A Designer’s Approach for Optimizing an End-Loaded Cantilever Beam while Achieving Structural Requirements by Timothy M. Demers 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 IN MECHANICAL ENGINEERING Approved: _________________________________________ Ernesto Gutierrez-Miravete, Project Adviser Rensselaer Polytechnic Institute Hartford, Connecticut December, 2009 © Copyright 2009 by Timothy M. Demers All Rights Reserved ii CONTENTS LIST OF TABLES ............................................................................................................ iv LIST OF FIGURES ........................................................................................................... v LIST OF SYMBOLS ....................................................................................................... vii ACKNOWLEDGMENT ................................................................................................ viii ABSTRACT ..................................................................................................................... ix 1. Introduction.................................................................................................................. 1 1.1 Background ........................................................................................................ 1 1.2 Methods of Optimization ................................................................................... 2 1.3 Problem Description........................................................................................... 6 2. Methodology ................................................................................................................ 8 2.1 Baseline Structure .............................................................................................. 8 2.2 Optimized Structure ......................................................................................... 10 2.3 Material Properties ........................................................................................... 10 2.4 COMSOL Procedure & Importation Method .................................................. 10 3. Results and Discussion .............................................................................................. 12 3.1 COMSOL Calibration & Mesh Size Determination ........................................ 12 3.2 Baseline Structure ............................................................................................ 14 3.3 Initial Eight Designs ......................................................................................... 16 3.4 Iteration to Successful Designs ........................................................................ 30 4. Conclusion ................................................................................................................. 35 5. References.................................................................................................................. 37 6. Appendix A................................................................................................................ 38 iii LIST OF TABLES Table 1: 304 Stainless Steel Material Properties [10] ..................................................... 10 Table 2: Number of Elements vs. y-Displacement .......................................................... 13 Table 3: Summary of Initial Beam Designs .................................................................... 30 Table 4: Summary of Design Iterations ........................................................................... 34 iv LIST OF FIGURES Figure 1: Optimal Chair [8] ............................................................................................... 4 Figure 2: Typical Homogenization Optimization Solution with Intermediate Densities [1]..................................................................................................................... 5 Figure 3: Typical Checkerboard Region in an Optimum Solution [1] .............................. 5 Figure 4: Thickness Calibration Solution ........................................................................ 13 Figure 5: COMSOL Analysis of Baseline Structure ....................................................... 15 Figure 6: von Mises Stress Along Top of Baseline Beam ............................................... 16 Figure 7: COMSOL Analysis of Design 1 ...................................................................... 17 Figure 8: von Mises Stress Along Top of Design 1......................................................... 18 Figure 9: von Mises Stress Along Top of Holes of Design 1 .......................................... 18 Figure 10: COMSOL Analysis of Design 2 .................................................................... 19 Figure 11: von Mises Stress Along Top of Design 2....................................................... 19 Figure 12: von Mises Stress Along Top of Larger Holes of Design 2 ............................ 19 Figure 13: COMSOL Analysis of Design 3 .................................................................... 20 Figure 14: von Mises Stress Along Top of Design 3....................................................... 21 Figure 15: von Mises Stress Along Top of Slot of Design 3 ........................................... 21 Figure 16: COMSOL Analysis of Design 4 .................................................................... 22 Figure 17: Close-up View of Slot Stress of Design 4 ...................................................... 22 Figure 18: von Mises Stress Along Top of Design 4....................................................... 23 Figure 19: von Mises Stress Along Top of Slot of Design 4 ........................................... 23 Figure 20: COMSOL Analysis of Design 5 .................................................................... 24 Figure 21: von Mises Stress Along Top of Design 5....................................................... 25 Figure 22: von Mises Stress Through Ends of Slot of Design 5 ..................................... 25 Figure 23: COMSOL Analysis of Design 6 .................................................................... 