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Contribution to the development of a tool to link Matlab constraint satisfaction platform to ANSYS structural… Villiard, Caroline 2011

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CONTRIBUTION TO THE DEVELOPMENT OF A TOOL TO LINK MATLAB CONSTRAINT SATISFACTION PLATFORM TO ANSYS STRUCTURAL MODELS  by CAROLINE VILLIARD B.A.Sc., LAVAL UNIVERSITY, 2009  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in The Faculty of Graduate Studies (Civil Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2011 © Caroline Villiard, 2011  ABSTRACT Engineers have to deal with uncertainties as a challenge when pursuing the safest and most economical design. Several methods have been used over the years to deal with these uncertainties and to make this conceptual design phase more straight forward. Safety goals have been achieved due to safety coefficients found in design codes, but there is a lack of software tools to carry out risk and reliability analysis. Within the framework of University research, the novel idea of combining wood and steel as two structural members of mid-rise hybrid buildings is being studied. The need for an optimized design to start off the hybrid structure project led to this thesis, where the goal is to link the ANSYS structural analysis program to a Matlab platform for integrating probabilistic constraints. The methodology proposed by Loewen seeks to remedy the uncertainty issues encountered by engineers, and his Probability Constraint Satisfaction Program (PCSP) is the foundation of the Matlab platform developed more extensively here. This method involves subdividing the design space and estimating the potential of meeting the constraints according to the probability distribution of the data in each interval. An advantage output of this technique are design spaces combining probabilities of various data. The structural component of this thesis involves formatted files serving as input for ANSYS program were created to automatically model multistorey hybrid steel-wood structures. Some frameworks and a literature review introduce the pros and cons of this type of construction. Many issues are dealt with during the modeling phase whereas some are left to the future developing stage. Simplified examples leading to an understanding of the methodology developed to build an efficient link are shown to illustrate the essence of the research done. Additional computations were added to the programs script files and resulted in a powerful tool that was put to the test with two different examples- a simple hybrid frame and a multistorey steel structure. As a result of this work on the example structures, the PCSP method produced design spaces, presented to the reader.  ii  TABLE OF CONTENTS ABSTRACT .................................................................................................................................... ii TABLE OF CONTENTS ............................................................................................................... iii LIST OF TABLES ......................................................................................................................... vi LIST OF FIGURES ...................................................................................................................... vii ACKNOWLEDGMENTS ............................................................................................................. ix 1  INTRODUCTION ................................................................................................................... 1  2  LITERATURE REVIEW ........................................................................................................ 3 2.1  Uncertainties for conceptual design ................................................................................. 3  2.1.1  Reliability methods ................................................................................................... 3  2.1.2  Conceptual design using probabilistic interval constraint satisfaction ..................... 4  2.2  Software overview .......................................................................................................... 10  2.2.1  ANSYS.................................................................................................................... 10  2.2.2  Matlab computations ............................................................................................... 13  2.2.3  Solution module ...................................................................................................... 14  2.3  Finite element method (FEM) software ......................................................................... 15  2.3.1 3  FRAMEWORKS ................................................................................................................... 22 3.1  Steel structures case studies ........................................................................................... 22  3.1.1 3.2  Scotia place hybrid structure ................................................................................... 22  Conceptual design using PCSP ...................................................................................... 23  3.2.1  Data processing ....................................................................................................... 23  3.2.2  Adaptation of CSP .................................................................................................. 26  3.2.3  Relevancy ................................................................................................................ 27  3.3  4  Modeling of shear walls .......................................................................................... 16  Hybrid steel-timber structures ........................................................................................ 27  3.3.1  Objective ................................................................................................................. 27  3.3.2  Flanged connection type ......................................................................................... 28  3.3.3  Hollow section connection type .............................................................................. 31  3.3.4  Relevancy ................................................................................................................ 32  PROBLEM STATEMENT & SOLUTION METHODS ...................................................... 33 4.1  Overview of methodology .............................................................................................. 33  4.1.1 4.2  Hypothesis and limitations ...................................................................................... 33  Proof calculations ........................................................................................................... 34 iii  4.2.1  Wood shear panel .................................................................................................... 35  4.2.2  Stiffness method ...................................................................................................... 38  4.3  Constraints satisfaction program .................................................................................... 40  4.3.1  Sections selection .................................................................................................... 40  4.3.2  Constrained equations ............................................................................................. 41  4.4  Nonlinear analysis .......................................................................................................... 41  4.4.1 5  ANSYS METHODOLOGY & APPLICATION ................................................................... 43 5.1  Steel sections generator ........................................................................................... 43  5.1.2  Automated structure creation .................................................................................. 45  5.1.3  Loop feature ............................................................................................................ 47  5.1.4  Wood panel patterns ................................................................................................ 48  Methodology .................................................................................................................. 48  5.2.1  Simple structure ...................................................................................................... 48  5.2.2  Multistorey structure ............................................................................................... 51  5.3  Further work ................................................................................................................... 51  MATLAB METHODOLOGY & APPLICATION ............................................................... 52 6.1  Simple example: Bolted assembly (end plate connection) ............................................. 52  6.1.1  Problem description ................................................................................................ 53  6.1.2  Results ..................................................................................................................... 54  6.1.3  Discussion ............................................................................................................... 57  6.2  Improvement to the CSP ................................................................................................ 57  6.2.1  Input module ........................................................................................................... 57  6.2.2  Solution module ...................................................................................................... 61  6.2.3  Output module......................................................................................................... 63  6.2.4  Plotting options ....................................................................................................... 64  6.3 7  Input ................................................................................................................................ 43  5.1.1  5.2  6  Plastic analysis ........................................................................................................ 42  Further work ................................................................................................................... 64  HYBRID STRUCTURE STUDY ......................................................................................... 65 7.1  Introduction .................................................................................................................... 65  7.2  Problem description ........................................................................................................ 65  7.2.1  Cost study ................................................................................................................ 66  7.2.2  Loading ................................................................................................................... 68  7.3  Results ............................................................................................................................ 69 iv  7.3.1  Simple hybrid frame................................................................................................ 69  7.3.2  Multistorey steel structure ....................................................................................... 72  7.3.3  Hybrid multistorey structure ................................................................................... 74  7.4 8  Analysis .......................................................................................................................... 76  CONCLUSIONS ................................................................................................................... 78  REFERENCES.............................................................................................................................. 80 APPENDICES .............................................................................................................................. 83 APPENDIX A : Lognormal distribution for CSP ..................................................................... 83 APPENDIX B : Proof calculations ........................................................................................... 85 APPENDIX C : Excel spreadsheets for PCSPs......................................................................... 87 APPENDIX D : Code references .............................................................................................. 93 APPENDIX E : ANSYS formatted files ................................................................................... 94  v  LIST OF TABLES Table 1 : Comparison of solution methods for determining valid ranges of probabilistic variables. ...... 7 Table 2 : List of ETABLE for steel elements outputs............................................................................ 12 Table 3 : List of ETABLE for wood panel elements outputs ................................................................ 12 Table 4 : CSA O86 tables for OSB capacities design value .................................................................. 17 Table 5 : Probabilistic parameters .......................................................................................................... 24 Table 6 : ANSYS SHELL types comparison ......................................................................................... 35 Table 7 : Results of hand calculated stiffness versus modeled stiffness for different frames ............... 39 Table 8 : Nonlinearity comparison for ANSYS and Dr.Frame programs.............................................. 42 Table 9 : Comparison of property values of W690x548 section ........................................................... 44 Table 10 : Structure, steel and wood properties for ANSYS input purposes......................................... 50 Table 11 : Parameters for design and probabilistic variables ................................................................ 54 Table 12 : Properties of the five chosen OSB wood panel .................................................................... 60 Table 13 : Cost of wood panels relative to their thickness .................................................................... 67 Table 14 : Equivalent static force [kN] for 6-10 storeys building with 3 bays square floor plan ......... 69 Table 15 : CSA O86 table 7.3C for OSB mechanical properties ........................................................... 93  vi  LIST OF FIGURES Figure 1 : Architecture of the PCSP – Input Module ............................................................................... 4 Figure 2 : Architecture of the PCSP – Solution Module.......................................................................... 5 Figure 3 : Architecture of the PCSP – Output Module ............................................................................ 6 Figure 4: OSB panel before the press .................................................................................................... 16 Figure 5 : Visual representation of shear types ...................................................................................... 19 Figure 6 : Comparison of cumulative distribution of Fy and Fu............................................................ 25 Figure 7 : Scatter plot of the set of data ................................................................................................. 26 Figure 8 : Snapshot view of a transformed spreadsheet ......................................................................... 27 Figure 9 : Flanged connector (version 1) ............................................................................................... 28 Figure 10 : Flanged connector (version 2) ............................................................................................. 29 Figure 11 : HSS connector ..................................................................................................................... 31 Figure 12 : Ux, Uy, Sx and Sy results for SHELL43 (modeled along with steel frame)....................... 36 Figure 13 : Ux, Uy, Sx and Sy results for SHELL181 (modeled along with steel frame)..................... 36 Figure 14 : Deflection in the lateral direction (X) of a wood panel ....................................................... 37 Figure 15 : Deflection results in lateral direction for a steel frame ....................................................... 39 Figure 16 : Deflection results in lateral direction for a hybrid frame .................................................... 40 Figure 17 : ANSYS input window & section properties........................................................................ 44 Figure 18 : Logically numbered keypoints representation ..................................................................... 46 Figure 19: Script from formatted files to run the loops ......................................................................... 47 Figure 20 : Simple hybrid frame result outputs ..................................................................................... 49 Figure 21 : End plate connection sketches ............................................................................................. 53 Figure 22 : Design plot without cost target ............................................................................................ 55 Figure 23 : Design plot with $500 cost target ........................................................................................ 56 Figure 24 : Total cost of the assembly ................................................................................................... 56 Figure 25 : Five first beam sections and thicknesses of the web and flanges (tw_b & tf_b)................. 59 vii  Figure 26 : Snapshot of Failure arrays from Matlab Workspace window ............................................. 