ratio 2D, H, nG, AC(e) yes no Wang and Foschi 1992 elastic modulus for each lamination ID, H, G, SR no yes Note: D = dimension, H = homogeneous, G = Gaussian (n = non-), I = independent, C = correlated, SR = spectral representation and AC(e) = autocorrelation structure (e= exponential type). Chapter 2 A Glued-Laminated Wood Beam-Column Model 2.1 Introduction Wood is a highly complex engineering material. It is, for example, anisotropic1, hetero-geneous, inelastic, rheologic, hygroscopic and biodegradable. Furthermore, because of its natural origin and organic composition it displays an unusually high degree of variability in its mechanical properties. It is not at all uncommon for full-size sawn lumber members to exhibit a between member coefficient of variation (COV) in bending strength and stiff-ness of 30% and 20%, respectively (Madsen and Buchanan 1986). This variability is not only observed between specimens but is also present within a specimen where strength and stiffness properties can fluctuate from point-to-point due to the presence of naturally occurring defects (knots, cracks, splits and checks) and fiber (grain) deviations. While in reality these defects introduce localized discontinuities in the material structure of wood, it remains advantageous to idealize the material simply as a heterogeneous continuum. It is then natural for the structural analyst to interpret the material properties of wood as stochastic fields. The versatility and performance of sawn lumber in structural applications can be greatly enhanced when it is used as the constituent material in a layered composite system 1 Wood is often idealized as an orthotropic material, denned in terms of its three anatomical directions: longitudinal (parallel to the grain), tangential (tangent to the annular growth rings) and radial (towards the pith). For simplicity it is also generally assumed that the growth rings are not cylindrical but planar. This permits a Cartesian coordinate system to be used to describe the material's orthotropy. 11 Chapter 2. A Glued-Laminated Wood Beam-Column Model 12 such as glued-laminated timber (glulam). This type of engineered system is comprised of a number of sawn lumber laminae of rectangular cross-section bonded together to form a laminated composite, as illustrated in Fig. 2.1. One of the obvious features of this structural form is that with the inclusion of end-joints within the lamina a member can be manufactured to any desired length; beams exceeding 15 meters are not at all uncommon. Furthermore, the lay-up of the lamina can be optimized so that the highest quality material is utilized throughout the member wherever the structural demand is the greatest; extremities of the cross-section, for example, in flexural applications. When contrasted with sawn lumber members this increase in performance is achieved under a reduced level of variability, which nonetheless is still significant; a recent test program of glued-laminated beams recorded, on average, a COV of 15% and 5%, respectively, in their bending strength and stiffness (Moody et al. 1990). The structural response of glued-laminated members up to failure is, however, more complex than with sawn lumber members. The ultimate load-carrying capacity of a glued-laminated beam-column is, in general, governed by several interdependent vari-ables: for example, the spatial variation in the strength and stiffness along and between the laminae (which, in turn, is related to the chosen lamination grades used in the beam-column lay-up), the strength and spatial distribution of end-joints, the size of the mem-ber, including the thickness of the laminae, and the load configuration. Because of their composite construction glued-laminated members generally have an ability to sustain load under progressive material failure through redistribution of the stresses. Damage accumulation up to the ultimate capacity of a member may not be confined to the lamina material alone (wood failure), as the end-joints (adhesive failure) are very often a weak link in the assembly. In order to formulate reliability-based design criteria and manufacturing guidelines which optimize the performance of glued-laminated beam-columns it is first necessary Chapter 2. A Glued-Laminated Wood Beam-Column Model 1 3 to have a clear understanding of their structural behaviour up to collapse and the con-tributing influence of the intervening variables. These objectives, obviously, cannot be achieved through experimental testing alone. To complement test programs, structural analysis models are required to facilitate a rational evaluation of this problem. In turn, these structural analysis models must reflect a degree of sophistication which is com-patible with the level of accuracy that can be achieved in the characterization of the stochastic material properties. In this chapter a nonlinear stochastic finite element model is presented for predicting the load-carrying capacity of glued-laminated beam-columns, with collapse resulting from progressive material failure and/or a general loss of structural stability. A simple one-dimensional finite element idealization is adopted, based on a higher order shear deformation beam theory. Material properties are assumed, in general, to vary spatially along each lamina, as well as between the laminae. The performance of the present model is then examined through a number of deterministic beam-column examples. 2.2 Formulation of the Problem At the outset, to simplify this problem, the rheologic and hygroscopic behaviour of wood is not taken into account. As well, it is assumed that the principal material axes of each lamina are coincident with those of the structural member. This allows the glued-laminated beam-column to be modeled as an unidirectional laminated composite. Fur-ther to this, since the longitudinal dimension of a lamina is generally at least an order of magnitude larger than its cross-sectional dimensions the stochastic material proper-ties within a lamina are modeled simply as one-dimensional stochastic fields along the longitudinal axis. Chapter 2. A Glued-Laminated Wood Beam-Column Model 14 2.2.1 Beam—Column Kinematics The type of laminated beam-column being considered is illustrated in Fig. 2.1. In the modeling of this type of structural member the following geometric, kinematic and load-ing restrictions have been imposed: the beam-column is prismatic, having a rectangular cross-section, initially straight, with perfectly bonded laminae (delamination is not con-sidered in this study), restrained against undergoing torsional and lateral deformations, and subject to monotonic loading confined to the x-z plane. The theoretical characterization of the kinematical response is based on a higher order shear deformation beam theory (HOSDBT), which, in a number of respects, is an attractive alternative to classical Bernoulli-Euler beam theory (BEBT) (Heyliger and Reddy 1988). Shear deformations become significant and should be taken into account when the span-to-depth ratio L/h of a bending member is small (say L/h < 10) and/or when the elastic modulus E in the axial direction is large relative to the transverse shear modulus G. With respect to the latter situation, E/G for softwoods, used as lamination stock in glued-laminated members, typically ranges between 10 and 20, as compared with 2.6 for steel. Figure 2.2 shows a portion of a beam-column under a general state of deformation, consistent with the imposed kinematic constraints noted above. A Total Lagrangian description is adopted with all quantities referred to a fixed Cartesian coordinate system having axes (Oxyz) or (0x-ix2x3). As shown in Fig 2.2, u(x) and w(x) denote, respectively, the axial and transverse displacements of the beam-column axis and 0(x) is the rotation of a transverse normal about this axis. With the HOSDBT the axial displacement of a point on the beam-column cross-section is assumed to be a complete cubic function of the thickness coordinate z, while the transverse displacement is taken to be constant Chapter 2. A Glued-Laminated Wood Beam-Column Model 15 through the thickness: UI(:E,2 / ,JZ) = u(x) + z^>(x) + z2((x) + z3((x) U2{x,y,z) = 0 u3(x,y,z) = w(x) (2.1) (2.2) (2-3) The functions ((x) and £(x) in Eq. (2.1) are determined from the required condition that the linear transverse shear strain vanishes on the top and bottom surface of the beam-column: ea3(x, y, -h/2) = e13(x, y, h/2) = 0 (2.4) Satisfaction of Eq. (2.4) results in the displacement field taking the form m(x,y,z) u 2 ( x , y , z ) u3(x,y,z) u + z 0 w (2-5) (2.6) (2.7) where the abbreviated notation of u = u(x), %p = tp(x) and w = w(x) has been adopted. Substitution of this displacement field into the nonlinear Green-Lagrange strain ten-sor 2 \dxj dxi dxi do yields two nonvanishing.components: the normal strain e = en and the transverse shear strain 7 = ei 3 , which, under the further assumption that e <C 1 and 7 C 1, are given by du dx dip 4 / 2 \ 2 / ^ d2wy dx 3 \hj V dx dx2 1 (dw + 2 [dx , 7 = U> + (2.9) (2.10) The limitation to small strains requires that the terms (du\/dx)2 and (dui/dz)2 are van-ishingly small, and hence do not appear in Eqs. (2.9) and (2.10), respectively. Under Chapter 2. A Glued-Laminated Wood Beam-Column Model 16 these imposed conditions geometric nonlinearity only arises in the normal strain compo-nent e. As formulated, Eq. (2.9) is a valid expression up to moderately large transverse deflections of the beam-column. Note that Eq. (2.10) correctly enforces a parabolic shear strain distribution over the depth of the cross-section. As a result this formulation eliminates the need for a shear correction coefficient which is required, for example, with Timoshenko beam theory (Cowper 1966). 2.2.2 Constitutive Model with Stochastic Material Properties The HOSDBT admits two nonzero stress components2: the normal stress a = crn and the transverse shear stress r = 013. Formulation of a constitutive model for wood, involving these two stress components requires, at a minimum, information on the response of specimens under the single stress states of uniaxial tension, compression and pure shear. Two different experimental approaches presently exist in North America to evaluate the required material properties of sawn lumber: on the one hand the basic properties can be obtained from small clear (defect-free) specimens, having cross-sectional dimensions of only a few centimeters (ASTM 1983), or, alternatively, tests can be performed on full size structural members (ASTM 1988). The results obtained will differ substantially. In the former case the specimens consist of essentially continuous parallel fibres and as a result pure wood properties are obtained from the tests. In the latter case the specimens have spatially distributed defects such as knots and slope of grain. These inherent defects will obviously influence the structural behaviour of the specimens as well as giving rise to a much higher variability in the material properties both between and within the specimens. This fact is made clear by considering Fig. 2.3 which shows the cumulative distri-butions for the tensile and compressive strength of 38 x 89 x 2000 mm spruce-pine-fir 2Under a Total Lagrangian formulation these stress components are of 2nd Piola-Kirchoff type. Chapter 2. A Glued-Laminated Wood Beam—Column Model 17 (SPF) specimens3 (Buchanan 1984). At the upper tail of each distribution the test records give the strength of nearly defect-free specimens, where it is observed that the tensile strength of the material exceeds the compressive strength. This is the same material behaviour which is obtained when testing small clear specimens. At the other end of the strength distributions the weak specimens, which have substantial strength reducing defects, exhibit the opposite behaviour with the specimens being stronger in compres-sion than in tension. This behaviour determines the dominant modes of failure under other forms of loading. For example, the failure of low quality material in bending will typically be initiated by brittle fracture in the tension zone of the beam. On the other hand, beams made from high quality material will generally exhibit ductile yielding over the compression zone before collapse. Also shown in Fig. 2.3 is an additional tensile strength distribution from specimens which were only 910 mm long. Comparing this data with the test results at 2000 mm gives evidence of a size (length) effect, whereby statistically a longer specimen is weaker than a shorter specimen when each is evaluated at the same probability level. Weibull brittle fracture theory (Bolotin 1969) has been adopted by many researchers to explain this phenomenon in wood (eg. Madsen and Buchanan 1986). The presence of a size effect can also be viewed as a natural consequence of the material strength being a stochastic process (Shinozuka 1972; Shinozuka and Lenoe 1976). It is this interpretation which is adopted in this study. A typical experimentally obtained stress-strain relationship from defect-free wood specimens in uniaxial tension and compression is reproduced in Fig. 2.4 (Malhotra and Bazan 1980). The same general trend in the stress-strain relationship has been obtained from full size specimens (Glos 1978). It is observed from Fig. 2.4 that the material 3The distributions shown in Fig. 2.3 are 3-P Weibull fits to the test data. The tensile strength has a mean of 30.7 MPa and a COV of 32.4%, while the compressive strength has a mean of 31.8 MPa and a COV of only 15.1%. Chapter 2. A Glued-Laminated Wood Beam-Column Model 18 response in tension is essentially linear elastic up to the point of brittle fracture. In compression the response is linear elastic to a proportional limit that is approximately 80% of the maximum compressive strength which is reached under ductile yielding of the specimen. Thereafter, under displacement control, the specimen undergoes strain soften-ing until failure at an ultimate compressive strain eCu of approximately 5%. A reasonably accurate idealization of this material behaviour is shown in Fig. 2.5. The primary pa-rameters required to define this stress-strain relationship are the tensile strength St, the compressive strength Sc, the elastic modulus E and the tangent modulus Ex = m^E. Secondary parameters are the strains et and ec which are, respectively, associated with the maximum obtainable tensile and compressive stresses. The material behaviour shown in Fig. 2.5 can be expressed mathematically as Ee + (E - ET)(ec - e)[l - h(ec - e)] if e < et 0 otherwise (2.11) where h(ec — e) is the Heaviside (unit step) function: . 0 if e < ec h(e e-e) = { (2.12) 1 if e > ec It is noted that there exists an ultimate compressive strain £ C u above which failure will occur; typically eCu is of the order of 5%. However, it is assumed herein that in flexu-ral applications a member will fail by progressive tensile fracture before eCu is reached anywhere in the member. Although readily defined, a state of pure shear in wood is extremely difficult to obtain experimentally in test specimens of structural size. Further to this, it is difficult to initiate a shear failure in these specimens as other modes of failure often dominate. Consequently, in this study, the material behaviour of a specimen in pure shear is simply assumed to be linear elastic, with shear rigidity G, up to a shear strength Ss. Chapter 2. A Glued-Laminated Wood Beam-Column Model 19 To this point the constitutive model requires the specification of 6 parameters: 3 material stiffnesses (E, Ej and G) and 3 strength values (St, Sc and Ss). To model each of these material properties as a stochastic field would be a daunting task; obviously some simplifications are necessary. For a lamina under a general state of loading, consistent with the constraints imposed earlier, the onset of material failure is expected to involve the interaction of the stress components a and T. Numerous strength criteria have been proposed to predict failure in anisotropic materials (Rowlands 1985). In particular the Tsai-Wu criterion, for which many of the other criteria can be expressed as a special or restricted case, has been used previously to evaluate the strength of wood (Liu 1984; Hasebe and Usuki 1989). For the special case of only two nonvanishing stress components a and T the Tsai-Wu strength criterion takes the form which defines a failure ellipse in the <J-T stress plane. For the particular problem under consideration this approach to failure analysis has a number of shortcomings. First, Eq. mode of failure or how material failure will progress within the structural element under redistribution of the stresses. Further to this, since Eq. (2.13) is being applied to a material for which the properties are stochastic one would expect that, in general, the strength parameters St, Sc and Ss would be statistically correlated to some degree. The exact nature of the cross-correlation between these three strength components would, undoubtedly, be very difficult to quantify. To make the stochastic material modeling tractable, it is assumed that failure is principally governed by brittle fracture in tension, and evaluated using the noninteractive (2.13) (2.13) only establishes the onset of failure at a point; it gives no indication as to the Chapter 2. A Glued—Laminated Wood Beam-Column Model 20 maximum stress failure criterion, which takes the simple form (2.14) where St = St(x) is assumed to be an one-dimensional homogeneous stochastic field. Tensile fracture, as indicated by Eq. (2.14), causes the formation of localized transverse cracks. In turn, post-failure material behaviour, as depicted in Fig. 2.5, results in the normal stress a and the elastic modulus E being reduced to zero at the crack locations. This behaviour represents the extreme case of a perfectly brittle material in which no tension softening occurs. It is further assumed that the shear rigidity G is unaltered by the tensile fracture. Note also that this failure criterion can be applied equally to evaluate the performance of the end-joints. To more fully capture the ability of the material to redistribute the stresses at dam-aged locations within a lamina the elastic modulus is modeled, in general, as a one-dimensional stochastic field E = E(x). To model the onset of ductile yielding throughout the member, the compression strength Sc — Sc(x) is assumed, in general, to be a one-dimensional stochastic field. 2.2.3 Nonlinear Equilibrium Equations The governing equations of motion for the laminated beam-column can be obtained via the Principle of Virtual Work: external work Wj and WE- The internal virtual work in Eq. (2.15) takes the form SWi + SWE = 0 (2.15) where <5(-) denotes the variational operator which is applied to both the internal and o (2.16) Chapter 2. A Glued-Laminated Wood Beam-Column Model 21 where in this application the strain and stress tensors reduce to the following 2-component vectors: e = [e,7]' <T = [*,T]* (2.17) with the superscript t denoting the transpose of a vector/matrix quantity. Consistent with a Total Lagrangian formulation the integration in Eq. (2.16) is performed over the undeformed volume V0 of the beam-column. In this beam-column model the most general load case to be considered involves a two parameter system consisting of independently applied axial and transverse loads. For example, for the beam-column shown in Fig. 2.1 the external virtual work is given by 6WE = -XpPo8u(L) - r Xqq08wdx (2.18) where p0 and q0 are, respectively, reference values for the axial end-load and the dis-tributed transverse load and A p and Xq the corresponding load factors. 2.3 Finite Element Formulation A displacement-based finite element formulation is adopted for the proposed glued-laminated beam-column structural analysis model. Spatial discretization is carried out using one-dimensional beam-column elements. To simplify the presentation herein the finite element equilibrium equations are derived at the element level. The variational statement, presented in the previous section, for the beam-column requires that the transverse displacement w be twice differentiable and Ci-continuous, wheras the axial displacement u and rotation t/> need only be once differentiable and C 0-continuous. Consequently, a conforming finite element is obtained using two nodes Chapter 2. A Glued-Laminated Wood Beam-Column Model 22 and employing linear (Lagrange) interpolation for u and ip and cubic (Hermitian) inter-polation for w. The resulting generalized nodal displacements are d = [ l l i ,^i , t i ; i ,0 i ,U2,02,U>2,02]* (2.19) where 0t- = dwi/dx for i = 1,2. The displacement field within an element is then given by u = [u^w]* = Nd (2.20) where N is the shape function array, which has been explicitly evaluated in Appendix B. In turn, the strain-displacement relationship within the finite element can be ex-pressed as e = B 0 d - f i B Z / d (2.21) The decomposition used in Eq. (2.21) is such that the matrix B D is independent of d while B^ depends linearly on d: B o with B 01 B o i B02 B1 +z B, BL1 0 (2.22) B02 = ( B 2 + B 4 ) BLI = d * B 4 B 4 (2.23) (2.24) (2.25) where the vectors B, (i — 1,..., 5) contain the appropriate derivatives of the interpolating shape functions for the element. The evaluation of these strain-displacement vectors are presented in Appendix B. The first variation of the strain vector in Eq. (2.21) is given by 6e = B Sd = [B0 + BL] 6d (2.26) Chapter 2. A Glued-Laminated Wood Beam-Column Model 23 Within this finite element context, the internal virtual work expression of Eq. (2.16) becomes 6Wi = I Se'adV = Sdf f BfadV = <5d*p (2.27) where p respresents the internal force vector and Ve the element volume. The external virtual work is given by SWE = -Sd'f (2.28) where f is the consistent load vector arising from the applied surface tractions acting on the element. Equating the internal and external virtual work, and noting that the virtual nodal displacements are arbitrary, results in the following nonlinear equilibrium equations being established at the element level: ip(d)= f B*trdv-f = p - f = 0 (2.29) J ve The global equations of equilibrium are obtained, in turn, by application of the direct stiffness method (Bathe 1982): !P(D) = P - F = 0 (2.30) where the notation used is obvious. 