Structural Comparison of a Composite and Steel Truss Bridge by Jeffrey Kinlan An Engineering Research Project Submitted to the Graduate Faculty of Rensselaer Polytechnic Institute in Partial Fulfillment of the Requirements for the degree of MASTER OF ENGINEERING IN MECHANICAL ENGINEERING Approved: _________________________________________ Professor Ernesto Gutierrez-Miravete, Engineering Project Adviser Rensselaer Polytechnic Institute Hartford, Connecticut April, 2012 © Copyright 2012 by Jeffrey Kinlan All Rights Reserved ii CONTENTS LIST OF TABLES ............................................................................................................ vi LIST OF FIGURES ........................................................................................................ viii LIST OF EQUATIONS .................................................................................................... ix LIST OF SYMBOLS ........................................................................................................ xi GLOSSARY ................................................................................................................... xiv ABSTRACT .................................................................................................................... xv 1. Introduction/Background ............................................................................................. 1 1.1 Truss Geometry .................................................................................................. 1 1.2 Truss Member Geometry ................................................................................... 2 1.3 Truss Gusset Plate Geometry ............................................................................. 3 2. Materials ...................................................................................................................... 5 2.1 Steel Material Selection ..................................................................................... 5 2.2 Composite Material Selection ............................................................................ 6 3. Composite Laminate Theory ....................................................................................... 9 3.1 CLT Assumptions .............................................................................................. 9 3.2 The ABD Matrix .............................................................................................. 10 3.3 Laminate Stress and Strain ............................................................................... 11 3.4 Laminate Failure Criterion ............................................................................... 12 3.5 Laminate Material Properties ........................................................................... 13 3.6 Member Laminate Layup ................................................................................. 14 4. Truss Loads ................................................................................................................ 16 4.1 Dead Load ........................................................................................................ 16 4.2 Live Load ......................................................................................................... 18 4.3 Dynamic Load .................................................................................................. 19 4.4 Truss Free Body Diagram ................................................................................ 19 5. Analysis Methodology ............................................................................................... 22 iii 5.1 5.2 5.3 2D ANSYS Finite Element Model................................................................... 22 5.1.1 Model Geometry, Element Type, and Mesh ........................................ 22 5.1.2 Material Properties, Loads, and Boundary Conditions ........................ 24 Method of Joints............................................................................................... 25 5.2.1 Naming Convention ............................................................................. 25 5.2.2 Nodal Free Body Diagrams.................................................................. 26 5.2.3 Matrix Equation ................................................................................... 26 3D ANSYS Finite Element Model................................................................... 27 5.3.1 Model Geometry and Mesh.................................................................. 28 5.3.2 Material Properties, Loads, and Boundary Conditions ........................ 29 6. Results........................................................................................................................ 30 6.1 2D ANSYS Finite Element Model Results ...................................................... 30 6.2 Method of Joints Results .................................................................................. 33 6.3 3D ANSYS Finite Element Model Results ...................................................... 34 7. Margin of Safety Calculation..................................................................................... 41 7.1 7.2 Steel Truss Margin of Safety............................................................................ 41 7.1.1 Axial Margin of Safety......................................................................... 41 7.1.2 Transverse Margin of Safety ................................................................ 42 7.1.3 Buckling Margin of Safety ................................................................... 43 Composite Truss Margin of Safety .................................................................. 45 7.2.1 Axial Margin of Safety......................................................................... 49 7.2.2 Transverse Margin of Safety ................................................................ 54 7.2.3 Buckling Margin of Safety ................................................................... 58 8. Conclusion ................................................................................................................. 62 9. References.................................................................................................................. 65 10. Appendix.................................................................................................................... 66 10.1 Appendix A – ANSYS Input File Code ........................................................... 66 iv 10.2 Appendix B – MATLAB Composite Laminate Stiffness Calculator .............. 70 10.3 Appendix C – MATLAB Composite Laminate Global Failure Calculator ..... 76 v LIST OF TABLES Table 1.1 - Truss Dimensions ............................................................................................ 2 Table 1.2 - Member Cross Section Dimensions ................................................................ 3 Table 2.1 - Steel Alloys [6]................................................................................................ 6 Table 2.2 - 5Cr-Mo-V Steel Properties [6] ........................................................................ 6 Table 2.3 - Composite Materials with Epoxy Resin System [3] ....................................... 7 Table 2.4 - 8552 IM7 Material Properties [3].................................................................... 8 Table 3.1 - Laminate Material Properties ........................................................................ 14 Table 3.2 - Candidate Composite Laminates ................................................................... 15 Table 4.1 - Member Weight Variables ............................................................................ 16 Table 4.2 - Gusset Plate Weight Variables ...................................................................... 17 Table 4.3 - Road Deck Weight Variables ........................................................................ 17 Table 4.4 - Vehicle Weight Variables ............................................................................. 18 Table 4.5 - Snow Load Variables .................................................................................... 19 Table 4.6 - Truss Loads ................................................................................................... 21 Table 6.1 - 2D FEM Member Axial Stresses .................................................................. 31 Table 6.2 - 2D FEM Member Axial Forces ..................................................................... 32 Table 6.3 - Method of Joints Forces ................................................................................ 33 Table 6.4 - Method of Joints Axial Member Stresses ..................................................... 34 Table 6.5 - Truss Group 1 3D FEM Transverse Member Stresses .................................. 35 Table 6.6 - Truss Group 2 3D FEM Transverse Member Stresses .................................. 36 Table 6.7 - Truss Group 3 3D FEM Transverse Member Stresses .................................. 36 Table 6.8 - Truss Group 4 3D FEM Transverse Member Stresses .................................. 37 Table 6.9 - Truss Group 1 3D FEM Transverse Member Forces .................................... 38 Table 6.10 - Truss Group 2 3D FEM Transverse Member Forces .................................. 38 Table 6.11 - Truss Group 3 3D FEM Transverse Member Forces .................................. 39 Table 6.12 - Truss Group 4 3D FEM Transverse Member Forces .................................. 40 Table 7.1 - Steel Material Allowable ............................................................................... 41 Table 7.2 - Steel Truss Axial M.S. .................................................................................. 42 Table 7.3 - Steel Truss Transverse M.S. .......................................................................... 43 Table 7.4 - Steel Truss Buckling M.S.............................................................................. 44 vi Table 7.5 - Composite Material Allowable ..................................................................... 45 Table 7.6 - Global Axial Ply Failure Load ...................................................................... 47 Table 7.7 - Global Transverse Ply Failure Load .............................................................. 48 Table 7.8 - Composite Laminate Group 1 Axial M.S...................................................... 50 Table 7.9 - Composite Laminate Group 2 Axial M.S...................................................... 51 Table 7.10 - Composite Laminate Group 3 Axial M.S.................................................... 52 Table 7.11 - Composite Laminate Group 4 Axial M.S.................................................... 53 Table 7.12 - Composite Laminate Minimum Axial M.S. ................................................ 54 Table 7.13 - Composite Laminate Group 1 Transverse M.S. .......................................... 55 Table 7.14 - Composite Laminate Group 2 Transverse M.S. .......................................... 56 Table 7.15 - Composite Laminate Group 3 Transverse M.S. .......................................... 57 Table 7.16 - Composite Laminate Group 4 Transverse M.S. .......................................... 57 Table 7.17 - Composite Laminate Minimum Transverse M.S. ....................................... 58 Table 7.18 - Composite Laminate Critical Buckling Loads ............................................ 59 Table 7.19 - Composite Composite Laminate Buckling M.S. ......................................... 60 Table 7.20 - Composite Laminate Minimum Buckling M.S. .......................................... 61 Table 8.1 - Steel vs. Composite Weight Comparison ...................................................... 64 vii LIST OF FIGURES Figure 1.1 - Warren Truss .................................................................................................. 1 Figure 1.2 - Truss Wireframe Schematic ........................................................................... 2 Figure 1.3 - Truss Member Cross Section ......................................................................... 2 Figure 1.4 - Node Gusset Plate Type ................................................................................. 3 Figure 1.5 - Gusset Plate Type a ........................................................................................ 4 Figure 1.6 - Gusset Plate Type b ....................................................................................... 4 Figure 1.7 - Gusset Plate Type c ........................................................................................ 4 Figure 3.1 - Composite Ply ................................................................................................ 9 Figure 3.2 - Through Thickness Coordinate System [4] ................................................. 10 Figure 3.3 - Tsai Wu Failure Criterion Ellipsoid [4] ....................................................... 13 Figure 4.1 - Dead and Live Truss Load FBD .................................................................. 20 Figure 4.2 - Dynamic Truss Load FBD ........................................................................... 20 Figure 5.1 - 2D ANSYS FEM Geometry ........................................................................ 23 Figure 5.2 - BEAM188 Element [5] ................................................................................ 23 Figure 5.3 - 2D ANSYS FEM Mesh ............................................................................... 24 Figure 5.4 - Member and Node Designation ................................................................... 25 Figure 5.5 - Edge, Top Side, and Internal Node Free Body Diagrams ............................ 26 Figure 5.6 - 3D CATIA Truss Model .............................................................................. 28 Figure 5.7 - 3D ANSYS FEM Mesh ............................................................................... 29 Figure 6.1 - Steel 2D ANSYS FEM Axial Stress Result ................................................. 30 Figure 6.2 - Tension Compression Members .................................................................. 34 Figure 8.1 - Axial M.S. Comparison ............................................................................... 62 Figure 8.2 - Transverse M.S. Comparison ....................................................................... 63 Figure 8.3 - Buckling M.S. Comparison.......................................................................... 63 viii LIST OF EQUATIONS Equation 3.1 - Q Matrix ................................................................................................... 