Frame Analysis and Design Computer Classroom/Laboratory Module for Machine Design Hodge Jenkins 1

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1 Frame Analysis and Design Computer Classroom/Laboratory Module for Machine Design Hodge Jenkins 1 Abstract Frames and trusses designs are often improperly handled by undergraduate mechanical engineering students as they design, model, and evaluate machine frames. A one-day, computer-based, classroom/laboratory module was created to address this situation. The module was developed for use in a junior-level machine design course. This module provides students with a means for improving understanding and knowledge about designing three-dimensional frames. Alternative modeling representations for obtaining stress and deflection results from static analysis of frames, utilizing idealized beam elements are presented. Limitations of using conventional finite solid elements to model long, thin-walled members are shown through a hands-on case study during class. Additional goals of the module are increased student interest in engineering design and analysis, increased student awareness of mechanical modeling implications, and improved student readiness for engineering practice. This frame modeling and analysis module is an advanced topic in a sequence of solid modeling and finite element modules that span the mechanics curriculum in the mechanical engineering specialty at Mercer University. Instructional materials developed for the laboratory session and related homework are provided and discussed. Multiple computer-generated solutions are compared with simplified, closed-form solutions to provide bounding cases for verification of results. While materials for this module were developed for specific software, the concepts and topical problem presented for investigation may be used with other design and analysis software. The materials may also be used as a self-paced tutorial. Keywords: Machine Design, Mechanical Engineering, Finite Elements Analysis, Computer Aided Design. INTRODUCTION AND BACKGROUND At Mercer University the Machine Design course is a junior-level mechanical engineering course and is the capstone of the solid mechanics sequence focusing on the design and analysis of machine components. Advanced solid mechanics and dynamics serve as prerequisites. Machine Design presents static and dynamic failure theories for ductile and brittle materials, and emphasizes the safe design of various machine elements, based on these failure theories and other design paradigms. Design components included in the course are: shafts, gears, bearings, cams, fasteners, chains, belts, flywheels, springs, brakes and clutches. To tie together the various machine elements, students have a semester-long project to apply their newly gained design knowledge to the design of a specific machine. An essential element of each machine is a structural frame. Analyzing the frame typically presents much difficulty for students, in that the frame is usually the most geometrically complicated structure they have attempted to analyze. In courses prior to machine design students have developed some skills in solid modeling and finite element analyses. Mercer University School of Engineering has an initiative to enhance student education and interest as well as readiness for modern engineering practice and graduate research, through the use of state-of-the-art engineering design and analysis software [1]. One educational goal of this effort has been to provide students with contiguous learning experiences in solid modeling and finite analysis via common software across the mechanics and design curriculum for students in mechanical engineering. Integration and use of mechanical engineering application software (such as Pro/Engineer and Pro/Mechanica [2]) throughout the solid mechanics courses provides 1 Mercer University, School of Engineering, 1400 Coleman Ave, Macon, GA 31207, jenkins_he@mercer.edu

