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Reliability-based optimization of plywood-web beams Menun, Charles Alexander 1988

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R E L I A B I L I T Y - B A S E D OPTIMIZATION OF P L Y W O O D - W E B B E A M S By Charles Alexander Menun B . A . Sc. (Civil Engineering) University of British Columbia, 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF MASTER OF APPLIED SCIENCE in T H E FACULTY OF GRADUATE STUDIES CIVIL ENGINEERING We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA October, 1988 © Charles Alexander Menun, 1988 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 C i v i l E n g i n e e r i n g The University of British Columbia Vancouver, Canada D a t e O c t o b e r 1 3 , 1988 DE-6 (2/88) Abstract The optimal design of a plywood-web beam, as for any structural element, is usually found by trial and error in which an initial design is modified until a solution which maximizes the beam's efficiency and meets a set of prescribed design criteria is found. To automate this process, a reliability-based optimization program which computes the optimal dimensions of a plywood-web beam is formulated and tested in this study. The program minimizes the cost of a plywood-web beam subject to constraints imposed upon its performance expressed in terms of acceptable levels of safety with respect to a set of limit states. A plywood-web beam model which incorporates the effects of shear deformations in the web components and the effects of non-rigid connections between the beam's flanges and webs is developed and used to compute a plywood-web beam's response under load. The beam's performance is evaluated by means of a reliability analysis in order to rationally account for any uncertainty associated with the beam's material properties and the loads acting on it. A n existing non-linear optimization routine computes the optimal continuous design of a plywood-web beam using the results of the structural and reliability analyses. A discrete solution, representing the optimal practical design which uses only available or allowable dimensions for the beam's components, is found by means of an exhaustive search of a restricted region of the design variable space near the optimal continuous solution. As an example, the program is used to optimize the design of a ply wood-web box girder. Sensitivity analyses are performed on the optimal design in order to identify and ii quantify the critical input parameters in the optimization process. The effects of errors in the problem's formulation, analytical errors arising from the structural analysis and statistical errors resulting in an inaccurate representation of the problem's probabilistic characteristics are studied. i i i Table of Contents Abstract ii List of Tables vii List of Figures ix Acknowledgement xv 1 Introduction 1 1.1 Motivation for the Study 1 1.2 Purpose and Scope of the Study 2 1.3 Thesis Organization 4 2 Design of Plywood—Web Beams 5 2.1 Description of Ply wood-Web Beams 5 2.2 Design Considerations 7 2.2.1 Configuration Selection 7 2.2.2 Material Selection 9 2.2.3 Component Dimensions , 12 3 The Plywood-Web Beam Model 14 3.1 Formulation of the F .A.P . Model 15 3.1.1 Displacement Functions 16 3.1.2 Strain Energy Computation 17 iv 3.1.3 Virtual Work and the System of Equations 20 3.1.4 Solving the System of Equations 22 3.2 Modelling the Plywood-Web Beams 23 3.2.1 Degrees of Freedom in the Plywood-Web Beam Model 24 3.2.2 Displacement Functions . 29 3.2.3 Strain Energy Computation . . 31 3.2.4 Computation of the Plywood-Web Beam's Response 35 4 Reliability Analysis 42 4.1 Background 42 4.2 Definition of Failure of a Ply wood-Web Beam 43 4.3 Assessing the Performance of a Structure 44 4.3.1 Level 3 Methods 45 4.3.2 Level 2 Methods 46 4.3.3 Monte Carlo Techniques 47 4.4 Reliability Analysis Program 47 4.4.1 Description of R E L A N 48 4.4.2 R E L A N Input 49 4.4.3 R E L A N Output 53 5 Optimization Procedure 55 5.1 Scope of the Optimization 55 5.2 Alternative Optimization Formulations 56 5.2.1 Unconstrained Optimization Problem 56 5.2.2 Constrained Optimization Problem 57 5.3 Adopted Optimization Procedure 57 v 5.4 Discrete-Value Optimization 59 6 Sensitivity Analysis 62 6.1 Purpose of Performing a Sensitivity Analysis 62 6.2 Problem Studied 62 6.2.1 Design Variables 63 6.2.2 Random Variables 65 6.2.3 Objective Function 68 6.2.4 Constraints 69 6.3 Sensitivity Analysis Results 70 6.3.1 Benchmark Optimization Problem 70 6.3.2 Sensitivity Analysis Procedure 72 6.3.3 Scope of the Sensitivity Analyses 74 6.3.4 Sensitivity Analyses of Problem Formulation Errors 74 6.3.5 Sensitivity Analysis of Analytical Errors 81 6.3.6 Sensitivity Analysis of Statistical Errors 85 7 Conclusions 95 Bibliography 98 A Sensitivity Analysis Results 100 B Estimated Tension Strength Distribution Parameters 129 VI List of Tables 6.1 Initial Values, Lower and Upper Bounds for Design Variables . . . . . . . 64 6.2 Allowable Discrete Values of Design Variables 64 6.3 Flange Material Properties 65 6.4 Web Material Properties 66 6.5 Web Stiffness Properties 66 6.6 Connection Stiffness Properties 67 6.7 Benchmark Optimization Problem Results 71 6.8 Optimal Design for Different Ultimate Limit States Included in the Reli-ability Constraint 76 6.9 Optimal Solution Obtained with the Moment-Axial Interaction Failure Function Assumed for Flexural Resistance Limit State 78 6.10 Optimal Solution Obtained with the Moment-Axial Interaction Failure Function Assumed for the Flexural Resistance Limit State and the Shear Stress Limit State Omitted 79 B . l Maximum Likelihood Estimators and Kolmogorov-Smirnov Goodness-of-Fi t Test Statistics for Normal Distributions Fit to Entire Samples of Sim-ulated Data 130 B.2 Maximum Likelihood Estimators and Kolmogorov-Smirnov Goodness-of-Fit Test Statistics for Logarithmic Normal Distributions Fit to Entire Sam-ples of Simulated Data 131 vii B.3 Maximum Likelihood Estimators and Kolmogorov-Smirnov Goodness-of-Fit Test Statistics for Two-Parameter Weibull Distributions Fit to Entire Samples of Simulated Data 132 B.4 Maximum Likelihood Estimators and Kolmogorov-Smirnov Goodness-of-Fit Test Statistics for Assumed Probability Distributions Fit to Censored Samples of Simulated Data 133 B.5 Critical Values of the Kolmogorov-Smirnov Goodness-of-Fit Test Statis-tics when the Distribution Parameters are Estimated from the Sample Data 134 vni List of Figures 2.1 Typical Plywood-Web Beam Configurations . 6 2.2 Basic Composition of a Plywood-Web Beam 6 2.3 Possible Configurations for a Plywood-Web Beam 8 2.4 Flange-Web Joint Geometries 12 2.5 Variable Web Thickness Along the Length of a Plywood-Web Beam . . . 13 3.1 Wood Floor Assembly Modelled by F .A .P 15 3.2 T-Beam Element Used in F .A .P 16 3.3 Four Degree of Freedom Model for a Plywood-Web Box Beam 24 3.4 Multi-Component Model for a Plywood-Web Box Beam 27 3.5 Lateral Displacement Contact-Type Problem 29 3.6 Longitudinal Slip in a Flange-Web Connector 33 3.7 Vertical Slip in a Flange-Web Connector 34 3.8 Freebody Diagram Used to Calculate the Shear Stresses at the Plywood-Web Beam's Neutral Axis 38 4.1 Moment-Axial Interaction Failure Surface 52 5.1 Region Near the Continuous Optimal Solution Searched by the Pseudo-Discrete Optimization Procedure 60 6.1 Box Girder to be Optimized 63 ix A . l Optimal Cost as a Function of the Cost Coefficients Used in the Objective Function 101 A.2 Optimal Flange Depth as a Function of the Cost Coefficients Used in the Objective Function 101 A.3 Optimal Web Height as a Function of the Cost Coefficients Used in the Objective Function 102 A.4 Reliability Indices of the Individual Failure Modes as a Function of the Material Cost Ratio 102 A.5 Optimal Cost as a Function of the Modelling Error Coefficients Associated with the Flexural and Shear Stress Computations 103 A.6 Reliability Index as a Function of the Modelling Error Coefficients Asso-ciated with Flexural and Shear Stress Computations 103 A.7 Optimal Flange Depth as a Function of the Modelling Error Coefficients Associated with Flexural and Shear Stress Computations 104 A.8 Optimal Web Height as a Function of the Modelling Error Coefficients Associated with Flexural and Shear Stress Computations 104 A.9 Reliability Indices of the Individual Failure Modes as a Function of the Modelling Error Coefficient for Flexural Stresses 105 A.10 Reliability Indices of the Individual Failure Modes as a Function of the Modelling Error Coefficient for Shear Stresses 105 A.11 Optimal Cost as a Function of the Number of Fourier Terms Used in the Plywood-Web Beam Model 106 A.12 Reliability Index as a Function of the Number of Fourier Terms Used in the Ply wood-Web Beam Model 106 x A . 13 Optimal Flange Depth as a Function of the Number of Fourier Terms Used in the Plywood-Web Beam Model . 107 A.14 Optimal Web Height as a Function of the Number of Fourier Terms Used in the Plywood-Web Beam Model 107 A . 15 Optimal Cost as a Function of the Mean Values of the Flange and Web Materials' Elastic Modulii 108 A . 16 Reliability Index as a Function of the Mean Values of the Flange and Web Materials' Elastic Modulii 108 A.17 Optimal Flange Depth as a Function of the Mean Values of the Flange and Web Materials' Elastic Modulii 109 A.18 Optimal Web Height as a Function of the Mean Values of the Flange and Web Materials' Elastic Modulii 109 A . 19 Optimal Cost as a Function of the Coefficients of Variation of the Flange and Web Materials' Elastic Modulii 110 A.20 Reliability Index as a Function of the Coefficients of Variation of the Flange and Web Materials' Elastic Modulii 110 A.21 Optimal Flange Depth as a Function of the Coefficients of Variation of the Flange and Web Materials' Elastic Modulii I l l A.22 Optimal Web Height as a Function of the Coefficients of Variation of the Flange and Web Materials' Elastic Modulii I l l A.23 Optimal Cost as a Function of the Mean Values of the Flange and Web Materials' Strength Distributions 112 A.24 Reliability Index as a Function of the Mean Values of the Flange and Web Materials' Strength Distributions 112 xi A.25 Optimal Flange Depth as a Function of the Mean Values of the Flange and Web Materials' Strength Distributions 113 A.26 Optimal Web Height as a Function of the Mean Values of the Flange and Web Materials' Strength Distributions 113 A.27 Optimal Cost as a Function of the Coefficients of Variation of the Flange and Web Materials' Strength Distributions . 114 A.28 Reliability Index as a Function of the Coefficients of Variation of the Flange and Web Materials' Strength Distributions 114 A.29 Optimal Flange Depth as a Function of the Coefficients of Variation of the Flange and Web Materials' Strength Distributions 115 A.30 Optimal Web Height as a Function of the Coefficients of Variation of the Flange and Web Materials' Strength Distributions 115 A.31 Reliability Indices of the Different Limit States as a Function of the Ten-sion Strength's Coefficient of Variation 116 A.32 Reliability Indices of the Different Limit States as a Function of the Com-pression Strength's Coefficient of Variation 116 A.33 Reliability Indices of the Different Limit States as a Function of the Shear-Through-Thickness Strength's Coefficient of Variation 117 A.34 Optimal Cost as a Function of the Mean Values of the Dead and Live Load Distributions 118 A.35 Reliability Index as a Function of the Mean Values of the Dead and Live Load Distributions 118 A.36 Optimal Flange Depth as a Function of the Mean Values of the Dead and Live Load Distributions 119 xii A.37 Optimal Web Height as a Function of the Mean Values of the Dead and Live Load Distributions 119 A.38 Optimal Cost as a Function of the Coefficients of Variation of the Dead and Live Load Distributions 120 A.39 Reliability Index as a Function of the Coefficients of Variation of the Dead and Live Load Distributions < . 120 A.40 Optimal Flange Depth as a Function of the Coefficients of Variation of the Dead and Live Load Distributions 121 A.41 Optimal Web Height as a Function of the Coefficients of Variation of the Dead and Live Load Distributions 121 A.42 Optimal Cost as a Function of the Correlation Coefficients Between the Flange or Web Components' Elastic Modulii 122 A.43 Reliability Index as a Function of the Correlation Coefficients Between the Flange or Web Components' Elastic Modulii 122 A.44 Optimal Flange Depth as a Function of the Correlation Coefficients Be-tween the Flange or Web Components' Elastic Modulii 123 A.45 Optimal Web Height as a Function of the Correlation Coefficients Between the Flange or Web Components' Elastic Modulii 123 A.46 Optimal Cost as a Function of the Correlation Coefficients Between the Flange or Web Components' Strength and Stiffness Properties 124 A.47 Reliability Index as a Function of the Correlation Coefficients Between the Flange or Web Components' Strength and Stiffness Properties 124 A.48 Optimal Flange Depth as a Function of the Correlation Coefficients Be-tween the Flange or Web Components' Strength and Stiffness Properties 125 xiii A.49 Optimal Web Height as a Function of the Correlation Coefficients Between the Flange or Web Components' Strength and Stiffness Properties . . . . 125 A.50 Optimal Cost Obtained with the Fitted Probability Distributions for the Douglas Fir Lumber's Tension Strength 126 A.51 Reliability Index Obtained with the Fitted Probability Distributions for the Douglas Fir Lumber's Tension Strength 126 A.52 Optimal Flange Depth Obtained with the Fitted Probability Distributions for the Douglas Fir Lumber's Tension Strength 127 A.53 Optimal Web Height Obtained with the Fitted Probability Distributions for the Douglas Fir Lumber's Tension Strength 127 A.54 Discrete Optimal Flange Depth Obtained with the Fitted Probability Dis-tributions for the Douglas Fir Lumber's Tension Strength 128 A.55 Discrete Optimal Web Height Obtained with the Fitted Probability Dis-tributions for the Douglas Fir Lumber's Tension Strength 128 xiv Acknowledgement I would like to express my gratitude to Dr. R. 0 . Foschi for his invaluable advice and guidance throughout the research and preparation of this thesis. The financial support from the Forintek Canada Corporation in the form of a Timber Engineering Fellowship is gratefully acknowledged. To the Department of Civi l Engineering faculty and staff, my friends and parents, I extend my sincerest thanks. Their encouragement, advice and assistance were greatly appreciated. xv Chapter 1 Introduction 1.1 Motivation for the Study The commercial and industrial construction market is a competitive one dominated by steel and concrete structures. Timber products are primarily used in residential construc-tion such as houses and low-rise apartment buildings. This trend results from the fact that traditional wood-based products, such as standard sizes of lumber and plywood are, when used alone, not efficient means of carrying the loads or attaining the spans often designed for in commercial or industrial structures. Furthermore, the variability of the strength and stiffness properties of timber deter designers from using timber in engineered structures. Plywood-web beams are composite structures constructed of commercially available sizes of lumber and plywood. These built-up beams are designed to make efficient use of the strength and stiffness properties of the wood components they are composed of, thereby creating a structural member which, when properly designed, can be competitive with the steel and concrete products typically used in commercial and light industrial buildings. 1 Chapter 1. Introduction 2 The design of plywood-web beams entails the following heirarchy of decisions: 1. the configuration of the plywood-web beam is chosen; 2. the materials to be used for the flange and web components are selected and 3. the dimensions of the beam's components are calculated. The merit of a given design is determined by its adequacy and efficiency. A n under-designed plywood-web beam will not provide an acceptable level of performance while an alternative which is over-designed will not be competitive with other more cost-efficient solutions. In general, the optimal design of a structural member is found by a trial and error method in which an initial design is modified until both the performance and efficiency objectives are met. Unfortunately, the complexity of the design process for plywood-web beams outlined above limits the number of iterations in the trial and error procedure that a structural engineer is willing to do. Consequently, a plywood-web beam is often designed to meet the performance requirements only with the efficiency objectives given secondary importance. Thus, the need for a method by which the optimal design of a plywood-web beam can be found becomes apparent. 1.2 Purpose and Scope of the Study In this thesis, an optimization procedure for the design of plywood-web beams is formulated and tested. Neither the actual loads acting on the plywood-web beam nor the actual strength and stiffness properties of the beam's components are known; hence, a reliability analysis is employed to evaluate the adequacy of a given design. A reliability analy-sis of a structure differs from a deterministic analysis in that the loads and material Chapter 1. Introduction 3 properties are treated as random variables from given population distributions rather than as fixed (deterministic) values. In this way, the variability and uncertainty of the loads and material properties can be accounted for in the design process. For a given limit state (failure mode), the reliability analysis computes the structure's reliability index, /?, which is related to its probability of failure, Pf, by: where $ is the cumulative distribution for the standard normal probability function. The reliability index can be used, as a measure of the structure's adequacy. In this study, each performance criterion is expressed in terms of a target reliability index, /?*, which the plywood-web beam must meet or exceed. The program developed is capable of analysing the linear-elastic behaviour of most conventional plywood-web beams and using the results of the structural analysis to eval-uate the reliability of the beam with respect to different limit states. In turn, the results of the reliability analysis are employed by a non-linear optimization algorithm which minimizes the cost of the plywood-web beam subject to both geometric and reliability constraints. The optimization procedure only addresses the lowest level of the decision heirarchy: the determination of the components' dimensions. The analyst must select the beam's configuration and the materials for its components. The program is not intended to be a design tool ready for use in a design office. Instead, it is a research tool to be used, to investigate some of the aspects of the reliability-based optimization of plywood-web beams. In particular, the program is used in this study to identify the critical parameters in the optimization process. Chapter 1. Introduction 4 1.3 Thesis Organization The formulation and testing of the optimization procedure developed in this thesis for plywood-web beams involves five stages: 1. Conventional design and construction practices for plywood-web beams are exam-ined in Chapter 2. The important aspects of the design and construction processes are identified and evaluated so that they can be properly incorporated into the optimization routine. 2. The numerical model used to perform the structural analyses of the plywood-web beams is formulated and discussed in Chapter 3. 3. Chapter 4 describes the important aspects of the reliability analysis employed in the optimization process. 4. The details of the optimization procedure developed in this study are presented in Chapter 5. 5. Chapter 6 examines the performance of the reliability-based optimization procedure with an example. Cost sensitivity analyses are conducted in order to identify and evaluate the critical input parameters in the optimization process. Chapter 2 Design of Plywood—Web Beams 2.1 Description of Plywood—Web Beams Traditionally, plywood-web beams are constructed of standard sizes of lumber which are fastened to the top and bottom edges of one or more vertical plywood webs. Some cross-sections of typical plywood-web beam configurations are illustrated in Figure 2.1. Along the length of a plywood-web beam, lumber stiffeners are attached to the webs between the top and bottom flanges to prevent web buckling and to distribute loads into the beam near heavy concentrated loads or reaction points. Typically, a plywood-web beam is designed to span a distance greater than the lengths of the lumber or plywood panels used in its construction; hence, splices are necessary at intervals along the beam to join adjacent members. Figure 2.2 illustrates the basic composition of a plywood-web beam. The efficient placement of the materials within a plywood-web beam allows stiff, light-weight sections to be constructed economically. In general, plywood-web beams weigh less than structurally comparable wood products such as heavy timber and glulam. This reduced weight facilitates their transportation and erection. Furthermore, plywood-web beams can be built at a construction site: often no special tools or equipment are necessary in their fabrication. However, plywood-web beams usually have a much higher surface area-to-volume ratio than solid timber members, making them more susceptible to fire and decay if they are not properly protected. 5 Chapter 2. Design of Plywood-Web Beams 6 Figure 2.1: Typical Plywood-Web Beam Configurations: (a) Box Beam, (b) I-Beam with Overlapped Joints, (c) I-Beam with grooved Joints Web Panels Figure 2.2: Basic Composition of a Plywood-Web Beam Chapter 2. Design of Plywood-Web Beams 7 2.2 Design Considerations A plywood-web beam is designed such that the flanges resist the flexural stresses in the beam while the webs primarily resist the shear stresses. The beam must satisfy the following design criteria: 1. the axial stresses in the flanges must not exceed the flange material's axial com-pression or tension strength, 2. the shear stresses in the webs at the beam's neutral axis must not exceed the web material's shear strength, 3. the shear stresses at a flange-web connection must not exceed the shear strengths of the web, the flange component or the connector, 4. the lateral stability of the beam must be ensured, 5. buckling of the web components must be prevented, 6. crushing of the webs near concentrated loads or supports must be avoided and 7. the deflection of the beam must be acceptable. The design process for a plywood-web beam involves (1) selecting the beam's config-uration, (2) choosing the materials to be used for the beam's components and (3) deter-mining the components' dimensions such that the design criteria listed above are satisfied. 