62 Figure 27 : Transformation from Excel check to Matlab syntax ........................................................... 63 Figure 28 : Shear forces in multistorey structure ................................................................................... 66 Figure 29 : Evaluation of a typical bolted connection cost .................................................................... 67 Figure 30 : 3D plot of the probability of success for a simple hybrid frame analysis on Matlab .......... 70 Figure 31 : 3D plot of total cost of a structure for a simple hybrid frame analysis on Matlab .............. 71 Figure 32 : 3D plot of total probability of success for a simple hybrid frame of less than $4000 ......... 72 Figure 33 : Two resulting plots from steel structures with a set cost target........................................... 73 Figure 34 : Two resulting plots from steel structures without a set cost target ..................................... 74 Figure 35 : ANSYS modeled structure with wood panel implemented “Perimeter & Centered” ......... 75 Figure 36 : ANSYS modeled structure with wood panel implemented “Linearly Centered” ............... 76 Figure 37 : Partial set of data with details of their origin....................................................................... 83 Figure 38 : Lognormal distribution calculations .................................................................................... 84 Figure 39 : Nonlinear analysis from Dr.Frame ...................................................................................... 85 Figure 40 : Stiffness of steel frame calculations .................................................................................... 86 Figure 41 : Spreadsheet for bolted connections ..................................................................................... 87 Figure 42 : Excel spreadsheet for steel sections check .......................................................................... 90 Figure 43 : Excel spreadsheet for steel sections check .......................................................................... 92 Figure 44 : CICS standard section generator for ANSYS coding.......................................................... 94 Figure 45 : Script from ANSYS to create lines when structures are taller than 9 storeys ..................... 95 Figure 46 : Script from ANSYS to install wood panels in a "Centered & Perimeter" pattern .............. 96  viii  ACKNOWLEDGMENTS I would like to thank my supervisors, Dr. Stiemer and Dr. Tesfamariam, for their support, their help and for the freedom they gave me throughout this project. I would also like to thank those who helped to keep the hybrid structure project going. I would like to acknowledge the help from my good friend, James Goulet, who gave me the head start I needed on ANSYS programming. Lastly, I couldn’t picture myself achieving all of these things without the presence of my parents in my life.  ix  1 INTRODUCTION Engineers require methods and software to help them deal with uncertainties in the first stage of structural design. To this end, this research aims to link the Probabilistic Constraint Satisfaction Problem (PCSP) method developed by Loewen (2009) to ANSYS structural analysis software. From experimental data, it is possible to obtain a reproductive probability distributions for different materials used in building construction. This was done by using data processing to find the actual distribution of steel strength, both tensile and yield. Following the same reasoning, the Constraint Satisfaction Problem (CSP) can be expanded into PCSP to include these uncertainties. This thesis also has the target of introducing the hybrid structure project which is comprised of multiple aspects that will be dealt with by many students over the next few years. Hence, this link between structural ANSYS model and the Matlab platform conceptualized for the scope of a master thesis is an important piece of preliminary work. An overview of the initial methodology is presented along with some information necessary to understand the general idea. As part of this project, two different examples were run to test the method and help in visualizing how the linking software works. Additional computation work was done to extend the versatility of the method for future users. Larger complex cases can also be tested with the newly developed link, therefore some methods will be presented to broaden the outcomes in order to start the hybrid structure project. In the second section (Chapter 2) of this thesis, an extensive literature review will be performed covering, all different fields of study that might relate to this complex project. This includes several papers, from the constraint satisfaction problem to the theory of modeling with finite element software. The core paper used is Loewen’s research, which is concerned with his platform to implement probabilistic data for uncertainties prediction analysis. Following is the frameworks section (Chapter 3) that aims to describe previously developed tools that are the foundations of this work. Previous studies were performed as part of term projects for courses, and are relevant to this current project. Hence, a description of the useful outcomes of these works is presented. The Chapter 4 shows the general theory behind the methodology followed during the research process. Computations that serve as tests to ensure relevant results are shown, along with other theory about 1  nonlinearity and CSPs. The last part of the section focuses on the connection design for a hybrid shear wall structure. Various types of possible connections were considered that would be design solutions for these walls. Chapters 5 and 6 focus on the two software programs that are intended to work in conjunction with each other. In both chapters, an extensive description of all new scripts written along with the thinking behind each step taken is shown for the sake of clarity. Finally, Chapter 7, titled “Hybrid Structure Study”, presents the application to the hybrid structure project at its current state. Two working examples were computed as part of this chapter; a simple hybrid frame and a multistorey steel structure. The design stages and the results are presented along with a discussion.  2  2 LITERATURE REVIEW A broad literature review was performed concerning multiple aspects related to this present project and will be summarized. It will cover the core paper used from Loewen’s research about his platform to implement probabilistic data for uncertainties prediction analysis. This will be followed by an overview of Matlab and ANSYS capabilities and finally, by some review of how the finite element method has been used in the past. The papers reviewed will be used as a foundation for this project.  2.1 Uncertainties for conceptual design Uncertainties in load and resistance have been studied for years and yet are largely implemented in design and building codes around the world. Load combinations, including coefficients applied to the live load, dead load and so forth have been subjected to reliability studies. This current sub section will highlight the previous research done to deal with these challenges. 2.1.1  Reliability methods  The Monte Carlo method is the most well known method for solving constrained problems, and was developed in the 1940s by physicists working on thermonuclear weapon projects in the Los Alamos National Laboratory (Metropolis, 1987). It operates by defining the domain of all possible inputs, and then generating inputs randomly from the previous specified domains and probabilistic distributions. From this point, a deterministic analysis is performed and the results are summed into the final distribution results. This method has its advantages and disadvantages, but Monte Carlo remains the most commonly used method. Along those lines, the basic principles of the constraint satisfaction problem was well explained and presented by Lhomme (1993) who defined the CSPs as “a set of variables each with an associated domain of possible values and a set of constraints on the variables”. Although not completely thorough, Lhomme’s work was the foundation for further development that occurred in the following years. With the B-consistency, he introduced an efficient way of constraining the optimal solution. Only this bounded interval containing any possible solution is subjected to the test, leading to a reduction of wasted time when testing extensive intervals. Although Monte Carlo is still being used these days, some improvements are needed.  3  2.1.2  Conceptual design using probabilistic interval constraint satisfaction  The most relevant paper used for this current work is the PhD. thesis of Loewen (2009). His work and extensive research on the CSPs matter resulted in a genius program using the Matlab platform. To give an overview of his work, it could be summarized as an integration of probabilistic distributions as implemented by Monte Carlo, complemented by a grid-search engine and an interval narrowing method to converge quickly and more efficiently to optimal solutions. His platform can be divided in three modules linked to each other. Figures 1, 2 and 3 show the architecture of the PCSP, with a clear distinction between modules. This present section will summarize the relevant features of his thesis while the following section will discuss some issues.  Figure 1 : Architecture of the PCSP – Input Module 4  Figure 2 : Architecture of the PCSP – Solution Module  5  Figure 3 : Architecture of the PCSP – Output Module  As seen in Figures 1, the input module is comprised of the variables definition and constraints determination. Their provenance is either code requirements or some limitation check imposed, for instance, by geometry restrictions. Linking to the solution module, the definition of the desired outputs is required. Within the solution module, the platform is conceptualized to search for the optimal solutions by grid-search method. This will be presented in details in the following sections. Finally, in the output module, it is generated different type of plots to visualize the solution. Depending on the number of variables and the objective function chosen in the input module, 2D, 3D or isosurface plotting modules are available. After reviewing the paper, numerous advantages for using his method for this current project were found and are shown in Table 1. Complementing the main case study, a few examples of simple structure were shown to the reader explaining the reasoning. It is well explained how to get from point A the problem statement to point B, at which point the design space is obtained, a crucial tool for preliminary design. In addition, the ideas were summarized succinctly, as shown at Table 1 which compares different types of CSPs.  6  Table 1 : Comparison of solution methods for determining valid ranges of probabilistic variables.  Initial Method  solution space  Dimensionality of  Size of results  results matrix  matrix  search  Narrowing  narrowing  ݊௉௅ை்  run of  ሺ‫ݏ‬௉௅ை் ሻ௡ುಽೀ೅  interval CSP  Search  of finding  results matrix  discontinuitie  entry  s in solution  (includes interval narrowing  = ݊௉௅ை் + ݊ே஺ோோைௐ  Hybrid:  x  ሺ‫ݏ‬௉௅ை் ሻ௡ುಽೀ೅  Interval  selected (check of CSP  resolution,  consistency)  systematic  1  Depends on  User defined: Naïve 1  x ݊௉௅ை் < ݊௎ௌாோ ≤ ݊  & Grid-  Not found  Depends on  1  ݊ Naïve  ሺ‫ݏ‬ே஺ோோைௐ ሻ௡ಿಲೃೃೀೈ  Narrowing  analysis per  iteration) ሺ‫ݏ‬௉௅ை் ሻ௡ುಽೀ೅  Grid-  Capabilities  1  Single  Interval  Number of  ሺ‫ݏ‬ே஺ோோைௐ ሻ௡ೆೄಶೃ  Search  selected (check of CSP consistency)  1  resolution, systematic  Depends on Monte Carlo  Naïve  ݊௉௅ை்  ሺ‫ݏ‬௉௅ை் ሻ௡ುಽೀ೅  Varies, typically  number of  101 to 107  analysis, random  ‫ݏ‬௉௅ை் = the number of subdivisions along the plotted variables ‫ݏ‬ே஺ோோைௐ = the number of subdivisions along the narrowed variables 1  The original content has been modified due to the lack of having the Interval Narrowing running as expected.  This lack is discussed in section 2.1.3  7  2.1.2.1 Hybrid method The following description discusses the main improvement that the method called the Hybrid Method has made to the CSP analysis method. As shown in Table 1, the hybrid method refers to the combination of the Interval Narrowing method and the Grid-Search method. To a great extent, the general formulation of a CSP is made by setting variables within their respective domains and by formulating constraints associated with those variables. Subsequently it is required to instantiate those variables by assigning a specific domain of values to the subset of variables. To ensure that the constraints of the problem are set locally, a partial solution for the total CSP is determined. For this purpose, consistency techniques are used to check if there are values that do not comply with the given constraint. This current work implemented the use of interval to define the variables. Bounded by the lowest and highest number of the intervals, a loop process is instigated to check the intervals either at their bounds or simply at their mid points to find the “hull” of solutions. Those intervals are an easier way of representing data, especially those uncertain data such as probabilistic data. The next two following sub-sections explain each separate method as whole. The hybrid method includes different traditional methodologies to find distributed probabilities for a given problem. First of all, the user selects their variables and specifies their types. There will be two primary classifications of variables: •  Design variables: These have to be chosen carefully as they are significant parameters while designing the structure to an optimal solution. They are referred to as the plotting variables since they are the top-level design variables that will later be assigned to the plot axes.  •  The probabilistic variables: These have uncertainties regarding their intensity or accuracy. Tolerances of scientific instruments or sets of data that occur within a certain probability distribution are both examples of these variables. They are referred to as the narrowing variables as they have a range of values that yields a solution.  This approach gives the freedom to the user to define any proportion of those two types of variables. Subsequently, they could choose to plot them in reference to each others as randomly as they want. Different axis could be defined, as well as different objective functions to illustrate different relationships. For instance, the cost for two different prototypes could be compare on two different 8  plots having the same axis. It is the user’s responsibility to define the problem with the appropriate variables. The objective function is also part of the requirements and can be changed quite easily to provide the designer with more options. 2.1.2.2 Interval narrowing method One could see the engineer’s judgment and experience as the first step of narrowing the intervals of the variables. To reduce the possible success intervals, the engineer has to apply general knowledge about the minimum and maximum values that could lead to a solution. The Narrowing Method takes out the intervals containing no valid solution. Up until now, the Matlab platform used in the hybrid method does not include this former part of the methodology; this work is still in progress. To compensate, the narrowing work has either to be done by the user within the selection criteria or has to be excluded from this method. One of the advantages of the method is to guarantee both a bounded solution for each entry and to exclude solutions outside the solution space. In the present case these irregularities might occur, so it is the user’s responsibility to choose variables and the number of boxes adequately. 2.1.2.3 Grid-Search method With the narrowed intervals the Grid-Search Method becomes important. First, the user defines the number of subdivisions for each variable, both the design and probabilistic one. The number of those divisions defines the dimension of the matrix generated by the method. A CSP test is carried out by instantiating those variables as mentioned previously. Matlab has the ability (advantage of being able to) compute such a matrix, which in some cases would have more than 3 dimensions. The summation depth wise, which will be explained later on, allows the user to get the total probabilities at each integration interval. Furthermore, these results can be displayed in a simpler and more meaningful way for the designer. 2.1.2.4 Deficiencies Upon further consideration, some deficiencies were found that needed to be addressed. First, although well explained, the probabilistic distributions were defined from educated guess rather than experimental data curve fitting. In other words, the mean and standard deviation were taken from past experience. However, the goal of the project was to develop a powerful tool, not to provide the 9  engineers with direct answers. Second, the idea of combining the grid-search engine with the interval narrowing engine was ideal and proved that his method was innovative. However, this was not yet implemented at the time of the thesis submission. As it will be demonstrated later with few examples, the lack of the interval narrowing engine can be negated by the use of experienced knowledge in the preliminary interval choosing process. Third, all input data, defined as design variables within his research, were continuous numbers. Not all section depths are easily found in the market, and the Matlab platform was built using continuous numbers rather than discrete ones. Slight modifications of the already build code are required to implement discrete values to get a precise analysis along with an accurate plot display.  2.2 Software overview Software programs constitute the foundation of this project. The goal of linking the ANSYS modeler tools with the Matlab reliability platform comes from the fact that those two programs possess many assets. The main ones are presented below. 