2.3.1 Solution of the Nonlinear Equilibrium Equations A numerical prediction of the entire load-deformation response of a laminated beam-column under progressive material failure necessitates the use of an incremental-iterative solution strategy. An outline of the proposed solution scheme follows. Assume that at a particular load level in the analysis that the nonlinear equilibrium equations are not satisfied for a given estimate of the global nodal displacements lnD; here the subscript n identifies the current load increment and the superscript i is the iteration Chapter 2. A Glued-Laminated Wood Beam-Column Model 24 counter within the load increment. Consequently, the global residual force vector is nonzero: LlP' = ^(^D) ^ 0. A better approximation of the equilibrium state can be achieved through a linearized Taylor series expansion of the global residual force vector about ' D: In Eq. (2.31) #U+ 1D) = O ^ D + AD) « #G,D) + dW _ DP _ <9D ~ SD ~~ T • A D = 0 (2.31) (2.32) where K j is the tangent stiffness matrix. Thus, the iterative correction to achieve equi-librium is given by A D = •> (2.33) or in terms of the total displacements as l D = ' D - ' K ; 1 ' ! ? (2.34) where it is assumed that ^K^ is nonsingular. Equation (2.34) defines the iterative Newton-Raphson method. The tangent stiffness matrix, at the element level takes the form , _dP N L dd = 6 E jf[J*'** [B%,3 + B 4 B 4 c r ] dx} dz (2.35) where iac is the stress-strain Jacobian matrix, Zj is the ^-coordinate of the lower face of the j-th lamina and is the x-coordinate of the first node of the i-th element. The depthwise integration in Eq. (2.35). is carried out over all NL laminae through the beam-column cross-section. The explicit form of J a £ is given by der de da de 0 dr . de dr 9~y . 0 G (2.36) Chapter 2. A GIued-Laminated Wood Beam-Column Model 25 with da de E-(E-ET)[l -h(e c -e)] if e < et (2.37) 0 otherwise as obtained from Eq. (2.11). The element tangent stiffness matrix of Eq. (2.35) is a function of the elastic moduli of the laminae through Eqs. (2.11) and (2.37). As noted previously, the elastic modulus along a lamina is to be modeled as a one-dimensional stochastic field E(x). To simplify the evaluation of the element tangent stiffness matrix it is convenient to assign a constant value to the elastic modulus for each lamina. Various schemes have been pro-posed for the discretization of stochastic fields within a finite element formulation (Liu and Der Kiureghian 1989; Li 1993). Obviously, the simplest approach is to assign the elastic modulus the corresponding value obtained from sampling the stochastic field at the mid-point along the element. This can be expressed as where Eij is the value to be used in the finite element analysis of the j-th lamina of the i-th element and Ej(x) is stochastic elastic modulus for the j-th lamina. Another, often adopted approach, is to use the local average of the stochastic field over the element as the assigned elastic modulus value (Vanmarcke and Grigoriu 1983): In both cases the resulting discretization of the stochastic elastic modulus along a lam-ina is a stepwise function, which is discontinuous at the element boundaries. The two approaches will agree under continued refinement of the finite element mesh. The tangent stiffness matrix in Eq. (2.35) is evaluated numerically using a combi-nation of Gauss and Lobatto quadrature. A Lobatto integration scheme is used in the (2.38) (2.39) Chapter 2. A Glued-Laminated Wood Beam-Column Model 26 spanwise direction so that the stress field can be evaluated along the interface between elements. Recall that n Gauss points exactly integrates a polynomial of order (2n — 1), where as only a polynomial of order (2n — 3) is integrated exactly with n Lobatto points (Burgoyne and Crisfield 1990). Thus, under the condition of linear elastic response, Eq. (2.35) is evaluated exactly using an integration grid consisting of 6 spanwise Lobatto points and 4 depthwise Gauss points per lamina. With constructed, Eq. (2.34) can be solved for ^ + 1D at the current load level. In turn, the stress field in the beam-column is updated over all of the integration points. This is followed by a check for material damage (tensile fracture of the lamination material and/or end-joints). If the average normal stress in a lamina of an element exceeds the tensile strength, evaluated at the midpoint along the element, then the elastic modulus and normal stress are set to zero within the lamina. In flexural applications, for example, this checking procedure implies that the outer fibres of the lamina can sustain a tensile stress in excess of the tensile strength without failing. This overestimation of the strength is permitted to compensate for the lamination effect4 (Foschi and Barrett 1980). End-joints are also checked for tensile fracture. In this model the end-joints are forced to be located only at element nodes. If the normal stress, averaged over the lamina thickness, exceeds the tensile strength of the end-joint then the elastic modulus and normal stress is set to zero within the lamina over the two elements which the node has in common. The accumulated loss in structural capacity resulting from localized material damage is reflected in the next update of the tangent stiffness matrix and residual force vector. Within a given load step the iterative process of Eq. (2.34) is terminated when the 4 Laminated construction assists in arresting the propagation of damage within the constituent lami-nae; for example, a strength determining edge crack which exists in one lamina may be suppressed from opening because it is bonded to an adjacent lamina. This phenomenon is distinct from the mechanism of load-sharing that exists in laminated construction. Chapter 2. A Glued-Laminated Wood Beam-Column Model 27 following displacement-based convergence criterion is satisfied I1 + 1 DI < eT (2.40) where | |(-)| |2 denotes the Eucildean norm, D is the vector of nodal displacements in which each component has been normalized and tj is a specified tolerance. The generalized displacement vector D has components of two types: displacements (UJ and Wj) and rotations (ipj and 9j). To make the Euclidean norm consistent each component of must be nondimensionalized by normalizing it with respect to the largest component of its type, which occurs during the i-th iteration of the n-th load step (Bergan and Clough 1972). To trace the complete equilibrium path (load-deflection curve) up to the collapse state for the beam-column a sequence of load steps must be specified. For simplicity of presentation consider the case when the beam-column is loaded proportionally, so that at the n-th load level n F = A„Fo where A n is the current load factor applied to the reference global load vector Fo = i F . Let the sequence of load parameters A l 5 A 2 , . . . , Afc_!, A*.,... , Ajv be specified in advance. The collapse load A C F 0 is of course not known a priori, but is less than A;vF0, a known upper bound. Say the beam-column fails at the load level corresponding to A ,^ as indicated by the tangent stiffness matrix becoming singular. Then an improved estimate to the collapse load can be established by applying a bisection search over the half open interval [\k-i,^k) until Ac is obtained to a desired level of accuracy. A judicious choice in the selection of the sequence of loading parameters will, of course, greatly reduce the computational effort required to obtain a solution. In this regard assistance may be obtained by utilizing an indicator function, such as the current stiffness parameter (Bergan 1978), to characterize the degrading of the tangent stiffness matrix under progressive material damage and adjust the load parameter accordingly so that changes a predetermined constant amount within each Chapter 2. A Glued-Laminated Wood Beam-Column Model 28 load step. This procedure, however, has not been implemented in this model. 2.3.2 Simplified Laminated Beam Model In the absence of applied axial loads collapse of a glued-laminated beam will generally result from progressive tensile fracture. This may occur without the beam material experiencing any ductile yielding in the compression zone, since as shown in Fig. 2.3 at low probability levels the tensile strength of the lamination stock is less than the compressive strength. Thus, for the failure analysis of beams the material can be assumed to be linear elastic up to brittle fracture in tension. Further to this, for typical span-to-depth ratios between 15 and 30 the beam failures will occur under a maximum transverse deflection which is only a fraction of the beam depth. Therefore, small deformation beam theory is also applicable. Consequently, up to first material failure in the beam the response is linear, and can be solved by a standard structural analysis of the form K A = Fo, where K is the linear global stiffness matrix. As before, let Fo denote a reference load system. Under this level of load the beam model determines the tensile stresses which result at the integration points within the beam. Let d0(i, j) denote the resulting average tensile stress in the j-th lamina of the i-th element. At this same location the load factor required to produce failure is given In turn, this calculation can be performed over the entire finite element mesh. The load factor, which when applied to F 0 , produces first failure is then obtained from X1 = min{A(l, 1),..., A(l, NL),...,X(k, /),...,\{NE, 1),..., X(NE, NL)} (2.42) Assume, for example, that A X — X(k,l). In general, the load-carrying capacity will not be exhausted at the first failure load level AJFQ . Subsequent failure loads can be by A(i,i) St(i,j) (2.41) Chap te r 2. A G l u e d - L a m i n a t e d W o o d B e a m - C o l u m n M o d e l 29 determined by repeating the above procedure after first setting the elastic modulus to zero at the failed location: i.e. E(k,l) = 0. Thus, through repeated linear analysis steps the progressive failure of the beam can be modeled and the ultimate load-carrying capacity A C F 0 determined, as illustrated in Fig. 4.3. Note that the load displacement envelope shown in Fig. 4.3 can be likened to that obtained from a test conducted under displacement control. For simplicity of presentation the procedure outlined above is applicable for laminated beams without end-joints. If end-joints are present in a beam then at each load level the ratio of strength to induced stress at every end-joint must be determined using an equation similar in form to Eq. (2.41). In turn, these ratios must be included in the evaluation of Eq. (2.42) to determine the load factor which produces either a lamina or end-joint failure in the beam. As already noted, for beams with end-joints the finite element discretization is adjusted so that end-joints are coincident with nodal locations. When an end-joint failure occurs along a lamina the elastic moduli over adjacent elements are set to zero. 2.3.3 Simpl i f ied Laminated C o l u m n M o d e l For a laminated column, which has a span-to-depth ratio that is sufficiently large to ensure that elastic buckling occurs, determination of the axial load-carrying capacity can be obtained as the solution of a generalized eigenvalue problem (Cook et al. 1989): (K - AKa)d> = 0 (2.43) where K and KC T are, respectively, the linear and geometric global stiffness matrices, A is an eigenvalue and the corresponding eigenvector. If Ka is formulated for an axial compressive load of unit magnitude then the lowest eigenvalue XCT in Eq. (2.43) equals the critical elastic buckling load PCT for the column. Extraction of A c r from Eq. (2.43) Chapte r 2. A Glued—Laminated W o o d B e a m - C o l u m n M o d e l 30 can be simply achieved by using the inverse iteration technique (Bathe 1982). Note that the elastic modulus is the only stochastic material property that presents itself in this problem and that its stochasticity is fully accounted for in the formulation of the linear global stiffness matrix. 2.4 Numerical Examples and Model Verification Several examples are now considered in order to evaluate the performance of the finite element structural analysis model which has just been presented. At this juncture in the presentation it is prudent to first consider problems which are statistically determinis-tic in nature. The first problem is a simply supported beam under sinusoidal loading. This rather elementary problem affords the opportunity to compare the finite element prediction of displacements and stresses against the exact solution obtained from solving the governing differential equations of HOSDBT. The second, more complex, problem examines the transverse deflection of an axially constrained asymmetric laminate, which because of bending-extension coupling requires the consideration of nonlinear geometric effects. The final problem evaluates the load-carrying capacity of a simply supported beam and beam-column. 2.4.1 Sinusoidal Load ing of a S imp ly Suppor ted B e a m Consider a simply supported beam subjected to sinusoidal loading of the form (2.44) The governing equations of motion of this problem, based on HOSDBT, can be obtained from the virtual work expression of Section 2.2.3: H£ + £) + iH M 2-£ ) - - (2-45) q0sm 7TX Chapter 2. A Glued-Laminated Wood Beam-Column Model 31 with the accompanying support conditions of w(0) = w(L) = 0. Under the given sinu-soidal load the displacement field takes the form w(x) = Wsin^j-) (2-47) 1)(x) = tfsin(^) (2.48) Upon substitution of Eqs. (2.44), (2.47) and (2.48) into (2.45) and (2.46) two algebraic equations result in terms of W and Solving these equations yields ^ / 7 0 G A ^ + 8 5 ^ 7 \ x4EI \ 70(7AL2 + 7r 2 £7 J V ' _3J^f^GAL^-2^El\ n3EI \ 70GAL2 + TT2EI J V 1 In turn, the deformation of the beam can be expressed in the convenient form 7TX « • > • - ^ - ( f ) ^ ( ™ S ^ ) « ( T ) where it is duely noted that the leading term in Eq. (2.51) corresponds to the solution given by BEBT. The problem is now made specific by considering a beam with the following properties: 6 = l m , L = 4.8m, L/h = 5, 10, 20, 30, or 40 E = 15 GPa, G = E/20 = 750 MPa, q0 = 100 kN/m Exploiting symmetry, the finite element analysis only requires modeling one half of the beam span. The results of the deflection analysis for this problem are presented in Table 2.1, for a select number of uniform finite element discretizations. It is observed that the Chapter 2. A Glued-Laminated Wood Beam-Column Model 32 finite element results are in very good agreement with the exact solutions obtained from Eq. (2.51). However, it is noted that the rate of convergence is relatively slow as the exact solution, to 5 significant figures, at each L/h value is not achieved even with 16 elements. This shortcoming was investigated further by considering an element which interpolates the ^-displacement field within an element with a cubic Hermitian polynomial, which is the same as that used for the ^-displacement. The deflection predictions based on this element formulation are presented in Table 2.2, where a much improved rate of convergence is displayed. As a further comparison, Table 2.1 also provides the deflection predictions obtained from BEBT, which are seen to be quite poor when the span-to-depth ratio of the beam is less than 20. A sample of the corresponding finite element stress analysis predictions are summa-rized in Table 2.3. Note that these tabulated shear force and bending moment resultants are, in all cases, evaluated at the mid-point within an element. These results are comple-mented by Figs. 2.6 and 2.7, which show, respectively, the normal and transverse shear stress distribution for the deep beam case of L/h = 5. From the normal stress distribu-tion shown in Fig. 2.6 the departure from the BEBT assumption of plane cross-sections remaining plane is readily apparent. From both Figs. 2.6 and 2.7 it is seen that with 8 elements excellent agreement is achieved between the finite element solution and the HOSDBT. Figures 2.6 and 2.7 also provide a comparison with a 2-dimensional elasticity solution (Pagano 1969), for which very good agreement is achieved. The results from this example have demonstrated that the basic finite element formu-lation based on HOSDBT provides a very good prediction of the displacement and stress state of beams over a full range of span-to-depth ratios. While a cubic interpolation of tp was seen to improve the element performance, it comes at the expense of two additional degrees-of-freedom per element. For this reason the element based on tjj being linear will be used in all subsequent analysis. Chap te r 2. A G l u e d - L a m i n a t e d W o o d B e a m - C o l u m n M o d e l 33 2.4.2 Geomet r ic Non l inear Response of a Laminate Consider the cylindrical bending of an asymmetric [ 9 O 4 / O 4 ] cross-ply laminated composite with axially constrained hinged support conditions under uniform transverse loading, as shown in Fig. 2.8. It will be seen that nonlinear geometric effects are manifested in this problem even under small transverse displacements as a result of the cross-sectional asymmetry and given support conditions. It is possible to formulate this problem in terms of a set of linear differential equations with nonlinear boundary conditions which, in turn, admits a simple numerical solution (Sun and Chin 1988). This problem is solved for a graphite/epoxy [ 9 O 4 / O 4 ] laminate with the following geometric and material properties: L = 9.0 in., b = 1.5 in., h = 0.04 in. Ex = 20 x 106 psi, E2 = 1.4 x 106 psi, G12 = 0.7 x 106 psi, vl2 = 0.3 The finite element modeling uses a uniform discretization of 18 elements over half of the beam span. The midspan displacement predictions from the finite element analysis are shown in Fig. 2.9 and compared against the numerical solution obtained by Sun and Chin. Excellent agreement between the two approaches is observed. Figure 2.9 also shows the inadequacy of linear (small deformation) theory when it is applied to this problem. It is also interesting to observe that the asymmetry of the cross-section gives rise to a substantially different load-displacement response under positive and negative loading. 2.4.3 U l t ima te Load Analys is As a final example the load-carrying capacity of a simply supported beam under uni-formly distributed loading is examined. The properties of the beam are the same as those Chapter 2. A Glued-Laminated Wood Beam-Column Model 3 4 given in Section 2.4.1 for the case when L/h — 20. The normal stress-strain behaviour of the beam material obeys the relationship depicted in Fig 2.5. With this constitutive model the load-displacement response of the beam up to failure is directly dependent on the strength ratio Rs = St/Sc and the tangent modulus ET = m E E in compression. In what follows Sc = 30 MPa and St = RsSc, with Rs < 4. The finite element modeling of the beam consists of 8 elements over its half span. Failure analysis in this example, in actuality, only requires the monitoring of the maximum tensile stress at the beam's midspan since first failure at this location determines collapse of the beam. Figure 2.10 shows the finite element prediction of the load-displacement response of the beam to failure for the cases when Rs = 1.0, 1.5, 2.0 and 4.0 and when the material behaviour is elastic perfectly-plastic in compression (obtained by setting m,£ = 0). The response is linear elastic to failure when Rs < 1. For Rs > 1 tensile fracture only occurs after some ductile yielding over the compression zone of the beam. It is observed, however, from Fig. 2.10 that only a slight deviation from linearity occurs up to Rs = 1.5, after which the nonlinear response becomes increasingly more pronounced. The ultimate uniformly distributed load to produce failure can also be determined analytically using a strength of materials approach (Buchanan, 1990): From Table 2.4 it is seen that the finite element predictions of load-carrying capacity, for all values of i?s, are in excellent agreement with Eq. (2.53). The amount of yielding which can be sustained by the beam before failure is also dependent upon the value of the tangent modulus Ex = TTIEE. Figure 2.11 shows, for the case when Rs = 4.0, the load-displacement responses to failure when the material in compression has a strain-hardening branch with = 0.05 and when the material is strain-softening with = —0.05. Previous research has suggested that for SPF 3 L2 ' [ Rs + 1 . (2.53) Chapte r 2. A G l u e d - L a m i n a t e d W o o d B e a m - C o l u m n M o d e l 35 material that ~ —0.02 (Buchanan 1982). As a final extension to this problem an axial compressive end load, of magnitude P c r/4, is applied to the beam. In this example the loading sequence involves instantaneous application of the end load followed by monotonically increasing the uniform transverse load. The corresponding load-displacement response to failure is illustrated in Fig. 2.12, for values of 0.5 < Rs < 1-7. For these values of the strength ratio significant ductile yielding occurs before tensile fracture of the beam-column material. For values of Rs > 1.7, St has no influence on the beam-column's response. 2.5 Concluding Remarks A relatively simple, yet comprehensive and versatile, finite element model has been presented for estimating the load-carrying capacity of glued-laminated wood beams, columns and beam-columns. This model has the capability of assigning stochastic ma-terial properties along the laminae of a member. Structural collapse occurs under pro-gressive material failure and/or an overall loss of stability. Material failure is governed by brittle tensile fracture, which may occur within a lamina or at an end-joint. This finite element model was validated by considering various deterministic beam-column problems, for which known solutions exist. Chap te r 2. A G l u e d - L a m i n a t e d W o o d B e a m - C o l u m n M o d e l 3 6 Table 2.1: Finite element solutions for the maximum deflection of simply supported beams (ip - linear). FEM Solution: HOSDBT ( 0 - linear) HOSDBT BEBT (L/h) NE = 2 NE = 4 NE = 8 NE = 16 (Exact) (Exact) 5 0.00087541 0.00087758 0.00087806 0.00087818 0.00087822 0.00049277 10 0.0046296 0.0047064 0.0047163 0.0047180 0.0047185 0.0039421 20 0.031837 0.032895 0.033088 0.033088 0.033092 0.031537 30 0.10390 0.10786 0.10875 0.10875 0.10877 0.10644 40 0.24322 0.25289 0.25535 0.25535 0.25541 0.25230 Note: NE = number of finite elements. Table 2.2: Finite element solutions for the maximum deflection of simply supported beams (tp - cubic). FEM Solution: HOSDBT ( 0 - cubic) HOSDBT (Exact) (L/h) NE = 2 NE = 4 NE = 8 5 10 20 30 40 0.00087868 0.0047205 0.033104 0.10880 0.25547 0.00087825 0.0047186 0.033093 0.10877 0.25542 0.00087822 0.0047185 0.033092 0.10877 0.25541 0.00087822 0.0047185 0.033092 0.10877 0.25541 Note: NE = number of finite elements. Chapter 2. A Glued-Laminated Wood Beam-Column Model 37 Table 2.3: Finite element solutions for the shear force and bending moments in simply supported beams. HOSDBT ( 0 - linear) (L/h) V @ x = 0.15 M @ x = 2.25 5 0.15133 0.23090 10 0.15277 0.23096 20 0.15432 0.23108 30 0.15595 0.23122 40 0.15750 0.23135 Equi Librium V ® x = 0.15 M @ x = 2.25 0.15352 0.23232 Table 2.4: Analytical and finite element predictions for the ultimate load-carrying ca-pacity of a simply supported beam. R s qu (kN/ m) Eq. (2.53) FEM 1.0 100.0 100.5 1.5 140.0 140.2 2.0 166.7 167.0 4.0 220.0 221.3 Chapter 2. A Glued—Laminated Wood Beam—Column Model 38 Chapter 2. A Glued-Laminated Wood Beam-Column Model 39 0 10 . 