2.2.1 Configuration Selection The selection of the plywood-web beam's configuration is primarily dependent upon stability considerations [22]. Both lateral stability of the beam and stability of the web Chapter 2. Design of Plywood-Web Beams 8 X X X X X X (a) (b) Figure 2.3: Possible Configurations for a Plywood-Web Beam: (a) Box Beam, (b) I-Beam must be ensured. As an example, consider the two configurations shown in Figure 2.3. Both beams are of the same depth and have the same flange and web areas. If the beam's lateral stability is deemed to be the governing failure mode, the box beam design shown in Figure 2.3 (a) should be selected since it will provide more resistance to lateral and torsional buckling than the I-beam shown in Figure 2.3 (b). However, the I-beam design should be chosen if web buckling is the critical failure mode. The single web, which has twice the thickness of the webs used in the box beam design, will provide greater resistance to web buckling. Other factors to consider when choosing the plywood-web beam's cross section are the material, production and labour costs associated with the different configurations. A design which has more components or a larger variety of different components will, in general, be more expensive to produce. For example, the I-beam configuration shown in Figure 2.3 (b) has the same volume of materials as the box beam shown in Chapter 2. Design of Plywood-Web Beams 9 Figure 2.3 (a); however, because the I-beam has four flange components compared the the box beam's two, additional labour and production costs are incurred due to the in-creased handling of materials required to construct the I-beam. Another situation in which increased labour and production costs may result is when the flange components on the compression edge of the beam have different dimensions or are made of a different material than the flange components on the tension edge. Because wood-based products tend to be stronger in compression than in tension, there is an incentive to use smaller dimensions or poorer quality material for the compression flanges, thus reducing the ma-terial costs. However, the fabrication of such designs is more complex and requires more organization than that of simpler designs. Consequently, higher labour and production costs result which offset some of the savings in material costs. 2.2.2 Material Selection Flange Components Either standard sizes of lumber or manufactured wood-based products such as glue-laminated timber or Parallam, are used for the flange components of a plywood-web beam. When determining the flange material to be used, material, labour and production costs must be considered. Although the material cost of lumber is less than that of the high-quality manufactured products, the reliability of a plywood-web beam constructed with lumber flanges will be less than that of one constructed with glulam or Parallam flanges of the same size. Since the performance of a plywood-web beam is the primary design criterion, flanges made of standard lumber must be, in general, larger than those Chapter 2. Design of Plywood-Web Beams 10 made of higher quality material in order to achieve the same level of reliability. Conse-quently, some of the savings realized by using poorer quality material are offset by the necessary increase in the flange's cross-sectional area. In addition, labour costs are often increased when poorer quality material is used since visual inspection of the flange mate-rial must be conducted to ensure that serious strength-reducing flaws are not present. If defects are found, they must be corrected. For example, large knots may have to be culled. Furthermore, standard lumber is available in lengths of up to approximately 25 feet while glulam and Parallam products are usually available in lengths ranging between 50 feet and 100 feet. The availability of the longer lengths of manufactured materials translates into a reduction in the production costs since fewer splices are necessary along the length of a flange component to achieve a desired span. Consequently, although the material costs of glulam and Parallam are greater than those of standard sizes of lumber, the re-duced labour and production costs and increased reliability associated with these higher quality flange materials can offset the cost differences. Web Components Douglas Fir plywood is traditionally used for the web components of plywood-web beams. However, other panel products such as waferboard and particleboard, can also used. The cost differences which arise due to the use of different web materials are primarily a function of the material costs alone which, in turn, are functions of the desired level of reliability to be attained. The material quality does not differ greatly between different panel products. Moreover, the available sizes of these products are the same. Consequently, unlike the selection process for the flange material, the labour and production costs associated with the different panel products do not significantly influence the selection of the web material. Chapter 2. Design of Plywood-Web Beams 11 Flange—Web Connectors The flange components can be fastened to the webs by either an adhesive, nails, or a com-bination of the two. In general, the connection can be described as being either "rigid" or "semi-rigid". A joint displays rigid behaviour if the lateral stiffness of the connector is significantly greater than the shear modulii of the components it is connecting. Adhesives commonly used to attach flange components to webs, such as casein and resorcinol glues, can be classified as rigid connectors. However, nailed con-nections and connections which use a combination of nails and an elastomeric adhesive (mastic) exhibit semi-rigid behaviour. Elastomeric adhesives are putty-like materials which are added to nailed connections to improve their strength and stiffness properties and to reduce squeaking and nail-popping problems. Plywood-web beams fabricated with rigid adhesives are stiffer than those with semi-rigid connectors. As a result, the beam's deflection is reduced if adhesive joints are used. In addition, other serviceability problems, such as squeaking, are eliminated in beams made with glued joints. However, most rigid adhesive connections must be fabricated under pressure and allowed time to cure before they can be installed. Furthermore, the performance of these adhesives is sensitive to such factors as moisture content of the components being joined, temperature and the texture of the bonding surfaces. In particular, the flange components must be dry before a rigid adhesive joint can be made; therefore, unseasoned lumber cannot be used. The bonding conditions for elastomeric adhesives are not as strict as those for rigid adhesives. Thus, the primary advantages of using semi-rigid flange-web connections are reduced production costs and the option of using unseasoned lumber for the flanges which, in turn, may reduce the material costs. Chapter 2. Design of Plywood-Web Beams 12 The two types of flange-web joint geometries commonly used are illustrated in Fig-ure 2.4. The overlap joint may be made with either a rigid or a semi-rigid connection. The grooved joint though, requires a rigid adhesive since the groove is usually not deep enough for nails or elastomeric adhesives to be used. Figure 2.4: Flange-Web Joint Geometries: (a) Overlap Joint (b) Grooved Joint 2.2.3 Component Dimensions Once the plywood-web beam's configuration and the materials for its components have been selected, the components' dimensions are calculated such that the design criteria listed earlier are satisfied. The optimal sizes of the components depend upon the relative costs of the different materials used in the beam. In general, the optimal dimensions will not be equal to the dimensions of available wood products. For example, lumber is only produced in certain standard sizes. Consequently, the final dimensions selected for the plywood-web beam's components are dictated by the sizes available for the materials used. Chapter 2. Design of Plywood-Web Beams 13 A n option to consider when sizing the plywood-web beam's components is to vary their dimensions along the length of the beam. In this way, the strengths of the com-ponents are used to their maximum capacity. For example, a beam which is designed to support a uniformly distributed load requires more shear resistance near its supports than at its midspan. Consequently, the web material can be economized by varying its thickness such that it follows the shear force diagram. This can be accomplished by varying the number of webs used along the length of the beam as illustrated in Figure 2.5. However, as discussed earlier with respect to the selection of the plywood-web beam's configuration, complex design details such as this usually have higher labour and production costs associated with them which may offset the material costs saved. Figure 2.5: Variable Web Thickness Along the Length of a Plywood-Web Beam Chapter 3 The Plywood-Web Beam Model Three objectives were considered when formulating the mathematical model for the plywood-web beams. First, the model must accurately predict the response of the plywood-web beam under load. Although this requirement may appear to be obvious, it should be seriously addressed in reliability studies. Errors arising from approximations and idealizations in the structural analysis have been found to be important factors to which the results obtained from reliability-based design procedures are sensitive [12]. Without an accurate representation of the structure's behaviour, confidence in the results cannot be realized. Second, the model should be computationally efficient. In general, reliability compu-tations require a large number of structural reanalyses to be performed before reaching a solution. Hence, it is desirable to have an efficient means of analyzing the structure's behaviour. This consideration is in direct conflict with the necessity of having an accu-rate model discussed above. Often the accuracy of the structural analysis is sacrificed in order to reduce the cost of the reliability analysis. Third, the model should be able to analyse the variety of plywood-web beam config-urations and construction techniques commonly used. This robustness would allow the analyst to study the behaviour of different combinations of flange and web components as well as the effects of using different types of connectors to fasten the beam's components together. 14 Chapter 3. The Plywood-Web Beam Model 15 A n existing program, F .A.P . ( Floor Analysis Program ), which uses a combination of finite elements and Fourier series in a finite strip formulation to model the behaviour of wood floor systems, has been found to be both an accurate and efficient means of analysing the types of structures to be considered in this study [7,8,9]. Consequently, F .A .P . was adopted and modified to analyse floor systems supported by plywood-web beams. 3.1 Formulation of the F . A . P . Model The type of wood floor assembly modelled by F .A.P . is shown in Figure 3.1. The floor system consists of sheathing which is nailed to standard lumber joists to produce an assembly which behaves as a stiffened plate. A detailed description of the mathematical formulation of the model used in F .A .P . is given in reference [7]. A summary of this formulation follows. Floor Sheathing Joist Figure 3.1: Wood Floor Assembly Modelled by F .A.P . Chapter 3. The Plywood-Web Beam Model 16 3.1.1 Displacement Functions The floor system is modelled as a series of T-beam finite elements, each composed of three basic components: (1) the floor sheathing, (2) a joist and (3) the connectors between the joist and the cover as shown in Figure 3.2. 0 Floor Sheathing- ~ 7 Joistr-3 Connectors Figure 3.2: T-Beam Element Used in F .A.P . The T-beam's displacements are described by a Fourier series approximation in the x-direction (parallel to the joist) and a finite element approximation in the y-direction (perpendicular to the joist) resulting in the following expressions: (i) for the joist component: N U(x) = V(x) W(x) N Chapter 3. The Plywood-Web Beam Model 17 N 'n-KX e(x) = E ( ^ J (3-1) (ii) for the cover component: N y) = £ Fln{y)sin (^ p) u(x,y) = J2F2MC0S\—) v(x,y) = f j ^ v ) ™ ^ ) (3.2) in which N is the number of terms in the Fourier series expansion, Un,Vn, Wn and 9n are unknown constants associated with the nth term in the Fourier series approximation and the functions F i n (y ) , F2n(y), Fzn(y) are obtained by means of a finite element polynomial approximation in the y-direction. Fln(y), F2n(y) and F3n(y) are expressed in terms of the cover's displacements and their derivatives at the points 0, 1 and 2 shown in Figure 3.2. As a result, for the nth term in the Fourier series expansion, the vector of unknowns, {£„}, corresponding to each T-beam element is: r* i r d w i n duln dvln dw0n \°ni = \Wln,—Q^- • S,Uin, —j^ • 3,t?in, • S,U0n,V0n, • S, U / TT T/ Q d W 2 n d U 2 n ^ U 2 n l T Wn, (/„, K n , 9n • 5, u>2 n, — s, u 2 n , - 5 — • s, v2n, • 5} oy ay dy Using the expressions denned in equations (3.1) and (3.2) for the displacements of a T-beam element, the strain energy stored in a T-beam element can be computed as a function of {6n}. 3.1.2 Strain Energy Computation The strain energy associated with a T-beam element is made up of contributions from its three basic components: (1) the cover, (2) the joist and (3) the connectors. Chapter 3. The Plywood-Web Beam Model 18 Strain Energy Stored in the Cover In general, the cover has different elastic properties in the directions parallel and perpen-dicular to the joists and may be considered as an orthotropic plate. Therefore, assuming small deflections, the strain energy stored in the portion of the cover included in a single T-beam element can be expressed as [18]: in which, for a plate of thickness d : Kx = E^fYlil - vxyuyx) Ky = Kx(Ey/Ex) v ~~~ ^xy^-x KG = GcP/12 where: Dx = Exd/(1 - vxyvyx) Dy = Dx (Ey/Ex) D„ = vxyDx DG = Gd Ex, Ey = elastic modulii in the x and y directions vXy = Poisson's ratio, strain in the x-direction when stress is applied in the y-direction vyx — Poisson's ratio, strain in the y-direction when stress is applied in the x-direction G = modulus of rigidity (shear modulus) in the x — y plane. Chapter 3. The Plywood-Web Beam Model 19 Strain Energy Stored in the Joist In terms of the displacements, U, V, W and 9 at node 3 on the joist component shown in Figure 3.2, the strain energy, Uj stored in the joist is given by: Jo EIy fcPW\2 EI, ('<PV\2. EAfduV GJ_fd£ 2 \dx2 ) + 2 \dx2) + 2 \dx) + 2 \dxy dx (3.4) in which: A = the joist's cross-sectional area Iy, Iz = the joist's moments of inertia about the y and z axes J = /3hb3, the joist's torsional moment of inertia E, G = the joist's modulus of elasticity and shear modulus respectively The torsional coefficient, /?, is a function of the joist's width-to-depth ratio, b/h, and is given by the formula [19]: ' - i 1-0.63- -In general, the modulii E and G will be different for each joist, but the ratio, E/G may be assumed to be constant. A value of E/G = 16.0 is often used for wood-based products. Strain Energy Stored in the Connectors The computation of the strain energy stored in the connectors includes the contributions from longitudinal (z-direction), lateral (y-direction) and rotational deformations. The strain energy associated with nail withdrawl other than that due to rotational deforma-tions is not considered. That is, the model assumes that the vertical displacement, WQ, of the cover directly over the joist equals the joist's vertical deflection, W. With these assumptions, the strain energy stored by the nails, Ujf, can be approximated by: NA :=1 ^ ( A u ) , 2 + ^ ( A , ) , 2 + ^ ( A ^ (3.5) Chapter 3. The Plywood-Web Beam Model 20 where: kx,ky and kg = nail stiffness for the longitudinal, lateral and rotational slips NA the number of nails per joist (Au) f-( A t * the slip in the x-direction for the ith nail the slip in the y-direction for the ith nail the rotational slip of the ith nail Equation (3.5), which evaluates the strain energy of the nails in a discrete manner, can be replaced by an equivalent continuous approximation by defining the slips, A u , Av and A9 as continuous functions along the length of the beam resulting in the following expression for the strain energy: in which e = the nail spacing along the joist. The slips can be expressed in terms of the T-beam element's displacements as follows: in which d = the cover thickness, h = the joist depth, UQ and VQ are the longitudinal and lateral displacements at node 0 in Figure 3.2 and dwo/dy is the rotation of the cover directly over the joist. 3.1.3 Virtual Work and the System of Equations The system of equations relating the applied loads to the unknown displacement func-tions, U(x), V(x), W(x), #(x), u(x,y), v(x,y) and w(x,y), is obtained by the principle (3.6) Chapter 3. The Plywood-Web Beam Model 21 of virtual work. The total potential energy, HT of a T-beam element is defined by: n r = Uc + Uj +. UN - UL (3.7) where UL is the load potential corresponding to the applied load function, p(x,y) : rs/2 rL UL= / p(x, y) • w(x, y) dxdy (3.8) J-s/2 JO The total potential energy associated with the nth term of the Fourier series approx-imation can be expressed in terms of the unknown coefficients, {£„}, by substituting the approximate displacement functions defined in equations (3.1) and (3.2) into the above expressions for Uc, Uj, UN and UL. The governing equations for the nth Fourier term are then found by taking the first variation of the total potential energy with respect to {£„}. That is: 6TlT = 6(Uc + Uj + UN-UL) = 0 (3.9) which leads directly to the system of equations: [ * . ] { * » } = { / n } in which: [kn] = the stiffness matrix of a T-beam element for the nth term in the Fourier series approximation {/„} = the load vector representing the load acting on a T-beam element for the nttl term in the Fourier series approximation Equation (3.9) is repeated for each T-beam element in the floor. The global stiffness matrix, [7^ n i„], for the nth Fourier series term can then be obtained by combining the Chapter 3. The Plywood-Web Beam Model 22 NJT T-beam matrices by standard methods. Similarly, the global load vector, {FN}, and the global vector of unknown displacement function coefficients, { A n } can also be found for the nth term in the Fourier series expansion. The above process is then repeated for each term in the Fourier series approximation resulting in N global stiffness matrices. In general, the global stiffness matrices associated with different Fourier series terms are uncoupled due to the orthogonal nature of the sine and cosine terms used in the Fourier series expansion. However, when the strain energy stored in the nails, UN, is evaluated in a discrete manner as shown in equation (3.5), orthogonality is no longer guaranteed and the global stiffness matrices associated with the mth and nih Fourier series terms are coupled by [-Km,*] resulting in the following system of equations: • [Ku] • ' {Ax} ' ' {^1} ' [*a.l] {A 2 } [Ki,i] < {A,} {Fi} [I<N,l] [KN,N}_ {FN} 3.1.4 Solving the System of Equations Once the global stiffness matrices, [KniTn],(m = 1,2- • • N), and the global load vector, {FN}, have been found for each term in the Fourier series expansion, the unknown coef-ficients of the assumed displacement functions, { A n } , can be found from the following iterative procedure: {A n}« = [K^]-11 N {FN} - J2[Kn<m]{Amy-1 m=l (i = l , 2 - - . ) Chapter 3. The Plywood-Web Beam Model 23 with the initial approximation: { A n } ° = [An.nl-1 {^»} The iterations are stopped when: I K A n l ' - I A n } - 1 ! ! < e lKAn}- 1 ! ! (n = 1,2---N) where e represents a convergence tolerance. 3.2 Modelling the Plywood-Web Beams In this study, the lumber joists modelled in F .A.P . are replaced by plywood-web beams. Only linear-elastic behaviour is modelled. It is assumed that load-distribution and web-stabilizing stiffeners are provided to prevent crushing of the web near concentrated loads or reaction points and to prevent web-buckling. Furthermore, adequate lateral support is assumed to be provided along the length of the beam, thereby ensuring lateral stability. Finally, non-linear contact-type problems are not modelled. The strain energy approach employed in F .A .P . is adopted for the plywood-web beam model because of its proven computational efficiency and so that floor and roof systems supported by plywood-web beams may be analysed. However, the plywood-web beam model is designed so that is may also be used to study the behaviour of a single plywood-web beam without sheathing attached to its top surface. Most conventional plywood-web beam configurations can be analysed with the present program. Beam configurations involving up to 10 flange components, 3 web components and 20 connectors can be modelled. In addition, the stiffness properties of each com-ponent and connector can be assigned individually. That is, there are no constraints on the materials selected for each component and connector in the beam. However, the composition of each plywood-web beam in a floor is assumed to be the same. Chapter 3. The Plywood-Web Beam Model 24 3.2.1 Degrees of Freedom in the Plywood-Web Beam Model Two alternatives were considered when formulating the numerical model to be used to perform the structural analysis of the plywood-web beams. Alternative 1: Four Degree of Freedom Model A simple method of modelling a plywood-web beam is to assume that the beam's re-sponse can be adequately described by only four degrees of freedom (U, V , W and 9) associated with each term in the Fourier series approximation. These degrees of freedom would be located at the centroid of the plywood-web beam as shown in Figure 3.3. In x (into page) y z T e Figure 3.3: Four Degree of Freedom Model for a Plywood-Web Box Beam this formulation, the effective stiffnesses of the beam, (EIy)eff, (EIz)efj, (EA)efj and (GJ)eff would be computed using the method of transformed sections and substituted into equation (3.4) in place of EIy, EIZ, EA and GJ respectively to yield the strain Chapter 3. The Plywood-Web Beam Model 25 energy associated with the plywood-web beam, U^, where: + ^ f ( £ ) 2 - ^ r ( g - . ( - 0 , This formulation of the plywood-web beam's contribution to the system's strain en-ergy should prove to be accurate provided two assumptions hold true: (1) the connectors joining the flange and web components together are rigid and hence prevent any relative slip between adjacent components and (2) the shear deformations in the webs have a negligible effect upon the resulting stresses in the beam's components. In many practical situations, the first assumption is satisfied. The flange components of a plywood-web beam are often fastened to the webs by means of an adhesive having a high shear modulus which essentially behaves as a rigid connection. However, the flanges can be fastened to the webs by semi-rigid connectors such as nails or elastomeric adhesives. For these types of connections, the first assumption is violated and the effective stiffness of the beam, in general, will be over-estimated. As a result, the computed response of the beam under load will not be accurate. Tables of adjustment factors which can be