2 students such a learning experience. Student have advanced their engineering and computing skills in computer aided design (CAD) and finite element modeling and finite element analysis (FEM/FEA) in each course. More detail on the effort may be found in an earlier paper [3]. One purpose of this learning module is to equip Machine Design students with modern and effective CAD tools for modeling and analyzing frames. CAD has been employed in student work for machine design and solid mechanics courses by many others [4]. What is significant about the effort described in this paper is the use of idealized beam elements to model a frame. Using FEA with idealized beams is an effective modeling approach for accurate structural analysis. Indeed the ideal beam elements create the most appropriate model for frame analysis, although the visualization of model results is not quite as appealing as the model using solid elements. Engineering students tend to be more visual and sequential learners; thus, visualization is an important means to engage students in active learning experiences [5]. Computer-based instruction has focused on improvement of conceptualization, visualization, and problem solving skills. It is apparent from several studies that spatial ability development for visualization is crucial to the success of an engineering student or professional engineer involved in designing, manufacturing, construction, and other graphically-related pursuits [6]. Furthermore, studies indicate that visualization skills can be improved through hands-on activities and innovative computer courseware. It has been documented that students who have received as little as one day of instruction on spatial strategies were significantly more successful in an engineering mechanics course. In another study that spanned four years and included over 500 students, Hsi et al [6] concluded that spatial strategy instruction contributes to confidence in engineering and improves problem solving ability. Sorby [7] suggests that spatial visualization instruction may also have long term benefits in terms of higher retention rates in engineering for students who participate in such instruction. Taken in total, the studies cited above suggest that multimedia modules should be considered as part of any course that is designed to improve students abilities to perform computer-aided design. MODULE APPROACH Machine Design has a compact course schedule because of subject learning objectives. Thus, the use of class time for this software tutorial has been limited to one in-class session for the learning module. A step-by-step tutorial document recapping the in-class tasks as well as an out-of-class assignment are provided with the module. The module is described in this text, including learning objectives, methods and assignments. Tutorials and assignments may be found at the web site of the Keck Engineering Analysis Center at Mercer University [8]. Integration of design and analysis is a common theme of the modules and is apparent from the in-class exercises and homework assignments. Students actively participate in the analysis as part of design. In Machine Design, the semester design project (part of the normal syllabus) emphasizes the use of the solid modeling and FEA software for the design and analysis of complicated of component geometry and loading, including structural frames. While this module was intended to be presented early in the course, the only course material required for completion prior to the module is failure theories. Problem for Investigation: Design a Frame for a 2-Ton Hoist Students are presented with a design situation to create a frame for a 2-ton overhead hoist. The design is based on standard structural steel shapes: a welded box beam (6-inches by 3-inches by 0.25-inches thick) and circular pipe (3.5-inches, outside diameter, by 3.0-inches, inside diameter). The legs are designed from the pipe, while the box beam is used for the ridge beam of the hoist. The included angle between the two legs on each end is 50-degrees. A solid model of the frame is shown in Figure 1.

3 Figure 1. Solid Model of Hoist Frame The nominal height of the hoist frame (ridge beam centerline) is 10-feet above the base surface of the vertical supports legs. The hoist has a 12-foot long ridge beam. While the solid model is a good physical representation of the geometry, it is not necessarily the optimal basis for creating a finite element model of the entire frame for stress analysis. Each structural member of the hoist is very thin and long. Using the standard auto-meshed finite solid element generation in Pro/Mechanica, the solution time may be substantial and not provide accurate results, based on poor convergence. Thus, the important of an alternative mathematical representation, an ideal beam, is necessary for frame design. General Background of Finite Element Analysis Information on the theory, development, and current state of finite element analysis (FEA) is plentiful in the literature and will not be presented here. The reader is referred to Bathe [9] for additional FEA background. Pro/Mechanica was selected as software platform because the finite element method applied in the software is relatively transparent to the students. The students learn about loads, boundary conditions, and such, but very little about meshing and element types, as that is done automatically by the software. Students have a basic understanding about element order and convergence, but only enough so that they can obtain reasonable results. Finite element theory is taught in a senior course on that topic. It is important to review the methods used by Pro/Mechanica software in calculating static stresses and deflections. Pro/Mechanica uses non-linear Geometric Element Analysis (GEA) with auto-meshing, as opposed to the more prevalent linear Finite Element Analysis, H-elements. GEA elements are essentially P-elements [10]. The difference between theses two types of elements is the that P-elements use higher order polynomials for element stress and strain functions to increase model accuracy, while leaving the mesh alone. H-elements have linear functions and require additional mesh refinement for verification of model convergence. Convergence of GEA model results is set prior to performing an analysis. Thus, it is possible to set desired convergence to provide a desired accuracy. This convergence (error) is the change in results from successive iterations of increasing polynomial order on non-converging sets of elements. The need for caution on accepting model results cannot be over emphasized to students. FRAME IDEALIZATION In virtually all general purpose FEA or GEA software, there are several types of modeling approaches to members. Most common is the solid element, which is computationally the least efficient of the modeling elements. For the solid model of Figure 1, a very large number of elements would be required to create a sufficiently accurate model.