10 Equation 3.2 - Through Thickness Vector ....................................................................... 10 Equation 3.3 - Transformation Matrix ............................................................................. 11 ฬ Matrix .................................................................................................. 11 Equation 3.4 - Q Equation 3.5 - [A] Matrix ................................................................................................ 11 Equation 3.6 - [B] Matrix ................................................................................................ 11 Equation 3.7 - [D] Matrix ................................................................................................ 11 Equation 3.8 - [ABD] Matrix........................................................................................... 11 Equation 3.9 - Laminate Mid-plane Strains ..................................................................... 12 Equation 3.10 - Global Ply Strains .................................................................................. 12 Equation 3.11 - Global Ply Stresses ................................................................................ 12 Equation 3.12 - Local Ply Strains .................................................................................... 12 Equation 3.13 - Local Ply Stresses .................................................................................. 12 Equation 3.14 - Tsai-Wu Failure Criterion ...................................................................... 13 Equation 3.15 - Tsai-Wu Failure Criterion Constants ..................................................... 13 Equation 3.16 - Laminate 0º Tensile Modulus ................................................................ 14 Equation 3.17 - Laminate 90º Tensile Modulus .............................................................. 14 Equation 3.18 - Laminate Through Thickness Tensile Modulus .................................... 14 Equation 3.19 - Laminate In-plane Poisson’s Ratio ........................................................ 14 Equation 3.20 - Laminate xz Poisson’s Ratio .................................................................. 14 Equation 3.21 - Laminate yz Poisson’s Ratio .................................................................. 14 Equation 3.22 - Laminate In-plane Shear modulus ......................................................... 14 Equation 3.23 - Laminate xz Shear Modulus .................................................................. 14 Equation 3.24 - Laminate yz Shear Modulus .................................................................. 14 Equation 4.1 - Member Weight ....................................................................................... 16 Equation 4.2 - Gusset Plate Weight ................................................................................. 16 Equation 4.3 - Road Deck Weight ................................................................................... 17 Equation 4.4 - Vehicle Weight ........................................................................................ 18 Equation 4.5 - Snow Weight ............................................................................................ 18 Equation 4.6 - Wind Drag Force...................................................................................... 19 ix Equation 5.1 - Beam Slenderness Ratio .......................................................................... 23 Equation 5.2 - Nodal Load to Force Relation .................................................................. 26 Equation 5.3 - Coefficient Matrix .................................................................................... 27 Equation 5.4 - Force Vector............................................................................................. 27 Equation 5.5 - Load Vector ............................................................................................. 27 Equation 6.1 - Member Force .......................................................................................... 31 Equation 7.1 - Margin of Safety ...................................................................................... 41 Equation 7.2 - Critical Buckling Load ............................................................................. 44 Equation 7.3 - Tsai-Wu Failure Criterion ........................................................................ 46 x LIST OF SYMBOLS LB – Truss Span [ft] Lm – Truss Member Span [in] LT – Truck Length [ft] Lm – Truss Member Length [in] wB – Truss Width [in] tr – Truss Road Deck Thickness [in] tg – Truss Gusset Plate Thickness [in] tk – kth Ply thickness [in] tp – Ply Thickness [in] A – Cross Sectional Area [in2] Am – Truss Member Cross Sectional Area [in2] Ag – Truss Gusset Plate Cross Sectional Area [in2] g – Weight of Gusset Plate [lb] m – Weight of Truss Member [lb] r – Weight of Road Deck [lb] v – Weight of Vehicles on Bridge [lb] WT – Weight of Heavy Truck [lb] s – Weight of Snow on Bridge [lb] Ps – Snow Design Load [lb/ft2] P – Weight of Vehicle Load, Snow Load, and Road Deck [lb] CD – Coefficient of Drag Fm – Member Force [lb] Pcr – Critical Buckling Load [lb] R – Reaction Force [lb] Fi – Force in Member i [lb] FD –Drag Force [lb] FDm – Truss Member Drag Force [lb] V – Wind Velocity (MPH) [C] – Coefficient Matrix {F} – Force Vector [lb] xi {P} – Load Vector [lb] [S] – Stiffness Matrix [1/psi] [Q] – Reduced Stiffness Matrix [psi] [๐ฬ ] – Transformed Reduced Stiffness Matrix [psi] [T] – Transformation Matrix [ABD] – Laminate Stiffness Matrix [A] – Top Left 3x3 Section of [ABD] Matrix [lb/in] [B] – Bottom Left and Top Right 3x3 Sections of [ABD] Matrix [lb] [D] – Bottom Right 3x3 Section of [ABD] Matrix [lb-in] [abd] – Inverse of Laminate Stiffness Matrix [a] – Top Left 3x3 Section of [abd] Matrix [in/lb] [b] – Bottom Left and Top Right 3x3 Sections of [abd] Matrix [1/lb] [d] – Bottom Right 3x3 Section of [abd] Matrix [1/lb-in] σ – Stress [psi] σm – Member Stress [psi] ε – Strain [in/in] τ – Shear Stress [psi] γ – Shear Strain [in/in] κ – Curvature [1/in] N – Force per unit length of laminate [lb/in] M – Moment per unit length of laminate [in-lb/in] ๏ฎ – Poisson’s Ratio E – Elastic Modulus [psi] G – Shear Modulus [psi] I – Moment of inertia [in4] ρ – Density [lb/in3] {z} – Through Thickness Laminate Coordinate Vector [in] H – Total Laminate Thickness [in] M.S. – Margin of Safety θ – Ply Orientation angle [radians] k – Laminate Ply Index xii t – Tensile c – Compressive u – Ultimate y – Yield f – Failure 1 – Local Axial Direction 2 – Local Transverse Direction 3 – Local Through Thickness Direction x – Global Axial Direction y – Global Transverse Direction z – Global Through Thickness Direction i – Matrix Row Index j – Matrix Column Index 0 – Indicates mid-plane when used as superscript s – Steel cm – Composite Material F1 – Tsai-Wu Failure Constant F2 – Tsai-Wu Failure Constant F11 – Tsai-Wu Failure Constant F22 – Tsai-Wu Failure Constant F66 – Tsai-Wu Failure Constant xiii GLOSSARY CAD – Computer Aided Design ANSYS Workbench and APDL – Finite element program used for structural analysis CATIA – Computer Aided Three-Dimensional Interactive Application FEM – Finite Element Model CLT – Composite Laminate Theory Composite Laminate – Two or more plies stacked in a sequence Symmetric Laminate – A laminate of plies that are symmetric about the mid plane Balanced Laminate – A laminate for which each positive orientation ply has a corresponding negative orientation ply of the same thickness and material. Ply – A single layer of composite material MATLAB – A programming language for technical computing from The MathWorks Cross Ply – A ply with an orientation between 0 and 90 degrees Unidirectional Ply – A ply with fibers running in only one direction 2D – Two Dimensional 3D – Three Dimensional xiv ABSTRACT The main purpose of this project is to design and analyze a truss structure bridge using composite materials that is stronger and lighter than an identical steel bridge. The geometry and loading conditions of the bridge are sized to mimic a real world environment. Every attempt was made to adhere to both state and federal regulations. The layup of the composite material bridge members was optimized to find the laminate that outperforms to the steel material the most. The analysis was performed using the method of joints and an ANSYS finite element model. Mesh studies were performed on all ANSYS finite element models to ensure solution convergence. An identical analysis was completed for the steel truss bridge. A comparison of the strength was made by evaluating the minimum margin of safety in all truss bridge members. To make a fair evaluation both composite material and steel truss bridges have identical geometries. The intent is to compare which material is more efficient when constructing a truss bridge. xv 1. Introduction/Background A bridge is the solution to a puzzle. It solves the common problem of how to span an obstacle through the use of basic engineering principles. This solution comes in many forms and the best is the most efficient, elegant, and safest. One of the more basic types of bridge is a truss structure. This is comprised of a collection of straight members organized in such a way that any load is transferred into the surrounding structure. The members of a truss bridge are connected at gusset plates. Numerous different geometries are possible in a truss bridge but the one to be analyzed in this report is the Warren truss, shown in Figure 1.1. This bridge design was first created by James Warren and Willoughby Monzoni in 1848 and is characterized by alternating equilateral triangles. Figure 1.1 - Warren Truss 1.1 Truss Geometry Dimensions of the Warren truss analyzed in this project were sized per the design standards of the United States Department of Transportation Federal Highway Administration [10]. This designates the minimum clearance of the bridge be no less than 16 feet. To meet this requirement and the requirement that the triangles in the bridge be equilateral the bridge members need to be 18.5 feet long. The design standard [10] also requires an absolute minimum lane width of 11 feet and strongly recommends wider. To be conservative and increase the safety of the bridge the lane width was set at 15 feet. The bridge design is for a two lane roadway which results in a total bridge width of 30 ft. To ensure that a bridge node coincided with the center of the span the number of triangles in the bridge was set to seven. This results in a bridge span of 74 feet. These dimensions are summarized in Table 1.1. 1 Dimension Span Lane Width Bridge Width Member Length Clearance Height Length (ft) 74 15 30 18.5 16 Table 1.1 - Truss Dimensions A wireframe schematic of these dimensions can be seen in Figure 1.2. Figure 1.2 - Truss Wireframe Schematic 1.2 Truss Member Geometry All members of the truss were modeled as long axial tension and compression beams with a hollow square cross section. This cross section type is shown in Figure 1.3. Figure 1.3 - Truss Member Cross Section The wall thickness and side length of the members was chosen to uphold the loads detailed in later chapters. The area and moment of inertia of these members are detailed in Table 1.2. 2 Dimension Wall Thickness (in) Side (in) Area (in2) Izz (in4) Magnitude 2 10 64 725 Table 1.2 - Member Cross Section Dimensions 1.3 Truss Gusset Plate Geometry The truss members are connected at the nodes by gusset plates. These gusset plates transfer the truss loads between the truss members. Each member is sandwiched between two gusset plates. The assembly of two gusset plates and a member is then fastened with by a bolted connection. In the truss there are three different gusset plate geometries. These are labeled gusset plate a, gusset plate b, and gusset plate c. The type of gusset plate at each node is shown in Figure 1.4. b a c c c c b c Figure 1.4 - Node Gusset Plate Type There are two instances of gusset plate a in the truss. Used only at the very bottom edges, gusset plate a is a connection between only three members. This type of gusset plate can be seen in Figure 1.5. 3 a Figure 1.5 - Gusset Plate Type a There are also only two instances of gusset plate b in the truss. This type is used at the top left and right of the truss. Gusset plate b connects four members of the truss. This type of gusset plate can be seen in Figure 1.6. Figure 1.6 - Gusset Plate Type b There are five instances of gusset plate c in the truss. This type is used on the interior nodes of the truss. Gusset plate c connects five members of the truss. This type of gusset plate can be seen in Figure 1.7. Figure 1.7 - Gusset Plate Type c 4 2. Materials Most truss bridges are constructed out of structural steel but wooden truss bridges are not uncommon if the loading is minimal. In bridge designs that utilize these materials the stresses are computed and then compared with a material allowable. If the stress is too high a designer has only two choices. One is to increase cross sectional area and the other is to redesign the geometry of the truss to more evenly distribute load. Each of these choices has unfortunate tradeoffs. Increasing cross sectional area increases weight which adds additional loading the truss has to carry. It can also cause other geometric problems which may violate the design parameters of the bridge. Redesigning the truss geometry adds to the number of connections needed in the truss and possible points of failure. When these two options are not available the designer has no choice but to change material which can lead to the need for larger or smaller cross sectional area. To avoid this problem the designer can choose the only kind of material that gives full customization ability of stiffness and strength without having to change geometry. These materials are composite materials. Composite materials are known for their ability to be tailored to any situation. A composite layup can have almost any strength and/or stiffness in any direction. In addition they often have better strength, stiffness, and corrosion properties as well as lower weight then standard metallic materials. These properties make composite materials ideal to use when designing truss bridge members. The layup of each member can be customized to meet the demands of each individual part without significantly increasing weight or cross sectional area. 2.1 Steel Material Selection Multiple types of steel were considered for the truss members. These included low, intermediate, and high alloy steels that are heat treated and otherwise processed into many different strength levels. These heat treatments and processing methods affect the alloy microstructure and thus the material properties. The material properties most important to the steel selected for the member are the tensile strength, stiffness, and density. These are important for the following reasons. A high tensile strength is important as it determines how much load the members can withstand before breaking. 