used to calculate the actual deflection of a plywood-web beam having semi-rigid connections from the beam's "rigid-connection" deflection have been published [21]; however, similar adjustment factors for the resulting stresses in the flanges and webs are not available. The effect of semi-rigid connectors on the behaviour of a plywood-web beam must be accounted for if the accuracy of the solution is not to be sacrificed. Alternatively, to restrict the plywood-web beam model to only rigidly-connected assemblies might be regarded as being short-sighted when one acknowledges that there are some economic benefits associated with using semi-rigid connectors. Chapter 3. The Plywood-Web Beam Model 26 The exclusion of shear strains from the plywood-web beam model makes the model stiffer than the actual beam it is representing. Consequently, the calculated deflection of the beam and the stresses in its components will not be correct. For most engineering materials, the assumption that shear deformations have a negligible effect on the stresses occurring in a structure is justified. However, the shearing modulus, G, of wood-based products is relatively small compared to its elastic modulus, E: for wood E/G « 16.0 while for steel, E/G « 2.5. Hence, the shear deformations and their effects on the stresses which result in a plywood-web beam may be significant. To ignore these effects would be detrimental to the accuracy of the results obtained from the structural analysis. Although this formulation would prove to be computationally efficient, it suffers from its inability to satisfactorily account for the two characteristics of plywood-web beams noted above. Alternative 2: Multi—Component Model A n alternative approach is to give each component of the plywood-web beam its own set of degrees of freedom (U, V , W and 9) corresponding to each term in the Fourier series expansion in the x-direction. As illustrated in Figure 3.4, each flange component of the beam is assigned four degrees of freedom, much like the joists in F . A . P . The web components, on the other hand, have five degrees of freedom. As for the flange components, the web's lateral, vertical and rotational degrees of freedom (Vw, Ww, 9W) are located at the centroid of the web element. However, two axial degrees of freedom, and are assigned to each web: one at the top edge of the web and the other at the bottom edge. The additional axial degree of freedom is included in order to approximately account for any shear deformations in the web. Consequently, the effects of these shear deformations upon the Chapter 3. The Plywood-Web Beam Model 27 c resulting stresses in the beam's components are better represented by this model than by the previously described model. Flange Web Figure 3.4: Multi-Component Model for a Plywood-Web Box Beam In addition, this formulation requires the inclusion of the strain energy associated with the connections which join the flange and web components. As a result, the stiffness properties of the flange-web connectors, enter the model's formulation and thus, the effect of their rigidity on the behaviour of the plywood-web beam can be accounted for. Obviously, the second formulation of the plywood-web beam model is not as efficient as the first due to the increased number of unknowns in the model and the necessity to calculate the flange-web connectors' contribution to the beam's strain energy. However, the second formulation better reflects the nature and composition of plywood-web beams. As discussed earlier, and illustrated here, a trade-off exists between a model's accuracy, efficiency and robustness. The choice of the model depends primarily upon the relative importance the analyst places on each of these considerations. For this study, which is investigative in nature, the accuracy and robustness of the model are more important Chapter 3. The Plywood-Web Beam Model 28 than its computational efficiency. Consequently, the second formulation of the plywood-web beam model was adopted. Contact-Type Problems In order to avoid the necessity of modelling the non-linear effects associated with contact-type problems, some of a plywood-web beam's degrees of freedom shown in Figure 3.4 must be constrained by others. Such an assumption is employed in F . A . P . to constrain the vertical displacement of the cover directly over the joist to be equal to the vertical displacement of the joist. A similar constraint is used in the formulation of the present model to force the vertical displacement of the top flange(s) of the built-up beam to be equal to the vertical displacement of the cover at node 0 in Figure 3.2. Consequently, when there is more than one flange component at the top edge of a beam, the vertical displacements of all of these top flange components are forced to be equal. Another contact-type problem which arises in this plywood-web beam model results from the lateral displacements of the flange and web components relative to each other at the flange-web connections. As illustrated in Figure 3.5 (a) for a box-beam, in order to avoid a relative displacement of the flange and web components in the y-direction, the lateral displacement of the top flange, V i , must be constrained as follows: v i = v3 + cee3 (3.11) v i = v4 + cce4 (3.12) similarly for the lateral displacement, V2, of the bottom flange: (3.13) V2 = V4-Ct04 (3.14) Chapter 3. The Plywood-Web Beam Model 29 Figure 3.5: Lateral Displacement Contact-Type Problem From equations (3.11) to (3.14) it follows that the lateral displacements of the webs, V-j and V4, must be equal and hence, so must the web rotations, 63 and #4. As a result, the degrees of freedom shown in Figure 3.5 (a) must be reduced to those shown in Figure 3.5 (b) in order to avoid contact-type problems of this nature. The program developed for this study automatically performs this operation. 3.2.2 Displacement Functions In a manner similar to that employed for the joists in F .A.P . , the unknown displacement functions of a plywood-web beam are approximated by a Fourier series expansion in the x-direction (parallel to the beam) and, for the axial displacements of the web components, by a linear approximation in the z-direction. The resulting expressions for a plywood-web beam's displacements are therefore: Chapter 3. The Plywood-Web Beam Model 30 (i) for the flange components: fnirx\ Uf(x) = E U f n c o s ( ^ ) (ii) for the web components: 'Ub -U* N Uw(x,z) = E N {nirx\ Vw(x) = E Vwnsin AT M * ) = E ( — J (3-16) in which Ufn, Wfn, and #/n are unknown constants associated with the nth term in the Fourier series expansion for a flange component's deflection while U^, U^, Vwn, Wwn and 0^ are the equivalent constants associated with a web component. The superscripts t and b indicate the top and bottom edges of a web element respectively; hw is the depth of the web component; Cz is the vertical distance between the flange's centroid and the web's centroid and z is measured from the top of the web component. Note that Vf(x) in equation (3.15) is denned in terms of Vw(x) and 9w(x). As discussed in Section 3.2.1, this is done to elirninate contact-type problems arising from the relative lateral displacements of the flanges and the webs. Chapter 3. The Plywood-Web Beam Model 31 3.2.3 Strain Energy Computation The strain energy associated with the plywood-web beam is made up of contribu-tions from (1) the flange components, (2) the web components and (3) the flange-web connectors. Flange Components The strain energy stored in a flange component, UF, is computed in terms of the four dis-placements, U/j Vf, Wf and Of, shown in Figure 3.4 (a) at the centroid of the component. Accordingly, which is identical to equation (3.4), the strain energy associated with the joists in F .A.P . , except that the stiffnesses of the flange component, (EIy)/, (EI2)/, (EA)/ and (GJ)/, replace the stiffnesses of the joist, EIy, EIZ, EA and GJ respectively. Web Components Like the floor cover in F .A.P . , the webs are, in general, made of an orthotropic sheathing material. Consequently, the expression used to compute the floor cover's strain energy in equation (3.3) can be used to compute the strain energy in the web, Uw- Thus: (3.17) Chapter 3. The Plywood-Web Beam Model in which the subscript, "w", indicates web material and, for a web of thickness tu ExtH\2{\ - vxzuzx) Kwx (Ez/Ex) ^xz^-wx KwG = Gt3J12 DWZ DWG - Extw/(l - vxzvzx) — DWX(EZ/EX) ^xz^wx = GU However, from equations (3.16) and geometric considerations: dWw n dVw Q d2Vw dz = 0 dz = 0 hence, the expression for the strain energy stored in the web simplifies to: Uw = Jo Jo Kwx (d2Vw dx2 + 2K, wG 'dV* L dx w -0. 'dUw . dx + IXG (dU^ + dWw dz dx dxdz Flange—Web Connectors (3. The strain energy, UFW, stored in the flange-web connections can be expressed as: (i) for adhesive connections: UFW = / Jo f (Aufw)2 + f(Awfw)2 + f(Aefw)2 dx (ii) for nailed connections (discrete formulation): NA r u F W = J2 t'=l ki ki ki -f (Au / W )f + f (Aw / w )J + -f (A9Jw)i dx (iii) for nailed connections (continuous formulation): u FW 'hi ki ' ki fe (AUfmy + ^  (Awfwy + fe (A9fwy dx where: Chapter 3. The Plywood-Web Beam Model 33 k%, k« and k% = the adhesive stiffness for longitudinal, vertical and rotational slip kdx, kdz and kd6 = the nail stiffness for longitudinal, vertical and rotational slip Aufw = the slip in the x-direction Awfw = the slip in the z-direction A0fw the rotational slip NA the number of nails along the length of the connection and e = the nail spacing along the length of the connection Figure 3.6: Longitudinal Slip in a Flange-Web Connector: (a) Along the Top Edge of the Web, (b) Along the Bottom Edge of the Web The slips can be expressed in terms of the displacements of the adjoining elements. Referring to Figure 3.6, the slip in the x-direction, Aufw, along the glue line (or nail line) can be expressed as: Chapter 3. The Plywood-Web Beam Model 34 (i) for flange-web components along the top edge of the web: \ tiwJ nw ax (ii) for flange-web connectors along the bottom edge of the web: A - / . = i D l + ( l - f ) Ul - V, -(y- «,) fiw \ fiw/ ax The vertical slip, as illustrated by Figure 3.7, is given by: Awfw = ww± ew-wf± (^j 9} (3.19) (3.20) and the rotational slip is: A#/u, = 9W — 9j (3.21) (3.22) The signs in front of the 9f and 9W terms in equation (3.21) depends upon the relative placement of the flange and web components as shown in Figure 3.7. Figure 3.7: Vertical Slip in a Flange-Web Connector: (a) Flange to the Left of the Web, (b) Flange to the Right of the Web Chapter 3. The Plywood-Web Beam Model 35 Total Strain Energy of the Plywood-Web Beam Having the strain energy associated with each of the plywood-web beam's components, the total strain energy of the beam, Upw},, can be computed from: NUMFLG NUMWEB NFWCON Upwb= £ UF+ £ Uw+ £ UFW (3.23) t=i t=i »=i where: N U M F L G = the number of flanges in the beam N U M W E B = the number of webs in the beam and N F W C O N = the number of flange-web connectors in the beam Substituting equations (3.15) and (3.16) into equation (3.23) allows the strain energy corresponding to the plywood-web beam to be expressed in terms of the unknown dis-placement coefficients, {Spwbn}. Finally, replacing Uj in equation (3.7) by Upwb yields the total potential energy of a T-beam element which has a plywood-web beam. The procedures outlined in Sections 3.1.3 and 3.1.4 can then be implemented to solve for the unknown coefficients of the assumed displacement functions. 3.2.4 Computation of the Plywood—Web Beam's Response Once the coefficients of the assumed displacement functions, { A n } , have been found for each term in the Fourier series approximation, the stresses in the plywood-web beam and the beam's deflections can be calculated. Five types of responses are required to be computed in this study: (1) the axial stresses in the flanges, (2) the bending stresses in the flanges, (3) the shear stresses in the web at the neutral axis of the beam, (4) the shear stresses at the flange-web connections and (5) the beam's deflection, including the deflection due to shear. The means by which the model computes each of these responses will now be described. Chapter 3. The Plywood-Web Beam Model 36 Axial Stresses in the Flanges The strain in a flange component due to the axial stresses acting in that component can be expressed as: dUf dx (3.24) Substituting the expression for Uj defined in equation (3.15) into equation (3.24) yields: Hence, the axial stress in a flange component is given by: Bending Stresses in the Flanges sin (3.25) (3.26) The strain in a flange component associated with the flexural stresses acting in that component is defined by: <PWf = z dx2 (3.27) where z = the distance from the flange's neutral axis to the point in the flange at which the stresses are to be calculated. Replacing Wf in equation (3.27) by the expression for Wj found in equation (3.15) gives: N r n = l nir ) Wfnsin {^jpj (3.28) Thus, the bending stress in a flange component is: TJ w^mn{—) (3.29) Chapter 3. The Plywood-Web Beam Model 37 Shear Stresses in the Webs at the Neutral Axis Unlike the axial and bending stresses in the flanges, the shear stresses in the web cannot be properly computed by this model from the definition of shear strain: ,. . du dw 7xz(x,z) = - + -Working with this definition, the assumed displacement functions adopted for the web elements in equations (3.16) can only approximate the shear stresses acting in the webs as being constant across the web's depth (in the z-direction): N r n = l L - ~ ( ¥ ) + ( ¥ ) ^ /nxx\ iCos [—) where ~llz{x) is an approximation to the actual shear strain, ixz(x,z) with: n = l dz 'Ub - U* ' COS (mrx\ and N fnirx\ Instead, to obtain an accurate estimate of the shear stresses acting at the neutral axis, the freebody diagram illustrated in Figure 3.8 is used to derive the following expression: (3.30) In equation (3.30), J2 Vfw is the total shear force transmitted to the top edge of the web by the flange-web connectors between points Xi and x 2 with V/w defined as: (i) for adhesive joints: Vfw = kax fX\Aufw)dx (3.31) (ii) for nailed joints: Vjw = — / (Aufw)dx e Jxi (3.32) Chapter 3. The Plywood-Web Beam Model 38 where Aufw is found from equation (3.19). The axial forces in the web, Fx and F2 in equation (3.30), are computed from: Fx = Dwxtm Jo dUw dx dz X=X\ (3.33) where: dUw dx N 'nir fUh — TP ' jjt I I wn _tun x=x\ n = l v Hence, after integrating equation (3.33) . (WKX\ ( n . ) + '£/ 6 - \ F 2 wn \ na 9 fmcxx^ A similar expression can be derived for the axial force, F2. A x 2 V f Fi Web Component 'na F 2 Neut ra l Axis of the Beam Figure 3.8: Freebody Diagram Used to Calculate the Shear Stresses at the Plywood-Web Beam's Neutral Axis Chapter 3. The Plywood-Web Beam Model 39 Shear Stresses at the Flange—Web Connections The shear stresses at the flange-web connections can be most accurately calculated by considering the relative slip in the x-direction, A u / W , between the adjoining com-ponents. The total shear force transmitted between adjacent components over a dis-tance, A x , between points xi and x2 along the beam's length is derived above in equa-tions (3.31) and (3.32). The flange-web shear stress is then calculated from: where djw is the depth of the contact area between the adjoining components. Strictly speaking, equation (3.34) is only valid for adhesive joints; for nailed flange-web connec-tions: T/-=W^j (3-35) where Anaii is the cross-sectional area of a nail and e is the nail spacing. Note that equation (3.35) represents the shear stress in the nails joining the two components. Deflections The deflection of a plywood-web beam component at a point, x i , along the length of the beam can be defined for flange components as: Wf = E Wfnsin (^ p) (3.36) and for web components as: Ww = W^sin (^p) (3.37) The computed deflections include shear deflections which arise from the approximate shear strain, 7*z, modelled in the web components. That is: W a ( Z l ) = V n ( X l ) + VV7(Xl) Chapter 3. The Plywood-Web Beam Model 40 where: Wa = the approximate total deflection computed by equation (3.36) or (3.37) Wb = the deflections due to bending and W* = the approximate shear deflections in the webs W*(xi) can be computed from: As discussed earlier when deriving the expression for the shear stress at the neutral axis, the model is too stiff to accurately predict the shear strains in the web components. Hence, W* underestimates the actual shear deflection of the beam. Since the deflection of a plywood-web beam due to shear can be significant, it is desirable to obtain an accurate estimate of the beam's shear deflections. A t any point, x i , along the length of the beam, the actual shear deflection, W3(xi) is given by: W.fa) = f * ina(x)dx (3.38) in which 7no(x) is the actual shear strain at the beam's neutral axis and can be defined as: 7 n a ( * ) = Ct • 7 a V 5 ( x ) (3.39) where 7 o v g ( x ) is the average shear strain in the beam and a is a numerical shape factor which depends on the beam's cross-section. Recall that when working from the definition of shear strain, the model predicts a constant shear strain, 7 * , across the depth of the web. This computed strain closely approximates the average shear strain, 7 a v a ; hence, lna{x) in equation (3.39) can be approximated by: 7 n a ( x ) » a • 7 : , ( x ) (3.40) Chapter 3. The Plywood-Web Beam Model 41 Since the shear stresses in the webs are directly proportional to the shear strains, equa-tion (3.40) can be restated as: Tna « Ct • T*vg (3.41) in which r n o is defined by equation (3.30) and r*vg can be found from: DwG avg (3.42) Therefore, the shape factor, a, can be evaluated at an arbitrary point, x2, along the beam from: Tna(x2) a « Ta%,(*2) Substituting equation (3.39) into equation (3.38) yields the actual shear deflection along the beam in terms of the average shear strain, ~favg: W, r a ( X l ) = a Javgdx Jo r 7* dx Tavg(X2) JO \9z dx J The actual total deflection of the component at X\ therefore becomes: Wt(Xl) = Wh{xx) + W,{xx) « Wb(x1) + aW:(x1) « W a (x 1 ) + ( a - l ) H C ( x 1 ) in which Wa(xi) is the approximate total deflection of the beam at a point x x along its length computed by equation (3.36) or (3.37). Chapter 4 Reliability Analysis 4.1 Background Until recently, structural design has been dominated by deterministic forms of analysis. Such methods assign fixed values to the material properties of the member being designed and to the loads it is subjected to. Stress and deflection calculations are then performed and the results are compared to the specified values of strength and stiffness assigned to the member. The member's adequacy is measured by its reserve strength and stiffness. This is the approach adopted by "working stress" design codes. Safety factors are used in deterministic design to account for any differences between the strength, stiffness and load parameters assumed in the design process and the actual values of these parameters. In this manner, the uncertainty associated with the mate-rial properties and loads is incorporated into the design. However, this is not a rational means of accounting for the variability and uncertainty of the design variables. Different materials exhibit different degrees of variability in their properties. Likewise, the uncer-tainty of the loads depends strongly upon their nature. Dead loads, for example, are generally less variable and better estimated than live loads. Consequently, the required safety factor to ensure an acceptable level of performance is problem dependent and not readily available. Hence, by employing a deterministic design format, some structures are over-designed while others may be dangerously under-designed. Clearly, a more rational means of incorporating the variability and uncertainty of the 42 Chapter 4. Reliability Analysis 43 design variables into the design procedure must be implemented in order to achieve more uniform levels of safety in engineered structures. Over the last twenty years, reliability-based analysis has been used increasingly in structural design to accomplish this goal. A reliability analysis of a structure differs from a deterministic analysis in that the material properties and loads are treated as random variables rather than as fixed (deterministic) values. Hence, the variability of the design parameters enters directly into the design pro-cess. Reliability-based design procedures usually require that the structure's behaviour satisfy each of a set of performance criteria (limit states) with a given level of probability which is deemed acceptable. Assessing the performance of a structural component in this manner has the advantage that the desired level of safety becomes input into the design process rather than a result. That is, in a reliability-based analysis, the probability of failure of the component with respect to a given failure mode is prescribed while, in a deterministic design procedure, the safety factor is specified which, in turn, sets an unknown level of safety for the structure. Consequently, the reliability-based approach gives the designer more control over a structure's performance. As a result, more uniform levels of safety can be attained between different structures and between different failure modes within a single structural system. 4.2 Definition of Failure of a Plywood—Web Beam Before a structure's performance can be evaluated, a definition of what constitutes a failure of the structure is required. Two classes of structural systems exist: (1) series systems, often described as "weakest link" systems and (2) parallel or redundant systems. A series system has no means of redistributing and carrying the loads acting on it after Chapter 4. Reliability Analysis 44 one of its elements has failed. Consequently, the failure of a series system occurs when any one of the limit states it is governed by is not satisfied. In contrast, a parallel system can redistribute the loads acting on it when one or more of its elements fail. Hence, failure of a parallel system does not occur until a set of limit states are violated. Most structures, including plywood-web beams, are better described as parallel or redundant systems rather than as series systems. For example, a plywood-web I-beam with two flange components at both edges of the web, would not necessarily collapse when one of its flange components fails. Instead, the load carried by that flange would be redistributed to the remaining flanges and the beam would continue to support the loads acting on it. However, although the beam does not collapse, its performance is reduced since its remaining components are subjected to higher levels of stress. Shinozuka and Itagaki [16] illustrated with numerical examples that the conditional probability of total failure of a redundant system subject to brittle failure modes or yielding given that some of the system's components had already failed is significantly high. Moses [12] concluded from these results that, for most practical cases, a redundant system may be treated as a series system in reliability studies. Based on this conclusion, the plywood-web beams considered in this study are treated as series systems. Hence, failure of the structure is assumed if any one of the limit states which govern the design of the plywood-web beam is not satisfied. 