Learning Module 5 Buckling Analysis

Learning Module 5 Buckling Analysis Title Page Guide What is a Learning Module? A Learning Module (LM) is a structured, concise, and self-sufficient learning resource. An LM provides the learner with the required content in a precise and concise manner, enabling the learner to learn more efficiently and effectively. It has a number of characteristics that distinguish it from a traditional textbook or textbook chapter: An LM is learning objective driven, and its scope is clearly defined and bounded. The module is compact and precise in presentation, and its core material contains only contents essential for achieving the learning objectives. Since an LM is inherently concise, it can be learned relatively quickly and efficiently. An LM is independent and free-standing. Module-based learning is therefore nonsequential and flexible, and can be personalized with ease. Presenting the material in a contained and precise fashion will allow the user to learn effectively, reducing the time and effort spent and ultimately improving the learning experience. This is the first module on thermal analysis and provides the user with the necessary tools to complete a thermal FEM study with different boundary conditions. It goes through all of the steps necessary to successfully complete an analysis, including geometry creation, material selection, boundary condition specification, meshing, solution, and validation. These steps are first covered conceptually and then worked through directly as they are applied to an example problem. Estimated Learning Time for This Module Estimated learning time for this LM is equivalent to three 50-minute lectures, or one week of study time for a 3 credit hour course. How to Use This Module The learning module is organized in sections. Each section contains a short explanation and a link to where that section can be found. The explanation will give you an idea of what content is in each section. The link will allow you to complete the parts of the module you are interested in, while being able to skip any parts that you might already be familiar with. The modularity of the LM allows for an efficient use of your time. 1

Table of Contents 1. Learning Objectives... 3 2. Prerequisites... 3 3. Pre-Test... 3 4. Tutorial Problem Statements... 4 5. Conceptual Analysis... 7 6. Abstract Modeling... 8 7. Software-Specific FEM Tutorials... 8 8. Post-Test... 8 9. Practice Problems... 8 10. Assessment... 9 Attachment A. Pre-Test... 10 Attachment B. Conceptual Analysis... 12 Attachment C1. SolidWorks-Specific FEM Tutorial 1... 15 Attachment C2. SolidWorks-Specific FEM Tutorial 2... 38 Attachment C3. SolidWorks-Specific FEM Tutorial 3... 55 Attachment D. CometSolution-Specific FEM Tutorials... 69 Attachment E. Post-Test... 70 Attachment F. Practice Problems... 73 Attachment G. Solutions to Practice Problems... 76 Attachment H. Assessment... 79 2

1. Learning Objectives The objective of this module is to introduce the user to the process of structural buckling analysis using FEM. Upon completion of the module, the user should have a good understanding of the necessary logical steps of an FEM analysis, and be able to perform the following tasks: Creating the solid geometry Assigning material properties Applying thermal boundary conditions Meshing Running the analysis Verifying model correctness Processing needed results 2. Prerequisites In order to complete the learning module successfully, the following prerequisites are required: By subject area: o Statics o Mechanics of Materials or Elasticity By topic: o Column end conditions o Column effective lengths o Euler s Formula o Young s modulus o Stress and critical stress o Loads and critical loads o Modification factors 3. Pre-Test The pre-test should be taken before taking other sections of the module. The purpose of the pretest is to assess the user's prior knowledge in subject areas relevant to mechanics of materials and buckling analysis. Questions are focused towards fundamental concepts including Euler s Formula, column end conditions, stress, and critical loads. The pre-test for this module given in Attachment I. Link to Pre-test 3

4. Tutorial Problem Statements A good tutorial problem should focus on the logical steps in FEM modeling and demonstrate as many aspects of the FEM software as possible. It should also be simple in mechanics with an analytical solution available for validation. Three tutorial problems are covered in this learning module. Tutorial Problem 1 A 3.5m long column made of AISI 304 steel has a square cross-section with dimensions of 100mm x 100mm. This column is used to support a 10 MPa pressure load in multiple setups with varying end conditions. Use FEM analysis to find the buckling load factor (BLF) and critical pressure load (P c ) of the column in each separate circumstance. The end conditions are as follows: a) One fixed end and one free end (fixed-free) b) Two fixed ends (fixed-fixed) c) One fixed end and one pinned end (fixed-pinned) d) Two pinned ends (pinned-pinned) 4

Tutorial Problem 2 A 3.6 m long L-shaped column made of AISI 304 steel has dimensions as shown in the figure below. The column contains sixty 60 mm diameter holes as seen in the figure below. This column is used to support a 6 MPa pressure load in multiple setups with varying end conditions. Use FEM analysis to find the buckling load factor (BLF) and critical pressure load (P c ) of the column in each separate circumstance. The end conditions are as follows: a) One fixed end and one free end (fixed-free) b) Two fixed ends (fixed-fixed) 5

Tutorial Problem 3 Honeycomb structures are known for their high strength to weight ratios. For simplicity, consider a single honeycomb component with a foil thickness (t) of 0.025mm, a cell size (l) of 3.175mm and a core height (h) of 12.7mm. The material used for the manufacturing of the honeycomb structure is Ti-8Al-1Mo-1V Titanium Alloy and the structure is loaded with a 200,000 Mpa pressure load. For this problem, compare the nominal stress and stress/weight ratio of this honeycomb data to a solid block of the same dimensions. Honeycomb Solid Block 6

5. Conceptual Analysis Conceptual analysis is the abstraction of the logical steps in performing a task or solving a problem. Conceptual analysis for FEM simulation is problem type dependent but softwareindependent, and is fundamental in understanding and solving the problem. Conceptual analysis for buckling analysis reveals the following general logical steps: 1. Pre-processing o Geometry creation o Material property assignment o Boundary condition specification o Mesh generation 2. Solution 3. Post-processing 4. Validation Attachment II discusses the conceptual analysis for the tutorial problem in this module. Link to Conceptual Analysis 7

6. Abstract Modeling Abstract modeling is a process pioneered by CometSolutions Inc. Abstract modeling enables all attributes of an FEM model (such as material properties, constraints, loads, mesh, etc.) to be defined independently in an abstract fashion, thus reducing model complexity without affecting model accuracy with respect to the simulation objective. It detaches attributes from one another, and emphasizes conceptual understanding rather than focusing on software specifics. Evidently, abstract modeling is independent of the specific software being used. This is a fundamental departure from the way most FEM packages operate. Conceptual analysis focuses on the abstraction of steps necessary for an FEM simulation, while abstract modeling focuses on the abstraction and modularization of attributes that constitute an FEM model. They are powerful enabling instruments in FEM teaching and learning. Link to Abstract Modeling 7. Software-Specific FEM Tutorials In software-specific FEM tutorial section, the tutorial problem is solved step by step in a particular software package. This section fills in the details of the conceptual analysis as outlined in previous section. It provides step by step details that correspond to the pre-processing, solution, post-processing and validation phases using a particular software package. Two commercial FEM packages are covered in this module: SolidWorks and CometSolution. Below are the two links: Link to SolidWorks FEM Tutorial 1 Link to SolidWorks FEM Tutorial 2 Link to SolidWorks FEM Tutorial 3 Link to CometSolution FEM Tutorials 8. Post-Test The post-test will be taken upon completion of the module. The first part of the post-test is from the pre-test to test knowledge gained by the user, and the second part is focused on the FEM simulation process covered by the tutorial. Link to Post-Test 9. Practice Problems The user should be able to solve practice problems after completing this module. The practice problems provide a good reinforcement of the knowledge and skills learned in the module, and 8

can be assigned as homework problems in teaching or self study problems to enhance learning. These problems are similar to the tutorial problems worked in the module, but they involve different geometries and thermal boundary conditions. Link to Practice Problems Link to Solutions for Practice Problems 10. Assessment The assessment is provided as a way to receive feedback about the module. The user evaluates several categories of the learning experience, including interactive learning, the module format, its effectiveness and efficiency, the appropriateness of the sections, and the overall learning experience. There is also the opportunity to give suggestions or comments about the module. Link to Assessment 9

Attachment A. Pre-Test 1. The internal force per unit area acting inside a body when forces are applied to the body is called: O Stress O Strain O Displacement O Reaction 2. A column will remain stable as long as the applied load does not exceed the: O Maximum load O Critical load O Weight of the column O Minimum load 3. The engineering principle commonly used for column analysis is: O Bernoulli s Equation O Fourier s Law O Newton s Law O Euler s Formula 4. The slenderness ratio of a column is defined as the column s: O Width divided by the length O Length divided by the radius of gyration O Area divided by the volume O Weight divided by the volume 5. The slenderness ratio of a column can also be roughly interpreted as the column s aspect ratio: O True O False 10

6. The constant C used in column buckling calculations is known as the: O Load constant O Stress constant O End condition constant O Material constant 7. The buckling of a column will only occur under compression loads. O True O False 8. The critical buckling load of a column depends on the: O Elastic modulus of the material O Slimness of the column O End restraint conditions O All of the above 9. In which of the following column categories is buckling in control rather than yielding? O Long columns O Short columns O Intermediate columns O None of the above 10. I long column is one considered to have a slenderness ratio greater than or equal to: O 10 O 50 O 100 O 200 Click to continue 11

