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Optimization of an Orthotropic Composite Beam by Brian Schmalberger An Engineering Project Submitted to the Graduate Faculty of Rensselaer Polytechnic Institute in Partial Fulfillment of the Requirements for the degree of MASTER OF Mechanical Engineering Approved: _________________________________________ Ernesto Gutierrez-Miravete, Engineering Project Adviser Rensselaer Polytechnic Institute Hartford, Connecticut May 2010 (For Graduation December 2010) 1 © Copyright 2010 by Brian Schmalberger All Rights Reserved 2 Table of Contents Optimization of an Orthotropic Composite Beam ............................................................. 1 Table of Contents ............................................................................................................... 3 List of Tables ..................................................................................................................... 4 List of Figures .................................................................................................................... 5 Nomenclature ..................................................................................................................... 6 Abstract .............................................................................................................................. 7 1. Introduction and Background ...................................................................................... 8 2. Methodology ................................................................................................................ 9 2.1 Kirchhoff’s Hypothesis .......................................................................................... 9 2.2 Stress-Strain Relationship for an Orthotropic Material ....................................... 10 2.3 Finite Element Modeling ..................................................................................... 11 3. Results and Discussion .............................................................................................. 13 3.1 Introduction.......................................................................................................... 13 3.2 Finite Element Model Validation ........................................................................ 14 3.2.1 Analytical Solution to Case 1: [0/90]S Under Axial Strain, εx ............. 14 3.2.2 Mesh Optimization ............................................................................... 18 3.2.3 Finite Element Solution to Case 1 ........................................................ 20 3.2.4 Analytical Solution to Case 2: [±30/0]S Under Axial Strain, εx........... 26 3.2.5 Finite Element Solution to Case 2 ........................................................ 27 3.3 Composite Beam Response to a Transverse Point Load ..................................... 29 3.4 Modal Analysis of a Composite Beam ................................................................ 33 4. Conclusion ................................................................................................................. 36 5. References.................................................................................................................. 38 3 List of Tables Table 1. Material properties for Graphite-Polymer Composite ...................................... 13 Table 2. Comparison of Mesh Refinements for the 90º-Layer ....................................... 19 Table 3. Comparison of Analytical and ANSYS Solutions for a [0/90]S Laminate ....... 26 Table 4. Comparison of Analytical and ANSYS Solutions for a [±30/0]S Laminate ..... 28 4 List of Figures Figure 1. Fiber Orientation in a Laminate ........................................................................ 8 Figure 2. Fixed-Free Beam .............................................................................................. 8 Figure 3. Normal Line Definition in Kirchhoff’s Hypothesis .......................................... 9 Figure 4. Definition of the SOLID95 Element in ANSYS ............................................. 12 Figure 5. Dimensions of a Single Lamina ...................................................................... 13 Figure 6. Through-Thickness Stress Distribution for a [0/90]S Laminate ..................... 16 Figure 7. Free Edge Stress Across the Width of the Laminate ...................................... 