26 Figure 24: von Mises Stress Along Top of Design 6....................................................... 26 Figure 25: von Mises Stress Along Top of Slot of Design 6 ........................................... 26 Figure 26: COMSOL Analysis of Design 7 .................................................................... 27 Figure 27: von Mises Stress Along Top of Design 7....................................................... 28 Figure 28: von Mises Stress Along Top of Slots of Design 7 ......................................... 28 Figure 29: COMSOL Analysis of Design 8 .................................................................... 29 v Figure 30: von Mises Stress Along Top of Design 8....................................................... 29 Figure 31: von Mises Stress Along Top of Larger Slots of Design 8.............................. 29 Figure 32: COMSOL Analysis of Design 2 Iteration ...................................................... 31 Figure 33: von Mises Stress Along Top of Design 2 Iteration ........................................ 32 Figure 34: von Mises Stress Along Top of Larger Holes of Design 2 Iteration .............. 32 Figure 35: COMSOL Analysis of Design 8 Iteration ...................................................... 33 Figure 36: von Mises Stress Along Top of Design 8 Iteration ........................................ 33 Figure 37: von Mises Stress Along Top of Fourth Slot of Design 8 Iteration................. 33 vi LIST OF SYMBOLS b base dimension of beam (m) c perpendicular distance from neutral axis to point farthest away from neutral axis, where σmax acts (m) E modulus of elasticity (Pa) h height dimension of beam (m) I moment of inertia of the cross-sectional area computed about the neutral axis (m4) L length dimension of beam (m) M moment (N-m) P applied load (N) v deflection x distance along beam from fixed end (m) density (kg/m3) σ normal stress (Pa) σy yield strength (Pa) Poisson’s Ratio vii ACKNOWLEDGMENT The author would like to thank his family who has been extremely supportive and patient during this Masters Project and the entire Master of Engineering curriculum. He would also like to thank the Rensselaer Polytechnic Institute faculty for sharing their knowledge and expertise. viii ABSTRACT This research utilized a detailed approach to designing an optimally-sized, load bearing structure. One of the simplest structures, a two-dimensional cantilever beam, was chosen to assess how geometric changes affect the load bearing capability of the beam. Eight beam designs were created and analyzed to verify their performance. Each of the beams was conceived from the designers’ point of view as being optimized for weight but still capable of sustaining an end load of 106 Newtons and capable of being manufactured with relatively simple and conventional techniques. The beams were created using Dassault Systems CATIA V5 and analyzed using COMSOL Multiphysics. Two of the successful designs were then iterated to create even lighter designs and to resolve structural inadequacies. Several recommendations are provided to assist the designer when generating a lightweight, manufactureable, and structurally-sound design. ix 1. Introduction 1.1 Background Optimization can be employed to find the best solution to problems that are quantifiable. This includes engineering design problems. Because many of the earlier optimization methods are gradient based and mathematically rigorous, computers have become a common tool in evaluating an optimum solution. Depending on the design criteria, an almost infinite number of quantifiable parameters exist which can be optimized. A general formulation of optimization determines the optimum solution by calculating the minimum or maximum value of a quantified parameter by varying design variables under the given design constraints [1]. “Optimum, the word derived from Ops, the name of the Sabine goddess of fertility and agricultural abundance, was first used by Leibniz in the 18th century, to mean the best of all possible” [1]. H. Kim, O. M. Querin, and G. P. Steven grouped structural optimization problems into three categories: sizing, shape, and topology. “Sizing optimization is typically applied to a truss-type structure to obtain the optimal cross-section areas of beams.” Plate thickness and beam cross-sectional areas are examples of the sizing design variables which can be modified. This type of optimization is relatively straightforward since it does not require changes to the Finite Element (FE) model of a structure as it is modified. Shape optimization determines the optimal boundaries of a structure for a defined fixed topology. Spline control points which define the shape of the structure are adjusted to provide an optimum shape. Unlike sizing optimization, shape optimization results in changes to the FE model, and therefore, adds difficulty due to combining mesh generation and finite element analysis into an optimization method. The downside to size and shape optimization is their dependency on the initial structure. Topology optimization overcomes this deficiency because the optimum topology is independent of the initial starting design. For this reason topology optimization methods are commonly utilized at the conceptual stage of a design process [1]. 1 In many industries achieving an optimum design is beneficial and sometimes critical to success. For example, in the aerospace field, achieving a minimum weight design for flight components is extremely important because a lighter component and system can directly reduce launch costs and reduce the number of missions required [2]. Traditionally, structural design optimization seeks to determine the optimal solution based on weight, volume, or compliance when exposed to the requirements of the design, such as displacements, stresses, and buckling [3]. This optimization provides either the most efficient or most effective use of the material in addition to minimizing the weight of the entire structure. In the studies examined by Fazil Sonmez, the goal of optimization “was to minimize the weight of the structure, in others to increase mechanical performance, e.g. to minimize stress concentration, maximize fracture strength, buckling strength, fatigue life, and heat flux, minimize peak contact stress, compliance, peak acceleration, and the probability of failure for brittle materials, and optimize dynamic behavior of structures” [4]. From this list, one realizes to the extent at which optimization can be applied. This increase in demand for lightweight and high performing structures drives the current research in the field of optimization [4]. 