Attachment B. Conceptual Analysis Conceptual Analysis of Buckling Simulation Conceptual analysis for a buckling problem using finite element analysis reveals that the following logical steps and sub-steps are needed: 1. Pre-processing (building the model) 1. Geometry creation 2. Material property assignment 3. Boundary condition specification 4. Mesh generation 2. Solution (running the simulation) 3. Post-processing (getting results) 4. Validation (checking) The above steps are explained in some detail as follows. 1. Pre-processing The pre-processing in FEM simulation is analogous to building the structure or making the specimen in physical testing. Several sub-steps involved in pre-processing are geometry creation, material property assignment, boundary condition specification, and mesh generation. The geometry of the structure to be analyzed is defined in the geometry creation step. After the solid geometry is created, the material properties of the solid are specified in the material property assignment step. The material properties required for the FEM analysis depends on the type of analysis. For example, in the buckling analysis of an isotropic material under isothermal conditions, only the modulus of elasticity, Poisson s ratio, and the mass density are needed. For most novice users of FEM, the boundary condition specification step is often the most challenging of all pre-processing steps. Two types of boundary conditions are possible. The first is prescribed displacement boundary condition which is analogous to holding or supporting the specimen in physical testing. The second is applied force boundary condition which is analogous to loading the specimen. Several factors contribute to the challenge of applying boundary conditions correctly: 1) Prescribed displacement boundary conditions expressed in terms such as u u const or const are mathematical simplifications, and frequently boundarya x boundaryb only represent supports in real structures approximately. As a result, choosing a good approximate mathematical representation can be a challenge. 2) How a boundary is restrained depends also on the element type. For example, for the "clamped" or "built-in" support, a boundary should be restrained as having zero nodal displacement if solid element is used, while for the same support, the boundary should be 12

restrained as having zero nodal displacement and zero nodal rotation if shell element is used. 3) Frequently, the structure to be analyzed is not fully restrained from rigid body motion in the original problem statement. In order to obtain an FEM solution, auxiliary restraints become necessary. Over-restraining the model, however, leads to spurious stress results. The challenge is then adding auxiliary restraints to eliminate the possibility of rigid body motion without over-restraining the structure. Because of the above challenges, one learning module will be devoted to boundary condition specification. Mesh generation is the process of discretizing the body into finite elements and assembling the discrete elements into an integral structure that approximates the original body. Most FEM packages have their own default meshing parameters to mesh the model and run the analysis while providing ways for the user to refine the mesh. 2. Solution The solution is the process of solving the governing equations resulting from the discretized FEM model. Although the mathematics for the solution process can be quite involved, this step is transparent to the user and is usually as simple as clicking a solution button or issuing the solution command. 3. Post-processing The purpose of an FEM analysis is to obtain desired results, and this is the goal of the postprocessing step. Typically, various components or measures such as the buckling load factors (BLFs) and displacements are available at any given location within the structure. The method for outputting these values is dependent upon the FEM software. 4. Validation Although validation is not a formal part of the FEM analysis, its inclusion in the process is very important. Blindly trusting an FEM simulation without first checking its correctness can be dangerous. The validation step usually involves comparing FEM results at one or more selected positions with exact or approximate solutions. These exact or approximate solutions are derived using classical approaches covered in mechanics of materials or elasticity courses. Carrying out the validation step also strengthens conceptual understanding and enhances learning. Conceptual Analysis of a Given Problem The goal of the FEM simulation is to analyze the behavior of the solid with the given forces and end conditions. The first tutorial problem shows an AISI 304 steel beam being utilized as a column. This column supports a pressure load on one end under four different conditions. Each of these end conditions will result in different BLFs and critical loads. Conceptual analysis of 13

the current problem is described below. Remember, because there are four separate column arrangements, the following steps must be completed four times. 1. Pre-processing (building the model) The geometry of the structure is first created using the design feature of the FEM package. Next, a material is assigned to the solid model. In the given problem, the material of the beam is given as AISI 304 steel. Depending on the software, the material is either directly selected as steel from the material library, or the properties of the material given in the problem are inputted directly. After assigning the material properties, the boundary conditions are specified. This problem has 2 different end conditions that need to be applied. The fixed end conditions require a fixed restraint which means zero displacement for all boundary nodes due to the solid mesh. The pinned end conditions will require a special restraint to allow rotation about the pin. The pressure load is applied to the top face of the column. These restraints and loads can be accomplished in different ways depending on the different software being used. The next step is to mesh the solid to discretize it into finite elements. Generally, commercial FEA software has automatic default meshing parameters such as average element size of the mesh, quality of the mesh, etc. Here the default parameters provided by the software are used. 2. Solution (running the simulation) The next step is to run the simulation and obtain a solution. Usually the software provides several solver options. The default solver usually works well. For some problems, a particular solver may be faster or give more accurate results. 3. Post-processing (getting results) After the analysis is complete, the post-processing steps are performed. Results such as buckling load factors and displacements can viewed at this point. 4. Validation (checking) Validation is the final step in the analysis process. In this step, the critical load and buckling load factor is calculated by hand. These analytical solutions are compared with the software generated results to check the validity of the analysis. This completes the Conceptual Analysis section. Click the link below to continue with the learning module. Click to continue 14

Attachment C1. SolidWorks-Specific FEM Tutorial 1 Overview: In this section, three tutorial problems will be solved using the commercial FEM software SolidWorks. Although the underlying principles and logical steps of an FEM simulation identified in the Conceptual Analysis section are independent of any particular FEM software, the realization of conceptual analysis steps will be software dependent. The SolidWorks-specific steps are described in this section. This is a step-by-step tutorial. However, it is designed such that those who are familiar with the details in a particular step can skip it and go directly into the next step. Tutorial Problem 1. A pressure loaded column is subjected to four different environments with different end conditions 0. Launching SolidWorks SolidWorks Simulation is an integral part of the SolidWorks computer aided design software suite. The general user interface of SolidWorks is shown in Figure 1. Main menu Frequently used command icons Help icon Roll over to display File, Tools and other menus Figure 1: General user interface of SolidWorks. In order to perform FEM analysis, it is necessary to enable the FEM component, called SolidWorks Simulation, in the software. 15

Step 1: Enabling SolidWorks Simulation o Click "Tools" in the main menu. Select "Add-ins...". The Add-ins dialog window appears, as shown in Figure 2. o Check the boxes in both the Active Add-ins and Start Up columns corresponding to SolidWorks Simulation. o Checking the Active Add-ins box enables the SolidWorks for the current session. Checking the Start Up box enables the SolidWorks for all future sessions whenever SolidWorks starts up. Check SolidWorks Simulation boxes Figure 2: Location of the SolidWorks icon and the boxes to be checked for adding it to the panel. Because this tutorial problem is divided into four parts, this tutorial will be formatted slightly different than in the other available learning modules. The problem requests four different column buckling studies; fixed-free, fixed-fixed, fixed-pinned and pinned-pinned. It is possible to create four separate studies within one part file. However, this greatly reduces the performance of the simulation software and leads to a significant increase in the time required to complete this LM. Thus, only one study will be created. This study will be modified throughout the module to simulate each individual setup and end conditions. The first setup to be evaluated is the fixed-free condition. 1. Pre-Processing Fixed-Free End Conditions Purpose: The purpose of pre-processing is to create an FEM model for use in the next step of the simulation, Solution. It consists of the following sub-steps: Geometry creation Material property assignment Boundary condition specification Mesh generation. 16

1.1 Geometry Creation The purpose of Geometry Creation is to create a geometrical representation of the solid object or structure to be analyzed in FEM. In SolidWorks such a geometric model is called a part. In this tutorial, the necessary part has already been created in SolidWorks. The following steps will open up the part for use in the FEM analysis. Step 1: Opening the part for simulation. One of the following two options can be used. o Option1: Double click the following icon to open the embedded part file, Plain_Column.SLDPRT, in SolidWorks. Click SolidWorks part file icon to open it ==> Plain_Column.SLDPR T o Option 2: Download the part file Plain_Column.SLDPRT from the web site http://www.femlearning.org/. Use the File menu in SolidWorks to open the downloaded part. The SolidWorks model tree will appear with the given part name at the top. Above the model tree, there should be various tabs labeled Features, Sketch, etc. If the Simulation tab is not visible, refer back to steps 1 and 2 in order to enable the SolidWorks Simulation package. Step 2: Creating a Study o Click the Simulation tab above the model tree o Under the drop-down menu select New Study o In the box under Name type in Plain Column Buckling Study o Select Buckling underneath Type as in Figure 3 o Click to create the study Figure 3: Creating a buckling study 17

1.2 Material Property Assignment The next step in FEM analysis is to apply the material properties to the column. The material is given in the problem as AISI 304 and the SolidWorks libraries can be used to apply the material properties. Step 3: Applying the material o Select in the upper left hand corner of the Simulation ribbon o In left-hand section, expand the SolidWorks Materials folder o Expand the Steel section and choose AISI 304 o Make sure the Linear Elastic Isotropic option is selected under Model Type and units are in SI o Verify the settings with Figure 4 and click OK Figure 4: Material property manager in SolidWorks 18

1.3 Boundary Condition Specification Since this is a buckling study, the boundary conditions will consist of fixed and/or pinned connections of the column known as fixtures. The following steps will apply the different end conditions to the column using fixtures. The first step will be to apply the boundary conditions for the column with fixed-free end conditions. Because the top end of the column is free, only one fixture will be applied to the bottom face of the column for this study. Step 4: Applying the fixed end condition to the bottom face o Click on the icon in the upper left corner of the simulation ribbon to drop down the fixture menu o Select Fixed Geometry o With the colored box highlighted, select the lower face of the column as seen in Figure 5 o Click to create the fixed boundary condition Figure 5: Applying a fixed boundary condition The next step is to load the column with the designated pressure load. Due to the nature of columns, the pressure load will be applied to the top face of the column. 19