18 Figure 8. Mesh Refinement Near the Free Edge ............................................................ 19 Figure 9. Symmetrical Boundary Conditions and Mesh Refinement ............................. 20 Figure 10. Strain Distribution in ANSYS for a [0/90]S Laminate .................................. 21 Figure 11. Stress Distribution (σx) in ANSYS for a [0/90]S Laminate ........................... 22 Figure 12. Stress Distribution (σy) in ANSYS for a [0/90]S Laminate ........................... 23 Figure 13. Strain in the Y-Direction (Deformation is 100X) ......................................... 24 Figure 14. Shear Stress Distribution (τxy) in ANSYS for a [0/90]S Laminate ................ 25 Figure 15. Stress Distribution for a [±30/0]S Laminate .................................................. 27 Figure 16. Stress Distributions for [±30/0]S, σx, σy, τxy .................................................. 27 Figure 17. Laminated Beam with Fixed-Free Boundary Conditions Loaded with Fz .... 29 Figure 18. Displacement in the Z-Direction Due to Fz=10N .......................................... 30 Figure 19. Von Mises Stress Due to Fz=10N.................................................................. 31 Figure 20. Von Mises Strain Due to Fz=10N ................................................................. 32 Figure 21. First Five Mode Shapes of [0/90]S Laminate Beam ...................................... 33 Figure 22. Mode Shape Frequency for [θ/90]S Laminate Beam ..................................... 34 Figure 23. Comparison of the Z-Displacement for +θ or –θ Due to Fz .......................... 35 5 Nomenclature E modulus of elasticity, MPa G shear modulus, MPa N Resultant force, N/m M Resultant moment, N-m/m Q Reduced stiffness, MPa γ shear strain ε strain ε0 strain of the reference surface θ fiber orientation angle, degree (°) κ curvature, m-1 κ0 curvature of the reference surface, m-1 μ density, kg/m3 σ normal stress, MPa τ shear stress, MPa ωn natural frequency, Hz υ Poisson’s ratio 6 Abstract This paper analyzes the response of an orthotropic composite beam as the fiber orienta- tion angle in the lamina increases from 0º to 90º. Specifically, the response to a point load and the modes of vibration are simulated. The beam has the properties of a graph- ite-polymer composite and will consist of four layers. The orientation angle fibers in the two outer layers vary while the fibers in the two inner layers remain constant at 90º. A finite element model was created in ANSYS and validated by solving a composite beam problem with a known solution. The finite element model was then used to analyze the response of the beam to a transverse point load. Specifically stress, strain, displacement, and vibration modes were analyzed. Lastly, a modal analysis was performed to find the frequencies of the vibration modes. This paper simulates a composite beam using the finite element method and analyzes the stress distribution, stiffness, displacement, and vibration modes as the fiber orientation and loading are changed. The beam is further optimized for given design constraints, such as maximum flexibility, maximum rigidity, and modes of vibration. 7 1. Introduction and Background Composites are widely used in the automobile, aerospace, and athletics industry. Examples of composites include bumpers, wings, bicycle frames, and downhill skis. Combined with the low weight and high strength characteristics of composite materials, the ability to optimize a composite structure for a specific property is useful to a design engineer. Properties of the structure, such as stiffness, stress distribution, and modes of vibration are affected by fiber and matrix material, fiber orientation, and the number of lamina. A composite is constructed of layers, called lamina, that are bonded together. Each layer is composed of fibers and a polymer matrix material that is required to hold the fibers in position. The orientation angle of the fibers can change, which will, in effect, change the response of the beam. The orientation angles of all of the layers are known a layup. For example, [0/90/90/0] is a composite layup with fibers in the top layer oriented at 0°, in the second layer at 90°, the third layer at 90°, and the bottom layer at 0°. This is also an example of a symmetric laminate because the layers are symmetric about the beam reference surface. It can be abbreviated as [090]S where the subscript refers to its symmetry. An example of a composite is shown in Figure 1. Figure 1. Fiber Orientation in a Laminate The composite beam simulated in this paper is a cantilever beam fixed at one end and free at the other end. A transverse point load was applied to the tip of the beam, as shown in Figure 2. Figure 2. Fixed-Free Beam 8 2. Methodology A simplified model often used to solve problems involving laminates is Kirchhoff’s hypothesis. The main assumption of this hypothesis is that there is no out-of-place strain. This point will be expanded in greater detail in the next section, but it is im- portant to note that the ANSYS model being used is three-dimensional, and therefore, does not consider this assumption. Nonetheless, Kirchhoff’s hypothesis can be used to validate the model and will be used in this paper to show the accuracy of the finite element model. 2.1 Kirchhoff’s Hypothesis The Kirchhoff hypothesis argues that a normal line perpendicular to the laminates will remain normal and will not change in length, regardless of the loading of the laminate (moments, distributed loads, point loads, etc.). This is illustrated in Figure 3. Figure 3. Normal Line Definition in Kirchhoff’s Hypothesis The laminated beam may bend or twist in a given direction due to the loading on the structure, but line AA’ will remain perpendicular and the same length. Line AA’ will only translate and rotate accordingly to remain perpendicular. Maintaining the same length of line AA’ is indicative of zero strain in the z-direction and the basis for Kirch- 9 hoff’s hypothesis. Two additional assumptions of the hypothesis are that the layers are perfectly bonded together and there is no slippage between them. Because of this no slip condition, the interfaces between the layers will remain parallel; and therefore, perpen- dicular to the normal line AA’. The mid-plane surface is known as the reference surface. The hypothesis states that the normal line AA’ only rotates and translates. Knowing the new location and position of line AA’ in reference to reference surface, all strains, displacements, and stresses along line AA’ can be determined. 2.2 Stress-Strain Relationship for an Orthotropic Material The stress-strain constitutive relationship can be simplified for orthotropic materials. The general form is shown below, where the Cij-terms in the matrix are the stiffness terms. The general form of the stress-strain relationship is shown below. 1 C11 C12 C13 C14 C15 C16 1 C C 22 C 23 C 24 C 25 C 26 2 2 21 3 C31 C32 C33 C34 C35 C36 3 23 C 41 C 42 C 43 C 44 C 45 C 46 23 13 C51 C52 C53 C54 C55 C56 13 12 C61 C62 C63 C64 C65 C66 12 This relationship can be simplified for an orthotropic material. In an orthotropic materi- al, there are three planes of symmetry. All three planes are orthogonal with respect to the other planes. One plane is perpendicular to the fiber direction and two planes are parallel to the fiber direction. In this case, the orthogonal planes coincide with the coordinate planes XY, YZ, and XZ. The orthogonal characteristic of the material properties reduces the number of constants required to describe the material. The stress- strain relationship for orthogonal materials is shown below. 10 1 C11 C12 C13 0 0 0 1 C 0 2 2 21 C 22 C 23 0 0 3 C 31 C 32 C 33 0 0 0 3 23 0 0 0 C 44 0 0 23 13 0 0 0 0 C 55 0 13 12 0 0 0 0 0 C 66 12 The stiffness terms can be written in terms of material properties. (1 23 32 ) E1 C11 1 ( 31 23 ) E1 (12 3212 ) E 2 C12 21 1 1 ( 31 21 32 ) E1 (13 12 23 ) E3 C13 1 1 (1 13 31 ) E 2 C 22 1 ( 12 31 ) E 2 ( 23 2113 ) E3 C 23 32 1 1 (1 12 21 ) E3 C 33 1 C 44 G 23 C 55 G13 C 66 G12 2.3 Finite Element Modeling ANSYS was used to simulate the response of the same [0/90] S laminate subjected to a known strain in the x-direction. One-quarter of the laminate beam was modeled to reduce the complexity of the simulation. Symmetry boundary conditions were defined for the XY- and XZ-planes. The SOLID95 element was chosen for the analysis because this is a 3-dimensional structural problem. SOLID95 is a 3-dimensional 20-node struc- tural solid element that is able to handle irregular shapes with a high degree of accuracy due to the allowable degrees of freedom. It is commonly used for curved boundaries, 11 which is convenient for composite analysis because there is often bending or curvature as a result of loadings. The SOLID95 element is