A Finite Element Analysis Method for Determining the Effect of Plate Distortion on Structural Analysis By Michael Lee An Engineering Project Submitted to the Graduate Faculty of Rensselaer Polytechnic Institute In Partial Fulfillment of the Requirements for the degree of MASTER OF ENGINEERING Major Subject: MECHANICAL ENGINEERING Approved: _________________________________________ Dr. Ernesto Gutierrez-Miravete, Engineering Project Adviser Rensselaer Polytechnic Institute Hartford, Connecticut December 2012 © Copyright 2012 By Michael R. Lee All Rights Reserved ii CONTENTS A Finite Element Analysis Method for Determining the Effect of Plate Distortion on Structural Analysis........................................................................................................ i LIST OF TABLES ............................................................................................................. v LIST OF FIGURES .......................................................................................................... vi ACKNOWLEDGMENT ................................................................................................. vii ABSTRACT ................................................................................................................... viii 1. Introduction.................................................................................................................. 1 2. Problem Description & Project Background ............................................................... 2 2.1 General Behavior of Plates when Deflections are Small ................................... 3 2.2 General Behavior of Plates when Deflections are Large ................................... 4 2.3 General Behavior of Shells ................................................................................ 5 2.4 Problem Discussion............................................................................................ 6 3. Analysis Methodology & Approach ............................................................................ 7 3.1 Plate Sizes .......................................................................................................... 7 3.2 Plate Material Selection ..................................................................................... 8 3.3 Finite Element Model Description ..................................................................... 8 3.4 Discussion Regarding Element Type Selection ................................................. 9 3.5 Using the von Mises Stress Criterion ............................................................... 10 3.6 Finite Element Analysis Procedure .................................................................. 10 4. Analysis Results......................................................................................................... 12 4.1 Finite Element Analysis Results ...................................................................... 12 4.2 Finite Element Analysis Results Discussion .................................................... 15 5. Future Work ............................................................................................................... 16 6. References & Relevant Literature ............................................................................. 17 7. Appendix: Finite Element Analysis Result Plots....................................................... 18 7.1 Von Mises Stress Results ................................................................................. 18 iii 7.2 Displacement Stress Results ............................................................................ 24 iv LIST OF TABLES Table 1: Material Properties of Structural A36 Steel Alloy .............................................. 8 Table 2: Finite Element Analysis Results Tabulated ....................................................... 12 Table 3: Finite Element Analysis Results - Deflection and Stress Ratios ....................... 13 v LIST OF FIGURES Figure 1: Generic Thin Plate Geometry from Reference (3) ............................................. 3 Figure 2: Finite Element Model of the Plate ..................................................................... 9 Figure 3: Deflection Ratio versus Initial Deflection........................................................ 14 Figure 4: Actual Deflection Ratio versus Initial Deflection ............................................ 14 Figure 5: Stress Ratio versus Initial Deflection ............................................................... 15 Figure 6: Iteration 1, Initial Deflection 0.0 inches - von Mises Stress ............................ 18 Figure 7: Iteration 2, Initial Deflection 0.006163 inches - von Mises Stress .................. 19 Figure 8: Iteration 3, Initial Deflection 0.012308 inches - von Mises Stress .................. 19 Figure 9: Iteration 4, Initial Deflection 0.0184 inches - von Mises Stress ...................... 20 Figure 10: Iteration 5, Initial Deflection 0.024406 inches - von Mises Stress ................ 20 Figure 11: Iteration 6, Initial Deflection 0.030297 inches - von Mises Stress ................ 21 Figure 12: Iteration 7, Initial Deflection 0.036051 inches - von Mises Stress ................ 