2.2.1  ANSYS  The robustness of ANSYS means that multiple options are offered to users, for example input data and functions in the software. The most efficient method is to use the formatted text file, most commonly saved as .inp or .txt files, rather than the “Menu Commands”. This has multiples advantages, such as allowing direct reading of text files from the main window called Graphical User Interface (GUI). By calling a text files out which was previously saved in folders, small variation of values clearly identified are permitted to be change at any point in the design process. These saved files represent the entire work done and can be kept in archives for further study and utilization. It is also possible to add any comments through the script to allow users to understand and modify files at will. 2.2.1.1 Elements creation When it comes down to the element definition, the options are broad. Being a finite element software, ANSYS could either choose the most appropriate amount of elements from previously created line, or the number could be user defined. To accelerate the computations while keeping full control on the program, a set numbers of elements were chosen for the beams and the columns. The function LMESH is used to mesh the lines and divide those up in the assigned number of elements. 10  The element type is also complex to define, and several iterations were needed to select the BEAM188 type. It was kept in mind that this project is just beginning, and more complex analysis will be run in the future. Some important features of the BEAM188 element type are listed in the help document for the ANSYS software program (2007). The three most relevant are shown below: •  “BEAM188 is suitable for analyzing slender to moderately stubby/thick beam structures. This element is based on Timoshenko beam theory. Shear deformation effects are included.”  •  “BEAM188 is… a quadratic beam element in 3-D. BEAM188 has six or seven degrees of freedom at each node, with the number of degrees of freedom depending on the value of KEYOPT(1). When KEYOPT(1) = 0 (the default), six degrees of freedom occur at each node. These include translations in the x, y, and z directions and rotations about the x, y, and z directions.”  •  “BEAM188 includes stress stiffness terms, by default, in any analysis with NLGEOM,ON. The provided stress stiffness terms enable the elements to analyze flexural, lateral, and torsional stability problems (using eigenvalue buckling or collapse studies with arc length methods).”  Where the function NLGEOM,ON refers to a geometric nonlinearity of the analysis turned on. For all of the reasons stated above, the BEAM188 was used for all the structure modeled for this project. 2.2.1.2 Automated areas creation The areas define the wood panel and SHELL element types are suitable. Because they could be created from keypoints as well as from lines, different scripts could be used. Although it seems straight forward, it is important to note that the keypoints and/or lines order does matter. Depending on the method used, the local axis of the areas will be assigned differently, and confusion could occur while processing the stress data. To keep the local axis oriented with the global axis, first the lower 2 keypoints or the lower line should be given to the program followed by the upper one. 2.2.1.3 Element table (ETABLE) outputs For a structural member, the following ETABLEs were found to be relevant for analysis on members behavior. Hence, further calculations with Matlab platform will be executed based on these values. 11  Table 2 : List of ETABLE for steel elements outputs  Name  Script (ETABLE,Name,…)  Descriptions  DX DY  …,U,X …,U,Y  Displacement in X-axis Displacement in Y-axis  AF  …, SMISC,1,14  Axial forces  Area  …, SMISC,13,26  Section area  ShearZ  …,SMISC,5,18  Section shear forces  MZ  …,SMISC,34,39  Stress on the +Y side of beam  Table 3 : List of ETABLE for wood panel elements outputs  Name  Script (ETABLE,Name,…)  Description  SEQV In-P11  …,S,EQV …,SMISC,1  Equivalent stress per unit length In-Plane force (per unit length)1  In-P22  …,SMISC,2  In-Plane force (per unit length)  In-P3  …,SMISC,3  In-Plane force (per unit length)  M11  …,SMISC,4  Moment out-of-plane (per unit length)  M22  …,SMISC,5  Moment out-of-plane (per unit length)  M12  …,SMISC,6  Moment out-of-plane (per unit length)  Q13  …,SMISC,7  Transverse Shear Forces (per unit length)  Q23  …,SMISC,8  Transverse Shear Forces (per unit length)  1  The unit length refers to the element length or heights.  An important thing to mention is the way the moment is calculated by ANSYS. The moment is seen as the maximum bending stress components (SBYT). Thus, to get a bending moment for the I-beam section in Y-axis, one needs to perform some calculations, ‫ = ܼܯ‬−  ‫ܫ‬௭௭ ∗ ܵ‫ܻܶܤ‬ ‫ݕ‬௠௔௫  ሺ1ሻ  In equation 1, the Izz refers to the moment of inertia around the Z axis, the SBYT is the maximum bending stress components and the ymax is the maximum Y coordinate in the cross section measured  12  from the centroid. These values would be extracted from the program by the use of the ETABLEs shown at Table 2. Similarly, the SDIR is the stress component due to axial load called out with “SMISC,31,36”. Therefore, the SDIR should be multiplied by the cross-sectional area to obtain the axial stress. However, for this project the axial stress as shown in Table 2 will be used. 2.2.2  Matlab computations  Matlab was chosen as the software platform for the development of the theory included in Loewen’s thesis. It is logical given that Matlab has all necessary features for formulating loops to build plots of higher dimensions. It can handle multi-dimensional arrays as well as representing these in easily visualized 2, 3 or 4D plots. The following sub-sections will the some specific coding used to generate the outputs. 2.2.2.1 Probabilistic function Two special function files to calculate the Cumulative Distribution Function (CDF) have been developed. Originally, in the Loewen (2009), the normal distribution was used for the steel strength. However, the log-normal distribution is selected in this project as a better and more suitable choice. The cdf_lognorm.m is written as follows: function c = cdflog(x,xmean,xsd) a=-8*(pi()-3)/(3*pi()*(pi()-4));  % parameter a  xsn=-1*(log(x)-xmean)/(xsd*sqrt(2)); % parameter for error function erfc=1-(sqrt(1-exp(-xsn^2*(4/pi()+a*xsn^2)/(1+a*xsn^2)))); % error function if log(x)>=xmean; c=1-(1/2*erfc);  % CDF  else c=1/2*erfc;  % CDF  end  13  As the program loops through the range of variables, arrays have to be used to define the minimum value, step number and the maximum value. In the simple case of design variables, the formulation looks like this for case where 6 different values are assigned to the first design variable (dv1) : dv1=[0.5,6.5,6];  % number of bolt rows  dv1a=dv1(1)+(dv1(2)-dv1(1))*(0:dv1(3))'/dv1(3);  However, in the most complicated case of a log normal variable, the range has to be well defined to get all possible values on both side of the mean. It has been calculated from a specific range of values that more than enough values fall into the interval bounded by the mean ± three standard deviations. The excel spreadsheet used for this purpose is shown in Appendix C. Here is the associated Matlab syntax: beta=3; mu1=5.938; % mu of ln(p1) sd1=0.033;  % stdev of ln(p1)  Ep1=exp(mu1+0.5*(sd1^2));  % mu of ln(p1)  Stp1=exp(mu1+0.5*sd1^2)*sqrt(exp(sd1^2)-1);  % stdev of ln(p1)  pv1=[Ep1-beta*Stp1,Ep1+beta*Stp1,20,mu1,sd1];  % LOGNORM Yield strength  The length of the arrays defines the number of loops in the function to calculate the probabilities of every combination. 2.2.3  Solution module  The solution module used to test all the constraints used by the Grid-Search solver module was developed in Matlab by Loewen (2009). All calculations are carried out according to the constraints established by the user, followed by a verification of the existence of the solution. If the constraints are not violated, the value of truth=1 previously assigned remains valid. However, if a lack is found, the truth value is turn to 0. After checking all constraints for each type of provenances as defined earlier, the total probabilities for each iteration are calculated and added up. Fortunately, various outcomes can to be generated if necessary. A general check to see if the cost target is met is done by assigning the value of either 0 or 1 to the truth variable with a set cost target value. As mention before, in addition to  14  the cost verification, throughout the loop a sub loop was implemented to verify extra constraints related to the design itself. The outcomes expected could take easily understandable shapes. In this case, 2D plots are suitable, since only 2 design variables were defined. The total probability compiled from each probabilistic variable is stored in a multi-dimensional array. In order to create a readable plot, a file function sumdepth.m adding up all probabilistic variables for each performed loop was developed by Loewen (2009), and is as follows: function matrix_out = sumdepth(matrix_in,ntot,nplot) for i=ntot:-1:nplot+1; matrix_out = sum(matrix_in,i); matrix_in = matrix_out; end  The matrix dimensionality is the number of the design variables increased by the number of probabilistic variables. The sumdepth.m function shown above use the total probability of each box throughout the probabilistic variables to display the total for each pixel related to the design variables. Logically, the sumdepth.m function should be used as many times as the number of probabilistic variables used for computation.  2.3 Finite element method (FEM) software Relevant documents written about the FEM programs ANSYS and Abaqus discuss their effectiveness and their broad range of applications. When implementing and analyzing orthotropic materials, those software programs offer many input possibilities, resulting in a thorough tool. Connections can sometimes be the most critical building part to design; this is especially true when connecting different types of materials. For instance, in the hybrid steel-wood structure innovative methods must be employed to account for the material discrepancies. Several options are offered in both FEM programs, but the program used throughout this project was ANSYS. Unfortunately no analytical study about connecting a steel frame with a wood panel has been found to date, making the programming more difficult. Although slightly different, the conclusions and educated guesses taken from the following papers helped to guide this study. 15  2.3.1  Modeling of shear walls  Much research was conducted to find an ideal and representative way to reproduce wood panels with finite element software, not only in the connections but also the material characteristics. The material in question is OSB selected for its high strength in shear due to the alignment of its strands. Its Module of Elasticity (MOE) is sometimes greater than 8300 MPa in the major axis and so its beneficial contribution to a shear wall system is significant. At the connection level, research has been done, but no satisfactory results were found to date. The apparatus required to transfer the shear and the compression stresses from the steel frame to the wood panel while dissipating energy in the structure has not been designed. Insights of some possible modeling techniques, along with some observations are given below. 2.3.1.1 Material characteristics OSB panels fall in the category of flakeboard, referring to the glued flakes of various type of wood, sometimes low-quality (Somayaji, 2001). Unlike plywood, which is made out of layers of veneers, the bending properties of the OSB tend to be lower due to a lack of a logical arrangement of its particles. However, it has been found to have a higher modulus of rupture than plywood.  Figure 4: OSB panel before the press Source: Wikipedia  A major drawback of OSB panels is the difficulty in accurately predicting its behavior and mechanical properties. As powerful as ANSYS software is, it needs precise inputs, especially when entering the 16  basis mechanical properties of the material. The difficulty comes from its non-homogeneous composition along with the complexity of modeling nonlinearity in each orthotropic direction. Researchers (Saliklis, 2000) commonly report that the nonlinearity is ignored for the sake an easier design process. When dealing with orthotropic panels, failure modes such as buckling can be directly related to their initial imperfections. Similarly, the failure point would be directly related to the effectiveness of the modeling. Simplifications falling on the conservative side allow the users to define to the best of their knowledge the panel characteristics, until more tests are done and more data is available to refine the model. As shown above, the mechanical properties of the plywood fall are unclear, and one’s interpretation vs. another’s test results could lead to noticeably different models. It is important to distinguish between speculation, and actual tests performed in order to determine realistic values that can later be implemented in an ANSYS program. Karacabeyli et al (1996) have done over 15 years of tests and calculations, providing the CSA O86 (CSA 2005) standards with data for engineering wood design. Properties for which the values can directly be obtained from CSA O86 are shown below in Table 4. Table 4 : CSA O86 tables for OSB capacities design value  The grades and types of OSB sometimes cited in the literature are defined by CSA O437 (CSA 1993) and are classified as OSB O-1 and O-2. Although it seems to be enough to differentiate between lower 17  and stronger panels, the “design rated OSB” referred to their qualifications and certification procedures. These design rated panels are split into 3 categories (A, B and C), called rating grades. Additionally, three types are available, Type 1 being “standard”, Type 2 being “plus” and Type 3 being “proprietary” without any certified properties. Type 2 is treated as the Type 1, but has one or more properties exceeding Type 1 by 10% or more. Detailed test results and conclusions of the analysis process lead to uniform results for grades A to C and Type 1 panels (Karacabeyli, 1996). The properties of interest for this project are: •  Modulus of Elasticity (E) for both bending and axial stress  •  Modulus of Rupture (MOR) for the planar and in-plane directions.  The MOR of the in-plane direction, referred to as the Shear-In-Plane (SIP) ultimate stress, is not given in tables. However, from testing results, the planar shear for the 5th percentile is provided for this particular stress. The British Standards for OSB design (BSI 2006) made the correlation between the planar shear and the MOR in-plane. When planar shear fluctuates between 0.8 to 1.2 MPa, the sought MOR is approximately 50 MPa. Moreover, the data from Somayaji (2001) was used to check this former assumption. The MOR of OSB is stated as 7000 Psi which is approximately 50 MPa. This value represents the shear in-plane of a wood panel. On Figure 6, this in-plane shear corresponds to the “Rolling shear” representation.  18  Figure 5 : Visual representation of shear types  However, the shear through the thickness (Figure 5) provides the necessary value to define the stiffness of a wood panel subjected to lateral loading. This value was found to be defined by G [MPa] in Karacabeyli et al. (1996) and will be assumed to be 1000 MPa throughout this project. Furthermore, Poisson’s ratio needs to be specified by a method other than by calculating it directly by the relationship between E and G because this equation is invalid when dealing with an orthotropic material. The value is fixed at 0.16 for the smaller Poisson’s ratio. Based on the paper by Thomas (2003), this value was found to generate reliable results while testing. Correlating these parameters to the experimental panel was being possible due to the formula given in the Timber Design Code (CSA 2005):  ∆=  2 ∙ ܸ௦ ∙ ℎଷ ℎ ݁௡ ℎ + ܸ௦ ∙ + 0.0025 ∙ ℎ ∙ + ∙∆ 3 ∙ ‫ܮ ∙ ܣ ∙ ܧ‬௪ ‫ܤ‬௩ ܷ݊݅‫݂ܱݏݐ‬ሺ݁௡ ሻ ‫ܮ‬௪ ௔  ሺ2ሻ  19  In Equation 2, ∆ is the horizontal in-plane deflection of top of shearwall, Vs is the maximum shear force per unit length, Hs and Lw represent the shearwall height and length, E and A refers to the crosssectional properties of the chords while en is the nail deformation and ∆௔ is the deflection due to the anchorage details. The first term determines the deflection of the shear wall due to the bending moment in the chord frame system. It is calculated from the inertia ‫ܣ = ܫ‬ௗ ଶ ⁄2 of the two chords at “Lw” distance with aid of the formula ݀ = ܸ௦ ∙ ‫ܮ‬௪ ଷ ⁄3 ∙ ‫ ܫܧ‬. Although it is easy to prove the first term of the equation, the added deflection due to the shear wall reaction is less clear. If the two materials work together, the deflection of each material alone should not be added up. Furthermore, the second term represents the shear wall deflection due to the panel shear resistance as mentioned above. The “Bv” is a term used by CSA 086 (2005), and it refers to the lateral stiffness of the OSB panel, shown in Table 2. The third term is for nail slipping, however in our case, where only comparison with the ANSYS model is needed, this term is neglected because a full connection between the panel and the frame is assumed. The fourth and last term predicts the extra deflection related to the hold down. Here again, there is no need for us to use this part of the equation. 