Step 5: Applying the pressure load o Click on the icon in the upper left corner of the simulation ribbon to drop down the external load menu o Select Pressure o With the colored box highlighted, select the upper face of the column as seen in Figure 6 o Ensure that units of N/mm^2 (MPa) are selected and enter 10 for the magnitude o If necessary, check the Reverse direction box such that the arrows are pointing down on the top face o Click to create the pressure load 1.4 Mesh Generation Figure 6: Applying the pressure load Purpose: The purpose of the Mesh Generation sub-step is to discretize the part into elements. The mesh consists of a network of these elements. Because a fine mesh is not needed in this example, large element sizes will be used to decrease the required solver running time. Step 6: Meshing the model o Right click on the icon in the model tree o Select Create Mesh o Drag the mesh density bar to the Coarse setting as shown in Figure 7 o Click to create the mesh 20

Figure 7: Meshing the model 2. Solution Purpose: The Solution is the step where the computer solves the simulation problem and generates results for use in the Post-Processing step. Step 1: Running the simulation o Within the simulation ribbon, click o When the analysis is finished, the icon will appear on the model tree 3. Post-Processing Purpose: The purpose of the Post-Processing step is to process the results of interest. For this problem, the buckling load factor (BLF) will need to be acquired in order to calculate the critical pressure load of the column. This BLF value can also help describe the presence of buckling in the column. The following table is SolidWorks interpretation of possible BLF values. BLF Value Buckling Status Notes 1 < BLF Buckling not predicted The applied loads are less than the estimated critical loads. Buckling is not expected. 0 < BLF < 1 Buckling predicted The applied loads exceed the estimated critical loads. Buckling is expected. BLF = 1 Buckling predicted The applied loads are exactly equal to the estimated critical loads. Buckling is expected. BLF = -1 Buckling not predicted The buckling occurs when the directions of the applied loads are all reversed. For example, if a bar is under tensile load, the BLF should be negative. The bar will never buckle. -1 < BLF < 0 Buckling not predicted Buckling is predicted if you reverse all loads. BLF < -1 Buckling not predicted Buckling is not expected even if you reverse all loads. 21

SolidWorks makes it very easy to acquire the BLF of any loaded part. Follow the next step in order to observe this data. Step 1: Displaying the Buckling Load Factor o Right click on the icon in the model tree o Select List Buckling Load Factors o Observe the BLF value o Select Save to save the BLF value Now that the buckling load factor has been determined, it can be used to calculate the critical load of the column. In order to do this, simply multiply the BLF by the applied load. This will give the critical load of the column. Therefore: Pcr = Critical Pressure Load = BLF x Applied Load = 3.1929(10MPa) = 31.93 MPa 4. Validation Purpose: The purpose of the Validation step is to compare FEM solutions with analytical solutions, or known published results, to validate the correctness of the FEM model. To check the validity of the SolidWorks answers, hand calculations must be carried out to determine the theoretical critical/buckling load of the column. Euler s formula will be used to carry out these calculations. This formula is as follows: Where C is the end condition constant, E is the material s modulus of elasticity, I is the second moment of inertia (or area moment of inertia), l is the column length, and A is the cross-sectional area. The end condition constant C can be determined from the table below. Column End Conditions Theoretical Value Conservative Value Recommended Value Fixed - Free ¼ ¼ ¼ Fixed - Fixed 4 1 1.2 Fixed - Pinned 2 1 1.2 Pinned - Pinned 1 1 1 From this table, it can be determined that the end condition constant C has a value of ¼ for the fixed-free boundary conditions. The modulus of elasticity (E) value for AISI 304 steel is also defined as 193 GPa while the length of the column is given in the problem statement as 3.5 m. The column s area moment of inertia (I) is calculated by the equation: 22

Where b is the base dimension and h is the height dimension of a rectangular cross-section. Because the column has a square cross section, b = h = 100 mm. Therefore: Therefore, the critical pressure load can be calculated as: The SolidWorks and hand calculated results are shown in the below table along with the percent difference in results. SolidWorks Hand Calculations Percent Difference P cr 31.93 32.40-1.45% Fixed-Fixed End Conditions 1. Pre-Processing Because many of the steps necessary for this study have already been covered in the fixed-free study, these steps will be skipped in the following three studies. 1.1 Geometry Creation These steps were completed in the fixed-free study and, therefore, need not be repeated. 1.2 Material Property Assignment The material properties were already assigned in the previous study. Because the same material is being used in each study, this step can be skipped. 1.3 Boundary Condition Specification The previous study was composed of fixed-free boundary conditions on the column. In this study, fixed-fixed end conditions are required. Since a fixed boundary condition was applied to the bottom face in the previous study, one of the fixed end conditions is completed. Next, it is time to apply the fixed end condition to the top face of the column. The top fixture must be modified in order to allow free vertical displacement. To allow only vertical displacement, a boundary condition must be applied to each side face to prevent horizontal 23

displacement. If a fixed geometry were used, each side would be completely restrained and the column would exhibit no buckling behavior. Therefore, the approach is to apply a reference geometry to a small portion of each side face near the top of the column to prevent horizontal movement. The length of each face to be fixed will be setl to half of the column s crosssectional dimension, or 50 mm. A new plane will be used to create a split line which will enable the use of this approach. Step 1: Creating a new plane for the split line o On the main menu, go to Insert -> Reference Geometry -> Plane o With the First Reference box highlighted, expand the model tree and select the Top Plane as shown in Figure 8 o Enter 50 for the offset distance o Check the Flip box if plane was created above the Top Plane o Click to create the plane Figure 8: Creating a new plane Step 2: Creating a split line o On the main menu, go to Insert -> Curve -> Split Line o Under the type of split, select Intersection o With the first colored box highlighted, expand the model tree and select the plane created in the previous step o With the second colored box highlighted, select each side of the column as shown in Figure 9 o Click to create the split line 24

Figure 9: Creating a split line Now that a split line has been created, it is possible to apply a reference geometry to each upper side face created by the split line. This reference geometry will allow vertical displacement but not horizontal displacement. Step 3: Applying the fixed end condition to the top face o Click on the icon in the upper left corner of the screen to drop down the fixture menu o Select Advanced Fixtures o With the first colored box highlighted, select the four upper side faces created by the split line o With the second colored box highlighted, select a top edge traveling in the z-direction as shown in Figure 10 o Under Translations, enter 0 for the allowable displacement distance o Click to create the reference geometry o Repeat these steps once more using another top edge that is perpendicular to the one used above (should travel in the x-direction) 25

Figure 10: Creating reference geometry The next step of the pre-processing procedure would typically be to apply any external loads on the member as required by the problem statement. However, the external load for this study is exactly the same as the previous study. This means that this step can be skipped as it has already been completed. Therefore, the next step in the process is mesh generation. 1.4 Mesh Generation Any time that changes are made within a study, the model must be re-meshed before the simulation can be run. This is done in the same way as described in the previous study. If needed, refer back to step 6 in the fixed-free study for a description on how to create the mesh. 2. Solution Step 1: Running the simulation o Within the simulation ribbon, click o When the analysis is finished, the icon will appear on the model tree 3. Post-Processing Just as in the previous study, use SolidWorks to display the BLF of the fixed-fixed column. Now that the buckling load factor has been determined, it can be used to calculate the critical pressure load of the column. Pcr = Critical Pressure Load = BLF x Applied Load = 49.892(10MPa) = 498.92 MPa 26

4. Validation Finally, hand calculations will be carried out and the solutions will be compared to the FEM results. Once again, Euler s formula will be used for these calculations. Because the cross-sectional dimensions and material properties have not changed since the previous study, the values of E, I and A will remain the same. However, due to the different end conditions, the value of the end condition constant C will change. This new value can be determined from the table below. Column End Conditions Theoretical Value Conservative Value Recommended Value Fixed - Free ¼ ¼ ¼ Fixed - Fixed 4 1 1.2 Fixed - Pinned 2 1 1.2 Pinned - Pinned 1 1 1 When designing a column, a recommended value of 1.2 should be used for C as it instills somewhat of a safety factor against failure. However, the theoretical value will be used in these calculations as it will yield results much closer to the actual critical pressure load. The theoretical C value from the table can be observed as 4. Also, because 50 mm of the column s length was used as a fixture, the column s length is no longer considered to be 3.5 m. The new length of the column is 3.5 m minus 50 mm, or 3.45 m. Therefore, the critical pressure load can be calculated as: The SolidWorks and hand calculated results are shown in the below table along with the percent difference in results. SolidWorks Hand Calculations Percent Difference P cr 498.92 533.47-6.48% 27

Fixed-Pinned End Conditions 1. Pre-Processing Because many of the steps necessary for this study have already been covered and completed in the previous two studies, these steps will not be described in detail and sometimes will be skipped altogether. 1.1 Geometry Creation These steps were completed in the fixed-free study and, therefore, need not be repeated. 1.2 Material Property Assignment Material properties were assigned in the first study. Because the same material is being used in each study, this step can be skipped. 1.3 Boundary Condition Specification The previous study was composed of fixed-fixed boundary conditions on the column. In this study, fixed-pinned end conditions are desired. Once again, the steps necessary to apply the fixed boundary condition to the bottom face can be ignored as they have already been completed. Therefore, the next logical step is to apply a pinned end condition to the top column face. Because advanced pinned/hinged connections are beyond the scope of this module, a much simpler procedure will be used. Perpendicular split lines will be created on the top face that intersect at the center of the face. A reference geometry will be applied to this intersection point allowing the connection to behave as if it were pinned. Granted, this procedure will not yield a true pinned connection, but the results will be sufficient for this problem. Step 1: Creating a new plane for the first split line o On the main menu, go to Insert -> Reference Geometry -> Plane o With the First Reference box highlighted, expand the model tree and select the Front Plane as shown in Figure 11 o Enter 50 for the offset distance o Check the Flip box if necessary to place the plane in the middle of the top face o Click to create the plane 28