21 Figure 13: Iteration 8, Initial Deflection 0.041648 inches - von Mises Stress ................ 22 Figure 14: Iteration 9, Initial Deflection 0.047077 inches - von Mises Stress ................ 22 Figure 15: Iteration 10, Initial Deflection 0.05233 inches - von Mises Stress ................ 23 Figure 16: Iteration 11, Initial Deflection 0.057403 inches - von Mises Stress .............. 23 Figure 17: Iteration 12, Initial Deflection 0.062297 inches - von Mises Stress .............. 24 Figure 18: Iteration 1, Initial Deflection 0.0 inches - Deflected Shape ........................... 24 Figure 19: Iteration 2, Initial Deflection 0.006163 inches - Deflected Shape ................. 25 Figure 20: Iteration 3, Initial Deflection 0.012308 inches - Deflected Shape ................. 25 Figure 21: Iteration 4, Initial Deflection 0.0184 inches - Deflected Shape ..................... 26 Figure 22: Iteration 5, Initial Deflection 0.024406 inches - Deflected Shape ................. 26 Figure 23: Iteration 6, Initial Deflection 0.030297 inches - Deflected Shape ................. 27 Figure 24: Iteration 7, Initial Deflection 0.036051 inches - Deflected Shape ................. 27 Figure 25: Iteration 8, Initial Deflection 0.041648 inches - Deflected Shape ................. 28 Figure 26: Iteration 9, Initial Deflection 0.047077 inches - Deflected Shape ................. 28 Figure 27: Iteration 10, Initial Deflection 0.05233 inches - Deflected Shape ................. 29 Figure 28: Iteration 11, Initial Deflection 0.057403 inches - Deflected Shape ............... 29 Figure 29: Iteration 12, Initial Deflection 0.062297 inches - Deflected Shape ............... 30 vi ACKNOWLEDGMENT To my family and friends, it’s been an adventure. Words simply can not express my gratitude to you all. Thank you for your patience, love, understanding, encouragement, and continued support in all my endeavours. To my family, I love you all. Specifically, to my parents and sister, the words thank you seems to fall short of the mark. I owe you three much. To my friends, you’ve helped me along throughout this whole process in more ways than one. For a specific few, they know who they are, thank you for answering my technical questions throughout this endeavour. But for all of you, “when you get the choice to sit it out or dance…. I hope you dance” – Lee Ann Womack (I hope you dance). So dance. vii ABSTRACT This project attempts to provide a viable method to establish the correlation between the deflection and stress results of a shaped (distorted) plate analyzed as a thin shell when compared to the original (flat) thin plate analytical solution. This project analyzes an elastic loading of plates with different shapes which are obtained from the loading of the previous (plate) iteration with less deflection. Thus, a relationship between the increasing plate distortion and the resulting stress and deflections can be established. Engineering calculations share a common assumption where the material being analyzed is assumed to be perfect. Material is far from perfect in the manufacturing environment. Manufacturing imperfection(s) that exceed out of tolerance condition(s) requires an engineering evaluation to determine if the end, final product is technically satisfactory. Analysis based upon (flat) thin plate, small deflection theory is invalidated when the asbuilt condition would dictate the performing a new calculation using thin shell, small deflection theory. Thin plate, small deflection theory assumes that the midplane remains unstrained subsequent to bending. As a result of the deformation (leads to bending), the midsurface of a shell is strained in comparison. This invalidates the (original) analysis based on thin plate, small deflection theory. The stress in a shell will differ from the equivalent plate as shell theory accounts for the strain at the midplane whereas plate theory does not. There is no correlation between the expected change in stress and plate deformation, where shell theory is the more valid analytical assumption. An exact solution requires a new calculation by the analyst. This project is of value to an analyst, as modeling the as-built condition and performing the calculations can be time consuming. This can be avoided if the correlation between a given plate distortion and the impact it has on stress and deflection is known. viii 1. Introduction A common assumption in engineering analysis and calculations is where the material being analyzed is perfect, free of material imperfections, free of residual stresses from manufacturing and fabrication. Material is far from perfect in the manufacturing environment. Basic tolerances are introduced into a product from the beginning of the production cycle. The material specifications allow for variations in part due to the tolerances that govern the technical acceptability of the ordered raw material (stock) used in later manufacturing process or processes. The as-machined condition is further governed by the drawing and manufacturing practice tolerances; for example, requirements set by techniques such as geometric dimensioning and tolerancing. Additional tolerances govern the installation