illustrated in Figure 4. Figure 4. Definition of the SOLID95 Element in ANSYS 12 3. Results and Discussion 3.1 Introduction Two cases were examined using both analytical and finite element methods. The first case is a four-layer symmetric laminate with a [0/90]S layup. The second case is a is a six-layer symmetric laminate with a [±30/0]S. The problems were first solved analytical- ly with classical lamination theory and then with the finite element method. The results were compared to show the accuracy of the model. For all analytical and finite elements models, the lamina material is a graphite-polymer composite with the following material properties: Table 1. Material properties for Graphite-Polymer Composite E1 155.0 GPa E2 12.10 Gpa E3 12.10 GPa ν23 0.458 ν13 0.248 ν12 0.248 G23 3.20 GPa G13 4.40 GPa G12 4.40 GPa For the finite element models, the laminate dimensions in millimeters are shown in Figure 5. Figure 5. Dimensions of a Single Lamina 13 3.2 Finite Element Model Validation 3.2.1 Analytical Solution to Case 1: [0/90]S Under Axial Strain, εx The first case is a four-layer laminate subjected to a loading such that there is only strain in the x-direction, εx. The thickness of each lamina is 0.150mm. The applied loading on the reference surface is defined as: and Strain through the thickness of the laminate is found using Equation 1. Substituting the values for strain and curvature, the global strain is There is only strain in the x-direction. There is no Poisson effect in either the y- or z- direction, nor is there in-plane shear strain or curvature. This is a result of the defined loading. Next, the global stresses are determined. 14 is defined as the reduced stiffness matrix and contains the stiffness values for each layer as a function of the fiber orientation angle, θ. elements are defined as , where For the [0/90]S laminate, Substituting the numerical values for into Equation 1 yields 15 The through-thickness stress distributions are plotted in Figure 6. The reference surface is located at z=0. This is the middle of the laminate. There is on shear stress and the stress in the y-direction is minimal when compared to the stress in the x-direction. The maximum stress in the x-direction is located in the top and bottom layers where the fiber orientation angle is 0º. The stress in these layers is considerably higher than the stress in the 90º layers because the fibers are parallel to the direction of strain, while the 90º fibers are perpendicular to the direction of the strain. In these layers the fibers experience no change in length because it is the matrix filler material that expands (or contracts). In the 0º, it is the fibers that expand (or contract). Figure 6. Through-Thickness Stress Distribution for a [0/90]S Laminate 16 Although the above problem had an applied strain, it is often more realistic to think in applied loads and then to analyze the stress and strain responses. To complete the classical lamination problem, the forces and moments required to develop the above strains and through-thickness stresses are calculated. A new matrix is introduced and is formally known as ABD matrix. The ABD matrix contains information to relate the forces and moments to the resultant strains and curvatures of the laminate. For this example, the ABD matrix is defined as 17 Carrying out the matrix multiplication, the forces and moments are calculated to be 3.2.2 Mesh Optimization The mesh density and the number of elements directly affect the accuracy of the solu- tions obtained by the finite element method. The in-plane loading of a symmetric laminate will create only in-plane stresses. In the coordinate system used for this example, in-plane stresses include σx, σz, and τxz. However, near the free edges of the laminate, inter-laminar stresses will develop due to the imbalance of the in-place stresses at the free edge. Here, inter-laminar stresses are σz, τxy, and τyz. To accurately calculate the free-edge stresses, the mesh is refined near the free edges. The free-edge stresses are plotted in Figure 7. Figure 7. Free Edge Stress Across the Width of the Laminate 18 The plot in Figure 7 shows that the stresses increase near the free edge of the beam. It is important to have a mesh that can accurately capture the increase in stress at the edge. The refined mesh is shown in Figure 8. Figure 8. Mesh Refinement Near the Free Edge This problem was solved