Figure 11: Creating a new plane Step 2: Creating a new plane for the second split line o On the main menu, go to Insert -> Reference Geometry -> Plane o With the First Reference box highlighted, expand the model tree and select the Right Plane o Enter 50 for the offset distance o Check the Flip box if necessary to place the plane in the middle of the top face o Click to create the plane Now that the planes have been created, they can be used to generate the intersecting split lines on the top face of the column. Step 3: Creating the first split line o On the main menu, go to Insert -> Curve -> Split Line o Under the type of split, select Intersection o With the first colored box highlighted, expand the model tree and select the plane created in the first step o With the second colored box highlighted, select the top face of the column as shown in Figure 12 o Click to create the split line 29

Figure 12: Creating a split line Step 4: Creating the second split line o On the main menu, go to Insert -> Curve -> Split Line o Under the type of split, select Intersection o With the first colored box highlighted, expand the model tree and select the plane created in the second step o With the second colored box highlighted, select both sections on the top face created by the first split line o Click to create the split line Now that the split lines have been created, it is possible to apply a reference geometry to the point created by the intersection of the split lines. This reference geometry will allow vertical displacement but not horizontal displacement. Step 5: Applying the pinned end condition to the top face o Click on the icon in the upper left corner of the screen to drop down the fixture menu o Select Advanced Fixtures o With the first colored box highlighted, select the intersection point between the previously created split lines o With the second colored box highlighted, select a top edge of the column as shown in Figure 13 o Under Translations, enter a value of zero for the allowable displacement o Click to create the reference geometry o Repeat this step once more using another top edge that is perpendicular to the one used above 30

Figure 13: Creating reference geometry The next step is to load the column with the designated pressure load. This step was skipped in the fixed-fixed study as no changes were made to the face on which the load was applied. However, now that the geometry of the top face has been altered, the pressure load must be modified to accommodate for these changes. One way to do this is to delete the existing load and create a new one. However, it will be easier to simply modify the existing external load. Step 6: Modifying the pressure load o Right-click on the existing external load and select Edit Definition o With the colored box highlighted, select all four of the top face sections created by the two split lines as shown in Figure 14 o Ensure that units are N/mm^2 (MPa) with a magnitude of 10 o If necessary, check the Reverse direction box such that the arrows are pointing down on the top face o Click to create the pressure load 31

Figure 14: Modifying the pressure load 1.4 Mesh Generation Any time that changes are made within a study, the model must be re-meshed before the simulation can be run. If needed, refer back to step 6 in the fixed-free study for a description on how to create the mesh. 2. Solution Step 1: Running the simulation o Within the simulation ribbon, click o When the analysis is finished, the icon will appear on the model tree 3. Post-Processing Just as in the previous two studies, use SolidWorks to display the BLF of the column under fixed-pinned end conditions. Now that the buckling load factor has been determined, it can be used to calculate the critical pressure load of the column. Pcr = Critical Pressure Load = BLF x Applied Load = 25.996(10MPa) = 259.96 MPa 32

4. Validation Finally, hand calculations will be carried out and the solutions will be compared to the FEM results. Euler s formula will once again be used for these calculations. Because the cross-sectional dimensions and material properties have not changed since the previous study, the values of E, I and A will remain the same. The given length of the column will also remain at 3.5 m. However, due to the different end conditions, the value of the end condition constant C will change. This new value can be determined from the table below. Column End Conditions Theoretical Value Conservative Value Recommended Value Fixed - Free ¼ ¼ ¼ Fixed - Fixed 4 1 1.2 Fixed - Pinned 2 1 1.2 Pinned - Pinned 1 1 1 From this table, it can be determined that the end condition constant C has a theoretical value of 2 for the fixed-pinned boundary conditions. Therefore, the critical pressure load can be calculated as: The SolidWorks and hand calculated results are shown in the below table along with the percent difference in results. SolidWorks Hand Calculations Percent Difference P cr 259.96 259.17 0.30% 33

Pinned-Pinned End Conditions 1. Pre-Processing Because many of the steps necessary for this study have already been covered and completed in the previous three studies, these steps will not be described in detail and sometimes will be skipped altogether. 1.1 Geometry Creation These steps were completed in the fixed-free study and, therefore, need not be repeated. 1.2 Material Property Assignment Material properties were assigned in the first study. Because the same material is being used in each study, this step can be skipped. 1.3 Boundary Condition Specification The previous study was composed of fixed-pinned boundary conditions on the column. In this study, pinned-pinned end conditions are desired. The steps necessary to apply the pinned boundary condition to the top face can be ignored as they have already been completed in the fixed-pinned study. Therefore, the next logical step is to apply a pinned end condition to the bottom face of the column. Because advanced pinned/hinged connections are beyond the scope of this module, a more simple procedure will be used on the bottom face. This procedure will be to create a split line on the bottom face and apply a fixed geometry to it. Normally, a new must be created in order to generate this split line. However, this plane created in the previous study and this step can be ignored. After the split line is generated, a fixed geometry will be applied to it, allowing the connection to behave as if it were pinned or hinged. Granted, this procedure will not yield a true pinned connection, but the results will be sufficient for this tutorial. Step 1: Creating the split line o On the main menu, go to Insert -> Curve -> Split Line o Under the type of split, select Intersection o With the first colored box highlighted, expand the model tree and select one of the two planes created in the previous study o With the second colored box highlighted, select the bottom face of the column o Click to create the split line Now that the split line has been created, it is possible to apply the fixed boundary condition. This fixed geometry will fix the line but allow displacement of the rest of the face. As a result, the connection will behave as if it were pinned. In the previous study, the bottom face was 34

completely fixed. One way to change the fixture is to delete the previous fixture and create a new one. However, a quicker procedure is to simply modify the existing fixture. Step 3: Applying the pinned end condition to the bottom face o Right-click on the existing fixed fixture and select Edit Definition o With the colored box highlighted, select the split line on the bottom face as shown in Figure 15 o Click to create the fixed geometry Figure 15: Modifying a fixed geometry The next step of the pre-processing procedure would typically be to apply any external loads on the member as required by the problem statement. However, the external pressure load was already modified in the previous study. This means that this step can be skipped as it has already been completed. Therefore, the next logical step in the process is mesh generation. 1.4 Mesh Generation Any time that changes are made to a study, the model must be re-meshed before the simulation can be run. If needed, refer back to step 6 in the fixed-free study for a description on how to create the mesh. 2. Solution Step 1: Running the simulation o Within the simulation ribbon, click o When the analysis is finished, the icon will appear on the model tree 35

3. Post-Processing Just as in the previous three studies, use SolidWorks to display the BLF of the column under pinned-pinned end conditions. Now that the buckling load factor has been determined, it can be used to calculate the critical pressure load of the column. Pcr = Critical Pressure Load = BLF x Applied Load = 12.723(10MPa) = 127.23 MPa 4. Validation Finally, hand calculations will be carried out and the solutions will be compared to the FEM results. Euler s formula will once again be used for these calculations. Because the cross-sectional dimensions and material properties have not changed since the previous study, the values of E, I and A will remain the same. The given length of the column will also remain at 3.5 m. However, due to the different end conditions, the value of the end condition constant C will change. This new value can be determined from the table below. Column End Conditions Theoretical Value Conservative Value Recommended Value Fixed - Free ¼ ¼ ¼ Fixed - Fixed 4 1 1.2 Fixed - Pinned 2 1 1.2 Pinned - Pinned 1 1 1 From this table, it can be determined that the end condition constant C has a theoretical value of 1 for the fixed-pinned boundary conditions. Therefore, the critical pressure load can be calculated as: The SolidWorks and hand calculated results are shown in the below table along with the percent difference in results. SolidWorks Hand Calculations Percent Difference P cr 127.23 129.58-1.81% 36

Results Summary Critical Pressure Loads SolidWorks Hand Calculations Percent Difference Fixed-Free 31.93 32.40-1.45% Fixed-Fixed 498.92 533.47-6.48% Fixed-Pinned 259.96 259.17 0.30% Pinned-Pinned 127.23 129.58-1.81% It can be seen from the table above that the SolidWorks results and hand calculated solutions are very close. With the exception of the fixed-fixed condition, all results and solutions are within 2% of one another. This ensures that the SolidWorks simulation results are accurate. The greater error in the fixed-fixed condition can be attributed the method of fixing a portion of the column s side faces. This may be the cause of some hand calculation inefficiencies. But, overall the results look relatively dependable. 37