5 Having a lower stiffness is important in reducing the buckling load of members and improving the ductility of the truss. Low density is important to reduce the overall weight of the truss which is a major factor affecting how much dead load the bridge must hold up. Table 2.1 is a list of all the candidate steel alloys. They are ranked on a ratio of ultimate tensile strength, σtu, to density, ρ. Since a high tensile strength and low density is desired the alloy with the highest ratio of these two properties was selected. The alloy with the highest ratio is 5Cr-Mo-V which is classified as an intermediate steel alloy. This means the amount of alloy elements in the steel are above those classified as low alloy steels but below those in stainless steels. Alloy Type Alloy Name Intermediate Alloy Low Alloy High Alloy High Alloy High Alloy High Alloy Low Alloy Intermediate Alloy High Alloy Low Alloy 5Cr-Mo-V 0.42C 300M 280 Maraging Ferrium S53 AerMet100 280 Maraging 0.40C 300M 5Cr-Mo-V AerMet100 AISI 4340 σtu (ksi) 280 280 280 280 275 275 270 260 262 260 ρ (lb/in3) 0.281 0.283 0.286 0.288 0.285 0.286 0.283 0.281 0.285 0.283 Et (Msi) 30 29 26.5 29.6 28 26.5 29 30 28 29 Ec (Msi) 30 29 28.6 30.7 28.1 28.6 29 30 28.1 29 ๐๐๐ ๐ 996 989 979 972 965 962 954 925 919 919 Table 2.1 - Steel Alloys [6] The full material properties of 5Cr-Mo-V are shown in Table 2.2. Property E (Msi) ๏ฎ G (Msi) σtu (ksi) σty (ksi) σcy (ksi) ρ (lb/in2) Value 30 0.36 11 280 240 -260 0.281 Table 2.2 - 5Cr-Mo-V Steel Properties [6] 2.2 Composite Material Selection Many materials fit the description of a composite material. Examples include concrete, wood, reinforced plastics, and many other materials. The definition of a composite 6 material is the combination of two different substances into one. The generic names for these substances are the fiber and the matrix. The fibers are a very long and thin material that is very strong and stiff in one direction while weak in the others. The matrix is an isotropic glue material that holds the fibers together. In order to achieve the highest and stiffest material properties all the fibers in a composite material are aligned in one direction. This direction is often times the direction of the highest load on the part. As a result composite materials lend themselves well to axially loaded materials. In this regard they make a very good material for truss members. As with other materials strength is the most important property of a composite material for structural applications. Other key properties of the material are density and stiffness. Low density helps alleviate the weight the truss must sustain and low stiffness helps members withstand possible buckling. Hexcel Corporation offers a wide selection of composite material prepreg and specifies their properties on its website [3]. Each uses a different resin system and fiber type. The resins available from Hexcel are Epoxy, BMI (Bismaleimide), Cyanate, and Phenolic. The one most appropriate for use in the composite truss is the Epoxy resin. This is because this type has the highest tensile strength and lowest density. Further selection of a type of Epoxy prepreg is achieved by looking at the individual properties of the materials. Table 2.3 is a list of possible candidate epoxy prepregs offered by Hexcel. This table is ordered by the ratio of tensile strength to density. HexPly Brand 8552 M73 EH04 F515 F593 M76 M74 F155 F263 F161 F185 Fiber Type IM7 IM7 M35J IM6 T2G145 M46J M55J T2C145 T3T145 Glass Fabric Kevlar σ1t (ksi) 395 364 377 243 220 315 319 266 198 66 74 Et (Msi) 23.8 23.5 26.5 21.7 18.3 39.6 52.8 18.3 19.2 3.1 3.5 ρ (lb/in3) 0.0470 0.0466 0.0484 0.0433 0.0441 0.0470 0.0470 0.0482 0.0458 0.0449 0.0465 Table 2.3 - Composite Materials with Epoxy Resin System [3] 7 ๐๐๐ ๐ 8404 7812 7789 5606 4992 6708 6794 5516 4316 1470 1584 The HexPly brand chosen from Table 2.3 for the truss was 8552 IM7 prepreg. This has a very high tensile strength and low density as well as a high stiffness. The full material properties of 8557 IM7 are shown in Table 2.4. Property Value E1 (Msi) 23.8 E2 (Msi) 1.7 E3 (Msi) 1.7 ๏ฎ๏ฑ๏ฒ 0.32 ๏ฎ๏ฑ๏ณ 0.32 ๏ฎ๏ฒ๏ณ 0.0229 G12 (Msi) 0.75 G13 (Msi) 0.75 G23 (Msi) σ1t (ksi) σ1c (ksi) σ2t (ksi) σ2c (ksi) τ12f (ksi) ρ (lb/in2) tp (in) 0.831 395 -245 16.1 -32.3 17.4 0.047 0.006 Table 2.4 - 8552 IM7 Material Properties [3] 8 3. Composite Laminate Theory A composite laminate is made up of two or more plies of composite material. Plies are a combination of fibers and matrix in a very thin sheet of material. In a unidirectional ply of material the fibers all align in the same direction. The angle these fibers make with the longitudinal direction of the entire laminate is referred to as the orientation of a ply. The matrix is a substance that holds the fibers together. The fibers and matrix of a single ply are depicted in Figure 3.1 Matrix Figure 3.1 - Composite Ply Combining plies of varying material, thickness, and orientation can create a laminate with any type of material properties the designer requires. The method to calculate the stress and strain in a laminate and in each individual ply when subject to a load is called Composite Laminate Theory or CLT [4]. 3.1 CLT Assumptions CLT assumes that each ply in a laminate is flawlessly bonded to any adjacent ones. This prevents any layers from slipping relative to each other and allows the laminate to act as a single layer of material when stressed and strained. Many of the assumptions of plate theory are also relevant in CLT. This includes the supposition that plane sections of the laminate remain plane under deformation; meaning lines perpendicular to the mid-plane remain so under deformation. This leads the second assumption that the shear strain perpendicular to the mid-surface, γxz and γyz, is zero. In addition, the stress through the thickness is assumed to be zero. These assumptions are in line with a plane stress condition. The third assumption is that the out of plane strain, εz, is also zero. One of the results of this assumption is that CLT is invalid at edges of laminates. In these regions the inter-laminar stresses are high and CLT becomes less accurate. 9 3.2 The ABD Matrix Before the stress and strain in a laminate can be calculated the material properties of each individual ply must be compiled into the Laminate Stiffness Matrix or [ABD] matrix. The first step in calculating this matrix is to calculate the [Q] matrix for each ply. This is done using equation [3.1]. ๐ธ1 Equation 3.1 - Q Matrix ๐12 ๐ธ2 ๐ธ 1 − ๐12 2 ๐ธ2 1 ๐12 ๐ธ2 [๐] = ๐ธ 1 − ๐12 2 ๐ธ2 1 [ 0 ๐ธ 1 − ๐12 2 ๐ธ2 0 1 ๐ธ2 ๐ธ 1 − ๐12 2 ๐ธ2 1 0 0 [3.1] ๐บ12 ] The [Q] matrix is also called the reduced stiffness matrix because it is the result of applying a plane stress condition, σ3 = 0, to the full stiffness matrix, [S]. The second step is to create a coordinate system through the thickness of the laminate. This coordinate system is input into the {zk} vector. In this vector z is the distance through the thickness of the kth ply. This can be seen in Figure 3.2. Figure 3.2 - Through Thickness Coordinate System [4] The equation used for each value of {zk} is shown in equation [3.2]. Equation 3.2 - Through Thickness Vector ๐ง๐ = − ๐ป + ๐ก๐ 2 [3.2] The third step is the transform the [Q] matrix into the [๐ฬ ] matrix. This is done with the transformation matrix [T]. The transformation matrix is computed for each ply 10 and is determined by a plies orientation angle θ. Equation [3.3] shows how to compute the transformation matrix. Equation 3.3 Transformation Matrix cos2 ๐ [๐] = [ sin2 ๐ − cos ๐ sin ๐ sin2 ๐ cos 2 ๐ cos ๐ sin ๐ 2 cos ๐ sin ๐ −2 cos ๐ sin ๐ ] cos 2 ๐ − sin2 ๐ [3.3] The [๐ฬ ] matrix can then be computed using equation [3.4]. ฬ ] Matrix Equation 3.4 - [๐ [๐ฬ ] = [๐]−1 [๐][๐] [3.4] Finally the fourth step is the computation of the [A], [B], and [D] matrix and compilation of them into the [ABD] matrix. The equations for these matrices can be seen in equation [3.5], [3.6], and [3.7]. These matrices are a compilation of each plies [Q] matrix and position in the laminate. ๐ Equation 3.5 - [A] Matrix ๐ด๐๐ = ∑(๐ฬ ๐๐ )๐ (๐ง๐ − ๐ง๐−1 ) [3.5] ๐=1 ๐ Equation 3.6 - [B] Matrix 1 ๐ต๐๐ = ∑(๐ฬ ๐๐ )๐ (๐ง๐ 2 − ๐ง๐−1 2 ) 2 [3.6] ๐=1 ๐ Equation 3.7 - [D] Matrix 1 ๐ท๐๐ = ∑(๐ฬ ๐๐ ) (๐ง๐ 3 − ๐ง๐−1 3 ) ๐ 3 [3.7] ๐=1 The [A], [B], and [D] matrix are combined into the 6x6 [ABD] in the following manner. The upper left 3x3 section of the [ABD] matrix is made up of the [A] matrix, the lower 3x3 section is made up of the [D] matrix, and the upper right and lower left 3x3 sections are made up by the [B] matrix. This can be seen in equation [3.8]. Equation 3.8 - [ABD] Matrix [๐ด๐ต๐ท] = [๐ด ๐ต ๐ต ] ๐ท [3.8] 3.3 Laminate Stress and Strain Using the laminate stiffness matrix the stresses and strains in the laminate can be calculated. The other pieces of information needed to compute these values are the forces and moments acting on the laminate. The laminate stiffness matrix and force moment vector can be combined to find the mid-plane strains of the laminate as shown in equation [3.9]. 11 Equation 3.9 - Laminate Mid-plane Strains ๐๐ฅ ๐๐ฅ๐ ๐ ๐ ๐๐ฆ ๐ฆ ๐ ๐ ๐ฅ๐ฆ ๐พ๐ฅ๐ฆ = [๐ด๐ต๐ท]−1 ๐ ๐ ๐ฅ ๐ฅ ๐๐ฆ ๐ ๐ฆ {๐ ๐ฅ๐ฆ } {๐๐ฅ๐ฆ } [3.9] From the mid-plane strains the global strain at the top of each ply in the laminate coordinate system can be calculated using equation [3.10]. Equation 3.10 - Global Ply Strains ๐๐ฅ๐ ๐๐ฅ ๐ ๐ฅ ๐ ๐ ๐ ๐ { ๐ฆ } = { ๐ฆ } + ๐ง๐ { ๐ฆ } ๐ ๐พ๐ฅ๐ฆ ๐ ๐ฅ๐ฆ ๐พ๐ฅ๐ฆ ๐ [3.10] Then, using Hooke’s Law the global stress at the top of each ply in the laminate coordinate system can be calculated using equation [3.11]. ๐๐ฅ ๐๐ฅ Equation 3.11 - Global Ply ๐ ๐ ฬ { ๐ฆ } = [๐ ] { ๐ฆ } Stresses ๐๐ฅ๐ฆ ๐ ๐พ๐ฅ๐ฆ ๐ [3.11] To calculate the global stress and strain at the bottom of each ply zk in equation [3.10] should be replaced with zk+1. To calculate the local stress and strain in the ply coordinate system the global stresses and strains are transformed using their transformation matrix. This is shown in equation [3.12] and [3.13]. Equation 3.12 - Local Ply Strains {1 2 Equation 3.13 - Local Ply Stresses ๐1 ๐2 ๐พ12 } = [๐] {1 ๐๐ฅ ๐๐ฆ ๐พ๐ฅ๐ฆ 2 ๐๐ฅ ๐1 ๐ ๐ { 2 } = [๐] { ๐ฆ } ๐๐ฅ๐ฆ ๐ ๐12 ๐ ๐ } [3.12] ๐ [3.13] 3.4 Laminate Failure Criterion Failure in composite materials is unique in that each ply of the laminate fails under a different load. This is due to the varying material properties and orientation of each layer. As a result each individual ply must be checked. This is done by using the TsaiWu failure criterion [4]. The equation for this criterion is shown in equation [3.14]. 12 Equation 3.14 - TsaiWu Failure 2 ๐น1 ๐1 + ๐น2 ๐1 + ๐น11 ๐12 +๐น22 ๐22 +๐น66 ๐12 − √๐น11 ๐น22 ๐1 ๐2 = 1 [3.14] Criterion The constants in equation [3.14] and are calculated using equation [3.15]. In this ๐น equation ๐1๐ , ๐1๐ถ , ๐2๐ , ๐2๐ถ , and ๐12 are properties of the material. 1 1 1 1 ๐น1 = ( ๐ + ๐ถ ) ๐น2 = ( ๐ + ๐ถ ) ๐1 ๐1 ๐2 ๐2 Equation 3.15 - Tsai-Wu 2 Failure Criterion Constants ๐น11 [3.15] 1 1 1 = − ( ๐ ๐ถ ) ๐น22 = − ( ๐ ๐ถ ) ๐น66 = ( ๐น ) ๐12 ๐1 ๐1 ๐2 ๐2 The Tsai-Wu failure criterion for orthotropic materials is similar to the von Mises failure criterion for isotropic materials. They both can be visualized in 3D space. Since unidirectional composite materials are very stiff in the longitudinal direction and weak in the transverse and through thickness direction the Tsai-Wu criterion takes the shape of an ellipsoid. This ellipsoid is shown in Figure 3.3. The stiff longitudinal direction is the 1 direction whereas the weaker transverse and through thickness directions are 2 and 3 respectively. Figure 3.3 - Tsai Wu Failure Criterion Ellipsoid [4] Inside the ellipsoid the ply will not fail but outside the boundary the ply will breakdown. 3.5 Laminate Material Properties The overall material properties of the laminate can be calculated using values from the laminate stiffness matrix, [ABD] matrix. These properties are useful to get a sense of how the laminate as a whole will respond to various forces and moments. The equations for these properties can be seen in Table 3.1 [4]. 13 Laminate Property Variable Equation 3.16 - Laminate 0º Tensile Modulus Ex Equation 3.17 - Laminate 90º Tensile Modulus Ey Equation 3.18 - Laminate Through Thickness Tensile Modulus Ez Equation 3.19 - Laminate In-plane Poisson’s Ratio ๏ฎxy Equation 3.20 - Laminate xz Poisson’s Ratio ๏ฎxz Equation 3.21 - Laminate yz Poisson’s Ratio ๏ฎyz Equation 3.22 - Laminate In-plane Shear modulus Gxy Equation 3.23 - Laminate xz Shear Modulus Gxz Equation 3.24 - Laminate yz Shear Modulus Gyz Equation 1 ๐11 ๐ป 1 ๐22 ๐ป 1 ๐22 ๐ป ๐12 −( ) ๐11 ๐12 −( ) ๐22 ๐12 −( ) ๐22 1 ๐33 ๐ป 1 ๐33 ๐ป ๐ธ๐ฆ 2(1 + ๐๐ฆ๐ง ) Table 3.1 - Laminate Material Properties 3.6 Member Laminate Layup The composite material layup of the members must be optimized to fit the loading conditions on the truss. This is done by varying the number of plies and their orientation to the longitudinal direction of the member. As will be described in the truss loads chapter of this report, each member must be strong enough to withstand all axial and transverse forces. Table 3.2 is a list of the laminates considered for the truss members in this project. This list contains laminates that have varying amounts of cross plies or nonzero orientation. They are broken down into four families. The first is laminates containing 45 and 0 degree plies, the second containing 30 and 0 degree plies, the third containing 90 and 0 degree plies, and the fourth containing a combination of orientations. Indicated next to each laminate is the percent of plies that are considered cross plies. The name of each laminate corresponds to its makeup. For example the [0340] 14 laminate contains 340 zero degree plies and the [3017/-3017/0136]S laminate contains seventeen 30 degree plies, followed by seventeen -30 degree plies, followed by 136 zero degree plies and then the mirrored about the mid-plane. The S denotes it is symmetric about the mid-plane. Table 3.2 - Candidate Composite Laminates It should be noted that the list of laminates includes only balanced and symmetric laminates. This is to avoid any unnecessary bending, twisting, stretching, or shearing that can be caused by having an anti-symmetric or unbalanced laminate layup. Included in the table of candidate layups is the material properties of the entire laminate calculated using the equations in Table 3.1. An analysis of each individual layup will be performed in later sections to determine which one is best. 15 4. Truss Loads There are three types of loads all bridges must withstand. These are a dead load, a live load, and a dynamic load. These three types are treated individually in the following sections. 