three times. First, it was solved with a sparse, uniform mesh. Next, it was solved with a heavily refined, uniform mesh. Lastly, it was solved with a mesh that was refined at the free edge only. The results for the 90º-layer are summa- rized in Table 2. Table 2. Comparison of Mesh Refinements for the 90º-Layer # Equations σx σy σz Δx Δy Δz Run Time Trial 1 4224 11.6 -0.377 -0.175 0.003 -0.716E-4 -0.133E-4 5s Trial 2 42641 11.9 -0.838 -0.089 0.003 -0.725E-4 -0.133E-4 59s Trial 3 23631 11.9 -0.894 -0.086 0.003 -0.722E-4 -0.133E-4 31s From Table 2, it is clear that a sparse mesh is not sufficiently accurate for the stresses, while it does accurately calculate the displacements. The heavily refined mesh of Trial 2 19 and the refined mesh in Trial 3 have the same results, but the computation time for Trial 3 is significantly longer. Therefore, the mesh used for the remainder of the ANSYS simulations is the mesh from Trial 3. It is optimized to only be refined at the free edges. This reduces computation time without sacrificing accuracy. The results from Trial 3 are shown below, with Figure 9 illustrating the boundary conditions and the applied strain. 3.2.3 Finite Element Solution to Case 1 Figure 9. Symmetrical Boundary Conditions and Mesh Refinement Figure 10 shows the applied strain in the x-direction. This is the strain that was applied to the model and is a quick way to validate that the load was correctly applied. 20 Figure 10. Strain Distribution in ANSYS for a [0/90]S Laminate The through-thickness stress, σz, is shown in Figure 11. It is important to note that the stress in the 0º layer is much higher than the stress in the 90º. The reason for this is that the fibers in the 0º are oriented in the direction of the applied strain, while the fibers in the 90º are perpendicular to the applied strain. Under the applied strain, the 0º-fibers are deformed, but the 90º-fibers are not. In the 90º-layers, it is the matrix material that is deformed. The modulus of the matrix material is significantly less that the modulus of the fibers; therefore, the stress in the 0º-layers is much higher than the stress in the 0º- layers. 21 Figure 11. Stress Distribution (σx) in ANSYS for a [0/90]S Laminate The out-of-plane stress distribution shown in Figure 12 shows the interaction of the layers well. The magnitude of the out-of-plane stress is higher in the 90º-layer than the 0º-layer. Again, this is because the fibers are oriented in the direction of the stress. It is the converse to the in-plane stress distribution in Figure 11. 22 Figure 12. Stress Distribution (σy) in ANSYS for a [0/90]S Laminate Also of note is the sign of the stress values. The 90º-layers have a positive stress and the 0º-layers have a negative stress. In an isotropic material, an applied load in strain the in x-direction would create a Poisson effect in the y- and z-directions. If the same x- direction strain were applied to an isotropic material, the laminate would contract in the y- and z-directions. In this case, however, the fibers in 90º-layers oppose any contrac- tion in the y-direction, while the fibers in the 0º-layers do not. This difference in deformation creates a bending effect where the out 0º-layers are in compression and the inner 90º-layers are in tension. The deformation is shown in Figure 13. 23 Figure 13. Strain in the Y-Direction (Deformation is 100X) The x-direction strain remains the same, but the y-direction strain clearly shows that reinforcing affect the 90º-fibers have on the middle layers. The free-edge effects are also very apparent, as is the reason for the refined mesh at the edges. The classical lamination theory showed that there is no out-of-place shear stress in the beam due to an applied strain. Shear stress in the XY-plane is plotted in Figure 14, and it shows that this calculation is correct. There is no out-of plane shear stress. 24 Figure 14. Shear Stress Distribution (τxy) in ANSYS for a [0/90]S Laminate In summary of this example, a four-layer [0/90]S laminate was loaded such that there was only strain in the x-direction. The through-thickness stresses were determined and compared to the solution from the finite element model. The results were equivalent. Next, the forces and moments required to obtain only stress in the x-direction were calculated. It is important to note, that the force in the x-direction is positive to obtain x- direction strain, while the force in the y-direction is also positive to prevent contraction in the y-direction due to the Poisson effect. The results of the analytical and finite element solutions are listed in Table 3. 