Attachment C2. SolidWorks-Specific FEM Tutorial 2 Tutorial Problem 2. A pressure loaded, L-shaped column is subjected to two different environments with different end conditions 0. Launching SolidWorks In order to perform FEM analysis, it is necessary to enable the FEM component, called SolidWorks Simulation, in the software. Step 1: Enabling SolidWorks Simulation o Click "Tools" in the main menu. Select "Add-ins...". The Add-ins dialog window appears, as shown in Figure 2. o Check the boxes in both the Active Add-ins and Start Up columns corresponding to SolidWorks Simulation. o Checking the Active Add-ins box enables the SolidWorks for the current session. Checking the Start Up box enables the SolidWorks for all future sessions whenever SolidWorks starts up. This tutorial problem requests two different column buckling studies; fixed-free and fixed-fixed. It is possible to create two separate studies within one part file. However, this greatly reduces the performance of the simulation software and leads to a significant increase in the time required to complete this LM. Thus, only one study will be created. This study will be modified throughout the module to simulate each individual setup and end conditions. The first setup to be evaluated is the fixed-free condition. Fixed-Free End Conditions 1. Pre-Processing Purpose: The purpose of pre-processing is to create an FEM model for use in the next step of the simulation, Solution. It consists of the following sub-steps: Geometry creation Material property assignment Boundary condition specification Mesh generation. 38

1.1 Geometry Creation The purpose of Geometry Creation is to create a geometrical representation of the solid object or structure to be analyzed in FEM. In SolidWorks such a geometric model is called a part. In this tutorial, the necessary part has already been created in SolidWorks. The following steps will open up the part for use in the FEM analysis. Step 1: Opening the part for simulation. One of the following two options can be used. o Option1: Double click the following icon to open the embedded part file, L- Shaped_Column.SLDPRT, in SolidWorks. Click SolidWorks part file icon to open it ==> L-Shaped_Column.SL DPRT o Option 2: Download the part file L-Shaped_Column.SLDPRT from the web site http://www.femlearning.org/. Use the File menu in SolidWorks to open the downloaded part. The SolidWorks model tree will appear with the given part name at the top. Above the model tree, there should be various tabs labeled Features, Sketch, etc. If the Simulation tab is not visible, refer back to steps 1 and 2 in order to enable the SolidWorks Simulation package. Step 2: Creating a Study o Click the Simulation tab above the model tree o Under the drop-down menu select New Study o In the box under Name type in L-Shaped Column Buckling Study o Select Buckling underneath Type as in Figure 1 o Click to create the study Figure 1: Creating a buckling study 39

1.2 Material Property Assignment The next step in FEM analysis is to apply the material properties to the column. The material is given in the problem as AISI 304 and the SolidWorks libraries can be used to apply the material properties. Step 3: Applying the material o Select in the upper left hand corner of the Simulation ribbon o In left-hand section, expand the SolidWorks Materials folder o Expand the Steel section and choose AISI 304 o Make sure the Linear Elastic Isotropic option is selected under Model Type and units are in SI o Verify the settings with Figure 2 and click OK Figure 2: Material property manager in SolidWorks 1.3 Boundary Condition Specification Since this is a buckling study, the boundary conditions can consist of fixed and/or pinned connections of the column known as fixtures. The following steps will apply the different end conditions to the column using fixtures. 40

The first step will be to apply the boundary conditions for the column with fixed-free end conditions. Because the top end of the column is free, only one fixture will be applied to the bottom face of the column for this study. Step 4: Applying the fixed end condition to the bottom face o Click on the icon in the upper left corner of the simulation ribbon to drop down the fixture menu o Select Fixed Geometry o With the colored box highlighted, select the lower face of the column as seen in Figure 3 o Click to create the fixed boundary condition Figure 3: Applying a fixed boundary condition The next step is to load the column with the designated pressure load. Due to the nature of columns, the pressure load will be applied to the top face of the column. Step 5: Applying the pressure load o Click on the icon in the upper left corner of the simulation ribbon to drop down the external load menu o Select Pressure 41

o With the colored box highlighted, select the upper face of the column as seen in Figure 4 o Ensure that units of N/mm^2 (MPa) are selected and enter 6 for the magnitude o If necessary, check the Reverse direction box such that the arrows are pointing down on the top face o Click to create the pressure load Figure 4: Applying the pressure load 1.4 Mesh Generation Purpose: The purpose of the Mesh Generation sub-step is to discretize the part into elements. The mesh consists of a network of these elements. Because a fine mesh is not needed in this example, large element sizes will be used to decrease the required solver running time. Step 6: Meshing the model o Right click on the icon in the model tree o Select Create Mesh o Drag the mesh density bar to the Coarse setting as shown in Figure 5 o Click to create the mesh 42

Figure 5: Meshing the model 2. Solution Purpose: The Solution is the step where the computer solves the simulation problem and generates results for use in the Post-Processing step. Step 1: Running the simulation o Within the simulation ribbon, click o When the analysis is finished, the icon will appear on the model tree 3. Post-Processing Purpose: The purpose of the Post-Processing step is to process the results of interest. For this problem, the buckling load factor (BLF) will need to be acquired in order to calculate the critical pressure load of the column. This BLF value can also help describe the presence of buckling in the column. The following table is SolidWorks interpretation of possible BLF values. BLF Value Buckling Status Notes 1 < BLF Buckling not predicted The applied loads are less than the estimated critical loads. Buckling is not expected. 0 < BLF < 1 Buckling predicted The applied loads exceed the estimated critical loads. Buckling is expected. BLF = 1 Buckling predicted The applied loads are exactly equal to the estimated critical loads. Buckling is expected. BLF = -1 Buckling not predicted The buckling occurs when the directions of the applied loads are all reversed. For example, if a bar is under tensile load, the BLF should be negative. The bar will never buckle. -1 < BLF < 0 Buckling not predicted Buckling is predicted if you reverse all loads. BLF < -1 Buckling not predicted Buckling is not expected even if you reverse all loads. 43

SolidWorks makes it very easy to acquire the BLF of any loaded part. Follow the next step in order to observe this data. Step 1: Displaying the Buckling Load Factor o Right click on the icon in the model tree o Select List Buckling Load Factors o Observe the BLF value o Select Save to save the BLF value Now that the buckling load factor has been determined, it can be used to calculate the critical pressure load of the column. In order to do this, simply multiply the BLF by the applied load. This will give the critical pressure load of the column. Therefore: Pcr = Critical Pressure Load = BLF x Applied Load = 2.3245(6 MPa) = 13.95 MPa 4. Validation Purpose: The purpose of the Validation step is to compare FEM solutions with analytical solutions, or known published results, to validate the correctness of the FEM model. To check the validity of the SolidWorks answers, hand calculations must be carried out to determine the theoretical critical/buckling pressure load of the column. Euler s formula will be used to carry out these calculations. This formula is as follows: Where C is the end condition constant, E is the material s modulus of elasticity, I is the second moment of inertia (or area moment of inertia), l is the column length, and A is the cross-sectional area. The end condition constant C can be determined from the table below. Column End Conditions Theoretical Value Conservative Value Recommended Value Fixed - Free ¼ ¼ ¼ Fixed - Fixed 4 1 1.2 Fixed - Pinned 2 1 1.2 Pinned - Pinned 1 1 1 From this table, it can be determined that the end condition constant C has a value of ¼ for the fixed-free boundary conditions. The modulus of elasticity (E) value for AISI 304 steel is also defined as 193 GPa while the length of the column is given in the problem statement as 3.6 m. Therefore, the only remaining value needed is that of the column s area moment of inertia. It is possible to calculate this value using the Parallel Axis Theorem, but these calculations are beyond the scope of this module. SolidWorks makes it very easy to obtain these types of values 44

using the Section Properties tool. Since the column s cross-section is not symmetrical, the column will bend about a principal axis as opposed to the x or y-axis. Because of this, the principal moments of inertia values are desired from the section properties. Due to the nature of the column s shape, there will be two different principal moments of inertia at the centroid. It is desirable to select the smaller value of the two as the column will bend about the axis with least resistance. Step 1: Displaying the section properties o Highlight the bottom face of the column o On the main menu, go to Tools -> Section Properties to open the section properties window as in Figure 6 o The lowest principal moment of inertia value can be observed as 1092220.93mm 4 o Record this number and close the properties window Figure 6: Displaying the section properties 45

Now that the value of I is known, the critical pressure load can be calculated as: However, this critical pressure load value does not take into consideration the holes in the column. To calculate the actual critical pressure load, the above value must be multiplied by a modification factor α that accounts for the reduction in load carrying capacity due to holes. This modification factor is defined as: There are two different ways that this factor can be calculated. One way is to calculate the volume of the column with and without holes and plug into the equation above. Another way is to simply used SolidWorks to determine these values and eliminate any calculation or rounding errors. The steps required to determine these values are described below. Step 2: Displaying the column s volume with holes o Ensure that no pieces of the column are selected/highlighted o On the main menu, go to Tools -> Mass Properties o With the mass properties window open, locate and record the volume value of 6602854.13mm 3 o Close the window Step 3: Displaying the column s volume without holes o In the feature design tree, click and drag the feature design line until it is positioned directly below the Boss-Extrude1 feature as shown in Figure 7 o This will suppress the hole features and display the column without holes o On the main menu, go to Tools -> Mass Properties o With the mass properties window open, locate and record the volume value of 8280000.00mm 3 o Close the window 46

Figure 7: Suppressing the hole features Now that the values of the column s volume with and without holes are known, the modification factor can be determined. Finally, the critical pressure load can be calculated by multiplying the modification factor by the critical pressure load calculated above. The SolidWorks and hand calculated results are shown in the below table along with the percent difference in results. SolidWorks Hand Calculations Percent Difference P cr 13.95 13.91 0.29% 47