1.2 Methods of Optimization There are many methods used to perform optimization. It would be impossible to list and summarize all of them. A few of the more common methods located during the author’s research is briefly summarized, along with one or two unique but relevant methods. One of the least sophisticated methods of optimization is by trial and error, which uses the expertise of an experienced engineer. Based on knowledge and familiarity with the performance of the component, the experienced engineer manipulates different parameters and examines numerical or graphical output data to choose between designs. As the complexity of the component and number of design parameters increase, this trial and error approach becomes a tedious and sometimes impossible task [5]. 2 Due to the advancement of computer technology, most of the current optimization techniques used employs some form of finite element analysis (FEA) software. One example is at the NASA Lewis Research Center. CometBoards, a structural design optimization program, has been developed specifically for design optimization of the Space Station components. “The CometBoard’s code can be used to optimize complex flight components under thermomechanical loads for typical behavior constraints consisting of stresses, displacements, buckling, crippling and frequencies” [2]. The CometBoards design is animated and examined with PATRAN FEA software. The output of this analysis is used to improve the design. The modified configuration is then optimized again with CometBoards, and the process is repeated until a satisfactory design is obtained. This procedure has proven itself successful because designs generated with CometBoards and PATRAN are more than 36% lighter than the manual designs obtained through traditional design methods [2]. In the design of steel frame structures, J. Isenberg, V. Pereyra, and D. Lawver combined the finite element structural analysis code FLEX with nonlinear programming optimization codes to minimize the total weight of the structure [5]. In the case of an optimal design of composite I-beams, S.K. Morton and J.P.H. Webber state that although most of the composite optimization methods are “extremely sophisticated and very efficient since they are designed for applications involving a large number of design variables and constraints, they are rather application-dependent” [6]. Raphael Haftka and Ramana Grandhi studied techniques related to shape optimization of the boundaries of two- and three-dimensional bodies. They focused their attention towards special problems of structural shape optimization which are due to a finite element model which must change during the optimization process. These problems, as previously mentioned, require sophisticated automated mesh generation techniques and careful choice of design variables. Most of the work in this area is “based on employing mathematical programming methods coupled with finite element analysis of the structure” [7]. 3 Relatively new software codes have been developed to optimize a design based on a load case and design requirements. Although these codes appear promising and possess potential, they are still too primitive for mass-production use. One such instance is the software codes being developed by Professor Grégoire Allaire and the Shape and Topology Optimization Group of the Centre de Mathématiques Appliquées de l'École Polytechnique (CMAP) in France. One example of the team’s work is the optimal chair show in Figure 1. As one can see, aside from the seat top and back, the entire chair takes on a shape which closely resembles the branches of a tree. From a manufacturing and cost point of view, this geometry may not be the most desirable. Therefore, although Professor Allaire’s software is very impressive and groundbreaking, it is not ideal for mass production [8]. Figure 1: Optimal Chair [8] Another software code developed for optimal topology utilizes the homogenization method to obtain an optimum solution. Solutions obtained from this method typically contain geometry with varying degrees of density values rather than a dual material distribution. One illustration of this optimal topology is presented in Figure 2, with 4 varying densities displayed as blurred outlines. This geometry presents difficulties encountered in manufacturing [1]. Figure 2: Typical Homogenization Optimization Solution with Intermediate Densities [1] Another undesirable feature commonly observed in an optimal topology is a checkerboard pattern. A checkerboard pattern is a region of alternating solid and void elements as shown in Figure 3. It has been proven that these patterns are due to numerical instabilities within the software code and do not represent an optimal feature. This discovery has lead to methods of suppressing these checkerboard patterns in an optimal topology [1]. Figure 3: Typical Checkerboard Region in an Optimum Solution [1] The last optimization method discussed is evolutionary structural optimization (ESO). The concept of ESO states that “by slowly removing inefficient material from a 5 structure, the shape of the structure evolves towards an optimum” [1]. Typically, the efficiency of material is determined by its stress level. ESO states that a reliable indication of inefficient material use is where the structure is low stressed. Therefore, the optimum design is reached when every element of a structure is approximately at the same stress level. This is termed a fully stressed design. To reach a fully stressed design, ESO is an iterative process where a small amount of low stressed material is removed during each iteration. One benefit of ESO is its simplicity. This has led to its application in a wide range of scenarios, such as natural frequency and buckling optimization [1]. 