or assembly of a component, affecting the final as-built condition. Manufacturing imperfections can exceed the product’s tolerance condition(s) which requires an engineering evaluation to ensure that the final condition is technically acceptable. This evaluation, depending on the nature of the problem, can be quite complex and time consuming to determine the technical acceptability of an asbuilt or as-received material condition. The intent of this project is to establish a viable method of determining an actual “tolerance” range based upon engineering analysis versus standard design practices. This “tolerance” range is based upon a correlation between the “imperfect” plate (both deflection and stress) and the original “perfect” plate solution. This analysis will examine a rectangular plate, fixed on all edges, subjected to a uniform pressure loading. The resulting deflected shape will be used as the distorted shape for the next iteration. The next iteration analyzes a deflected shape with no pre-existing, residual stresses in the plate. The same loading and boundary conditions are retained from one analysis step to the next. This process was chosen to simply the analytical method to one (1) Finite Element Model (FEM). 1 2. Problem Description & Project Background The project is based on thin plate, small deflection theory. Engineering calculations, using the equations from References (1) and (2), are often used to perform quick verifications of a large Finite Element Analysis (FEA) or provide a technical evaluation of a plate under the applicable boundary and loading conditions. These calculations demonstrate in either case the technical acceptability of a given design. As-built, out of tolerance conditions require recalculations to verify that the work accomplished meets the design requirements. In some cases, the resulting as-built condition analysis is performed in accordance with shell theory in lieu of the original thin plate analysis. The intent of this project is to determine a tolerance range or a scaling factor where, in-plane tensile and membrane stresses and forces are introduced. These “internal” stresses and forces will skew the initial thin plate, small deflection theory, analytical results. A plate is defined as an initially flat structural element, such as table tops, roofs, panels, and disks. The thickness of a plate is much smaller than the other dimensions. Thin shells are curved surface structures, such as wings, air foils, pipes, and pressure vessels. Manufacturing imperfections are typically defined a surface cracks, dents, or notches to a given part or component material condition. For this purposes of this project, manufacturing imperfections are considered to be a result of distortion from fabrication processes, such as welding, which results in the plate exceeding the flatness requirement from the material and/or design specification(s) or extreme variations in fabrication methods which leads to distortion of the plate, among other concerns. The following information regarding plate and shell theory presented in Sections 2.1 through 2.4 is taken in part from Reference (3). 2 2.1 General Behavior of Plates when Deflections are Small Figure 1 below shows a schematic representation of a typical plate. The schematic on the t right illustrates the behavior of the midplane plate cross sections at the initial condition (top) and after some load is applied (bottom). Figure 1: Generic Thin Plate Geometry from Reference (3) The four (4) fundamental assumptions of thin plate (refer to Figure 1), small deflection theory of bending (also known as the classical [plate] theory, Kirchhoff hypotheses) are as follows: 1. Deflection of the midplane is small compared with thickness, w << t. Therefore, w 1 and the square if the slope is the slope of the deflected surface is small x w 2 negligible 0 . x 2. Midplane remains unstrained subsequent to bending. 3. Plane sections (mn, see Figure 1) initially normal to the midplane remain plane normal after bending, therefore out of plane shear strains are zero xz yz 0 . Deflection is associated principally with bending strains. Out of plane normal strain z is also omitted. 4. Out of plane normal stress z is neglected. Thus, the assumption is unreliable near concentrated transverse loads. 3 Note that the midplane is parallel to the faces of the plate, dividing the thickness in half. Plate thickness (t) is measured normal to the midplane. Flexural properties of the plate depend upon thickness (t). If the resulting deflections are not small, Assumptions 1 and 2 are invalid. Assumption 1 is invalid since the deflection is on the order of the plate thickness (w ≈ t). Assumption 2 would be invalid since plate bending results in midplane strains. If the plate is thick, Assumptions 3 and 4 are invalid. Assumption 3 is invalid since shear strains are important. Assumption 4 is invalid since out-of-plane normal stress is important. The one exception, where deflection can be large where the classical (Kirchhoff) assumptions still hold true, is when a plate bends into a developable surface, such as cones and cylinders, and the midplane can remain unstrained. There are two conditions that govern this exception. 