Fixed-Fixed End Conditions 1. Pre-Processing Because many of the steps necessary for this study have already been covered in the fixed-free study, these steps will be described in less detail or skipped altogether. 1.1 Geometry Creation These steps were completed in the fixed-free study and, therefore, need not be repeated. 1.2 Material Property Assignment The material properties were already assigned in the previous study. Because the same material is being used in each study, this step can be skipped. 1.3 Boundary Condition Specification The previous study was composed of fixed-free boundary conditions on the column. In this study, fixed-fixed end conditions are required. Since a fixed boundary condition was applied to the bottom face in the previous study, this end condition has been satisfied and can be skipped. Next, it is time to apply the fixed end condition to the top face of the column. The top fixture must be modified in order to allow vertical displacement. To allow only vertical displacement, a boundary condition must be applied to each side face to prevent horizontal displacement. However, applying a fixed geometry to each side would completely restrain the column and result in no buckling behavior. Therefore, the approach is to apply a reference geometry to a small portion of each side face near the top of the column. The length of each face to be fixed must be small enough that it doesn t interfere with the buckling shape of the column. Fixing 15mm of the sides near the top face should be sufficient for this study. In order to to this, a new plane will be used to create a split line. This split line will form 15mm sections at the top of the side faces. These sections can then be restrained to resemble a fixed-fixed column condition. Step 1: Creating a new plane for the split line o On the main menu, go to Insert -> Reference Geometry -> Plane o With the First Reference box highlighted, expand the model tree and select the Top Plane as shown in Figure 8 o Enter 15 for the offset distance o Check the Flip box if plane was created above the Top Plane o Click to create the plane 48

Figure 8: Creating a new plane Step 2: Creating a split line o On the main menu, go to Insert -> Curve -> Split Line o Under the type of split, select Intersection o With the first colored box highlighted, expand the model tree and select the plane created in the previous step o With the second colored box highlighted, select each side face of the column as shown in Figure 9 o Click to create the split line Figure 9: Creating a split line 49

Now that a split line has been created, it is possible to apply a reference geometry to each side face section created by the split line. This reference geometry will allow vertical displacement but not horizontal displacement. It will take two steps to accomplish this task. One step will restrict translation in the z-direction and one will restrict translation in the x-direction Step 3: Applying the fixed end condition to the top face (z-direction) o Click on the icon in the upper left corner of the screen to drop down the fixture menu o Select Advanced Fixtures o With the first colored box highlighted, select all of the upper side sections created by the split line on the sides of the column o With the second colored box highlighted, select the a top edge running in the z- direction as shown in pink in Figure 10 o Under Translations, enter 0 for the distance o Click to create the reference geometry Figure 10: Creating reference geometry 50

Step 4: Applying the fixed end condition to the top face (x-direction) o Click on the icon in the upper left corner of the screen to drop down the fixture menu o Select Advanced Fixtures o With the first colored box highlighted, select all of the upper side sections created by the split line on the sides of the column o With the second colored box highlighted, select the a top edge running in the x- direction as shown in pink in Figure 11 o Under Translations, enter 0 for the distance o Click to create the reference geometry Figure 11: Creating reference geometry The next step of the pre-processing procedure would typically be to apply any external loads on the member as required by the problem statement. However, the external load for this study is exactly the same as the previous study. This means that this step can be skipped as it has already been completed. Therefore, the next step in the process is mesh generation. 51

1.4 Mesh Generation Any time that changes are made within a study, the model must be re-meshed before the simulation can be run. This is done in the same way as described in the previous study. If needed, refer back to step 6 in the fixed-free study for a description on how to create the mesh. 2. Solution Step 1: Running the simulation o Within the simulation ribbon, click o When the analysis is finished, the icon will appear on the model tree 3. Post-Processing Just as in the previous study, use SolidWorks to display the BLF of the fixed-fixed column. Now that the buckling load factor has been determined, it can be used to calculate the critical pressure load of the column. P cr = Critical Pressure Load = BLF x Applied Load = 28.148(6MPa) = 168.89 MPa 4. Validation Purpose: The purpose of the Validation step is to compare FEM solutions with analytical solutions, or known published results, to validate the correctness of the FEM model. To check the validity of the SolidWorks answers, hand calculations must be carried out to determine the theoretical critical/buckling pressure load of the column. Euler s formula will be used to carry out these calculations. This formula is as follows: Because the cross-sectional dimensions and material properties have not changed since the previous study, the values of E, I and A will remain the same. However, due to the different end conditions, the value of the end condition constant C will change. This new value can be determined from the table on the following page. 52

Column End Conditions Theoretical Value Conservative Value Recommended Value Fixed - Free ¼ ¼ ¼ Fixed - Fixed 4 1 1.2 Fixed - Pinned 2 1 1.2 Pinned - Pinned 1 1 1 When designing a column, a recommended value of 1.2 should be used for C as it instills somewhat of a safety factor against failure. However, the theoretical value will be used in these calculations as it will yield results much closer to the actual critical pressure load. The theoretical C value from the table can be observed as 4. Also, because 15 mm of the column s length was used as a fixture, the column s length is no longer considered to be 3.6 m. The new length of the column is 3.5 m minus 15 mm, or 3.585 m. Therefore, the critical pressure load can be calculated as: Once again, this value for the critical pressure load is that of the column without holes. The critical pressure of the column with holes can be found by multiplying by the modification factor. The SolidWorks and hand calculated results are shown in the below table along with the percent difference in results. SolidWorks Hand Calculations Percent Difference P cr 168.89 224.38-24.73% 53

Results Summary Critical Pressure Loads SolidWorks Hand Calculations Percent Difference Fixed-Free 13.95 13.91 0.29% Fixed-Fixed 168.89 224.38-24.73% It can be seen from the table above that the SolidWorks results and hand calculated solutions for the fixed-free study are very close. An error of less than half a percent ensures that the SolidWorks simulation results are accurate. The greater error in the fixed-fixed condition can be attributed to the broader range of end constant values. The value for C can range anywhere from 1 to 4, leading to the possibility of inconsistencies. But because the calculated value falls within the allowable range of C values, it can be assumed that the solutions are relatively accurate. Modifying the end constant value for this particular problem would yield much more desirable results. 54

Attachment C3. SolidWorks-Specific FEM Tutorial 3 Tutorial Problem 3. A titanium honeycomb component is subjected to a pressure load and compared to a solid column of identical dimensions. 0. Launching SolidWorks In order to perform FEM analysis, it is necessary to enable the FEM component, called SolidWorks Simulation, in the software. Step 1: Enabling SolidWorks Simulation o Click "Tools" in the main menu. Select "Add-ins...". The Add-ins dialog window appears, as shown in Figure 2. o Check the boxes in both the Active Add-ins and Start Up columns corresponding to SolidWorks Simulation. o Checking the Active Add-ins box enables the SolidWorks for the current session. Checking the Start Up box enables the SolidWorks for all future sessions whenever SolidWorks starts up. This tutorial problem requests two different buckling studies; one on a honeycomb structure and one on a similar solid block. Because this tutorial problem requires studies on two separate parts, the simulation on each part will be performed individually. The results of the two studies will then be compared at the end of the tutorial. The first setup to be evaluated is the titanium honeycomb structure. Honeycomb Column 1. Pre-Processing Purpose: The purpose of pre-processing is to create an FEM model for use in the next step of the simulation, Solution. It consists of the following sub-steps: Geometry creation Material property assignment Boundary condition specification Mesh generation. 55

1.1 Geometry Creation The purpose of Geometry Creation is to create a geometrical representation of the solid object or structure to be analyzed in FEM. In SolidWorks such a geometric model is called a part. In this tutorial, the necessary part has already been created in SolidWorks. The following steps will open up the part for use in the FEM analysis. Step 1: Opening the part for simulation. One of the following two options can be used. o Option1: Double click the following icon to open the embedded part file, Honeycomb_Column.SLDPRT, in SolidWorks. Click SolidWorks part file icon to open it ==> Honeycomb_Column. SLDPRT o Option 2: Download the part file Honeycomb_Column.SLDPRT from the web site http://www.femlearning.org/. Use the File menu in SolidWorks to open the downloaded part. The SolidWorks model tree will appear with the given part name at the top. Above the model tree, there should be various tabs labeled Features, Sketch, etc. If the Simulation tab is not visible, refer back to steps 1 and 2 in order to enable the SolidWorks Simulation package. Step 2: Creating a Study o Click the Simulation tab above the model tree o Under the drop-down menu select New Study o In the box under Name type in Honeycomb Buckling Study o Select Buckling underneath Type as in Figure 1 o Click to create the study Figure 1: Creating a buckling study 56

1.2 Material Property Assignment The next step in FEM analysis is to apply the material properties to the column. The material is given in the problem as a Ti-8Al-1Mo-1V titanium alloy and the SolidWorks libraries can be used to apply the material properties. Step 3: Applying the material o Select in the upper left hand corner of the Simulation ribbon o In left-hand section, expand the SolidWorks Materials folder o Expand the Titanium Alloys section and choose Titanium Te-8Al-1Mo-1V o Make sure the Linear Elastic Isotropic option is selected under Model Type and units are in SI o Verify the settings with Figure 2 and click OK Figure 2: Material property manager in SolidWorks 57