4.1 Dead Load The dead load on a bridge is weight due to its structure. This is made up of the weight of the truss members, gusset plates, and road deck. These loads never change during the life of the bridge. The weight of one truss member was calculated using equation [4.1]. Equation ๐ = ๐๐ด๐ ๐ฟ๐ 4.1 Member [4.1] Weight There are 15 members per side of the bridge and 9 cross members in the truss for a total of 39 members. The weight from all members in the truss was calculated using the values shown in Table 4.1. Member Variable ρs (lb/in3) ρcm (lb/in3) Am (in2) Lm (in) # Members # Cross Members Magnitude 0.281 0.047 64 222 15 9 Table 4.1 - Member Weight Variables Using equation [4.1] and the values of Table 4.1 the weight of each steel member equals 3,992 lbs while the weight of each composite member equals 668 lbs. All 39 members in the truss add up to a total member weight of 155,705 lbs for the steel truss and 26,043 lbs for the composite truss. The weight of each gusset plate was calculated using equation [4.2]. Equation 4.2 - Gusset Plate ๐ = ๐๐ด๐ ๐ก๐ [4.2] Weight Each node of the truss contains 2 gusset plates. Since there are 9 nodes per side of the truss there are 18 gusset plates per side. This means the entire truss has a total of 36 16 gusset plates. Of the 18 gusset plates per side 4 of them are type a, 4 are type b, and 10 are type c. The weight due to all of these gusset plates was calculated with the values shown in Table 4.2. Gusset Plate Variable ρs (lb/in3) ρcm (lb/in3) tg (in) Ag Type a (in2) Ag Type b (in2) Ag Type c (in2) # of Type a Plates # of Type b Plates # of Type c Plates Magnitude 0.281 0.047 1 500 824 1097 8 8 20 Table 4.2 - Gusset Plate Weight Variables Using equation [4.2] and the values in Table 4.2 the weight of steel gusset plates a, b, and c equal 140 lbs, 231 lbs, and 308 lbs. The weight of the corresponding composite gusset plates equal 23 lbs, 39 lbs, and 52 lbs. All 18 gusset plates in the steel truss weigh a total of 9,140 lbs and all gusset plates in the composite truss weigh 1,529 lbs. The weight of the road deck was calculated using the density of asphalt and the volume of the road as shown in equation [4.3]. The volume of the road deck was calculated using the length and width of the bridge in combination with the thickness of the road deck. Equation 4.3 - Road Deck ๐ = ๐๐ด๐ ๐โ๐๐๐ก ๐ฟ๐ต ๐ค๐ต ๐ก๐ [4.3] Weight The values used to calculate the weight of the road deck are shown in Table 4.3. Road Deck Variable ρasphalt (lb/ft3) LB (ft) wB (ft) tr (ft) Magnitude 45 [7] 74 30 1 Table 4.3 - Road Deck Weight Variables Using equation [4.3] and the values in Table 4.3 the weight of the road deck equals 99,900 lbs. 17 4.2 Live Load The live load on a bridge is weight due to items traveling over the bridge or weights that may temporarily put load on the bridge. This is a combination of the weight of the vehicles using the bridge and the snow that can accumulate on the road deck. These loads change and get redistributed over the life of the bridge. The vehicle weight on the bridge was calculated by assuming each lane of the bridge is packed end to end with the heaviest allowed vehicles. According to the department of transportation the heaviest truck allowed on a highway weighs 80,000 lbs and measures 51 feet in length [11]. The weight from these vehicles is calculated using equation [4.4]. ๐ฟ๐ต ๐ฃ = 2 ( ๐๐ ) ๐ฟ๐ Equation 4.4 Vehicle Weight [4.4] The values used to calculate the weight of the vehicles are shown in Table 4.4. Vehicle Weight Variable LB (ft) LT (ft) WT (lb) Magnitude 74 51 80,000 Table 4.4 - Vehicle Weight Variables Using equation [4.4] and the values in Table 4.4 the weight of the vehicles equals 232,157 lbs. A snow load must be accounted for in case the bridge is covered with snow. This is not an insignificant amount and should not be overlooked when calculating bridge loads. The State of Connecticut building code specifies a minimum snow load all structures must meet. In Connecticut the most stringent snow load is 40 lb/ft2 as found in the Connecticut Building Code [8]. Using this design parameter the snow load on the bridge is calculated using equation [4.5]. Equation 4.5 - Snow ๐ = ๐๐ ๐ฟ๐ต ๐ค๐ต Weight The values used to calculate the weight of the snow are shown in Table 4.5. 18 [4.5] Snow Load Variable PS (lb/ft2) LB (ft) wB (ft) Magnitude 40 74 30 Table 4.5 - Snow Load Variables Using equation [4.5] and the values in Table 4.5 the weight of the snow equals 88,800 lbs. 4.3 Dynamic Load The dynamic load on a bridge is due to temporary loads on a bridge that might perturb the structure momentarily. The most common type of dynamic load is wind load which acts in possibly any direction but most often as against the side faces of the truss. This is generated in the form of a drag force on the truss. The equation to calculate the drag force is equation [4.6]. Equation 4.6 - Wind Drag Force ๐น๐ท = 1 2 ๐๐ ๐ด๐ถ๐ท 2 [4.6] In this equation V is the wind speed. The United States Department of Transportation designates that all structures be able to withstand a 110 MPH wind in the county of Hartford, Connecticut [9]. The A is the area of the item withstanding the wind load. The area of the gusset plates can be found in Table 4.2 and the area of the members is 185 in2. CD is the coefficient of drag [1]. For the gusset plates this is assumed to be a flat plate which has a CD of 2.0 and the members are assumed to be a cylinder with a CD of 1.2. The result of inputting these values into equation [4.6] was that members are subject to a drag force of 48 lb and gusset plates are subject to a drag force of 215 lb, 354 lb, and 472 lb respectively for gusset plate type a, b, and c. 4.4 Truss Free Body Diagram The loads calculated in the previous sections are applied to the truss in the following manner. The dead and live load are distributed evenly over each side of the truss while the dynamic load acts only on only one face of the truss. All loads are assumed to be reacted by the nodes of the truss. Half the weight of each member is distributed between its connecting nodes. The weight of one member is denoted by the variable m. The 19 weight of each gusset plate is reacted at the node to which it is attached and denoted by the variable g. The subscript letter for each g denotes the type of gusset plate weight. The road deck, vehicle, and snow load are denoted by the variable P. P is assumed to be reacted by the lower three middle nodes. The wind drag force on the gusset plates are reacted at each node and denoted by the variable FDa, FDb, and FDc. The wind drag force on each member is distributed evenly between the two nodes it is connected to and denoted by the variable FDm. Finally, the truss itself is assumed to be simply supported. The free body diagram of these dead and live bridge loads on the each half of the truss is shown in Figure 4.1. Ray 2m 2.5m 2.5m 2m 2gb 2gc 2gc 2gb Ri Rax 1.5m 2.5m 2.5m 2.5m 1.5m 2ga 2gc 2gc 2gc 2gc P/3 P/3 P/3 Figure 4.1 - Dead and Live Truss Load FBD The free body diagram of the dynamic loads on the front of the bridge can be seen in Figure 4.2. 1.5FDm FDb 2FDm FDa FDm FDc 2FDm FDc 2FDm FDc 1.5FDm FDc 2FDm FDb FDc 2FDm FDa FDm Figure 4.2 - Dynamic Truss Load FBD The values of all the loads on the steel and composite truss in Figure 4.1 and Figure 4.2 are summarized in Table 4.6. 20 Load P m ga gb gc FDa FDb FDc FDm Steel Truss Weight (lb) 3,992 140 231 308 Composite Truss Weight (lb) 210,428 668 23 39 52 215 354 472 48 Table 4.6 - Truss Loads 21 5. Analysis Methodology Application of the dead, live, and dynamic loads on the truss structure will generate stress and strain in the members. These stresses are distributed throughout the geometry based on the axial and transverse forces that develop. The calculation of member stresses and forces was performed using three methods. The first method was a 2D ANSYS finite element model, the second was by using the method of joints [2] and third was a 3D ANSYS finite element method. Computation of the stresses and forces using these three methods are described in this section. The 2D ANSYS FEM and method of joints were used to calculate the axial stresses in the members under the dead and live load while the 3D ANSYS FEM calculates the transverse member stresses. 5.1 2D ANSYS Finite Element Model As previously described a 2D ANSYS FEM was used to calculate the axial stresses in each truss member. This method breaks down a CAD model of the truss geometry into discrete elements and nodes. The stresses inside these elements are then calculated based on the specific loads and boundary conditions on each individual node. Plotting the stress in all the elements of a FEM gives an overall picture of how the load is distributed throughout the geometry. In this analysis the finite element software that was utilized was ANSYS APDL. The results were axial stresses in each member. To convert the axial stresses into axial member forces the axial stresses are multiplied by the cross sectional area of each member. In the case of the composite truss members the resulting axial forces were input into CLT as global axial forces on the laminate to compute local ply stresses. 5.1.1 Model Geometry, Element Type, and Mesh The CAD model of the truss was generated using lines in ANSYS. The members were modeled as lines and the nodes were modeled as key points. Figure 5.1 shows this CAD model. 22 Figure 5.1 - 2D ANSYS FEM Geometry Once the CAD model was created an element type was selected. BEAM188 was chosen as the most appropriate element for this analysis. Figure 5.2 shows a BEAM188 element. Figure 5.2 - BEAM188 Element [5] BEAM188 elements are used for slender to moderately thick beam structures. This type of element is based on Timoshenko beam theory and is an ideal type of element for bridge truss geometry. The members fit the definition of slender beam structures very well. ANSYS recommends the slenderness ratio, defined in equation [5.1], must be greater than 30 for beams to be adequately modeled using a BEAM188 element type. Equation 5.1 - Beam Slenderness Ratio Slenderness Ratio ๏ฝ GAL2 EI [5.1] The slenderness ratio of each steel truss member is 1594 while each composite truss member is 137. These slenderness ratios are significantly above the recommended minimum. In Timoshenko beam theory the higher the slenderness ratio of a beam, the more accurate the results of the structural analysis. BEAM188 element type also allows the definition of beam cross sectional shape. Possible shapes include quadrilaterals, circles, rings, c-channels, I beams, z shapes, L brackets, T brackets, box shapes, and many others. Since the beams have a hollow rectangular cross section the box shape was chosen. 23 The 2D FEM was meshed using quadrilateral mapped meshing. This resulted in a very nice mesh. The mesh density was refined until it could be shown that adding more elements to the model did not change the resulting solution. This convergence point was reached on all 2D finite element models before results were trusted. Figure 5.3 is a view of the converged mesh. Note the model looks to be 3D because the BEAM188 elements have the graphical ability to show a cross sectional view of the members but in reality acts only in two dimensional space. Figure 5.3 - 2D ANSYS FEM Mesh 5.1.2 Material Properties, Loads, and Boundary Conditions The steel truss material was defined as linear isotropic and the composite truss material was defined as linear orthotropic. The inputs for the isotropic material are elastic modulus and Poisson ratio while the orthotropic material requires elastic modulus, Poisson ratio, and shear modulus in each of the three material directions. Loads were applied to the truss per the free body diagram in the previous chapter. The application point of these loads was at the nodes connecting the beams of the truss. Dead and live loads were applied in the vertical y direction. The result is stress in the axial x direction. The boundary conditions were set such that the truss is simply supported. This means that node A is fixed in space and node I is set as a pinned support. This translates to restraining node A from movement in the x, y, and z direction as well as a rotationally constraining it around the x and y axis. It also sets node I as restrained from movement in the y and z directions and the rotationally restrained are around the x and y axis. 24 5.2 Method of Joints The second method of calculating the axial forces due to the dead and live loads on the truss is the method of joints. Calculating forces and stresses in this manner is a good check that the FEM is performing as it should. In this method a free body diagram is drawn at each joint or node. Each side of the Warren truss in this analysis has a total of nine nodes and fifteen members. Since the truss is simply supported this results in 15 unknown member forces and 3 unknown reaction forces. A sum of forces in the x and y direction at each of the nine nodes results in a total of eighteen equations. The system of eighteen unknowns and eighteen equations can be solved using matrix algebra. The results were axial forces in each member. To convert the axial forces into axial member stresses the axial forces are divided by the cross sectional area of each member. In the case of the composite truss members the resulting axial forces were input into CLT as global axial forces on the laminate to compute local ply stresses. 5.2.1 Naming Convention In order to draw all nine nodal free body diagrams the naming convention, shown in Figure 5.4, was created. All members are assigned a number and all nodes a letter. B D F 4 3 1 8 5 2 A 7 12 9 6 C H 11 13 10 E 15 14 G I Figure 5.4 - Member and Node Designation The convention employed in depicting the free body diagrams at each node was that each member force is drawn positive in tension. Designating this convention is important when interpreting the results. Free body diagrams at nodes also include reactions forces which at node A is a fixed support and at node I is a roller support. 