4 Nodes are associated with each point of the element. Using the mechanics of materials linear constitutive relationships for mass and stiffness, matrices are determined at and between the nodes, respectively. While this standard approach works well in monolithic-type components, it is not usually very successful in structures which are made of numerous members that are long-length and relatively small in area. Simpler elements can be used more effectively to idealize the physical model. In Pro/Mechanica these elements include mass elements, spring elements, beam elements, and shell elements. This module focuses on using idealized beam elements. Beam elements are very computationally efficient. Beams are graphically represented as lines. In Pro/Mechanica these idealized beams are generally recommended for use where the member length is 10 times greater than the width dimensions. Other constraints/attributes of beams in this software package include: constant cross section, planar geometry between nodes. It should be noted that the idealized beams do not provide all the functions of the solid elements. Only bending stress, axial stress, and torsional stress results are considered in the beam. Local stress concentrations are not included. Shear stress results are not well handled by beam elements. An idealized beam model for the previously depicted hoist frame is given in Figure 2. While not as visually appealing as the solid model of Figure 1, the ideal beam model is computationally more efficient. Results from this model are obtained in seconds, not hours. A close-up inspection of one end of the model reveals the geometry crosssection, for visual confirmation of the appropriate section. Figure 2. Idealized Beam Model of Hoist Frame IN-CLASS TUTORIAL To introduce the concept of an idealized beam, students are first given a solid element model of the hoist ridge beam. Using solid elements and students perform a static analysis, fixed constraints at each end. The students then create an idealized beam model of the same ridge beam and repeat the static analysis. As part of the homework assignment students compare the results of the idealized beam model, the solid element model and actual 3-D frame model with legs to hand calculations for the ridge beam alone (fixed-fixed and simple supports). The idealized beam model for the entire frame of the hoist is created, starting from the single idealized ridge beam. A pre-existing model of the frame will be available if class time is running short. A procedure for detailed solid model and idealized beam analyses in Pro/Mechanica Wildfire 2, are available as separate documents [8].

5 Modeling Considerations and Geometry Modeling considerations for analysis of the frame must include the effects of the support legs. The legs are attached to the beam in the middle of the leg cross-section. As the legs are 3.5-inches in diameter, and are connected to the 144-inch long ridge beam, the modeled ridge beam is thus inches between the two support leg centers. The 50 included angle between the legs indicates the legs run horizontally inches from the center of the ridge beam, and vertically 120-inches from the ground. As the legs are anchored to the ground, fixed constraints are added at those nodes. Section Properties Section properties of the ideal beams are determined by the software and listed in Table 1, which shows typical program data for the hollow rectangular section of the ridge beam. Cross-sectional properties are calculated by the program. Other standard or custom cross-sections are available in the software. Table 1. Ideal Beam Properties of Hollow Rectangular Section, Used for the Ridge Beam Variable/Property Value b 3 in d 6 in bi 2.5 in di 5.5 in Area 4.25 in 2 Iyy in 4 Izz in 4 Cz 3 in SOLID ELEMENT MODEL To improve the time effectiveness of the module, a solid model of the ridge beam alone with both ends fixed as constraints was provided. (Note: Students have developed skills to create solid element extrusions in prior courses.) The ridge beam solid model is shown in Figure 3 with the applied 2-ton load. Note: the figure reveals that the loads was applied over an area (6-inches by 6-inches), as opposed to a line or point contact force to better approximate the actual loading condition and reduce localized stress concentration near the load. Figure 3. Detailed Ridge Beam Model Using Solid Elements Stress (vonmises) and deflection results for this model are given in Table 2 as well as presented in Figures 4 and 5, respectively. As expected, the largest magnitude stresses are at the top and bottom surfaces of the beam, at the farthest distance from the neutral axis of the cross-section. Colored plots of the deformed beam are very descriptive

6 for showing stresses and deformations. The solid model also serves to demonstrate the computational inefficiencies of using solid elements for this long and relatively thin ridge beam. Figure 4. VonMises Stress for Static Analysis of the Ridge Beam Model Using Solid Elements Figure 5. Displacement for Static Analysis of the Ridge Beam Model Using Solid Elements Idealized Beam Model After performing a static analysis for the solid element model, students are instructed on building an ideal model for the same ridge beam. A tutorial is provided for the students to follow. Students run a static analysis on the ideal beam model with the applied 2-ton load applied as a point force. The idealized beam model of the ridge beam with point loading and constraints is presented in Figure 6. Note: that static loading for beam elements is placed at nodes, thus a node was required in the middle of the beam. Localized stress concentration is not determined by the elements, so it was not a consideration for load application. Figure 6. Idealized Ridge Beam Model Using Beam Elements

7 Stress and deflection results from this ideal beam model for static loading are given in Table 2. Figure 7 shows the resulting displacement visualization. Clearly this depiction is less appealing than that of the solid model of Figure 5. Students compare the computational convergence times of this idealized beam model to that of the solid element model for this long and relatively thin ridge beam. The idealized beam models achieve a converged result over an order of magnitude faster than the solid element model. Stress results are available in several component forms. For comparison purposes vonmises stresses are obtained. Figure 7. Displacement for Static Analysis of the Ridge Beam Model Using Idealized Beam Elements Frame Model Students expand the ridge beam to create a complete model of the entire frame, using the ridge beam as the basis. Points are added for the base points of the legs at both ends. Fours beams of a circular pipe cross-section are added to the represent the legs. Note: Figure 8 depicts the completed whole frame model with loading and supports. Figure 8. Hoist Frame Modeled with Idealized Beams Figure 9 (left and right, respectively) depict the resulting displacements and stresses of all the members in the frame. The stress results plotted are the vonmises stresses.