1. Pure bending of plate to a cylindrical surface, which occur via moments, requires that the thickness to be much less than the radius of curvature (t << radius of curvature). 2. Bending of a plate to a cylindrical surface by lateral load requires the deflection to be much less than the width of the plate (w << plate dimensions other than thickness). 2.2 General Behavior of Plates when Deflections are Large In the case of large displacements (w ≥ t), the midplane of the plate stretches. In-plane tensile stresses develop within the plate stiffen and add considerable load resistance to it. Large deflection theory assumes that the resulting deflections (w) are no longer small in comparison with the thickness (t) but are small when compared with the remaining plate dimensions. An analytical result indicating large deflections would be recomputed to ensure that the derived solution is indeed both mathematically and technically correct. 4 Large deflections of plates can lead to the plate forming both developable surfaces, such as cylinders and cones, and non-developable surfaces, such as spheres and saddles. A developable surface will recover its original flat shape and dimensions and implies the absence of any deformation. When a plate bends into a cylindrical geometry, tension in the midplane can only be produced if the end supports are immovable. The requirement here is that the deflection (w) must be less than the plate thickness (t) does not hold in cases where a simplysupported plate bends into a developable surface. The classical formulas are still valid for this case, at least until yielding occurs or when the deflection (w) approaches the plate dimensions. Otherwise, the relationship between a given load and the resulting deflection is no longer linear for large deflections. The relationship is linear for small deflections or large deflections that result in a developable surface. It is not possible to apply a unit load and scale to large deflection solutions. The linear theory neglects the membrane action and is satisfactory for cases where the resulting maximum deflection (wmax) is less than half the t shell thickness wmax . 2 2.3 General Behavior of Shells Thin shells are assumed to have a constant thickness. The shell thickness is small in comparison to the shell dimensions in the other two directions. The shell midsurface is the plane that bisects the shell thickness. A shell is considered thin if the thickness (t) to 1 t radius of curvature (r) ratio is small . Shell theory is broken into two distinct, r 20 commonly applied theories. 1. Membrane Theory: A membrane is a body with the same shape as a shell but incapable of conveying the moments or shear forces. 5 2. Bending (or general) theory: Includes the effect of bending. Membrane theory is corrected where discontinuity effects are pronounced (changes in thickness, slop, curvature). Information relative to shell membrane stresses is usually of greater practical significance than the knowledge of bending stresses. The membrane stresses are considered uniform through wall thickness assuming that the shells have no abrupt changes in thickness, slope, or curvature. Bending theory is equivalent to membrane theory and is corrected in areas of discontinuity. Membrane forces are independent of bending and completely definite by conditions of static equilibrium. No material properties are used in its derivation. Therefore, membrane theory applies to all shells made of any material. The assumptions of shell theory, assuming small deflections, are as follows: t 1. Thickness (t) to radius of curvature (r) ratio is much less than 1 1 . r 2. Deflection of the midplane is small compared with thickness, w << t. 3. Plane sections normal to the midsurface remain plane and normal after bending occurs xz 0, yz 0, z 0 . 4. Out of plane normal stress z is negligible. Load resisting action of a shell differs from other structural forms. In-plane forces are developed to form a primary resistance action. 2.4 Problem Discussion Assumption 2 from thin plate, small deflection theory is invalidated when compared to thin shell theory. The mid-surface of a shell is strained, which differs from plate theory. Therefore, in the original analysis performed using thin plate, small deflection theory is invalidated when the resulting fabricated (as-built) condition would dictate the repeating the engineering analysis utilizing thin shell, small deflection theory. 6 3. Analysis Methodology & Approach A Finite Element Model (FEM) will be built and is representative of a perfect, flat rectangular plate. The FEM “plate” has fixed boundaries on all sides with a uniform pressure loading. The software used for this investigation is Abaqus 6.10. The FEA in this study will be an iterative process. The initial FEA result data set will provide a baseline deflection and stress values. This first analytical iteration is the original thin plate, small deflection solution. The resulting deflected shape will be used as the distorted shape for the next calculation. This next calculation analyzes a deflected shape with no pre-existing, residual stresses in the plate. The results of the following iterations will be compared to this initial FEA solution. Load resistance from in plane forces is developed as a result of the increasing deflections, which are accounted for in the shell solution. As the plate deflection increases, the plate will behave more like a shell. The FEA results will show the strained midsurface of the shell and membrane stresses. The FEA results will be summarized and compared against one another. The change in stress will be compared between the different FEM and the individual iterations. FEA results that satisfy the criterion for large deflection of plates (described in Section 2.2) will be noted but not considered as a part of this correlation. 