25 5.2.2 Nodal Free Body Diagrams There are three types of connection nodes in a Warren truss. They include edge nodes, upper side nodes, and internal nodes shown in Figure 5.5. Nodes A and I are edge nodes, nodes B and H are top side nodes, and nodes C, D, E, F, and G are internal nodes. The difference between the three types is the amount members connected at each node. Edge nodes connect two axial members, upper side nodes connect three, and internal nodes connect five. A sample of the free body diagram for each type of node is shown in Figure 5.5. Note that the internal node can be flipped about the y axis to represent nodes on the lower side of the truss. y Fii y y Fi Fi Fi x x Fiii P P Fiv x Fiii Fii P Fii Figure 5.5 - Edge, Top Side, and Internal Node Free Body Diagrams 5.2.3 Matrix Equation After all free body diagrams have been drawn, equilibrium equations for the sum of the forces in the x and y directions can be written. This results in a total of 18 equations which can then be combined into matrix form as seen in equation [5.2]. Equation 5.2 - Nodal Load to Force Relation ๏C๏๏ปF๏ฝ ๏ฝ ๏ปP๏ฝ [5.2] In this equation [C] is the coefficient matrix, {F} is the vector of member forces, and {P} is the vector of nodal loads. The coefficient matrix, force vector, and load vector are shown in equations [5.3], [5.4], and [5.5]. Matrix equation [5.2] was solved using Microsoft Excel. 26 Equation 5.3 Coefficient [C] = Matrix Equation 5.4 - Force Vector Equation 5.5 - Load Vector cos(60 sin(60) -cos(60) -sin(60) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 cos(60 -sin(60) -cos(60) sin(60) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 cos(60 sin(60) -cos(60) -sin(60) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 cos(60 -sin(60) -cos(60) sin(60) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 cos(60 sin(60) -cos(60) -sin(60) 0 0 0 0 0 0 {F} = {P} = 0 0 0 0 0 0 0 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 cos(60 -sin(60) -cos(60) sin(60) 0 0 0 0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 Ra1 Ra2 Ri 0 1.5m + 2g a 0 2m + 2g b 0 2.5m + 2g + + P/3 0 2.5m + 2g c 0 2.5m + 2g + + P/3 0 2.5m + 2g c 0 2.5m + 2g + + P/3 0 2m + 2g b 0 1.5m + 2g a 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 cos(60 sin(60) -cos(60) -sin(60) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 cos(60 -sin(60) -cos(60) sin(60) 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 [5.3] [5.4] [5.5] 5.3 3D ANSYS Finite Element Model The method of joints and BEAM188 element used in the 2D FEM cannot calculate transverse forces and stresses. In order to calculate these values a 3D FEM was needed. This method expands upon the previous 2D FEM geometry. In the 2D FEM the CAD model was made up of lines and beam elements. In the 3D FEM this was expanded to rectangular members and solid elements. In this analysis the finite element software used was ANSYS workbench. The results were transverse stresses in each member. To 27 convert the transverse stresses into transverse member forces the transverse stresses are multiplied by the cross sectional area of each member. In the case of the composite truss members the resulting transverse forces were input into CLT as global transverse forces on the laminate to compute local ply stresses. 5.3.1 Model Geometry and Mesh The geometry for this model is more complex than it is for the 2D model so it was created using CATIA V5. In this model the full cross section of each of the members is reproduced. Figure 5.6 shows the CAD geometry of this model. This model is comprised of members and connection solids at the interfaces of the members. Figure 5.6 - 3D CATIA Truss Model The 3D FEM was meshed using bricks elements that resulted in a very fine mesh. Just like in the 2D model the mesh density was refined until it could be shown that adding more elements to the model did not change the resulting solution. This convergence point was reached before results were trusted. Figure 5.7 is a view of the converged mesh. 28 Figure 5.7 - 3D ANSYS FEM Mesh 5.3.2 Material Properties, Loads, and Boundary Conditions The material property inputs for the 3D ANSYS FEM are identical to those of the 2D ANSYS FEM. Steel was defined as a linear isotropic material and the composite material was defined as a linear orthotropic material. The vertical loads on the truss have an effect on the transverse stresses. To account for this effect both vertical and transverse forces were applied to the 3D FEM. Vertical loads were applied to the each side of the truss per the vertical free body diagram in the previous chapter. Transverse loads were applied to only the front of the truss as shown in the transverse free body diagram in the previous chapter. The application point of these loads was at the connection plates of the members of the truss. To get results from this FEM stresses were plotted in the transverse z direction. The boundary conditions were set so the truss is simply supported. In the 3D ANSYS model this meant applying a fixed constraint to the left side of the truss and a zero displacement constraint in the y and z direction to the right side. 29 6. Results The results of applying the analysis methodologies detailed in the previous chapter are presented in the following sections. Results of the steel material truss are presented along with those from each candidate member laminate layup. The results of this study will help choose the ideal laminate layup to most efficiently endure the loading on the truss. 6.1 2D ANSYS Finite Element Model Results The results of the 2D FEM are stresses in each member in the axial direction. The distribution of stresses throughout this model did not vary significantly between the steel and composite models. Members in compression in the steel model were also in compression in the composite models. This was also true for members in tension. The only difference was the magnitude of the stresses in each truss. A color plot of the 2D finite element model axial stress results for the steel truss is shown in Figure 6.1. Figure 6.1 - Steel 2D ANSYS FEM Axial Stress Result The difference in stress magnitude between the models can be seen in the summary of results shown in Table 6.1. ANSYS outputs a minimum and maximum stress for each member. To be conservative the maximum tensile stress of the members in tension and the minimum compressive stress of the members in compression are reported in this table. 30 Table 6.1 - 2D FEM Member Axial Stresses To convert axial member stresses into axial member forces equation [6.1] was used. Equation 6.1 - Member Force ๐น๐ = ๐๐ ๐ด๐ This results in the axial member forces shown in Table 6.2. 31 [6.1] Table 6.2 - 2D FEM Member Axial Forces 32 6.2 Method of Joints Results The outputs from the method of joints are an axial force in each truss members. The inputs are the boundary conditions and loading conditions detailed in previous chapters. The difference between the steel and composite truss member forces are the result of the decreased weight of the composite truss. The loading on the composite truss is lower so the axial force in each member is as well. The results for each material are shown in Table 6.3. Included in this table are the reaction forces at node A and I. Steel Truss Composite Truss Axial Force (lb) Axial Force (lb) -161,838 -128,239 80,919 64,120 152,084 126,608 -156,961 -127,424 -58,853 -43,567 186,387 149,207 46,616 41,520 -209,695 -169,967 46,616 41,520 186,387 149,207 -58,853 -43,567 -156,961 -127,424 152,084 126,608 80,919 64,120 -161,838 -128,239 -146,426 -112,107 0 0 -146,426 -112,107 Force F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 Ry Rx Ri Table 6.3 - Method of Joints Forces As can be seen in Table 6.3 all the composite laminates have identical axial forces. This is because the only properties that affect the outcome are material density and geometric shape. These properties are equal in all of the laminates. The results show that eight of the members are in tension and seven are in compression. This is visualized in Figure 6.2 where red members are in tension and blue members are in compression. 33 4 1 8 3 5 2 7 12 9 6 11 10 13 15 14 Figure 6.2 - Tension Compression Members To convert the axial member forces into axial member stresses equation [6.1] was used. This results in the axial member stresses in Table 6.4. Member 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Steel Truss Composite Truss Axial Stress (psi) Axial Stress (psi) -2,529 -2,004 1,264 1,002 2,376 1,978 -2,453 -1,991 -920 -681 2,912 2,331 728 649 -3,276 -2,656 728 649 2,912 2,331 -920 -681 -2,453 -1,991 2,376 1,978 1,264 1,002 -2,529 -2,004 Table 6.4 - Method of Joints Axial Member Stresses 6.3 3D ANSYS Finite Element Model Results The results of the 3D ANSYS FEM are stresses in each member in the transverse direction. Each member was investigated to find the maximum tensile and minimum compressive transverse stress. The results of all 19 truss models are split up into multiple tables based on model groupings. The first grouping contains the steel, single, and combined ply orientations laminate models is shown in Table 6.5. The second grouping, containing the +/-45 degree cross ply laminates, is shown in Table 6.6. The third grouping, containing the +/-30 degree cross ply laminates, is shown in Table 6.7. 34 Finally, the fourth grouping, containing the 90 degree cross ply laminates, is shown in Table 6.8. Member 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Steel Min Max -2,081 360 -859 1,535 -318 1,875 -1,397 269 -1,230 316 -405 1,248 -491 988 -1,333 320 -488 982 -405 1,248 -1,169 310 -1,373 239 -318 1,878 -879 1,739 -2,028 354 Member Transverse Stress (psi) [3042/-3042/9042/042]S [4542/-4542/9042/042]S [0340] Min Max -405 403 -418 366 -241 235 -123 140 -107 244 -254 159 -185 134 -120 125 -184 133 -255 159 -103 246 -123 142 -243 242 -927 895 -433 673 Min -893 -598 -163 -579 -457 -298 -204 -522 -204 -298 -434 -566 -163 -636 -872 Max 296 705 810 181 291 550 146 230 414 550 290 180 812 863 294 Min -408 -432 -198 -191 -144 -274 -150 -206 -150 -274 -145 -184 -199 -685 -428 Table 6.5 - Truss Group 1 3D FEM Transverse Member Stresses 35 Max 339 415 311 119 192 244 138 146 140 244 194 119 310 799 337 Member 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 [4517/-4517/0136]S Min -551 -466 -191 -208 -174 -243 -151 181 -152 -243 -167 -203 -193 -783 -540 Max 341 656 374 113 217 273 168 116 170 273 219 114 382 992 482 Member Transverse Stress (psi) [4534/-4534/0102]S [4543/-4543/084]S Min -740 -536 -136 -482 -430 -274 -165 -436 -165 -274 -415 -474 -134 -640 -755 Max 257 962 732 126 186 452 308 169 308 453 187 126 733 1,137 256 Min -876 -553 -181 -685 -628 -303 -226 -636 -225 -303 -605 -675 -178 -647 -899 [4551/-4551/068]S Max 202 1,172 997 159 201 609 415 200 415 610 212 150 998 1,257 210 Min -1,033 -622 -238 -917 -869 -328 -293 -873 -292 -329 -837 -906 -234 -738 -1,070 Max 241 1,404 1,311 209 232 793 546 228 544 793 240 181 1,311 1,395 254 Table 6.6 - Truss Group 2 3D FEM Transverse Member Stresses Member [3017/-3017/0136]S 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Min -496 -540 -194 -175 -127 -241 -143 -142 -144 -241 -123 -169 -196 -923 -550 Max 350 659 307 111 199 244 162 94 162 244 202 112 311 1,091 658 Member Transverse Stress (psi) [3034/-3034/0102]S [3043/-3043/084]S [3051/-3051/068]S [3068/-3068/034]S Min -602 -626 -155 -346 -284 -237 -110 -291 -111 -237 -274 -342 -156 -891 -667 Max 303 954 524 113 186 363 201 123 200 364 189 113 524 1,288 576 Min -669 -693 -158 -455 -389 -252 -140 -390 -140 -252 -376 -450 -156 -860 -743 Max 257 1,134 673 130 216 443 250 139 250 445 221 130 673 1,414 513 Min -738 -746 -199 -558 -503 -267 -171 -494 -171 -267 -486 -552 -196 -823 -824 Max 195 1,316 830 149 250 527 305 153 302 529 258 148 830 1,550 441 Table 6.7 - Truss Group 3 3D FEM Transverse Member Stresses 36 Min -1,057 -834 -341 -808 -846 -302 -250 -806 -250 -303 -814 -808 -333 -797 -1,064 Max 369 180 1,293 201 369 764 485 192 478 769 383 199 1,292 1,959 356 Member 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 [9017/0153]S Member Transverse Stress (psi) [9034/0136]S [9043/0127]S [9051/0119]S [9068/0102]S [9085/085]S Min -520 -498 -258 -218 -115 -293 -244 -182 -244 -293 -109 -218 -257 -621 -374 Min -618 -488 -263 -279 -127 -314 -277 -211 -278 -314 -127 -280 -261 -479 -356 Max 452 380 241 173 302 186 182 215 182 185 304 172 245 665 450 Max 476 420 297 232 335 211 235 283 235 210 338 232 303 590 475 Min -660 -474 -266 -303 -133 -321 -291 -232 -290 -321 -133 -304 -263 -478 -352 Max 486 437 324 260 349 231 259 315 259 230 352 259 332 563 485 Min -694 -460 -270 -321 -138 -325 -301 -264 -300 -325 -138 -322 -266 -493 -353 Max 494 452 347 284 360 262 279 343 279 261 363 283 356 543 493 Min -754 -429 -280 -249 -145 -340 -319 -337 -317 -339 -146 -349 -284 -546 -405 Max 575 502 293 333 380 331 316 403 317 330 383 333 404 570 537 Min Max -797 698 -401 543 -321 437 -367 384 -161 398 -392 411 -333 349 -421 466 -330 350 -393 409 -163 400 -367 384 -325 451 -627 670 -474 579 Table 6.8 - Truss Group 4 3D FEM Transverse Member Stresses To convert transverse member stresses into transverse member forces equation [6.1] was used. This results in the transverse member forces shown in Table 6.9, Table 6.10, Table 6.11, and Table 6.12. 37 Member Member Transverse Force (lb) [3042/-3042/9042/042]S [4542/-4542/9042/042]S [0340] Steel Min -133,171 -55,000 -20,323 -89,402 -78,720 -25,912 -31,453 -85,338 -31,215 -25,890 -74,784 -87,866 -20,377 -56,244 -129,792 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Max 23,035 98,214 120,013 17,205 20,226 79,885 63,210 20,454 62,860 79,866 19,839 15,265 120,186 111,277 22,685 Min -25,951 -26,778 -15,407 -7,867 -6,826 -16,265 -11,823 -7,700 -11,773 -16,292 -6,618 -7,844 -15,583 -59,349 -27,724 Max 25,814 23,425 15,021 8,943 15,604 10,195 8,545 7,983 8,533 10,147 15,767 9,085 15,508 57,281 43,053 Min -57,156 -38,301 -10,425 -37,052 -29,235 -19,057 -13,067 -33,385 -13,025 -19,047 -27,798 -36,253 -10,445 -40,730 -55,832 Max 18,938 45,130 51,812 11,590 18,653 35,226 9,322 14,739 26,500 35,216 18,552 11,540 51,962 55,263 18,796 Min -26,112 -27,648 -12,672 -12,224 -9,216 -17,536 -9,600 -13,184 -9,600 -17,536 -9,280 -11,776 -12,736 -43,840 -27,392 Max 21,696 26,560 19,904 7,616 12,288 15,616 8,832 9,344 8,960 15,616 12,416 7,616 19,840 51,136 21,568 Table 6.9 - Truss Group 1 3D FEM Transverse Member Forces Member 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 [4517/-4517/0136]S Member Transverse Force (lb) [4534/-4534/0102]S [4543/-4543/084]S [4551/-4551/068]S Min -35,291 -29,801 -12,208 -13,328 -11,148 -15,556 -9,672 11,597 -9,734 -15,544 -10,687 -12,974 -12,337 -50,115 -34,591 Min -47,363 -34,319 -8,735 -30,872 -27,542 -17,557 -10,547 -27,919 -10,534 -17,556 -26,533 -30,349 -8,596 -40,984 -48,351 Max 21,819 41,965 23,927 7,261 13,888 17,453 10,777 7,397 10,859 17,484 14,022 7,316 24,431 63,460 30,829 Max 16,472 61,558 46,846 8,070 11,935 28,899 19,690 10,823 19,734 28,981 11,985 8,049 46,901 72,762 16,355 Min -56,090 -35,363 -11,599 -43,815 -40,195 -19,370 -14,442 -40,703 -14,391 -19,377 -38,750 -43,206 -11,414 -41,421 -57,553 Max 12,907 74,995 63,800 10,177 12,845 38,999 26,567 12,822 26,586 39,031 13,578 9,594 63,853 80,429 13,430 Min Max -66,131 15,393 -39,820 89,824 -15,244 83,878 -58,671 13,395 -55,590 14,852 -21,014 50,723 -18,745 34,913 -55,864 14,597 -18,659 34,806 -21,025 50,734 -53,554 15,355 -57,983 11,575 -14,964 83,904 -47,204 89,254 -68,474 16,261 Table 6.10 - Truss Group 2 3D FEM Transverse Member Forces 38 Table 6.11 - Truss Group 3 3D FEM Transverse Member Forces 39 Table 6.12 - Truss Group 4 3D FEM Transverse Member Forces 40 7. Margin of Safety Calculation The ultimate goal of using composite materials in place of steel for the construction of a truss bridge is achieving a lower weight structure that has higher strength. Strength can be quantified by calculating a margin of safety in each truss member. Margin of safety is defined as in equation [7.1]. Equation 7.1 - Margin of Safety ๐. ๐. = ๐ด๐๐๐๐ค๐๐๐๐ ๐๐ก๐๐๐๐๐กโ −1 ๐ด๐๐ก๐ข๐๐ ๐๐ก๐๐๐๐๐กโ [7.1] When the M.S. reaches a value less than or equal to zero the structure will fail. An alternate definition is that it is equal to the factor of safety minus one. This factor is used to compare the steel and composite material trusses. This is done by computing the M.S in every member of every truss model for both the axial and transverse directions. Then a comparison between the lowest axial and transverse M.S. in each truss is made. The truss with the highest M.S. is deemed the strongest and the one with the lowest is ranked as the weakest. This results in a simple quantitative parameter to compare between each truss model. 