25 Table 3. Comparison of Analytical and ANSYS Solutions for a [0/90]S Laminate Analytical Solution ANSYS Solution Error 0º layers σx 155.7 155.6 0.06% σx 3.02 2.5 1.3% τxy 0 0 0% 90º layers σx 12.16 12.05 0.90% σx 3.02 2.5 1.3% τxy 0 0 0% 3.2.4 Analytical Solution to Case 2: [±30/0]S Under Axial Strain, εx To further prove that the ANSYS model is accurate, a second case is solved. The material and dimensions of the laminate remain the same, as well as the initial strain. This example differs in the number of layers and the fiber orientation angle. The calcu- lation performed in Example 1 will be omitted, with only the results shown here. The through-thickness stresses are shown in Figure 15. This stress distribution is quite different than the distribution in Example 1 because of the fiber orientation angle. In Example 1, the laminate was constructed as a cross-ply, meaning that the fibers were perpendicular (at 0º and 90º). In Example 2, the fibers are not perpendicular. The stress in the x-direction is for the 0◦ layers is the same as Example 1, but different for the ±30º layers because a component of the fiber is in the x-direction. This is the same reason that shear stress and y-direction stresses result are present. The fibers are non- orthogonal and will all contract or expand to certain amount regardless of the direction of loading. The ANSYS model results are show in Figure 16, and the results are com- pared in Table 3. 26 Figure 15. Stress Distribution for a [±30/0]S Laminate 3.2.5 Finite Element Solution to Case 2 The ANSYS model results are show in Figures 16, and the results are compared in Table 3. It’s important to note that the +30º and -30º have similar responses to an applied strain in the x-direction. Figure 16. Stress Distributions for [±30/0]S, σx, σy, τxy 27 To further illustrate the accuracy of the finite element model, the stresses are listed in Table 3. The out-of-place shear stress is the only value that changes between +30º and - 30º. This is because the fibers in these two layers are 60º apart and cause the layer to curl around different axes. Additionally, the stresses in the 0º layer are equivalent to the stresses in the same layer of Example 1. Table 4. Comparison of Analytical and ANSYS Solutions for a [±30/0]S Laminate Analytical Solution ANSYS Solution Error 30º layers σx 92.8 93.0 0.22% σy 30.1 29.8 1.0% τxy 46.7 46.3 0.86% -30º layers σx 92.8 93.0 0.22% σy 30.1 29.8 1.0% τxy -46.7 -46.3 0.86% 0º layers σx 155.7 155.6 0.06% σy 3.02 2.98 1.32% τxy 0 0 0% 28 3.3 Composite Beam Response to a Transverse Point Load The above examples show that the finite element model developed in ANSYS is accu- rate. The next step is to analyze the response of the laminate beam under different loadings and with different fiber orientations. The beam used is a four-layer symmetric laminate with the dimensions show in Figure 4. The material will remain graphite- polymer. The layup is defined as [θ/90]S, where θ ranges from 0º to 90º. Figure 17 shows the laminated beam constrained at one end and free at the other end. There is a force of 10N in the z-direction at the free edge of the beam. Figure 17. Laminated Beam with Fixed-Free Boundary Conditions Loaded with Fz To better understand the effects of fiber orientation on the beam, the orientation of the fibers in the outer layers increased from 0º to 90º. The orientation of the inner layers remained constant at 90º. As the orientation angle increases in the outer layers, the beam will behave more like an transversely isotropic beam. Because the beam was loaded with a force in the z-direction, displacement in the z-direction was used to compare how the stiffness of the beam changed with orientation angle. Figure 18 shows a plot of z- displacement for the different angles. 