3.1 Plate Sizes Initial plate sizes were generated partly from applying the equations from Table 11.4, Section 8a from Reference (1). This was done to verify that the initial solution from the FEA are valid within reason and to ensure that the initial condition meets the requirements of the small deflection theory of bending (see Section 2.1). Additionally, this provides a verification of the FEM by comparing the handbook solution (Reference (1)) with the FEA results. The solution presented in Reference (1) is based on plate theory where as the FEA used in this investigation is based on shell theory. The chosen plate size to model is 12 inches long (“a” dimension in the equations from Reference 7 (1)), 6 inches wide (“b” dimension in the equations from Reference (1)), and 0.125 inches thick. 3.2 Plate Material Selection The material chosen for this analysis is Structural A36 Steel Alloy. The relevant material properties, from Reference (4), are summarized below in Table 1 below. Table 1: Material Properties of Structural A36 Steel Alloy Modulus of Elasticity (E) 29,000,000 psi Poisson’s Ration (υ) 0.32 Yield Strength (σy) 36,000 psi Ultimate Tensile Strength (σUTS) 58,000 psi Density (ρ) 0.284 lbs/in3 Structural A36 Steel Alloy was chosen as it is an isotropic, linear elastic material. The use of an isotropic, linearly elastic material is consistent with the formulae derivations presented in engineering handbooks and design texts such as References (1), (2), and (5). It should be noted though, that this investigation is a comparison of the initial results case and the deflected load case. Actual material property values will influence the behavior of the plate under loading and the acceptance criteria of a given design. 3.3 Finite Element Model Description Each plate was modeled using quad4 shell elements at the midsurface. An image that is representative FEM is shown in Figure 2 below. 8 Figure 2: Finite Element Model of the Plate All four edges of the plate were fixed, allowing no displacements or rotations in any directions as noted previously. The thickness of the plate is entered via the crosssectional property assigned to the meshed plate. A 10 psi load was distributed uniformly over the entire, topside of the plate. 3.4 Discussion Regarding Element Type Selection The majority of the commercially available FEA software packages or codes can handle plate, shell geometry, and in plane and out of plane loading as long as enough elements are used to represent the physical geometry. The number of elements required is based upon a reasonable approximation of the physical object being modeled, such as the relationship or fit between the modeled curvatures in a FEM to the physical “part” shape or curvature. The typical definition of a plate is an element capable of supporting both in-plane and lateral loads. The plate definition also includes the assumption of being flat initially. A shell is often considered to be a plate with either two dimensional curvatures or as a complete shell of revolution that supports both in-plane and lateral loads. For most 9 element derivations used in commercial codes, classical plate element types do not consider the development of membrane stresses where as shell element types do. Of course in reality, plates will develop such membrane stresses but the classical formulations for these are not considered, in part due to the rationale discussed previously in Section 2. Thus, the use of plane or plate elements alone cannot be used to properly represent the response of a shell subject to arbitrary loadings. Hence, the FEM generated for this investigation utilizes shell elements in lieu of plate elements as the analyst may have done in the initial, baseline analysis. 3.5 Using the von Mises Stress Criterion The von Mises stress is used to perform the comparison of the FEA stress results. This allows for measuring stress at one single scalar variable. Examining the principal stresses or the stress states within different sections of the plate may provide additional information, for example show the different layers through the plate thickness in a state of compression or tension. Alternatively, the analysis can examine the stresses based on Cartesian coordinates. This would however increase the number of different stress variables (or directions) to examine and compare. Most structural evaluations are based on using the von Mises stress criterion for a convenient comparison with the material yield strength. Therefore, the stress results presented by this investigation are based upon the von Mises stress. 