1.3 Boundary Condition Specification Due to the nature and applications of honeycomb structures, this study will be treated as a column buckling study with fixed-free end conditions. Because the top end of the column is free, only one fixture will be applied to the bottom face of the column for this study. Step 4: Applying the fixed end condition to the bottom face o Click on the icon in the upper left corner of the simulation ribbon to drop down the fixture menu o Select Fixed Geometry o With the colored box highlighted, select the lower face of the column as seen in Figure 3 o Click to create the fixed boundary condition Figure 3: Applying a fixed boundary condition The next step is to load the column with the designated pressure load. Due to the nature of columns, the pressure load will be applied to the top face of the column. Step 5: Applying the pressure load o Click on the icon in the upper left corner of the simulation ribbon to drop down the external load menu o Select Pressure 58

o With the colored box highlighted, select the upper face of the column as seen in Figure 4 o Ensure that units of N/mm^2 (MPa) are selected and enter 200,000 for the magnitude o If necessary, check the Reverse direction box such that the arrows are pointing down on the top face o Click to create the pressure load Figure 4: Applying the pressure load 1.4 Mesh Generation Purpose: The purpose of the Mesh Generation sub-step is to discretize the part into elements. The mesh consists of a network of these elements. Because a fine mesh is not needed in this example, the default element sizes will be used to decrease the required solver running time. Step 6: Meshing the model o Right click on the icon in the model tree o Select Create Mesh o Use the default mesh setting as seen in Figure 5 o Click to create the mesh 59

Figure 5: Meshing the model 2. Solution Purpose: The Solution is the step where the computer solves the simulation problem and generates results for use in the Post-Processing step. Step 1: Running the simulation o Within the simulation ribbon, click o When the analysis is finished, the icon will appear on the model tree 3. Post-Processing Purpose: The purpose of the Post-Processing step is to process the results of interest. For this problem, the buckling load factor (BLF) will need to be acquired in order to calculate the critical pressure load of the column and observe the nominal stress and stress/weight ratio. This BLF value can also help describe the presence of buckling in the column. The following table is SolidWorks interpretation of possible BLF values. BLF Value Buckling Status Notes 1 < BLF Buckling not predicted The applied loads are less than the estimated critical loads. Buckling is not expected. 0 < BLF < 1 Buckling predicted The applied loads exceed the estimated critical loads. Buckling is expected. BLF = 1 Buckling predicted The applied loads are exactly equal to the estimated critical loads. Buckling is expected. BLF = -1 Buckling not predicted The buckling occurs when the directions of the applied loads are all reversed. For example, if a bar is under tensile load, the BLF should be negative. The bar will never buckle. -1 < BLF < 0 Buckling not predicted Buckling is predicted if you reverse all loads. BLF < -1 Buckling not predicted Buckling is not expected even if you reverse all loads. 60

SolidWorks makes it very easy to acquire the BLF of any loaded part. Follow the next step in order to observe this data. Step 1: Displaying the Buckling Load Factor o Right click on the icon in the model tree o Select List Buckling Load Factors o Observe the BLF value o Select Save to save the BLF value Now that the buckling load factor has been determined, it can be used to calculate the critical pressure load of the column. In order to do this, simply multiply the BLF by the applied load. This will give the critical pressure load of the column. Therefore: Pcr = Critical Pressure Load = BLF x Applied Load = 677.1(200 GPa) = 135.42 TPa 4. Validation Purpose: The purpose of the Validation step is to compare FEM solutions with analytical solutions, or known published results, to validate the correctness of the FEM model. However, due to the complex nature of honeycomb buckling calculations, the results will only be compared to those of the solid column. This section will simply address the nominal stress and stress/weight ratio of the honeycomb structure. The nominal stress of a column is defined as the critical pressure load multiplied by the ratio of the net cross-sectional area to the total cross-sectional area. Thus, the nominal stress is calculated as follows: The next goal of the problem is to determine the column s stress/weight ratio. However, because this structure is so small in size, SolidWorks will display a reading of 0 for the mass. Therefore, it is acceptable to use the structure s volume in the calculations as the volume of a structure is directly proportional to its weight. It is very easy to acquire these values from SolidWorks using the following method. Step 1: Displaying the part s mass properties o Ensure that no pieces of the column are selected/highlighted o On the main menu, go to Tools -> Mass Properties o With the mass properties window open, locate and record the volume value of 4 mm 3 o Close the window Now that structure s volume is known, it is possible to calculate the stress/volume ratio. This is done as follows: 61

The next step is to determine the critical pressure load and calculate the nominal stress and stress/volume ratio of a similar solid column and compare. Solid Column 1. Pre-Processing 1.1 Geometry Creation The purpose of Geometry Creation is to create a geometrical representation of the solid object or structure to be analyzed in FEM. In this tutorial, the necessary part has already been created in SolidWorks. The following steps will open up the part for use in the FEM analysis. Step 1: Opening the part for simulation. One of the following two options can be used. o Option1: Double click the following icon to open the embedded part file, Solid_Column.SLDPRT, in SolidWorks. Click SolidWorks part file icon to open it ==> Solid_Column.SLDPR T o Option 2: Download the part file Solid_Column.SLDPRT from the web site http://www.femlearning.org/. Use the File menu in SolidWorks to open the downloaded part. The SolidWorks model tree will appear with the given part name at the top. Above the model tree, there should be various tabs labeled Features, Sketch, etc. If the Simulation tab is not visible, refer back to steps 1 and 2 in order to enable the SolidWorks Simulation package. Step 2: Creating a Study o Click the Simulation tab above the model tree o Under the drop-down menu select New Study o In the box under Name type in Solid Column Buckling Study o Select Buckling underneath Type o Click to create the study 62

1.2 Material Property Assignment The next step in FEM analysis is to apply the material properties to the column. The material is given in the problem as a Ti-8Al-1Mo-1V titanium alloy and the SolidWorks libraries can be used to apply the material properties. Step 3: Applying the material o Select in the upper left hand corner of the Simulation ribbon o In left-hand section, expand the SolidWorks Materials folder o Expand the Titanium Alloys section and choose Titanium Te-8Al-1Mo-1V o Make sure the Linear Elastic Isotropic option is selected under Model Type and units are in SI o Click OK to apply the material 1.3 Boundary Condition Specification Due to the nature of column applications, this study will be treated as a column buckling study with fixed-free end conditions. Because the top end of the column is free, only one fixture will be applied to the bottom face of the column for this study. Step 4: Applying the fixed end condition to the bottom face o Click on the icon in the upper left corner of the simulation ribbon to drop down the fixture menu o Select Fixed Geometry o With the colored box highlighted, select the lower face of the column as seen in Figure 1 o Click to create the fixed boundary condition 63

Figure 1: Applying a fixed boundary condition The next step is to load the column with the designated pressure load. Due to the nature of columns, the pressure load will be applied to the top face of the column. Step 5: Applying the pressure load o Click on the icon in the upper left corner of the simulation ribbon to drop down the external load menu o Select Pressure o With the colored box highlighted, select the upper face of the column as seen in Figure 2 o Ensure that units of N/mm^2 (MPa) are selected and enter 200,000 for the magnitude o If necessary, check the Reverse direction box such that the arrows are pointing down on the top face o Click to create the pressure load 64

Figure 2: Applying the pressure load 1.4 Mesh Generation Purpose: The purpose of the Mesh Generation sub-step is to discretize the part into elements. The mesh consists of a network of these elements. Because a fine mesh is not needed in this example, the default element sizes will be used to decrease the required solver running time. Step 6: Meshing the model o Right click on the o Select Create Mesh o Use the default mesh setting icon in the model tree o Click to create the mesh 2. Solution Purpose: The Solution is the step where the computer solves the simulation problem and generates results for use in the Post-Processing step. Step 1: Running the simulation o Within the simulation ribbon, click o When the analysis is finished, the icon will appear on the model tree 65

3. Post-Processing Purpose: The purpose of the Post-Processing step is to process the results of interest. For this problem, the buckling load factor (BLF) will need to be acquired in order to calculate the critical pressure load of the column and observe the nominal stress and stress/weight ratio. This BLF value can also help describe the presence of buckling in the column. The following table is SolidWorks interpretation of possible BLF values. BLF Value Buckling Status Notes 1 < BLF Buckling not predicted The applied loads are less than the estimated critical loads. Buckling is not expected. 0 < BLF < 1 Buckling predicted The applied loads exceed the estimated critical loads. Buckling is expected. BLF = 1 Buckling predicted The applied loads are exactly equal to the estimated critical loads. Buckling is expected. BLF = -1 Buckling not predicted The buckling occurs when the directions of the applied loads are all reversed. For example, if a bar is under tensile load, the BLF should be negative. The bar will never buckle. -1 < BLF < 0 Buckling not predicted Buckling is predicted if you reverse all loads. BLF < -1 Buckling not predicted Buckling is not expected even if you reverse all loads. SolidWorks makes it very easy to acquire the BLF of any loaded part. Follow the next step in order to observe this data. Step 1: Displaying the Buckling Load Factor o Right click on the icon in the model tree o Select List Buckling Load Factors o Observe the BLF value o Select Save to save the BLF value Now that the buckling load factor has been determined, it can be used to calculate the critical pressure load of the column. In order to do this, simply multiply the BLF by the applied load. This will give the critical pressure load of the column. Therefore: Pcr = Critical Pressure Load = BLF x Applied Load = 7428.4(200 GPa) = 1,485.68 TPa 4. Validation Purpose: The purpose of the Validation step is to compare FEM solutions with analytical solutions, or known published results, to validate the correctness of the FEM model. However, due to the complex nature of honeycomb buckling calculations, the results will only be compared to those of the solid column. This section will simply address the nominal stress and stress/weight ratio of the honeycomb structure. 66