20 30 40 50 60 70 Strength (MPa) Figure 2.3: Cumulative distribution of tensile and compressive strength of S P F test specimens (Buchanan 1984). Figure 2.4: Experimental axial stress-strain curve from a defect-free wood specimen (Malhotra and Bazan 1980). Figure 2.5: Idealized axial stress-strain behaviour of softwood. Chapter 2. A Glued-Laminated Wood Beam-Column Model 42 Figure 2.6: Normal stress distribution for beam under sinusoidal loading. 0.5 -i 0.4 : 0.3 : E : ^ 0.2 : £ 0.1 : C L -CD '-HOSDBT. Flnsticity Theorv. Q 0-0 3 / f E -o. i \ o : ^ - 0 2 -0 0 : - 0 . 3 z L / h = 5 x = 0.15 m N E = 8 ^ - 0 . 4 z -n - 1 1 1 1 1 r—- l 1 1 1 1 1 1 1 1 1 i 1 1 1 1 i 1 1 i 0.10 0 .15 Shear Stress (MPa) Figure 2.7: Shear stress distribution for beam under sinusoidal loading. Chapter 2. A Glued-Laminated Wood Beam-Column Model 43 Figure 2.8: Asymmetric [9O4/O4] laminated beam configuration. JO 1 o-1 inpnr Theory. Large Deflection Theory. f " ~7 f m / f m ' -o - 1 -X -2-" ° - 3 -O —' - 4 --2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5. 2.0 2.5 Normal ized Midspan Displacement ( w / h ) Figure 2.9: Load-displacement response of an asymmetric [904/04] laminated beam. Chapter 2 . A Glued-Laminated Wood Beam-Column Model 4 4 q II Rs S c s t / s c 30 MPa or i n q II c/) or -i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i i i 50 100 150 200 250 Midspan Displacement (mm) Figure 2.10: Load-displacement beam response to failure for varying Rs. 300-250 E 200^ '150 T3 O O 100 50 H R s = 4.0 S c = 30 MPa mE = 0.05 mE = 0.00 mE = -0.05 — 1 1 1 1 1 1 50 100 150 200 250 Midspan Displacement (mm) Figure 2.11: Load-displacement beam response to failure for varying rriE-Chapter 2. A Glued-Laminated Wood Beam-Column Model 45 Figure 2.12: Load-displacement beam-column response to failure for varying Rs-Chapter 3 Elastic Modulus of Wood 8LS a Stochastic Process 3.1 Introduction The numerical modeling of material properties as stochastic fields must, in some way, be supported by experimental data. The availability of appropriate data, however, is in general extremely scarce. This results from the fact that standardized testing proce-dures, almost without exception, only provide statistical information on the variability in mechanical properties between test specimens, ignoring the within specimen variability. In support of this approach to testing, codified design procedures are generally based on mean stiffness and a characteristic (5th-percentile) strength value of the test speci-mens. Testing procedures which in some way could give an estimation of within specimen variability are seen to be too complex and labour intensive and are at present deemed unnecessary. An exception to this practice is provided by the testing procedure to pro-duce machine stress rated (MSR) lumber. In this test structural size wood members are passed through a grading machine which facilitates an approximation to the spatial variation in the modulus of elasticity (MOE) along the length of the test specimen. The scope of work covered in this chapter includes: an overview of this test procedure as a means to measuring the spatial stochasticity in the MOE in wood; evaluation of the ensemble and sample-to-sample statistics from the experimentally obtained MOE profiles; simulation of MOE profiles following the spectral approach outlined in Appendix 46 Chapter 3. Elastic Modulus of Wood as a Stochastic Process 47 A; and, structural application of this MOE information to the response variability in the elastic buckling of laminated wood columns. 3.2 Experimental Evaluation of the M O E of Wood The lengthwise variability in the elastic modulus of a wood specimen can be determined, in ah approximate manner, by means of a stiffness-based grading machine, a schematic of which is shown in Fig. 3.1. This machine measures, at a number of discrete points along the length of a member, the force required to produce a prescribed midspan deflection A over a short test span L0. For each position of the specimen as it is fed through the machine, a nominal or average MOE value En(xi) over the test span is computed by the machine's data acquisition system following elementary beam theory: *•<*<> = P-^KF ™ where P(xi) is the applied force and I is the moment of inertia of the cross-section. When the specimen has passed through the machine the output is a record, or profile, which approximates the true lengthwise variability in the elastic modulus as a sequence of averages over the test span. Discussion on the accuracy of this approach is deferred to Section 3.4. It is noted that the commercial application of this test procedure is to stress rate the material through the statistical correlation between either the average or minimum stiffness of the specimen and its tensile strength over the test length (Bodig and Jayne 1982). Obtaining the MOE profile is only a by-product of this test procedure, albeit of key importance to the problem at hand. An experimentally obtained data set consisting of 179 MOE records from spruce and pine specimens was used in this study (Karacabeyli 1992). Each specimen was Chapter 3. Elastic Modulus of Wood as a Stochastic Process 4 8 designated as Grade C lamination stock by visual inspection in accordance to the rules of a national grading standard (CSA 1989) The specimens were of rectangular cross-section, dimensioned either 38 mm x 89 mm (102 pes.) or 38 mm x 140 mm (77 pes.) and having a total specimen length Ls = 4.27 m, from which test records of length Lt = 2.865 m were obtained. Note, any variation which may be present in the elastic modulus along the width or depth of the specimen is ignored in this study. The particular grading machine used to produce the MOE profiles had a test span L0 = 0.91 m and enforced a center point deflection of A = 4.45 mm. The machine was set up so that bending of the specimens took place about the weak axis, with the specimens traveling through the machine at a rate of 48 meters per minute. Any dynamic effects which might arise from this rate of loading have been ignored in the processing of the MOE profiles. Under this setup a total of 291 MOE point-estimates, spaced 9.88 mm apart, were obtained from each specimen. In what follows let E(x) denote the ensemble MOE corresponding to the test data and En(x) a generic test record. This nomenclature is used to distinguish test data and results derived from test data from the underlying quantities which they approximate; so that, for example, E(x) is a experimental measure of the actual MOE process E(x). Figure 3.2 shows two typical MOE profiles obtained from the grading machine. Note that a five point moving average (mild low-pass) filter was used to remove high frequency experimental noise from the records. Figure 3.3 shows the ensemble mean and standard deviation of E(x), from which it is apparent that these statistics do not show substantial variation with respect to the sampling location. The analysis thus far suggests that the MOE process is, at least, homogeneous in the weak sense. For subsequent analysis each record is decomposed as follows: En(x) = E*n + en(x) (3.2) Chapter 3. Elastic Modulus of Wood as a Stochastic Process 49 where E* is the specimen mean over the test length Lt and en(x) is the zero-mean fluctuating part. Over the ensemble the specimen to specimen variability in E* results in a nonparametric mean of 11.56 GPa and a standard deviation of 1.04 GPa. Thus, the £(a;)-process is nonergodic, when examined over this test length. The specimen means are assumed, without loss of generality, to obey a 2-P Weibull distribution: P(E*) = 1 - exp [-(E*/m)k] (3.3) where m and k are, respectively, the scale and shape parameters. The 2-P Weibull fit to the test data, which is shown in Fig. 3.4, resulted in m = 12.0 GPa and k = 13.8. Figure 3.5 shows the ensemble mean and standard deviation of the zero-mean fluc-tuating part e(x). At this level of examination the variability in the ensemble statistics, as a function of the sampling location, is more pronounced. Nevertheless, homogeneity for the e(x)-process is assumed as it is expected that this variability will diminish with increasing ensemble size. Further evaluation of the e(a:)-process will be carried out via its spectral representation. To this end, the finite length one-sided spectral density ( ? e „ ( ' c ) corresponding to the en(x) record can be evaluated as GiM = ~ \ ^ M \ 2 (3-4) TCLt where J~en(K) 1 S the finite Fourier transform. Employing a FFT algorithm allows for an efficient evaluation of Eq. (3.4) at discrete wave numbers = I A K (i = 1,2,3,- - •). The resolution of the individual spectrums, however, is relatively poor, with A/c = 2.19 rads/m; this being a direct consequence of the chosen sampling rate of the MOE records in the spatial domain. The ensemble one-sided spectral density of the e(x)-process, as shown in Fig. 3.6, is obtained from (3.5) Chapter 3. Elastic Modulus of Wood as a Stochastic Process 50 with £[•] denoting the operation of ensemble expectation. From Fig. 3.6 it is noted that there exists, over the ensemble, a dominant spectral amplitude at «i = 2.19 rads/m. In fact, the dominant spectral amplitude could occur at a wave number less than K 1 } but cannot be captured from the data due to the coarse spectral resolution. This long wave length component may simply be a feature of the £^(x)-process. An alternate explanation, however, may be that a linear or higher order trend exists in the process, arising possibly from the growth characteristics of the wood, with one end of a lumber piece consisting of more mature, and hence stiffer, wood compared to the other end. This conjecture has some support from a previous experimental investigation (Wangaard and Zumwalt 1949). If this is true the ^(x)-process is rendered nonhomogeneous. Unfortunately, the chosen test length for the specimens is not long enough and the procedure of specimen collection was inadequate to provide a judgment on this matter. Consequently, it will be assumed herein that the £'(x)-process is homogeneous up to second order moments. Finally, it must be reemphasized that all of the test data from the grading machine was obtained at a fixed test span (L0 = 0.91 m). Thus, any influence this parameter may have on the measured elastic modulus, through Eq. (3.1), cannot be ascertained from this data set. In what follows a tilde will be used to distinguish a simulated quantity from a test quantity or the actual underlying quantity it approximates. Adopting the procedure to simulate one-dimensional univariate homogeneous stochastic processes presented in Appendix A, e(x) can be written as a summation of sinusoids: 3.3 Simulating M O E Profiles (3.6) 71=1 Chapter 3. Elastic Modulus of Wood as a Stochastic Process 51 where, to be consistent with the FFT algorithm, nn = n- Art. Note that this discretization of the spectrum results in a simulated process which is periodic with wave length L P = 2TT/AK. In order to relate Ge(n) to GZ(K) in Eq. (3.6), it is advantageous to fit G e ( K ) to an analytical expression. From examination of Fig. 3.6 an appropriate form is GZ(K) = KC1 jc 2 exp(—c3/c2) + c4 exp(—C5K)| (3-7) where the five nonnegative constants ci, c2, c3, c4 and C5 can be determined by regression analysis ofthe data, as obtained from Eq. (3.5), with Eq. (3.7). The regression procedure based on nonlinear function minimization1, in a least squares sense, takes the form N * i = E {GsM - GB(KN)}2 -> min (3.8) 71=1 Given the value of A/c obtained from the data (2.19 rads/m) and from examining Fig. 3.6, N was set equal to 15. The regression analysis can be complemented with two constraint conditions. First, the mean sample variance from the data must be met; that is, the integral of Eq. (3.7) must equal £[<xfj = 715 MPa. The second condition requires that Ge(rt) be unimodal with a maximum value at «i = A/c. The results of the fit are given in Table 3.1. For subsequent reference let this approach be designated as Method A. It is of interest, at this point, to evaluate the spectral moments2 corresponding to Gi(rz). The location of the centroid Kx along the /c-axis equals 4.51 rads/m, while the radius of gyration K2 equals 5.66 rads/m. Each of these geometric quantities is identified in Fig 3.6. These two quantities combine, according to Eq. (A.56), to give a spectral bandwidth 8 = 0.61 for the e(x)-process. 1 Wherever the use of nonlinear function minimization is mentioned or implied throughout this thesis it has been performed using the B F G S algorithm (Press et al. 1986). 2See Appendix A for the definition of these quantities. Chapter 3 . Elastic Modulus of Wood as a Stochastic Process 52 Two simulated e„(x) records, based on Method A, are shown in Fig. 3.7. Note that each record simulated in this manner is periodic with wave length Lp = Lt = 2.865 m. As a consequence of this periodicity it is seen that it is advantageous to start with test specimens of the longest length possible so that profiles can be simulated for the full range of lengths encountered in practice. Figure 3.8 presents a comparison of the cumulative distribution of sample root mean square (RMS) a^n and crgn as obtained, respectively, from the data and simulated using Eq. (3.6), with the fitted parameters. The feature that Eq. (3.6) produces ergodic samples over the record length Lt is readily apparent. One means of achieving better agreement for the sample mean square is to assume that the five spectral parameters are random variables. Considering the form of Eq. (3.7), it is assumed that each is an independent log-normal variate. Fitting of the mean and standard deviation of each c,- can, again, be achieved using nonlinear function minimization, which in this case takes the form with the variance evaluated at the quantiles qj = 0.05j>. The fitted parameter values are given in Table 3.1 and the resulting cumulative distribution of sample RMS a^n is shown in Fig. 3.8, both under the designation of Method B. Note that within the accuracy of the regression analysis the mean parameter values for the c; equal those obtained by The final step in simulating an jBn(a;)-record is to add a sample mean value E* to e„(x): (3.9) Method A. En(x) = E*n + en(x) (3.10) For the case when E* is 2-P Weibull distributed it can be simulated as E*n = m.[-\n(l-p)} l/k (3.11) Chapter 3. Elastic Modulus of Wood as a Stochastic Process 53 with p a uniformly distributed random number on the interval [0,1]. In turn, it must be checked that En(x) > 0 for all x € [0,L(]. Passing this requirement the simulated record can be added to the ensemble. Otherwise, another realization must be generated. 3.4 Accuracy of the Experimental Data An important and as yet unanswered question underlying the analysis in this study is the accuracy of the £^n(x)-profiles generated by the stiffness-based grading machine. A brief discussion of this matter follows. From beam theory it can be shown that the general equation for the deflection A under a center point load P of a simply supported beam of length L with moment of inertia I and varying elastic modulus E(n) is given by a ' J / ^ ^ / ' M * , } ( 3 . 1 2 ) For the beam configuration illustrated in Fig. 3.1 this expression can be rewritten, after a coordinate change of the second integral, as where one is to recall that for the application being considered A is a known quantity. In Eq. (3.13), let En(x) denote the actual localized elastic modulus of a specimen, which, if assumed known, allows one to compute the corresponding load profile Pn(x), which the grading machine would measure. With this information, the corresponding -En(x)-profue from the grading machine is produced using Eq. (3.1): * • ( * ) = ( 3 - 1 4 ) which can then be compared against En(x) for accuracy. Chapter 3 . Elastic Modulus of Wood as a Stochastic Process 54 In the absence of real i?n(x)-profiles, pseudo profiles will be simulated according to Eq. (3.10). In order to recover _En(x)-profiles which are of length Lt = 2.865 m the £ n(x)-profiles must be, at least, of length (Lt + L0), which in turn puts a constraint on the value of A/c used in Eq. (3.6). Two i£n(x)-profiles (obtained using AAC = 0.5 rads/m and N = 40) are shown in Fig. 3.9, along with the corresponding .En(x)-profiles for comparison. From this figure, the smoothing, or averaging, effect of the deflection analysis over the test span, as expressed by Eq. (3.1), on the resulting i?„(x)-profiles is readily apparent. This fact is further quantified in Fig. 3.10 which shows a comparison between the distributions of within specimen RMS <re and a^. from a simulation of 200 profiles. It is noted from this figure that, on average, the grading machine reduced the RMS of the pseudo specimens, over all probability levels, by approximately 20%. A filtering scheme has been proposed to improve the correlation of the grading machine output to its input, however the approach is not numerically robust (Foschi 1987). Finally, upon a second look at the experimental i?n(x)-profiles shown in Fig. 3.2, it must be concluded that a certain amount of noise is still present in the results. This may be attributable, in part, to the transverse vibration of the specimens induced by their longitudinal motion through the grading machine (Samson 1987). 3.5 Response Variability in the Buckling Load of Columns Structural application of this stochastic MOE information is now made to the elastic buckling of laminated wood columns. The structural analysis is performed using the finite element laminated column model presented in Section 2.3.3. Recall that the governing equation for the elastic buckling analysis can be posed as a linear eigenvalue problem: ( K - A K a ) $ = 0 (3.15) Chapter 3. Elastic Modulus of Wood as a Stochastic Process 55 where K and KCT are, respectively, the linear and geometric global stiffness matrices, A is an eigenvalue and $ the corresponding eigenvector. The lowest eigenvalue A c r is extracted from Eq. (3.15) using an inverse iteration technique. In this formulation, the linear stiffness matrix K, must reflect the spatial stochasticity in the elastic modulus along each lamina. For each lamina within a finite element, of the discretized column, the assignment of an elastic modulus value is obtained by sampling the stochastic field at the midpoint along the element. As an example, for the j-th lamina in the i-th element E(i,j) = E* + ~ej(xi) (3.16) where X{ locates the midpoint of the i-th element and E* and <3j(x,) are determined according to Eqs. (3.11) and (3.6), respectively. Note that if the overall length of the column exceeds Lt each lamina must be constructed from more than one MOE record. The next step in the analysis is to determine the response variability in the elastic buckling load of laminated wood columns as a function of their stochastic stiffness prop-erties. Monte Carlo simulation is used in conjunction with the stochastic finite element model described previously. Attention is restricted to columns under the action of a concentric end load and simple support conditions. The columns are of rectangular cross-section, each having a width of 130 mm and depths of either 38 mm, 76 mm, 152 mm or 304 mm, corresponding to 1, 2, 4 or 8 laminae (each 38 mm thick), with perfect bond existing between the laminae. Each column has a span-to-depth ratio of 40 and buckling is constrained to take place parallel to the depth dimension of the cross-section. The lengths of available lamination material were taken to be random and uniformly distributed between 1.0 m and 2.75 m, so that in all cases the wave length of the simulated elastic modulus of the lamination piece is longer than its length. The ratio of the mean elastic modulus to the shear rigidity was set equal to 15, which is a typical value for softwoods. Note that this resulted in Chapter 3 . Elastic Modulus of Wood as a Stochastic Process 5 6 only a 1% reduction in the mean buckling load when compared with the classical Euler load. In the analysis which follows three material models are considered: Material Model 1: E(x) stochastic, with E(x) = E* + e(x); Material Model 2: E(x) random, with E(x) = E*; Material Model 3: E(x) stochastic, with E(x) = £[E*] + e(x). Model 1 utilizes the full information available for the elastic modulus, as presented in the previous sections. Model 2, on the other hand, ignores totally the fluctuating e(x)-process and only considers the mean elastic modulus as a random quantity. With Model 3 the mean elastic modulus is taken to be constant between lamination pieces (equal to the mean of the sample means), with variability only in the e(x)-process. A Model 3 approach has been adopted in many studies which do not rely directly on supporting experimental data; as for example, the research of Shinozuka and Astill (1972), Deodatis (1989) and Zhang and Ellingwood (1995). The simulation results for each ofthe material models are presented in Fig. 3.11, with the quantity of interest being the coefficient of variation (COV) in the elastic buckling load PCT. All of the results are based on a simulation size of 250 and a column discretization of 10 x NL finite elements, where NL equals the number of laminae in the member. From a preliminary study it was found that these values produced stable numerical results. In order to make a full comparison between the three material models it is of benefit to note that the variability in the specimen means, expressed by the standard deviation CTE*, is of nearly the same magnitude as that in a^, numerically given as 1040 MPa and 715 MPa, respectively. At the outset, it is noted that the Model 1 and 3 results were insensitive to the method by which the MOE profiles were generated (i.e. using either Method A or B). From Fig. 3.11, it is observed that when the number of laminae exceeds Chap te r 3 . E las t ic Modu lus of W o o d as a Stochast ic Process 5 7 one Models 1 and 2 yield results which are in close agreement with each other. Thus, for this structural application it is sufficient to use the simpler approach of Model 2. From the Model 3 results it is observed that the stochastic fluctuations in the MOE process have a less pronounced influence on the response than that produced by the variability between laminae. As expected for all of the material models the COV in Pcr decreases with an increase in the number of laminae. In particular, when the number of laminae reaches three the COV in Pcr is reduced to less than 5%, which suggests that at this point a deterministic analysis may be adequate. 