4.3 Assessing the Performance of a Structure The performance criteria for a structural component can be expressed in the form: Ri > Si (4.1) Chapter 4. Reliability Analysis 45 in which i? t is the resistance of the member and 5,- is the load effect for the ith limit state. In general, Ri and Si are functions of a set of p basic random variables, X = (X\,X2 • • • Xp) whose outcomes are x = (x i , x2 • • • xp). Equation (4.1) can be restated as: Gi{X) = Ri(X) - Si(X) > 0 (4.2) where Gi(X) is denned as the failure function. From equation (4.2), it follows that the structure's behaviour complies with the ith limit state when is positive. Therefore, G{ = 0 defines a boundary in the p-dimensional space, called the failure surface, which separates all the combinations of x that satisfy the ith limit state from those combinations which do not. The objective of a reliability analysis is to evaluate the probability that a given design will lie in the failure region defined by Gi < 0 in which all combinations of x do not satisfy the iih limit state. There are three basic ways in which the probability content of the failure region can be assessed: (1) by exact level 3 methods, (2) by approximate level 2 methods and (3) by Monte Carlo methods. 4.3.1 Level 3 Methods As discussed in Section 4.1, the reliability-based approach to structural design specifies a tolerable level of probability, P/,-, of the ith limit state not being satsified. This probability can be computed from: Pu- J J ••• J fx1,X2-xp(xi,x2---xp)dx1dx2---dxp (4.3) G?<0 where fxi,X2~xp(xi> x2 • • • xp) is the joint probability density function of the random variables X. This form of reliability analysis is known as a "level 3" method. Such Chapter 4. Reliability Analysis 46 methods utilize complete descriptions of the probabilistic nature of the problem and the failure region to calculate the exact probability of failure, P/,-. Although, in theory, equation (4.3) can be used to find the probability that a structure will fail to satisfy a given performance criterion, practical limitations arise which make this formulation difficult to employ successfully. First, information is usually unavailable or inadequate to properly define the joint probability density function. Second, even if a suitable approximation to the joint probability density function can be found, integrating equation (4.3) will often prove to be time-consuming and expensive since, in general, analytical solutions do not exist and numerical integration must be performed. Finally, in the case of this study where a numerical procedure is used to compute the response of the plywood-web beam under load, the failure region, Gi < 0 , is not explicitly defined; hence, the limits of the integrals in equation (4.3) are unknown. 4.3.2 Level 2 Methods To overcome the problems associated with the formulation of level 3 reliability analysis procedures, level 2 methods have been developed. These methods evaluate the failure probability, P/,-, in an approximate manner by simplifying the probabilistic properties of the problem and idealizing the failure region. Usually, an iterative calculation scheme is employed in a level 2 analysis to evaluate a structure's reliability, thereby avoiding the expensive and time-consuming numerical integration necessary in the level 3 method described in Section 4.3.1. Consequently, the loss in accuracy of the results obtained from a level 2 reliability analysis is compensated for by decreased computational costs. Chapter 4. Reliability Analysis 47 4.3.3 Monte Carlo Techniques A n alternative approximate method of evaluating the integral in equation (4.3) is by means of a Monte Carlo simulation. This method involves using a random number generator to "select" N combinations of outcomes, x, according to the probability dis-tributions which describe the random variables, X. With each set of outcomes, Xj, the realization of the ith failure function, g,, is evaluated and it is determined whether or not Xj lies in the failure region, g% <0 . Hence, in a "hit and miss" fashion, the topography of the p-dimensional random variable space is revealed and the probability content of the failure region can be approximated by: where n / is the number of combinations of x which yielded negative values for g,. The number of trials, N, required for this means of estimating P/,- to provide reliable results depends upon the value of Pj{. For the probabilities of failure typically encoun-tered in structural design (0.001 to 0.0001), between 10,000 and 100,000 trials are usually necessary. The load effect term, Si, in the failure function is provided by the numerical plywood-web beam model formulated in Chapter 3. Consequently, evaluation of the failure function is a relatively expensive and time-consuming operation in this study; the number of reanalyses performed should be minimized. Hence, the number of trials required by a Monte Carlo simulation to confidently evaluate P/,- is prohibitive. 4.4 Reliability Analysis Program R E L A N , a reliability analysis program developed by Foschi, is used in this study to assess the reliability of alternative plywood-web beam designs. Chapter 4. Reliability Analysis 48 4.4.1 Description of R E L A N A brief summary of the reliability analysis implemented by R E L A N follows. Detailed mathematical formulations of the algorithms used in the program can by found in the relevant references cited in this section. The reliability analysis is performed by means of a level 2 method originally formu-lated by Hasofer and Lind [11] for problems involving uncorrelated, normally distributed random variables. The Hasofer and Lind procedure first normalizes the random variables, X, yielding a new set of random variables, Y = (Yi, Y2 • • • Yp), such that: YiSSXiZ±*L (.' = l ,2 - .p) where fixi and crxi are the mean and standard deviation of X{. The algorithm approx-imates the failure surface by a hyperplane which is tangent to the failure surface at the point, {y*}, nearest the origin of the standard normal space. A n iterative procedure is used to locate {y*} from which the reliability index, /?, is computed for the limit state under consideration. By definition, /3 is the minimum distance from the origin to the failure surface in the standard normal space; hence, /3 can be computed from: /?=ll{y*}| | Furthermore, if the failure surface is linear and hence the hyperplane approximation is in fact the true failure surface, f3 is related to the probability of failure by: 0 = - 4 " 1 (Pf) where $ is the cumulative distribution for the standard normal probabilty function. The restrictions imposed on the probabilistic nature of a problem by the Hasofer and Lind procedure however are unrealistic: the material properties and loads are often Chapter 4. Reliability Analysis 49 better represented by non-normal distributions and, in general, the strength and stiff-ness properties of wood-based products are correlated. To overcome these restrictions of the Hasofer and Lind procedure, two modifications to the original algorithm were made. First, a transformation formulated by Rackwitz and Fiessler [13] was introduced so that non-normal variables could be included in problems. The Rackwitz and Fiessler algorithm uses a "normal tail approximation" [5,13] to transform non-normal variables into "equivalent" normal variables before executing any computations in an iteration of the Hasofer and Lind procedure. Second, to incorporate correlation between random variables into the analysis, a procedure proposed by Der Kiureghian [4] is used to trans-form correlated normal variables (from the Rackwitz and Fiessler transformation) into uncorrelated normal variables. 4.4.2 R E L A N Input To use R E L A N , the analyst must supply data about the marginal probability distribu-tions and covariances of the random variables in the problem. Although information on these probabilistic properties is often scarce, it is usually more readily available than the information required to define the joint probability density function used in the level 3 method described in Section 4.3.1. The analyst must also provide the failure function, G and a means of calculating its gradient with respect to the basic random variables. Unlike the level 3 method described by equation (4.3), the failure region, G < 0 , need not be predetermined. Consequently, the difficulties encountered when evaluating a structure's reliability by means of a level 3 method are either reduced or eliminated by using the level 2 method employed by R E L A N . Chapter 4. Reliability Analysis 50 Limit State Failure Functions As noted in Section 2.2, there axe seven design criteria which a plywood-web beam's design must satisfy. However, in this study, three of these criteria: lateral stability, web-buckling and web crushing, are assumed to always be satisfied and therefore are not addressed. Failure functions are derived for the remaining four limit states: (1) flexural stresses in the flanges, (2) shear stresses in the webs at the neutral axis of the beam, (3) shear stresses at the flange-web connections and (4) deflection of the beam. Flexural Stresses in the Flanges There are two alternatives for the failure function which describes the plywood-web beam's flexural resistance. First, the form of failure function currently used by the Canadian Timber Design Code, C AN-086-M84, compares the flange's axial compression or tension strength to the maximum axial stress in the flange. Thus (i) for the compression flange: G = Fc + (<ra + <rb)(D + Q) (4.4) (ii) for the tension flange: G = Ft - (cra + <jb)(D + Q) (4.5) where: Fe,Ft the axial compression and tension strengths of the flange material the axial stress in the flange due to a unit load as found by equation (3.26) the bending stress at the appropriate face of the flange component due to a unit load as found by equation (3.29) and D,Q the dead load and live load acting on the beam Chapter 4. Reliability Analysis 51 This formulation assumes that the flanges are subjected to the maximum axial stress across their entire depths. Although this may not be a poor assumption for deep plywood-web beams in which the axial stress term, <7a, dominates the bending stress, <r&, it may be overly conservative for shallower beams. A n alternative formulation of this failure function incorporates both the axial resis-tance and the bending resistance of the flange into the limit state and assumes a moment-axial interaction failure mechanism. The linear moment-axial interaction failure surface shown in Figure 4.1 is used in this study, yielding the following failure functions: (i) for the compression flange: G = FcFb + (Fc<rb + Fbaa){D + Q) (4.6) (ii) for the tension flange: G = FtFh - {Ftab + Fbaa)(D + Q) (4.7) in which Fb is the modulus of rupture of the flange material and all other terms are as denned for equations (4.4) and (4.5). Shear Stresses in the Webs at the Neutral Axis The performance of a plywood-web beam's resistance to the shear stresses at its neutral axis in a web component is evaluated with the failure function: G = Vp - rna(D + Q) (4.8) in which: Vp = the shear through thickness strength of the web component and Tna = the shear stress, due to a unit load, acting at the beam's neutral axis in the web component as computed by equation (3.30). Chapter 4. Reliability Analysis 52 Figure 4.1: Moment-Axial Interaction Failure Surface Flange—Web Connection Shear Stresses The shear stresses which are developed in the beam's flange-web joints are accounted for by the following failure functions: (i) for shear failure in the web component: G = Vpf- rfw(D + Q) (4.9) (ii) for shear failure in the flange component: G = FV- rfw(D + Q) (4.10) in which: Vpf = the rolling shear strength of the web material Fv = the longitudinal shear strength of the flange material and Tfw = the shear stress in the flange-web connection as computed by equation (3.34) Chapter 4. Reliability Analysis 53 Deflections The failure function used in this study for the deflections of a plywood-web beam is: G=j-Wt(D + Q) (4.11) where: k = the ratio of the beam's length, X , which the beam is allowed to deflect and Wt = the total deflection of the beam as found from equation (3.43) 4.4.3 R E L A N Output For a given limit state, R E L A N computes a structure's reliability index, /?, which is related to the true probability of failure by: /3 « - f c - 1 (Pf) (4.12) Hence, /3 can be used as an indicator of a structure's probability of failure. The relationship between the reliability index and the probability of failure shown in equation (4.12) is not exact due to the normal tail approximation employed by the Rackwitz and Fiessler algorithm and because, in general, the failure surface is non-linear and must be represented by a hyperplane approximation. In general, a structure's behaviour must satisfy more than one limit state. Under this set of conditions, it is rational and desirable to design the structure so that its overall probability of failure is fixed at a specified acceptable level rather than assigning an allowable probability of failure to each failure mode. Thus, the overall safety of the structure becomes the measure of the structure's adequacy. A means of estimating a structure's total probability of failure for a set of limit states is needed. Assuming that failure of the structure occurs if any one of m specified limit states is not satisfied, the Chapter 4. Reliability Analysis 54 structure's overall probability of failure can be expressed as: (4.13) where U indicates the union of the m limit states and # is the realization of the ith limit state. One method of evaluating equation (4.13), developed by Ditlevsen [6], can be im-plemented using the results of R E L A N to calculate the upper and lower bounds of the structure's reliability index for a set of failure modes. The bounds calculated by this algorithm are very narrow for most situations and hence, have practical value in the reliability analysis of structures subject to several possible failure modes. As with any level 2 reliability analysis, the reliability index computed by R E L A N is an approximation to the structure's true reliability index. The idealizations of the prob-abilistic nature of the problem and the simplifications in the representation of the failure domain imposed by the program all contribute to the reduced accuracy of the results. Nevertheless, the algorithms used in R E L A N have been found to compute sufficiently accurate estimates of a structure's true reliability index. Consequently, there is reason to believe that the reliability index computed by R E L A N can serve adequately as a measure of a structure's performance. (Pf) total -P U (in < o) i = l Chapter 5 Optimization Procedure A reliability-based optimization procedure was formulated to compute the optimal design of plywood-web beams. The routine attempts to minimize the cost of a beam subject to constraints on its performance with respect to a predefined set of limit states (design criteria). The beam's reliability index corresponding to this set of limit states is used as the measure of its performance. 5.1 Scope of the Optimization As discussed in Section 2.2, the design process for plywood-web beams involves the following heirarchy of decisions: 1. selection of the beam's configuration 2. selection of the materials to be used and 3. calculation of the beam's dimensions. Ideally, each of these aspects of the design should be incorporated into the optimization problem. However, most studies performed to date on deterministic optimization formu-lations deal only with the calculation of a structure's optimal dimensions. Moreover, no literature on reliability-based optimization of either configurations or material selection exists [10]. Although the inclusion of these aspects of a plywood-web beam's design may significantly influence the optimal solution, they are not addressed in the remainder of 55 Chapter 5. Optimization Procedure 56 this study. The analyst must select the beam's configuration and the materials to be used for its components. The optimization routine is then used to compute the optimal dimensions of the beam which minimize its cost subject to the constraints imposed on its performance. 5.2 Alternative Optimization Formulations In reliability-based optimization studies, two different formulations of the optimization problem are often employed. 5.2.1 Unconstrained Optimization Problem A n unconstained optimization formulation may be adopted in which the cost of the plywood-web beam is expressed as: C = C0({z}) + CfPf (5.1) where: Co = the initial cost of the beam as a function of the design variables, {z} Cf = the cost of failure and Pf = the probability of failure computed by the reliability analysis Hence, the program searchs for the optimal trade-off between the plywood-web beam's initial cost and its probability of failure. Thus, a rational means of establishing the optimal level of safety which a structure should possess is provided. However, this for-mulation suffers from the disadvantage that the cost of failure, Cf, in equation (5.1) is difficult to assess since it would require placing a value on human life and/or the pre-diction of any finacial consequences of a business being inoperative for a period of time while repairs are undertaken. Chapter 5. Optimization Procedure 57 5.2.2 Constrained Optimization Problem The alternative formulation of the optimization problem commonly used in reliability-based studies is posed as: minimize C = CQ ({Z}) (5.2) such that 0i > $ (i = l ,2 , - - -n) in which: CQ = the initial cost of the beam as a function of the design variables, {z} 0i = the reliability index associated with the ith limit state or set of limit states /?* = a target reliability index for the ith limit state or set of limit states deemed to reflect adequate performance of the structure Although some rationality in the optimal design is sacrificed due to the use of predefined target safety levels, /?*, the constrained optimization problem in equation (5.2) is easier to define than the unconstrained problem in equation (5.1) since the evaluation of the cost of failure in not necessary. 5.3 Adopted Optimization Procedure The second formulation of the optimization problem described above in Section 5.2 was adopted for this study. A non-linear optimization routine for constrained problems, NLPQLO, which is available from the general library on M T S at the University of British Columbia Computing Centre, was used to find the optimal plywood-web beam design. Chapter 5. Optimization Procedure 58 The NLPQLO algorithm is based on Schittkowski's implementation of the quadratic ap-proximation method of Wilson, Han and Powell [14,15]. Details of the routine's imple-mentation on M T S can be found in reference [23]; however, some features of the program that are attractive for this study are the following. 1. The objective function to be minimized (the cost of the plywood-web beam) may be a non-linear function of the design variables. Since the material cost is, in part, a function of the volume of material used, it will generally be a non-linear function of the design variables (ie. the components' dimensions). 2. The constraints on the optimal solution may be non-linear. This will usually be the case with the reliability-based constraints in the problem. 3. Although the initial design point, {z0}, must be good, NLPQLO does not require that it be feasible. 4. The NLPQLO routine has been shown to be both reliable and efficient. The analyst must provide NLPQLO with a means of evaluating the objective function, constraints and their first partial derivatives with respect to the design variables. In this study, the evaluation of the reliability-based constraints is performed by R E L A N with their first partial derivatives approximated by central differencing. For example, to compute dPi/dzj where /?,- is the reliability index of the ith limit state and Zj is the jth design variable, Zj is perturbed by ± AZJ and the resulting reliability indices, ft and ft axe recorded. Thus, dfit = ft ~ ft dzj 2AZJ The objective function and any geometric constraints imposed upon the optimization problem (eg. maximum depth of the plywood-web beam) are generally simple algebraic Chapter 5. Optimization Procedure 59 functions which axe trivial to evaluate. 5.4 Discrete—Value Optimization In general, only discrete sizes of wood products such as lumber and plywood are com-mercially available. Hence, although the dimensions for a plywood-web beam computed by NLPQLO may be optimal, they are not necessarily practical. Several discrete variable, non-linear optimization methods could be used to solve a problem of this nature [3]. However, Siddall [17] suggests that "no really satisfactory methods appear to be available . . . for directly using a non-linear strategy which treats the variables as discrete." Instead, a more practical way of solving the problem is to treat the design variables as continuous and round-off the optimal solution to a nearby discrete solution. Such approaches are often called "pseudo-discrete" methods. Although they will generally return a solution, there is no guarantee that the solution will be the optimal one for the discrete problem. In this study, a pseudo-discrete method is employed to find the "optimal" discrete values for the plywood-web beam's dimensions from lists of allowable sizes provided by the analyst. First, starting from an initial design point, {z0} provided by the analyst, NLPQLO is used to calculate the continuous solution to the optimization problem de-scribed by equation (5.2). Once the optimal values for the design variables, {z*}, are found, the analyst specifies a "radius", r, for each design variable which indicates the maximum number of discrete values above and below the continuous solution which are to be considered as candidates for the optimal discrete solution. A systematic, exhaus-tive search of this restricted discrete solution space is then conducted to find the optimal combination of discrete values for the design variables. Figure 5.1 illustrates the solution Chapter 5.' Optimization Procedure 60 z l J i O p t i m a l C o n t i n u o u s S o l u t i o n Reg ion S e a r c h e d by P s e u d o - D i s c r e t e ~V77~~' Discre te Value O p t i m i z a t i o n P r o c e d u r e C o m b i n a t i o n s • y C o n s t r a i n t iL Z 2 Figure 5.1: Region Near the Continuous Optimal Solution Searched by the Pseudo-Discrete Optimization Procedure space searched by the above pseudo-discrete procedure for an optimization problem of two design variables, each having a search radius of r = 2. The number of discrete value combinations to be considered, M, is approximately: M = n(2r<) (5.3) where nz is the number of design variables in the optimization problem and r,- is the radius associated with the ith design variable. Equation (5.3) shows that the number of discrete value combinations to be checked grows rapidly as either the radii or the number of design variables is increased. To improve the efficiency of the pseudo-discrete algorithm, the aspects of the design which are the least expensive to calculate are checked first followed by those which require greater computational effort. Should a given discrete solution fail to satisfy either the minimum cost condition of any of the constraints, the solution is immediately rejected and the next candidate design is examined. The order in which a trial design is checked is as follows. Chapter 5. Optimization Procedure 61 1. The discrete design's objective function is computed. If it is less than the value associated with the continuous solution, the design is assumed to be infeasible and rejected. If, on the other hand, the objective function value is greater than that of a previously examined feasible discrete design, the design is rejected on the basis that it is inefficient. 