2.3.1.2 Connection Research by Hite (2002) confirmed their analysis by experimental testing. The specimen tested consisted of a wood frame sheathed with wood panels. The panels had to be connected so that the shear stress is taken such that small directional strengths are affecting its structural integrity. This study, along with this thesis, aims to determine the behavior of a lateral resisting wall subjected to vertical loading. Experimental results proved consistency with predicted results. The stiffness of the wall was found to be within 6% of the expected wall behavior. However, the ultimate load was 20% different than that calculated. Another interesting and surprising finding was that the ultimate capacity slightly decreases with added vertical load. At the same time, there is an increase of 8% in the shear wall rigidity. Although the connections are designed for wood to wood interfaces, it is interesting to follow the methodology of the research performed. Obviously, the principles behind this type of connection differ from those suitable for wood-steel interface. However, this research studied some new ways to connect shear walls to dissipate energy. Each fastener consisted of two springs acting in the two longitudinal axes. Before plasticity is reached, the two springs act as a single spring in the displacement direction. 20  After yielding occurs, the springs elongate creating a stiffer and thus stronger connection. The details of spring modeling are relevant to this project, and the basic concepts about the FEM program helped to get this work started. No significant difference was noted whether the wood panels were offset or not, due to the modeling software’s connection design. However, it has a physical representation and it is an interesting feature to use. In the paper, the wood properties were taken as E=8.3 GPa, G=640 MPa and Poisson’s ratio as 0.3. Along with previous references cited, it can be seen again how confusing the literature can be for the modulus of wood panels, both plywood and OSB. If the MOR represents the planar shear, it is amplified by ten compared to what was shown before. If it was meant to be the shear throughthickness modulus, the strength of the wood panels is underestimated .  21  3 FRAMEWORKS This section will begin with the presentation of past research on steel structures along with the strategies used to remedy deficiencies and achieve genius performance. Additionally, work done first on the reliability method applied to conceptual design and second, on the preliminary concept of the hybrid wood and steel structural system will be introduced.  3.1 Steel structures case studies Most of the research done up to this point on light hybrid wood-steel structures has been driven by the desire to find new structure to resist seismic loading. Since the mass of the structure affects its period and thus its behavior when subject to dynamic loading, low weight structure ensures better behavior. The Scotia Place in Auckland, New Zealand, is an innovative state-of-the-art building design, making use of these cost effective structures. As well, other wood shear walls with steel frames have been tested and designed at UBC. Along the same lines, a study of a light structure made out of only steel was done by Yousuf (2009) Interesting guidelines for steel section selections, along with references for seismic load applications, are among the main outcomes of this paper. However, in the scope of this project only a simplified load will be applied to the model structure, because the main goal is to get Matlab and ANSYS to work in conjunction. 3.1.1  Scotia place hybrid structure  The utilization of wood and steel combined take place in different state in a structure and in the formal case, it is called “Building system” hybridization (Khorasani, 2010) where the two components are combined in the resisting system of the overall structure. The other recognized type of hybridization is at the “Component level”, whereas the member itself is built from different materials, to take advantage of the strength of each material. In the Scotia Place paper written by Moore (2000), multiples advantages were listed for having a lightweight structure. Among these were the ease for carrying material around in a crowded city location and the improved performance when subjected to seismic loading. The benefits were enough to motivate further studies on this new type of structure.  22  3.1.1.1 Structural specifications The hybridization was done using a steel frame for the structure and a wood floor for diaphragm members. Although promoted as a hybrid steel-wood structure, a concrete slab was used at ground level along with a masonry wall in the basement for the foundation. It is common to design a concrete foundation due to the necessity of having strong foundation. In addition to the wall and the slab, reinforced concrete cast-in-drilled-hole (CIDH) piles work with the lateral load resisting system and are designed for resisting tension loads. 3.1.1.2 Design notes This research highlights the design load requirements along with providing a useful comparison study with an alternative structure using concrete floors. It is predictable that the alternate structure using concrete floors weights five times more than with wood floors and is governed by a different type of load. As far as the lightweight structure is concerned, the wind load governs and the maximum lateral deflection is the main design check to perform. The building code of NZ (NZS 1992) has a drift limit of 0.2% of the storey height, similar to the National Building Code of Canada (NBCC 2005). The lateral system was thus designed to achieve the lateral stiffness necessary to limit the drift to the code requirement. The same objective is set for the hybrid structure designed as follow, and thus the bracing will not be located at floor level.  3.2 Conceptual design using PCSP An early study of the work previously presented by Loewen (2009) was performed. The purpose of this work was to become familiarize with his work, and in addition to incorporate new features. The relevant details of this project are presented in this subsection. 3.2.1  Data processing  Data processing is an essential step for any user if an accurate probabilistic analysis is sought. It ensures that the data used in the analysis to produce outputs is reliable enough to base new principles on. In order to demonstrate the general method, below is shown a discrete example, made of a set of steel mechanical proprieties. The yield and tensile strengths of any material have to be considered as random variables since they are imprecise parameters. A set of data was available for an analysis for 23  this project. The details of the steel test results are available at the Appendix A. The following results were found for the mean and the standard deviation of both the yield strength and the tensile strength. (Fy expected: 345 MPa, Fu expected: 450 MPa). Table 5 : Probabilistic parameters  Parameters of Fy  Data set result  Parameters of Fu  Data set result  373 13  Mean (µFu) Standard deviation (σFu)  506 10  Mean (µFy) Standard deviation (σFy) Mean log(µFy)  5.938  Mean log(µFu)  6.227  Standard dev. log(σFy)  0.033  Standard dev. log(σFu)  0.019  From comparison of the cumulative distribution function (CDF) diagram, it was seen that either the normal distribution or the log-normal distribution will fit the data. These two plots are shown in Figure 6. Note that the curve of the tensile strength does not fit very well, that is probably due to the error generated by a too small set of data. The initial assumption made by Loewen (2009) while developing his method was normally distributed yield strength. However, in the literature (Alpsten, 1972), the steel strength is generally a log normally distributed variable. Another reliable source outlining this theory is the code calibration of the National Building code (NRC, 2005), based on some other data coming from the literature. In addition, considering that the strength of any material is strictly positive, a natural logarithmic distribution makes more sense. Taking those facts into consideration, the following work will assume that the steel yield and tensile strength are log-normally distributed with parameters taken from the sample of data.  24  Figure 6 : Comparison of cumulative distribution of Fy and Fu  I t should be clarified that the computations made in the thesis imply that these variables are not correlated with each other. This is seen with the output matrix, which includes every possible combination regardless of the physical meaning of the values. If the two parameters are highly correlated, their opposite extreme values are not likely to occur at the same instant of time. Even though the yield strength and the tensile strength may seem to fall into that category, this is actually not the case. The plot shown at Figure 8 shows how small the index “R2” and therefore how poor the correlation is for the present set of data. This can be explained by the large blend of metal contained in steel. All these metals have different mechanical proprieties, hence the steel sample could stay elastic longer but necking could occur sooner, for example.  25  Figure 7 : Scatter plot of the set of data  3.2.2  Adaptation of CSP  The PCSP platform requires the formulation of constraint equations in certain way. Thus, all constraint problems could be formulated using a spreadsheet method. The steps to achieve an efficient spreadsheet are: •  Defining design variables that will iteratively be changed by the designer.  •  Entering fixed variables that will not be changed and referencing them.  •  Computing the formulas and constraints related to those design variables.  •  Executing an iterative process to optimize the design variables.  However, this method has some weaknesses. An engineer has to spend a lot of time to find a suitable design value, and frequently the method would never converge to the optimal solution. In the case where probabilistic variables are involved, this method cannot incorporate them. Fortunately, the same spreadsheet used by engineers in practice can be translated to the Matlab language. This way, they could optimize their previous work building a spreadsheet and finding all the applicable constraints. A snapshot view is shown below from a bolted connection design example performed previously. Additionally, the full spreadsheet is displayed in Appendix C. 26  Figure 8 : Snapshot view of a transformed spreadsheet  In the previous figure, on the left panel, the syntax used in Matlab is shown. The “PV” refers to the probabilistic variables which are, in this example, the yield strength (Fy), the tensile strength (Fu), and the factored shear force (Vf). These variables are transformed to “pv1a” to “pv3a” for simplification purposes. Same idea is applied to the design variables “DV” which are the number of row of bolts (n) and the bolt diameter (d). The transformation process will be further discussed in the following section. 3.2.3  Relevancy  This past work was useful in starting the current work, due to the relevant information. Understanding the uncertainty implementations, as well as the Matlab coding can be quite difficult without some guidance. This past section presented the technique used for implementing new code section block in the original Matlab platform, and this knowledge can be carried over to the subsequent user. 3.3  Hybrid steel-timber structures  Studying new ways to connect steel frames and wood panels was the topic of a previous term project (Villiard, 2010). The objectives of that project, along with the resulting prototypes are described in this section. Lastly, the potential of using this previous work for the current research is discussed. 3.3.1  Objective  The work described here was undertaken in order to get an understanding of the issues with a hybrid structure involving steel frames and wood shear walls. The central problem, yet to be resolved, is how to overcome the incompatibility of these two materials by finding an efficient, inexpensive and 27  practical connection. The following sections describe the options, including their specifications, design calculations and advantages. 3.3.2  Flanged connection type  A system of back-to-back flanges allows free movement of the panel while acting as a guide. The prototypes are shown in Figures 10 and 11, with the green parts being the Fiber Reinforced Polymer (FRP) and the black being the rubber pad, explained later. The steel section used for the two prototypes was chosen to be W150x18, which has a plastic section modulus of 136x103 mm3.  Figure 9 : Flanged connector (version 1)  28  Figure 10 : Flanged connector (version 2)  3.3.2.1 Structural specifications The FRP sections were designed to be as simple and as cheap as possible. The use of rubber or polyurethane springs allows the structure to dissipate energy, which results in a stress reduction in the wood panel. This type of material is usually designed to absorb energy in small strokes to improve the life span of equipment or machinery, exposed to vibrations. A post cycled resistance evaluation has to be done to prove that the elastomer can perform as required. According to Smith (1998), natural rubber (NR) would be more suitable for the shear wall since the rubber pad requirements are: •  To be soft enough to allow the wood panel swags without restriction while dissipating energy.  •  To have the ability to restore the original shape quickly, even after many cycles.  The pad thickness is evaluated the same way as the flange length. Ultimately, when the maximum deflection is reached, the wood panel would be leaning against a fully compressed pad. Two deflections are of interest for the thickness determination: 29  •  Δmax: Maximum deflection allowed or maximum deflection of a braced frame subjected to the maximum expected load.  •  ΔP: Calculated deflection after the first plastic hinge formation.  For instance, if we opt for a soft natural rubber filled with polyester fibers, which allows dispersion of the compression stress in the wood panel while maintaining the stiffness in the other directions, the thickness of the rubber pad can be calculated as follows: ࢚ ሾ‫ܕܕ‬ሿ = ሺ∆࢓ࢇ࢞ − ∆ࡼ ሻ + ૞  ሺ૛ሻ  According to research done on elastomers, the shape factor (S) defined as the ratio between the loaded area and the lateral surface area free to bulge is the characteristic parameter (Smith, 1998). The second flanged connector prototype aims to allow for more lateral deformation of the elastomer pad. In both cases, the need to have a device with a high vertical stiffness and low shear stiffness requires that S assumes values greater than 5 and less than 30, since most of the deformation will be horizontal, i.e. perpendicular to the pad surface. The low shear stiffness will allow for greater vertical displacement of the wood panel, leading to long-term integrity of the elastomer. The layers of the elastomer pad will have to be further designed when more information is known. It can be seen in Figures 9 and 10 that the wood panel is connected at the center line of the steel section, thus allowing the designer to fill the rest of the space. Moreover, it results in evenly distributed lateral loads and shear stresses, a major advantage. 3.3.2.2 Design calculations The flanges play a guiding role in the lateral deflection movement. They could also be seen as helping with the out-of-frame deformations, but this would require more careful calculation. Their only purpose for the scope of this project is to confine the wood panel. The length of the flanges was studied with the help of the Dr.Frame software. The load was applied until plastic deformation of the first joint, and the deflections of the top two joints were marked down. Different results were obtained as the steel section was modified. Hand written calculations were made based on the generalized method to predict the moment expected to occur at the hold-down. In addition, it was noticed that the vertical loading, simulating the load from the higher floors, did not significantly change the load at which plastification occurred. However, the frame shifted less without the horizontally distributed 30  load. Therefore, it is conservative to use the deflection of about 25mm obtained when the first load pattern is applied. A quick and economic way should be chosen to connect the W-section to the FRP materials. The sprayed-on method would allow the builder to perform the connection between steel and FRP quickly. Glued connections are often used for FRP materials, and this is an alternative. Whichever method is chosen, an analysis to calculate the friction force needed to keep the FRP connectors in place will be required. 3.3.3  Hollow section connection type  This type of connection was designed similar to the double-flanged one. The hollow section used for calculations is a HSS 102x102x6.4.  Figure 11 : HSS connector  31  Similar to the flanged connector, a section of elastomer is included in the system for both wood panels. Here again, the lengths of the FRP legs will be defined according to the structural analysis. The section chosen for this project results in a deformation similar to the I-beam section. Recall that this deformation is taken at the instant when the first plastic hinge is developed in the structure. For the present design, an average of 25 mm is required to ensure an effective connection between the wood panels and the FRP flanges. The main advantage of this connection is in the installation sequence it could be made in such a way that it would “snap on” and allow for on-site assembly. The connectors would be designed according to the HSS dimensions required for structural integrity. It can be seen that the eccentricity is a concern for the shear stress distribution. Although this is a legitimate concern, every hollow section has a strong torsional resistance, and hence no problems are expected. This type of connector should be chosen if HSS sections are preferable for the steel section. Nevertheless, the similarities between the last three connection prototypes rears that none of the sections are clearly preferable. They all have same advantages of being economic, easy to assemble, and light. 