3.6 Concluding Remarks A straight forward procedure, based on a spectral approach, has been presented for simulating, in a realistic manner, the lengthwise variability in the elastic modulus of wood specimens, as a one-dimensional, homogeneous, nonergodic stochastic field. Central to this study is the use of actual test data, from lumber specimens, to calibrate the parameters which define the spectral density function. As a structural application, Monte Carlo simulation in combination with a stochastic finite element model was used to analyze the variability in the elastic buckling load of laminated wood columns with stochastic stiffness properties. It was observed, however, that consideration of only the variability in the elastic modulus between lamination pieces (i.e. a random variable approach) was sufficient in providing an accurate estimate of the response variability. Furthermore, as the number of laminae in a column exceeds three a deterministic analysis may even be deemed adequate. Chapter 3. Elastic Modulus of Wood as a Stochastic Process 58 Table 3.1: Spectral parameter values for GC(K). Cl c2 c 3 c4 c5 Method A 2.39 61700 0.271 8660 0.507 Method B Mean value 2.39 61700 0.271 8660 0.507 Standard deviation 0.26 57000 0.020 7300 0.051 Figure 3.1: Schematic of a stiffness-based grading machine. Chapter 3. Elastic Modulus of Wood as a Stochastic Process 59 14000 .13000 D Q_ V 12000 A O x 1 1000 10000 0.0 x —coordinate (m) 12000-11000 H o _<P. 10000 o CL X < L J 9000 H 8000 • - i 1 1 1 1 i 1 1 1-2.0 2.5 0.0 - i — i — i — i — i — i — i — i — 0.5 1.0 ~ i i 1.5 x - c o o r d i n a t e (m) 3.0 Figure 3.2: Experimental £)n(a;)-profiles. Chapter 3. Elastic Modulus of Wood as a Stochastic Process 6 0 1 4 0 0 0 1 2 0 0 0 O C L 1 0 0 0 0 $ 8 0 0 0 CD U O Q_ I X <LU 6 0 0 0 4 0 0 0 2 0 0 0 0 . 0 Mean Value Std. Dev. 0 . 5 1 .0 1.5 L e n g t h ( m ) 2 . 0 2 . 5 3 . 0 Figure 3.3: Ensemble statistics for E(x). 1.0 0 . 9 -] >>0.8 'J5 0 . 7 : a | 0 . 6 £ o . 5 : CD > 0 . 4 : o 0 . 2 -0.1 -0 . 0 Data. 2 - P Weibull Fit. r i i — ! — i — i — i — i — i — | — i — i — i — i — i — r — T — i — i — | — i — i — i — i — i — i — i — i — i — | — i — 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 M e a n MOE ( M P a ) - i — i — i — i — i — i — r 1 6 0 0 0 Figure 3.4: Cumulative distribution of specimen means E* and 2-P Weibull fit to data. Chapter 3. Elastic Modulus of Wood as a Stochastic Process 61 1200 1000 O Q_ 800H 600 in g 400 o Q_ 200 M e a n V a l u e S t d . D e v . - 4 0 0 - ] — i — i — ' — ' — r 0.0 0.5 - i — i — r -1.0 -i i i i i i n 1.5 2.0 x - c o o r d i n a t e (m) Figure 3.5: Ensemble statistics for e(x). - i 1 1 1 1 1 r -2.5 3.0 140000 120000 0 • • • « • D a t a & E q . (3.5). Fi t t o E q . (3.7). 1 I | 1 1 1 r—i r * — i r ^ - r i — i — i — . — i — — i i i i i i i r 0.0 5.0 10.0 15.0 20.0 25.0 Wave Number /c (m ) 30.0 Figure 3.6: Ensemble spectral density G6(K). Chapter 3. Elastic Modulus of Wood as a Stochastic Process 62 - 2 0 0 0 T 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1——l 1 1 1 i I I r-O.'O 0.5 1.0 1.5 2.0 2.5 3.0 x - c o o r d i n a t e (m) Figure 3.7: Simulated en(x) profiles. Chapter 3. Elastic Modulus of Wood as a Stochastic Process 63 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0 1 6 0 0 o v (MPa) Figure 3.8: Cumulative distribution of erg and erg-Chapter 3. Elastic Modulus of Wood as a Stochastic Process 64 13000 12500 O Q_ 12000 if) "O O 1 1500 11000 H LU 10500 E n ( x ) -P ro f i l e . E n ( x ) -P ro f i l e . 1 0000 - |— i—i—i—i—I—i—i—i—i—r—i—i—i—r 0.0 0.5 1.0 1.5 2.0 2.5 D i s t a n c e (m). i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i i i 1 1 1 r 3.0 3.5 4.0 13000 12500 O C L 12000 in "a o 11500 11000 H 10500H E n ( x ) -P ro f i l e . E„ (x ) -Pro f i le . 10000H—i—i—i—i—|—i—i—i—i—|—r—r—i—i—I—i—i—i—i I ' i i i I ' 1 — r - " I 1 1 1 ' L ' ' ' ' „ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 D i s t a n c e (m) Figure 3.9: Comparison between £ n(x)-profiles and E n(x)-profiles. Chapter 3 . Elastic Modulus of Wood as a Stochastic Process 6 5 1.0 0.9 _>,0.8 ' 3 0.7 D . - O 0 . 6 C L 0 . 5 > 0.4 ^| ° 0 . 3 Z5 o 0.2 0.1 0:0 I i i i I T i i i i i f \ i | i l i i i i i i i I i I I I I I ' i i I ' ' I 1 1 1 1 1 1 l - „ 200 400 600 800 1000 1200 RMS (MPa) Figure 3.10: Cumulative distribution of and ae. Q _ 10-9-8-7-6H »•• Material Model 1. Material Model 2. Material Model 3. c 5-> 4 o Number of Laminae Figure 3.11: COV in PCR for the laminated columns. Chapter 4 SFEA — A Sensitivity Study 4.1 Introduction In this chapter, the stochastic finite element method is employed to examine the effect of spatially varying strength and stiffness properties on the load-carrying capacity of a glued-laminated wood beam. In this presentation the stochastic material modeling is limited to only the elastic modulus and the tensile strength of the lamination material. As has been shown in the previous chapter, test data are readily available to facil-itate modeling of the elastic modulus along a lamina as a one-dimensional stochastic field. Equivalent tensile strength data, on the other hand, are essentially nonexistent. Consequently, the within lamina spatial correlation between the two processes is also unknown. A test program to collect data for these two quantities would, no doubt, be complex, labour intensive and expensive. Before contemplating such an undertak-ing some important questions should first be considered: How important are the spatial fluctuations in the elastic modulus in redistributing the stresses throughout a laminated beam? Is it necessary to model this property as a stochastic field, or is specifying a characteristic (mean or 5th-percentile) value sufficient? At what level of within lamina variability in the tensile strength does it become expedient to model it as a stochastic field? Does the cross-correlation structure between the elastic modulus and the tensile strength have to be quantified? To address these questions three material models are 66 Chapter 4. SFEA — A Sensitivity Study 67 considered in this study (listed with decreasing level of complexity): Material Model 1 stochastic elastic modulus and tensile strength; Material Model 2 stochastic tensile strength only; Material Model 3 random tensile strength only. A response variability study, using Monte Carlo simulation, is conducted to evaluate the influence of each of these material modeling assumptions on a glued-laminated beam's load-carrying capacity. The particular glued-laminated beam under consideration has a rectangular cross-section, is simply supported and subjected to a monotonically increas-ing uniformly distributed transverse load to failure. The beam has a span of 6.08 m, a cross-sectional breadth and depth of 130 mm and 304 mm respectively, with a horizontal lay-up of 8 laminae, each 38 mm thick. It is assumed that the laminae are perfectly bonded one to another and free of end-joints. Thus, the accumulation of damage under increasing load is completely confined to the lamination material. 4.2 Stochastic Material Modeling In this section, the underlying theory of Material Model 1 is first presented, followed by the simplifying assumptions which produce Material Models 2 and 3. Spanwise along each lamina, the modulus of elasticity E(x) and the tensile strength S(x) are assumed to constitute homogeneous one-dimensional stochastic fields: E{x) = E* + e(x) (4.1) S{x) = S* + s{x) (4.2) where E* and S* are the corresponding ensemble mean values and e(x) and s(x) are stochastic fields representing the fluctuations of each material property about its mean Chapter 4 . SFEA — A Sensitivity Study 68 value. Note that the constraint conditions e(x) > —E* and s(x) > —S* must be enforced in order to avoid physically unrealizable material behaviour. Adopting a spectral approach, as presented in Appendix A, the zero-mean stochastic fields e(x) and s(x) are characterized by their one-sided spectral densities, having the plausible form Ge(/e) = i a e 2 6 ^ 2 e x p ( - 6 e « ) (4.3) G.(«) = ^a2sb3sK2exp{-bsK) (4.4) with K, denoting the wave number, be and bs spectral constants and ae and as denoting the standard deviation of e(x) and s(x), respectively. As shown in Fig. 4.1, the parame-ters be and bs determine the shape of each spectrum, as well as the scale of correlation of each field: with a larger parameter value implying a longer distance over which a signif-icant level of correlation will persist in the stochastic field. Note, to facilitate numerical implementation of this spectral approach it is assumed that there exists upper cut-off wave numbers KUC and K U s , such that Ge(n) — 0 for K > KUE and GS(K) = 0 for K > KU). The values assigned to /cUe and KUS are directly dependent upon the values of be and 6S, respectively, and the truncation error defined by Eq. (A.45). Corresponding to the spectral density functions given by Eqs. (4.3) and (4.4) are the autocorrelation functions . gj. (4.5) fl.(0 = g ; . (4.6) The normalized form of this autocorrelation structure is illustrated in Fig. 4.2. Chapter 4. SFEA — A Sensitivity Study 69 It is to be expected that, in general, some statistical dependence will exist between the two processes E(x) and S(x). This dependence can be expressed in terms of the cross-correlation function RES(0'-REs(0 = E*S* + Res(0 (4-7) with £ denoting the lag between sampling points along the x-axis and Res(0 the cross-correlation function between e(x) and s(x). It is assumed that over the ensemble Res(i) is an even function: Res(0 = Rea{—0- Equivalently, this cross-correlation structure can be expressed in terms of the one-sided cross-spectral density function Ges(K), which by virtue of the above assumption is a real even function. Finally, the interdependence between GE(K), GS(K) and GES(K.) are related by the coherence function 7 ^ : ^w-sSSw- (4'8) Realizations of the e(x) and s(x) processes can then be generated in accordance with the following formulae (Shinozuka 1987): N E ( X ) = J 2 V 2 G ^ K A A ^ • cos(/e,-x + (4.9) t=i N s(x) ~ Y \JlGs(Ki)l\Ks • \jils(Ki) COS(K,X + <j>u) t '=l + V / l-7e 2 s(^)cos(K t-x + hi)} (4-10) in which AK = KU/N and /c; = (2i — 1)AK/2. The variables 0lt- and (/>2; are independent random phase angles uniformly distributed on the interval 0 to 2TT. Note that by virtue of the central limit theorem the simulated fields given by Eqs. (4.9) and (4.10) are asymptotically Gaussian as N —• 0 0 . Furthermore, from these equations, it is obvious that Jes(K) — 1 implies total coherence and 7 g S ( « ) = 0 total incoherence in the simulation of the stochastic fields. Lacking knowledge about an appropriate form for 7 e S ( « ) > the two Chapter 4. SFEA — A Sensitivity Study 70 extreme values of the coherence function will be applied in the subsequent application of Material Model 1. Material Model 2, in which only the tensile strength is modeled as a stochastic field, is obtained using Eqs. (4.2) and (4.10), with JIS(K) = 0.0. Material Model 3 is obtained by assigning to each lamina the minimum strength value obtained over its length as determined by Material Model 2: i.e. Smin — S* + sm,-n. Thus, for each lamina, Smin is a random variable. With both Material Models 2 and 3 the elastic modulus is simply assigned the constant value E(x) = E*. 4.3 Stochastic Finite Element Beam Model It is assumed in this study that failure of the beam is governed by brittle tensile fracture of the lamination material. Further to this, it is assumed that the material response in compression remains linear elastic and the deformations before impending failure are small. With these provisos one is justified in using the simplified finite element beam model presented in Section 2.3.2. This model uses repeated linear analysis steps to capture the progressive failure of the beam up to its ultimate load-carrying capacity qu. The load-displacement response of an example beam, using this finite element model, is illustrated in Fig. 4.3. For the most general material model considered in this study both the elastic modulus and tensile strength of each lamina are modeled as stochastic fields. For each lamina within an element the assignment of an elastic modulus and tensile strength value is based on a simple collocation scheme whereby the stochastic fields are sampled at the midpoint along the element. As an example, for the j-th lamina in the i-th element, material properties are assigned as follows E(iJ) = Ej + ejixi) (4.11) Chapter 4. SFEA — A Sensitivity Study 71 S(i,j) = S* + S j ( X i ) (4.12) where xt- locates the midpoint of the i-th element and ej(xj) and Sj(x,) are determined according to Eqs. (4.9) and (4.10), respectively. Figures 4.4 to 4.8 provide, for various values of bs, a comparison between the tensile strength process generated using Eqs. (4.2) and (4.10) and a piecewise constant approximation to it using Eq. (4.12). 4.4 Response Variability Analysis Monte Carlo simulation is used to evaluate the response variability in the beam's load-carrying capacity as a consequence of the chosen material model. To simplify the analysis each lamina in the beam is selected from the same population; this corresponds to all of the lamination stock being from the same species and grade. In an effort to approximate the properties of spruce lamination material (Karacabeyli, 1992) the elastic modulus is assigned the values E* = 12 GPa and 8E = cre/E* = 0.10. Lacking experimental data on the tensile strength process a plausible range of values are assigned to bs (0.25 m, 0.50 m, 1.0 m, 2.0 m and 4.0 m) and 8S = as/S* (0.10, 0.20 and 0.30). Furthermore, it is assumed that S* = 50 MPa. Material Model 1 is simulated using either -fls(rt) = 0 or 1. With this material model it is also assumed that be equals to 6S. In the sensitivity study which is to follow a reference beam is defined with the material properties be = bs = 1.0m, Ss = 0.20 and 7 2 s ( K ) = 0. The nonparametric statistics of the minimum strength Smin which result from the stochastic tensile strength model, using the various values of bs and 8s, are presented in Table 4.1. Note that obtaining these 2nd order statistics is a principal objective of a conventional test program. Available data on spruce lamination material (Karacabeyli, 1992) gives Smin = 26 MPa and 8Smin = 24% for a test length of 3.0 m. This confirms that the assumed parameter values for the tensile strength process used in this study are Chapter 4. SFEA — A Sensitivity Study 72 not unrealistic. Before conducting the sensitivity study for the material models it was necessary to first determine the appropriate finite element discretizations for the beam and the simulation sizes to produce stable numerical results. Recalling Figs. 4.4 to 4.8 and as shown in Fig. 4.9 convergence of the finite element predictions are strongly dependent upon the values of be and bs. From this preliminary study it was decided, for this beam, to use 10, 25, 50, 75 and 100 equal length finite elements when be — bs — 4.0 m, 2.0 m, 1.0 m, 0.50 m and 0.25 m, respectively. Simulations with 250, 500 and 1000 replications for the reference beam were performed using 10 different random seeds to initialize the trial simulations. The results from this exercise are presented in Table 4.2. It was decided from this study to use a simulation size of 500 replications. Also of question is the accuracy of the simplified finite element model for the beam problem under consideration. To address this point, a simulation of 500 replications was performed for the reference beam using the nonlinear finite element model presented in Chapter 2. To eliminate the possibility of ductile yielding in the compression zone the compressive strength was set to an extremely high value (100 MPa). This finite element model predicted qu = 20.4 kN /m and 8qu = 10.8 %, whereas the simplified model yielded qu = 20.1 kN /m and Squ = 11.7 %. The agreement between these two models is deemed acceptable, especially when one considers that a 32-fold computational effort is required to produce the fully nonlinear finite element simulation: using a current technology microcomputer (100 MHz Pentium CPU) required 90 seconds for the simplified model simulation compared with 2,850 seconds for the full nonlinear model. Table 4.3 presents the 2nd-order statistics (mean and COV) for the beam's load-carrying capacity resulting from the chosen material model and the assigned material parameters. From Table 4.3 the following general observations can be made: Chapter 4. SFEA — A Sensitivity Study 73 1. The differences in the response statistics using Model 1 with jls(rz) = 0 and with les(K) = 1 a r e relatively small. Over all cases, the maximum difference in the prediction of qu is less than 10% using these two extreme values for Jes(K)- m most cases, the difference is less than 5%. 2. In all cases the response statistics obtained using Model 2 are bounded between the two results from Model 1. In general, the response statistics obtained using Model 1, with 7eS(«) = 0, and Model 2 are in close agreement one with the other. 3. The response statistics obtained with Model 3 are significantly different than those obtained with the other material models, especially when 8s > 0.2. Observations 1 and 2, taken in combination with the fact that within lamina cross-correlation data between the elastic modulus and the tensile strength would be difficult to obtain experimentally, suggests that a viable approximation is to consider only the tensile strength as a stochastic field in this type of beam problem. Figure 4.10 shows the cumulative distribution of load-carrying capacity predicted by each of the 3 material models, using bs — 1.0 m and 8s — 0.20, and provides further support for this conclusion. Model 3 can be dismissed as woefully inadequate for accurate prediction of the load-carrying capacity of the beam when the within lamina variability in the tensile strength is of the order of 20% or more. For all of the simulation results presented in Table 4.3 be = bs. Enforcing this equality causes the greatest difference in the Model 1 results between 7eS(«) = 0 a n d 7es(K) = 1-Additional simulations were performed keeping bs = 1.0 m and varying be from 0.25 m to 4.0 m. As shown in Table 4.4, when be ^ bs the Model 1 results are in much closer agreement with each other. This fact provides further support to use Material Model 2. This response variability study was undertaken with an arbitrary decision to use the spectral densities of Eqs. (4.3) and (4.4) to model the stochasticity of the elastic modulus Chapter 4. SFEA — A Sensitivity Study 74 and the tensile strength. These particular spectral density functions were obtained from the following family of density functions: Gn(«) = -,a2nb^KN exp(-bNK) n = 2,4,6, • • • (4.13) by setting n = 2. It is important to investigate the impact of this decision. To this end, additional simulations were performed using spectral density functions given by Eq. (4.13) with n = 4 and 6. In order to make a proper comparison each spectral density function must have the same apparent wave length La, as discussed in Appendix A. This requires that b4 = yj5/2b2 and 66 = ^U/3b2. The simulation results are presented in Table 4.5, where it is observed that the order of the spectral density function has very little effect on the response statistics of the beam. Finally, the simulation results presented in Table 4.3 for Material Model 2 are illus-trated in Figs. 4.11 and 4.12, which show, respectively, the variability in qu and 8qu as a function of bs. The general trend is that both qu and 8qu increase as bs increases. Further to this, it is observed at constant values of bs that qu increases in value for decreasing values of Ss. The opposite trend is seen to occur for 8qu. 4.5 Size Effect Study An investigation was also undertaken to determine what, if any, statistical size (length and depth) effect would result from the stochastic tensile strength model for this beam and loading pattern. To this end, additional simulations were performed with the beam span and cross-sectional depth halved and doubled from the reference values of LQ — 6.08 m and h0 = 0.304 m. This study was undertaken using Material Model 2, with 8S = 0.20 and varying values of bs. To facilitate proper comparison, the response quantity of interest is now the modulus of rupture (MOR), as determined by the elastic flexure formula from Chapter 4. SFEA — A Sensitivity Study 75 elementary beam theory: where R is the bending strength (MOR) of the beam. The simulation results, presented in Tables 4.6 and 4.7, confirm both a length and depth effect at the mean MOR level R and the 5th percentile probability level i?o.os- Illustrated in Figs. 4.13 and 4.14 is a fit of Weibull's brittle fracture theory (Bolotin 1969) to these simulation results. To describe a length effect Weibull's theory takes the form R = R0 ( Y) (4-15) and for a depth effect it is given by - (h V/kh R = Ro If) (4.16) where R is the mean MOR at length L or depth h, R0 is the mean reference MOR at length L0 and depth ha and kj_, and k^ are the fitted size effect parameters for length and depth, respectively. It is observed from Figs. 4.13 that with the length effect good agreement is obtained with the simulation results and Eq. (4.15), for all values of bs < 2.0 m. From Fig. 4.14 it is seen that for the depth effect very good agreement is achieved between the simulation results and Eq. (4.16) when bs < 1.0 m. It is of interest to note that from a recent experimental test program of glued-laminated beams that values of ki = 10 and kh = 10 were recommended (Moody et al. 1990). 