2. The designs which fall in between these bounds on the objective function are then checked to see if they violate any geometry constraints imposed on the problem. 3. Finally, if no geometry constraints are violated, the reliability-based constraints are evaluated. A design which successfully passes all of the above checks becomes the current optimal discrete design. The process is repeated until every candidate combination has been examined. Chapter 6 Sensitivity Analysis 6.1 Purpose of Performing a Sensitivity Analysis The optimal plywood-web beam design computed by the reliability-based optimization procedure described in Chapter 5 is influenced by the input design parameters used in the analysis. In general, the value of each design parameter has some error associated with it due to its inexact measurement or computation. Consequently, the adequacy and efficiency of the computed optimal design are also erroneous. Sensitivity analyses are used in this study to ascertain the influence of the different input parameters upon the computed optimal design and, in particular, identify the critical parameters in the optimization procedure. This information, in turn, can be used to establish the required degree of accuracy for each of the input parameters, thereby providing a rational means of determining the necessary sample sizes when gathering statistical data and the required complexity of the numerical or analytical models which provide the computed input parameters. 6.2 Problem Studied In this study, sensitivity analyses were performed on the optimization of a box girder's design. The simply-supported girder has a length of 25 feet and supports a conventional timber roof system consisting of purlins, decking and roofing materials. The box girders 62 Chapter 6. Sensitivity Analysis 63 3.5 i n . i r Figure 6.1: Box Girder to be Optimized supporting the roof are spaced at 6 foot centres. Acting on the girder is a uniformly distributed load. The dead load component consists of the box girder's self weight and the weight to the roof structure it supports. The live load component is assumed to be a snow load. 6.2.1 Design Variables The cross-section of the box girder is shown in Figure 6.1 with the dimensions to be computed indicated by z 1 } z2 and z3. A summary of the initial values assumed for the design variables and well as the lower and upper bounds on the allowable values of these dimensions is given in Table 6.1. The upper and lower bounds represent maximum and minimum sizes of the materials available or limits on the design imposed by the manufacturing process. For the purposes of finding the pseudo-discrete optimal solution as described in Sec-tion 5.4, lists of allowable discrete dimensions for the design variables must be provided. Chapter 6. Sensitivity Analysis 64 Table 6.1: Initial Values, Lower and Upper Bounds for Design Variables Dimension (see Figure 6.1) Description Initial Value (inches) Lower Bound (inches) Upper Bound (inches) Z\ Flange Depth 5.50 3.50 13.25 zi Web Height 24.00 12.00 48.00 zz Flange-Web Overlap 3.50 3.50 13.25 Table 6.2: Allowable Discrete Values of Design Variables Z\ z 2 zz Flange Depth Web Height Flange-Web Overlap (inches) (inches) (inches) 1.50 12.00 1.50 3.50 13.00 2.00 5.50 ; : 7.25 1 inch intervals 0.5 inch intervals 9.25 : 11.25 47.00 13.00 13.25 48.00 13.25 Table 6.2 summarizes the lists used in this optimization problem. The allowable discrete values of the flange depth are based on the standard sizes of lumber commercially available while the discrete values of the web height and the flange-web overlap are based on conventional design and construction practice of using convenient dimensions. It could be argued that the allowable discrete web height dimen-sions should only be those which are evenly divisible into 48 inches, thus allowing no waste from a standard 48 inch wide sheet of plywood or waferboard. However, for this study, it is assumed that any strips of plywood too narrow to be used as web material, can be used for other purposes such as splice plates to join adjacent web components Chapter 6. Sensitivity Analysis 65 Table 6.3: Flange Material Properties Material Weibull Distribution Mean C.O.V. a Property Parameters Scale Shape (psi) ( % ) (psi) Modulus of Elasticity 1,973,600 5.800 1,827,500 0.20 Modulus of Rupture 8,983 2.700 7,988 0.40 Compression Strength 8,022 2.700 7,134 0.40 Tension Strength 6,309 2.700 5,611 0.40 Shear Strength 652 2.700 580 0.40 "Coefficient of Variation along the girder's length. 6.2.2 Random Variables The box girder's flange components are made of select structural Douglas Fir lumber and the web material is 5/8 inch, 5 ply regular unsanded Douglas Fir plywood. Both the strengths and stiffnesses of these materials are treated as random variables having the probability distributions shown in Tables 6.3 and 6.4. In all cases, a two-parameter Weibull distribution is assumed. Furthermore, the random variables are assumed to be uncorrelated. Plywood is an orthotropic material. As indicated by the strain energy formulation for the webs in equation (3.18), there are four stiffnesses associated with the webs in this study: Kwx, KwG-, DVUX and Dwq- The modulus of elasticity of the web material, Eweb, listed in Table 6.4 is calculated from: where tw is the web thickness; hence, from the definition of D W X found below Chapter 6. Sensitivity Analysis 66 Table 6.4: Web Material Properties Material Property Weibull Distribution Parameters Mean C.O.V. ° Scale (psi) Shape (psi) ( % ) Modulus of Elasticity 1,024,400 5.800 948,500 0.20 Shear through Thickness Strength 598 3.700 540 0.30 Rolling Shear Strength 304 3.700 274 0.30 "Coefficient of Variation Table 6.5: Web Stiffness Properties Stiffness Mean Value Property 23,000 lb • in 1,500 lb • in 571,000 lb / in 49,500 lb / in equation (3.17): Ex 1 - VxzVzx In addition, it is assumed that all four stiffnesses found in equation (3.18) are perfectly correlated to the modulus of elasticity, Eweb. Therefore, Eweb is the only random variable required to describe the stiffness properties of the web material entirely. Table 6.5 lists the assumed mean values of the four web stiffnesses corresponding to the modulus of elasticity listed in Table 6.4. The flange and web components are connected together by means of an adhesive. In Chapter 6. Sensitivity Analysis 67 Table 6.6: Connection Stiffness Properties Stiffness Mean Value Property Longitudinal Stiffness 300,000 ( lb/ in) / in 2 Lateral Stiffness 300,000 ( lb/ in) / in 2 Rotational Stiffness 900,000 (lb/rad)/in 2 a manner similar to that employed for the different stiffnesses of the web material, the lateral and rotational stiffnesses of the adhesive are assumed to be perfectly correlated to the longitudinal stiffness which was originally modelled by a normally distributed random variable having a mean of 300,000 l b / i n 3 and a coefficient of variation of 15 %. However, preliminary analyses indicated that the reliability index of the girder is insensitive to this random variable. Hence, for the remainder of the study, the flange-web connection stiffnesses are assumed to be constants and assigned their mean values shown in Table 6.6. The uniformly distributed loads acting on the box girder are treated as random vari-ables. The dead load, which includes the girder's self weight and the weight of the roof structure it supports, is assumed to be normally distributed with a mean value of 0.086 psi and a coefficient of variation of 10 %. The live load component is assumed to be a snow load and is represented by a Extreme Type I (Gumbel) distribution having a mean value of 0.145 psi and a coefficient of variation of 50 %. The dead load and the live load are assumed to be uncorrelated. Chapter 6. Sensitivity Analysis 68 6.2.3 Objective Function The optimization program is used to minimize the cost of the box girder defined by: C = 2 • Cfbfdf + 2 • Cwtwhw + 4 • Caha (6.1) where: C the cost per linear foot of the box girder Cf . = the cost per unit cross-sectional area per linear foot of flange material Cw — the cost per unit cross-sectional area per linear foot of web material ca the cost per unit width of flange-web overlap per linear foot of adhesive 6/, df = the width and depth of the flange components the thickness and height of the web components K the width of the flange-web overlap Referring to Figure 6.1, the component dimensions found in equation (6.1) can be replaced by: bf = 3.500 inches tw = 0.602 inches df = zi hw = z-i ha = z3 Hence, equation (6.1) becomes: C = 7.0 • Cfzx + 1.204 • Cwz2 + 4.0 • Caz3 (6.2) Although it is not imperative that the exact material and labour costs associated with the flange and web components be used in this study, it is desirable to have reasonable Chapter 6. Sensitivity Analysis 69 values so that the effects of a material's relative cost with respect to the other materials are properly represented. For the purposes of this study, the cost coefficients, Cf, Cw and Co, are estimated, from the current material costs of Douglas Fir lumber, plywood and construction adhesives. Thus the cost coefficients: Cf = ($0.030/in2) /linear ft. Cw = ($0.080/in2) /linear ft. Ca = ($0.005/in )/linear ft. were assumed and substituted into equation (6.2) to yield the objective function: C = 0.21 • zx + 0.09632 • z2 + 0.02 • z3 (6.3) 6.2.4 Constraints The optimal design's performance with respect to a prescribed set of design criteria serves as the constraint in the optimization problem. Both ultimate and serviceability limit states are considered. For this example, the ultimate limit states are the plywood-web box girder's resistance to (1) axial stresses in its flange components and (2) shear stresses at its neutral axis in the web components. The shear resistance of the flange-web connections are not included. The failure functions used to evaluate the resistance of the flange material to the resulting axial stresses are those derived in equations (4.4) and (4.5). These failure functions are of the form currently assumed by the Canadian Timber Design Code for this limit state. Equation (4.8) is used as the failure function for the shear stress limit state in this example. The girder's performance with respect to the ultimate limit states is measured by its overall probability of failure or, alternatively, its reliability index with respect to Chapter 6. Sensitivity Analysis 70 these failure modes. The bounds on the reliability index are calculated using Ditlevsen's method [6]. The lower bound, which is a conservative estimate of the girder's true reliability index, is used as the constraint's value. The serviceability limit state, which is treated separately from the ultimate limit states, is the girder's deflection. A midspan deflection of L/360 is to be allowed. Target reliability indices, fl* and /?*, which the plywood-web box girder's actual ultimate and serviceability reliability indices, /3U and fla, must meet or exceed, are used as the performance criteria and hence act as the constraints on the optimal design. In this example, ft = ft = 3.0 {P} « 0.0013). The girder's geometry is constrained in two ways. First, the design variables, Z\, Z-I and ZZ must assume values within the lower and upper bounds defined in Table 6.1. Second, to prevent the box girder's top and bottom flanges from mathematically "over-lapping", the design variables, z 2 and Z3 must satisfy the following constraint: Z2 - 2 • Z3 > 0 Referring to Figure 6.1, it can be seen that this relationship forces the webs' height, z 2 , to be large enough that the desired flange-web overlap, Z 3 , at the webs' top and bottom edges can be provided. 6.3 Sensitivity Analysis Results 6.3.1 Benchmark Optimization Problem The problem description outlined in Sections 6.2.1 to 6.2.4 serves as a benchmark in this study against which the effects of changes in the various design parameters on the computed optimal design can be evaluated. The structural analysis for this benchmark Chapter 6. Sensitivity Analysis 71 case uses the first three odd terms, (n = 1,3,5), in the Fourier series approximation of the plywood-web girder's displacements. In practice, it is impossible to exactly ascertain the actual material properties of the plywood-web girder and the optimization problem formulation which yields the true op-timal solution. However, for purposes of comparison, it is assumed for the remainder of this study that the material properties, optimization problem constraints, objective function and structural analysis assumed for this benchmark case in fact represent the actual properties and behaviour of the plywood-web girder. Thus, the resulting continu-ous and pseudo-discrete optimal designs obtained when the benchmark case is assumed can be thought of as the "true" optimal girder designs. These solutions are summarized in Table 6.7. The optimization was performed several times assuming different initial design points, {z0}. In all cases, the continuous optimal solution found by NLPQLO was within 1.0 % of the solution shown in Table 6.7. Table 6.7: Benchmark Optimization Problem Results Solution Design Variables Optimal Reliability Indices Type Z\ (inches) z2 (inches) zz (inches) Cost ($/linear ft.) A , A Continuous 6.187 35.904 3.500 4.83 3.000 6.184 Pseudo-Discrete 7.250 34.000 3.500 4.87 3.010 6.325 The total depth of the girder is approximately 41 inches for both the continuous and pseudo-discrete solutions, resulting in a depth-to-length ratio of about 1/7 for the optimal design. This ratio is unusually high: timber members are usually designed to have depth-to-length ratios of between 1/15 and 1/20. However, in this example, neither the flanges' width nor the plywood webs' thickness are treated as variables. Hence, in order Chapter 6. Sensitivity Analysis 72 to reduce the stresses in the flange and web components to levels which can be resisted by the materials used, the optimization procedure has no option except to increase the depth of the girder until the target reliability can be met. Furthermore, because neither the lateral stability of the girder nor the buckling of its web components are addressed in this study, there is no incentive for the optimization algorithm to reduce the girder's depth. Thus, unless a constraint is imposed upon the girder's depth, the depth-to-length ratio of the optimal design will not necessarily fall within the usual limits for timber members. In this study, the only constraints imposed upon the girder's depth are those which result indirectly from the lower and upper bounds assigned to the design variables in Table 6.1. Table 6.7 shows that for this example, the serviceability limit state imposed upon the girder's design does not influence the optimal solution. Consequently, this limit state is not addressed in the remainder of the study. Subsequent references to the girder's performance or reliability index are associated with the girder's ultimate limit states only. 6.3.2 Sensitivity Analysis Procedure The sensitivity of the cost, dimensions and performance of the computed optimal design with respect to the parameters in the optimization process was studied. To observe the influence of a given parameter on each of these attributes of the optimal design, the parameter was varied over a reasonable range of values while holding all other parameters in the problem constant at their values assumed in the benchmark case. The effects of perturbing a given input parameter on the computed optimal cost and dimensions of the plywood-web girder are readily available from the results of the optimization procedure: altering a parameter's value will change the computed dimen-sions and hence the cost of the girder. However, the performance of the girder's design, Chapter 6. Sensitivity Analysis 73 represented by its reliability index, acts as the constraint in the optimization problem. Consequently, because the optimal solution found when one of the input parameters is altered will generally lie on the reliability constraint, the design's reliability index will remain /? = j3*. Hence, the girder's performance will appear to be unaffected by changes made to the input parameters. However, in fact, a relative change in the girder's performance does result when its dimensions are altered. It was assumed in Section 6.3.1 that the optimization problem defined in Sections 6.2.1 to 6.2.4 exactly represents the actual composition and behaviour of the plywood-web girder being studied. Hence, the "actual" reliability index of a given design having values {z**} for its design variables can be evaluated by performing a reliability analysis upon a plywood-web girder having dimensions {z**} but material properties, performance criteria and material cost coefficients assumed in Sections 6.2.1 to 6.2.4. Using the results of these reliability analyses, it is possible to measure the relative difference between the reliability indices of girders having different dimensions. Any deviation from the optimal dimensions computed by the benchmark problem due to a change in a design parameter's value, causes a corresponding change in the girder's measured performance. The reliability index of a girder computed in this manner is an indicator of where in the design variable space, {z}, that design lies in relation to the reliability constraint used in the benchmark problem. In this study, a perturbed design parameter is thought of as an error. Therefore, a design, {z**}, resulting from the use of a perturbed design parameter is not the true optimal solution to the problem. If an input parameter is altered such that a less conservative design than that obtained by the benchmark case results, the computed reliability index will be less than the target reliability index, /3* indicating that the design, {z**}, is actually infeasible. Alternatively, if a more conservative design is obtained, the corresponding reliability index will be Chapter 6. Sensitivity Analysis 74 greater than /?*, indicating that the design, {z**}, is not optimal. 6.3.3 Scope of the Sensitivity Analyses The errors associated with the optimization problem's input parameters fall into three basic catagories: (1) problem formulation errors, (2) analytical errors and (3) statistical errors. Sensitivity analyses were conducted in order to ascertain the influence of these errors on the computed optimal design. Graphs illustrating the sensitivity of the problem to the various input parameters con-sidered are presented in Appendix A . Both the continuous and pseudo-discrete solutions were considered; however, for clarity in.presentation, only the continuous solution results are shown for the cost and performance sensitivity analyses. In general, the pseudo-discrete results corresponding to these aspects of the computed optimal design differ by less than 2.0 % from the continuous solutions. 