3.3.4  Relevancy  Although these previous connections types have not been studied exhaustively, understanding the complications and issues of the structure has been very important for this project. The combination of such different but complimentary materials poses a challenge in shear wall designs partly due to the incompatibility of their lateral stiffness. To overcome this problem, the connection is designed to dissipate energy in the system, leaving the steel to take the large compression force. As a starting point, this previous work presented an overview of what has been done in the past and what is left to be done in future work.  32  4 PROBLEM STATEMENT & SOLUTION METHODS This section summarizes methodology of the entire thesis. First of all, some hypotheses are posed and assumptions are made, resulting in a simpler project, where the objective is just to build a link between ANSYS and Matlab. These assumptions were reasoned in such a way that they could be subjected to future tests, evaluating if they are still justifiable. This project, however, aims to extensively explain the methodology used to make both software programs work in conjunction. Details of the theory of both programs are presented separately (Chapters 5 and 6). This chapter will also present the proof calculations performed to ensure a reliable set of outcomes. The constraint satisfaction program will be discussed in enough detail to be able to implement it into a Matlab platform. Lastly, the issue of nonlinearity will be considered along with the related particulars of ANSYS, and the associated problems that had to be overcome.  4.1 Overview of methodology The main goal, around which this research is oriented, is to build a link between two programs in order to give the user a useful engine to improve and accelerate the first design step. This link will be developed focusing on the hybrid structure issue. From multiple options of steel sections and wood panel thicknesses, an economically optimized configuration is to be found for a hybrid steel frame and wood panel shear wall system. From variations of sizes at the member level, and with the dimensions of the structure itself (number of bays and storeys), suitable sections for various percentiles of success, different cost targets and other satisfaction constraints can be found and shown on a user-friendly readable plot. 4.1.1 •  Hypothesis and limitations The type of hybridization studied will be the “Building system” hybridization. Only I-Beam steel sections along with OSB panels are used. The optimization process aims to find the best steel sections among the 24 selected, complemented by a few OSB panel thicknesses. For the single bay study, five different potential thicknesses are selected and checked against their respective strengths whereas in the multistorey structure, one panel thickness is chosen, and six different panel lay-out are studied.  33  •  No connection details are studied. Perfectly fixed joints are used between steel members and between steel members and wood panel shear walls. The reason is that this study is done to compare different sections and find the optimized combination of members rather than to find the overall optimized structure type. The only time connections will be detailed is when dealing with the cost study.  •  At this level, a typical beam to column connection will be studied to get a price estimate for the material, including the labor required for each steel connection. As for the connection between the wood and steel components, a fixed price per panel will be defined in order for the price to reflect how many panels there are.  •  Square plan structures (i.e. same amount of bays in both directions) are used for the sake of simplicity. The width of each bay will remain the same for this preliminary study, along with a constant storey height for storeys above the ground floor, with the ground floor assigned to a specific height. Six to ten storeys structures are modeled and compared to each other both economically and based on their resistance to load.  •  The load definition will remain hypothetical as no details of the structure itself have been studied up to now. As a simplifying assumption, load calculations will thus be limited to the worst case scenario dictated by the NBCC 2005. The use equivalent static load procedure is preferable in this case because many analyses have to be done on ANSYS to facilitate the comparison between all different member combinations.  4.2 Proof calculations As mentioned before, to ensure good performance of the program, and be able to predict the results, hand calculations were performed for some of the components. It was simple to perform them for a simple steel frame and for a single shear panel. As far as the hybrid steel frame and wood sheathing are concerned, assumptions were made for the expected deflection since is it difficult to analyze such a system by hand. One way to calibrate the ANSYS program was to separately test both materials and calculate the deflections by hands. The connection details have not yet been designed at this stage of the hybrid structure project, therefore it would be meaningless to perform extensive calculations for the shear wall deflection. However, the method of adding the stiffness of the frame to the stiffness of the panel remains a good option and will be used in subsection 4.2.2.  34  4.2.1  Wood shear panel  Carefully selecting the right element is crucial to ensuring reliable results from the ANSYS analysis. An extensive review of available SHELL type elements was performed and checked against literature recommendations and code requirements. If a poor SHELL type is chosen, subtle changes in the specifications could lead to incorrect results that would misrepresent the behaviour of the wood shear wall panels. For example, some SHELL elements are defined as shear walls without taking into account the transverse shear force whereas some of them are more suitable for curved surfaces. Table 4 shows the advantages and drawbacks of few of them. Table 6 : ANSYS SHELL types comparison  Tension &  FEATURES Transverse shear  compression  resistance  strength enabled.  enabled.  SHELL28  NO  NO  NO  NO  SHELL43  YES  YES  NO  NO  SHELL63  YES  NO  YES  YES  SHELL181  YES  YES  YES  YES  ELEMENTS  Nonlinear capability  Layered element capability  Although SHELL 181 is the most thorough, it is also the most complex to use. It allows for layered materials, and takes into account more properties, controlled through the assignment of the “KEYOPT” options. Considering the possibility of further studies about the behavior of the shear wall, and even though the next two figures shows similar results for (ordered from top to bottom, left to right) lateral displacement (UX), vertical displacement (UY), stresses in the X direction (SX), and stresses in the Y direction (SY), the SHELL181 element will be used for this entire project.  35  Figure 12 : Ux, Uy, Sx and Sy results for SHELL43 (modeled along with steel frame)  Figure 13 : Ux, Uy, Sx and Sy results for SHELL181 (modeled along with steel frame)  36  From Figures 13 and 14, it can be seen that SHELL43 and SHELL181 act similarly even though nonlinearity is taken into account for the wood panel in the latter case. Note that the nonlinearity feature will allow the more complex structures to behave according to reality, which is one of the reasons why SHELL181 is preferred over SHELL43. To define the MOR of interest for the ANSYS program, multiple iterative tries have been performed. In compliance with the assumption of modeling the OSB panel as a single layer element with representative mechanical properties, the in-plane shear resistance is excluded from this analysis. The in-plane shear force act to try to rip layers apart from each other, and is also called rolling shear, as shown in the previous chapter. The most suitable value to be used for Gxy, governing the deflection of the wood panel in the lateral direction, is the “G” for the shear-through-thickness (STT) (Karacabeyli et al, 1996).  Figure 14 : Deflection in the lateral direction (X) of a wood panel  37  Figure 14 shows the deflection graph for a panel resisting shear stress alone. The force is applied at the top of the panel, and in order to represent this shear force, a 2x4” wood member is attached to the 25 top nodes of the panel. Maximum deflection still occurs at the corner where the load is applied, but it is more representative to look at the dispersed orange or yellow scale rather than the confined red spot to compare to the results from the code equation for shear wall deflection. The equation was studied before in detail, and it was found that only the second term is of interest in this present comparison process. ∆௦௛௘௔௥ ௣௔௡௘௟ = ܸ௦ ∙  ℎ௦ 50݇ܰ 5݉ = ∙ = 4.17݉݉ 5݉ 12݇ܰ/݉݉ ‫ܤ‬௩  ሺ3ሻ  Comparing to the limit between the yellow and orange scale of 4.42mm, the result is only 5.6% higher. This is satisfactory and conservative if the selected mechanical properties are kept constant for each panel implemented. 4.2.2  Stiffness method  Using the stiffness method to compute the deflection of a simple frame with fixed supports at ground level and with fixed connections at rigid beams is straight forward. The deflection along with the stiffness of the frame is presented in Table 5. The same steel frame was implemented in ANSYS for the sake of comparison, and the result is satisfactory. The stiffness of the frame turns out to be approximately 4.01x106 N/m, taking the average of the hand calculations and the analysis preformed by the software. The stiffness method by hand assumes a constant displacement of the two top nodes, whereas the ANSYS analysis results in different lateral displacement along the beam, but the maximum difference from corner to corner is less than 1mm, which is insignificant. With efficiently designed connections between the steel members and the wood panel, it would be expected that the total stiffness would be the addition of the two material stiffnesses. As presented earlier in this section, the modeling of the wood panel itself was studied in depth. From the literature review, the ANSYS help document, and comparisons, the final model gives reliable results for the deflection of a panel subjected to a 50,000 N lateral load. From this, the stiffness of the panel was determined to be 11.31x106 N/m. The addition gives us a total stiffness of about 16.00x106 N/m for this particular case of W250x101 beam and column sections with an 18mm thick shear wall. In the next sections, the input and assumptions made during the structural modeling process are 38  presented to explain the different stiffnesses obtained, but the discrepancies are likely explained by the fact that the connections are assumed to be fixed, which adds stiffness to the hybrid frame. Table 7 : Results of hand calculated stiffness versus modeled stiffness for different frames  Steel frame  Lateral Deflection  Wood panel  Hybrid frame  By hand  ANSYS  By hand  ANSYS  By hand  ANSYS  11.48  12.2  4.17  4.42  3.19  2.09  4.36  4.10  12.00  11.31  16.36  23.92  (Ux [mm])  Stiffness [x106 N/m] Avg.: 4.23  Avg.: 11.66  Diff.: > 30%  Figure 15 : Deflection results in lateral direction for a steel frame  39  Figure 16 : Deflection results in lateral direction for a hybrid frame  4.3 Constraints satisfaction program The theory of the constraint satisfaction program was explained briefly within the literature review. However these following sub-sections will explain the guidelines for designing the proposed program. To reach a certain level of efficiency, one needs to define constraint equations and make sure each of them is satisfied. From a wide range of steel sections, only a few will fit suitably to the loads that are to be undertaken. A narrowing process also had to be performed to first reduce the scripting time and second the computation time. In addition, the narrowing process ensures that the analysis outcome has considered all possible combinations so that no structure that could potentially be optimal would be ignored. 4.3.1  Sections selection  For this first project stage, 24 I-beam steel sections were selected. This is not intended to represent all possible sections, because only I-beam sections are considered whereas hollow sections or C-shaped sections could be a better option. Nonetheless, the variation went further by selecting different suitable 40  sections for each type or component. From knowledge about structural behavior, the beams are governed by flexural failure whereas the columns are selected based on their compression resistance. From the section guidelines provided by the paper written by Yousuf (2009), more precise section selection was made possible by selecting beam and column sections from the moment and compression tables provided by the Handbook of Steel Construction (HSC) (CISC, 2006). The column sections were ordered from the most resistant to the least to allow the section to vary from the first 5 storeys to the upper ones. The script details will be shown in the following section. The rationale behind this decision was based on the larger load carried by lower storeys of a mid-rise building. 4.3.2  Constrained equations  The constraint satisfaction problem gets its foundation from the constrained equations implemented in Matlab. Some Excel spreadsheets checking all types of stresses (tension, compression, moment, shear, combined one, drift and cost) were created. Specific commands convert the typical language of the spreadsheets into Matlab suitable equations to allow for a simple copy-paste process. These spreadsheets are shown in Appendix C and were described partly in Chapter 2, and will be reviewed more extensively in Chapter 6. Although the help of the code requirements shaped the equations, it is required to keep a smart engineering thinking behind. The use of specific references from the Canadian Standards Association was crucial in ensuring that all checks were performed. As far as the wood panels were concerned, the code CSA-O84 (CSA 2005) was the main reference and is shown in Appendix B.  4.4 Nonlinear analysis A non-linear analysis accounting for P-delta effect along with the formation of plastic hinges are needed as the structure studied increases in complexity, referred to as a highly hyperstatic structure. ANSYS uses a function called NLGEOM,ON to perform a nonlinear analysis, where the nonlinearity is defined as occurring at the geometric level. Comparing the analysis from ANSYS to Dr.Frame, which is another tool where non-linear analysis and plastic hinges are implemented, showed that the results were satisfactory. For the 1 bay 1 storey structure used for this comparison, the effects of the nonlinearity on the stresses are small, but consequently with the general belief, the results were slightly higher than the one from a linear analysis. This can be seen from the data compiled in Table 8. 41  Table 8 : Nonlinearity comparison for ANSYS and Dr.Frame programs.  Bending Moment [N*m]  BEAMS stresses from ANSYS 755299 NLGEOM,OFF 757603 NLGEOM,ON BEAMS stresses from Dr.Frame 709403 NLGEOM,OFF 724420 NLGEOM,ON  Shear at node[N] Axial force [N]  Max  Min  461570 461615  -521070 -521992  -115220 -113257  470445 469547  -529554 -530453  -114719 -113491  However, and as will be described later in this report, some convergence issue were encountered with the function “NLGEOM,ON” when dealing with a higher hyperstatic structure. However, although some options of ANSYS will enhance the overall performance for complex structures and help deal with the convergence issue, it lengthens the computation time by a significant amount. A large number of nonlinear model and functions are available for the user to refine the analysis. The consequences when the nonlinearity is turned on will be discussed later. For the sake of greater precision, as is recommended by the program, time steps were assigned to the load pattern to most closely match the true load pattern. 4.4.1  Plastic analysis  At its most basic, plastic behavior of an element means that the stress remains constant as the strains increase. The distribution of the plastic hinges indicates the load path through the structures. Plastic hinges form in all loaded structures when the resistance reaches the elastic limit of the materials. To apply this feature in ANSYS, the materials themselves have to be defined as nonlinear. The formation of plastic hinged is not critical, but being aware of them and taking advantage of them can lead to the most optimized structural design. The plastic analysis will be ignored until further studies are done that deal with dynamic loading patterns for seismic design purposes.  42  5 ANSYS METHODOLOGY & APPLICATION This chapter (and the following one) contains information about how the two main software programs used in this work were programmed to perform as required. These sections will refer to the theory discussed briefly as part of Chapter 2. From the input definition to the possible outputs, the large possibilities are shown to allow for further developments. More coding could be performed to automate the link between the ANSYS-Excel-Matlab software, although it will be shown that only a minimum amount of effort is required by the user to operate these programs in conjunction. As mentioned earlier, ANSYS software is the most suitable software to model efficient and representative hybrid structures. Due to its versatility in creating almost every type of material, the trustable results from hybrid combinations of wood and steel along with their connections are expected to be trustworthy. All parts related to the structure definition, from members properties and material characteristics to the overall structure dimensions and loads were defined in the coding files read by the ANSYS software. The outputs are also explained in the following sections as they are a major part of the success of this software conjunction.  5.1 Input This sub-section will present to the reader the basic functions used in the formatted files, along with the specific files created for this project. Within these files, all required commands are called and organized in such a way that further modifications could be done quickly. Basic controls such as DO and IF exist and are quite useful, for instance when dealing with multiple similar storeys and types of materials. In addition, functions displaying the title along with the view orientation are added to these files to allow for an easy visualization of the generated structure. More complex files are put together as more coding sections are created. From the steel sections generator to the lay-out of the wood panel coded text, most of the subsections were re-used as the project was progressing. 