3.6 Finite Element Analysis Procedure The FEA results are post processed within the Abaqus software. From the “Results Context Space” within Abaqus, a report of the iteration’s nodal displacement results are generated. These are imported into a Microsoft Excel spreadsheet and added to the nodal locations from the iteration’s input deck (the “.inp” files). The next iteration’s input file is then created from the revised, updated nodal locations. This is imported back into Abaqus as a new FEM and submitted as a new analysis. A second method is to directly submit the input deck to the Abaqus software utilizing the Abaqus Command Window 10 option. Regardless of methodology, the process repeats itself by opening or otherwise opening the new results database (the “.odb” files) to generate the nodal displacements report for the next iteration. 11 4. Analysis Results The information presented within this Section is derived from the FEA result plots, showing the von Mises stress and maximum deflection results, are provided in Section 7. This analysis takes a uniform pressure loading of plates with different shapes where the shapes are obtained from the loading of a parent plate with less deflection. The deformed shapes begin to exhibit the behavior of shell theory versus thin plate, small deflection theory as the initial plate deflection increases. 4.1 Finite Element Analysis Results This section presents the ratios between the stress and deflections in the initial condition and as a result of the deflected shape as the (initial) plate deflection increases. Note that the stress results presented here is the maximum stress, at the center of the long edge of the plate (see Section 7.1). This is expected in accordance with the handbook solution provided by Reference (1). It should be noted that the stresses at the center of the plate increased with each iteration. This behavior was also expected. Table 2: Finite Element Analysis Results Tabulated Initial Iteration Deflection (inches) 1 2 3 4 5 6 7 8 9 10 11 12 0.0 0.006163 0.012308 0.0184 0.024406 0.030297 0.036051 0.041648 0.047077 0.05233 0.057403 0.062297 Von Mises Stress (psi) 7711 7550 7356 7135 6893 6638 6374 6110 5848 5594 5351 5119 Maximum Deflection (inches) 0.006163 0.006145 0.006092 0.006006 0.005891 0.005754 0.005597 0.005429 0.005253 0.005073 0.004894 0.004718 12 Actual Deflection (inches) 0.006163 0.012308 0.0184 0.024406 0.030297 0.036051 0.041648 0.047077 0.05233 0.057403 0.062297 0.067015 Meets Small Deflection Theory YES YES YES YES YES YES YES YES YES YES YES NO The initial deflection presented in Table 2 above is the shape of the plate at the beginning of the analysis iteration, with 0.0 inches representing the perfectly flat condition. The “Actual Deflection” is the summation of the initial deflection with the resulting maximum deflection obtained from the FEA iteration results. Deflection is based upon the nodal displacements. Note that the last iteration failed to meet small deflection theory as discussed in Section 2.1. Any following initial deflected condition would automatically invalidate the initial baseline analysis, in this case the first iteration. No ratio or scaling factor can be applied to the thin plate, small deflection theory initial results as the large deflection theory (see Section 2.2) becomes applicable. Table 3: Finite Element Analysis Results - Deflection and Stress Ratios Iteration 1 2 3 4 5 6 7 8 9 10 11 12 Initial Deflection (inches) 0.0 0.006163 0.012308 0.0184 0.024406 0.030297 0.036051 0.041648 0.047077 0.05233 0.057403 0.062297 Deflection Ratio 1.0 0.9971 0.9885 0.9745 0.9559 0.9336 0.9082 0.8809 0.8523 0.8231 0.7941 0.7655 Actual Deflection Ratio 1.0 1.9971 2.9856 3.9601 4.9160 5.8496 6.7577 7.6387 8.4910 9.3141 10.1082 10.8738 Stress Ratio 1.0 0.9791 0.9540 0.9253 0.8939 0.8608 0.8266 0.7924 0.7584 0.7255 0.6939 0.6639 Table 3 presents the correlation between the initial analysis step, Iteration 1, and the follow on analytical steps or iterations for the FEA von Mises Stress and the nodal displacement or deflection. The “Deflection Ratio” is based on the maximum deflection from a given iteration versus the maximum deflection from the first iteration. The “Actual Deflection Ratio” is similar, except it is a comparison of the “Actual Deflection” (refer to the discussion of the results presented in Table 2) versus the maximum deflection from the first iteration. The “Stress Ratio” is the comparison between the von Mises stress of a given iteration versus the von Mises stress obtained by first iteration. 13 1.20 Deflection Ratio 1.00 0.80 0.60 0.40 0.20 0.00 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.06 0.07 Initial Deflection (inches) Deflection Ratio Figure 3: Deflection Ratio versus Initial Deflection Actual Deflection Ratio 12 10 8 6 4 2 0 0 0.01 0.02 0.03 0.04 0.05 Inital Deflection (inches) Actual Deflection Ratio Figure 4: Actual Deflection Ratio versus Initial Deflection 14 1.2 Stress Ratio 1 0.8 0.6 0.4 0.2 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Initial Deflection Stress Ratio Figure 5: Stress Ratio versus Initial Deflection The “Deflection”, “Actual Deflection”, and “Stress” ratios presented in Table 3 are plotted in Figure 3, Figure 4, and Figure 5 above respectively. The comparison with in each figure is based on the changing plate dimensions, with respect to its increasing curvature. 