8 Figure 9. VonMises Stresses and Displacements for Static Analysis of the Hoist Frame Model Using Idealized Beam Elements RESULTS AND DISCUSSION Maximum deflections and stresses for the various ridge beam models using three methods (hand calculations, solid element models and ideal beam models), and the entire frame ideal beam model are provided in Table 2. Since the effects of the support legs could not be included in standard closed-form solutions, both simply supported and fixedend supported conditions of the ridge beam were calculated using stand beam deflection and stress tables [11]. All cases modeled a static loading of 2 tons in the mid-span of the ridge beam. As expected the maximum stresses are located at the point of loading (mid-span) of the ridge beam. Table 2. Frame Stress and Deflection for Various Models. No. Case Method Maximum VonMises Stress (psi) 1 Simply supported ridge beam, Hand theoretical calculation 2 Fixed-end supported ridge Hand beam, theoretical calculation 3 Fixed-end supported ridge beam, 4% convergence error solid model 4 Fixed-end supported ridge beam, <1% convergence error ideal beam model 5 Entire frame model, Ridge beam with legs <1% convergence error Solid element model Ideal Beam model Ideal Beam model Maximum Deflection (inch) 21, , , , , Students should be able to make several comparisons from the data of Table 2. First, the fixed-end supported ridge beam cases demonstrate the validity of all the methods: hand calculations, solid element model, and idealized beam model (cases 2, 3, and 4 from Table 2). Nearly identical maximum stresses and maximum deflections were obtained for those methods/models. Results from the frame model (case 5) may be compared to theoretical hand-calculated solutions for simply supported and fixed-end supported cases of the ridge beam (cases 1, 2) to assess the quality of the results. It is apparent that flexibility of the leg supports in the frame model yielded a maximum stress and a maximum displacement in the ridge beam that are different from the static analysis of the ridge beam model alone. The maximum stress and deflection obtained from the complete frame model lie somewhere between the results for the

9 simply supported and fixed-end supported conditions of the ridge beam alone. This can be attributed to the effective linear and rotational stiffness provided by the legs, as end conditions for the ridge beam. It can also be noted that the simply supported end condition of the ridge beam alone yielded a higher maximum stress than the complete frame model. From standard moment diagram analysis the simply supported case generates a higher internal resistive moment in the ridge beam. Thus, a conservative approach for stress analysis would be to analyze the simply supported end case. This may not be apparent to students. However, it is important to not over generalize, as this may not always be the case in other frame designs. Convergence error for the results of the Pro/Mechanica GEA approach was less than 1% error for the ideal beam analyses and less than 4% error for the solid element model results. (This convergence is the change deflection, strain energy, and stress in successive iterations for increasing polynomial order of stress and displacement functions in non-converged elements.) Students also compare the elapsed computer time to obtain converged results for the GEA model composed of many solid elements to that of the idealized beam model. On a 2.8 GHz Pentium 4 computer with 512Mb RAM, running Windows XP, with a 4% limit convergence, solution times of 409 seconds and 2 seconds were obtained for the solid element model and ideal beam model respectively for the ridge beam model (cases 3 and 4). Clearly the ideal beam has a significant computational advantage. This module was intended for use in the spring of 2006, therefore student feedback on the module and homework was not available at time of publication. It is anticipated that upon completion of the module and associated homework, students should have a reasonable understanding of the usefulness and limitations of ideal beams to model frames. Learning Objectives of Module: Students will (as a result of this module): 1. Improve software familiarization with finite element modeling alternatives such as the ideal beam 2. Be able to analyze more complicated 3-D geometry 3. Use and understand physical representations of idealized beams 4. Understand the differences of modeling beam end conditions, with various constraints 5. Better visualize 3-D loads and resulting stresses and deformations 6. Understand modeling assumptions affect solution results and computational time 7. Understand design in action: changing beam cross-sections and materials for stress and deflection 8. Attain improved confidence in using CAE tools Methods of Assessment It is important to assess the effectiveness of this module on student learning. Several means of assessment are planned. First, a homework assignment related to the ideal beam model reinforces the learning objects of the inclass tutorial. The assignment is due within 1-week after the in-class tutorial. For homework students must complete the data in Table 2. Also students are asked to create an aluminum ridge beam design with similar stress and deflection to the original steel shape, and then create a steel design that has a lower weight and similar stress than their aluminum design. The homework assignment provides one means of assessment. However, more detailed assessment of this module may also be accomplished by evaluating student surveys and quizzes, given before and after completion of the module with homework. In the planned survey students will self-rank their confidence in their ability on several tasks (using a modified Likert scale from 1 to 5, 5 being very confident, 1 being not confident at all). Typical abilities surveyed included: 1. Understanding software capabilities 2. Specifying problem details 3. Generating problem solutions 4. Analyzing problem results 5. Discussing meaning of results 6. Conducting analysis independently