5.1.1  Steel sections generator  Standard steel sections have to be created based on the shapes proposed by the ANSYS library. From a database containing all sections and their properties of interest, loop functions have been created to produce the so called “CISC Steel Sections Generator”. As a first step, one loop through all sections 43  was performed. It could be modified easily if one needed to examine in depth the result of only one beam/column combination, by looping through a constrained number of sections. The coded script is now an integral part of all formatted files using steel section members. Detailed code is shown in Appendix E for the sake of clarity. The input window, shown in Figure 17, displays the properties needed by ANSYS to create standard I-beam sections. Calculations are later automatically carried out by the program to generate the section properties. There are slight differences between the automated results and the CISC standard properties. A difference of less than 2% is negligible for purpose of this study, which is to compare load distributions throughout the structures. See Table 9 for details about the calculated percentages.  Figure 17 : ANSYS input window & section properties  Table 9 : Comparison of property values of W690x548 section  ANSYS calculator values CISC Handbook values  Difference (%)  Area [mm2] Iyy [mm4]  69547 6.69E+09  69900 6.73E+09  0.51% 0.59%  Js [mm4]  7.12E+07  7.07E+07  0.71% 44  Up to now, only I-beams sections have been implemented, but the method developed is not limited to this type of section. For instance, if one wanted to implement hollow sections, the first step would be to create a text file readable by ANSYS. The number of sections is saved under the parameter “nbSect”. This helps indicate to the program how many loops are required to consider with accuracy all combinations. Each step, various sections are attributed to a line and later, meshed into nodes and elements of this type of section. 5.1.2  Automated structure creation  5.1.2.1 Keypoints To allow a future user to quickly change the dimensions of the structure, a totally automated file was written for multistorey structure. The goal is to provide ANSYS with the number of floors (nf) and the number of bays (nb), and then it will generate a structure with logically named keypoints. These define the joints of the structure, specifically the location where the beam and column lines intersect. A logical notation is needed, since the following assignments throughout the project are based on certain pattern. For instance, when diverse wood panel patterns are implemented, it is easily done by referring to numbered bays and storeys. Figure 18 shows a simple pattern of a three storeys by three bays structure where the sequence of the numbering can be seen; the first part on the Z-axis, the second part on the Y-axis and the last part of the keypoint number is on the X-Axis.  45  Figure 18 : Logically numbered keypoints representation  5.1.2.2 Lines and elements Using the logical numbering method, the generation of the lines is not complicated. Some challenging issues were solved by adding a minimum of lines to the code. The required height of possibly 9 and 10 storeys caused problems when looping through the keypoints. The code shown in Appendix E demonstrates the methodology employed for the first storeys up to the 9th one. A different loop was implemented for the 9th storey alone whereas, all the upper storeys up to the 18th one could use the script following the logic shown therein. A special script would be needed if one would want control over the orientation of the column sections. This was done in the simple frame structure by adding a 5th keypoint only for orientation. ANSYS uses this point as a reference and it is called K, the orientation node. It is also important to mention the order of the creation of the lines. For the ease of separating column and beam sections, it was found to be important to create all beams before creating any column lines. This way, when 46  assigning specific sections to be columns or beams, all first line numbers refer to the same category. Denote that the process of assigning a section to a line is in other words to create elements along the line. 5.1.2.3 Areas The areas define the wood panel and are suitable for a SHELL element type assignment. Because they could be created from keypoints as well as from lines, different coded scripts could be used. Although it seems straight forward, it is important to note that the order of the keypoints and/or lines does matter. Depending on the method used, the local axes of the areas will be assigned differently and confusion could occur while processing the stress data. To keep the local axes oriented with the global axes, first the lower 2 keypoints or the lower line should be given to the program, followed by the upper one. 5.1.3  Loop feature  Multiple loop functions are necessary as part of a fully automated formatted file. All chosen steel and wood sections are randomly combined to represent different potential combinations when seeking the optimal one. Three loop assignments (only two are needed if wood panels are excluded) are performed to complete a thorough analysis of possible combinations.  Figure 19: Script from formatted files to run the loops  To ensure a smooth run, prior to terminating each loop a coded script erases all elements and refreshes all keypoints. It was found that this was necessary for good performance of the program. For each loop, the element result tables, called “ETABLE”, are stored. For each structural combination, the relevant ETABLEs presented in Section 2.2.3 are recorded for each selected member element.  47  5.1.4  Wood panel patterns  As far as this project is concerned, the wood panels are attached to the steel members by the “sharing nodes” method. This leads to the creation of “areas” based on keypoints already defined by the steel structure. This eases the programming when one wants to analyze different patterns of wood panels. These could be identified by name representing their most distinctive distribution characteristic such as “Centered”, “Core”, and “Perimeter” braced. The script assigning the wood panel a “Perimeter & Centered” pattern is show in Appendix E. In order to find the optimal distribution of the OSB panels integrated with different structures, various patterns must be defined. The continuation of this work is left for a future researcher. However, note that the line comment character should be added at the beginning of each line containing the ETABLEs command for the wood component if not relevant.  5.2 Methodology First of all, the simple structure presented in the last section was created to learn about the different steps necessary to get the desired outputs. ANSYS software carries out computations quickly due to the fact that only user defined outputs are accounted for. Although this functionality is a good in keeping the runtime to a minimum, it also implies that the user is required to call for every output they might need in the analysis. This partly explains why it was necessary to begin with a simple structure for preliminary modeling and analysis. The simple structure also allows for comparisons with proof calculations that can be done by hand. These hand calculations and discussion about the results were shown in Chapter 4. Additionally, this simple structure (SS) was also modeled and subjected to different loads so the link with the Matlab software could be verified. 5.2.1  Simple structure  As mentioned earlier, one of the most important issues that this project dealt with is the hybrid structure compatibility. The aim was to develop effective connections between the steel frame and the wood frame panel in this type of structure. From the literature exposed in Chapter 2, multiple choices were offered to the developer (COMBIN40, MCP184, ect..). As far as this project is concerned, sharing nodes was used to accelerate the learning process while accurately reproducing one kind of connection. In other words, this represents a full connected frame and thus generates tension and shear fields in the sheathing elements. Some preliminary modeling work (Figure 20) was performed for steel frame with the infill panel offset to match the available space within the steel members. 48  Presented below in Figure 20 are four windows showing from top to bottom and left to right: UX (displacement in X-axis), SX (stresses in X-direction), SY (stresses in Y-direction) and SEQV (Equivalent stresses) of this first studied structure followed by Table 10 with the specific characteristics of the structure itself, namely the steel and wood properties.  Figure 20 : Simple hybrid frame result outputs  49  Table 10 : Structure, steel and wood properties for ANSYS input purposes  Structure properties  Value [unit]  Width of bay Height of column  10 000 [mm] 10 000 [mm]  DOF at supports  One pinned and one fixed  DOF at nodes  Fixed frame  Sheathing connection  Nodes sharing/ Fixed  Offset parameters  Column offset to outside of beam section  Steel section properties  Value [unit]  Section dimensions Young’s modulus (E)  Variable 1 200 000 [MPa]  Density  7.850 E+06 [kg/mm3]  Poisson’s ratio  0.3  Wood OSB section properties  Value [unit]  Section thickness Young’s modulus (EX)  Variable 2 8300 [MPa]  Young’s modulus (EY)  4800 [MPa]  Young’s modulus (EZ)  4800 [MPa]  Poisson’s ratio (NUXY)  0.16  Poisson’s ratio (NUXZ)  0.16  Poisson’s ratio (NUYZ)  0.16  Shear modulus (GXY)  1000 [MPa]  Shear modulus (GXZ)  200 [MPa]3  Shear modulus (GYZ)  200 [MPa]3  Density  0.640 E+06 [kg/mm3]  1  Steel sections are designated through a loop and generated via the “standard sections generator” explained earlier in this chapter.  2  Sheathing panels are designated according to CSA 086 and follow the dimensions for OSB structural panels.  3  Values for GXZ and GYZ are still being debated as introduced in the previous section. Values of 200 MPa were chosen to stay on the conservative side.  50  5.2.2  Multistorey structure  The steps taken to create multi-storey structures are such that a future user and programmer could take on this work where it was left. Only simple changes in the “nb_bay” and “nb_storey” have to be performed and only a few further adjustments have to be done to ensure generation of the output matrices. These adjustments are discussed below: •  As mentioned above, the “nb_loop” and “update” variables indicate to the program the dimensions of the output matrices. Therefore, before running any files it is essential to first ensure that the Loop through sections function is still contained within the input range. Thereafter, the “nb_loop” variable can be modified accordingly.  •  The largest possible number of each type of element should be entered in the output generator commands to ensure the right format of the matrices. This is done by using the FORTRAN language as shown in this example, where 10000 is the number of columns, E means exponential, 17 stands for how many digits per value and 6 defines the number precision.:  •  *MWRITE,result_beam,result_beam70,txt (10000E17.6) Depending in whether the user wants wood sheathing panel according to one of the cases studied, or requires a new pattern for further study, some loops may have to be modified. An example of one type is presented in Appendix E. However, if a steel only structure is used, a line comment character should be added to the textfile at the beginning of all lines containing wood related commands.  •  For comparative studies, a loop could be used that goes through the number of storeys and bays directly. This could be done by adding two more *DO loops through all of the coding.  5.3 Further work ANSYS software is a powerful tool offering many options for design and solution purposes. The work done as part of this project acts as a good head start for further multistorey modeling work. However, more attention to detail in the connections is needed, along with more work on determining the optimal solver and solving options. From the files included on the DVD, referring to their specific text files, a future user could understand their basis, and easily expand those to the desired extent.  51  6 MATLAB METHODOLOGY & APPLICATION The Matlab program follows the work done in ANSYS by implementing the PCSPs previously discussed. The first section of this chapter shows a simple example and aims to introduce the method behind this type of constraint satisfaction program. Further development on the input and output parameters had to be performed to implement the hybrid structure and the multistorey structure and these will be shown subsequently. Here again, note that this work aims to build a foundation for future work on a more complex multistorey hybrid structure. It is in the best interest of prospecting users to consider the work that has been done developing this platform. Therefore, the code formulations and the accompanying explanations will be discussed in Section 6.2.  6.1  Simple example: Bolted assembly (end plate connection)  This section will introduce the Matlab probabilistic constraint satisfaction platform to the reader with the aid of a simple problem. This first example is based on a structural design problem often encountered by engineers- an end plate connection. Some assumptions have been made to simplify the example and focus on the probabilistic issues: •  It is assumed that the weld is excluded from the resistance calculations. However, the cost will include a full length fillet weld.  •  The bolt hole pitch and edge distances are assumed to be constant to a value equivalent to the extreme cases involved in this example.  •  The end plate thickness is chosen to be 8mm.  52  Figure 21 : End plate connection sketches  6.1.1  Problem description  As mentioned before, the first step is to define the variables and transformed the spreadsheet used for calculations to the Matlab language. It is the responsibility of the user to ensure that all constraints and formulas are well documented. The spreadsheet used is shown in Appendix C, and it includes all comments about assumptions made. The design variables are as follows: •  “n” The number of bolt rows in the assembly (i.e. rows of 2, see Figure 21)  •  “d” The diameter of the bolts,  and the probabilistic variables are: •  “Fy” The yield strength of the steel (considering that all the steel in the assembly is the same)  •  “Fu” The tensile strength of the steel (same consideration as Fy)  •  “Vf” The factored shear stress applied to the assembly.  The range of values used and the parameters for the probabilistic variables (along with their distribution type) are displayed in Table 11.  53  Table 11 : Parameters for design and probabilistic variables  Design  n  variables  Intervals  [1,6]  d  Probabilistic  [mm]  variables  [16,32]  Fy  (lognormal)  Fu  Vf  (log-normal)  (Normal)  Mean µ  5.938  6.227  850 kN  St. dev. σ  0.033  0.019  100 kN  The strength parameters have been discussed in Section 3.2 of this report. However the shear load (Vf) is assumed to be normally distributed since at the preliminary step of a design, one usually knows approximately how much load will be applied to the assembly. It is assumed that this load might be lower or higher depending on the assembly location relative to the applied load. The assembly has to be in some way prepared for these uncertainties. The choice of a standard deviation equal to 100kN leading to a coefficient of variation of more than 10%, intends to represent the large uncertainties in the future loading. These design variables are chosen by the user depending on how the optimization is to be done. The present example focuses on the bolt selection, assuming they are of grade A490 with a Fub=1035 MPa. Their size and quantity will be the display on each axis of the outcome. The design is made to fall within the limits establish by CAN/CSA S16-01. The Code clause number has been removed from the spreadsheet for the sake of clarity. The computations will lead to a probabilistic design that respects all structural constraints. Moreover, it could lead to another interesting output- the total cost of the design. To get this output, another constraint is added stating that success occurs if the total cost calculated falls under a specific target cost. For instance, this constraint could be developed according to the budget and the importance of the assembly for structural stability. 6.1.2  Results  The main output is a 2D plot that contains with the probability of meeting the constraints. Three different plots are shown below to illustrate the versatility of the method. Small divisions are used for the two first plots in this example, and the purpose will be explained in the following section. In Figure 54  22, the probability of having a design that would resist the structural loads is plotted. No limit for the cost is enabled in the Matlab formulation. Probability of a resistant design 6  0.9 0.8  5 Number of rows (n)  0.7 0.6  4  0.5 3  0.4 0.3  2 0.2 0.1  1 16  18  20  22 24 26 28 bolt diameter (d) [mm]  30  32  Figure 22 : Design plot without cost target  Figure 23 shows the probability of meeting a $500 cost target. The upper bound of feasible space that represents the cost limit is a sharp line. Above this line, probabilistic variables are no longer applied to the costs due to the assignment of the value 0 to the truth parameter.  