Again, the nominal stress of a column is defined as the critical pressure load multiplied by the ratio of the net cross-sectional area to the total cross-sectional area. However, for a solid block the net area is equal to the total area. Thus, the nominal stress is equal to the value of the critical pressure load. The next goal of the problem is to determine the column s stress/weight ratio. However, because this structure is so small in size, SolidWorks will display a reading of 0 for the mass. Therefore, it is acceptable to use the structure s volume in the calculations as the volume of a structure is directly proportional to its weight. It is very easy to acquire these values from SolidWorks using the following method. Step 1: Displaying the part s mass properties o Ensure that no pieces of the column are selected/highlighted o On the main menu, go to Tools -> Mass Properties o With the mass properties window open, locate and record the volume value of 128.02mm 3 o Close the window Now that structure s volume is known, it is possible to calculate the critical load/volume ratio. Remember that the critical load is simply the critical pressure load multiplied by the area acted upon by the pressure load. 67

Results Summary P cr σ nom Honeycomb Column 135.42 4.23 1.06 Solid Column 1,485.68 1,485.68 11.61 As can be seen from the data, the solid column has a much higher critical pressure load. However, this larger load results in a higher amount of stress in the column. This also leads to a much higher stress to volume ratio for the solid column as opposed to the honeycomb column. In other words, there is less stress per unit volume in the honeycomb column as opposed to the solid column. This data helps explain why honeycomb structures are becoming more popular in today s manufacturing world. 68

Attachment D. CometSolution-Specific FEM Tutorials Overview: In this section, the tutorial problem will be solved using commercial FEM 69

Attachment E. Post-Test 1. A column will remain stable as long as the applied load does not exceed the: O Maximum load O Critical load O Weight of the column O Minimum load 2. The engineering principle commonly used for column analysis is: O Bernoulli s Equation O Fourier s Law O Newton s Law O Euler s Formula 3. The slenderness ratio of a column is defined as the column s: O Width divided by the length O Length divided by the radius of gyration O Area divided by the volume O Weight divided by the volume 4. The constant C used in column buckling calculations is known as the: O Load constant O Stress constant O End condition constant O Material constant 5. The buckling of a column will only occur under compression loads. O True O False 70

6. The critical buckling load of a column depends on the: O Elastic modulus of the material O Slimness of the column O End restraint conditions O All of the above 7. In which of the following column categories is buckling in control rather than yielding? O Long columns O Short columns O Intermediate columns O None of the above 8. What is a restraint? O Restriction for motion O Force acting on the body O Stress in the body O A type of mesh 9. In which of the following column categories is buckling in control rather than yielding? O Long columns O Short columns O Intermediate columns O None of the above 10. The modification factor used when a column contains holes is defined as the column s: O Volume with holes O Volume with holes divided by the volume without holes O Volume without holes O Volume without holes divided by the volume with holes 11. The BLF of a column is defined as it s: O Buckling load factor O Basic load factor O Big load factor O Buckling length factor 71

12. The critical load of a column can be calculated by multiplying the BLF by the: O Anticipated load O Applied load O Theoretical load O Desired load 13. What is a method for simulating a pinned connection on the bottom face of a column? O Apply a fixed geometry to the face O Apply a fixed geometry to a point on the face O Apply a fixed geometry to a split line on the face O None of the above 14. What is the best method for simulating a fixed connection on the top face of a column? O Apply a fixed geometry to the face O Apply a reference geometry to the face O Apply a reference geometry to a portion of the side faces O Apply a fixed geometry to the side faces 15. What is a mesh? O A type of restraint O A type of material O A material property O A grid of finite elements 16. What are the advantages and disadvantages of using a fine mesh as opposed to a coarse mesh? 72

Attachment F. Practice Problems 1. A 3.5m long column made of AISI 304 steel has a circular cross-section with a diameter of 100mm. This column is used to support a 10 MPa pressure load and is exposed to pinnedpinned end conditions. Use FEM analysis to find the buckling load factor (BLF) and critical pressure load (P c ) of the column and compare these results to those of the square cross-section in Tutorial #1. Problem #1.SLDPRT Solution to Problem 1 2. A 3.5m long column made of AISI 304 steel has a circular cross-section with a diameter of 100mm. This column is used to support a 10 MPa pressure load and is exposed to fixed-fixed end conditions. Use FEM analysis to find the buckling load factor (BLF) and critical pressure load (P c ) of the column and compare these results to those of the square cross-section in Tutorial #1. Also calculate the column s critical load to weight ratio for comparison with Problem 3. Problem #2.SLDPRT Solution to Problem 2 73

3. The column from Problem 2 is now hollowed out and exhibits new cross-sectional dimensions as shown below. This column is again used to support a 10 MPa pressure load and is exposed to fixed-fixed end conditions. Use FEM analysis to find the buckling load factor (BLF), the critical pressure load (P c ) and the column s critical load to weight ratio for comparison with Problem 2. Problem #3.SLDPRT Solution to Problem 3 4. A 4.0m long column made of AISI 1020 steel has cross-sectional dimensions as shown below. This column is used to support a 10 MPa pressure load and is exposed to fixed-free end conditions. Use FEM analysis to find the buckling load factor (BLF) and critical pressure load (P c ) of the column. Problem #4.SLDPRT Solution to Problem 4 74

5. A 2.2m long column made of 1060 aluminum alloy has cross-sectional dimensions as shown below. This column is used to support a 1 MPa pressure load and is exposed to fixed-free end conditions. Use FEM analysis to find the buckling load factor (BLF) and critical pressure load (P c ) of the column. Problem #5.SLDPRT Solution to Problem 5 6. A 3.72m long column made of AISI 304 steel has a square cross-section with dimensions of 100mm x 100mm. The column contains fifteen 60mm diameter holes as seen in the figure below. This column is used to support a 10 MPa pressure load and is exposed to pinnedpinned end conditions. Use FEM analysis to find the buckling load factor (BLF) and critical pressure load (P c ) of the column and compare the results to that of the pinned-pinned column in Tutorial #1. Problem #6.SLDPRT Solution to Problem 6 75

Attachment G. Solutions to Practice Problems Column End Conditions Theoretical Value Conservative Value Recommended Value Fixed - Free ¼ ¼ ¼ Fixed - Fixed 4 1 1.2 Fixed - Pinned 2 1 1.2 Pinned - Pinned 1 1 1 Problem 1: Pcr = BLF x Applied Load = 9.5891(10MPa) = 95.89 MPa SolidWorks Hand Calculations Percent Difference P cr 95.89 97.19-1.34% Problem 2: Pcr = BLF x Applied Load = 37.918(10MPa) = 379.18 MPa SolidWorks Hand Calculations Percent Difference P cr 379.18 400.09-5.23% Fcr = Pcr x Ac = 379.18MPa(0.0079m 2 ) = 2.99 MN Load/Weight = 2.99 MN/27,488.94 g = 108.77 N/g 76

Problem 3: Pcr = BLF x Applied Load = 64.012(10MPa) = 640.12 MPa SolidWorks Hand Calculations Percent Difference P cr 640.12 656.15-2.44% Fcr = Pcr x Ac = 640.12MPa(0.0028m 2 ) = 1.80 MN Load/Weight = 1.80 MN/9,896.02 g = 181.89 N/g Problem 4: Pcr = BLF x Applied Load = 1.817(10MPa) = 18.17 MPa SolidWorks Hand Calculations Percent Difference P cr 18.17 18.15 0.11% 77

Problem 5: Pcr = BLF x Applied Load = 1.2311(1MPa) = 1.231 MPa SolidWorks Hand Calculations Percent Difference P cr 1.231 1.228 0.24% Problem 6: Pcr = BLF x Applied Load = 10.602(10MPa) = 106.02 MPa Pcr cr = 0.908(112.88MPa) = 102.49 MPa SolidWorks Hand Calculations Percent Difference P cr 106.02 102.49 3.44% 78

Attachment H. Assessment 1. Do you feel it was bad to not have a teacher there to answer any questions you might have? O It didn t matter O It would have been nice O I really wanted to ask a question 2. How did the interactivity of the program affect your learning? O Improved it a lot O Improved it some O No difference O Hurt it some O Hurt it a lot 3. The six levels of Bloom s Taxonomy are listed below. Rank how well this learning module covers each level. 5 meaning exceptionally well and 1 meaning very poor. i. Knowledge (remembering previously learned material) O 5 O 4 O 3 O 2 O 1 ii. iii. Comprehension (the ability to grasp the meaning of the material and give examples) O 5 O 4 O 3 O 2 O 1 Application (the ability to use the material in new situations) O 5 O 4 O 3 O 2 O 1 79

iv. Analysis (the ability to break down material into its component parts so that its organizational structure may be understood) O 5 O 4 O 3 O 2 O 1 v. Synthesis (the ability to put parts together to form a new whole) O 5 O 4 O 3 O 2 O 1 vi. Evaluation (the ability to judge the value of the material for a given purpose) O 5 O 4 O 3 O 2 O 1 4. Do you think the mixed text and video format works well? O Yes O Indifferent O No 5. Do you think the module presents an effective method of learning FEA? O Yes O Indifferent O No 6. Did you prefer this module over the traditional classroom learning experience? Why or why not. 80

7. How accurate would it be to call this module self-contained and stand-alone? O Very accurate O Accurate O Indifferent O Inaccurate O Very inaccurate 8. What specifically did you like and/or dislike about the module. 9. How useful were the practice problems? O Very helpful O Helpful O Indifferent O Unhelpful O Very unhelpful 10. Was there any part of the module that you felt was unnecessary of redundant? Was there a need for any additional parts? 11. Please list any suggestions for improving this module. 12. Overall, how would you rate your experience taking this module? O Excellent O Fair O Average O Poor O Awful 81