10 Students will also be asked a series of yes or no questions, and to provide additional comments, to gain specific insight into the some other aspects of the module efficacy. Evaluation of quizzes on related material before and after completion of the module and associated homework will also serve as an assessment tool. OBSERVATIONS AND CONCLUSIONS While it is difficult to ascertain whether all the educational goals of the software modules will be achieved, similar experiences have improved the perception student have of their abilities [1]. In general students have significant enthusiasm for learning new CAE tools. The purpose of this module is to provide a means for students to continue learning and to continue usage of the CAE software throughout the course. The primary goals for the module are 1) increase student understanding of frame design and analysis by improving visualization and modeling idealization, and 2) provide an apparent link between design and analysis. From past experiences visualization of stresses and deformation, especially in 3-D, is generally helpful to students. The colorful graphics in three-dimensions are important for students to see bending stresses and deflections. In this module, the CAE module provides students useful design and analysis experience. ACKNOWLEDGEMENT This work has been supported, in part, by a grant from the W. M. Keck Foundation. Any opinions, findings and conclusions or recommendations are those of the authors and do not necessarily reflect the views of the W. M. Keck Foundation. REFERENCES [1] Jenkins, H.E. Increasing Student Interest and Understanding in a First Mechanics Course Through Software Modules, 2004 ASME International Mechanical Engineering Congress [2] Pro/Engineer release, Wildfire 2, PTC Corp.: Nov. 28, [3] Jenkins, H.E. and Mahaney, J.M., Improving Mechanical Design using Solid Modeling and FEA, ASEE- SE Conference, April, [4] Steif, P. S. and Naples, L.M., Design and Evaluation of Problem Solving Courseware Modules For Mechanics of Materials, Journal of Engineering Education, v. 92, n. 7, 2003, pg [5] Kolari, S. and Savander-Ranne, C., Visualization Promotes Apprehension and Comprehension, Int. J. Eng. Educ., v. 20, n.3, 2004, pg [6] Hsi, S., Linn, M.C., and Bell, J.E. The role of spatial reasoning in engineering and the design of spatial instruction, J. of Eng. Educ.,, v.86, n.2, 1997, pg [7] Sorby, S. A. and Baartmans, B. J., The Development and Assessment of a Course for Enhancing the 3-D Spatial Visualization Skills of First Year Engineering Students, J. of Eng. Educ., v. 89, n. 3, 2000, pg [8] Keck Engineering Analysis Center at Mercer University, Dec. 9, [9] Bathe, K. J., Finite Element Procedures, Prentice-Hall, Englewood Cliffs, [10] Beckers, P., Cugnon, F., and Darnhaut, L. "P and H Elements," SAMCREF User Conf., Liege, France, 1996 [11] Shigley, J.E., Mischke, C.R., and Budynas, R, Machine Design, McGraw-Hill, Hodge Jenkins Dr. Hodge Jenkins is an Assistant Professor of Mechanical Engineering in the Department of Mechanical and Industrial Engineering at Mercer University in Macon, Georgia. Prior to coming to Mercer in 2002, Dr. Jenkins was engaged in optical fiber product development with Bell Laboratories of Lucent Technologies. He is a registered professional engineer, and with over 20 years of design and development experience in high-precision design, dynamic structural analysis, process automation, control, and robotics. Dr. Jenkins holds a Ph.D. in Mechanical Engineering from Georgia Institute of Technology in (1996), as well as BSME (1981) and MSME (1985) degrees from the University of Pittsburgh. His professional affiliations include ASME, IEEE, and ASEE.