1.3 Problem Description The conventional and prevalent approach to designing a lightweight but structurally adequate structure is the iterative method. The design engineer starts with a basic shape and then analyzes the design using FEA software. Once the analytical results are obtained, changes are made, as required, to lighten the structure in some areas and strengthen the structure in other areas. This iterative process continues until an optimally-sized structure capable of withstanding the load conditions and design parameters is attained. In addition, since typically many constraints and variables exist in a design problem, a large number of analyses may be necessary to determine the adequacy of the design [1]. This process is very time-consuming, costly, and tedious. Utilizing FEA software for shape optimization also has unfavorable aspects. During the process, geometry of the structure may undergo substantial changes which result in an impractical structure. Examples include the geometry may become unfeasible, the area or volume may become too large, and manufacturing of the geometry may become too difficult. In these cases, the interaction of an experienced engineer is required to resolve the concerns. How well the resulting optimum shape reflects the best possible shape is another concern [4]. Ordinarily, the structural shape of the final design depends on the engineer’s criteria, and the “design depends partly on economical, aesthetical, construction techniques and environmental aspects” [9]. 6 From a business point of view, if this process can be shortened, it will free up manpower and assets to work other tasks. One major method to shorten this process is for the design engineer to initially construct a design that is very close to the finished design. This will facilitate the task of the computer software, and, if successful, fewer iterations will be required and the sooner the design will come to fruition. 7 2. Methodology The methodology of this process was based on the same idea as the evolutionary structural optimization method. It began with a simple, rectangular, two-dimensional, cantilever beam with a downward end load of 106 Newtons. First the beam height was calculated based on the highest stressed section. A beam with this height was generated using Dassault Systems CATIA V5 solid modeling software. To create a baseline structural representation, this beam was imported and analyzed using COMSOL Multiphysics software to locate the inefficient material. From this, eight different beam designs were generated. Each of the eight designs was conceived from a design engineer’s intuition of being capable of satisfying the design requirements but also achieving a low weight solution. These beams were independent deigns that were not iterations on each other. They were used to compare different manufacturing methods to determine the best method to lighten a structure. These eight designs were then analyzed using COMSOL Multiphysics software to determine their load carrying ability. The eight designs were compared to each other based on weight and strength. The successful designs were iterated to create even lighter designs or to resolve minor structural inadequacies. The structural analyses of these revisions were then re-ranked for final consideration. As Lluis Gil and Antoni Andreu stated, cost may be a better measure of optimization from the point of view of real engineering construction since cost is not directly related to the quantity of material but to other factors such as labor costs and construction difficulties. This may be true, but for simplicity, minimum weight was chosen as the primary objective and measure of structural optimization. The optimum shape based on structural considerations with the lowest weight will be the design that ensures the structure works close to the structural limit [9]. 2.1 Baseline Structure To simplify this investigation, a cantilever beam with an end load was chosen as the base model. A downward end load of 106 Newtons was applied to the end of a five meter 8 long cantilever beam. To begin with an initial, roughly optimized design, the stress of the beam was set equal to the yield stress of the material. This defined the height of the beam. The material of the beam was chosen to be 304 Stainless Steel. From basic Strength of Materials knowledge, the highest stressed section of an endloaded cantilever beam is at the fixed end and is due to the moment. This moment is calculated using the equation M P x (Equation 1) where P is the end load and x is the length of the beam. To calculate the height of the beam at the fixed end, the maximum stress equation is used: MAX M c . I Since I 1 3 h bh and c , 12 2 the maximum stress equation becomes 12 Mh 2bh 3 6M 2 bh MAX MAX which can then be rearranged and solved for ‘h’ to result in: h 6M b MAX (Equation 2) From equation 2 an initial, roughly optimized design can be generated. This beam geometry was created utilizing Dassault Systems CATIA V5 solid modeling software. This model was then saved as a .dxf file format and imported into COMSOL Multiphysics software to generate a baseline structural model. 9 2.2 Optimized Structure Once the baseline structural model was established, the beam was examined for low stressed regions. Utilizing a designer’s intuition, eight different beam designs were created. The goal of these designs was to remove the inefficient material resulting in a lightweight and fully stressed structure where the stress across the entire beam was close to or equal to the yield stress of the material. The most promising designs were iterated once to further refine and lighten the structures. 