Printed Edition - Radioss For Linear 10 0

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A Platform for Innovation TM RADIOSS for Linear Static, Modal, Buckling and Inertia Relief Analysis HyperWorks is a division of Altair Engineering Contact Information Web site www.altair.com FTP site Address: ftp.altair.com or ftp2.altair.com or http://ftp.altair.com/ftp Login: ftp Password: Location Telephone e-mail North America 248.614.2425 [email protected] China 86.21.6117.1666 [email protected] France 33.1.4133.0992 [email protected] Germany 49.7031.6208.22 [email protected] India 91.80.6629.4500 1800.425.0234 (toll free) [email protected] Italy 39.800.905.595 [email protected] Japan 81.3.5396.2881 [email protected] Korea 82.31.716.4321 [email protected] Scandinavia 46.46.286.2052 [email protected] United Kingdom 44.1926.468.600 [email protected] Brazil 55.11.3884.0414 [email protected] Australia 64.9.413.7981 [email protected] New Zealand 64.9.413.7981 [email protected] The following countries have distributors for Altair Engineering: Asia Pacific: Indonesia, Malaysia, Singapore, Taiwan, Thailand Europe: Czech Republic, Hungary, Poland, Romania, Spain, Turkey. ©2009 Altair Engineering, Inc. All rights reserved. No part of this publication may be reproduced, transmitted, transcribed, stored in a retrieval system, or translated to another language without the written permission of Altair Engineering, Inc. To obtain this permission, write to the attention Altair Engineering legal department at: 1820 E. Big Beaver, Troy, Michigan, USA, or call +1-248-614-2400. Trademark and Registered Trademark Acknowledgments ® ® © Listed below are Altair HyperWorks applications. Copyright Altair Engineering Inc., All Rights Reserved for: ® ® ® ® HyperMesh 1990-2009; HyperView 1999-2009; OptiStruct 1996-2009; RADIOSS 1986-2009; HyperCrash™ ® ® ® ® 2001-2009; HyperStudy 1999-2009; HyperGraph 1995-2009; MotionView 1993-2009; MotionSolve 2002® ® 2009; TextView™ 1996-2009; MediaView™ 1999-2009; HyperForm 1998-2009; HyperXtrude 1999-2009; ® HyperView Player 2001-2009; Process Manager™ 2003-2009; Data Manager™ 2005-2009; Assembler™ 2005-2009; FEModel™ 2004-2009; BatchMesher™ 2003-2009; Templex™ 1990-2009; Manufacturing Solutions™ 2005-2009; HyperDieDynamics™ 2007-2009; HyperMath™ 2007-2009; ScriptView™ 2007-2009. In addition to HyperWorks® trademarks noted above, GridWorks™, PBS™ Gridworks®, PBS™ Professional®, PBS™ and Portable Batch System® are trademarks of ALTAIR ENGINEERING INC., as is patent # 6,859,792. All are protected under U.S. and international laws and treaties. All other marks are the property of their respective owners. Table of Contents RADIOSS for Linear Analysis, Concept and Optimization Table of Contents.................................................................................................................... II Chapter 1: Introduction............................................................................................ 1 1 – HyperWorks Overview............................................................................................... 1 1.1 – HyperWorks Tool Descriptions ............................................................................... 2 1.2 – RADIOSS Integration with HyperWorks.................................................................. 4 2 – RADIOSS Overview .................................................................................................. 5 2.1 – RADIOSS process.................................................................................................. 6 3 – Guide lines ................................................................................................................ 7 Chapter 2: Linear Static Analysis ........................................................................... 9 1 – What is a static analysis? ........................................................................................ 9 2 – Model Definition Structure ....................................................................................... 12 2.1 – Input/output section .............................................................................................. 12 2.2 – Subcase information section................................................................................. 13 2.3 – Bulk data section .................................................................................................. 14 3 – Linear Static Analysis Setup.................................................................................... 22 Exercise 2.1: Stress and displacement analysis in a Simple supported beam. .............. 26 Exercise 2.2: Static analysis of a solid bracket. ............................................................. 40 Chapter 3: Modal Analysis .................................................................................... 49 1 – Definitions ............................................................................................................... 49 1.1 – Natural frequency (Eigenvalue) ............................................................................ 50 1.2 - Mode shape (Eigenvector) .................................................................................... 50 HyperWorks 10.0 II Proprietary Information of Altair Engineering, Inc. 2 - Why Modal Analysis Is Important ............................................................................. 51 3 – Eigenvalue Solution Methods .................................................................................. 52 4 – Modal Analysis using FEM ...................................................................................... 53 4.1 - FEA eigensystem .................................................................................................. 53 5 – How to Setup a Modal Analysis ............................................................................... 54 Exercise 3.1: Shell Clamped BEAM model.................................................................... 56 Exercise 3.2: Compressor Bracket Modal Analysis ....................................................... 