4.2 Finite Element Analysis Results Discussion The results presented in Section 4.1 above illustrate the affect of the increasing plate curvature or deflection on the analytical results. Due to both the size and resulting shape of the plate, the stress and maximum deflection values obtained from the FEA decreased over each analysis step. This is due to the resulting shape forming into the conical shape as discussed in Sections 2.2 and 2.3. As the plate bends, the in-plane forces are developing and forming a primary resistance action to the pressure loading. It should be noted that over each iteration, the stresses at the center of plate increase. Refer to Section 7.1 for the von Mises Stress plots. 15 5. Future Work The results from this investigation are limited to one loading condition and demonstrates the analytical method that can be used to determine the correlation between a given plate deformation and stress when compared to the original solution. To expedite this process, a software code should be written to automate the process of generating the next initial condition. Additional work is required to address the other load cases covered by handbooks such as “Roark’s Formulas for Stress and Strain” (Reference (1)). Future studies should consider producing deformed shapes by using these different types of loading conditions on a first iteration. The resulting deformed plate shapes should then be subjected to a second FEA iteration under the same loading conditions. This should be repeated until the results violate thin plate, small deflection theory. Further analysis is required of additional plate sizes (and thicknesses) from what was performed within this study. It should be further noted that this investigation is an analytical in nature only. Testing of instrumented, physical specimens should be performed as a validation and verification of the analytical results. Another variation of this project would be the examination of the behavior but utilizing a non-isotropic materials like composites, which for a variety of reasons, are increasingly being used in product design. However, it should be noted that the shell and plate theories outlined in Section 2 are in part based on isotropic materials. 16 6. References & Relevant Literature (1) Young, Warren C. and Budynas, Richard G.; Roark’s Formulas for Stress and Strain, Seventh Edition; McGraw-Hill, New York, 2002. (2) Timoshenko, Stephen P. and Woinowsky-Krieger, S.; Theory of Plates and Shells, Second Edition; McGraw-Hill, New York, 1959. (3) Vasko, Thomas J. and Bak, Michael; Course Notes from MANE-6200 Plates and Shells; Fall Semester 2010. (4) Hibbler, R. C.; Mechanics of Materials, Sixth Edition; Pearson Prentice Hall, New Jersey, 2005. (5) Shigley, Joseph E., Mischke, Charles R., and Budynas, Richard G.; Mechanical Engineering Design, Seventh Edition; McGraw-Hill, New York, 2004. 17 7. Appendix: Finite Element Analysis Result Plots 7.1 Von Mises Stress Results This section presents the FEA von Mises stress results. Figure 6: Iteration 1, Initial Deflection 0.0 inches - von Mises Stress 18 Figure 7: Iteration 2, Initial Deflection 0.006163 inches - von Mises Stress Figure 8: Iteration 3, Initial Deflection 0.012308 inches - von Mises Stress 19 Figure 9: Iteration 4, Initial Deflection 0.0184 inches - von Mises Stress Figure 10: Iteration 5, Initial Deflection 0.024406 inches - von Mises Stress 20 Figure 11: Iteration 6, Initial Deflection 0.030297 inches - von Mises Stress Figure 12: Iteration 7, Initial Deflection 0.036051 inches - von Mises Stress 21 Figure 13: Iteration 8, Initial Deflection 0.041648 inches - von Mises Stress Figure 14: Iteration 9, Initial Deflection 0.047077 inches - von Mises Stress 22 Figure 15: Iteration 10, Initial Deflection 0.05233 inches - von Mises Stress Figure 16: Iteration 11, Initial Deflection 0.057403 inches - von Mises Stress 23 Figure 17: Iteration 12, Initial Deflection 0.062297 inches - von Mises Stress 7.2 Displacement Stress Results This section presents the FEA maximum deflection results. Figure 18: Iteration 1, Initial Deflection 0.0 inches - Deflected Shape 24 Figure 19: Iteration 2, Initial Deflection 0.006163 inches - Deflected Shape Figure 20: Iteration 3, Initial Deflection 0.012308 inches - Deflected Shape 25 Figure 21: Iteration 4, Initial Deflection 0.0184 inches - Deflected Shape Figure 22: Iteration 5, Initial Deflection 0.024406 inches - Deflected Shape 26 Figure 23: Iteration 6, Initial Deflection 0.030297 inches - Deflected Shape Figure 24: Iteration 7, Initial Deflection 0.036051 inches - Deflected Shape 27 Figure 25: Iteration 8, Initial Deflection 0.041648 inches - Deflected Shape Figure 26: Iteration 9, Initial Deflection 0.047077 inches - Deflected Shape 28 Figure 27: Iteration 10, Initial Deflection 0.05233 inches - Deflected Shape Figure 28: Iteration 11, Initial Deflection 0.057403 inches - Deflected Shape 29 Figure 29: Iteration 12, Initial Deflection 0.062297 inches - Deflected Shape 30