7.1 Steel Truss Margin of Safety The intermediate alloy 5Cr-Mo-V steel is an isotropic material. This means it has the same strength in every direction. The material allowable for this strength only varies with the type of stress applied. Table 7.1 details the material allowable for 5Cr-Mo-V steel in compression and tension. 5Cr-Mo-V σty σcy Allowable (ksi) 240 -260 Table 7.1 - Steel Material Allowable It should be noted that this material is stronger in compression then it is in tension. 7.1.1 Axial Margin of Safety The axial M.S. of the steel truss was calculated using the results of the 2D ANSYS finite element model and method of joints. The maximum axial stress in each member as computed by these methods was compared to the appropriate tension or compression 41 material allowable. The member stresses used in this calculation can found in the previous chapter. The resulting axial M.S. for each method is shown in Table 7.2. Member 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 MOJ 2D ANSYS Axial M.S. 102 93 189 138 100 96 105 88 282 215 81 74 329 275 78 73 329 275 81 74 282 215 105 88 100 96 189 138 102 93 Table 7.2 - Steel Truss Axial M.S. An example of the M.S. as calculated for member 1 from the method of joints result is shown below. As can be seen in this example calculation since the stress is compressive the compressive material allowable is used. ๐. ๐. = ๐ด๐๐๐๐ค๐๐๐๐ ๐๐ก๐๐๐๐๐กโ −260,000[๐๐ ๐] −1= − 1 = 102 ๐ด๐๐ก๐ข๐๐ ๐๐ก๐๐๐๐๐กโ −2,529[๐๐ ๐] Table 7.2 shows that the lowest M.S. occurs in member 8. In the method of joints this M.S. is 78 and is 73 in the 2D ANSYS finite element model. Member 8 is the top center member of the truss. 7.1.2 Transverse Margin of Safety The transverse M.S. of the steel truss is calculated using the results of the 3D ANSYS finite element model. These results are in the form of a maximum and minimum transverse stress in each member. Since the maximum stress is always a tensile stress and the minimum stress is always a compressive stress the maximum stress is compared to the tensile material allowable and the minimum stress is compared to the compressive material allowable. This results in a M.S. for both tension and compression. The lower 42 of these two results is treated as the M.S. for the member. The member stresses used in this calculation can found in the previous chapter. The minimum transverse M.S. in each member is presented in Table 7.3. Member 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Transverse M.S. 124 155 127 185 210 191 242 194 243 191 222 188 127 137 127 Table 7.3 - Steel Truss Transverse M.S. An example of the M.S. as calculated for member 1 from the 3D FEM results is shown below. ๐๐๐๐ ๐๐๐ ๐. ๐. = ๐ด๐๐๐๐ค๐๐๐๐ ๐๐๐๐ ๐๐๐ ๐๐ก๐๐๐๐๐กโ 240,000[๐๐ ๐] −1= − 1 = 666 ๐ด๐๐ก๐ข๐๐ ๐๐๐๐ ๐๐๐ ๐๐ก๐๐๐๐๐กโ 360[๐๐ ๐] ๐ถ๐๐๐๐๐๐ ๐ ๐๐ฃ๐ ๐. ๐. = ๐ด๐๐๐๐ค๐๐๐๐ ๐ถ๐๐๐๐๐๐ ๐ ๐๐ฃ๐ ๐๐ก๐๐๐๐๐กโ −1 ๐ด๐๐ก๐ข๐๐ ๐ถ๐๐๐๐๐๐ ๐ ๐๐ฃ๐ ๐๐ก๐๐๐๐๐กโ ๐ถ๐๐๐๐๐๐ ๐ ๐๐ฃ๐ ๐. ๐. = −260,000[๐๐ ๐] − 1 = 124 −2,081[๐๐ ๐] Since the compressive M.S. is much lower than the tensile this is the minimum M.S. of the member. Table 7.3 shows that the lowest margin is 124 in the first member of the truss which is the left most diagonal member. 7.1.3 Buckling Margin of Safety A potential failure mode of the truss members in addition to yielding is axial buckling of the members in compression. To check members for buckling the axial force in each is 43 compared with the critical buckling load. The equation for the critical buckling load is shown in equation [7.2]. Equation 7.2 - Critical Buckling Load ๐๐๐ = ๐ 2 ๐ธ๐ผ ๐ฟ2 [7.2] This is the buckling load for a pinned-pinned connection of an axially loaded compression member. If the force in any compression member exceeds this value the member will buckle. The buckling load of each steel member is 4.36 million pounds because each member has the same length, stiffness, and moment of inertia. The forces calculated in the method of joints are used as the actual strength in the M.S. calculation for buckling. The buckling M.S. for each member is shown in Table 7.4. Members in tension are N/A since they are not subject to buckling. Member 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Buckling M.S. 26 N/A N/A 27 73 N/A N/A 20 N/A N/A 73 27 N/A N/A 26 Table 7.4 - Steel Truss Buckling M.S. An example of the buckling M.S. as calculated for member 1 from the method of joints results is shown below. ๐๐๐๐ ๐๐๐ ๐. ๐. = ๐ด๐๐๐๐ค๐๐๐๐ ๐๐๐๐ ๐๐๐ ๐๐ก๐๐๐๐๐กโ 4,357,653[๐๐] −1= − 1 = 26 ๐ด๐๐ก๐ข๐๐ ๐๐๐๐ ๐๐๐ ๐๐ก๐๐๐๐๐กโ 161838[๐๐] Table 7.4 shows that the lowest buckling M.S. is 20 in member 8 which is the top center member of the truss. 44 7.2 Composite Truss Margin of Safety The carbon fiber epoxy IM7/8552 composite material is an orthotropic material. This means it has a different strength in each material direction. This is because all the fibers are aligned in one direction which results in a high strength in the fiber direction and a very weak strength in the transverse matrix direction. This can be seen in the material allowable for the IM7/8552 carbon fiber epoxy shown in Table 7.5. 8552/IM7 Carbon Fiber Epoxy σ1t σ1c σ2t σ2c Allowable (ksi) 395 -245 16.1 -32.3 Table 7.5 - Composite Material Allowable The difference in strength between the axial and transverse direction of this material is the reason plies are stacked up in a laminate at varying orientations. Varying the orientation of some of the plies in a laminate is a way to strengthen the laminate in the desired orientation angle direction. This allows for the ability to customize the strength of a laminate in any direction. Determining failure in a composite material laminate requires checking every ply for failure. This is done by employing classical laminate theory and Tsai-Wu failure criterion. The process is to first find the global axial and global transverse force that will cause failure in each ply of the laminate. This is done by separately applying an axial and transverse unit force per length of laminate to the laminate. The length of laminate used is measured as the length of the laminate in the direction perpendicular to the force. In the case of the axial direction this is the circumference of the box member which is equal to 40 inches. In the case of the transverse direction this is equal to 222 inches which is the length of each member. The resulting local stresses, from the unit forces, in each ply are calculated using CLT. These local stresses are inserted into the Tsai-Wu failure criterion equation along with the Tsai-Wu constants. The Tsai-Wu failure criterion equation and the Tsai-Wu criterion constants can be found in the CLT chapter. The equation is repeated here in equation [7.3]. 45 Equation 7.3 - TsaiWu Failure Criterion 2 ๐น1 ๐1 + ๐น2 ๐1 + ๐น11 ๐12 +๐น22 ๐22 +๐น66 ๐12 − √๐น11 ๐น22 ๐1 ๐2 = 1 [7.3] To find the force to cause failure in each ply the unit force on the laminate is increased until the term on the left hand side of the Tsai-Wu equation is equal to 1. When this occurs the failure load has been found. Due to the quadratic nature of the Tsai-Wu failure criterion there is both a tensile and compressive unit force which will cause failure. The process is repeated until the tensile and compressive failure load has been found in each ply of the laminate for both the axial and transverse directions. This procedure was followed for each of the candidate laminates considered in this project. The resulting axial failure loads are shown in Table 7.6 and the resulting transverse failure loads are shown in Table 7.7 for each ply of each laminate. 46 Laminate [0340] [4517/-4517/0136]S [4534/-4534/0102]S [4543/-4543/084]S [4551/-4551/068]S [3017/-3017/0136]S [3034/-3034/0102]S [3043/-3043/084]S [3051/-3051/068]S [3068/-3068/034]S [9017/0153]S [9034/0136]S [9043/0127]S [9051/0119]S [9068/0102]S [9085/085]S [3042/-3042/9042/042]S [4542/-4542/9042/042]S Ply Orientation 0 45 -45 0 45 -45 0 45 -45 0 45 -45 0 30 -30 0 30 -30 0 30 -30 0 30 -30 0 30 -30 0 90 0 90 0 90 0 90 0 90 0 90 0 30 -30 90 0 45 -45 90 0 Axial Tension Failure Load (lb) 32,232,000 21,776,300 21,776,300 24,602,600 16,594,100 16,594,100 18,504,400 14,026,000 14,026,000 15,579,100 11,807,300 11,807,300 13,080,400 25,843,600 25,843,600 25,205,800 19,909,100 19,909,100 19,135,900 17,154,900 17,154,900 16,427,100 14,912,100 14,912,100 14,248,600 10,657,100 10,657,100 10,156,800 16,208,400 30,165,900 14,782,300 27,126,400 13,949,700 25,485,600 13,189,200 24,024,300 11,536,800 20,919,200 9,856,600 17,818,000 16,894,600 16,894,600 8,887,400 17,103,000 10,421,300 10,421,300 6,332,600 12,186,500 Table 7.6 - Global Axial Ply Failure Load 47 Compression Failure Load (lb) -19,992,000 -24,045,000 -24,045,000 -13,953,500 -17,819,700 -17,819,700 -10,216,600 -14,931,600 -14,931,600 -8,536,100 -12,496,100 -12,496,100 -7,131,400 -18,785,300 -18,785,300 -13,923,900 -13,647,700 -13,647,700 -10,094,400 -11,544,100 -11,544,100 -8,549,100 -9,911,700 -9,911,700 -7,351,000 -6,955,000 -6,955,000 -5,175,500 -32,862,100 -20,559,200 -29,862,000 -19,068,100 -28,144,000 -18,079,300 -26,585,800 -17,146,100 -23,222,200 -15,065,200 -19,820,700 -12,908,600 -13,938,400 -13,938,400 -18,071,700 -10,607,900 -12,353,000 -12,353,000 -12,876,800 -7,558,500 Laminate [0340] [4517/-4517/0136]S [4534/-4534/0102]S [4543/-4543/084]S [4551/-4551/068]S [3017/-3017/0136]S [3034/-3034/0102]S [3043/-3043/084]S [3051/-3051/068]S [3068/-3068/034]S [9017/0153]S [9034/0136]S [9043/0127]S [9051/0119]S [9068/0102]S [9085/085]S [3042/-3042/9042/042]S [4542/-4542/9042/042]S Transverse Ply Tension Failure Orientation Load (lb) 0 7,291,400 45 17,564,200 -45 17,564,200 0 11,538,700 45 23,172,800 -45 23,172,800 0 14,798,300 45 25,390,700 -45 25,390,700 0 15,925,200 45 26,676,100 -45 26,676,100 0 16,434,200 30 10,163,600 -30 10,163,600 0 8,203,400 30 10,928,000 -30 10,928,000 0 8,740,600 30 11,110,000 -30 11,110,000 0 8,839,100 30 11,115,500 -30 11,115,500 0 8,799,200 30 10,529,700 -30 10,529,700 0 8,238,900 90 30,225,800 0 16,824,400 90 47,372,000 0 26,341,000 90 56,453,700 0 31,370,400 90 64,529,200 0 35,834,600 90 81,700,400 0 45,295,000 90 98,889,800 0 54,704,300 30 38,684,200 -30 38,684,200 90 57,414,600 0 30,486,700 45 57,838,000 -45 57,838,000 90 67,635,100 0 35,145,900 Table 7.7 - Global Transverse Ply Failure Load 48 Compression Failure Load (lb) -14,628,000 -22,289,600 -22,289,600 -23,288,600 -28,796,300 -28,796,300 -30,030,600 -31,073,600 -31,073,600 -32,382,100 -32,091,800 -32,091,800 -33,438,200 -16,949,000 -16,949,000 -16,529,500 -18,206,700 -18,206,700 -17,683,800 -18,491,000 -18,491,000 -17,917,500 -18,475,500 -18,475,500 -17,863,100 -17,420,600 -17,420,600 -16,760,100 -22,158,000 -33,763,000 -34,659,600 -52,879,200 -41,253,500 -62,989,400 -47,096,800 -71,968,700 -59,440,900 -91,018,500 -71,642,700 -110,005,100 -64,201,500 -64,201,500 -38,022,600 -61,953,800 -68,558,900 -68,558,900 -41,949,800 -71,466,100 The failure loads were calculated using the MATLAB code in appendix C. It should be noted that plies with equivalent orientation, material, and thickness in different laminates will have different failure loads. This is because the failure load of a ply is more determined by its place in the laminate and by the material properties of the surrounding plies. This is because the stiffness of each laminate is different resulting in the load on the laminate flowing through it in a different fashion. 7.2.1 Axial Margin of Safety Using the failure loads in Table 7.6 as the material allowable for each ply and the forces calculated using the 2D ANSY FEM, in the previous chapter, an axial M.S. can be calculated. The following tables present the minimum M.S. each ply of each member. The results of all 18 laminates are split up into multiple tables based on model groupings. The first grouping contains the single and +/-45 degree cross ply laminates. The results of which are shown in Table 7.8. The second grouping contains the +/-30 degree cross ply laminates and the results are shown in Table 7.9. The third grouping contains the 90 degree cross ply laminates and the results are shown in Table 7.10. Finally, the forth grouping contains the combined orientation laminates and the results are shown in Table 7.11. 