4.6 Concluding Remarks A stochastic finite element analysis of the load-carrying of a glued-laminated beam, un-der progressive tensile fracture, has been presented. Lacking sufficient experimental data, numerous assumptions were made with respect to the stochastic material modeling of the Chapter 4. SFEA — A Sensitivity Study 76 elastic modulus and tensile strength. A response variability study, based on Monte Carlo simulation, showed that, for the particular beam under consideration, sufficient accuracy was achieved in predicting the response statistics when only the tensile strength was mod-eled as a stochastic field. This observation has significant simplifying consequences on the amount of experimental data that must be acquired to model and calibrate the stochastic material properties. In particular, it is not necessary to quantify the cross-correlation structure between the elastic modulus and tensile strength processes. Further to this, modeling the tensile strength of the laminae as random variables resulted in collapse load statistics which varied widely from those obtained when the within lamina variation of strength was accounted for in the beam model. For the levels of variability in the strength process which were considered in this study it is concluded that a stochastic analysis is required to accurately evaluate the load-carrying capacity of the laminated beams. It was also shown that this tensile strength model caused a statistical size (length and depth) effect to be present in the bending strength of the laminated beams. Chapter 4. SFEA — A Sensitivity Study 77 Table 4.1: Material Model 3 tensile strength statistics. Ss = 0.10 Ss = 0.20 <$s = 0.30 b. Smin Smin '-'min c . '-'mm 8s (m) (MPa) (%) (MPa) (%) (MPa) (%) 0.25 37.4 5.6 24.8 16.8 12.9 40.2 0.50 38.7 6.1 27.4 17.2 16.6 38.6 1.00 40.3 6.6 30.7 17.3 21.3 35.5 2.00 42.0 7.1 34.0 17.5 26.2 33.2 4.00 43.9 7.6 37.8 17.7 31.8 31.5 Simulation size = 1000 laminae. Table 4.2: Load-carrying capacity statistics for the reference beam as a function of simulation size. NR = 250 NR = 500 NR = 1000 Trial # qu q~u q~u (kN/m) (%) (kN/m) (%) (kN/m) (») 1 20.31 11.4 20.19 11.7 20.09 12.1 2 20.24 12.2 20.27 11.8 20.25 11.7 3 20.09 12.3 20.10 11.8 20.08 11.7 4 20.08 10.5 20.05 11.1 19.96 11.5 5 20.20 11.5 20.34 11.5 20.20 11.5 6 20.13 11.7 20.16 11.7 20.17 11.6 7 20.04 11.4 20.06 11.3 19.98 11.4 8 19.90 12.0 19.83 12.2 19.99 12.0 9 20.32 10.9 20.19 11.4 20.19 11.4 10 19.88 11.3 20.07 11.7 20.09 11.5 AVG. 20.12 11.5 20.13 11.6 20.10 11.6 COV (%) 0.72 4.6 0.66 2.5 0.48 2.0 NR = simulation size. Chapter 4. SFEA — A Sensitivity Study 78 Table 4 .3 : Load-carrying capacity statistics obtained from the various material models. Ss = 0 .10 Ss = 0 20 8s = 0 .30 Material Model q~u Qu q~u ( k N / m ) (%) ( k N / m ) (%) ( k N / m ) (%) be = bs = = 0 .25 m ( N E = 100) Stochastic E(x) and S(x) les = 0 2 0 . 7 5.7 18.1 9 .7 15 .2 15.1 Stochastic E(x) and S(x) 7es = 1 22 .5 3.1 19 .0 8 .0 16 .1 12 .8 Stochastic S(x) 2 1 . 0 4 .9 18 .3 9.0 15 .3 15 .6 Random Smin 18.6 5.3 12 .6 13 .6 6 .0 43 .1 bs = 0 .50 m ( N E = 75) Stochastic E(x) and S{x) 7es = 0 2 1 . 4 5.6 19.1 10 .0 16 .3 14 .6 Stochastic E(x) and S(x) les = 1 23 .1 3.3 19 .8 8.9 17 .1 1 4 . 8 Stochastic S(x) 2 1 . 7 5.3 19.2 9.9 1 6 . 6 16 .2 Random Smin 19.2 5.8 13 .6 15 .7 7.2 4 4 . 2 be = --bs = 1.0 m ( N E = 50) Stochastic E(x) and S(x) 7es = 0 22 .2 7.3 20.1 1 1 . 7 18 .0 17.1 Stochastic E(x) and S(x) > 7es - 1 2 3 . 7 3 .9 2 0 . 9 10 .2 18 .6 15 .2 Stochastic S(x) 2 2 . 5 6.9 20 .2 11 .4 18 .0 17 .1 Random Sm{n 20 .0 6.4 15 .3 14 .5 10 .2 3 4 . 7 be---bs = 2.0 m ( N E = 25) Stochastic E(x) and S{x) ' Tes = 0 23 .4 8.9 2 1 . 8 13 .7 2 0 . 0 18 .4 Stochastic E(x) and S(x) ) 7es = 1 2 4 . 3 4 .7 22 .3 12 .2 2 0 . 8 17 .0 Stochastic S(x) 23 .4 8.4 2 1 . 8 13 .9 2 0 . 3 19 .1 Random Smin 2 0 . 8 7.4 16 .7 17 .9 12 .1 35 .1 K = = 4 .0 m ( N E = 10) Stochastic E(x) and S{x) 5 7es = 0 24 .3 11.3 23 .4 17 .5 2 2 . 4 2 2 . 0 Stochastic E(x) and S{x) ' les = 1 2 4 . 6 5.6 2 3 . 9 14 .7 2 3 . 3 2 1 . 0 Stochastic S(x) 2 4 . 6 10.0 23 .6 16 .8 2 2 . 8 2 0 . 6 Random 5 m t n 22 .2 7.7 19 .5 15 .0 16 .6 2 4 . 3 1) Simulation size = 500 beams. 2) NE = number of finite elements used to discretize the beam. Chapter 4. SFEA — A Sensitivity Study 79 Table 4.4: Load-carrying capacity statistics for be ^ bs (Material Model 1, with 6$ — 0.20 and bs — 1.0 m). / es 0 1 es 1 K (m) (kN/m) (%) (kN/m) (%) 0.25 19.7 11.3 19.8 11.1 0.50 19.9 11.2 20.1 11.5 1.00 20.1 11.4 20.9 10.2 2.00 20.0 11.7 20.1 12.0 4.00 20.2 12.0 20.3 11.9 Simulation size = 500 beams. Table 4.5: Load-carrying capacity statistics using different spectral density functions (Material Model 2, with 6S = 0.20). 62 64 = yfifih b6 = JujJb2 b2 qu 8q» qu qu (m) (kN/m) (%) (kN/m) (%) (kN/m) (%) 0.25 18.3 9.0 18.4 8.6 18.4 8.7 0.50 19.2 9.9 19.1 9.8 19.2 9.6 1.00 20.2 11.4 20.1 10.9 20.0 10.6 2.00 21.8 13.9 21.7 13.5 21.6 13.3 4.00 23.6 16.8 23.9 16.5 23.9 16.4 Simulation size = 500 beams. Chapter 4. SFEA — A Sensitivity Study 80 Table 4.6: MOR statistics for beams of varying length (Material Model 2). L0, h0 L0/2, h0 2L0, h0 bs R Ro.05 R -Ro.os R Ro.05 (m) (MPa) (MPa) (MPa) (MPa) (MPa) (MPa) 0.25 42.1 35.1 44.8 37.4 39.9 33.6 0.50 44.3 36.4 47.1 38.1 41.5 34.3 1.00 46.6 37.4 50.3 38.8 43.8 35.8 2.00 50.2 38.6 54.6 40.0 46.4 37.5 4.00 54.5 40.3 56.3 40.9 49.8 38.3 Simulation size : = 500 b earns. Table 4.7: MOR statistics for beams of varying depth (Material Model 2). L0, h0 h0/2 L0, 2h0 K R Ro.os R Ro.05 R Ro.05 (m) (MPa) (MPa) (MPa) (MPa) (MPa) (MPa) 0.25 42.1 35.1 44.8 35.8 39.3 33.4 0.50 44.3 36.4 47.8 37.9 41.2 34.2 1.00 46.6 37.4 51.5 39.6 43.5 35.8 2.00 50.2 38.6 57.2 42.1 46.4 36.4 4.00 54.5 40.3 63.4 44.3 49.3 38.2 Simulation size = 500 beams. Chapter 4. S F E A — A Sensitivity Study 81 g(/c) = G(K)/O2 b b b b b 0.25 m 0.50 m 1.00 m 2.00 m 4.00 m - i r — r — | r 10 -1 I [ — T -15 20 Wave N u m b e r AC (m ) 25 30 Figure 4.1: Normalized spectral density function: #(/c) = G(/c)/cr2 0.25 m 0.50 m 1.00 m 2.00 m 4.00 m Lag £ (m) 5" 6 Figure 4.2: Normalized Autocorrelation function: r(£) = R(£)/CT2. Chap te r 4. S F E A — A Sensi t iv i ty S tudy 82 152 F.E. Mesh - Failure Sequence CD D C TD i_ O 0 a 1 N 0 •152-1824 X —Coordinate Beam Axis Node, typ. 3040 140 Midspan Deflection (mm) Figure 4.3: Progressive failure analysis to determine qu. Chapter 4. S F E A — A Sensitivity Study 83 1 0 0 -o 9 0 -Q_ 8 0 -r~ 7 0 -C n 6 0 -c -CD i _ 5 . 0 -CO 4 0 -CD -.-^ 3 0 -'in Fen 2 0 -1 0 -S i m u l a t e d P r o c e s s . F . E . D i s c r e t i z a t i o n ( 1 0 0 E l e m e n t s ) . Ft, fl S * = 5 0 M P a , <5 S =0 .20 , b s = 0 . 2 5 m 0 1 4 2 3 x —coord inate (m) —r-5 Figure 4.4: Realization of tensile strength process (8s = 0.20 and bs = 0.25 m). 1 0 0 -9 0 S i m u l a t e d P r o c e s s . F . E . D i s c r e t i z a t i o n ( 7 5 E l e m e n t s ) . S * = 5 0 M P a , <5 S =0 .20 , b s = 0 . 5 0 m 0 ~r 4 x —coordinate (m) Figure 4.5: Realization of tensile strength process (8s = 0.20 and 6S = 0.50 m). Chapter 4. S F E A — A Sensitivity Study 84 100-90-10 Simulated Process . F.E. Discretization (50 Elements). S*=50MPa, 6 S =0.20, b ,= 1.00m 0 1 ~T~ 2 T 4 5 x — c o o r d i n a t e ( m ) Figure 4.6: Realization of tensile strength process (8s = 0.20 and bs = 1.00 m) Simulated Process . F.E. Discretization (25 Elements). 10 - S*=50MPa, <5S=0.20, b s =2.00m 0 1 2 "T 4 x — c o o r d i n a t e ( m ) Figure 4.7: Realization of tensile strength process (8s = 0.20 and bs = 2.00 m). Chapter 4. S F E A — A Sensitivity Study 85 D CL 100-• 9 0 -^ 8 0 -7 0 ~CP 6 0 P 5 0 CO 4 0 -CD — 3 0 CO 5 2 0 1 0 S i m u l a t e d P r o c e s s . F . E . D i s c r e t i z a t i o n ( 1 0 E l e m e n t s ) . S * = 5 0 M P a , <5S=0.20, b s = 4 . 0 0 m 0 1 2 3 4 x — c o o r d i n a t e ( m ) Figure 4 . 8 : Realization of tensile strength process (6$ = 0 . 2 0 and bs = 4 . 0 0 m). o o 2 5 ^ 2 4 ' 2 3 2 2 _ 2 1 • CD 2 0 O E 5 c • CD 19 1 8 1 7 1 5 * ^ - h - h - 4 . 0 0 \ . • - b e - b s - 1 . 0 0 ~~ — . b e = b s = 0 . 5 0 • . b e = b s = 0 . 2 5 2 5 5 0 7 5 1 0 0 N u m b e r of F i n i t e E l e m e n t s 2 5 Figure 4 . 9 : Convergence of the stochastic finite element beam model. Chapter 4. S F E A — A Sensitivity Study 86 Ultimate Load ( k N / m ) Figure 4.10: Distribution of load-carrying capacity qu (6$ = 0.20 and bs = 1.0 m). Chapter 4. SFEA — A Sensitivity Study 87 Figure 4.11: Response variability in qu. 25 Figure 4.12: Response variability in 6qu. Chapter 4. SFEA — A Sensitivity Study 8 8 6 0 ^ 5 5 -D Q _ 5 0 O C 4 5 O 0) 4 0 ^ , . * k L = 8 . 5 b s = 4 . 0 0 - , - " ' * k L = 9 9 b s = 2 . 0 0 ^ ' ' ' . ^ " " " _ - - — - - k L = ' k L = 1 1 . 8 b s = 1 . 0 0 * - . •— b s = 0 .50* ' b s = 0 . 2 5 . . . S i m u l a t i o n R e s u l t s . W e i b u l l T h e o r y . 3 5 ~1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i i r -0 . 0 0 . 5 1 .0 1 .5 2 . 0 2 . 5 Length Ratio ( L 0 / L ) Figure 4.13: Size effect - mean MOR versus beam span ratio (8s = 0.20). 6 5 6 0 H o CL . ^ 5 5 H o 5 0 -o3 45H 4 0 -3 5 _ , - " k h = 7 : 8 b s = 4 . 0 0 / _ _ _ _ _ _ . k h = 9 . 2 b s = 2 . 0 0 " - " ^ - " " ^ ^ - - ' " ~ " \ _ _ _ - - , . k h = 1 0 . 7 b s = 1 . 0 0 * - " ^ - - " ' ^ _ - - ~ ~ ~ ~ ^s ^ ' ^ 2 ^ ' " . . . S i m u l a t i o n R e s u l t s . b s = 0 . 2 5 * W e i b u l l T h e o r y . T 1 1 1 1 1 1 1 1 1 1 1 1 1 j 1 1 1 1 1 1 1 1 1 0 . 0 0 . 5 1 .0 1 .5 2 . 0 2 . 5 Depth Ratio ( h 0 / h ) Figure 4.14: Size effect - mean MOR versus beam depth ratio (6s = 0.20). Chapter 5 SFEA — Calibration, Verification and Application 5.1 Introduction In the previous chapter, in the absence of experimental data, the tensile strength along a lamina was modeled as a homogeneous stochastic field, having the form S(x) = S* + s{x) (5.1) where S* is the mean strength and s(x) is the stochastic field representing the fluctuations of the tensile strength about its mean value. This model was designated Material Model 2. From this model an ensemble of tensile strength profiles were generated over a given specimen length. As shown in Fig. 5.1, for each realization of the strength process a minimum value of strength, denoted by Smin, c a n he identified. Over the ensemble these values were collected and ranked to give the distribution of tensile strength for the specimen population. This model was designated Material Model 3. It was noted at that point that obtaining this distribution and its defining statistics are the principal objectives of a conventional test program. These testing procedures are relatively simple to perform and consequently there exists an extensive collection of this type of test data. To facilitate a stochastic finite element analysis of a glued-laminated beam Mate-rial Model 2 information is required. An experimental database with this information does not exist. It would be ideal if one could start with Material Model 3 and from that generate Material Model 2. That is, starting with only the distribution of tensile 89 Chapter 5. SFEA — Calibration, Verification and Application 90 strength, over a given specimen length, generate the necessary parameters which define the stochastic process. This approach is investigated in this chapter using the concept of barrier crossing analysis of stochastic field theory, as presented in Appendix A. Utilizing this approach the parameters which define Eq. (5.1) are then calibrated against avail-able test data. After which, the stochastic finite element beam model is verified against test data on the load-carrying capacity of glued-laminated beams. Finally, application is made to optimizing the performance of these beams against a prescribed reliability constraint. 5.2 Barrier Crossing Analysis of the Tensile Strength Process Figure 5.1 shows a realization of a tensile strength process S(x) and identifies the mini-mum strength Smin achieved over a specimen length L (set at 6.0 m in Fig. 5.1), with <W = min {<?(*)} (5.2) 0<x<L Over the ensemble, Smin is a random variable. Associated with the realization shown in Fig. 5.1 is an arbitrarily assigned strength barrier, denoted by a. The probability that Smin lies below this barrier can be expressed as PSmin(a) = PriS^ < a] = Pr[La < L] = 1 - Pa(L) (5.3) where La locates the first down-crossing of the barrier a, as shown in Fig. 5.1, and Pa(L) is the probability of not down-crossing this barrier over the specimen length (0,L]. It is to be noted that in Eq. (5.3) the first equality is a statement related to the probability distribution for the random variable Smin, while the second and third equalities are statements related to the stochastic process S(x). This inter-relationship defines the first passage problem of stochastic field theory1 (Nigam 1983; Ochi 1990). As noted in 1The first passage problem is generally presented within the context of random vibration theory, with the problem posed in terms of the first up-crossing, over time, of a specified threshold. J Chapter 5 . SFEA — Calibration, Verification and Application 91 Appendix A, a general analytical solution does not exist for Pa(L). It is therefore often assumed, for mathematical expediency, that Pa(L) follows a Poisson distribution: Pa{L) = exp(-u-L) (5.4) where v~ represents the mean rate of down-crossing the barrier a. Assuming that the strength process S(x), given by Eq. (5.1), is ergodic and Gaussian allows v~ to be expressed as i / K 2 G s ( K ) d K \ 1 / 2 r i r (a - s*)21 \ " ° ~ 27T { S™Gs{K)dK J 6 X P \ 2 l/o~G,(K)dK|j ( • ' where Gs(K) is the spectral density function associated with s(x) and S* is the ensemble mean of S(x). As discussed in Appendix A, the Poisson assumption made in Eq. (5.4) is not with-out its shortcomings. For example, no allowance is made for the process being below the barrier a at x = 0. As an alternative, Vanmarcke (1975, 1983) has proposed an approximate solution based on a two-state Markov crossing assumption: P„(£) = P . ( 0 ) e x p { - _ ^ } (5.6) with - - { a ~ S * ] (5.7) yJlo° Gs{rt)dK In Eq. (5.6) Pa(0) represents the probability that the process is above the barrier a at x = 0 and $(•) is the cumulative Gaussian distribution function. It is shown in Appendix A that Pa(0) = 1 - $(T?) (5.8) Combining Eqs. (5.6) and (5.8) allows Eq. (5.3) to be approximated by ftu. = i - [i - * ( , ) ] = c p { - i r ^ n } Chapter 5 . SFEA — Calibration, Verification and Application 9 2 with va and 77 defined by Eqs. (5.5) and (5.7), respectively. In what follows it is assumed that the spectral density function for s(x) takes the previously adopted form GS(K) = ^a2sb3sri2exV(-bsK) (5.10) As presented, Eq. (5.9) involves three parameters which define S(x): the ensem-ble mean S* and o~s and bs which determine s(x). The objective is to calibrate these three parameters so that the cumulative distribution of Smin, as predicted by Eq. (5.9), agrees with the distribution obtained from actual test data. Using nonlinear function minimization this problem can be formulated as ND „ * = £ 0 * 5 • -Pi ) - m i n (5.11) 71=1 where Pfmin denotes the cumulative distribution obtained from the barrier crossing anal-ysis and Psmin 1S the cumulative distribution from the test data. In Eq. (5.11) the least squares fit is performed over all ND data points from the cumulative distribution of the test data. To test this approach a stochastic tensile strength process with known parameters was simulated, using S* = 50 MPa, as = 10 MPa and bs = 1.0 m. With these parameter values 500 specimens, of length L — 6.0 m, were generated using Eq. (5.1) and the simulation formula for s(x): N I s ( x ) = XI \/2C7 s(/cn)A«:cos(«;nx + </>„) (5.12) 71 = 1 A sample realization of this process is shown in Fig. 5.1. Over the 500 specimen ensemble the cumulative distribution of minimum tensile strength was determined, producing a mean value of 30.4 MPa and a COV of 17.6%. The resulting distribution is shown in Fig. 5.2. Next, using Eq. (5.9) in Eq. (5.11) resulted in the barrier crossing analysis producing fitted parameter values of S* = 47.5 MPa, as = 9.05 MPa and bs = 0.919 m. In Chapter 5. SFEA — Calibration, Verification and Application 93 turn, using these parameter values an ensemble of 500 specimens was simulated using Eq. (5.1) and Eq. (5.14), from which was extracted the cumulative distribution of minimum tensile strength. This distribution, which is also shown in Fig. 5.2, has a mean value of 29.5 MPa and a COV of 16.3 %. It is observed from Fig. 5.2 that the cumulative distribution of Smin that resulted from the barrier crossing analysis is in good agreement with the original distribution particularly at lower probability levels. Further to this, it is noted that the nonlinear function minimization of Eq. (5.11), which produced the fitted parameters, is robust, being generally insensitive to the initial estimates of these parameters in starting the search algorithm. Before closing this section, it is necessary to discuss why the tensile strength process S(x) was assumed ergodic. From the experimental data obtained from the MOE of wood it was observed that this process was nonergodic, with the mean of each specimen being random. It is reasonable to expect that the same would be true for the tensile strength process. A consequence of nonergodicity, however, is that four parameters must be calibrated by Eq. (5.11): the mean and standard deviation of S*, denoted by /xs* and as* respectively, and the spectral parameters crs and bs which determine s(x). A preliminary investigation, for this case, resulted in Eq. (5.11) producing two minima: one with as* = 0 and the other when as = 0. In the latter case there is no fluctuating strength process. This exercise showed that there is not enough information to fit all four parameters. In particular, one cannot determine both of the distribution parameters which define the mean strength. Assuming the tensile strength process to be ergodic allows the calibration to be performed. 5.3 U .B .C . Laminating Stock and Beam Test Program In order to provide a consistent experimental database for the calibration of the stochastic tensile strength process and the verification of the stochastic finite element beam model _ Chapter 5. S F E A — Calibration, Verification and Application 94 a test program was devised and commissioned (Timusk et al. 1994). All lamination pieces in this test program had a cross-sectional size of 38 mm x 127 mm and were selected from the spruce-pine-fir (SPF) species group. The following test data were collected for the lamination stock database: tensile strength and mean elastic mod-ulus from a) 100 Grade B specimens, each 4.88 m (16 ft) long; b) 100 Grade B specimens, each 2.44 m (8 ft) long; c) 100 Grade C specimens, each 4.88 m (16 ft) long; d) 100 Grade C specimens, each 2.44 m (8 ft) long; e) 100 Grade B specimens, each 2.44 m (8 ft) long with an end-joint at mid-length; f) 100 Grade C specimens, each 2.44 m (8 ft) long with an end-joint at mid-length. Two types of full scale beam tests were conducted: a) 30 beams, each 4.88 m (16 ft) long, comprised of 8 Grade C laminae from 4.88 m long lamination stock, without end-joints; b) 30 beams, each 7.32 m (24 ft) long, comprised of 10 Grade C and 2 Grade B laminae from 4.88 m long lamination stock, with end-joints. In an effort to ensure that the group of test specimens used to establish the lamination stock properties and those used in the construction of the test beams were statistically similar a selection process based on matching MOE between the specimens in the two groups was enforced. Testing to determine the mean MOE of each specimen was performed in a continuous feed testing machine similar to that described in Chapter 3. The tension testing of the Chapter 5. S F E A — Calibration, Verification and Application 95 lamination stock was performed in a linear tension testing machine, with each specimen tested to destruction while clamped at its ends by hydraulic grips. The grip length at each end of the specimen was 610 mm, which reduced the effective length of the test specimens to 3.625 m (12 ft) and 1.220 mm (4 ft). Table 5.1 summarizes the finding from the U.B.C. test program for the lamination stock properties. This information is complemented by Figs. 5.3 and 5.4 which show the cumulative distributions of tensile strength for Grades B and C lamination stock, respectively. It is this information that will be used in calibrating the tensile strength models. Also shown in Figs. 5.5 and 5.6 are scatter plots of tensile strength and MOE for Grades B and C lamination stock, respectively. It is observed from both Figs. 5.5 and 5.6 that for these two data sets the two material properties are weakly correlated: for Grade B the correlation coefficient is 0.27 and for Grade C it is only 0.11. Obviously, this weak correlation between MOE and tensile strength lessens the effect of the matched sampling that was carried out in the test program. Finally, Fig. 5.7 shows the distributions of end-joint strength for Grade B and C lamination stock. It is to be noted that of the 100 Grade B test specimens only 66 failed at the end-joint and of the 100 Grade C test specimens only 43 failed at the end-joint. The end-joint strength statistics presented in Table 5.1, for both Grades B and C, are based on specimens with end-joint failures only. The true strengths of the end-joints for Grade B and C, which are not known, may be higher than that reported in Table 5.1.2 2 A better estimation of the end-joint strength can be obtained by applying a Bayesian updating rule (Ang and Tang 1975) to the full set of test data. This requires the maximization of the likelihood function m= n f(s'\°) n end-joints laminae where f(S \ 9) and F(S \ 9) are, respectively, the probability density function and cumulative distribution function of end-joint strength, 5,- are the measured strength (end-joint or lamina) and 9 is the set of distribution parameters. The second product term in this expression states the probabilities for the tested end-joints to have strengths greater than the measured lamina strengths. This Bayesian approach has not been utilized in this study. Chapter 5. SFEA — Calibration, Verification and Application 96 As noted above, two sets of 30 full-scale beams were tested to destruction: 4.88 m (16 ft) beams, without end-joints in the lamination material and 7.32 m (24 ft) beams with end-joints (hereafter, reference to these beam tests will be given by their span length in feet). In both tests the beams were simply supported and subjected to two symmetrically placed concentrated loads about the beam's midspan, as shown in Fig. 5.8. This loading pattern produces a region of constant moment between the loads. In order to prevent crushing ofthe lamination material 150 mm long bearing plates were provided at the supports and the loading locations. Note that the resulting center to center clear span of the 16 ft beam was shortened to 4.725 m and the 24 ft beam was reduced to 7.050 m. Further discussion of the beam test results will be deferred to Section 5.6. 