55  Probability of a total cost < 500$ 6  0.9 0.8  5 number of rows (n)  0.7 0.6  4  0.5 3  0.4 0.3  2 0.2 0.1  1 16  18  20  22 24 26 28 Bolt diameter (d) [mm]  30  32  Figure 23 : Design plot with $500 cost target  Finally, Figure 24 shows the cost for all possible assemblies. Bigger sizes for the pixels help seeing in the plausible price for a specific combination of “n” and “d”. Total cost of the assembly ($) 6  500 450  number of rows (n)  5  400 350  4 300 250 3 200 150  2  100 50  1 16  18  20 22 24 26 28 Diameter of bolts (d) [mm]  30  32  0  Figure 24 : Total cost of the assembly  56  6.1.3  Discussion  The previous section displayed various output plots produced using the improved Loewen’s method (Loewen, 2009). Sizes of the pixel could be changed by the designer, and different choices could be made depending on the cases. For the first two plots, the pixels are three times more refined than the last one. This can be justified as follows. One could have an allowable success probability ranging between 60 and 100%, depending on the nature of the project and the sensitivity to failure. For this reason, the first two plots display smaller pixels to see all the possible combinations related to the previous probabilistic range. Smaller and smaller pixels would lead to smoother transition of colors, but would need more computation time. The purpose of the last plot was to show the likely cost for each specific assembly, therefore, the same number of boxes as the number of possible combinations was chosen. These outputs could be used for design during the first phase of a structural design project. It is interesting to observe in Figure 22 which assembly resists the applied force regardless of its cost. Budget managing and low cost structures are important issues, so the Figure 23 would probably be more useful since it is possible to enter any cost target in the computations. This example introduced the possibilities of the PCSP to the reader. The fundamentals of the method will be excluded from the next section, and it is assumed that it is well understood already.  6.2 Improvement to the CSP This next section presents the new code added to the original file to accommodate this current project’s particularities. Starting with the steps taken to import the inputs from excel as well as from ANSYS, the following sub-sections will present the excel spreadsheet used to generate the right format to copy-paste into “.m” files. The different types of possible outcomes will be presented along with their special coding requirements. The CSP’s architecture remains accurate so one can refer to Figures 1, 2 and 3 for more information. 6.2.1  Input module  Following the same logic as the simple example, the variables are selected based on their relevancy for design purposes. Three design variables are defined if the structure is hybrid, and the aim is to  57  optimize the wood panel along with the steel members. However, the multistorey structure presented below will only contain two design variables- the beam and the column sections. The probabilistic variables are easily defined for this project, as the steel members are yet again the main design component. Presented in Chapter 5 of this thesis, the wood mechanical properties could also be implemented using this kind of variable, as they are also unpredictable. In addition, for this project the cost of steel members is defined as following a Gaussian probabilistic distribution due to the variation of the cost of steel weight over time and between locations. The variables are defined as follows: % Design variable data [low value, high value, number of boxes] dv1=[0.5,5.5,5]; plotted)  % 5 Steel sections number referring to sections (to be  dv2=[0.5,5.5,5]; dv3=[1,5,5]; plotted)  % Wood panels number referring to 5 thicknesses (to be  comb=dv1(3)*dv2(3)*dv3(3);  % Number of combinations of both materials.  The “comb” parameter is defined at the beginning and will help the entire coding process. Its value is identical to the “nb_loop” defined in the formatted files for the ANSYS program. 6.2.1.1  Combination counter “cc”  Similarly to the “update” parameter set in ANSYS to determine which rows to fill with the specific output values, the CC is used in Matlab to ensure the right output values are read from the result matrices. The CC counter is therefore added following the “end” of all probabilistic variables to ensure the total probability is accurately calculated for each material combination. 6.2.1.2 Input of Geometry Properties From excel spreadsheets built to help organize the steel sections properties that are of interest, textfiles were created. Some of these files were presented before in the ANSYS section of this thesis since they 58  were used to define the same sections in the building process. To simplify the process and prevent the designer from making mistakes in his calculations, the same directory, C:\ANSYS is used. This will ensure that the same sections are being checked by the two programs and there is no variation in the section selection, which would result in unreliable results. As mentioned earlier, sections suitable for beam and column members were selected, and this distinction is kept throughout the entire process. To enhance the performance of the checking system, the properties of each standard section are implemented directly from the “CISC standard sections spreadsheet” A rigorous verification should always be performed prior to importing the .txt files, as these values will not be seen by the user at a later stage. Following the importation process, arrays of zeros are created for each variables with a specific length of dv1(3), although the value of dv2(3) should be the same if the same number of beam and column sections are selected. In addition, the indicator “_b” or “_c” stand for beam or column properties respectively. Due to the variation of the column sections for the 5th storey and up, a distinction will be made with the index “l” for lower and “h” for higher. Then the data from the matrices created in Matlab are assigned to the variables, and as a result each section has its own row in the property arrays.  Figure 25 : Five first beam sections and thicknesses of the web and flanges (tw_b & tf_b)  A similar and slightly simpler process was done for the wood panel properties. Only the useful properties were chosen limiting the matrices to a 5 X 11 size. In Table 1, the original file from CSA O84-09 (2009) for engineering design of wood structural panel was seen. The chosen wood panel types and their properties are shaded grey in the following table.  59  Table 12 : Properties of the five chosen OSB wood panel  2R24 1R24/2F16 2R32/2F16 2R40/2F20 2R48/2F24 1F16 1F20 1F24 1F32 1F48  Nb.  th  1 2 3 4 5 6 7 8 9 10  9.5 11 12 15 18 15 15 18 22 28.5  m_p0 t_p0 t_p90 p_p0 p_p90 v_p 180 240 270 460 630 310 360 480 640 1200  53 60 65 67 92 60 67 77 92 130  18 30 38 48 59 43 48 59 75 110  62 71 77 92 110 87 92 110 140 180  54 54 67 87 94 78 87 94 130 150  42 46 50 55 60 47 54 59 64 85  v_pb0 v_pf0 Ba0 3.8 4.4 4.8 6.1 7.8 5.2 6.1 7.8 9.2 14  0.6 0.6 0.6 0.61 0.65 0.52 0.61 0.65 0.63 0.73  Ba90  44 48 55 66 77 56 56 75 99 108  33 36 36 38 44 36 38 44 55 61  6.2.1.3 Cost properties As part of the material properties, the cost of the steel per unit weight, the cost of the wood panel, and of the joints and connections were implemented in order to define the cost target. As mentioned before, the cost of steel fluctuates therefore the probabilistic features of the program were used to account for this variation. As for the wood panel, the cost was set for each type of panel, depending on their thickness and type. The use of a textfile, called “Cost_panel.txt”, to define the cost of each selected panel thickness was useful for this purpose. The value is then simply extracted from the textfile and assigned to the appropriate panel. 6.2.1.4 Input of ANSYS solicitation results Here again the same method behind the importation process was used. However, the textfiles were produced directly from ANSYS as output matrices rather than being manually created by Excel. Those stresses had to be assigned to the respective variables. For instance, the six ETABLES created for steel sections resulted in the six variables DX, DY, Tf, Cf, Vf and Mf, which are the two deflections, the tension, compression, shear and moment stresses respectively. Due to the existence of two different types of column sections, the matrices created for this purpose have three columns and as many rows as there are combinations. In order from the first to the third column, the assignments are done starting  60  with the lower storey column, then the upper storey column and finally with the beam. A part of the script is shown below for the sake of clarity. % Lower storey columns(5 first one) Dx(:,1)=max(abs(DX_c_l),[],2); Vf(:,1)=max(abs(SZ_c_l),[],2); Cf(:,1)=abs(min((AF_c_l),[],2)); if max(AF_c_l,[],2)>0 Tf(:,1)=max((AF_c_l),[],2); else Tf(:,1)=0; end Dy(:,1)=max(abs(DY_c_l),[],2); Mf(:,1)=max(abs(MZ_c_l),[],2);  6.2.2  Solution module  6.2.2.1 Failure inspection system When dealing with a series of elements for each structural member, single elements or combinations of these elements subjected to different stresses could fail. One could easily predict the failure of a simple connection, especially when hand calculations are performed, however when it comes to a hyperstatic and more complex stress distribution, it is sometimes impossible to tell which component would fail. For the sake of a more robust optimization platform, a failure inspection system was created and implemented in Matlab. It was kept simple, and thus multiples zeros arrays of dimensions “comb X 1” are created before the looping sections. While looping and checking each section of each combination, if the “truth” parameter is assigned the value of 0, then the failure variable (Fail_Bend for instance), would be assigned a value of 1. Figure 26 shows a snapshot from the Workspace window of Matlab. In this example, the number of combinations was set to 9, hence the arrays for all resistance check 61  (bend=bending, Comp=Compression, Cost=Cost target met, Shear=shear, Slend=Slenderness ratio, Tens=Tensions, Drift=drift) had nine columns and one row. Figure 26 shows the case where the bending resistance failed for every combination while all other checks resulted in positive outcomes.  Figure 26 : Snapshot of Failure arrays from Matlab Workspace window  Without the help of the failure inspection system, the output plots of the probability of success would have simply shown that no combinations were satisfying the constraints. This system could be seen as a helpful proof check for coding, as well as being a good indicator of the failure mode of the structure studied. 6.2.2.2 Excel spreadsheet As part of the scope of this project, a CSA O325-07 (2007) standard OSB panel spreadsheet was created to facilitate the input and resistance check of the wood panel used as sheathing. In addition to this, a combination of some existing spreadsheets and some new work resulted in the “CISC standard sections spreadsheet”. The spreadsheet, shown in Appendix C, was written with the first step being to implement the standard wood panel properties in the second tab, called the master sheet. The same macros used in the CISC spreadsheet were used in this spreadsheet as well. It was more of a challenge to verify the resistance of the steel sections mainly due to the complexity of the stresses that they undergo. All beams were checked for bending and shear, whereas the main focus for the columns was their compression and tension resistance. However multiples other combined stresses could cause failure and/or plastification of the members. Therefore, more cases were implemented in the spreadsheet to extend the resistance checks to more cases:  62  •  Section classification class for compression and flexural compression members. (Clause CSA S16 [11.2], Tables 1 and 2)  •  Flexural compression of both beam and columns members. (Clause CSA S16 [13.8])  •  Slenderness ratio for compression and tension members. (Clause CSA S16 [10.4])  6.2.2.3 Constraints implementation From these two spreadsheets introduced above, adequate constraint equations are written to the file, including the design and probabilistic variables. Once again, the method developed by Loewen (2009) to identify whether the statement is a constant “C” or an equation “E” whereby the proper Matlab syntax is generated was useful. Only slight modification is necessary subsequent to the copy-paste process. One example is shown below of the applied modifications that are easily done and understandable by the user.  Figure 27 : Transformation from Excel check to Matlab syntax  6.2.3  Output module  6.2.3.1 Axis identification system To ensure readable outputs that are easy to understand, the section names for both the beam and the column sections were imported from textfiles created previously by the same excel spreadsheet that defined their properties. Thereafter, there were some specific assignments to define the “tick labels” and divide the axes accordingly. Details of this code can be seen in the last section of the “m” files located in the DVD provided along with this thesis.  63  6.2.4  Plotting options  As mentioned in the simple example section, different plots could be created to suit the needs of the designer. Some 3D sliced plots were created to represent the simple hybrid structure using different beam and column sections along with different wood panel thicknesses. These plots could also display probability of success, as well as the total cost of different structural combinations for economical optimization.  6.3 Further work From the two spreadsheet files provided and coming from this project, the implementation of additional resistance checks for wood panels as component of a multistorey structure is left to a subsequent researcher. However, the checks would be similar, if not identical, to the checks performed for the simple hybrid frame. Another improvement for the PCSPs would be to implement probabilistic distributions for the wood mechanical properties. Useful information about this subject can be found in the paper by Karacabeyli et al. (1996). On another subject, the existing connection details would need some resistance/design checks. As far as this project is concerned, the connections are simplified, ensuring no constraint equations for this component.  64  7 HYBRID STRUCTURE STUDY 7.1 Introduction The need for an optimized design method to begin the hybrid structure project is the reason for this study. Steel structures, assumed to be moment resisting frames, remain the base of the hybrid building. Nevertheless, the need for efficient bracing combined with the advantages of a light material directed the choice towards wood based bracing members. A broad range of options were available for this case study. Presently, the PCSPs platform built using Matlab software can handle one type of steel member and one type of wood panel at a time. Although it could be extended, the platform was designed in a way such that steel I beam sections along with different thicknesses and patterns of OSB panels, were implemented. The first simple frame model consisted of one bay of various steel members braced by 5 different thicknesses of wood panels. Furthermore, the multistorey steel structure was automated with ANSYS formatted files and then checked using the probabilistic constraint platforms for all resistances and code limitations. The process went further by implementing a pattern of wood panels to brace a pinned structure. However, the results from this structure have not converged at this point, thus no results from the PCSPs are available.  7.2 Problem description From the initial problem, which was to find a cost optimized and stress resistant combination of steel members and wood panels, some design decisions were made to narrow the options. Some of these were previously discussed in the Chapter 4 of this thesis such as the limitation to use only I-beam steel members throughout this project. The narrowing was done to achieve three specific objectives: •  While looping through steel sections and wood panel thicknesses; Find the most economical and resistant section combination for a simple wood braced frame (i.e., W ???x??? and Wood thickness =?[mm])  •  While looping through steel sections; Refine the resistance checks from Excel to Matlab  •  With different wood panel patterns implemented and while looping through steel sections; Find the most economical combination of steel members while having a constant wood panel thickness and structural dimensions.(i.e., wood thickness, number of bays & number of floor = cst, W ???x??? and wood pattern = ? [mm]) 65  Unfortunately at this stage only one pattern has been implemented into ANSYS. Additionally, the multistorey hybrid structure presents convergence issues, thus no optimization of this structure could be completed. The functionality of the ANSYS-Matlab link is mostly illustrated by the simple frame and the multistorey steel building examples, and these examples can lead to further developments. 7.2.1  Cost study  Based on the program developed in the previous work done, and given a random shear stress with a rear of 45kN at the joints, the bolted connection program built in Matlab was re-run for the sake of better approximations for the cost input. The following plot shows the shear stress distribution in a multistorey steel-only structure. It can be seen that the highest value is not likely to be more than 45 or 50kN. Therefore, a normally distributed variable was created having a mean of 45kN and a standard deviation of 10kN in order to get a good approximation of the total cost of the assembly. Scanning over the design space and seeking conservative results, a total cost of $400 will be used for the beam to column connections.  