6.3.4 Sensitivity Analyses of Problem Formulation Errors Problem formulation errors result primarily from the analyst's incomplete understanding of the nature of the optimization problem. Sources of this type of error include, for example (1) the exclusion of critical failure modes from the reliability-based constraints imposed upon the problem, (2) the improper formulation of a limit state's failure function and (3) the use of inaccurate cost coefficients in the objective function. Failure Modes Included in the Design Criteria The failure modes which govern the design of a plywood-web beam are problem depen-dent. In general, preliminary analyses should be performed in order to determine which Chapter 6. Sensitivity Analysis 75 failure modes are critical in the design. However, even after such preliminary investiga-tions, it is possible for the analyst to mistakenly identify a critical limit state as being unimportant to the optimal solution. Moreover, the computational cost of the optimiza-tion program increases with the number of failure modes included in the design criteria. Therefore, there is an incentive for the analyst to omit limit states from the assessment of the structure's performance. The plywood-web girder can be subjected to four ultimate limit states in this study: (1) compressive stresses in the top flange, (2) tensile stresses in the bottom flange, (3) shear stresses in the web components at the girder's neutral axis and (4) shear stresses at the flange-web connections. The consequences of excluding any of these failure modes from the optimization procedure for the girder design in this study are outlined in Ta-ble 6.8. The results in Table 6.8 indicate that the webs' resistance to the shear stresses at the girder's neutral axis is the dominant limit state in the reliability constraint: the presence or absence of this failure mode in the constraint's formulation had the greatest influence on the resulting optimal design's cost, performance and dimensions. The tensile resistance of the bottom flange was the second most critical failure mode followed by the limit states associated with the top flange's compression resistance and the flange-web connections' shear resistance. If a given failure mode is sensitive to a parameter, p, but the failure mode does not contribute significantly to the girder's overall probability of failure as calculated by Ditlevsen's method [6], the optimization problem may not be sensitive to p. Consequently, the influence which a particular failure mode has on the reliability constraint affects the results of the sensitivity analyses performed on the problem's input parameters. According to this hypothesis, the exclusion of a critical failure mode should strongly Chapter 6. Sensitivity Analysis 76 Table 6.8: Optimal Design for Different Ultimate Limit States Included in the Reliability Constraint Limit States Included" Solution Type Design Variables Optimal Cost ($/linear ft.) Reliability Index6 Z\ (inches) Z2 (inches) ZZ (inches) 1,2,3,4 Continuous Discrete 5.629 5.500 37.663 38.000 3.500 3.500 4.88 4.89 3.015 3.016 1,2,3C Continuous Discrete 6.187 7.250 35.904 34.000 3.500 3.500 4.83 4.87 3.000 3.010 2,3 Continuous Discrete 5.210 7.250 36.976 33.000 3.500 3.500 4.73 4.77 2.967 2.980 1,3 Continuous Discrete 4.074 3.500 38.187 40.000 3.500 3.500 4.60 4.66 2.919 2.923 3 Continuous Discrete 3.500 3.500 37.402 38.000 3.500 3.500 4.40 4.47 2.845 2.864 1,2 Continuous Discrete 11.334 9.250 12.000 17.000 3.500 3.500 3.60 3.65 2.395 2.472 3,4 Continuous Discrete 3.500 3.500 37.785 38.000 3.500 3.500 4.44 4.47 2.857 2.864 1,2,4 Continuous Discrete 7.767 7.250 21.758 23.000 3.500 3.500 3.80 3.81 2.583 2.598 4 Continuous Discrete 3.500 3.500 15.497 16.000 3.500 3.500 2.37 2.42 1.505 1.569 "1 = compression stresses in top flange 2 = tension stresses in bottom flange 3 = shear stresses at neutral axis 4 = flange-web connection shear stresses 'Computed as described in Section 6.3.2 c Benchmark problem Chapter 6. Sensitivity Analysis 77 affect the results of sensitivity analyses. Table 6.8 shows that the optimal design is altered only slightly when the flange-web shear failure mode is included. Hence, the omission of this limit state from the benchmark problem should not affect the results of the following sensitivity analyses significantly. When the flange-web shear resistance limit state is not included in the reliability con-straint, it can be shown that both goals of the optimization routine: increased reliability and decreased cost, are achieved when the flange-web overlap design variable, z3, assumes its lower bound. Only when the problem's constraint includes the girder's flange-web shear resistance is there motivation for z3 to assume a value other than its lower bound in order to reduce the shear stresses in the flange-web connections. However, Table 6.8, shows that even when this failure mode is accounted for in the reliability constraint, the overlap did not vary from its lower bound. In the remainder of the study, the flange-web shear resistance was not included in the optimization problem's constraint. It was found, as a result, the the design variable z3 did not vary from its lower bound in any of the sensitivity analyses conducted; it was insensitive to every parameter considered and thus behaved as a constant. Formulation of the Flexural Resistance Limit State As discussed in Section 4.4.2, the limit state associated with a plywood-web beam's flexural resistance can be described in two ways. The benchmark problem employs the limit state formulation which compares a flange's axial compression or tension strength to the maximum stress in the flange. The alternative limit state formulation proposed in Section 4.4.2 assumes that a linear moment-axial interaction failure mechanism governs a plywood-web beam's flexural resistance. Because it includes an additional random variable (modulus of rupture), this alternative formulation incurs greater computational Chapter 6. Sensitivity Analysis 78 costs than the formulation used in the benchmark problem. However, it may be argued that the moment-axial interaction failure mode better represents the girder's flexural resistance. Hence, a tradeoff exists between the analysis' cost and accuracy. The optimal girder design obtained when this alternative flexural resistance limit state is employed is outlined in Table 6.9. Table 6.9: Optimal Solution Obtained with the Moment-Axial Interaction Failure Function Assumed for Flexural Resistance Limit State Solution Design Variables Optimal Reliability Type Cost Index0 (inches) (inches) (inches) (S/linear ft.) P Continuous 6.056 34.746 3.500 4.69 2.957 Pseudo-Discrete 5.500 36.000 3.500 4.69 2.958 "Computed as described in Section 6.3.2 With the moment-axial interaction failure mechanism assumed for the girder's flexu-ral resistance limit state, a less conservative optimal design compared to that obtained by the benchmark problem results. However, the optimal designs found using the different limit state definitions are very similar. Two explanations for the similarity between the computed optimal designs are available. First, the shear resistance limit state dominates the reliability constraint; hence, any modifications made to the flexural limit state cannot be expected to affect the optimal design greatly. Second, as discussed in Section 4.4.2, because the girder's cross-section is relatively deep, the axial component of the stress in a flange is much larger than the bending component. Thus the maximum axial stress formulation of the flexural limit state, employed by the benchmark problem is a close approximation to the moment-axial interaction failure mode formulation. Chapter 6. Sensitivity Analysis 79 The above explanations however are only valid for this example. In general, the effect of using different failure functions for a particular limit state is problem dependent. To illustrate this, the optimization problem was repeated with the moment-axial interaction failure function assumed for the flexural stress limit state and the shear stress limit state omitted from the reliability constraint. In this case, as deduced from the results found in Table 6.8, the tension flange limit state governs the girder's reliability and the optimal design will be shallower than that obtained when the shear stress limit state was included. The results of the trial are shown in Table 6.10. Table 6.10: Optimal Solution Obtained with the Moment-Axial Interaction Failure Function Assumed for the Flexural Resistance Limit State and the Shear Stress Limit State Omitted Solution Design Variables Optimal Reliability Type Cost Index" (inches) (inches) (inches) ($/linear ft.) Continuous 7.784 12.000 3.500 2.36 1.922 Pseudo-Discrete 7.250 14.000 3.500 2.94 2.020 "Computed as described in Section 6.3.2 Comparing the optimal design obtained when only the maximum axial stress limit states were considered (see Table 6.8, row # 6) to that shown in Table 6.10 illustrates that the optimal design can be significantly affected by the forms of its limit states' functions. Cost Coefficients Section 2.2 discussed the difficulties of estimating the actual cost associated with a par-ticular component used in a plywood-web beam. Along with the material costs, labour Chapter 6. Sensitivity Analysis 80 and production costs must also be accounted for. Consequently, there will generally be some discrepancy between the assumed cost coefficients used in the objective function and the true cost coefficients. Figures A . l to A.3 illustrate the effects of errors in the objective function's cost coefficients on the optimal computed design. The costs in Figure A . l are computed using the benchmark cost coefficients, Cf, Cw and Ca listed in Section 6.2.3, applied to the girder designs, {z**}, found when the perturbed cost coefficients were assumed. This procedure, which is analogous to the reliability index computations described in Section 6.3.2, serves as an indicator of where in the design variable space, {z}, a given design lies in relation to the objective function assumed in the benchmark problem. The performance of a computed optimal design does not change when the cost co-efficients are perturbed since the reliability constraint in not affected by changes to the objective function. However, the resulting flange and web component dimensions are af-fected significantly (see Figures A.2 and A.3). Consequently, as illustrated in Figure A . l , the solution is not optimal if the benchmark cost coefficients are in fact accurate. The reliability indices of the different ultimate limit states included in the optimization problem's formulation are functions of the design variables, Z\, z2 and z3. Furthermore, Figures A.2 and A.3 show that the optimal dimensions of the girder's components are dependent upon the assumed cost coefficients used in the objective function. Hence, the relative influence of the different failure modes on the computed optimal design changes as the ratio of the flange material's cost to the web material's cost changes. Figure A.4 illustrates the effect of this material cost ratio upon the reliability indices of the individual limit states which make up the reliability constraint. The smaller the value of a given failure mode's reliability index relative to the other failure modes considered, the more influential that failure mode is on the girder's overall safety. Chapter 6. Sensitivity Analysis 81 As the ratio of the flange material's cost to the web material's cost increases, the influence of the shear stress failure mode on the optimal solution is reduced relative to the influence of the flexural failure modes. As discussed in Section 6.3.4, the influence which a particular failure mode has on the optimal design affects the results of sensitivity analyses. Therefore, the use of incorrect cost coefficients will lead to inaccurate evaluations of the optimization problem's sensitivity to its input parameters. For this example however, Figure A.4 shows that the shear stress limit state dominates the reliability constraint over a wide range of material cost ratios. As a result, the relative influence of the different limit states on the optimal solution does not change significantly with the assumed cost coefficients. Hence, the use of incorrect cost coefficients should not affect the remaining sensitivity analyses performed in the study significantly. 6.3.5 Sensitivity Analysis of Analytical Errors Even if the optimization problem is correctly formulated, errors may still be introduced into the computed optimal design through analytical errors. Two sources of analytical errors in this study are (1) modelling errors which arise from idealizations in the struc-tural analysis and (2) convergence errors which result when the girder's response is not computed to a sufficient degree of accuracy. Modelling Errors The plywood-web beam model used in this study assumes linear elastic behaviour, ho-mogeneous material properties along the length of the beam and ideal simply-supported reaction points. These assumptions and approximations all contribute to errors in the computed optimal design. The only way these errors can be reduced is through the use of more complex computational procedures which better reflect the actual characteristics Chapter 6. Sensitivity Analysis 82 of the plywood-web girder. To study the effects of an error in a calculated load effect, S{, on the optimal design, a modelling error coefficient, c m , was introduced into the limit state function associated with 5,-: Gt = Ri- emSi (6.4) In general, em should be represented by a random variable since the composition of different girders is never identical and hence, em varies accordingly. However, in this study, e m is assumed to be constant. The modelling error coefficient was varied over a range of values and the attributes of the resulting optimal designs were noted. Errors in the calculated flexural stresses and in the calculated shear stresses were investigated. The results of the analyses are illustrated in Figures A.5 to A.8. Figures A.5, A.6 and A.8 show that the optimal cost, performance and web height of the girder are more sensitive to errors in the shear stress computations than to errors in the flexural stress computations. The results of Section 6.3.4 indicated that the shear stress limit state influenced the computed optimal design more than any other failure mode which the girder was subjected to. Hence, it is reasonable that errors in the shear stress computations should, as a result, have a greater effect on the computed optimal design than errors made in the flexural stress calculations. Both the continuous and pseudo-discrete solution results in Figure A.7 however reveal that a similar trend does not exist for the optimal flange depth which is more sensitive to errors in the flexural stress computations. This observation suggests that the flange depth design variable is influenced to a much greater degree by the changes in the flanges' axial stresses than by changes in the webs' shear stresses. So much greater that, even after the relative influence of the different failure modes on the optimal design are accounted for, Chapter 6. Sensitivity Analysis 83 the flange depth is still more sensitive to errors in the flexural stress computations than to errors in the shear stress computations. Hence, the hypothesis presented in Section 6.3.4 is not always valid. It is possible for a parameter which does not affect the dominant limit state to significantly affect the optimal design. In a manner similar to that described in Section 6.3.4 with respect to deviations in the objective function's cost coefficients, errors in the stress computations determine the relative influence which the different failure modes have upon the computed optimal girder design. Figures A.9 and A.10 illustrate how the reliability indices of the ultimate limit states included in the optimization problem's constraint are controlled by errors in the flexural and shear stress computations. Over the range of modelling error coefficients studied, the shear stress limit state remained the critical failure mode in the reliability constraint; however, the relative influence of the different failure modes does change significantly as a result of errors in the shear stress computations (see Figure A.10). This observation suggests that the effects of analytical errors should not be ignored. Such errors, by altering the relative influence which each failure mode has on the optimal solution, will , in turn, have a noticeable effect on the influence of other input parameters to the optimization problem as hypothesized in Section 6.3.4. Convergence Errors The number of terms, N, used in the Fourier series approximations of the girder's dis-placements controls the structural analysis' accuracy. Unlike modelling errors which can only be reduced through the formulation of more sophisticated models, convergence errors can be reduced by simply using more terms in the Fourier series expansion. However, the computational cost of the structural analysis is directly proportional to this parameter; Chapter 6. Sensitivity Analysis 84 hence, there it is desirable to use as few Fourier terms as possible. The ways in which the number of terms used in the Fourier series approximation of the girder's response affect the computed optimal design's cost, performance and dimensions are illustrated in Figures A.11 to A.14. Figures A.11 and A.12 show that the sensitivity of the optimal design's cost and per-formance decreases as more terms are used in the Fourier series. This trend is indicative of the numerical model's convergence to the "true" solution with increased refinement of the assumed displacement functions. However, similar trends are not apparent in the continuous solution results for the optimal flange depth and web height shown in Fig-ures A.13 and A . 14. There appears to be a lag between the convergence of the girder's optimal cost and performance and the convergence of the design variables to their op-timal values. The pseudo-discrete results though do display some convergence in the optimal values of the design variables. Except for the trial in which 5 terms were used in the Fourier series expansion, the optimal discrete flange depth assumed its benchmark value while the web height converged to its benchmark value as the number of Fourier terms used was increased from N = 1 to N = 4. The optimal pseudo-discrete solution obtained using 3 and 4 terms in the Fourier series are identical. However, when 5 terms were used in the Fourier series, the benchmark discrete solution was infeasible and an alternative solution was found. Three Fourier terms were used in the structural analysis employed by the benchmark problem. Figures A . 11 to A . 14 show that using 4 terms in the Fourier series approxi-mation causes changes to the computed optimal design of the same magnitude as those incurred if the flexural or shear stress calculations are in error by approximately ± 5 % (see Figures A.5 to A.8). A n error of this magnitude in the stress computations is not unreasonable; hence, the use of more than 3 terms in the structural analysis is perhaps Chapter 6. Sensitivity Analysis 85 unwarranted as the improved accuracy with respect to convergence errors will be nullified by the effects of modelling errors. 6.3.6 Sensitivity Analysis of Statistical Errors Statistical errors refer to those inaccuracies associated with the representation of the random variables in a problem. These errors should not be confused with the physical uncertainty associated with the random variables themselves which is accounted for in the reliability analysis. Physical uncertainty is a characteristic of a given population described by the population's probability distribution; statistical errors though arise from the inexact measurement of the parameters which describe the probability distribution. Hence, statistical errors are directly attributable to the sampling techniques used to estimate these parameters. The sensitivity of the computed optimal design to the parameters which describe the probabilistic nature of the problem was examined. Included in the study were the influences of (1) the distribution parameters associated with each of the random variables in the problem, (2) the correlation coefficients between material properties of the girder's components and (3) the assumed distribution type used to model a material property. Distribution Parameters Information about a probability distribution's parameters can only be obtained through physical testing or examination. However, such testing is expensive to conduct and, even if it is done, the results of one test can only be used as estimates for the actual distribution parameters of the materials used in the construction of a plywood-web beam and the loads acting on it. Consequently, it is desirable to evaluate the sensitivity of the optimal design with respect to the assumed distribution parameters which, for a given distribution Chapter 6. Sensitivity Analysis 86 type, can be denned by the distribution's mean value and coefficient of variation (C.O.V.). In the following sensitivity analyses, the mean value or C .O.V. of the distribution under consideration was held constant at its benchmark value while the effects of changes in the other parameter on the computed optimal design were recorded. Figures A.15 to A.41 summarize the influence of the mean and C .O.V. of the stiffness, strength and load random variables on the computed optimal design's cost, performance and dimensions. Stiffness Proper t ies The results presented in Figures A.15 and A.16 indicate that the girder's optimal cost and performance are insensitive to changes in the mean values of the flange and web materials' elastic modulii over a range of ± 30 % from their benchmark values. In contrast, the continuous solution results in Figures A.17 and A.18 show that the optimal flange depth varies by about ± 17 % over the range of mean values studied while the optimal web height varied by approximately ± 7 %. The pseudo-discrete solutions for the optimal flange depth and web height however did not vary from their benchmark values over the entire range of mean values examined. Ultimately, a designer would have to use available sizes of lumber and plywood for the girder's components; hence, the discrete solution results suggest that, for practical purposes, the computed optimal design is insensitive to the mean values of the elastic modulii. As illustrated in Figures A.19 to A.22, all aspects of the girder's design are more sensitive to the C.O.V. of the webs' elastic modulus than to that of the flanges. This observation may be explained in part by the relative importance of the different limit states included in the problem. The findings of Section 6.3.4 revealed that for this example, the shear stress limit state dominates the reliability constraint. Thus, an input parameter which affects this limit state more than other input parameters will, in turn, Chapter 6. Sensitivity Analysis 87 also affect the optimal design to a greater degree, regardless of how these parameters affect the other limit states considered. The plywood's modulus of elasticity influences the shear stress limit state more the the lumber's modulus of elasticity as evidenced by the shear stress computations described in Section 3.2.4. Hence, it is reasonable that the computed optimal design should exhibit greater sensitivity to the C.O.V. of the web material's elastic modulus than to that of the flange material. S t rength Proper t ies For both the lumber and plywood material properties, the opti-mal design was equally or more sensitive to deviations in the parameters which describe the strength distributions than to deviations in,the respective parameters used to describe the stiffness distributions. Thus, for a given level of accuracy in the computed optimal design, the strength distributions' parameters must be measured to a higher degree of accuracy than the stiffness distributions' parameters. In general, Figures A.23 to A.30 indicate that, of the three strength distributions studied, the parameters associated with the shear strength of the webs influenced the optimal design to the greatest degree followed by those associated with the tension and compression strengths of the flanges. This observation agrees with the hypothesis put forth in Section 6.3.4 that parameters associated with the more influential failure modes in the reliability constraint will , in turn, have a more significant effect on the optimal girder design. The effects of a failure mode's relative influence on the sensitivity of the optimal design can also be seen in another aspect of Figures A.27 to A.30. For the C.O.V. of a given strength distribution, the sensitivity of the problem (defined by the slope of the graph) changes over the range of C .O.V. values studied. As a strength distribution's C.O.V. approaches zero, the distribution becomes narrower and thus the reliability index of the Chapter 6. Sensitivity Analysis 88 corresponding limit state increases. As a result, the influence which that limit state has on the optimal design decreases with respect to the other limit states considered. The problem's sensitivity to a distribution's C.O.V. is dependent upon the relative importance of the limit state with which it is associated as discussed in Section 6.3.4; hence, it is reasonable that the optimal design's sensitivity to a strength distribution's C .O.V. should decrease as the C .O.V. decreases. In fact, the results in Figures A.27 to A.30 show that as a given strength distribution's C .O.V. approaches zero, the optimal design converges to that found in Table 6.8 for the trial in which the corresponding limit state was excluded from the reliability constraint's formulation. Figure A.31 illustrates how the reliability indices of the different failure modes at the optimal solution are affected by the tension strength distribution's C .O.V. Although the C .O.V. of the tension strength only directly affects the corresponding limit state, this in turn affects the optimal design and hence, indirectly affects the other limit states in the problem. The smaller the value of a given limit state's reliability index, the greater its influence on the girder's overall probability of