5.4 Calibration of the Tensile Strength Process As presented in Section 5.1, a tensile strength process S(x) is to be characterized by S(x) = S* + s(x) (5.13) with S* the ensemble mean strength and s(x) a homogeneous zero-mean process simu-lated by the formula N I s(x) - \I^Gs(Kn)lS.K cos(nnx + <j>n) (5-14) 71 = 1 having a spectral density function of the form Gs(K) = ±a2sb3sK2exp(-bsrz) (5.15) The three parameters S*, crs and bs will determine the distribution of minimum tensile strength, over a given specimen length, through the barrier crossing analysis expressed by Eq. (5.9). These strength parameters can be calibrated so that the distribution given Chapter 5. SFEA — Calibration, Verification and Application 97 by Eq. (5.9) agrees with the distribution of tensile strength from test data, through application of Eq. (5.11). Table 5.2 summarizes the calibration results for Grade B and C tensile strength processes, for a specimen length of 3.658 m. Based on these parameter values the resulting spectral density functions are illustrated in Fig. 5.9. Using the parameter values given in Table 5.2, 500 Grade B and 500 Grade C tensile strength specimens were simulated. Sample realizations from these processes are shown in Fig. 5.10 and 5.11. The resulting cumulative distributions of simulated tensile strength for Grade B and C lamination stock are shown in Figs. 5.12 and 5.13, respectively. As expected, it is observed that the agreement between the cumulative distribution from the test data and the fitted distribution from the barrier crossing analysis is best at lower probability levels (< 0.50). The calibration procedure, which was just presented, was based on the underlying assumption that the tensile strength processes are Gaussian. Consequently, there is the possibility of generating realizations of these processes which violate the requirement that S(x) > 0 over the specimen length. For the Grade B and C lamination stock the probability associated with simulating nonpositive tensile strength values is very low: Pr[S{x) < 0] = < 0.845 x 10"8 for Grade B 0.156 x 10"4 for Grade C given the calibrated parameters in Table 5.2. In the event that a nonpositve tensile strength value occurs during a simulation the realization is simply discarded and replaced with another generated specimen. It is noted from the summary of the U.B.C. test program given in the previous section that the tensile strength was obtained for two lengths of Grade B and C lamination stock (16 ft and 8 ft). The intention for this was to calibrate the strength models at two lengths Chapter 5. SFEA Calibration, Verification and Application 98 using a modified form of Eq. (5.11): ND r „ L2=W m m Li =8' mm (5.16) This approach would enforce a better fit for all lengths of lamination stock between Li and L2, if the test data at the two lengths is suitably well behaved. This means that the data should properly display a length effect at all probability levels. Closer examination of the Grade B test data, as presented in Fig 5.3, shows that at low probability levels (< 0.05) the longer specimens are stronger than the shorter specimens. Also, as seen from Fig. 5.4, for the Grade C material the length effect is more pronounced at lower probability levels. Both of these behaviours confounded the calibration procedure. It was therefore decided to calibrate the tensile strength model at only one length, using the longer test specimens. 5.5 U L A G Computer Program With a calibration procedure for the tensile strength process established, various modifi-cations and enhancements were made to the simplified laminated beam model presented in Chapter 2 and utilized for the sensitivity study in Chapter 4. The resulting computer program is entitled ULAG: Ultimate Load Analysis of Glulam (Folz and Foschi 1995). An overview, highlighting the various features of ULAG follows: 1. The requisite input material properties for ULAG include for each species, grade and size of lamination stock (eg. SPF, No.2, 2x6), the tensile strength and the corresponding mean elastic modulus from each lamination test specimen as well as the end-joint strength and the mean elastic modulus from each end-joint specimen. A utility program to ULAG processes this test data producing the tensile strength parameters (S*, as and bs, along with KU) for each type of lamination stock. In Chapter 5. SFEA — Calibration, Verification and Application 99 addition, the elastic modulus and the tensile strength test data are fitted to 2-P Weibull distributions. Also, the nonparametric correlation coefficient between the tensile strength and the mean elastic modulus is determined. This utility program efficiently produces a database from all the available test data for subsequent access by the ULAG program. 2. As determined by the sensitivity study conducted in Chapter 4 the modeling of the elastic modulus as a stochastic field had only a secondary influence on the estimation of the load-carrying capacity of a laminated beam. Consequently, it was deemed sufficient to only take account of the variation in the mean elastic modulus between lamination pieces. That is, ULAG models the elastic modulus of each lamination type (different species, grade or size) as a random variable, which, as already noted is assumed to be 2-P Weibull distributed. As programmed, ULAG assigns to each lamination piece a realization of the tensile strength process. In addition, over the length of the lamination piece, the minimum strength value is determined. Corresponding to this strength value a mean elastic modulus is assigned to the lamination piece which preserves the correlation structure between these two random variables. To achieve this the strength value is first transformed to a standard normal variable zs'-where, $(•) and Psmin are, respectively, the cumulative distribution functions ofthe standard normal variate and the minimum tensile strength. In standard normal space the conditional expectation and variance between Smin and E are given, respectively, by (5.17) S(zE\zs) = p*SEzs (5.18) Chapter 5. SFEA — Calibration, Verification and Application 100 Vax(sE|*s) = l-pfE (5.19) with p*SE the correlation coefficient between z$ and zE. Under incomplete proba-bility information (Der Kiureghian and Liu 1986) p*SE can be approximated by the formula pSE = 1.064 - 0M9pSE + 0.005,4s (5-20) where PSE is the correlation coefficient obtained from the test data. In standard normal space the elastic modulus is simulated by ZE = P*SEZS + R-Ny/l - PSE (5-21) where is a uniformly distributed random number on the interval (0,1). Finally, the assigned mean elastic modulus for the lamination piece is given by E = P£1(*(zE)) (5.22) At the location of an end-joint the mean elastic modulus is first determined by av-eraging the elastic moduli for the two lamination pieces which are joined together. An end-joint strength value is then assigned which preserves the correlation struc-ture obtained from the test data by following a procedure similar to that outlined above. 3. ULAG includes various quality control options that can be imposed on the con-struction of the simulated beams. For example, lower limits can be specified for the mean elastic modulus of the lamination material in order to adhere to grading requirements set out in national standards for Stress Grades of glulam (CSA 1989; CSA 1994). Also, proof loading limits can be placed on the minimum tensile strength of the lamination stock and the end-joints. Chapter 5. SFEA — Calibration, Verification and Application 101 4. A second utility program to ULAG processes information on the availability of lengths of lamination stock from which to manufacture beams. For each lamination type the proportion of lamination stock at each available length is recorded in a database (eg. say for SPF No.2 2 x 6 30% of the available material is 3.0 m long, 50% is at 3.5 m and the remaining 20% is 4.5 m long). This database, which is accessed by ULAG, directly determines the distribution (number and location) of end-joints throughout the simulated beams. 5. The method by which a beam is assembled within ULAG follows the procedure gen-erally used in the actual manufacturing process of glued-laminated beams. First, ULAG identifies the number of lamination types used in the beam lay-up. Then for each type ULAG selects, at random, lengths of lamination pieces, which obey the distribution of available lamination stock lengths, until the total required length of lamination stock of that type is met. The lamination pieces are then laid out end to end and end-jointed to form a continuous piece, after which they are cut into lengths equal to the length of the beam. This assembly process is illustrated in Fig. 5.14 for a beam comprised of 5 laminae, having a lay-up of grades B - C - D - C - B . 6. As output, ULAG predicts the distribution of load-carrying capacity of laminated beams or tension members under the failure criterion of either first or progressive tensile fracture. These results are ranked and fitted with a 2-P Weibull distri-bution using the lower quartile of the simulation results. ULAG also records the distribution of maximum deflection under the specified loads. 5.6 Verification of the U L A G Computer Program In an effort to evaluate the ability of ULAG to predict the load-carrying capacity of glued-laminated beams simulation results were compared with the data from the U.B.C. Chapter 5. S F E A — Calibration, Verification and Application 102 full-scale beam test program. 5.6.1 16 ft Beams The cumulative distribution of load-carrying capacity from the 16 ft beam tests and the simulation results by ULAG are presented in Fig. 5.15 and the corresponding distribution statistics are summarized in Table 5.3. The predictions given by ULAG are based on a simulation size of 200 beams and a finite element discretization of 25 elements. As can be observed from Fig. 5.15, the ULAG simulation estimates are in very good agreement with the beam test data, especially at probability levels below 0.50. In particular, as determined from Table 5.3, the 5-th percentile ULAG estimate is only 2.5% greater than the interpolated 5-th percentile test result. Poorer agreement at probability levels above 0.50 is attributable to the fact that the tensile strength calibration procedure for the lamination stock degrades as the probability level increases (or, equivalently, as the barrier crossing decreases). The consequence of this is minimal, however, since any reliability assessment of the load-carrying capacity of the beam is based on the lower tail of the distribution (Foschi et al. 1989). Figure 5.16 shows an expanded view of the ULAG simulation results and the test data up to the first quartile probability level. Also shown in Fig. 5.16 is a 2-P Weibull fit to the lower quartile of the ULAG simulation results; visual inspection confirms good agreement between the two distributions. The statistics from this fit are reported in Table 5.3. These fitted results could be used in a subsequent reliability study. To confirm that a full stochastic characterization of the tensile strength ofthe lamina-tion stock was warranted for this glued-laminated beam problem an additional simulation was performed with the tensile strength modeled only as a random variable. This ap-proach is equivalent to assuming for the tensile strength Material Model 3, as presented in Chapter 4. The cumulative distribution of load-carrying capacity of the beam using this Chapter 5. SFEA — Calibration, Verification and Application 103 material model is compared against the stochastic strength model and the beam test data in Fig. 5.17. It is observed from Fig. 5.17 that, over all probability levels, a random vari-able approach for modeling the tensile strength grossly underestimates the load-carrying capacity of the beam. This conclusion applies, in general, to the strength evaluation of all glued-laminated beams and this fact has been acknowledged by researchers for some time (Foschi 1980). 5.6.2 24 ft Beams The cumulative distribution of load-carrying capacity from the 24 ft beam tests and the ULAG simulation results are presented in Fig. 5.18, with the corresponding distribution statistics summarized in Table 5.4. The predictions given by ULAG are based on a beam discretization of 50 finite elements and a simulation size of 200 beams. Figure 5.18 shows that the ULAG simulation results are in relatively good agreement with the beam test data, underestimating the load-carrying capacity by approximately 5% to 10% over all probability levels. From Fig. 5.18 it is seen that for 7 of the 30 test beams collapse was initiated by end-joint failures. It was observed during the test program (Timusk et al. 1994) that in all these instances the end-joints which failed were situated close to midspan in the bottom lamina of the beam. In addition, it is observed from Fig. 5.18 that of these 7 beams 5 had load-carrying capacities below the median value of the test group. It is suspected that the difference between the test results and the ULAG predictions are partially attributable to the modeling of the end-joint strength. To investigate this further, an additional simulation was conducted with an artificially high strength assigned to the end-joints to eliminate the possibility of any end-joint failures within the beams. The resulting cumulative distribution of load-carrying capacity is shown in Fig. 5.19 and the accompanying statistics are given in Table 5.4. It is seen from these results that under the condition of no end-joint failures the simulation estimates Chapter 5. SFEA — Calibration, Verification and Application 104 are in better agreement with the censored test data, which consists of the 23 beams which did not have end-joint failures. This censored test data has the following nonparametric statistics for load-carrying capacity: mean of 128.3 kN, COV of 10.9% and 5th percentile of 103.9 kN. It is noted that the 5th percentile estimate by ULAG is now within 3% of the corresponding interpolated test value. From this exercise it can be concluded that there exists an increased discrepancy between ULAG predictions and the test results when end-joints failures occur in the beams. This finding is, in part, not unexpected given that the accuracy of the end-joint test data was brought into question in Section 5.3. It is suggested that further test data from beams constructed with end-joints is required in order to determine more fully the predictive capability of ULAG for beams with end-joints. Figure 5.20 shows an expanded view of the two ULAG simulation results and the test data up to the first quartile probability level. As can be seen from Fig. 5.20 the ULAG simulation with end-joint failures eliminated is an upper bound estimate to this beam's load-carrying capacity. Furthermore, this simulation result provides the best agreement with the uncensored test data. Also shown in Fig. 5.20 are 2-P Weibull fits to the lower quartile of each ULAG simulation result. The statistics from these fits are given in Table 5.4. These fitted results could form the basis for a reliability study, with the understanding that these distributions provide a conservative estimate to the beam's true load-carrying capacity. 5.7 Size Effect Study using U L A G The statistical size effect that manifests itself in the strength of glued-laminated beams is a complex phenomenon that involves a number of intervening variables: location and intensity of the load application as this determines the critically stressed volume of the Chapter 5. SFEA — Calibration, Verification and Application 105 beam; spatial distribution and severity of strength reducing defects within the lamination material; spatial distribution of end-joints throughout the beam, which is directly related to the available length of lamination material used in the construction of the beam. The contributing influence of each of these variables cannot be economically determined by experimental testing alone. Obviously, the majority of the investigation must be conducted using a simulation model such as ULAG. Experimental testing can then be limited to calibrating and verifying the simulation model. As an example of a size effect study ULAG is used to determine the length and depth effect of a simply supported glued-laminated beam subjected to a uniformly distributed load over the entire span up to failure. The reference beam has a nominal 16 ft (4.725 m) span, comprised of 8 Grade C laminae, each dimensioned 38 mm x 126 mm, with 60% of the lamination stock 4.0 m long and the remainder at 2.5 m. Consequently, each beam will contain end-joints in each lamina. Simulations were conducted for this reference beam configuration and for beams with the span and the cross-sectional depth halved and doubled. Comparison of the simulation results is based on the modulus of rupture (MOR) of each beam, as discussed previously in Section 4.5. A summary of the findings is presented in Table 5.5, where the stated statistics are given in terms of 2-P Weibull fits to the lower quartile of the simulation results. The cumulative distributions of MOR, up to the first quartile, for changing values of beam span and depth are shown, respectively, in Figs. 5.21 and 5.22. In addition, the equations of Weibull's brittle fracture theory were fitted to the 5th percentile strength values in order to calibrate the length and depth size parameters. According to Weibull's brittle fracture theory the length and depth effect can be described by Eqs. (4.15) and (4.16), respectively. In this study these equations were altered to reflect evaluation of the beam strength at the 5th percentile probability level which corresponds to the characteristic strength value specified in the Canadian design code Chapter 5. SFEA — Calibration, Verification and Application 106 (CSA 1994). The resulting fitted size effect parameters for length and depth are given in Table 5.5. It must be appreciated that these parameter values are only applicable to this particular beam configuration and loading. For example, if the available length of lamination stock changed then the spatial distribution of end-joints throughout the beams would be altered; in turn, this would effect the strength of the beams as a function of size. In closing this section it is interesting to note that the Canadian design code (CSA 1994) makes no provision for a size effect adjustment factor to be applied to the specified bend-ing strength values of glued-laminated beams. 5.8 Reliability Analysis using U L A G and R E L A N The manufacture of glued-laminated beams in Canada is carried out in compliance with a national standard (CSA 1989) in order to ensure that the structural properties of the constructed beams are commensurate with the corresponding values specified in the Canadian design code (CSA 1994). Codes, by their very nature, however, are comprised of simple conservative design equations. Consequently, the glued-laminated beam manu-facturer, who is fabricating beams to these criterion, may not be utilizing the full capacity of the beams. Ideally, the manufacturing process should be carried out to optimize the beam's load-carrying capacity against an acceptable reliability constraint. This can be achieved by first using ULAG to determine the load-carrying capacity of the beam that is proposed for a particular application. With the capacity of the beam statistically quantified its reliability can then be evaluated so that the probability of nonperformance (failure) is below an accepted level. This failure analysis can be performed using a general reliability program such as RELAN: REliability ANalysis (Foschi and Folz 1992), which utilizes FORM, SORM and response surface solution strategies. An example of how these Chapter 5. SFEA — Calibration, Verification and Application 107 two programs could be used in concert to optimize the performance of glued-laminated beams follows. Consider the case of manufacturing nominal 16 ft (4.725 m) long glued-laminated beams comprised of 8 laminae. The available lamination stock is a combination of Grade B and C material, with each lamina having a cross-sectional size of 38 mm x 126 mm. The Grade C lamination stock has a length of 4.725 m while the Grade B material is available in two packets: for the first all of the lamination material is 4.725 m long and for the second 60% is 4.0 m long and the remainder is 2.5 m long. Consequently, end-joints will only appear in the beams if the second packet of Grade B lamination stock is used. The material properties of the lamination stock are taken to be identical to those obtained from the U.B.C. test program. In their use the beams are to be simply supported and must carry both a uniformly distributed occupancy load and a dead load, equal to 25% of the mean live load. The design situation for this beam can be described in term of a performance function: G = (j)KDqu-[1.25D+ 1.50L) • s (5.23) where <j> — 0.90 is the Code assigned resistance factor (CSA 1994), KD = 0.85 is a load duration adjustment factor3, qu is the load-carrying capacity of the beam as determined by ULAG, 1.25D and 1.50L are, respectively, the factored dead and live loads (expressed as pressures) applied to the beam and s is the spacing between beams (or the tributary loading width for the beam). With reference to Eq. (5.25), nonperformance of the beam is given by G < 0; conversely, G > 0 implies performance as intended and G = 0 defines the limit state between performance and nonperformance. The probability of 3The load-carrying capacity qu is obtained using ULAG, for which the input material properties are based on short-term testing. In application, the dead and live loads are sustained over the service life of the beam, which in this case is taken to be 50 years. Due to creep rupture the load-carrying capacity of the beam degrades with time. Consequently, a load duration adjustment factor must be applied. From studies on sawn lumber beams, using a damage accumulation model, it was determined that KD — 0.85 for this loading (Foschi et al. 1989). This same value will be assumed appropriate in this study. Chapter 5. SFEA — Calibration, Verification and Application 108 nonperformance can be determined by calculating the probability of the event G < 0. This probability calculation, which is commonly given in terms of the reliability index /?, will be obtained using the first order reliability method (FORM) within RELAN. The manufacturer's objective is to economically construct a beam that performs its intended function under the constraint of achieving a target reliability level of /3 = 3.0, which is commensurate with the implied reliability in the Canadian design code equations. In this particular study optimizing the beam's load-carrying capacity will be restricted to only changing the number of Grade B laminae and whether or not they will contain end-joints. In Eq. (5.25), qu, D and L are assumed to be independent random variables. The occupancy load L applies to offices and is assumed to follow a Gumbel distribution for maxima over a 50 year return period (Foschi et al. 1989). The dead load D is taken to be normally distributed. The statistical information for the loads is summarized in Table 5.6. ULAG determines qu; which, in turn, is fitted to a 2-P Weibull distribution using the lower quartile of the simulation results. Table 5.7 summarizes the statistical information from the simulation results of beams with all Grade C laminae to beams with up to four Grade B laminae. As already noted, two cases are considered for beams with Grade B laminae; the first has no end-joints, while the second has end-joints based on the available lengths of Grade B lamination stock. A summary of the 2-P Weibull load-carrying capacity statistics from the ULAG simulations is given in Tables 5.7 and 5.8. It is interesting to note that inclusion of end-joints in the beams decreases the mean load-carrying capacity and increases the COV, as compared to beams without end-joints. With the probabilistic description of qu, D and L established, RELAN was employed to determine the reliability of the glued-laminated beams as a function of the number of Grade B laminae, whether or not end-joints were present and the spacing between adjacent beams (set at s — 1.0 m, 2.0 m, 3.0 m and 4.0 m). The reliability analysis is Chapter 5. S F E A — Calibration, Verification and Application 109 summarized in Tables 5.9 and 5.10 for glued-laminated beams without and with end-joints, respectively. These tabulated results are also displayed graphically in Figs. 5.23 and 5.24. In turn, from these graphs, allowable values of beam spacing can be determined under the criterion of ensuring a reliability level of f3 = 3.0. These results are presented in Table 5.11. Now, as an application of this analysis, assume that these beams must be placed such that s = 2.30 m. Table 5.11 shows the manufacturer that he can either construct beams which have one Grade B lamina without end-joints or use two Grade B laminae with end-joints. 