2.3 Material Properties As previously stated, the beam material chosen was 304 Stainless Steel. The properties of this material are: Table 1: 304 Stainless Steel Material Properties [10] 2.4 COMSOL Procedure & Importation Method All of the beams in this study were analyzed using COMSOL as a 2D Plane Stress, Static Analysis model within the Structural Mechanics Module. Aside from the calibration model, which was created directly in COMSOL, all of the beams were generated in CATIA V5 as space geometry. A side view of these beams was then projected onto a 2D drawing. This drawing was saved as a .dxf file and imported into COMSOL. Because importation into COMSOL proved to be somewhat tricky, the following process was successfully followed for each of the beams. First, a new 2D Plane Stress Static Analysis file was opened. The beam geometry was then imported under File Import CAD Data From File and selecting the .dxf file. By clicking the Options button, the author checked Edge entities, try forming solids and unchecked Solid entities. At 10 this point, an imported beam geometry was displayed on the screen. For some reason unknown to the author but presumed to be a units conversion issue (meters to millimeters), the beam imported one thousand times larger than created in CATIA V5. To correct this, the beam was scaled down by a factor of 0.001 in both the x and y directions. This resulted in beams matching the geometry generated in CATIA V5. When the .dxf file was imported, holes within the beams were imported as geometric bodies. Therefore, the parent beam geometry was separated from the holes by using the Split Object function. Lastly, the holes were deleted resulting in the final, imported geometry. 11 3. Results and Discussion 3.1 COMSOL Calibration & Mesh Size Determination Prior to analyzing 2D geometry in COMSOL, the thickness parameter first needed to be calibrated. To do so, the deflection of a rectangular cantilever beam, five meters long by one meter high by one meter thick, was used. First, the deflection of the free end was determined using Strength of Material knowledge. This beam’s properties are found in Table 1. An end load of -106 N was applied. The deflection of a cantilever beam with an end load is determined by v PL2 3L x . 6 EI The deflection at the free end, where x=L, will be the maximum; therefore, PL3 3EI PL3 bh 3 3E 12 v MAX v MAX v MAX 10 6 5 3 1 13 3 193 10 9 12 vMAX 0.002591 m Next, a rectangular beam, five meters long by one meter high, was created in COMSOL. Under Subdomain Settings, the material properties were entered (see Table 1 for values). The “thickness” is the parameter that was being verified for accuracy; it was entered as 1. Under Boundary Settings, the left vertical edge was constrained both in the x and y directions. This simulated a fixed wall condition. Under Point Settings, a load of -106 N was applied at the bottom node of the free end. The model was then meshed once and solved. The below figure is the resulting analysis color coded for y-displacement. 12 Figure 4: Thickness Calibration Solution Utilizing the Data Display function, the y-displacement at (5,0) was determined to be -0.002683 meters. This is approximately equal to the displacement calculated using Strength of Material equations. Therefore, a thickness of one was sufficient and accurate for solving these beam models. Next the appropriate mesh size must be determined. Again the calibration model was used. The original mesh solved for in Figure 4 consisted of 122 elements. This model was then re-meshed four times, and the y-displacement was measured at the point (5,0). This data is shown in Table 2 below. Table 2: Number of Elements vs. y-Displacement 13 The change in y-displacement with increasing mesh elements occurred at the hundredthousandths decimal place. Due to the closeness of these results, all of the following models were meshed to an element number of 7,000 or greater. This ensured relatively high accuracy but kept the models to a reasonable number of elements. For the purpose of this research, the results from these models were more than adequate. 3.2 Baseline Structure Using Equation 1, the moment at the fixed end is calculated to be: M Px M 10 6 5 M 5 10 6 N m Entering this moment into Equation 2 and setting the maximum stress equal to the yield stress results in: h 6M b y h 6 5 10 6 b 207 10 6 h .145 b For this 2D analysis, ‘b’ is one meter, resulting in h .145 1 h 0.381 m Hence, the baseline beam with a height of 0.381 meters was generated using CATIA V5 and imported into COMSOL Multiphysics. 14 The CATIA V5 software determined the beam weight to be 14,973.3 kg. The material properties were entered into the Subdomain Settings as defined in Table 1, and a load of -106 N was applied to the bottom node of the free end. The opposite end was constrained in the x and y directions to simulate a fixed condition. The beam was then meshed several times until 7,000 elements or more were achieved. The final element count was 15,360 elements. Figure 5 below depicts the results of the COMSOL analysis, color coded according to von Mises stress. Figure 5: COMSOL Analysis of Baseline Structure Stress concentrations occur at the application of the load and at the wall; these unrepresentative values will be ignored. Since the maximum stress of a rectangular beam occurs at the top and bottom surfaces of the beam, the stress and displacement was measured at the top of the beam to avoid the stress concentration at the application of the load on the bottom. Utilizing the Cross-Section Plot Parameters function, the stress at the top of the beam along its length is depicted in Figure 6. All stresses, excluding the spike at the wall due to the stress concentration, are below the yield stress of 207 MPa. As can also be seen in Figure 5 above, much of the material on the inside of the structure is stressed much lower than the yield stress, depicted as the color blue. The following alternative designs strived to eliminate this inefficient, low-stressed material while 15 keeping the maximum stresses below the yield stress. The displacement measured at the top of the free end, at the coordinate (5,0.381), was -0.046994 meters. The only criterion for this analysis is the maximum stress