49 -45 172 252 168 161 418 133 456 134 456 133 418 161 168 252 172 0 100 284 190 93 242 151 516 77 516 151 242 93 190 284 100 45 127 191 127 119 308 101 347 99 347 101 308 119 127 191 127 -45 127 191 127 119 308 101 347 99 347 101 308 119 127 191 127 0 72 213 142 68 176 113 387 56 387 113 176 68 142 213 72 45 106 161 108 99 257 85 293 83 293 85 257 99 108 161 106 -45 106 161 108 99 257 85 293 83 293 85 257 99 108 161 106 [4551/-4551/068]S 45 172 252 168 161 418 133 456 134 456 133 418 161 168 252 172 [4543/-4543/084]S 0 233 379 249 135 354 198 679 111 679 198 354 135 249 379 233 [4534/-4534/0102]S [4517/-4517/0136]S 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 [0340] Member 0 60 179 120 56 147 95 326 47 326 95 147 56 120 179 60 Table 7.8 - Composite Laminate Group 1 Axial M.S. 50 45 89 135 90 83 215 72 247 69 247 72 215 83 90 135 89 -45 89 135 90 83 215 72 247 69 247 72 215 83 90 135 89 0 50 150 100 47 122 80 273 39 273 80 122 47 100 150 50 30 97 229 153 91 236 122 417 75 417 122 236 91 153 229 97 -30 97 229 153 91 236 122 417 75 417 122 236 91 153 229 97 0 72 220 147 67 174 117 401 56 401 117 174 67 147 220 72 30 82 197 132 77 199 105 359 64 359 105 199 77 132 197 82 -30 82 197 132 77 199 105 359 64 359 105 199 77 132 197 82 0 60 189 126 56 147 100 344 47 344 100 147 56 126 189 60 30 70 171 114 66 171 91 312 54 312 91 171 66 114 171 70 -30 70 171 114 66 171 91 312 54 312 91 171 66 114 171 70 Table 7.9 - Composite Laminate Group 2 Axial M.S. 51 [3068/-3068/034]S 0 99 292 194 93 242 154 529 77 529 154 242 93 194 292 99 [3051/-3051/068]S -30 135 299 199 126 327 158 542 104 542 158 327 126 199 299 135 [3043/-3043/084]S 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 30 135 299 199 126 327 158 542 104 542 158 327 126 199 299 135 [3034/-3034/0102]S [3017/-3017/0136]S Member 0 52 163 109 48 126 87 298 40 298 87 126 48 109 163 52 30 49 102 81 46 119 65 222 38 222 65 119 46 81 102 49 -30 0 49 36 102 97 81 78 46 34 119 88 65 62 222 212 38 28 222 212 65 62 119 88 46 34 81 78 102 97 49 36 90 203 163 107 190 496 85 293 157 293 85 496 190 107 163 203 0 130 298 197 122 318 156 536 100 536 156 318 122 197 298 130 90 192 153 101 179 468 80 277 148 277 80 468 179 101 153 192 0 123 280 185 115 301 147 505 95 505 147 301 115 185 280 123 90 167 134 89 156 407 70 242 129 242 70 407 156 89 134 167 Table 7.10 - Composite Laminate Group 3 Axial M.S. 52 [9085/085]S 0 137 317 210 129 336 166 571 106 571 166 336 129 210 317 137 [9068/0102]S 90 216 173 114 202 527 90 311 166 311 90 527 202 114 173 216 [9051/0119]S 0 148 354 233 139 364 185 635 114 635 185 364 139 233 354 148 [9043/0127]S 90 238 190 125 222 582 99 341 183 341 99 582 222 125 190 238 [9034/0136]S 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 [9017/0153]S Member 0 108 243 161 101 264 128 439 83 439 128 264 101 161 243 108 90 142 114 75 133 347 60 206 110 206 60 347 133 75 114 142 0 91 203 136 85 221 108 369 71 369 108 221 85 136 203 91 Member 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 [3042/-3042/9042/042]S [4542/-4542/9042/042]S 30 98 192 129 92 238 102 350 76 350 102 238 92 129 192 98 45 87 118 79 81 211 63 216 68 216 63 211 81 79 118 87 -30 98 192 129 92 238 102 350 76 350 102 238 92 129 192 98 90 128 101 67 119 309 53 184 99 184 53 309 119 67 101 128 0 75 195 130 70 181 103 355 58 355 103 181 70 130 195 75 -45 87 118 79 81 211 63 216 68 216 63 211 81 79 118 87 90 91 71 48 85 220 38 131 70 131 38 220 85 48 71 91 0 53 138 92 49 128 73 252 41 252 73 128 49 92 138 53 Table 7.11 - Composite Laminate Group 4 Axial M.S. An example calculation of the M.S. in the 0 degree ply of the first member in the [0 340]S laminate model is shown below. In this example the load is tensile so the tension allowable is used as the allowable strength. ๐. ๐. = ๐ด๐๐๐๐ค๐๐๐๐ ๐๐ก๐๐๐๐๐กโ 32,232,000[๐๐] −1= − 1 = 233 ๐ด๐๐ก๐ข๐๐ ๐๐ก๐๐๐๐๐กโ 137,501[๐๐] The minimum M.S. in each laminate model is shown in Table 7.12. Included in this table is the member and ply orientation type that results in the minimum M.S. 53 Laminate [0340] [4517/-4517/0136]S [4534/-4534/0102]S [4543/-4543/084]S [4551/-4551/068]S [3017/-3017/0136]S [3034/-3034/0102]S [3043/-3043/084]S [3051/-3051/068]S [3068/-3068/034]S [9017/0153]S [9034/0136]S [9043/0127]S [9051/0119]S [9068/0102]S [9085/085]S [3042/-3042/9042/042]S [4542/-4542/9042/042]S Minimum M.S. 111 77 56 47 39 77 56 47 40 28 99 90 85 80 70 60 53 38 Ply Orientation Type Member # Containing Containing Minimum M.S. Minimum M.S. 0 8 0 8 0 8 0 8 0 8 0 8 0 8 0 8 0 8 0 8 90 6, 10 90 6, 10 90 6, 10 90 6, 10 90 6, 10 90 6, 10 90 6, 10 90 6, 10 Table 7.12 - Composite Laminate Minimum Axial M.S. The results shown in Table 7.12 that for axial strength the [0340] laminate is the strongest followed by the [9017/0153]S layup. This is because these laminates contain many plies oriented in the direction of the axial load. 7.2.2 Transverse Margin of Safety Using the failure loads in Table 7.7 as the material allowable for each ply and the forces on each member, calculated using the 3D ANSYS FEM in the previous chapter, a transverse M.S. can be calculated. This analysis method outputs a maximum tensile force and a minimum compressive force. It was found that the minimum M.S. is calculated when using the tensile force so the M.S. results use these forces. The results of all 18 laminates are split up into multiple tables based on model groupings. The first grouping contains the single and +/-45 degree cross ply laminates. The results of which are shown in Table 7.13. The second grouping contains the +/-30 degree cross ply laminates and the results are shown in Table 7.14. The third grouping contains the 90 54 degree cross ply laminates and the results are shown in Table 7.15. Finally, the forth grouping contains the combined orientation laminates and the results are shown in Table 7.16. An example calculation of the M.S. in the 0 degree ply of the first member in the [0340]S laminate model is shown below. ๐. ๐. = ๐ด๐๐๐๐ค๐๐๐๐ ๐๐ก๐๐๐๐๐กโ 7,291,400[๐๐] −1 = − 1 = 281 ๐ด๐๐ก๐ข๐๐ ๐๐ก๐๐๐๐๐กโ 25,814[๐๐] Table 7.13 - Composite Laminate Group 1 Transverse M.S. 55 Table 7.14 - Composite Laminate Group 2 Transverse M.S. 56 Member 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 [9017/0153]S 90 1,044 1,242 1,958 2,737 1,565 2,532 2,594 2,191 2,593 2,553 1,552 2,742 1,928 709 1,048 0 580 691 1,089 1,523 871 1,409 1,443 1,219 1,443 1,421 863 1,526 1,073 394 583 [9034/0136]S 90 1,553 1,761 2,492 3,191 2,207 3,501 3,151 2,618 3,150 3,520 2,187 3,195 2,439 1,254 1,558 0 863 979 1,385 1,774 1,226 1,946 1,752 1,455 1,751 1,957 1,216 1,776 1,356 697 866 [9043/0127]S 90 1,813 2,017 2,722 3,394 2,525 3,813 3,406 2,800 3,403 3,836 2,503 3,398 2,659 1,565 1,818 0 1,007 1,120 1,512 1,886 1,403 2,119 1,892 1,555 1,890 2,131 1,391 1,888 1,477 869 1,010 [9051/0119]S 90 2,039 2,228 2,905 3,554 2,798 3,845 3,617 2,939 3,612 3,863 2,775 3,558 2,834 1,856 2,043 0 1,132 1,237 1,613 1,973 1,554 2,135 2,008 1,632 2,006 2,145 1,541 1,975 1,573 1,030 1,134 [9068/0102]S 90 2,220 2,541 4,352 3,831 3,354 3,855 4,035 3,170 4,026 3,866 3,331 3,833 3,155 2,237 2,375 0 1,231 1,408 2,412 2,123 1,859 2,137 2,236 1,757 2,232 2,143 1,846 2,125 1,749 1,240 1,316 [9085/085]S 90 2,211 2,846 3,531 4,019 3,881 3,761 4,425 3,316 4,410 3,774 3,861 4,020 3,423 2,305 2,668 Table 7.15 - Composite Laminate Group 3 Transverse M.S. Member 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 [3042/-3042/9042/042]S 30 2,042 856 746 3,337 2,073 1,097 4,149 2,624 1,459 1,097 2,084 3,351 743 699 2,057 -30 2,042 856 746 3,337 2,073 1,097 4,149 2,624 1,459 1,097 2,084 3,351 743 699 2,057 90 3,031 1,271 1,107 4,953 3,077 1,629 6,158 3,895 2,166 1,629 3,094 4,974 1,104 1,038 3,054 0 1,609 675 587 2,629 1,633 864 3,270 2,067 1,149 865 1,642 2,641 586 551 1,621 [4542/-4542/9042/042]S 45 2,665 2,177 2,905 7,593 4,706 3,703 6,548 6,189 6,454 3,703 4,657 7,593 2,914 1,130 2,681 -45 2,665 2,177 2,905 7,593 4,706 3,703 6,548 6,189 6,454 3,703 4,657 7,593 2,914 1,130 2,681 90 3,116 2,546 3,397 8,880 5,503 4,330 7,657 7,237 7,548 4,330 5,446 8,880 3,408 1,322 3,135 0 1,619 1,322 1,765 4,614 2,859 2,250 3,978 3,760 3,922 2,250 2,830 4,614 1,770 686 1,629 Table 7.16 - Composite Laminate Group 4 Transverse M.S. The minimum M.S. in each laminate model is shown in Table 7.17. Included in this table is the member and ply orientation type that results in the minimum M.S. 57 0 1,223 1,574 1,953 2,223 2,146 2,080 2,448 1,834 2,439 2,087 2,135 2,223 1,893 1,275 1,475 Laminate [0340] [4517/-4517/0136]S [4534/-4534/0102]S [4543/-4543/084]S [4551/-4551/068]S [3017/-3017/0136]S [3034/-3034/0102]S [3043/-3043/084]S [3051/-3051/068]S [3068/-3068/034]S [9017/0153]S [9034/0136]S [9043/0127]S [9051/0119]S [9068/0102]S [9085/085]S [3042/-3042/9042/042]S [4542/-4542/9042/042]S Minimum M.S. 126 181 202 197 182 117 105 97 88 65 394 697 869 1030 1231 1223 551 686 Ply Orientation Type Member # Containing Containing Minimum M.S. Minimum M.S. 0 14 0 14 0 14 0 14 0 2 0 14 0 14 0 14 0 14 0 14 0 14 0 14 0 14 0 14 0 1 0 1 0 14 0 14 Table 7.17 - Composite Laminate Minimum Transverse M.S. The results shown in Table 7.17 show that for transverse strength the [9085/9085]S laminate is the strongest followed by the [9068/90102]S layup. This is because these laminates contain the most cross plies oriented in the direction of the transverse load. 7.2.3 Buckling Margin of Safety Similar to the steel truss the members of the composite truss subject to compression forces must be checked against their critical buckling load. Using equation [7.2] and the global material properties of each laminate the critical buckling load was calculated and is shown for each candidate laminate in Table 7.18. 58 Layup [0340] [4517/-4517/0136]S [4534/-4534/0102]S [4543/-4543/084]S [4551/-4551/068]S [3017/-3017/0136]S [3034/-3034/0102]S [3043/-3043/084]S [3051/-3051/068]S [3068/-3068/034]S [9017/0153]S [9034/0136]S [9043/0127]S [9051/0119]S [9068/0102]S [9085/085]S [3042/-3042/9042/042]S [4542/-4542/9042/042]S Pcr (lb) 3,457,072 2,867,772 2,256,407 1,929,424 1,637,708 3,070,417 2,629,742 2,381,341 2,153,799 1,653,541 3,148,085 2,828,713 2,658,648 2,507,205 2,184,913 1,862,258 1,856,259 1,323,274 Table 7.18 - Composite Laminate Critical Buckling Loads This results in the buckling M.S. for each member of each truss shown in Table 7.19. N/A indicates the member is in tension and is not subject to buckling. An example of the M.S. as calculated for the first member of the [0340]S laminate is shown below. ๐. ๐. = ๐ด๐๐๐๐ค๐๐๐๐ ๐๐ก๐๐๐๐๐กโ 3,457,072 [๐๐] −1= − 1 = 26 ๐ด๐๐ก๐ข๐๐ ๐๐ก๐๐๐๐๐กโ 128,239 [๐๐] 59 Table 7.19 - Composite Composite Laminate Buckling M.S. 60 The minimum M.S. for member buckling in each laminate is shown in Table 7.20. As can be seen in Table 7.19 this minimum always occurs in member number eight. Laminate [0340] [4517/-4517/0136]S [4534/-4534/0102]S [4543/-4543/084]S [4551/-4551/068]S [3017/-3017/0136]S [3034/-3034/0102]S [3043/-3043/084]S [3051/-3051/068]S [3068/-3068/034]S [9017/0153]S [9034/0136]S [9043/0127]S [9051/0119]S [9068/0102]S [9085/085]S [3042/-3042/9042/042]S [4542/-4542/9042/042]S Minimum Buckling M.S. 19 16 12 10 9 17 14 13 12 9 18 16 15 14 12 10 10 7 Table 7.20 - Composite Laminate Minimum Buckling M.S. The laminate that resists buckling the most is the [0340]S laminate. This is a result of this laminate having the highest stiffness in the axial buckling direction. 61 8. Conclusion The degree of success in using composite materials in the truss members of a bridge is measured by the ability of the application to decrease weight and increase strength. The strength change was found to be very dependent on the layup of the composite members. Some laminates are stronger while others are weaker. Laminates that have shown themselves to be stronger than the steel in one direction but are much weaker in another were also among the candidates. What follows is a summary of all the truss models. The minimum M.S. in the axial direction for each truss model is shown in Figure 8.1. Minimum Axial M.S. of Truss Material/Layup 120 100 Axial M.S. 80 60 40 20 0 Truss Material/Layup Figure 8.1 - Axial M.S. Comparison As expected the [0340]S layup is the strongest because it has all its fibers aligned in the axial direction. A similar figure of the minimum M.S. in the transverse direction is shown in Figure 8.2. 62 Minimum Transverse M.S. of Truss Material/Layup 1,400 1,200 Transverse M.S. 1,000 800 600 400 200 0 Truss Material/Layup Figure 8.2 - Transverse M.S. Comparison This shows that the layups with the 90 degree orientation plies are easily the strongest. In particular are the [068/90102]S and the [085/9085]S truss laminates. This is because 90 cross plies have all there fibers aligned with the transverse direction. Finally, a figure comparing the minimum buckling M.S. is shown in Figure 8.3. Minimum Buckling M.S. of Truss Material/Layup 25 Buckling M.S. 20 15 10 5 0 Truss Material/Layup Figure 8.3 - Buckling M.S. Comparison 63 This shows the only layup to compare to the steel is the [0340]S layup. The lower margins across the board are because the axial modulus of elasticity of the composite material is lower than steel. Since this variable is an important factor in determining buckling these results are to be expected. The final important comparison between a steel and composite truss is calculating the reduction in weight for the composite material truss. Table 8.1 shows this difference. The structure weight is the sum of the members, gusset plates, and road deck weight. Truss Material/Layup Steel Composite Structure Weight 264,746 127,472 % Difference -52% Table 8.1 - Steel vs. Composite Weight Comparison As can be seen there is over fifty percent in weight savings when switching to a carbon fiber epoxy composite material in the overall truss. This is due to an 83% reduction in density of composite material over the steel. The results show that switching to IM7/8552 carbon fiber epoxy results in a higher strength and lower weight truss bridge. The ideal layup is the [9043/0127]S composite laminate. This layup has much higher axial strength than the steel and a much improved transverse strength. It only takes a 26% decrease in buckling strength to achieve these higher axial and transverse margins. It can therefore be concluded that the switch to composite materials over traditional steel will overall be beneficial to the construction of a Warren truss bridge structure. 64 9. References [1] Array Solutions. Wind Loads. 23 March 2002. 31 March 2012 <http://www.arraysolutions.com/Products/windloads.htm>. [2] Camp, Charles. Trusses - Method of Joints. Lecture. Memphis: The University of Memphis, 2012. [3] Hexcel Corporation. Prepreg Data Sheets. 31 March 2012 <http://www.hexcel.com/Resources/prepreg-data-sheets>. [4] Hyer, Michael W. Stress Analysis of Fiber-Reinforced Composite Materials. Lancaster: DEStech Publications, Inc., 2009. [5] Kxcad.net. BEAM188 Element. <http://www.kxcad.net/ansys/ANSYS/ansyshelp/Hlp_E_BEAM188.html>. [6] Metallic Materials Properties Development and Standardization (MMPDS). Washington: Federal Aviation Administration, 2010. [7] Reade Advanced Materials. Weight Per Cubic Foot And Specific Gravity. 11 January 2006. 31 March 2012 <http://www.reade.com/Particle_Briefings/spec_gra2.html>. [8] State of Connecticut. Connecticut Building Code. 2005. 31 March 2012 <http://www.archive.org/stream/gov.ct.building/ct_building_djvu.txt>. [9] U.S. Department of Housing and Development. Wind Zone Comparisons. Washington, 2006. [10] U.S. Department of Transportation Federal Highway Administration. Application of Design Standards, Uniform Federal Accessibility Standards, and Bridges. 