29 4.5 Displacement in the Z-Direction (mm) 4 3.5 3 2.5 2 1.5 1 0.5 0 0 10 20 30 40 50 60 70 80 90 Fiber Orientation Angle (degree) Middle-Top Middle-Bottom End-Top End-Bottom Figure 18. Displacement in the Z-Direction Due to Fz=10N Four locations on the beam are plotted: the top and bottom of the beam at L/2 and the top and the bottom of the beam at L. The bottom of the beam, both at the middle and end locations, displaces more than the top of the beam until the orientation angle is approximately 65º. At this point, the beam begins to behave as if all of the fibers are aligned in the same direction. The converging line illustrates this behavior. The bottom edge of the beam leads the top edge in z-displacement because the 90º-layers are in the center of the laminate and the angled fibers are on the outside of the laminate. Due to the force in the z-direction, the angled fibers will begin to curl around the x-axis. This curling effect will cause the bottom edge of the layer to displace more than the top edge. Next the stresses were analyzed for the same beam and point load. Figure 19 shows the effect of fiber orientation on stress in the laminate. Here, Von Mises stress was plotted instead of the three principal stresses to simplify the comparison against the orientation angle. 30 80 8000 70 7000 Von Mises Stress - Outer Layers (MPa) Von Mises Stress - Inner Layers (MPa) 60 6000 50 5000 40 4000 30 3000 20 2000 10 1000 0 0 0 20 40 60 80 Fiber Orientation Angle (degree) End-Inner Beginning-Outer Middle-Outer Figure 19. Von Mises Stress Due to Fz=10N There are a number of important observations from Figure 19: 1. The stress in the inner layers was constant over the length of the beam for a given fiber orientation in the outer layers. As discussed in Section 3.2.1 this is because the fibers are perpendicular to the applied force and are not stressed axially. It is the matrix material that experiences the higher levels of stress. 2. Conversely, the stress in the outer layers was not constant over the length of the beam for a given orientation angle. In the above graph, the stress in the outer layers is plotted in two locations: at the fixed end of the beam at that middle of the beam. The two locations reach a relative maximum and minimum stress at opposite locations. For example, at 45º the stress at the beginning of the beam is at its maximum, while the stress in the middle of the beam is at its minimum. 3. The stress in the outer layers at the free end of the beam was equivalent to the stress at the inner layers because it was far enough away from the fixed con- 31 straint that the stress induced in the layers was small. This stress was not plotted in Figure 19. Finally, the strains were analyzed and are plotted in Figure 20. 0.25 0.2 Von Mises Strain 0.15 0.1 0.05 0 0 20 40 60 80 Fiber Orientation Angle (degree) Beginning-Inner Middle-Inner Middle-Outer Figure 20. Von Mises Strain Due to Fz=10N As with the Von Mises stresses, there are a few important observations from the strain plot: 1. The strain in the inner layers is not constant over the length of the beam. Strain at the fixed end of the beam reaches a maximum value at a low orientation angle and remains elevated until all of the fibers of the beam begin to align with the 90º-fibers. However, further away from the fixed edge, the strain in the inner layers is not affected by the angle of the outer layers. At approximately L/4 away from the fixed edge, the strain in the inner layers becomes constant. 32 2. At the middle of the beam, the stress in the outer layers repeats a distinctive pat- tern. The strain plateaus at a strain level for approximately 20º and then increases. This pattern is repeated for 0-20º, 20-40º, and above 40º. 3. The strain at the end of the beam is not plotted because it is the same as the strain in the inner layers of the beam away from the fixed end. Strain at the end is min- imal because it is displacing more than it is changing length. 3.4 Modal Analysis of a Composite Beam Also of interest are the modes of vibration for the laminated beam. Modal analysis is used in ANSYS to extract the first five modes of vibration for [θ/90] S, where θ ranges from 0-90° The Block Lanczos method of extraction is used because this method is well suited for large eigenvalue problems and typically has a shorter run time than other methods. The model has the same degree-of-freedom constraints as the model used in Section 3.3.1. Figure 21 shows the first five modes of vibration for [0/90]S. Figure 21. First Five Mode Shapes of [0/90]S Laminate Beam 33 The effective stiffness of the beam changes as the fiber orientation of the outer layers increase from 0° to 90°. Figure 22 is a plot of the frequency at which the first five mode shapes appear. 