of the beam, but the displacement was also measured as additional information. Figure 6: von Mises Stress Along Top of Baseline Beam 3.3 Initial Eight Designs 3.3.1 Design 1 The first attempt to reduce the weight of the baseline structure was through the use of circular lightning holes. The sole purpose of lightning holes is to remove material through the simple operation of machining holes through the structure. Thirteen, 0.1 meter diameter holes were drilled through the baseline beam along the centerline; see Appendix A for beam dimensions. The remaining structure weighed 14,170.78 kg. The 16 beam was meshed into 24,192 elements and solved. The results of the stress analysis are shown in Figure 7. The y-displacement at the top of free end was measured to be -0.047334 meters. Figure 7: COMSOL Analysis of Design 1 Again utilizing the Cross-Section Plot Parameters function, the stress at the top of the beam along its length was obtained and is depicted in Figure 8. The stress along the tops of the holes was also measured and is illustrated in Figure 9. Based on these two graphs, the maximum stress is approximately 2.03x108 Pa, which is below the yield stress of 207 MPa. The left side of the beam, closest to the wall, is higher stressed than the free end. 17 Figure 9: von Mises Stress Along Top of Holes of Design 1 Figure 8: von Mises Stress Along Top of Design 1 3.3.2 Design 2 The next design was similar to the first design except the inner holes were larger in diameter compared to the outer holes. This model was generated to remove more material in the middle region of the beam, where the stresses in the baseline model appeared less than at the ends of the beam. The inner holes were increased from 0.1 meter to 0.15 meter diameter holes. See Appendix A for specific beam geometry. The structure weighed 13,321.96 kg. The beam was meshed into 18,920 elements. The results of the analysis are shown in Figure 10. The y-displacement at the top of free end was measured to be -0.048301 meters. 18 Figure 10: COMSOL Analysis of Design 2 Again utilizing the Cross-Section Plot Parameters function, the stress at the top of the beam along its length was measured and is shown in Figure 11. The stress along the tops of the larger holes was extracted and is displayed in Figure 12. Based on these two graphs, the maximum stress is again adjacent to the fixed end at the top of the beam and is approximately 2.03x108 Pa. Figure 12: von Mises Stress Along Top of Larger Holes of Design 2 Figure 11: von Mises Stress Along Top of Design 2 19 3.3.3 Design 3 The third design utilized a horizontal slot through the center of the beam in place of the lightning holes. The weight of the structure was 13,142.60 kg; the beam was meshed into 15,344 elements. The results of the analysis are shown in Figure 13. The ydisplacement at the top of free end was measured to be -0.098964 meters. Figure 13: COMSOL Analysis of Design 3 The stress at the top of the beam along its length was measured and is presented in Figure 14. The stress along the top of the slot is exhibited in Figure 15. Based on these two graphs, the maximum stress occurs at the right end of the slot and is approximately equal to 3.55x108 Pa. This value exceeds the yield stress of the material. 20 Figure 14: von Mises Stress Along Top of Design 3 3.3.4 Figure 15: von Mises Stress Along Top of Slot of Design 3 Design 4 The next concept started with Design 3 but extended the slot through the fixed end. This was an attempt to remove low stressed material near the fixed end. The weight of the structure was 13,075.45 kg. The beam was meshed into 8,848 elements. The results of the analysis are shown below in Figure 16. The y-displacement at the top of the free end was measured to be -0.102780 meters. 21 Figure 16: COMSOL Analysis of Design 4 Just as in Design 3, the highest stressed location, based on the color coding of the results in Figure 16, appears to be the right end of the slot. This region is enlarged below in Figure 17. Figure 17: Close-up View of Slot Stress of Design 4 22 The stress at the top of the beam along its length is depicted in Figure 18. The stress along the top of the slot is shown in Figure 19. Based on these two graphs, the maximum stress does in fact occur at the right end of the slot and is approximately equal to 3.55x108 Pa. Again, this value exceeds the yield stress of the material. Figure 18: von Mises Stress Along Top of Design 4 3.3.5 Figure 19: von Mises Stress Along Top of Slot of Design 4 Design 5 This design was another attempt at removing material in the middle region of the beam. It utilized the slot concept from the previous two designs but increased the slot height in the middle of the beam. The weight of the structure was 11,326.48 kg. The beam was meshed into 21,376 elements. The results of the analysis are depicted in Figure 20. The y-displacement at the top of free end was measured to be -0.116815 meters. 23 Figure 20: COMSOL Analysis of Design 5 The stress at the top of the beam along its length is portrayed in Figure 21. The stress at the tops of the ends of the slot is displayed in Figure 22. Due to the slot geometry, the horizontal line used to measure the stress at the tops of the ends of the slot passes through the open section of the slot; therefore, the middle of Figure 22 does not register any stress. Based on these two graphs, the maximum stress occurs at both the left end of the slot and at the top of the left end of the beam. Both values are approximately equal to 3.53x108 Pa. 24 Figure 21: von Mises Stress Along Top of Design 5 3.3.6 Figure 22: von Mises Stress Through Ends of Slot of Design 5 Design 6 Design 6 used the slot concept but increased the height of the slot at the free end. This was an effort to remove the inefficient material found close to the free end in the baseline beam. The weight of the structure was 11,265.39 kg. The beam was meshed into 10,992 elements. The results of the analysis are exhibited in Figure 23. The ydisplacement at the top of free end was measured to be -0.148161 meters. 