failure. Figure A.31 indicates that the tension strength limit state becomes the dominant limit state in the reliability constraint when its C .O.V. is approximately 50 %. This value coincides with the value in Figures A.27 to A.30 at which the optimal design's sensitivity to the tension strength's C .O.V. increases most rapidly. Similar results for the C.O.V.s of the lumber's compression strength and the plywood's shear strength are shown in Figures A.32 and A.33. The sensitivity exhibited by the benchmark problem to deviations in the mean value of the shear strength distribution is approximately equal to that found for the shear stress modelling error coefficient (compare Figures A.5 through A.8 to Figures A.23 through A.26) Similar observations can be made with respect to the mean values of the axial strengths of the flanges and the flexural stress modelling error. Thus, it is asserted that the accuracy Chapter 6. Sensitivity Analysis 89 of the optimal solution may be affected equally by analytical and statistical errors. Loads Figures A.34 to A.41 show that all aspects of the computed optimal design are more sensitive to changes in the live load's probability distribution than to that of the dead load. This result is probably due to the live load's greater magnitude and variability compared to the dead load; thus causing its distribution parameters to have a greater influence on the girder's reliability. At their benchmark values, the optimal design's sensitivity to the mean value of the live load is approximately equal to its sensitivity to the mean value of the web's shear strength. However, the optimal solution is much less sensitive to the live load's C .O.V. than to that of the shear strength. Deviations in the dead and live loads' C.O.V.s affected the girder's optimal design by approximately the same amount as deviations in the C.O.V .S of the flange and web materials' elastic modulii. Correlation Coefficients Two types of correlation between random variables were studied: (1) correlation between the stiffness properties of two components which come from the same population and (2) correlation between the strength and stiffness properties of a given material. Correlation Between the Stiffness Properties of Two Components from the Same Population In most practical situations, the material properties of wood-based products selected from the same population are correlated. This phenomenon is often referred to as the "lot" effect and arises due to the fact that a shipment of lumber, plywood or other wood product from a mill contains members that, in general, have similar physical characteristics such as moisture content, density and flaws which control Chapter 6. Sensitivity Analysis 90 the strength and stiffness properties of the wood. Consequently, the members in the shipment may come from an interval of the population's probability distribution, not from the entire range of the distribution. The degree of correlation in a given shipment is difficult to estimate without testing. As discussed earlier, testing is expensive and, at best, can only provide estimates of the probabilistic properties of the material used. Thus, the need for evaluating the effects of the correlation between the stiffness properties on the optimal solution becomes apparent. The sensitivity of the optimization problem to deviations in the correlation coefficients, p&, used to describe the degree of correlation between the stiffnesses of the flange components and between the stiffnesses of the web components are shown in Figures A.42 to A.45. Although, for both the lumber and plywood populations, the continuous solution results display some sensitivity to the assumed correlation coefficient, ps, the discrete optimal flange depth and web height assumed their benchmark values regardless of the degree of correlation between the stiffness properties. For practical purposes, the optimal design is insensitive to pE- In light of the sensitivity analyses obtained earlier for the elastic modulii distributions' parameters, this observation is reasonable. Correlation Between the Strength and Stiffness of a Component The strength and stiffness properties of a wood-based product are often governed by the same physical properties of the wood. For example, a strength-reducing flaw in a piece of lumber will often also act to reduce the lumber's stiffness. Consequently, the strength and stiffness properties of timber are usually correlated; the extent of the correlation depending pri-marily upon how strongly the critical physical characteristics of the timber affect both the strength and stiffness properties. Chapter 6. Sensitivity Analysis 91 The influences on the optimal design of the correlation coefficients, p$, between the strength and stiffness properties of the box girder's flanges and webs are also illustrated in Figures A.46 to A.49. In general, all aspects of the optimization problem exhibited greater sensitivity to the strength-stiffness correlation coefficient, p$, than to the stiffness-stiffness correlation coefficient, p£- Furthermore, as should be expected, the computed optimal design was more sensitive to the assumed correlation between the webs' stiffness and shear strength than to that between the flanges' stiffness and strengths. For all three strength distributions considered, the optimal cost and performance decreased as ps increased. Hence, from a design point of view, in the absence of informa-tion about the correlation between a material's strength and stiffness, it is conservative to assume ps = 0.0. Assumed Distribution Type The type of probability distribution used to model a material property or load in a relia-bility analysis is generally selected after studying sample data and making use of physical reasoning about the nature of the property. For example, a material's strength distri-bution is often modelled by a logarithmic normal distribution or a Weibull distribution since these distribution types can preclude the occurence of negative strength values [20]. However, there is always some degree of uncertainty associated with the selection of a random variable's distribution type. The behaviour of the optimization problem with respect to the distribution type cho-sen for the flange's tensile strength was investigated. The Weibull distribution assumed in the benchmark problem was used as the underlying parent distribution from which 20 samples of 500 tension strength values were selected randomly. Each sample was ranked Chapter 6. Sensitivity Analysis 92 and normal, logarithmic normal and two-parameter Weibull distributions were fitted to the sample data from the entire distribution using the method of maximum likelihood estimation [2]. Hence, 60 different representations of the Douglas Fir lumber's tension strength distribution were available. The fitted distribution parameters and the results of Kolmogorov-Smirnov goodness-of-fit tests [1] on each fitted distribution are shown in Tables B . l to B.3 in Appendix B . The optimization problem was performed with each of these distributions assumed for the flanges' tension strength. The ranked computed optimal costs, performances and dimensions are shown in Figures A.50 to A.53. Histograms of the discrete optimal flange depth and web height are shown in Figures A.54 and A.55. The optimization results obtained using the fitted two-parameter Weibull distribu-tions are approximately centered around the benchmark optimal solution. The difference between the optimal cost computed by the benchmark problem and that found with a fitted Weibull distribution assumed for the Douglas Fir lumber's tension strength varied over a range of ± 4 % for the 20 trials performed. The continuous solutions for the optimal flange depth varied within +20 % and —14 % of the benchmark value while the computed optimal web height ranged from +2.5 % to —3.3 % of its benchmark solu-tion. The histograms of the discrete optimal flange depth and web height show that the discrete solutions are also centered approximately at the benchmark results. The optimal designs obtained with the fitted logarithmic normal distributions as-sumed for the lumber's tension strength are centered near the optimal solution found when the axial tension strength limit state was excluded from the reliability constraint (see Table 6.8). In fact, as indicated in Figures A.54 and A.55, for each of the 20 tri-als performed, the discrete solution was exactly that obtained for this case. Hence, the use of a logarithmic normal distribution to model the lumber's tension strength has the Chapter 6. Sensitivity Analysis 93 undesirable effect of increasing the reliability index of the corresponding limit state and thus reducing its influence on the design. In contrast, feasible solutions were not found for the trials in which the tension strength was modelled by a normal distribution. In these cases, the reliability index of the tension strength's limit state was decreased substantially, causing it to dominate the optimization problem. The larger the Kolmogorov-Smirnov goodness-of-fit test statistic, the poorer the as-sumed distribution fits the sample data. Hence, the results of the Kolmogorov-Smirnov tests presented in Tables B . l to B.3 suggest that, in general, the two-parameter Weibull distributions fit the sample data best followed by the normal and logarithmic normal distributions. The critical values of the Kolmogorov-Smirnov test statistic for a fitted probability distribution to a sample are shown in Table B.5 for several levels of signifi-cance. Referring to these critical values, on average, the fitted two-parameter Weibull distributions should not be rejected at the 20 % significance level while the fitted normal and logarithmic normal distributions should be rejected at the 5 % and the 1 % signifi-cance levels respectively. These results suggest that the normal and logarithmic normal distribution assumptions provide poor approximations to the tension strength's assumed true probability distribution. Typically in reliability analyses, the strength random variable assumes a final value in the lower tail of its distribution at the point, {y*}, on the failure surface, G, = 0 (see Section 4.4.1). Hence, for the strength distributions, it is more important that the lower tail be accurately represented rather than the distribution as a whole. To further investigate the effects of the tension strength's assumed distribution type on the computed optimal design, normal, logarithmic normal and two-parameter Weibull distributions were fit to the lower tails of the simulated sample data. Only those tension Chapter 6. Sensitivity Analysis 94 strengths below 2750 psi were assumed to be available. This strength value corresponds to approximately the 10th percentile of the parent distribution. Maximum likelihood estimation was used to fit the distributions to these censored samples, the results of which are shown in Table B.4. The cost, performance and dimensions of the optimal designs found when these probability distributions were assumed for the tension strength random variable are shown in Figures A.50 toA.55. The two-parameter Weibull distributions fit to the entire sample and to the censored data yielded, similar results. However, when the Weibull distribution's parameters are estimated from the censored data, the results exhibit greater variability. This observation is attributable to the reduced number of data points in the censored sample. A similar trend can be seen with the logarithmic normal distribution results. However, in addition to increased variability, the designs obtained using the logarithmic normal distributions fit to the censored data also display increased reliability compared to those found with the logarithmic normal distributions fit to the entire sample data. This result reflects the improved fit of the logarithmic normal distributions to the strength distribution's lower tail when the censored sample data is used to estimate the distribution parameters. On average, when the censored data was used, the logarithmic normal distribution assumption yielded unconservative designs compared to those found with the Weibull distribution. Meanwhile, 16 of the 20 trials in which a normal distribution was assumed for the tension strength did not produce a feasible solution. Although estimating the distributions' parameters from the lower tail of the sample data tends to reduce the effects of the assumed distribution type, significant differences in the computed optimal design still result. Hence, to obtain a valid optimal design, the results shown in Figures A.50 to A.55 indicate that appropriate distribution types must be assumed for the strength random variables. Chapter 7 Conclusions A reliability-based optimization program which computes the optimal dimensions of a plywood-web beam's components was formulated and tested in this study. The optimization program attempts to minimize the cost of a plywood-web beam subject to constraints imposed on its performance expressed in terms of acceptable levels of safety with respect to a prescribed set of limit states. The performance of the struc-ture is evaluated by means of reliability analyses in order to rationally account for the uncertainty and variability associated with the beam's material properties and the loads acting on it. In the evaluation of the limit states' failure functions, the plywood-web beam's response under load is computed by means of a linear-elastic structural analysis which incorporates the shear deformations of the web components and the stiffnesses of the flange-web connections. The program uses an existing non-linear optimization routine for constrained prob-lems to compute the optimal continuous design of the ply wood-web beam. However, although this continuous solution may be optimal, it is not necessarily practical since, in general, the plywood-web beam's dimensions must coincide with the available or allow-able sizes of commercially produced lumber and plywood. The optimal discrete design, which assigns only allowable values to the beam's dimensions, is found by means of a pseudo-discrete procedure which conducts an exhaustive search of a restricted region of the design variable space near the computed optimal continuous solution. 95 Chapter 7. Conclusions 96 Sensitivity analyses were performed on the continuous and pseudo-discrete optimal designs in order to identify the critical input parameters in the optimization process. Such information is useful in establishing the required degree of accuracy in the measurement or computation of these parameters. The following conclusions can be made from the results of the sensitivity analyses: 1. The sensitivity of the computed continuous optimal design to a given input pa-rameter is affected by two aspects of the reliability constraint: (i) the influence of the parameter on each of the limit states considered in the problem and (ii) the relative influence each of these limit states has on the reliability constraint. It was found that if, for example, a given failure mode was sensitive to a parameter, p, but the failure mode did not significantly affect the reliability constraint, the optimal solution was not sensitive to p. 2. In general, the computed optimal design was most sensitive to changes in the prob-abilistic nature of the strength distributions followed by changes to the probability distributions of the live and dead load random variables and, lastly, the elastic modulii random variables. 3. Analytical errors, resulting in the inaccurate computation of the plywood-web beam's response under load, were found to have a significant effect on the rel-ative influence which each limit state in the reliability constraint had upon the optimal design. As a result, such errors would, in turn, affect the sensitivity of the optimization problem with respect to other input parameters as discussed in item (1) above. Furthermore, modelling errors were found to affect the optimal design to the same degree as errors in the mean values of the strength probability distributions. Therefore, it is asserted that the representation of the plywood-web Chapter 7. Conclusions 97 beam's structural behaviour is as important as the representation of the random variables in the problem. 4. The assumed distribution type of the lumber's tension strength random variable was found to significantly affect the reliability index of the corresponding limit state and, in turn, the computed optimal design. Normal, logarithmic normal and two-parameter Weibull distributions were fit to simulated sample data from the entire strength distribution and to censored data from approximately the lower 10 % of the distribution. Significant differences in the optimal designs computed using the different probability distributions were observed in both cases. Hence, appropriate distribution types must be assumed for the strength random variables in order to compute a valid optimal design for the plywood-web beam. Bib l iography [1] Benjamin, J.R. and Cornell, C .A. , Probability, Statistics and Decision For Civil Engineers, McGraw-Hil l Book Co., Toronto, Ont., 1970, pp. 466-469. [2] Bury, K . V . , Statistical Models in Applied Science, John Wiley, Toronto, Ont., 1975, pp. 161-168. [3] Cella, A . and Soosaar, K . , "Discrete Variables in Structural Optimization," in: Optimal Structural Design: Theory and Applications, R . H . Gallagher and O.C. Zienkiewicz, Eds., John Wiley, Toronto, Ont., 1972, pp. 210-222. [4] Der Kiureghian, A . and Lui , P.L. , "Structural Reliability Under Incomplete Proba-bility Information," Journal of Engineering Mechanics, A S C E , Vol. 112, No. 1, Jan., 1986, pp. 85-104. [5] Ditlevsen, 0. , "Principle of Normal Tail Approximation," Journal of Engineering Mechanics, A S C E , Vol. 107, No. EM6, 1981, pp. 1191-1208. [6] Ditlevsen, O., "Narrow Reliability Bounds for Structural Systems," Journal of Struc-tural Mechanics, A S C E , Vol. 7, 1979, pp. 435-451. [7] Foschi, R.O. , "Structural Analysis of Wood Floor Systems," Journal of Structural Engineering, A S C E , Vol. 108, No. ST7, July, 1982, pp. 1557-1574. [8] Foschi, R.O. , "Reliability of Wood Structural Systems," Journal of Structural En-gineering, A S C E , Vol . 110, No. ST12, Dec , 1984, pp. 2995-3013. [9] Foschi, R.O. , "Wood Floor Behaviour: Experimental Study," Journal of Structural Engineering, A S C E , Vol. I l l , No. ST11, Nov., 1985, pp. 2497-2508. [10] Frangopol, D . M . , "Structural Optimization Using Reliability Concepts," Journal of Structural Engineering, A S C E , Vol. I l l , No. 11, Nov., 1985, pp. 2288-2301. [11] Hasofer, A . M . and Lind, N .C . , "Exact and Invariant Second-Moment Code Format," Journal of Engineering Mechanics, A S C E , Vol. 100, No. E M I , Feb., 1974, pp. I l l— 121. [12] Moses, F. , "Sensitivity Studies in Structural Reliability," in: Structural Reliabil-ity and Codified Design, N . C . Lind, Ed. , Solid Mechanics Division, University of Waterloo, Study No. 2, Waterloo, Ont., 1970, pp. 1-17. [13] Rackwitz, R. and Fiessler, B . , "Structural Reliability Under Combined Random Load Sequences," Computers and Structures, Vol. 9, 1978, pp. 489-494. [14] Schittkowski, K . , "The Nonlinear Programming Method of Wilson, Han and Powell with an Augmented Lagrangian-Type Line Search Function. Part 1: Convergence Analysis," Numerische Mathematik, Vol. 38, Fasc 1, 1981, pp. 83-114. 98 Bibliography 99 [15] Scbittkowski, K . , "The Nonlinear Programming Method of Wilson, Han and Powell with an Augmented Lagrangian-Type Line Search Function. Part 2: A n Efficient Implementation with Linear Least Squares Subproblems," Numerische Mathematik, Vol. 38, Fasc 1, 1981, pp. 115-128. [16] Shinozuka, Q. and Itagaki, U . , "On the Reliability of Redundant Structures," Annals of Reliability and Maintainability, 1966. [17] Siddall, J . , Optimal Engineering Design, Marcel Dekker, Inc., New York, N . Y . , 1982. [18] Timoshenko, S.P. and Woinowsky-Krieger, S., Theory of Plates and Shells, McGraw-Hil l Book Co., Toronto, Ont., 1959. [19] Timoshenko, S.P. and Goodier, J .N. , Theory of Elasticity, McGraw-Hil l Book Co., Toronto, Ont., 1970, pp. 309-313. [20] Thoft-Christensen, P. and Baker, M . J . , Structural Reliability Theory and Rs Appli-cations, Springer-Verlag, New York, N . Y . , 1982, pp. 44-58. [21] U.S. Forest Products Laboratory, Adhesives in Building Construction, Forest Service Agriculture Handbook, No. 516, U.S. Dept. of Agriculture, 1978, pp. 32-34. [22] U.S. Forest Products Laboratory, Wood Handbook: Wood as an Engineering Mate-rial, Forest Service Agriculture Handbook, No. 72, U.S. Dept. of Agriculture, 1974, pp. 10.10-10.12. [23] Vaessen, W. , UBC NLP: Nonlinear Function Optimization, Computing Centre, Uni-versity of British Columbia, 1984. Appendix A Sensitivity Analysis Results 100 Appendix A. Sensitivity Analysis Results 101 8.00 7.00 -u CO v 3 6.00 H o - 5.00 -o 4.00 3.00 Flange Material Cost Web Material Cost -40.0 -20.0 0.0 20.0 40.0 Percentage Difference from Benchmark Cost Coefficient Figure A . l : Optimal Cost as a Function of the Cost Coefficients Used in the Objective Function 16.0 14.0 ^ 12.0 GO CD g 10.0 o. CD o u ax C CO 8.0 6.0 -4.0 -2.0 0.0 Upper Bound on Flange Depth Lower Bound on Flange Depth -B- -B-Flange Material: Continuous • Flange Material: Discrete Web Material : Continuous A Web Material : Discrete -40.0 -20.0 0.0 20.0 40.0 Percentage Difference from Benchmark Cost Coefficient Figure A.2: Optimal Flange Depth as a Function of the Cost Coefficients Used in the Objective Function Appendix A. Sensitivity Analysis Results 102 0) JS o a .2? S3 60.0 50.0 40.0 30.0 -20.0 10.0 -0.0 Upper Bound on Web Height A A Lower Bound on Web Height Flange Material: Continuous • Flange Material: Discrete — Web Material : Continuous A Web Material : Discrete -40.0 -20.0 0.0 20.0 40.0 Percentage Difference from Benchmark Cost Coefficient Figure A.3 : Optimal Web Height as a Function of the Cost Coefficients Used in the Objective Function 4.0 v a >i 2 CO 4) si 2.0 Compression Strength Limit State Tension Strength Limit State Shear Strength Limit State 0.30 0.40 0.50 Ratio of the Flange Material's Cost to the Web Material's Cost 0.0001 0.001 ° CO a — 0.01 Figure A.4: Reliability Indices of the Individual Failure Modes as a Function of the Material Cost Ratio Appendix A. Sensitivity Analysis Results 103 8.00 7.00 to tu 3 6.00 o " 5.00 a. o 4.00-3.00 0.60 Flexural Stress Computations Shear Stress Computations 0.80 1.00 1.20 Modelling Error Coefficient 1.40 Figure A.5: Optimal Cost as a Function of the Modelling Error Coefficients Associ-ated with the Flexural and Shear Stress Computations 4.0 X e 3 3.0 -a; 2.0 0.60 Flexural Stress Computations Shear Stress Computations 0.80 1.00 1.20 Modelling Error Coefficient - - 0.0001 a 0.001 e> •a o 0.01 1.40 Figure A.6: Reliability Index as a Function of the Modelling Error Coefficients As-sociated with the Flexural and Shear Stress Computations Appendix A. Sensitivity Analysis Results 104 16.0 14.0 - Upper BouncLon Flange Depth s \ v _ 1 2 -° " w <U A S IO.O A • Flexural Stress Computations: Continuous Flexural Stress Computations: Discrete —— Shear Stress Computations : Continuous A Shear Stress Computations : Discrete a o a c 8.0-6.0-4.0-2.0-Lower Bound on Flange Depth 0.0 0.60 0.80 1.00 1.20 Modelling Error Coefficient 1.40 Figure A.7: Optimal Flange Depth as a Function of the Modelling Error Coefficients Associated with the Flexural and Shear Stress Computations a A o a '33 s A <u IS 60.0 50.0 -40.0 30.0 -20.0 -0.0 Upper Bound on Web Height • • .