Figure 28 : Shear forces in multistorey structure  66  Figure 29 : Evaluation of a typical bolted connection cost  In addition, two different companies from a Vancouver material suppliers’ network provided some insight into the pricing of OSB wood panels and steel sections (Curlis Lumber Inc., Russel Metals Inc.). OSB panels are inexpensive on their own compared to how much the installation could cost, depending on the connection types. Therefore a set cost was added per panel along with nine steelwood connections at a price of 30$/unit. Even though the quotes assumed that the wood panels would be ordered in bulk, the prices in Table 13 are shown per unit. Note that there are 9 panels per braced frames due to the dimensions being 7500mm wide per 3600mm high. Table 13 : Cost of wood panels relative to their thickness  Thicknesses [mm]  12.5  15.5  18.5  22  28.5  Price/unit [$]  10.00  14.00  20.00  30.00  40.00  67  In regards to the steel sections, the large variability of the steel cost over different locations and times was taken into account a third probabilistic variable. The mean value was taken as near to the price obtained from AJ Forsyth (Russel Metals Inc.) as 0.924$/kg (7253$/m3 for steel). 7.2.2  Loading  The loading definition differs from the simple to the multistorey structure. Some simplifications were made for the analysis to remain conservative while still obtaining results to demonstrate the link between ANSYS and Matlab. As far as the simple structure is concerned, it was assumed that the loads, both vertical and horizontal, did not have to represent any real pattern. The reason for this is due to the fact that this frame could either stand alone or be an element of any floor on a seven storey by four bays structure. The loads were determined based on educated guesses and are -100N/mm and 70,000 N for the distributed vertical load and the horizontal point load respectively. The first set of output plots (Figures 30, 31 and 32) shown in next section illustrates the efficacy of loading pattern. The multistorey structure however, requires a better defined load pattern. The reference used for the load definition is the BCBC (2006), the British Columbia version of the NBCC (2005). The most challenging load to apply to a structure is the one related to the seismic activity due to the uncertainty. According to the weight of an average size beam and column section based on a 3x3 bay square floor plan, the total weight of such a structure for different heights can be calculated. The simplification of having a uniform weight over the entire structure could be modified if one wants more accurate equivalent static load while looping through the various sections. The Table 13 shows the storey load distribution of the base shear, calculated from the Equation 3 shown below. (BCBC, clause [4.1.8.11.]).  ܸ௕ =  ௌሺ்ೌ ሻ∙ெೡ ∙ூಶ ∙ௐ ோ೏ ∙ோ೚  (5)  As the code recommends, the weight (W) is the self-weight of the structure with 25% of the snow load added. The base shear adjustment (Mv) and te importance factor (IE)factors were assumed to be one, while the ductility factor (Rd)and overstrength factor (Ro) are values from the category “Ductile moment-resisting frames”, found in Table 4.1.8.9. The spectral acceleration for a period Ta was based on values for steel moment frames in the Vancouver area. From the resulting base shear, the distribution over the height of the structure is proportional to the height of each storey. 68  Table 14 : Equivalent static force [kN] for 6-10 storeys building with 3 bays square floor plan  These loads were imported from ANSYS using the function “*TREAD”. A special script was created to distribute the load to the top of each column joint throughout the floor. The specific load was then applied according to the height of the studied structure. As far as the vertical loads are concerned, the excel spreadsheet conceived by Hassan (2010) was used, and for the maximum combination the distributed load was determined to be 18 N/mm for the ANSYS application.  7.3 Results In this section the plots of the results for the simple hybrid frame and the multistorey steel structure are presented. 7.3.1  Simple hybrid frame  From all the possible design spaces available to display, only few are presented in this section. The first is a sliced 3D plot showing the probability of success for different structural combinations for a simple hybrid frame, shown in Figure 30. The X and Y axes show the sections used for the beam and the column elements respectively. On the Z-axis, the wood thickness for all selected panels is shown. It is interesting to see that there are many possible outcomes, corresponding to the number of red pixels on each slice. Logically, the bottom slice for a 28.5 mm thick panel is mostly red as the lateral stiffness is higher. 69  Figure 30 : 3D plot of the probability of success for a simple hybrid frame analysis on Matlab  Note that lists of all the beam and column sections are available in a textfile for clarity. Moreover, it is possible to display only the results for a certain range of sections as the design process converges. One could decide to exclude the column sections that present no hull of solutions, such as W360x147, as shown on Figure 30. If the outcome of the cost does not impact the choice, and referring to the figure above, several combinations would lead to a 100% reliable system. Examples are numerous, such as a column sections of W360x196, a beam section of W410x85, and a 22mm thick panel. Alternatively, the following figure shows the cost analysis for the simple frame. No resistance checks were performed for these plots. This figure could be useful for the designer to get a good idea of the expected cost of the structure using this range of sections.  70  Figure 31 : 3D plot of total cost of a structure for a simple hybrid frame analysis on Matlab  71  Figure 32 : 3D plot of total probability of success for a simple hybrid frame of less than $4000  7.3.2  Multistorey steel structure  Due to the choice of many values for the loading of the steel structure, and to the lack of significance for the hybrid project, no extensive analyses were performed in search of relevant outcomes. It would be irrelevant to spend times on the design of an unbraced steel structure when many are built and designed every day. However, to show the efficient performance of the Matlab platform, Figure 33 and 34 demonstrate possible comparisons using the coded script built for this thesis. Figure 33 first shows the probability of success for a series of structural combinations given that their total cost falls under $250 000. For comparison, Figure 34 shows the probability of success for the same series regardless of the cost for a 7 storey x 4 bay structure. Fewer pixels are colored red due to some of the structures not respecting the total cost. Note that the cost per unit weight of steel was defined as one of the 72  probabilistic variables, which explains the change of color in the probable structural combinations. If the weight was set as a fixed value, the red pixel would become blue, showing a 0% chance of meeting the total cost target. Note that the column sections presented on the Y-axis refer to the lower storey section. Therefore, the graph should be read carefully. For instance, the first pixel in the left bottom corner represents a structure with the lower column section beingW360x287, the upper storey section being W310x313, and the beam section being W530x74.  Figure 33 : Two resulting plots from steel structures with a set cost target  73  Figure 34 : Two resulting plots from steel structures without a set cost target  7.3.3  Hybrid multistorey structure  At this stage of the research, two wood panel patterns have been implemented into ANSYS. The structures did not yield any stress results but the structure models are shown at Figures 34 and 35 to show all the possibilities offer to the user. These patterns are called “Perimeter & Centered” and “Linearly Centered”.  74  Figure 35 : ANSYS modeled structure with wood panel implemented “Perimeter & Centered”  75  Figure 36 : ANSYS modeled structure with wood panel implemented “Linearly Centered”  7.4 Analysis The results obtained from the simple hybrid frame show that the columns are the sections of interest for frame design. In other words, whether the beam sections are changed or not, the probability of success remains almost constant as long as the column section stays the same. Comparatively, for the multistorey steel structure, the inverse is seen - the beam sections are the controlling section. In the 76  analysis presented here, and to achieve convergence in ANSYS, the structure was assumed to be linear. However, the columns were not affected by the P-Delta effect, and thus they were not subjected to a realistic load application. Even though the hybrid multistorey structure was successfully modeled in ANSYS, some convergence issues prevented the production of any output plots. Although it is possible to get results for the stresses in elements, they are not reliable. It was evaluated that using ANSYS to run and analyze such a structure is beyond the scope of this thesis. However, the spreadsheet built remains all that is needed to check the wood panels as part of a multistorey hybrid structure. Moreover, the scripts have already been transformed to their equivalent Matlab language. The modeling and analysis for both structures were performed adequately as shown in the previous subsection. The results for both structures are conclusive and match well with the expected values. The optimization could be done by future users if all required inputs were implemented. The hybrid project is expected to produce further relevant outcomes, and those were reflected in these thesis in a theoretical way.  77  8 CONCLUSIONS From the literature review extensively performed, it has been shown that the hybrid structure can perform very well. However, more studies will have to be performed on the connection types, and if some of the connections presented in this thesis are selected, they should be implemented into a real shear wall for testing. This would allow the designer to properly model them using ANSYS software, resulting in reliable mechanical properties. As far as the programming part of this thesis is concerned, both methods employed in separate software programs proved themselves to be useful and easily done. As was promoted by Loewen (2009), using the PCSPs platform in structural design can result in more efficient decisions by engineers at early stages. More probabilistic variables could be used along with changing design variables to allow for flexibility in design. As well, to learn and develop the formatted files created for ANSYS, the help document along with the existing files are useful for a good understanding. Moreover, the first and main goal stated in the introduction is stated here as a reminder: “..to build a link between ANSYS and Matlab program..”. This was successfully done as demonstrated by the output plots shown in Chapter 7 of this thesis. All specific coded scripts designed to ease the work of prospective users were introduced to the reader. Some additional coded files are available in electronic form along with this thesis. Some preliminary work was also done regarding the proper installation of the wood sheathing panel for a certain chosen pattern, which unfortunately did not converge. As mentioned earlier, the simple frame with I-beams and OSB panels was checked by the probabilistic constraint satisfaction program built in Matlab, and the coding is available and ready for the next user to implement for the multistorey platform. A short list of further research is presented below: •  The convergence issue is something that should be considered. However, the assumption made of sharing nodes in the connections is probably the source of this issue. Therefore, by studying some ideal connection types to allow both materials to work optimally, this issue should be resolved. If it persists, more learning and further refinements to ANSYS would be necessary. 78  •  OSB panel resistance checks could be based on probabilistic values from the tests performed at FPInnovations-Forintek. Mean values and coefficients of variation were given by Karacabeyli et al. (1996). Only the value given in the code to ensure a safe design was implemented here, but since the Matlab software can easily handle a large range of probabilistic distributions, further studies should be done to implement the correct values for more reliable results.  •  The wood panel patterns were briefly studied as part of this project. Although it is not challenging to implement them in ANSYS, it could be difficult to define them in the first place. The convergence issue discussed earlier might be due to some lack of resistance in the hybrid structure as a whole. Therefore, a trial and error approach along with some educated guesses should aid the future researcher considerably.  79  REFERENCES Alpsten, G.A (1972). “Variation in mechanical and cross sectional properties of steel.” National Research Council of Canada, Lehigh University, Bethlehem. Pp775-805. American Institute of Steel Construction. (2005) “Specifications for Structural Steel Buildings.” Chicago, Illinois: American Institute of Steel Construction. British Standard Institution,BSI (2006). “BS EN 300 Oriented strand boards (OSB. Definitions, classification and specifications”. Bartlett, F.M., Hong, H.P., Zhou, W. (2003). “Load factor calibration for the proposed 2005 edition of the National Building Code of Canada: Statistics of loads and load effect”. The University of Western Ontario. pp. 429-439 Branston et al. (2006). “ Light-gauge steel-frame – wood structural panel shear wall design method.” Canadian Journal of Civil Engineering 33, NRC Canada. Canadian Standards Association.(2001). “CAN/CSA S16-01 Limit States Design of Steel Structures.” Canadian Standards Association.(2005). “CAN/CSA S8-06 Timber Design Code.” Canadian Standards Association.(2005). “National Building Code of Canada.” Canadian Standards Association.(1993). “CSA O437 Standards on OSB and Waferboard” Canadian Standards Association.(2009). “CSA O86-09 Engineering design in wood” CISC. (2006). “Handbook of Steel Construction”. Quadratone Graphics Ltd, Toronto, Ontario. Hite, M.C. and Shenton, H.W., III. (2002). “Modeling the non-linear behavior of wood frame shear walls.” Proceedings, 15th ASCE Engineering Mechanics Division Conference, American Society of Civil Engineers (ASCE), New York, New York.  80  Judd, J.P., and Fonseca, F.S. (2005). “Analytical model for sheathing-to-framing connections in wood shear walls and diaphragms.” Journal of Structural Engineering, American Society of Civil Engineering, Vol. 131, No. 2, pp. 345-352. Karacabeyli, E, Lau, P, Henderson, C.R, Meakes, F.V. and Deacon, W. (1996). “Design rated oriented strandboard in CSA standards”. Canadian Journal of Civil Engineering 23. Pp431-443. NRC Canada. Khorasani, Y. (2011). “Feasibility study of hybrid wood steel Structures”. Master thesis, University of British Columbia, Canada.  Lange,J. Naujoks,B. (2006).”Thin-Walled Structures”. 44 ELSEVIER, Technishe Universitat Darmstadt, Germany. Lhomme,O. (1993). “Consistency techniques for numeric CSPs” Proceedings of Thirteenth Joint Conference on Artificial Intelligence, IJCAII 1993, Morgan Kaufmann, San Mateo, CA. pp.232238 Loewen, N. (2009). “Conceptual Design using Probabilistic Interval Constraint Satisfaction”, PhD Thesis, University of British Columbia, Canada. Madenci, E., and Guven, I., (2006) “The Finite Element Method and Applications in Engineering Using Ansys.” University of Arizona, Chapter 7. URL: Metropolis,N. (1987). “THE BEGINNING of the MONTE CARLO METHOD” Los Alamos Science, Special Issue. Pp 125-130 Moore, M. (2000). “Scotia Place – A Case Study of Hugh-Rise Construction Using Wood and Steel.” New Zealand Timber Design Journal. Issue 1 – Volume 10. New Zealand Standard Loadings Code (1992).” NZS 4203” NBCC. 2005.”National Building Code of Canada”. Institute for Research in Construction, National Research Council of Canada. Ottawa, Ont. 81  Ministry of Forests and Range and Ministry responsible for housing.“British Columbia Building Code 2006.” Institute for Research in Construction, National Research Council of Canada. BC. http://www.bccodes.ca Pirayesh, H. (2010). “Design of a Simple Building using NBCC_IBC”. http://www.sigi.ca/engineering/timber_design.html Release 11.0 Documentation for ANSYS. (2007) © SAS IP, Inc. Saliklis, E.P., Mussen, A.L. (2000). “Investigating the buckling behavior of OSB Panels.” Wood and Fiber Science. 32. Pp 259-268. Smith, Edward H. (1998). “Mechanical Engineer’s Reference Book”. 12th Edition. Elsevier. Chapter 7.5, 7.6 and 7.7 Somayaji,S. (2001). “Civil Engineering Materials”. 2nd Edition. California Polythenic State University, San Luis Obispo.Pp 331-348 Stalnaker, J, Harais, E. (1999), “Structural Design in wood”. 2nd Ed. Kluwer Academic Publisherrs, Massachusetts.  Thomas, W.H. (2003). “Poisson’s ratio of an oriented strand board”. Wood Sci Technol 37. Pp259-268. UK Villiard, C. (2009). “Conceptual Design using Probabilistic Interval Constraint Satisfaction”. CIVL 518 course by Dr. Haukaas. UBC. Villiard, C. (2010). “Composite Connections for Hybrid Steel-Timber Structures”. CIVL 510 course by Dr. Stiemer. UBC. Yousuf, M., and Baghchi, A. (2009). “Seismic design and performance evaluation of steel-frame buildings designed using the 2005 National building code of Canada”. Canadian Journal of Civil Engineering 36. Pp280-294. NRC Canada. 82  APPENDICES APPENDIX A : Lognormal distribution for CSP  Figure 37 : Partial set of data with details of their origin.  83  Figure 38 : Lognormal distribution calculations  84  APPENDIX B : Proof calculations  Figure 39 : Nonlinear analysis from Dr.Frame  85  Figure 40 : Stiffness of steel frame calculations  86  APPENDIX C : Excel spreadsheets for PCSPs  Figure 41 : Spreadsheet for bolted connections  87  88  89  Figure 42 : Excel spreadsheet for steel sections check  90  91  Figure 43 : Excel spreadsheet for steel sections check  92  APPENDIX D : Code references Table 15 : CSA O86 table 7.3C for OSB mechanical properties  93  APPENDIX E : ANSYS formatted files  Figure 44 : CICS standard section generator for ANSYS coding  94  Figure 45 : Script from ANSYS to create lines when structures are taller than 9 storeys  95  Figure 46 : Script from ANSYS to install wood panels in a "Centered & Perimeter" pattern  96  

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