15 4 2009. 31 March 2012 <http://www.fhwa.dot.gov/design/0625sup.cfm>. [11] U.S. Department of Transportation Federal Highway Administration Freight Management and Operations. Bridge Formula Weights Calculator. 27 February 2012. 31 March 2012 <http://www.ops.fhwa.dot.gov/freight/sw/brdgcalc/calc_page.htm>. 65 10.Appendix 10.1 Appendix A – ANSYS Input File Code ! Initialize the model /FILENAME,file /TITLE, 2D Steel Truss /UNITS,BIN ! USCS system of units ! Define parameters for future use Line_Div = 128 ! Line divisions for meshing Length_T = 74*12 ! Sets length of truss Side = 10 ! Sets cross section side length t = 2.016 ! Sets cross section wall thickness Member_Length = 18.5*12 ! Sets the length of each member Truss_Height = ((Member_Length)**2-(Member_Length/2)**2)**(1/2) !Truss Height m = 667.8 ! Weight of one member ga = 23.5 ! Weight of gusset plate a gb = 38.7 ! Weight of gusset plate b gc = 51.6 ! Weight of gusset plate c P = 210428.4 ! Weight of vehicles, road deck, and snow ! Model Preprocessor /PREP7 ! Enters the general input data preprocessor ! Element Type ET,1,BEAM188 ! 2D Beam Element SECTYPE,1,BEAM,HREC ! Selects hollow rectangle as cross section SECOFFSET,CENT ! Sets centroid SECDATA,Side,Side,t,t,t,t ! Sets dimension of cross section ! Material properties 66 MP,EX,1,9.11E6 ! Elastic modulus x MP,EY,1,9.11E6 ! Elastic modulus y MP,EZ,1,9.11E6 ! Elastic modulus z MP,PRXY,1,0.3201 ! Poisson's ratio xy MP,PRXZ,1,0.3201 ! Poisson's ratio xz MP,PRYZ,1,0.3201 ! Poisson's ratio yz MP,GXY,1,3.45E6 ! Shear modulus xy MP,GXZ,1,3.45E6 ! Shear modulus xz MP,GYZ,1,3.45E6 ! Shear modulus yz ! Geometry ! Creates keypoints at nodes for truss K,1,0,0,0 K,2,Member_Length*1/2,Truss_Height,0 K,3,Member_Length,0,0 K,4,Member_Length*3/2,Truss_Height,0 K,5,Member_Length*2,0,0 K,6,Member_Length*5/2,Truss_Height,0 K,7,Member_Length*3,0,0 K,8,Member_Length*7/2,Truss_Height,0 K,9,Member_Length*4,0,0 ! Creates lines for truss members L,1,2 L,1,3 L,2,3 L,2,4 L,3,4 L,3,5 L,4,5 L,4,6 L,5,6 67 L,5,7 L,6,7 L,6,8 L,7,8 L,7,9 L,8,9 ! Meshing LESIZE, all,,,Line_Div ! Sets divisons of lines to variable Line_Size MAT,1 ! Selects Material 1 to mesh with TYPE,1 ! Sets the element type to mesh to type 1 (BEAM188) MSHAPE,0,2D ! Shape of elements are Quad elements MSHKEY,1 ! Sets to a mapped Mesh LMESH, all ! Meshs all lines Finish ! Ends preprocessor ! Solution processor /SOLU ! Enters the solution processor ANTYPE,STATIC ! Sets the analysis type to static analysis ! Boundary Conditions KSEL,S,,,1 ! Selects node on bottom left NSLK,S ! Selects node on bottom left D,All,UX,0 ! Applies a pinned constraint to left hand side D,All,UY,0 ! Applies a pinned constraint to left hand side D,All,UZ,0 ! Applies a pinned constraint to left hand side D,All,ROTX,0 ! Applies a pinned constraint to left hand side D,All,ROTY,0 ! Applies a pinned constraint to left hand side KSEL,S,,,9 ! Selects node on bottom right NSLK,S ! Selects node on bottom right D,All,UY,0 ! Applies a roller constraint to right hand side 68 D,All,UZ,0 ! Applies a roller constraint to right hand side D,All,ROTX,0 ! Applies a roller constraint to right hand side D,All,ROTY,0 ! Applies a roller constraint to right hand side ALLSEL,ALL ! Reselects everything ! Dead and Live Loads KSEL,S,,,1 ! Selects KP for node a NSLK,S ! Selects node a F,all,FY,-(1.5*m+2*ga) ! Applies force to node a KSEL,S,,,2 ! Selects KP for node b NSLK,S ! Selects node b F,all,Fy,-(2*m+2*gb) ! Applies force to node b KSEL,S,,,3 ! Selects KP for node c NSLK,S ! Selects node c F,all,Fy,-(2.5*m+2*gc+P/3) ! Applies force to node c KSEL,S,,,4 ! Selects KP for node d NSLK,S ! Selects node d F,all,Fy,-(2.5*m+2*gc) ! Applies force to node d KSEL,S,,,5 ! Selects KP for node e NSLK,S ! Selects node e F,all,Fy,-(2.5*m+2*gc+P/3) ! Applies force to node e KSEL,S,,,6 ! Selects KP for node f NSLK,S ! Selects node f F,all,Fy,-(2.5*m+2*gc) ! Applies force to node f KSEL,S,,,7 ! Selects KP for node g NSLK,S ! Selects node g F,all,Fy,-(2.5*m+2*gc+P/3) ! Applies force to node g KSEL,S,,,8 ! Selects KP for node h NSLK,S ! Selects node h F,all,Fy,-(2*m+2*gb) ! Applies force to node h KSEL,S,,,9 ! Selects KP for node i 69 NSLK,S ! Selects node i F,all,Fy,-(1.5*m+2*ga) ! Applies force to node i ALLSEL,ALL ! Reselects everything ! Solve SOLVE ! Solves the current load step FINISH ! Ends solution processor ! Post processor /POST1 ! Enters the post processor /ESHAPE,1.0 ! Plots beam elements PLNSOL,S,X,0,1.0 ! Plots stress in the x direction 10.2 Appendix B – MATLAB Composite Laminate Stiffness Calculator % Clears the screen and variables clear all; clc; % Material properties for Graphite E1 = 23.8E6; E2 = 1.7E6; v12 = 0.32; G12 = .750E6; thickness = 0.006; % Computes the Q Matrix Q(1,1) = E1/(1-v12*v12*E2/E1); Q(1,2) = v12*E2/(1-v12*v12*E2/E1); Q(1,3) = 0; Q(2,1) = Q(1,2); 70 Q(2,2) = E2/(1-v12*v12*E2/E1); Q(2,3) = 0; Q(3,1) = Q(1,3); Q(3,2) = Q(2,3); Q(3,3) = G12; % # of plies in layup Num_plies = 336; theta_1 = 30; theta_2 = -30; theta_3 = 90; theta_4 = 0; % Creates a matrix of plies of size equal to previous input for i = 1:Num_plies/8 Ply(i,1) = theta_1; end for i = Num_plies/8+1:2*Num_plies/8 Ply(i,1) = theta_2; end for i = 2*Num_plies/8+1:3*Num_plies/8 Ply(i,1) = theta_3; end for i = 3*Num_plies/8+1:5*Num_plies/8 Ply(i,1) = theta_4; end 71 for i = 5*Num_plies/8+1:6*Num_plies/8 Ply(i,1) = theta_3; end for i = 6*Num_plies/8+1:7*Num_plies/8 Ply(i,1) = theta_2; end for i = 7*Num_plies/8+1:Num_plies Ply(i,1) = theta_1; end % Laminate Thickness H = Num_plies * thickness; % Creates the z matrix for i = 1:Num_plies + 1 z(i,1) = -H/2 + (i - 1) * thickness; end % Zeroes out the A,B,D,a,b,d matrices for i = 1:3 for j = 1:3 A(i,j) = 0; B(i,j) = 0; D(i,j) = 0; a(i,j) = 0; b(i,j) = 0; d(i,j) = 0; end end 72 % Calculates the stresses for each ply for i = 1:Num_plies % Calculates m and n for each ply m = cos(Ply(i,1)/180*pi); n = sin(Ply(i,1)/180*pi); % Calculates the Qbar matrix based on Q, m, and n QBar(1,1,i) = Q(1,1) * m^4 + 2 * (Q(1,2) + 2* Q(3,3)) * n^2 * m^2 + Q(2,2) * n^4; QBar(1,2,i) = (Q(1,1) + Q(2,2) - 4 * Q(3,3)) * n^2 * m^2 + Q(1,2) * (n^4+m^4); QBar(1,3,i) = (Q(1,1) - Q(1,2) - 2 * Q(3,3)) * n * m^3 + (Q(1,2) - Q(2,2) + 2 * Q(3,3)) * n^3 * m; QBar(2,1,i) = QBar(1,2,i); QBar(2,2,i) = Q(1,1) * n^4 + 2 * (Q(1,2) + 2* Q(3,3)) * n^2 * m^2 + Q(2,2) * m^4; QBar(2,3,i) = (Q(1,1) - Q(1,2) - 2 * Q(3,3)) * n^3 * m + (Q(1,2) - Q(2,2) + 2 * Q(3,3)) * n * m^3; QBar(3,1,i) = QBar(1,3,i); QBar(3,2,i) = QBar(2,3,i); QBar(3,3,i) = (Q(1,1) + Q(2,2) - 2 * Q(1,2) - 2 * Q(3,3)) * n^2 * m^2 + Q(3,3) * (n^4 + m^4); % Calculates the ABD matrices for j = 1:3 for k = 1:3 A(j,k) = A(j,k) + QBar(j,k,i) * (z(i+1) - z(i)); B(j,k) = B(j,k) + QBar(j,k,i) * (z(i+1)^2 - z(i)^2); D(j,k) = D(j,k) + QBar(j,k,i) * (z(i+1)^3 - z(i)^3); end end % Calculates the transformation matrix 73 T(1,1,i) = m^2; T(1,2,i) = n^2; T(1,3,i) = 2*m*n; T(2,1,i) = n^2; T(2,2,i) = m^2; T(2,3,i) = -2*m*n; T(3,1,i) = -m*n; T(3,2,i) = m*n; T(3,3,i) = m^2 - n^2; end % Applies the correction coefficient to fully calculate the B & D matrix B = B/2; D = D/3; % Calcutes the 6x6 ABD matrix for i = 1:3 for j = 1:3 ABD(i,j) = A(i,j); ABD(i+3,j) = B(i,j); ABD(i,j+3) = B(i,j); ABD(i+3,j+3) = D(i,j); end end % Calculates the 6x6 abd matrix abd = inv(ABD); % Calculates the a, b, and d matrices for i = 1:3 for j = 1:3 74 a(i,j) = abd(i,j); b(i,j) = abd(i,j+3); d(i,j) = abd(i+3,j+3); end end % Calculates the material properties of the laminate E_x = 1 / (a(1,1) * H); E_y = 1 / (a(2,2) * H); E_z = E_y; nu_xy = -a(1,2)/a(1,1); nu_yz = -a(1,2)/a(2,2); nu_xz = nu_xy; G_xy = 1 / (a(3,3) * H); G_xz = G_xy; G_yz = E_y / (2 * (1 + nu_yz)); % Prints out the properties of the laminate fprintf('Ex = %.2E Msi\n',E_x/1E6) fprintf('Ey = %.2E Msi\n',E_y/1E6) fprintf('Ez = %.2E Msi\n\n',E_z/1E6) fprintf('nu_xy = %.4f \n',nu_xy) fprintf('nu_xz = %.4f \n',nu_xz) fprintf('nu_yz = %.4f \n\n',nu_yz) fprintf('Gxy = %.2E Msi\n',G_xy/1E6) fprintf('Gxz = %.2E Msi\n',G_xz/1E6) fprintf('Gyz = %.2E Msi\n\n',G_yz/1E6) fprintf('H = % in\n\n',H) 75 10.3 Appendix C – MATLAB Composite Laminate Global Failure Calculator % Clears the screen and variables clear all; clc; % Length of Laminate in axial direction L = 10*4; % Material properties for Graphite E1 = 23.8E6; E2 = 1.7E6; v12 = 0.32; G12 = .75E6; thickness = .006; sigma_1_T = 395E3; sigma_1_C = -245E3; sigma_2_T = 16.1E3; sigma_2_C = -32.3E3; Tau_12_F = 17.4E3; % Computes the Q Matrix Q(1,1) = E1/(1-v12*v12*E2/E1); Q(1,2) = v12*E2/(1-v12*v12*E2/E1); Q(1,3) = 0; Q(2,1) = Q(1,2); Q(2,2) = E2/(1-v12*v12*E2/E1); Q(2,3) = 0; Q(3,1) = Q(1,3); Q(3,2) = Q(2,3); Q(3,3) = G12; 76 % Creates a matrix of plies of size equal to previous input % # of plies in layup Num_plies = 336; theta_1 = 45; theta_2 = -45; theta_3 = 90; theta_4 = 0; % Creates a matrix of plies of size equal to previous input for i = 1:Num_plies/8 Ply(i,1) = theta_1; end for i = Num_plies/8+1:2*Num_plies/8 Ply(i,1) = theta_2; end for i = 2*Num_plies/8+1:3*Num_plies/8 Ply(i,1) = theta_3; end for i = 3*Num_plies/8+1:5*Num_plies/8 Ply(i,1) = theta_4; end for i = 5*Num_plies/8+1:6*Num_plies/8 Ply(i,1) = theta_3; end for i = 6*Num_plies/8+1:7*Num_plies/8 Ply(i,1) = theta_2; end for i = 7*Num_plies/8+1:Num_plies Ply(i,1) = theta_1; 77 end % Information for the Forces N_M_matrix(1,1) = 1/L; N_M_matrix(2,1) = 0; N_M_matrix(3,1) = 0; N_M_matrix(4,1) = 0; N_M_matrix(5,1) = 0; N_M_matrix(6,1) = 0; % Laminate Thickness H = Num_plies * thickness; % Creates the z matrix for i = 1:Num_plies + 1 z(i,1) = -H/2 + (i - 1) * thickness; end % Zeroes out the A,B,D,a,b,d matrices for i = 1:3 for j = 1:3 A(i,j) = 0; B(i,j) = 0; D(i,j) = 0; a(i,j) = 0; b(i,j) = 0; d(i,j) = 0; end end % Calculates the stresses for each ply 78 for i = 1:Num_plies % Calculates m and n for each ply m = cos(Ply(i,1)/180*pi); n = sin(Ply(i,1)/180*pi); % Calculates the Qbar matrix based on Q, m, and n QBar(1,1,i) = Q(1,1) * m^4 + 2 * (Q(1,2) + 2* Q(3,3)) * n^2 * m^2 + Q(2,2) * n^4; QBar(1,2,i) = (Q(1,1) + Q(2,2) - 4 * Q(3,3)) * n^2 * m^2 + Q(1,2) * (n^4+m^4); QBar(1,3,i) = (Q(1,1) - Q(1,2) - 2 * Q(3,3)) * n * m^3 + (Q(1,2) - Q(2,2) + 2 * Q(3,3)) * n^3 * m; QBar(2,1,i) = QBar(1,2,i); QBar(2,2,i) = Q(1,1) * n^4 + 2 * (Q(1,2) + 2* Q(3,3)) * n^2 * m^2 + Q(2,2) * m^4; QBar(2,3,i) = (Q(1,1) - Q(1,2) - 2 * Q(3,3)) * n^3 * m + (Q(1,2) - Q(2,2) + 2 * Q(3,3)) * n * m^3; QBar(3,1,i) = QBar(1,3,i); QBar(3,2,i) = QBar(2,3,i); QBar(3,3,i) = (Q(1,1) + Q(2,2) - 2 * Q(1,2) - 2 * Q(3,3)) * n^2 * m^2 + Q(3,3) * (n^4 + m^4); % Calculates the ABD matrices for j = 1:3 for k = 1:3 A(j,k) = A(j,k) + QBar(j,k,i) * (z(i+1) - z(i)); B(j,k) = B(j,k) + QBar(j,k,i) * (z(i+1)^2 - z(i)^2); D(j,k) = D(j,k) + QBar(j,k,i) * (z(i+1)^3 - z(i)^3); end end % Calculates the transformation matrix T(1,1,i) = m^2; 79 T(1,2,i) = n^2; T(1,3,i) = 2*m*n; T(2,1,i) = n^2; T(2,2,i) = m^2; T(2,3,i) = -2*m*n; T(3,1,i) = -m*n; T(3,2,i) = m*n; T(3,3,i) = m^2 - n^2; end % Applies the correction coefficient to fully calculate the B & D matrix B = B/2; D = D/3; % Calcutes the 6x6 ABD matrix for i = 1:3 for j = 1:3 ABD(i,j) = A(i,j); ABD(i+3,j) = B(i,j); ABD(i,j+3) = B(i,j); ABD(i+3,j+3) = D(i,j); end end % Calculates the 6x6 abd matrix abd = inv(ABD); % Calculates the a, b, and d matrices for i = 1:3 for j = 1:3 80 a(i,j) = abd(i,j); b(i,j) = abd(i,j+3); d(i,j) = abd(i+3,j+3); end end % Calculates the epsilon_kappa matrix epsilon_Kappa_matrix = abd * N_M_matrix; % Calculates the global strains at the top and bottom of each layer for i = 1:Num_plies for j = 1:3 epsilon_global_top(j,:,i) = epsilon_Kappa_matrix(j) + z(i) * z(i+1) * epsilon_Kappa_matrix(j+3); epsilon_global_bot(j,:,i) = epsilon_Kappa_matrix(j) epsilon_Kappa_matrix(j+3); end end % Calculates the global stresses at the top and bottom of each layer for i = 1:Num_plies sigma_global_top(:,:,i) = QBar(:,:,i) * epsilon_global_top(:,:,i); sigma_local_top(:,:,i) = T(:,:,i) * sigma_global_top(:,:,i); sigma_global_bot(:,:,i) = QBar(:,:,i) * epsilon_global_bot(:,:,i); sigma_local_bot(:,:,i) = T(:,:,i) * sigma_global_bot(:,:,i); end % Calcualtes the Tsai-Wu Failure Criterion Coefficients F_1 = 1 / sigma_1_T + 1 / sigma_1_C; F_2 = 1 / sigma_2_T + 1 / sigma_2_C; 81 + F_11 = -1 / (sigma_1_T * sigma_1_C); F_22 = -1 / (sigma_2_T * sigma_2_C); F_66 = (1 / Tau_12_F)^2; count = 1; for i = 1:Num_plies p = eval(solve('F_1 * x * sigma_local_top(1,1,i) + F_2 * x * sigma_local_top(2,1,i) + F_11 * (x * sigma_local_top(1,1,i))^2 + F_22 * (x * sigma_local_top(2,1,i))^2 + F_66 * (x * sigma_local_top(3,1,i))^2 - sqrt(F_11 * F_22) * x * sigma_local_top(1,1,i) * x * sigma_local_top(2,1,i) = 1')); Answer(count,1) = p(1); Answer(count + 1,1) = p(2); count = count +2; end for i = 1:Num_plies Ply_Num(i,1) = i; end % Creates table of bottom sigma x, sigma y, and tauxy for each ply Table_Top_Stress_global(:,1) = Ply_Num; Table_Top_Stress_global(:,2) = Ply; count = 1; for i = 1:Num_plies Table_Top_Stress_global(i,3) = Answer(count)/1E6; Table_Top_Stress_global(i,4) = Answer(count+1)/1E6; count = count + 2; end disp('Failure Load of Each Ply (Mlb)') 82 disp(' Ply # Angle Tension Compression ') disp(Table_Top_Stress_global) % Creates vector of ply numbers for i = 1:Num_plies Ply_Num(i,1) = i; end % Creates table of bottom sigma x, sigma y, and tauxy for each ply Table_Top_Stress_global(:,1) = Ply_Num; Table_Top_Stress_global(:,2) = Ply; for i = 1:Num_plies for j = 1:3 Table_Top_Stress_global(i,j+2) = sigma_global_top(j,:,i) / 1E6; end end % Creates table of top sigma 1, sigma 2, and tau12 for each ply Table_Bot_Stress_global(:,1) = Ply_Num; Table_Bot_Stress_global(:,2) = Ply; for i = 1:Num_plies for j = 1:3 Table_Bot_Stress_global(i,j+2) = sigma_global_bot(j,:,i) / 1E6; end end % Creates table of bottom sigma 1, sigma 2, and tau12 for each ply Table_Top_Stress_local(:,1) = Ply_Num; Table_Top_Stress_local(:,2) = Ply; for i = 1:Num_plies for j = 1:3 83 Table_Top_Stress_local(i,j+2) = sigma_local_top(j,:,i) / 1E6; end end % Creates table of top sigma 1, sigma 2, and tau12 for each ply Table_Bot_Stress_local(:,1) = Ply_Num; Table_Bot_Stress_local(:,2) = Ply; for i = 1:Num_plies for j = 1:3 Table_Bot_Stress_local(i,j+2) = sigma_local_bot(j,:,i) / 1E6; end end % Displays the tables %disp(' Global Stress Top of Plys (Msi)') %disp(' Ply # Angle Sigmax Sigmay Tauxy') %disp(Table_Top_Stress_global) %disp(' Global Stress Bottom of Plys (Msi)') %disp(' Ply # Angle Sigmax Sigmay Tauxy') %disp(Table_Bot_Stress_global) %disp(' Local Stress Top of Plys (Msi)') %disp(' Ply # Angle Sigma1 Sigma2 Tau12') %disp(Table_Top_Stress_local) %disp(' Local Stress Bottom of Plys (Msi)') %disp(' Ply # Angle Sigma1 Sigma2 %disp(Table_Bot_Stress_local) 84 Tau12')