25 Figure 23: COMSOL Analysis of Design 6 The stress at the top of the beam along its length is shown in Figure 24. The stress along the top of the slot can be seen in Figure 25. Based on these two graphs, the maximum stress occurs at the top of the right end of the slot and is approximately equal to 5.5x108 Pa. Figure 24: von Mises Stress Along Top of Design 6 Figure 25: von Mises Stress Along Top of Slot of Design 6 26 3.3.7 Design 7 This next design replaced the single slot with three smaller slots. The result of these slots is two inner members that connect the upper portion of the beam to the lower portion, similar to a truss structure. The weight of this beam was 10,194.31 kg. The beam was meshed into 24,032 elements. The results of the analysis are depicted in Figure 26. The y-displacement at the top of free end was measured to be -0.063271 meters. Figure 26: COMSOL Analysis of Design 7 The stress at the top of the beam along its length is presented in Figure 27. The stress along the top of the slots is illustrated in Figure 28. Based on these two graphs, the maximum stress occurs at the top of the left end of the beam and is approximately equal to 3.2x108 Pa. 27 Figure 27: von Mises Stress Along Top of Design 7 3.3.8 Figure 28: von Mises Stress Along Top of Slots of Design 7 Design 8 The last initial design is similar to the previous design except a fourth slot was added and the two inner slot heights were increased to remove more material in the middle region of the beam. See Appendix A for beam dimensions. The weight of this beam was 10,425.98 kg. The beam was meshed into 8,296 elements. The results of the analysis are depicted in Figure 29. The y-displacement at the top of free end was measured to be -0.059411 meters. 28 Figure 29: COMSOL Analysis of Design 8 The stress at the top of the beam along its length is shown in Figure 30. The stress along the top of the larger slots is depicted in Figure 31. Based on these two graphs, the maximum stress again occurs at the top of the left end of the beam and is approximately equal to 3.0x108 Pa. Figure 30: von Mises Stress Along Top of Design 8 Figure 31: von Mises Stress Along Top of Larger Slots of Design 8 29 3.3.9 Summary of Initial Beam Designs Table 3 below summarizes the results of the design cases. Of the previous eight designs, only two kept the maximum stress in the beam under the yield stress; these were Designs 1 and 2. The lightest designs were Designs 7 and 8. The designs containing one long slot were the least successful. These were Designs 3, 4, 5, and 6. They were neither the lightest designs nor the lowest stressed designs. In fact, these four designs were the top four highest stressed designs. Table 3: Summary of Initial Beam Designs Based on these results, Designs 2 and 8 were iterated once to further refine the designs. Design 2 was chosen because of the beams with the lowest stress, it was the lightest design; Design 8 was selected because of the beams with the lowest weight, it was the beam with the lowest stress. Since the stress in Design 2 is below the yield stress, the lightning holes will be modified in size to further lighten the beam. The slots in the Design 8 beam will be adjusted to decrease the maximum stress in the beam to a level below the yield stress while attempting to keep the weight low. 3.4 Iteration to Successful Designs 3.4.1 Iteration to Design 2 Based on the color coding of Figure 10, the author decided to increase the diameter of the lightning holes starting with the fifth hole from the left. These holes were increased 30 from 0.15 meter diameter holes to 0.25 meter. This was an attempt to remove more material where the stresses in Design 2 were low. See Appendix A for dimensions of this new beam. The weight of the structure decreased to 11,022.43 kg. The beam was meshed into 12,564 elements. The results of the analysis are shown in Figure 32. The y-displacement at the top of free end was measured to be -0.052593 meters. Figure 32: COMSOL Analysis of Design 2 Iteration The stress at the top of the beam along its length is shown in Figure 33. The stress along the top of the larger holes is depicted in Figure 34. Based on these two graphs, the maximum stress occurs at the top of fifth hole from the left and is approximately equal to 2.3x108 Pa. The stress along the top of the beam adjacent to the fifth hole also exceeds the yield stress and is approximately equal to 2.15x108 Pa. 31 Figure 33: von Mises Stress Along Top of Design 2 Iteration 3.4.2 Figure 34: von Mises Stress Along Top of Larger Holes of Design 2 Iteration Iteration to Design 8 Utilizing Figure 29 as a starting point, the author decreased the height of the second slot to match the first slot, decreased the height of the third slot to 0.15 meter, and increased the height of the fourth slot to 0.2 meter. This was an effort to decrease the stresses at and above the second and third slots while removing more material around the fourth slot. See Appendix A for dimensions of this new beam. The weight of the beam increased to 10,907.36 kg. The beam was meshed into 9,004 elements. The results of the analysis are depicted in Figure 35. The y-displacement at the top of free end was measured to be -0.054740 meters. 