1400 1200 1000 Frequency (Hz) 800 600 400 200 0 0 20 40 60 80 100 120 140 160 180 Fiber Orientation (degree) Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Figure 22. Mode Shape Frequency for [θ/90]S Laminate Beam The frequency plot shows important characteristics of the orientation angle: 1. The natural frequencies of the beam are symmetric around θ = 90º. This shows that the beam behaves similarly regardless of whether the fibers in the outer layer are oriented at +θ or –θ. Figure 23 illustrates this behavior by comparing the dis- placement of the beam in the z-direction due to a force in the z-direction (same as Section 3.3.1). The stress and displacement are equivalent, but the direction of the stress and displacement change. 34 Figure 23. Comparison of the Z-Displacement for +θ or –θ Due to Fz 2. The beam begins to behave like a transversely isotropic beam as θ approaches 90°. At this point the fibers in all four layers are oriented in the same direction. 3. This frequency plot gives an indication as to when the beam is the stiffest. The equation for the natural frequency of a cantilever beam is defined below. It shows that the natural frequency of the beam increases proportionally with the stiffness of the beam. The beam is, therefore, stiffest when the angle of the outer layers is close to 0°. At this orientation, the fibers in the inner and outer layers are perpendicular. k 3EI ωn = where, k = m L3 35 4. Conclusion ANSYS was used to simulate the behavior of a four-layer, symmetric laminate that was constrained as a cantilever beam. The finite element model was verified by showing the solutions for two problems using classical lamination theory. This theory invokes the Kirchhoff hypothesis, which states that there is no through-thickness strain. In actuality, there is through-thickness strain, as shown by the ANSYS model, but the hypothesis is robust enough to validate the finite element model. The beam used in the simulations was constructed as a [θ/90]S layup for simplification. The response of the beam was investigated as a function of the orientation of the fibers in the outer layers. Specifically, stress, strain, displacement, and modes of vibration were investigated. The ANSYS simulations show that the beam has a maximum stiffness when the fibers in the outer layers are oriented perpendicular to the fibers of the inner layers. This is an intuitive conclusion, and it is supported by the following figures: 1. Figure 18 shows the minimum displacement occurs θ=0°. 2. Figure 19 shows the maximum Von Mises stress occurs at θ=0°. 3. Figure 20 shows that minimum Von Mises strain occurs at θ=0°. 4. Figure 22 shows that maximum natural frequencies for the first five mode occurs at θ=0°. In terms of laminate design, this information is useful. This paper shows a subset of the possible fibers orientations because only the outer layers varied. However, the afore- mentioned figures give insight into the optimal orientation angle of the outer layers. Two design intent examples are described: 1. The laminate is to be designed for maximum flexibility, such as a downhill ski. The recommended orientation angle for the outer layers is 40-50°. In this range, strain and displacement are maximized and stress is minimized. This should provide for minimum stiffness. 36 2. The laminate is to be designed for maximum rigidity, such as a bicycle frame. The recommended orientation angle for the outer layers is 0-10°. In this range, strain, displacement, and stiffness are minimized. In this range or orientation an- gles, however, stress is very high, so a more thorough failure analysis must be done. 37 5. References 1. Adams, R. D. and M. R. Maheri. “Dynamic Flexural properties of Anisotropic Fibrous Composite Beams.” Composites Science and Technology 50 (1994) 497-514. 2. Assarar, Mustapha, Jean-Marie Berthelot, Abderrahim El Mahi, and Youssef Sefrani. “Damping analysis of Orthotropic Composite Materials and Laminates.” Composites: Part B 39 (2008) 1069-1076. 3. Benchekchou, B., M. Coni, and R. G. White. “The Structural Damping of Com- posite Beams with Tapered Boundaries.” Composites Structures 35 (1996) 207- 212. 4. Chawla, Krish, Uday K. Vaidya, and Ashutosh Goel. “Fatigue and Vibration Re- sponse of Long Fiber Reinforced Thermoplastics.” Univ. of Birmingham. 5. Della, Christian N. and Dongwei Shu. “Free Vibration of Composite Beams with Two Non-overlapping Delaminations.” International Journal of Mechanical Sci- ences 46 (2004) 509-526. 38