--A" A Flexural Stress Computations: Continuous • Flexural Stress Computations: Discrete Shear Stress Computations : Continuous A Shear Stress Computations : Discrete 10.0 -| Lower Bound on Web Height 0.60 0.80 1.00 1.20 Modelling Error Coefficient 1.40 Figure A.8: Optimal Web Height as a Function of the Modelling Error Coefficients Associated with the Flexural and Shear Stress Computations Appendix A. Sensitivity Analysis Results 105 4.0 x CD •a e s 3.0 -| .2 2.0 Compression Strength Limit State Tension Strength Limit State Shear Strength Limit State 0.0001 0.001 ° <d o - - 0.01 0.60 0.80 1.00 1.20 Modelling Error Coefficient 1.40 Figure A.9: Reliability Indices of the Individual Failure Modes as a Function of the Modelling Error Coefficient for Flexural Stresses 4.0 CD a >. •2 3.0 IS cs 05 2.0 Compression Strength Limit State Tension Strength Limit State Shear Strength Limit State 0.60 0.80 1.00 1.20 Modelling Error Coefficient 0.0001 at 6-0.001 ° a) .o o 0.01 1.40 Figure A . 10: Reliability Indices of the Individual Failure Modes as a Function of the Modelling Error Coefficient for Shear Stresses Appendix A. Sensitivity Analysis Results 106 8.00 7.00 -u to v 3 6.00 in O " 5.00 to s a o 4.00 -3.00 2 3 4 Number of Terms in Fourier Series Approximation Figure A.11: Optimal Cost as a Function of the Number of Fourier Terms Used in the Plywood-Web Beam Model CO 4.0 X V •a c - 3.0 -I 2.0 0.0001 c8 0.001 o to XI o - - 0.01 2 3 4 Number of Terms in Fourier Series Approximation Figure A . 12: Reliability Index as a Function of the Number of Fourier Terms Used in the Plywood-Web Beam Model Appendix A. Sensitivity Analysis Results 107 16.0 14.0 - Upper Bound on. Flange Depth 0) to A o C 12.0 10.0 - • Continuous Solution Pseudo-Discrete Solution o. O) Q SI M G CO 8.0 6.0 • 4.0 -2.0 -Lower Bound on Flange Depth 0.0 Number of Terms in Fourier Series Approximation Figure A.13: Optimal Flange Depth as a Function of the Number of Fourier Terms Used in the Plywood-Web Beam Model to v A o S A SB '3 ts A V 60.0 50.0 30.0 40.0 -I] 20.0 10.0 -0.0 Upper Bound on Web Height • Continuous Solution Pseudo-Discrete Solution Lower Bound on Web Height 1 1 1 2 3 4 Number of Terms in Fourier Series Approximation Figure A. 14: Optimal Web Height as a Function of the Number of Fourier Terms Used in the Plywood-Web Beam Model Appendix A. Sensitivity Analysis Results 108 8.00 7.00 -3 6.00 H o " 5.00 a. © 4.00 3.00 M.O.E. of the Flange Material M.O.E. of the Web Material i | | | | | r -40.0 -20.0 0.0 20.0 Percentage Difference from Benchmark Mean Value 40.0 Figure A.15: Optimal Cost as a Function of the Mean Values of the Flange and Web Materials' Elastic Moduli i 4.0 M.O.E. of the Flange Material M.O.E. of the Web Material a 5 3.0 XI a OS 2.0 0.0001 0.001 3 '3 o XI a £> o u O H 0.01 -40.0 -20.0 0.0 20.0 Percentage Difference from Benchmark Mean Value 40.0 Figure A.16: Reliability Index as a Function of the Mean Values of the Flange and Web Materials' Elastic Moduli i Appendix A. Sensitivity Analysis Results 109 16.0 14.0 ^ 12.0 -A 2 10.0 -a, u a v OB a a 8.0 -6.0 4.0 2.0 0.0 Upper Bound on. Flange Depth • M.O.E. of the Flange Material : Continuous M.O.E. of the Flange Material : Discrete — M.O.E. of the Web Material : Continuous A M.O.E. of the Web Material : Discrete Lower Bound on Flange Depth -40.0 -20.0 0.0 20.0 Percentage Difference from Benchmark Mean Value 40.0 Figure A.17: Optimal Flange Depth as a Function of the Mean Values of the Flange and Web Materials' Elastic Modulii 60.0 50.0 -A o a A W> 33 S3 X I CO 40.0 -30.0 20.0 -10.0 0.0 Upper Bound on Web Height M.O.E. of the Flange Material : Continuous • M.O.E. of the Flange Material : Discrete M.O.E. of the Web Material : Continuous A M.O.E. of the Web Material : Discrete Lower Bound on Web Height -40.0 -20.0 0.0 20.0 Percentage Difference from Benchmark Mean Value 40.0 Figure A.18: Optimal Web Height as a Function of the Mean Values of the Flange and Web Materials' Elastic Moduli i Appendix A. Sensitivity Analysis Results 110 8.00 7.00 -co a 6.00 o " 5.00 to s o 4.00 -3.00 0.0 M.O.E. of the Flange Material M.O.E. of the Web Material 20.0 40.0 60.0 Coefficient of Variation (%) 80.0 100.0 Figure A.19: Optimal Cost as a Function of the Coefficients of Variation of the Flange and Web Materials' Elastic Modulii 4.0 M.O.E. of the Flange Material M.O.E. of the Web Material x c a 3.0 2 CO en - - 0.0001 u 3 '3 6. 0.001 o O u a. 0.01 2.0 0.0 20.0 40.0 60.0 Coefficient of Variation (%) 80.0 100.0 Figure A.20: Reliability Index as a Function of the Coefficients of Variation of the Flange and Web Materials' Elastic Modulii Appendix A. Sensitivity Anedysis Results 111 16.0 14.0 -A u a 12.0 10.0 -a <o a i> c 8.0-6.0 4.0-2.0 0.0 Upper Bound on. Flange Depth M.O.E. of the Flange Material : • M.O.E. of the Flange Material: M.O.E. of the Web Material A M.O.E. of the Web Material Continuous Discrete Continuous Discrete 6 A 6 A '"A"- -- A -Lower Bound on Flange Depth 0.0 20.0 40.0 60.0 Coefficient of Variation (%) 80.0 100.0 Figure A.21: Optimal Flange Depth as a Function of the Coefficients of Variation of the Flange and Web Materials' Elastic Moduli i A o d A .2? B A V 5E 60.0 50.0 40.0 30.0 20.0 0.0 Upper Bound on Web Height A £T"A"' - A -M.O.E. of the Flange Material • M.O.E. of the Flange Material M.O.E. of the Web Material A M.O.E. of the Web Material Continuous Discrete Continuous Discrete 10.0 H Lower Bound on Web Height 0.0 20.0 40.0 60.0 Coefficient of Variation (%) 80.0 100.0 Figure A.22: Optimal Web Height as a Function of the Coefficients of Variation of the Flange and Web Materials' Elastic Moduli i Appendix A. Sensitivity Analysis Results 112 8.00 7.00 -Compression Strength of Flange Material Tension Strength of Flange Material Shear Strength of Web Material S 6.00 o - 5.00 o 4.00 -3.UU -40.0 -20.0 0.0 20.0 Percentage Difference from Benchmark Mean Value 40.0 Figure A.23: Optimal Cost as a Function of the Mean Values of the Flange and Web Materials' Strength Distributions 4.0 e J 3 (8 OA 3.0 2.0 Compression Strength of Flange Material Tension Strength of Flange Material Shear Strength of Web Material -40.0 -20.0 0.0 20.0 Percentage Difference from Benchmark Mean Value - - 0.0001 0.001 ° •a et) X ) O u a. - - 0.01 40.0 Figure A.24: Reliability Index as a Function of the Mean Values of the Flange and Web Materials' Strength Distributions Appendix A. Sensitivity Analysis Results 113 16.0 12.0 v A S I O . O -o a 8.0 -6.0 -4.0 -2.0 0.0 Upper BouryLon Flange Depth \ ^ . Flange Compression Strength: Continuous • Flange Compression Strength: Discrete Flange Tension Strength : Continuous A Flange Tension Strength : Discrete Web Shear Strength : Continuous * Web Shear Strength : Discrete A * & & • s _ .& Lower Bound on f Flange Depth -40.0 -20.0 0.0 20.0 40.0 Percentage Difference from Benchmark Mean Value Figure A.25: Optimal Flange Depth as a Function of the Mean Values of the Flange and Web Materials' Strength Distributions A o a A OB s A 60.0 50.0 -40.0 30.0 20.0 -10.0 -0.0 Upper Bound on Web Height • -* A Lower Bound oiy-^  Web Height Flange Compression Strength: Continuous • Flange Compression Strength: Discrete Flange Tension Strength : Continuous A Flange Tension Strength : Discrete Web Shear Strength : Continuous 3(< Web Shear Strength : Discrete -40.0 -20.0 0.0 20.0 Percentage Difference from Benchmark Mean Value 40.0 Figure A.26: Optimal Web Height as a Function of the Mean Values of the Flange and Web Materials' Strength Distributions Appendix A. Sensitivity Analysis Results 114 8.00 7.00 is 3 6.00 H o " 5.00 -CO s o 4.00 -3.00 0.0 Compression Strength of Flange Material Tension Strength of Flange Material Shear Strength of Web Material 20.0 40.0 60.0 Coefficient of Variation (%) 80.0 100.0 Figure A.27: Optimal Cost as a Function of the Coefficients of Variation of the Flange and Web Materials' Strength Distributions 4.0 "0 c 3 3.0 -CO CD 0. 2.0 Compression Strength of Flange Material Tension Strength of Flange Material Shear Strength of Web Material 0.0001 CD u 3 '3 fa 0.001 ° £ > CO Xi o u a. 0.01 0.0 20.0 40.0 60.0 Coefficient of Variation (%) 80.0 100.0 Figure A.28: Reliability Index as a Function of the Coefficients of Variation of the Flange and Web Materials' Strength Distributions Appendix A. Sensitivity Analysis Results 115 CD A o a a. v Q <u 6 0 a a 16.0 14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0 Upper Bound on Flange Depth * # * \5|C • '* Lower Bound on Flange Depth Flange Compression Strength: Continuous • Flange Compression Strength: Discrete — - Flange Tension Strength : Continuous A Flange Tension Strength : Discrete Web Shear Strength : Continuous * Web Shear Strength : Discrete 0.0 20.0 40.0 60.0 Coefficient of Variation (%) 80.0 100.0 Figure A.29: Optimal Flange Depth as a Function of the Coefficients of Variation of the Flange and Web Materials' Strength Distributions 60.0 50.0 A o a A b O '3 a A 4) 40.0 -30.0 -20.0 -10.0 0.0 Upper Bound on Web Height /* A A A-A ; i i * # * # * • Flange Compression Strength: Continuous Flange Compression Strength: Discrete Lower Bound onX Web Height A O Flange Tension Strength : Continuous Flange Tension Strength : Discrete Web Shear Strength : Continuous Web Shear Strength : Discrete 0.0 20.0 40.0 60.0 Coefficient of Variation (%) 80.0 100.0 Figure A.30: Optimal Web Height as a Function of the Coefficients of Variation of the Flange and Web Materials' Strength Distributions Appendix A. Sensitivity Analysis Results 116 8.0 x CD e XI 6.0 4.0 -2.0 Compression Strength Limit State Tension Strength Limit State Shear Strength Limit State 0.0 20.0 40.0 60.0 Coefficient of Variation of the Flange's Tension Strength Figure A.31: Reliability Indices of the Different Limit States as a Function of the Tension Strength's Coefficient of Variation 8.0 CD OS Compression Strength Limit State Tension Strength Limit State Shear Strength Limit State 8 6.0 H a Xi a) 4.0 2.0 0.0 20.0 40.0 60.0 Coefficient of Variation of the Flange's Compression Strength Figure A.32: Reliability Indices of the Different Limit States as a Function of the Compression Strength's Coefficient of Variation Appendix A. Sensitivity Analysis Results 117 8.0 x v a Compression Strength Limit State Tension Strength Limit State Shear Strength Limit State 6.0 CO V en 4.0 2.0 0.0 10.0 20.0 30.0 Coefficient of Variation of the Web's Shear Strength 40.0 Figure A.33: Reliability Indices of the Different Limit States as a Function of the Shear-Through-Thickness Strength's Coefficient of Variation Appendix A. Sensitivity Analysis Results 118 8.00 7.00 -3 6.00-o o o 5.00 -4.00 -3.00 Dead Load Live Load -40.0 -20.0 0.0 20.0 Percentage Difference from Benchmark Mean Value 40.0 Figure A.34: Optimal Cost as a Function of the Mean Values of the Dead and Live Load Distributions 4.0 73 S X I CO 05 3.0 2.0 Dead Load Live Load 0.0001 CO 0.001 o X I CO X I o i -BU 0.01 -40.0 -20.0 0.0 20.0 Percentage Difference from Benchmark Mean Value 40.0 Figure A.35: Reliability Index as a Function of the Mean Values of the Dead and Live Load Distributions Appendix A. Sensitivity Anedysis Results 119 16.0 14.0 ^ 12.0 GO CD x! a 10.0 a. v a CD a as 8.0 -6.0 -4.0 2.0 0.0 Upper Bound Flange Depth on Dead Load Dead Load Live Load Live Load Continuous Discrete Continuous Discrete • A A • • £1 . a - O Lower Bound Flange Depth on -40.0 -20.0 0.0 20.0 40.0 Percentage Difference from Benchmark Mean Value Figure A.36: Optimal Flange Depth as a Function of the Mean Values of the Dead and Live Load Distributions 60.0 50.0 -CD X ! o a xi .£? '33 s CD 40.0 30.0 20.0 10.0 0.0 Upper Bound on Web Height Dead Load : Continuous • Dead Load : Discrete Live Load : Continuous A Live Load : Discrete Lower Bound on Web Height -40.0 -20.0 0.0 20.0 Percentage Difference from Benchmark Mean Value 40.0 Figure A.37: Optimal Web Height as a Function of the Mean Values of the Dead and Live Load Distributions Appendix A. Sensitivity Analysis Results 120 8.00 7.00-CD a 6.00 o " 5.00 a a. o 4.00 -3.00 Dead Load Live Load 0.0 20.0 40.0 60.0 Coefficient of Variation (%) 80.0 100.0 Figure A.38: Optimal Cost as a Function of the Coefficients of Variation of the Dead and Live Load Distributions 4.0 x c § 3.0 -| 3 2.0 Dead Load Live Load 0.0 20.0 40.0 60.0 Coefficient of Variation (%) - - 0.0001 CD h 3 "a 0.001 o a o i~ a. - - 0.01 80.0 100.0 Figure A.39: Reliability Index as a Function of the Coefficients of Variation of the Dead and Live Load Distributions Appendix A. Sensitivity Analysis Results 121 16.0 14.0 12.0 H in V 1 10.0 xi o. 8.0 0) D CD g> 6.0 to 4.0 -2.0 0.0 Upper Bound on_ Flange Depth Dead Load : Continuous • Dead Load : Discrete Live Load : Continuous A Live Load : Discrete • • • a - A A Lower Bound on Flange Depth 0.0 20.0 40.0 60.0 Coefficient of Variation (%) 80.0 100.0 Figure A.40: Optimal Flange Depth as a Function of the Coefficients of Variation of the Dead and Live Load Distributions 60.0 50.0 Upper Bound on Web Height Xi o a xi .2? '33 X Xi <u 40.0 30.0 20.0 -• • a . A • "A~ -A" - A Dead Load: Continuous Dead Load: Discrete Live Load : Continuous Live Load : Discrete 10.0 H Lower Bound on Web Height 0.0 0.0 20.0 40.0 60.0 Coefficient of Variation (%) 80.0 100.0 Figure A.41: Optimal Web Height as a Function of the Coefficients of Variation of the Dead and Live Load Distributions Appendix A. Sensitivity Analysis Results 122 8.00 7.00 -<s i> a 3 6.00 o - 5.00 a B a. o 4.00 3.00 0.0 M.O.E. of the Flange Material M.O.E. of the Web Material I I I : I I I 0.2 0.4 0.6 Correlation Coefficient — i r 0.8 1.0 Figure A.42: Optimal Cost as a Function of the Correlation Coefficients Between the Flange or Web Components' Elastic Moduli i 4.0 a X) 10 cu OS 3.0 2.0 M.O.E. of the Flange Material M.O.E. of the Web Material 0.0 0.2 0.4 0.6 Correlation Coefficient 0.8 0.0001 o u a S 0.001 o a XI o u a. — 0.01 1.0 Figure A.43: Reliability Index as a Function of the Correlation Coefficients Between the Flange or Web Components' Elastic Modulii Appendix A. Sensitivity Analysis Results 123 16.0 v XI o a 14.0 - Upper Bound on. Flange Depth _ 12.0 -10.0 A M.O.E. of the Flange Material : Continuous M.O.E. of the Flange Material : Discrete M.O.E. of the Web Material : Continuous M.O.E. of the Web Material : Discrete a 1) a 8.0-6.0-" 6 • 6 4.0-2.0-Lower Bound on Flange Depth 0.0-0.0 0.2 0.4 0.6 Correlation Coefficient 0.8 1.0 Figure A.44: Optimal Flange Depth as a Function of the Correlation Coefficients Be-tween the Flange or Web Components' Elastic Moduli i CD XI O a x\ .2? '35 93 Xi CD 60.0 Upper Bound on 50.0 H Web Height 40.0 30.0 20.0 -10.0 0.0 • • f l " M.O.E. of the Flange Material : Continuous • M.O.E. of the Flange Material : Discrete M.O.E. of the Web Material : Continuous A M.O.E. of the Web Material : Discrete Lower Bound on Web Height 0.0 0.2 0.4 0.6 Correlation Coefficient 0.8 1.0 Figure A.45: Optimal Web Height as a Function of the Correlation Coefficients Be-tween the Flange or Web Components' Elastic Moduli i Appendix A. Sensitivity Analysis Results 124 o o a a, o 8.00 7.00 -3 6.00-5.00 -4.00 -3.00 0.0 Compression Strength of Flange Material Tension Strength of Flange Material Shear Strength of Web Material 0.2 0.4 0.6 Correlation Coefficient 0.8 1.0 Figure A.46: Optimal Cost as a Function of the Correlation Coefficients Between the Flange or Web Components' Strength and Stiffness Properties X C CD 4.0 3 3.0 2.0 0.0 Compression Strength of Flange Material Tension Strength of Flange Material Shear Strength of Web Material 0.0001 CO fa 0.001 ° CO X I o 0.01 0.2 0.4 0.6 Correlation Coefficient 0.8 1.0 Figure A.47: Reliability Index as a Function of the Correlation Coefficients Between the Flange or Web Components' Strength and Stiffness Properties Appendix A. Sensitivity Analysis Results 125 16.0 14.0 ^ 12.0 CO to A | IO.O -i A a. 8.0 tu O 6.0 4.0 2.0 -0.0 Upper Bound on Flange D e p t h \ • A Flange Compression Strength: Continuous Flange Compression Strength: Discrete Flange Tension Strength : Continuous Flange Tension Strength : Discrete Web Shear Strength : Continuous Web Shear Strength : Discrete Lower Bound on Flange Depth - A -0.0 0.2 0.4 0.6 Correlation Coefficient 0.8 1.0 Figure A.48: Optimal Flange Depth as a Function of the Correlation Coefficients Be-tween the Flange or Web Components' Strength and Stiffness Properties 60.0 50.0 -tu A u «3 A '3 s A ii 40.0 -30.0 -20.0 -10.0 -0.0 Upper Bound on Web Height \ ^ A • • A * * • Flange Compression Strength: Flange Compression Strength: Continuous Discrete Lower Bound o^/ Web Height A * • Flange Tension Strength : Flange Tension Strength Web Shear Strength : Web Shear Strength Continuous Discrete Continuous Discrete 0.0 0.2 • 0.4 0.6 Correlation Coefficient 0.8 1.0 Figure A.49: Optimal Web Height as a Function of the Correlation Coefficients Be-tween the Flange or Web Components' Strength and Stiffness Properties Appendix A. Sensitivity Analysis Results 126 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 Computed Optimal Cost ($/linear ft.) Figure A.50: Optimal Cost Obtained with the Fitted Probability Distributions for the Douglas F i r Lumber's Tension Strength Reliability Index Figure A.51: Reliability Index Obtained with the Fitted Probability Distributions for the Douglas F i r Lumber's Tension Strength Appendix A. Sensitivity Analysis Results 127 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Optimal Flange Depth (inches) Figure A.52: Optimal Flange Depth Obtained with the Fitted Probability Distribu-tions for the Douglas F i r Lumber's Tension Strength Computed Optimal Web Height (inches) Figure A.53: Optimal Web Height Obtained with the Fitted Probability Distributions for the Douglas F i r Lumber's Tension Strength Appendix A. Sensitivity Analysis Results 128 20 18 -16 D O C cj u 3 o o o 14 -12 _ 10 3 2 8 -6 -4 -2 -• 0 Weibull Weibull Lognormal Lognormal Normal Entire Sample Lower Tail of Sample Entire Sample Lower Tail of Sample Lower Tail of Sample 3.50 5.50 7.25 9.25 11.25 Optimal Discrete Flange Depth (inches) 13.25 Figure A.54: Discrete Optimal Flange Depth Obtained with the Fitted Probability Distributions for the Douglas F i r Lumber's Tension Strength 20 18 16 -14 3 12 -10 -3 2 H Weibull : Entire Sample I | Weibull : Lower Tail of Sample l \ l Lognormal : Entire Sample \ / \ Lognormal : Lower Tail of Sample ^2 Normal : Lower Tail of Sample i I it 28.0 30.0 32.0 34.0 36.0 38.0 40.0 Optimal Discrete Web Height (inches) 42.0 44.0 Figure A.55: Discrete Optimal Web Height Obtained with the Fitted Probability Dis-tributions for the Douglas F i r Lumber's Tension Strength Appendix B Estimated Tension Strength Distribution Parameters 129 Appendix B. Estimated Tension Strength Distribution Parameters 130 Table B . l : Maximum Likelihood Estimators and Kolmogorov-Smirnov Goodness-of-Fit Test Statistics for Normal Distributions Fit to Entire Samples of Sim-ulated Data Sample Mean Standard Kolmogorov-Number Deviation Smirnov (psi) (psi) Test Statistic 1 5786.6 2162.1 0.0378 2 5435.2 2316.9 0.0543 3 5563.8 2242.9 0.0326 4 5753.4 2266.9 0.0422 5 5603.3 2201.5 0.0528 6 5628.4 2313.0 0.0459 7 5739.3 2203.3 0.0481 8 5379.6 2166.8 0.0426 9 5654.5 2236.9 0.0379 10 5594.0 2230.9 0.0378 11 5571.6 2271.0 0.0299 12 5510.4 2223.4 0.0482 13 5751.0 2372.4 0.0269 14 5631.3 2260.5 0.0504 15 5513.0 2180.2 0.0508 16 5551.0 2273.6 0.0249 17 5497.9 2279.9 0.0512 18 5718.6 2156.6 0.0269 19 5573.2 2314.6 0.0386 20 5562.5 2274.0 0.0445 Average 5600.9 2247.4 0.0412 Appendix B. Estimated Tension Strength Distribution Parameters 131 Table B.2: Maximum Likelihood Estimators and Kolmogorov-Smirnov Goodness-of-Fit Test Statistics for Logarithmic Normal Distributions Fit to Entire Sam-ples of Simulated Data Sample Mean Standard Kolmogorov-Number Deviation Smirnov (psi) (psi) Test Statistic 1 5873.3 2755.7 0.0668 2 5544.1 3018.8 0.0746 3 5645.6 2827.4 0.0909 4 5847.4 • 2925.0 0.0739 5 5673.4 2710.8 0.0572 6 5763.6 3127.1 0.0754 7 5805.8 2699.4 0.0653 8 5467.7 2763.4 0.0958 9 5751.9 2901.9 0.0946 10 5683.4 2846.2 0.0761 11 5688.2 3016.4 0.0819 12 5598.8 2825.2 0.0770 13 5852.1 3072.7 0.0937 14 5746.5 3002.8 0.0738 15 5601.2 2778.6 0.0866 16 5647.2 2913.9 0.0927 17 5597.2 2938.8 0.0691 18 5813.4 2811.0 0.0995 19 5708.3 3172.8 0.0948 20 5672.7 2961.2 0.0727 Average 5699.1 2903.4 0.0806 Appendix B. Estimated Tension Strength Distribution Parameters 132 Table B.3: Maximum Likelihood Estimators and Kolmogorov-Smirnov Goodness-of-Fit Test Statistics for Two-Parameter Weibull Distributions Fit to Entire Samples of Simulated Data Sample Scale Shape Kolmogorov-Number (psi) Smirnov Test Statistic 1 6488.2 2.909 0.0235 2 6125.3 2.509 0.0340 3 6260.9 2.680 0.0218 4 6468.4 2.772 0.0401 5 6300.7 2.753 0.0336 6 6330.7 2.602 0.0259 7 6447.0 2.825 0.0313 8 6049.9 2.662 0.0342 9 6354.1 2.730 0.0324 10 6290.2 2.705 0.0167 11 6267.7 2.642 0.0215 12 6199.7 2.668 0.0262 13 6474.9 2.615 0.0219 14 6331.9 2.694 0.0399 15 6198.0 2.733 0.0323 16 6246.0 2.611 . 0.0303 17 6192.3 2.596 0.0277 18 6410.5 2.877 0.0310 19 6271.8 2.597 0.0313 20 6258.9 2.619 0.0204 Average 6298.4 2.690 0.0288 Appendix B. Estimated Tension Strength Distribution Parameters 133 Table B.4: Maximum Likelihood Estimators and Kolmogorov-Smirnov Goodness-of-Fit Test Statistics for Assumed Probability Distributions Fit to Censored Samples of Simulated Data Sample Normal Logarithmic Normal Weibull Number Mean Std. Dev. Mean Std. Dev. Scale Shape (psi) (psi) . (Psi) (psi) (psi) 1 5029.1 1585.6 15100.2 16794.4 7001.8 2.734 2 4469.1 1517.7 10390.8 10661.9 5915.9 2.595 3 4655.6 1540.4 9783.8 8882.8 6154.0 2.717 4 4544.6 1405.2 9901.9 8831.6 5750.3 3.051 5 4704.2 1493.6 9914.4 8706.6 6069.5 2.922 6 5189.0 1832.1 24714.2 40559.3 8599.4 2.070 7 4827.3 1500.8 10064.4 8561.9 6202.0 3.017 8 4527.9 1494.9 9842.8 9289.4 5887.1 2.746 9 4964.8 1716.3 13464.4 15137.7 7206.3 2.378 10 4931.2 1664.0 12876.1 13900.1 6889.0 2.519 11 4549.9 1491.8 11467.1 12092.2 6024.4 2.699 12 4730.6 1566.0 11659.3 11982.0 6398.2 2.642 13 4568.4 1515.6 9297.8 8318.9 5915.4 2.755 14 4863.6 1636.6 14911.3 18080.8 6942.8 2.456 15 4822.7 1562.1 12969.1 13896.3 6567.8 2.688 16 4611.3 1541.7 10419.4 10183.6 6136.2 2.649 17 4601.3 1534.7 11018.1 11253.5 6169.7 2.629 18 5167.4 1772.3 16182.5 19676.0 7835.5 2.327 19 4667.3 1631.4 12823.9 15054.2 6592.0 2.367 20 4811.9 1589.0 15238.6 18623.1 6817.8 2.520 Average 4762.3 1579.6 12602.0 14024.3 6553.8 2.624 Appendix B. Estimated Tension Strength Distribution Parameters 134 Table B.5: Critical Values of the Kolmogorov-Smirnov Goodness-of-Fit Test Statistic when the Distribution Parameters are Estimated from the Sample Data Level of Significance Critical Value (%) (Sample size = 500) 20 0.0320 15 0.0335 10 0.0355 5 0.0387 1 0.0449 

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