32 Figure 35: COMSOL Analysis of Design 8 Iteration The stress at the top of the beam along its length is shown in Figure 36. The stress along the top of the right hand slot is depicted in Figure 37. Based on these two graphs, the maximum stress occurs at the top of the left end of the beam and is approximately equal to 2.4x108 Pa. Other than this location, the other stresses are below the yield stress of the material. Figure 36: von Mises Stress Along Top of Design 8 Iteration Figure 37: von Mises Stress Along Top of Fourth Slot of Design 8 Iteration 33 3.4.3 Summary of Design Iterations Table 4 below summarizes the results of the iterations. Although the modifications to Design 2 reduced the weight further, they caused the stress to increase above the yield stress. Just as in the initial design, the stresses decrease from left to right along the beam. The revision to Design 8 lowered the maximum stress but not to a level below the yield stress. In addition, these changes increased the weight of the beam. Aside from this, the high stresses in the rest of the beam were relatively consistent, spread evenly throughout the structure, and below the yield stress. Table 4: Summary of Design Iterations 34 4. Conclusion Based on the analytical results from the first eight designs, one long horizontal slot did not appear to be beneficial to a cantilever beam. The tight radii at the ends of the slot seem to be a stress riser. If the beam was taller and a more gradual radius could be used, a slot may be viable, but for the geometry of this beam, a slot did not appear to be an effective method for removing low stressed material in the center of the beam. Designs 2 and 8 were chosen as the initial successful designs. Design 2 was selected because it was the lightest design among the lowest stressed beams. The iteration reduced the weight by 2,299.53 kg to a final weight of 11,022.53 kg. Unfortunately in the process, the maximum stress increased to 2.3x108 Pa. Design 8 was chosen because it was the lowest stressed design among the lightest beams. The iteration decreased the maximum stress by 6.0x107 Pa to a final maximum stress of 2.4x108 Pa. To reduce the stresses, material was actually added to the beam; therefore, the iteration increased the weight by 481.38 kg to a final weight of 10,907.36 kg. Of these two beams, the lightest weight structure was Design 8. Unfortunately, after this one iteration, both designs contained stresses above the yield stress of the material. Further iterations would be required to achieve lower stressed designs and determine which is in fact the lighter design. As a result of this study, the author makes several suggestions when attempting to design a lightweight structure similar to a cantilever beam with an end load. First, when utilizing simple, two-dimensional manufacturing techniques, such as machined holes and slots, it will be nearly impossible to obtain a fully stressed design. The abrupt changes in geometry and stiffness due to holes and slots generate low and high stressed regions which will be very difficult to smooth out and equalize. Second, long horizontal slots do not appear to be favorable when attempting to remove inefficient material from rectangular cantilever beams. The slots become too narrow and cause stress risers at the 35 ends. Next, several slots seem to be more effective at obtaining a more consistently loaded structure compared to many lightning holes. This obviously makes sense since the slots remove more material than circular holes. Also, the slots reduce the number of stress spikes due to changes in stiffness and geometry, as seen in Figure 33. Slots still experience these but they are fewer in number and were easier to level out. The sheer number of lightning holes made it difficult to level out the stresses. Each hole would need to be adjusted separately to achieve a specific stress level. This would take many more iterations than the slot design would. And since the slot design was lighter than the lightning hole design, it does not seem worthwhile to spend time and money performing this task. Lastly, several short slots appear to be more effective at removing inefficient material compared to both one long slot and many circular lightning holes. All three techniques can be adjusted to keep the stress level at a certain level, but the slots are better at more evenly distributing the stresses and keeping the weight to a minimum. For simplicity this research employed a simple, rectangular, end loaded cantilever beam to explore different options in optimizing a structural member. Unfortunately it is rare when a real world problem is subjected to such simple circumstances. As pointed out by H. Kim, O. M. Querin, and G. P. Steven, “a structure may be subjected to multiple loadings and multiple support conditions during its life; may behave non-linearly; and is often subjected to more than one optimality criterion, often of conflicting nature” [1]. It is the hope of the author that the lessons learned through this research can be adapted and applied to real world situations and structures. 36 5. References [1] Kim, H., Querin, O.M., Steven, G.P. January 2002. On the development of structural optimization and its relevance in engineering design. Design Studies 23 (1): 85102. [2] Patnaik, S.N., Gendy, A.S., Hopkins, D.A., Berke, L. November 1996. Weight minimization of flight components. Computers & Structures 61 (4): 597-616. [3] Tao, L., Zichen, D. September 2006. Design optimization for truss structures under elasto-plastic loading condition. Acta Mechanica Solida Sinica 19 (3): 264-274. [4] Sonmez, F. July 15, 2007. Shape optimization of 2D structures using simulated annealing. Computer Methods in Applied Mechanics and Engineering 196 (3536): 3279-3299. [5] Isenberg, J., Pereyra, V., Lawver, D. January 2002. Optimal design of steel frame structures. Applied Numerical Mathematics 40 (1-2): 59-71. [6] Morton, S.K., Webber, J.P.H. 1994. Optimal design of a composite I-beam. Composite Structures 28 (2): 149-168. [7] Haftka, R.T., Grandhi, R.V. August 1986. Structural shape optimization – A survey. Computer Methods in Applied Mechanics and Engineering 57 (1): 91-106. [8] Centre de Mathématiques Appliquées de l'École Polytechnique. “Shape and Topology Optimization Group of the CMAP.” Centre de Mathématiques Appliquées. http://www.cmap.polytechnique.fr/~optopo/index.php?lang=en (accessed October 16, 2009). [9] Gil, L., Andreu, A. March 2001. Shape and cross-section optimization of a truss structure. Computers & Structures 79 (7): 681-689. [10] Hibbeler, R.C. 1997. Mechanics of Materials. Upper Saddle River, NJ: Prentice Hall. 37 6. Appendix A Beam Geometry for Baseline Beam Beam Geometry for Design 1 Beam Geometry for Design 2 Beam Geometry for Design 3 Beam Geometry for Design 4 38 Beam Geometry for Design 5 Beam Geometry for Design 6 Beam Geometry for Design 7 Beam Geometry for Design 8 Beam Geometry for Iteration to Design 2 39 Beam Geometry for Iteration to Design 8 40