5.9 C o n c l u d i n g R e m a r k s It has been shown that the parameters defining an ergodic tensile strength process (based on a mean value plus a summation of sinusoids) can be fitted through a barrier crossing analysis to reproduce the cumulative distribution of tensile strength of lamination stock of a given length. This approach was adopted to fit tensile strength processes to data obtained from a U.B.C. testing program of lamination stock. In turn, this procedure was incorporated in the stochastic finite element program ULAG: Ultimate Load Analysis of Glulam. This stochastic beam model uses Monte Carlo simulation to predict the cumulative distribution of load-carrying capacity of a beam. Comparison was made between the ULAG estimates and data obtained from a U.B.C test program of glued-laminated beams. It was found that for beams without end-joints in the laminae the ULAG predictions were in very good agreement with the test data. For beams with end-joints the correlation with the test data was not as close but deemed within an acceptable level. Question was raised as to the accuracy of the end-joint test data in calibrating the database used by ULAG. Further investigation is required in this area to ascertain ULAG's predictive capabilities. Chapter 5. S F E A — Calibration, Verification and Application 110 Finally, the use of ULAG as an analysis and design tool was demonstrated through a size effect study and an example of optimizing a beam's load-carrying capacity under a reliability constraint during the manufacturing stage. Chapter 5. S F E A — Calibration, Verification and Application 111 Table 5.1: Summary of U.B.C. test program - lamination stock. Sample Tensile Strength Modulus of Elasticity Test description size Mean COV Mean COV (MPa) (%) (GPa) (%) Grade B - 8 ft specimens 100 38.39 22.70 13.29 6.48 Grade B - 16 ft specimens 100 33.01 18.57 13.33 6.15 Grade C - 8 ft specimens 100 30.80 20.85 11.57 4.88 Grade C - 16 ft specimens 100 23.97 24.90 11.59 4.38 Grade B - end-joints f 100 34.48 20.34 12.81 2.70 Grade C - end-joints f 100 28.39 21.86 11.12 2.72 fin the Grade B end-joint tests 66 of the 100 specimens failed at the end-joint. $In the Grade C end-joint tests 43 of the 100 specimens failed at the end-joint. Table 5.2: Calibration results for Grades B and C lamination stock. Grade S* os bs (MPa) (MPa) (m) B 45.07 7.99 0.962 C 38.48 9.24 0.958 Chapter 5. S F E A — Calibration, Verification and Application 112 Table 5.3: Load-carrying capacity statistics from the 16 ft U.B.C. beam tests and the ULAG predictions. Statistics U.B.C Tests ULAG 2-P Weibull Fit f ULAG (R.V)t Mean load (kN) 87.16 82.66 79.48 58.41 COV in load (%) 17.89 14.72 8.16 24.34 5th percentile load (kN) 57.85 59.32 61.77 35.09 f Fit is made to the first quartile of the simulation results. JIn this simulation the tensile strength is modeled as a random variable. Table 5.4: Load-carrying capacity statistics from the 24 ft U.B.C. beam tests and the ULAG predictions. Load U.B.C ULAG 2-P ULAG 2-P Statistics Tests Weibull Fit t (No end-joints) f Weibull Fit t Mean (kN) 125.6 119.3 121.5 122.0 122.6 COV (%) 10.95 9.98 11.00 10.04 11.98 5th percentile (kN) 104.3 97.06 97.13 101.1 99.81 fin this simulation no end-joint failures are allowed to occur. |Fit is made to the first quartile of the simulation results. Table 5.5: Summary of a size effect study using ULAG. Length Effect Depth Effect L0/2 L0 2L0 hQ/2 ha 2h0 i?o.o5 (MPa) f 27.88 26.33 25.01 27.63 26.33 23.49 Fitted Parameters kL = 12.7 h = 9.1 f All i?o.o5 values given above are based on 2-P Weibull fits to the lower quartile of the ULAG simulation results. Chapter 5. S F E A — Calibration, Verification and Application 113 Table 5.6: Dead and live load statistics. Load Type Probability Mean COV Model (kPa) (%) Occupancy Load (L) G umbel 2.40 23.0 Dead Load (D) Normal 0.60 10.0 Table 5.7: 2-P Weibull load-carrying capacity statistics for 16 ft beams, varying the number of Grade B laminae, with no end-joints. Grade B Scale Shape 5th Percentile Laminae (kN/m) (kN/m) 0 24.54 9.704 18.07 1 27.55 11.70 21.37 2 29.23 12.38 23.00 3 29.75 12.82 23.60 4 30.07 12.63 23.76 Table 5.8: 2-P Weibull load-carrying capacity statistics for 16 ft beams, varying the number of Grade B laminae, with end-joints. Grade B Scale Shape 5th Percentile Laminae (kN/m) (kN/m) 0 24.54 9.704 18.07 1 27.19 10.57 20.53 2 27.95 11.35 21.52 3 28.65 12.17 22.45 4 29.14 12.33 22.34 Chapter 5. S F E A — Calibration, Verification and Application 114 Table 5.9: Reliability analysis of 16 ft beams, varying the beam spacing and the number of Grade B laminae, with no end-joints. Grade B /3-values Laminae s = 1.0 m s = 2.0 m s = 3.0 m s = 4.0 m 0 4.48 2.81 1.48 0.32 1 5.12 3.35 2.02 0.88 2 5.37 3.59 2.26 1.15 3 5.48 3.69 2.35 1.24 4 5.47 3.70 2.38 1.28 Table 5.10: Reliability analysis of 16 ft beams, varying the beam spacing and the number of Grade B laminae, with end-joints. Grade B /^ -values Laminae s = 1.0 m s = 2.0 m s = 3.0 m s = 4.0 m 0 4.48 2.81 1.48 0.32 1 4.87 3.20 1.91 0.80 2 5.08 3.36 2.05 0.93 3 5.28 3.51 2.18 1.06 4 5.35 3.58 2.25 1.14 Chapter 5. SFEA — Calibration, Verification and Application 115 Table 5.11: Allowable spacing for the 16 ft beams, ensuring a reliability level of 8 = 3.0. Beam Spacing 5 (m) Grade B Without W l Laminae End-joints End-joints 0 1.90 1.90 1 2.30 2.15 2 2.45 2.30 3 2.55 2.40 4 2.56 2.45 Chapter 5. S F E A — Calibration, Verification and Application 116 80 L e n g t h ( m ) Figure 5.1: Realization of tensile strength process showing 5m t-„ and strength barrier a. Figure 5.2: Cumulative distribution of tensile strength obtained from barrier crossing analysis. Chapter 5. S F E A — Calibration, Verification and Application 117 Figure 5.3: Cumulative distribution of tensile strength of Grade B lamination stock. Tensile Strength (MPa) Figure 5.4: Cumulative distribution of tensile strength of Grade C lamination stock. Chapter 5. S F E A — Calibration, Verification and Application 118 1 4 0 0 0 CL 1 3 0 0 0 > v 1 2 0 0 0 -t—• O 1 1 0 0 0 • 10000H in =3 9 0 0 0 "a o Test Data. Regression Line. 8 0 0 0 i i i i i i i i i i ' ' • ' i ' i ' ' i ' ' ' ' i ' ' 1 1 i ' • ' • i ' ' ' ' i ' 1 1 1 i ' 1 ' 0 5 10 15 2 0 2 5 3 0 3 5 4 0 4 5 5 0 Tens i l e S t r e n g t h ( M P a ) Figure 5.5: Scatter plot of Grade B tensile strength versus modulus of elasticity. 1 4 0 0 0 r2 1 3 0 0 0 > » 1 2 0 0 0 --4—' ' u O 1 1 0 0 0 Lu T5 1 0 0 0 0 <f> 3 9 0 0 0 o 8 0 0 0 Test Data. Regression Line. i i i i i i i i i i i i i i i i i i i i i i i i i i I I i i i ' ' ' i ' ' ' ' i 1 1 1 1 i 1 1 1 • _ 0 5 10 15 2 0 2 5 3 0 3 5 4 0 4 5 5 0 Tens i l e S t r e n g t h ( M P a ) Figure 5.6: Scatter plot of Grade C tensile strength versus modulus of elasticity. Chapter 5. S F E A — Calibration, Verification and Application 119 Figure 5.7: Cumulative distribution of end-joint strength of Grade B and C lamination stock. Chapter 5. S F E A — Calibration, Verification and Application 120 kd 126 mm 16 Ft Beams 2712 mm , 1626 mm . 2712 mm >k >< > 10 Grade C Lams "2 Grade B Lams kd 126 mm 24 Ft Beams Figure 5.8: Full-scale glued-laminated beam testing configurations. Chapter 5. SFEA — Calibration, Verification and Application 121 25.0 Figure 5.9: Spectral density functions for Grades B and C lamination stock. Chapter 5. S F E A — Calibration, Verification and Application 122 80 70-tn 30-1 w20-c I— 10 Simulated Specimen No. 1 Simulated Specimen No. 2 0 H—i—i—i—i—I—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—I i—i i r-0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 x - coordinate (m) Figure 5.10: Realizations of Grade B tensile strength process. 80 70-O 0- 60 50-Simulated Specimen No. 1 Simulated Specimen No. 2 0 "I—i—I—i—i—I—i—I—i—I—|—I—i—I—i—|—I—i—i—i—|—i—i—i—I—|—i—i—i—i—|—i—i—i—i—I—i—i—i r 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 x — coordinate (m) Figure 5.11: Realizations of Grade C tensile strength process. Chapter 5. S F E A — Calibration, Verification and Application 123 10 15 20 25 30 35 40 45 50 55 60 Tensile Strength (MPa) Figure 5.12: Cumulative distribution of simulated tensile strength of Grade B lamination stock (L = 3.658 m). Tensile Strength (MPa) Figure 5.13: Cumulative distribution of simulated tensile strength of Grade C lamination stock (L = 3.658 m). Chapter 5. S F E A — Calibration, Verification and Application 124 B C D 1 I 1 L L L Figure 5.14: Assembly procedure for a glued-laminated member. Chapter 5. SFEA — Calibration, Verification and Application 125 U l t i m a t e L o a d 2 P U ( kN) Figure 5.15: Cumulative distribution of load-carrying capacity for the 16 ft beams -comparison of test results with ULAG predictions. 40 45 50 55 60 65 70 75 80 U l t i m a t e L o a d 2 P U (kN) Figure 5.16: First quartile cumulative distribution of load-carrying capacity for the 16 ft beams. Chapter 5. S F E A — Calibration, Verification and Application 126 Figure 5.17: Cumulative distribution of load-carrying capacity for the 16 ft beams -comparison of test results with ULAG predictions, assuming tensile strength to be a random variable. Chapter 5. S F E A — Calibration, Verification and Application 127 70 80 90 100 110 120 130 140 150 160 U l t i m a t e L o a d 2 P U ( kN) Figure 5.18: Cumulative distribution of load-carrying capacity for the 24 ft beams -comparison of test results with ULAG predictions. 70 80 90 100 110 120 130 140 150 160 U l t i m a t e L o a d 2 P U ( kN) Figure 5.19: Cumulative distribution of load-carrying capacity for the 24 ft beams -comparison of test results with ULAG predictions, with no end-joint failures. Chapter 5. S F E A — Calibration, Verification and Application 128 120 Figure 5.20: First quartile cumulative distribution of load-carrying capacity for the 24 ft beams. Chapter 5. S F E A — Calibration, Verification and Application 129 0.25 >s0.20 -O O - g O . 1 5 > 0.10 o | 0.05 O 0.00 15 L = U / 2 = 8 ft. — L = L 0 = 16 ft. ~ + L = 2L 0 = 32 ft. T 1 1 1 r-30 20 25  35 M o d u l u s of R u p t u r e ( M P a ) 40 Figure 5.21: Cumulative distribution of MOR for beams of various lengths ( L 0 = 4.725 m). 0.00 I r r^T^i—i—i—'—i—'—i—i—i—i— 1 — 1 15 20 25 30 35 M o d u l u s of R u p t u r e ( M P a ) 40 Figure 5.22: Cumulative distribution of MOR for beams of various depths (h0 = 304 mm). Chapter 5. S F E A — Calibration, Verification and Application 130 X 4 T J c > s -0 2 rr • • • • • 8 Grade C. 1 Grade B & 0 0 0 0 0 2 Grade B & * * * * * 3 Grade B & * - * - * ^ - K 4 Grade B & Target Reliability (3 Grade Grade Grade Grade C. C. C. C. 0 -|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—r—i—i—i—i—|—t—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—T—T-0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 B e a m Spacing (m) Figure 5.23: Reliability analysis of 16 ft beams, varying the beam spacing and the number of Grade B laminae, with no end-joints. 0 H—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—n—r-|—i—r-r-0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 B e a m Spacing (m) Figure 5.24: Reliability analysis of 16 ft beams, varying the beam spacing and the number of Grade B laminae, with end-joints. Chapter 6 Conclusions 6.1 Summary A relatively simple, yet comprehensive, stochastic finite element model has been presented herein for predicting the response variability in the load-carrying capacity of glued-laminated wood columns, beams and beam-columns. In this problem spatial stochasticity occurs in the elastic modulus and the compressive and tensile strengths of the lamination material. Central to this investigation was the calibration of the stochastic material models to actual test data. The finite element modeling was based on a one-dimensional higher order shear de-formation beam theory. It was shown that this model yielded an accurate representation of deformations and stresses in a variety of statistically deterministic beam problems. Material behaviour was, in general, assumed nonlinear; being perfectly brittle in tension and yielding in compression. Material failure occurred in a lamina or at an end-joint where ever the tensile strength was exceeded. Delamination was not considered. The beam-column model employed a nonlinear geometric and material Newton-Raphson so-lution strategy, tracing the load-displacement response up to collapse. Determination of the collapse load was directed by a bisection search of the prescribed loading sequence. Collapse of the beam-column occurred as a result of progressive material failure and/or an overall loss of structural stability. For the load-carrying analysis of laminated beams 131 Chapter 6. Conclusions 132 the finite element model was simplified. In this case failure was restricted to brittle tensile fracture of the lamination material. The finite element beam model employed repeated linear analysis steps applied to the progressively degrading beam up to the attainment of the collapse load. It was found that this simplified model provided more than a thirty-fold reduction in computational effort, timewise, when compared with the full nonlinear beam-column model. Further to this, the two models yielded collapse load predictions that were in good agreement one with the other. Column analysis was formulated as an eigenvalue problem with the model determining the critical elastic buckling load. In general, the compressive and tensile strength and the elastic modulus of the lam-ination material were modeled as one-dimensional homogeneous stochastic fields. A spectral approach was adopted to characterize all stochastic quantities. Realizations of each material property were simulated using a series representation of the stochastic field as a summation of sinusoids, each with a random phase, and weighted according to the spectral density function. For a given finite element discretization of a glued-laminated member material properties were assigned by using, for each lamina, the stochastic field values at the mid-point of each element. The collapse load response statistics of glued-laminated members (columns, beams or beam-columns) were determined through a Monte Carlo simulation, which employed, repeatedly, the stochastic finite element models discussed above. Initially, the stochastic modeling of the elastic modulus of the lamination material was characterized by a one-dimensional nonergodic process with its defining parameters calibrated against available test data. Good agreement was obtained between the cali-brated model and the test data. As a structural application, the response variability in the elastic buckling load of glued-laminated columns was investigated. From this study, it was determined that this problem did not warrant stochastic modeling of the elastic Chapter 6. Conclusions 133 modulus; a random variable approach was deemed sufficiently accurate. This conclu-sion implies that the between lamina variability in stiffness influences the response of a glued-laminated column more than the within lamina variability. This was followed by a sensitivity study on the response variability in the failure load of glued-laminated beams. In this investigation both the elastic modulus and the tensile strength of each lamina were modeled as one-dimensional correlated stochastic fields. The material models were not calibrated to experimental data. Rather, the defining parameters of the stochastic material models were varied, over a prescribed range, to determine their influence on the response of the beam. It was observed that the specification of the stochastic tensile strength model was of primary importance in influencing the beam response, whereas the elastic modulus and its cross-correlation with the strength had only a secondary influence. This finding has significant implication with respect to collection of test data; it eliminates the necessity to quantify the cross-correlation between the tensile strength and the elastic modulus. However, calibration of the tensile strength process remains an outstanding issue. Finally, this investigation concluded with a statistical size effect study. It was shown that the stochastic tensile strength model caused both a length and depth effect to be present in the bending strength of the laminated beams. It was acknowledged at the outset of this study that experimental determination of the tensile strength of the lamination material as a stochastic process would be an extremely difficult and expensive undertaking. Consequently, a simpler approach was sought. To this end, barrier crossing analysis was used to calibrate the defining param-eters of the tensile strength process so that simulated strength profiles reproduced, to a reasonable level of accuracy, the cumulative distribution of minimum tensile strength obtained from test specimens. With this approach the tensile strength was modeled as a one-dimensional ergodic process. Conventional tensile strength testing provides the Chapter 6. Conclusions 134 necessary information to perform this calibration procedure. This tensile strength model was utilized in the stochastic finite element beam model ULAG: Ultimate Load Analysis of Glulam. ULAG was based on the simplified stochastic finite element beam model. Numerous enhancements (eg. MOE-rating and proof loading of the lamination material) were incorporated within ULAG to make it a more complete analysis tool. The collapse load predictions of ULAG were compared against recent test results on full-scale glued-laminated beams. For the 16 ft beams, without end-joints in the laminae, the agreement was very good. For the 24 ft beams, with end-joints in the laminae, the agreement was deemed acceptable. Further testing is required in this area to ascertain ULAG's predictive capabilities for glued-laminated beams with end-joints. ULAG was then employed to demonstrate and quantify the statistical size effect in beam strength. As a final application, ULAG was combined with a first order reliability assessment procedure to optimize, at the manufacturing stage, the load-carrying capacity of a glued-laminated beam. Each of these applications demonstrated the usefulness of ULAG as an analysis and design tool. ULAG could be used by Code developers to carry out parameteric studies on the numerous variables that influence the load-carrying capacity of glued-laminated beams. For glued-laminated beam manufacturers ULAG allows them to optimize beam performance for the end use application of their products. In both these cases ULAG should be used in concert with experimental test programs. 6.2 Concluding Remarks To a large extent this thesis has focused on the application of stochastic finite element methodology to determine the response variability in the load-carrying capacity of glued-laminated wood beams. It was shown that this problem did in fact warrant the need for a stochastic finite element analysis; modeling of the tensile strength as a random Chapter 6. Conclusions 135 variable yielded unacceptable results. Calibration of the material models to actual test data was requiste for this stochastic finite element analysis to produce meaningful results that agreed with experimental testing. In a larger context these two fundamental points (one, that certain problems do require a stochastic finite element analysis and, two, that calibration of stochastic material models to test data is paramount to yielding usable results) formed the impetus for this thesis; the particular problem that was studied was, in one respect, just a vehicle to provide substantiation of the thesis. At the outset of this study it was acknowledged that much meaningful research has been carried out under the umbrella of stochastic finite element analysis. In many cases, however, this research has focused on refining the underlying methodology without making a real connection with applications that may require such a sophisticated level of analysis. In this regard, it is humbly contended that more research which marries this theory with realistic applications is overdue. 6.3 Further Areas of Research A straightforward extension of the present work is to evaluate the response variability in glued-laminated beam-columns. The required stochastic finite element model has been presented in study. The representation of the stochasticity of the compressive strength is identical in form to the stochastic tensile strength model. All that remains is the calibration of the compressive strength process to available test data. It would be of interest to extend the present work to consider delamination as a contributing mechanism to the progressive collapse of the glued-laminated members. 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YAMAZAKI, F., SHINOZUKA, M. and DASGUPTA, G. (1988). Neumann Ex-pansion for Stochastic Finite Element Analysis. Journal of Engineering Mechanics, ASCE, 114(8), 1335-1354. 70. ZHANG, J. and ELLINGWOOD B. (1995). Effects of Uncertain Material Properties on Structural Stability. Journal of Structural Engineering, ASCE, 121(4), 705-716. Appendix A Theory and Simulation of Stochastic Fields A . l Introduction This appendix presents an overview of the basic theory of stochastic fields1 which will be applied throughout this study to develop spatially stochastic material models for imple-mentation within a finite element analysis: The presentation is restricted to considering only one-dimensional homogeneous stochastic fields. Key to any stochastic analysis is the discretization of the parameter space of a process into a denumerable set of random variables. The approach adopted herein, is to use a cosine series involving random phase angles and amplitudes weighted according to a spectral representation of the process (Shinozuka 1987; Shinozuka and Deodatis 1991). The underlying theory and properties implied by this type of stochastic characterization (eg. Gaussianness and ergodicity) are highlighted for the univariate and bivariate cases. The numerical simulation of realizations of a stochastic process further requires that the process be represented by a finite number of random variables. Formulae by which this can be achieved are also investigated. In addition, the basic underlying theory of spectral moments and barrier crossing analysis is presented. 1 The term stochastic field is generally used to describe random spatial variations in a physical quan-tity. This permits one to make a distinction from a stochastic process which quantifies the random temporal variations of a physical quantity. In this study these two terms will be taken as synonymous. 144 Appendix A. Theory and Simulation of Stochastic Fields 145 A.2 Spectral Representation of Stochastic Processes A.2.1 One-dimensional univariate homogeneous stochastic processes Let f0{x) be a one-dimensional univariate homogeneous stochastic process parameterized with respect to the spatial variable x and having, without loss of generality, a zero-mean. Then the following relationships hold: S[f0(x)} = 0 (A.l) £[fo(x)f0(x + t)] = Rfofo(0 (A-2) where the operator £[•] denotes ensemble expectation and £ is the lag distance between sampling points along the z-axis. Equation (A.2) defines the auto-correlation function Rf0f0(£) for a process which is homogeneous in the weak sense. It follows from Eq. (A.2) that Rf0f0(£) is an even function: Ruuti) = €[fo(x)f0(x + 0] = £[fo{x - Ofo(x)) = Rfofo(-0 (A3) Under the condition that Rj0}0{£) l s absolutely integrable, its Fourier transform exists and defines the spectral density function >?/„/„(«;): 1 r°° Z7T J-<x> where K is the wave number, having the dimensions of radians per unit length. It is necessary to impose the condition that Sfo/o(0) = 0 to satify the requirement that f0(x) is a zero-mean process. It follows from Eqs. (A.3) and (A.4) that SJ0J0(K) is also a real-valued even function: 1 f°° SjoM = SMo(-K) = — RUJ£) c o s « ) di (A.5) Appendix A. Theory and Simulation of Stochastic Fields 146 In turn, Eq. (A.5) motivates the introduction of the one-sided spectral density function GjDf0(n) such that GMO(K) = 2SFEJO(K), K > 0 (A.6) and 2 r°° GSoM = - i2/„/ o(0cos( K0^ (A.7) 7T JO The inverse Fourier cosine transform of Eq. (A.7) yields ' GMo(K)cD*(Kt)dK (A.S) 0 Equations (A.7) and (A.8) constitute one form of the well-known Weiner-Khinchine relations. By setting £ = 0 in Eq. (A.8) the following important result is obtained /•oo = */./.(<>) = / GJofo(K)dK (A.9) J 0 where a"jo is the ensemble variance of f0(x). For future reference, the normalized (unit area) one-sided spectral density function gj0j0(ri) is introduced: 1 r°° 9uM = ~T GSoJo{*)dK (A.10) CTf JO Jo Now consider the cosine series representation of a one-dimensional homogeneous stochas-tic process f(x) given by N f(x) = lim V An cos(Knx + <£n) (A l l ) ]\—*-0O n=l with wave numbers Kn = A/e(2n —1)/2, amplitudes An and mutually independent random phase angles <f>n, which are uniformly distributed over the interval (0,27r). Implicit in the formulation of Eq. (A.ll) is the condition that as N —• oo and A/c —> 0 the product • A K remains constant. A p p e n d i x A . Theory and Simulat ion of Stochast ic F ie lds 147 The objective in this presentation is to determine the form of the An such that f(x) agrees with the target process f0(x) up to second order statistics. To this end, consider the ensemble variance of f(x): *} = *//(()) = S[f(xf) N N = x i i m £ £ AnAm£[cos(nnx + <f>n) cos(Kmx + <f>m)] N-*oo —' —' n=l m=l N { f27r 1 lim YsAn\ 7T COS2(KUX + <f>n) d(f)n = Hm E Ul (A.12) n = l In obtaining the above result use was made of the orthogonality of the cosine functions and the independence of the random phase angles. Now from Eq. (A.9) a ) o = / Gf<,fo(K)dK = lim^o E G /o/ 0 (««)A/c (A13) JO N->oo n = 1 Equating these results yields An = yjlGsjA**)** Thus, Eq. (A.ll) can be rewritten as N N-+oo f(x) = lim J2 v2G / o / o(Kn)A/ccos(fi;nx + (j)n) (A.15) 71 = 1 It can be shown (Shinozuka and Deodatis 1991) that the cosine series representation given by Eq. (A. 13) has the following properties: £ [ / ( * ) ] = TO]=0 (A.16) RJAO = RuM) (A.l?) GJM = G/o/„(/c) (A.18) A p p e n d i x A . Theory and Simulat ion of Stochast ic F ie lds 148 It is also immediately apparent from the form of E q . (A.15) that, by virtue of the central l imit theorem, f(x) is Gaussian. Thus, if f0(x) is also Gaussian no distinction is required between f(x) and f0(x)- Furthermore, it can be shown (Shinozuka and Deodatis 1991) that each realization of the stochastic process produced by E q . (A.14) is ergodic up to second order moments. Let f^l\x) denote a generic sample function, then 1 rL < / « ( x ) > = l im - / fW{x)dx = £ [ /„ (*) ] = 0 L-+00 L Jo = </(tW°(*+o>=*/./.(o (A.19) (A.20) where the operator < • > defines spatial expectation. It is also directly apparent that the sample functions are bounded in magnitude: /<•>(*) |< J i m J2 ^2Gufo(Kn)AK (A.21) n=l A.2 .2 One-d imens iona l bivariate homogeneous stochastic processes Consider next a pair of one-dimensional homogeneous stochastic processes fj(x) (j = 1,2) each with zero-mean £ [ / ; ( * ) ] = 0 (A22) and having cross-correlation and cross-spectral matrices R-(0 = R°2l(0 R°22(0 s°(«) S^K) S°M S , 2 1 ( K ) 5"|2(/C) (A.23) where S°k(n) is the Weiner-Khintchine tranform of R°k(£). It follows from the homogen-ity of the stochastic processes that (A.24) and therefore (A.25) Appendix A. Theory and Simulation of Stochastic Fields 1 4 9 where (•) denotes complex conjugation. Thus, in general, Sjk(n) is a complex quantity for j ^ k. Now consider the cosine series representation of a pair of one-dimensional homoge-neous, stochastic process given by N /i(x) = lim V Aln cos(Knx + <j>ln) (A.26) 7V-+oo — ' n—1 { N N "| J2 A2n cos(nnx + an + (f>ln) + V A3n cos(/c„x -f <f>2n) \ (A.27) 71 = 1 71 = 1 y with K N , Ain and </>jn (j = 1,2; / = 1,2,3) denoting quantities which have been previously defined in the univariate case. The objective here is to determine the form of the A\n and the phase angle an such that each fj(x) agree with the target process f°(x) up to second order statistics. For brevity of presentation, the following results are given without derivation: N , i2 u(0 = l i m ^ - A l n c o s ( K n O (A.28) 71 = 1 N : ^12(0 = lim £ - A i „ A 2 n COS(AC„£ + a„) (A.29) N^°°n=l 2 ^ 2 ( 0 = JimEj(4 + 4)«»(«.0 (A-3°) TV—»oo ' Z 71 = 1 Now, N 1 -oo N i?„(0) = lim V -A l n = ^ ( 0 ) - / G^Md/c = lim £ G 1 X ( K „ ) A « (A.31 .31) which implies that A l n - ^ 2 G ° U ( ^ ) A K (A32) as was obtained before for the univariate case. Next, N " 1 roo R12{0) = lim £ - A l n A 2 n cos <*n = i2J2(0) = / S°2{K) 6.K (A.33) Tl=l A p p e n d i x A . Theory and Simulat ion of Stochastic F ie lds 1 5 0 To advance the derivation it is advanageous to write S°2(K) in complex polar form as S°2(K) =| SI2(K) I elB where | ^ ( K ) | is the modulus of S^2(K) and 9 is the phase angle obtained from tan0 = Im {S^2(K)}/ Re {SI2(K)}, where Re{-} and Im{-} denote, respectively, the real and imaginary parts of a complex quantity. Then, W 1 , C O #12(0) = lim J2 - A i „ A 2 n c o s a n = / | G°12(K) \ cos 6 dK N = lim I Gl2Un) I cos 6n AK (A.34) By setting an = 6n, the following result is obtained A K — 0 N - > C O n = 1 where 7 2 ( K ) defines the coherence function: It can be shown (Bendat and Piersol, 1986) that 0 < 7 2 (K ) < 1 (A.37) Next, i?22(0) = lim £ \(A22n + A23n) = R°2(0) = f°° G°22(rz) dK = lim £ G°22(rtn)AK (A38) from which A 3 n = sj2G°22(Kn)Art - A \ n = v / 2C?§ 2 (« n )A« • yj\ - 7 2 K ) (A39) Substituting Eqs. (A.32), (A.35) and (A.39) into Eqs. (A.26) and (A.27) yields the final result: N N—»oo I = lim J2Gf1(Acn)A/cncos(/cnx + ^ i „ ) (A.40) n=l Appendix A. Theory and Simulation of Stochastic Fields 151 f2(x) - lim < V J2G°22(KN)&KN • V 7 2 ( A c n ) cos{nnx + 9n + <f>ln) N—too I ., v Kn=l N 1 + J2 v /2^ 2(/c n)A/c n • yjl - 7 2 ( « n ) cos(Knx + 4>2n) \ (A.41) n=l J where, 72(/cn) is defined in Eq. (A.36) and 9n = tan - 1 Im {G°12(Kn)} (A.42) [Re {G°12(Kn)} The implications of total incoherence, for which 72(K) = 0, and perfect coherence, ob-tained with 72(K) = 1, are immediately obvious in the above series equations. The verification that the above series representation of one-dimensional bivariate homogeneous stochastic processes agrees with the target processes fj(x) up to second order statistics can be found in the literature (Shinozuka 1987). Hereafter, the nomenclature will not distinguish between the series representation of a process, say f(x), and its target process f0{x). A.3 Simulation of Stochastic Processes A procedure to numerically simulate one-dimensional univariate homogeneous stochastic processes will now be outlined; the bivariate case will not be considered as it follows by direct extension. From the infinite series representation of f(x) given by Eq. (A. 15), the following approximation makes this formulation amenable to numerical simulation: N f(x) = z\Z y2Gff{nN)AK cos{Knx + <j>n) (A.43) 71=1 with KN = A/c(2n - l)/2, A K = K U / N (A.44) where KU is an upper cut-off wave number at which the one-sided spectral density func-tion Gff(nu) is assumed to vanish. The error e involved in trunctating (?//(«) as a Appendix A. Theory and Simulation of Stochastic Fields 152 function of be quantified as follows: / • K u e(/c„) = 1 - / gfj(K)dK (A.45) Jo with the objective being that e <C 1. A number of important consequences result from this simulation formula (Shinozuka and Deodatis 1991). First, f(x) is periodic with wave length L P = 2TV/AK. In turn, each realization f^(x) is now ergodic to second order when the length of the simulated sample function equals L P . Second, the simulated stochastic process is asymptotically Gaussian as N ^ oo. The rate of convergence to Gaussianness has been investigated in the above cited reference. Finally, the rate of convergence of the simulated stochastic process to match the truncated target spectral density function G / 0 / 0 , given the conditions set out in Eq. (A.44), is l/N2. The computational effort required to numerically simulate sample functions from Eq. (A.43) can be drastically diminished by application of transform techniques; particular attention will be given herein to the fast Fourier transform (FFT) and the fast Hartley transform (FHT). In order to take advantage of the FFT technique, Eq. (A.43) can be rewritten as f{x) = Re ( £ B N exp(i/cnx)l (A.46) I n=0 J with KN = UAK, AK = KU/N, B N = ^2Gjj(KN)AK • exp(z^n), B 0 = 0 (A.47) By construction, Eq. (A.46) is to be evaluated at x = pAx with (p — 0 , 1 , . . . , M — 1). As defined, Eq. (A.46) is periodic with wave length L P = 2%/'AK — MAx. Noting that the following relationship holds A x = (A48) MAK Appendix A. Theory and Simulation of Stochastic Fields 153 Eq. (A.46) can be rewritten as f(pAx) = Re j J2 Bnexp(i2wnp/M)\ (A.49) I 71=0 J to which the FFT technique is readily applicable. In order to avoid aliasing it is necessary that Ax < — = — A.50 ~ Ku An v ' In turn, this requires that M > 2N. With this approach the greatest computational efficiency is achieved when M is an integer power of 2. The complex arithemetic implicit in Eq. (A.49) can be avoided by application of the Hartley transform (Bracewell 1986; Winterstein 1990), which takes the form M - l M - l f{PAx) = { J2 H n {cos(i2nnp/M) + sm(i2Tmp/M)} = ^ ^"ncas (i2Tnp/M) (A.51) 71=0 71=0 with HN = ^2Gff(Kn)AKCOs6n + \j2G}j{nM-n) sin 6M_n, 6n = <f)n + 7r/4, HQ = 0 (A.52) The computational advantages afforded by Eqs. (A.49) and (A.51) do not come without some limitations. In particular, simulation of sample functions by either FFT or FHT algortihms requires that the entire record of length Lp be generated and that the spacing between points in the spatial domain must satisfy Eqs. (A.48) and (A.50). This latter point is somewhat restrictive when one is simulating sample functions of a particular quantity for a finite element analysis in which the element discretization is nonuniform. Of course this problem does not present itself with Eq. (A.43) Attention has focused thus far only on Gaussian stochastic processes. An iterative simulation procedure, which is an extension of the method presented above, has been de-veloped to generate non-Gaussian stochastic processes (Yamazaki and Shinozuka 1988). Appendix A. Theory and Simulation of Stochastic Fields 154 Implementation of such a procedure, however, presumes that one has available sufficient data that mandates the use of a non-Gaussian model. A . 4 Spectral Moments Various important characteristics of a homogeneous stochastic process can be expressed in terms of the first few moments of its one-sided spectral density function (Vanmarcke 1972). For a homogeneous stochastic process f(x) with spectral density function G(K), the m-th spectral moment is defined by roo Xm = KMG(rt)dK m = 0,1,2,- •• (A.53) Jo In particular, the moment of order zero equals the variance of the process; A0 = cr2,. If the process is at least once mean square differentiable then the second moment equals the variance of the derived process; that is, A2 = a2,, with / ' = df/dx. Furthermore, to each spectral moment a characteristic wave number is defined by *™=MM , m = l,2,.-. (AM) If G(K) is viewed as a spectral mass, then by geometrical interpretation K\ gives the abscissa of the centroid of G(K) and A 2^ equals the radius of gyration of G(K) about the origin. By this analogy, K2 indicates where the spectral mass is concentrated along the wave number axis. In turn, an apparent wave length LA for the process can be defined by La = %l (A.55) Combining the above results, the degree of dispersion or spread of G(K) about the cen-troidal wave number K\ can be quantified in nondimensional terms by the spectral band-width parameter 6: 5 = 1 -X0Xi 1 K2 1/2 0 < 6 < 1 (A.56) Appendix A . Theory and Simulation of Stochastic Fields 155 As an application to what has just been presented consider the family of spectral density functions of the form <?»(*) = ^2nb^Knexp(-bnK) n = 2,4,6,- •• (A57) Then, by straight forward evaluation, for a given value of n + m)\ a] n! bl n K m = i J M ^ O l ( A . 5 9 ) On V n! y V + 3n + 2 I a = oTTI ( A - 6 0 ) 27TO n It is noted from Eq. (A.61) that the bandwidth parameter has a maximum value of 6 = 0.5 when n = 2 and tends to zero as n becomes large. In an investigation under incomplete information a consistent comparison between competing forms of the spectral density function can be achieved by enforcing that each one has the same apparent wave length L A . For example, if one is considering to use either G^^) o r GS(K) in Eq. 1/2 (A.43) then a comparison can be made under the condition that 68 = ^/l5 / 2 62 a n a " o|(/c) = CXI(K). A.5 Barrier Crossing Analysis Consider a one-dimensional ergodic process F(x), which is at least once differentiable, and has the form F(x) = F* + f(x) (A62) Appendix A. Theory and Simulation of Stochastic Fields 156 with F* the ensemble mean and S[f(x)] = 0. A typical sample function obtained from this process is depicted in Fig. A . l . Let the probability that the process down-crosses the barrier F(x) = a in the interval (x, x -j- dx] be denoted by v~dx. It then follows that v~dx = Pr[{F(x) > a} D {F'(x)dx > a- F(x)}\ = Pr[{a - F* < f(x) <a-F* - f'dx}} ra-F'-f'dx P(fj')df, (with F' = — = / ' < 0) (A.63) = dx /0 r  — F df •oo Ja-F* dx where Pr[-] denotes the probability associated with the given event and p(f,f) is the joint probability density function. By application ofthe mean-value theorem, Eq. (A.63) can be expressed as " « = - f f'-(a-F*J')df (AM) J—oo Given that the process f(x) is homogeneous it follows that £[f(x)f'(x)] = 0. Under the further assumption that f(x) is zero-mean Gaussian then 2 / ' ) = ? ( / ) •?( / ' ) and Eq. (A.64) becomes 1 1 exp zircr ]0~ p I 2 afJ \as., (A.65) In turn, Eq. (A.66) can be rewritten as 1 ( ^ rl2Gf(K)dKV12 (a - F*)2 (AM) 2TT V Io°° Gf(n)dK 1/2 f exp *\2 2 [f~Gj(K)dK = Lib.) • exp{^-2T\XO I 2 with V = (a-jn AX (A.67) (A.68) Appendix A. Theory and Simulation of Stochastic Fields 157 In summary, Eqs. (A.67) and (A.68) gives the mean rate of down-crossing the barrier a by a realization of the ergodic Gaussian process F(x), expressed in terms of the ensemble mean F* and the spectral density function GJ(K). It is noted, in closing this section, that Eq. (A.64) is a particular form of Rice's Formula (Nigam 1983), which, in general, is applicable to any homogeneous process. Specialization of this formula to various non-Gaussian processes is available in the liter-ature (Grigoriu 1984). A.6 Probability Distribution of Minimum Values The minimum value of a stochastic process over a fixed interval of length L is obviously a random variable. For the ergodic process F(x) of Eq. (A.62) let denote this quantity: Fmin= mm{F(x)} (A.69) 0<x<L The probability distribution iV m i n is related to the barrier crossing analysis developed in the previous section in the following way PFrrin(a) = FriF^ <a} = Fr[La < L) = 1 - Pa(L) (A70) where La locates the first down-crossing of the barrier a, as shown in Fig. A . l , and Pa(L) is the probability of not down-crossing this barrier over the interval (0,Z/j. The formulation of this problem is equivalent to the first passage problem of random vibration theory (Nigam 1983). A general analytical solution to Eq. (A.70) is not available. However, if one accepts the widely used assumption that the barrier crossings follow a Poisson process then Pa(L) = exp(-^L) (A71) Appendix A. Theory and Simulation of Stochastic Fields 158 and ^ F m . „ H = l - e x p ( - ^ L ) (A.72) with v~ given by Eq. (A.67). While analytically appealing this approach suffers in a number of respects, with the resulting error dependent upon the bandwidth of the parent process F(x), when barrier crossing levels of practical interest are being considered. It is interesting to note, however, that the result is asymptotically exact as a —»• —oo (Cramer and Leadbetter 1967). The Poisson process assumption implies that each crossing of a barrier a is an independent event. For a narrow-band process this is a particularly poor assumption given that crossings tend to occur in clumps. On the other hand, for wide-band processes no allowance is made for the distance over which the process is below the barrier. In addition, this model implies that there is zero probability associated with the barrier being down-crossed at x = 0 and certain probability as x —> co. A.6.1 Two-State Markov Crossing Assumption An approximate analytical solution to Eq. (A. 70) has been proposed which addresses some of the deficiencies inherent in the Poisson model (Vanmarcke 1975, 1983). In particular, this formulation associates a finite probability to the event of being below the barrier a at x = 0. This model assumes an exponential form for the probability of not down-crossing the barrier within the interval [0,X]: Pa(L) = Pa{0) exp(-^ 0L), L > 0 (A73) where Pa(0) is the probability that the process is above the barrier a at x = 0 and p,a is the limiting decay rate of the first crossing probability. As a —• —oo, Eq. (A.73) should asymptotically approach the Poisson model; so that in the limit P„(0) = 1 and ua — u~. Appendix A. Theory and Simulation of Stochastic Fields 159 With reference to Fig. A . l , let L0 and L\ denote, respectively, the length over which the process is below and above the barrier between successive barrier down-crossings. From this perspective the pattern of barrier crossings can be viewed as a two-state Markov process. Based on the theory of recurrent events, the expected length between successive barrier down-crossings is given by S[L0 + L1] = ~ (A.74) and the expected length over which the process is below the process is given by J[L2]T , = f P(F)dF = 4= f eM-V2/2)dv = *(»?) (A.75) C [J-IQ ~r -kij J—oo \ / Z T T J—OO where 77 is defined by Eq. (A.68) and $(•) is the cumulative Gaussian distribution function. It then follows directly that Wo] = ( O " 1 •*(>/) (A.76) = ( 0 - 1 • [1 - ( A - 7 7 ) If it is further assumed that the length L\ is exponentially distributed with expectation S[LX) = = • [1 " HV)} (A.78) then the probability of initially starting above the barrier is given by Finally, making the necessary substitutions allows Eq. (A.72) to be rewritten as PF^(°) = 1 - [1 - * ( » / ) ] « p { - ^ 4^)]} (A-80) with v~ and 77 defined, respectively, by Eqs. (A.67) and (A.68). In summary, Eq. (A.80) gives the form of the cumulative distribution of minimum values for the Gaussian process Appendix A. Theory and Simulation of Stochastic Fields 160 F(x) of Eq. (A.62) over a length L, using the two-state Markov barrier crossing model. The dependence of this result on the length L, which is to be expected, is nevertheless duly noted. Appendix A . Theory and Simulation of Stochastic Fields 161 Appendix B Finite Element Shape Functions From Eq. (2.20) the displacement field of the beam-column element is given by u = N d (B.l) where u = [u, ij}, w]1 (B.2) and d = [u1,1p1,W1,01,U2,t/j2,W2,02]t (B.3) with 0i = dwi/dx for i = 1,2. The shape function array N in Eq. (B.l) has the explicit form r iVi 0 0 0 N2 0 0 0 N = 0 i V \ 0 0 0 i V 2 0 0 (BA) 0 0 #i H2 0 0 H3 HA where i V ; and Hj are, respectively, linear Lagrange and cubic Hermitian interpolating polynomials. In a natural (intrinsic) coordinate system, with axial coordinate £ E [—1,1], the shape function are given by ^2 = + 0 (B.5) (B.6) 162 Appendix B. Finite Element Shape Functions 163 H2 # 4 x ( 2 - a e + e3) (B.7) (B.8) (B.9) (B.10) with le the element length. The strain displacement vectors given in Eqs. (2.23) to (2.25) take the form dNt n n n dN2 n n n -^,0,0,0,-^,0,0,0, Bi — J(.,x B2 = [^,0,0,0,^2,0,0,] 0,-^,0,0,0,-^,0,0, B4 B5 ' n dHx dH2 dHi dH2 0 0 — - — - 0 0 d(' di d( ' di de ' ^e2 #2' de (B.ll) (B.12) (B.13) (B.14) (B.15) where the Jacobian J^iX is the given by J i ' x ~ dx ~ 2 (£.16) 

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