60 Chapter 4: Linear Buckling Analysis .................................................................... 67 1 – Definitions ............................................................................................................... 67 1.1 – Linear buckling and offset elements ..................................................................... 68 2 – How to Setup a Linear Buckling Analysis ................................................................ 69 Exercise 4.1: Wing Linear Buckling Analysis. ................................................................ 71 Chapter 5: Inertia Relief Analysis ......................................................................... 75 1 – Definitions ............................................................................................................... 75 2 – How to Setup an Inertia Relief Analysis................................................................... 76 Exercise 5.1: Satellite Inertia Load Test. ....................................................................... 78 III HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 1: Introduction Chapter 1 Introduction 1- HyperWorks Overview HyperWorks®, A Platform for Innovation™, is an enterprise simulation solution for rapid design exploration and decision-making. As one of the most comprehensive CAE solutions in the industry, HyperWorks provides a tightly integrated suite of best-in-class tools for: o Modeling o Analysis o Optimization o Visualization o Reporting o Performance data management. Based on a revolutionary “pay-for-use” token-based business model, HyperWorks delivers increased value and flexibility over other software licensing models. Below we list the applications that are part of HyperWorks, for extra information about them go to www.altairhyperworks.com web page or go to HyperWorks online documentation. HyperWorks 10.0 RADIOSS for Linear 1 Proprietary Information of Altair Engineering, Inc. Chapter 1: Introduction 1.1 – HyperWorks Tool Descriptions Finite Element Meshing and Modeling HyperMesh Universal finite element pre- and post-processor HyperCrash Finite element pre-processor for automotive crash and safety analysis BatchMesher Geometry cleanup and auto-meshing in batch mode for given CAD files Multi-body Dynamics Modeling MotionView Multi-body dynamics pre- and post-processor Solvers RADIOSS Finite element solver for linear and non-linear problems MotionSolve Multi-body dynamics solver OptiStruct Design and optimization software using finite elements and multi-body dynamics Post-processing and Data Analysis HyperView High performance finite element and mechanical system post-processor, engineering plotter, and data analysis tool HyperGraph Engineering plotter and data analysis tool HyperGraph 3D Engineering 3-D plotter and data analysis tool HyperView Player Viewer for visualizing 3-D CAE results via the Internet or desktop Study and Optimization HyperStudy Integrated optimization, DOE, and robustness engine Data Management and Process Automation Altair Data Manager A solution that organizes, manages, and stores CAE and test data throughout the product design cycle RADIOSS for Linear 2 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 1: Introduction Process Manager Process automation tool for HyperWorks and third party software; Processes can be created with the help of Process Studio. Assembler A tool that enables CAE analysts to manage, organize, and control their CAE mesh data Manufacturing Environments Manufacturing Solutions A unified environment for manufacturing process simulation, analysis, and design optimization HyperForm A unique finite element based sheet metal forming simulation software solution HyperXtrude An hp-adaptive finite element program that enables engineers to analyze material flow and heat transfer problems in extrusion and rolling applications Molding Provides a highly efficient and customized environment for setting up models for injection molding simulation with Moldflow Forging Provides a highly efficient and customized environment for setting up models for complex three-dimensional forging simulation with DEFOM3D Friction Stir Welding Provides an efficient interface for setting up models and analyzing friction stir welding with the HyperXtrude Solver HyperWorks Results Mapper Process Manager-based tool that provides a framework to initialize a structural model with results from a forming simulation HyperWorks 10.0 RADIOSS for Linear 3 Proprietary Information of Altair Engineering, Inc. Chapter 1: Introduction 1.2 – RADIOSS Integration with HyperWorks RADIOSS is part of the HyperWorks toolkit, as described early this is a finite element solver designed to solve linear and non-linear simulations. It can be used to simulate structures, fluid, fluid-structure interaction, sheet metal stamping, and mechanical systems. Multi-body dynamics simulation is made possible through the integration with MotionSolve. The solvers consist of loosely integrated executables (see picture below). To the user the integration is seamless thru the run script provided. Based on the file naming convention the right executable or combination of executables is chosen. Solver Overview The pre-processing for RADIOSS is made using HyperMesh and the post-processing using HyperView and HyperGraph. During the next exercises the HyperWorks integration with RADIOSS will be showed in detail, and for more about it the user should go to our online documentation. RADIOSS for Linear 4 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 1: Introduction 2 – RADIOSS Overview Altair RADIOSS is a next-generation implicit and explicit finite-element solver for linear statics and dynamics as well as complex nonlinear transient dynamics and multi-body dynamics. This robust, multidisciplinary solution allows manufacturers to maximize durability, NVH, crash, safety, manufacturability and fluid-structure interaction performance in order to bring innovative products to market faster. RADIOSS’ comprehensive analysis capabilities for linear and non-linear finite element analysis, sheet metal stamping, and multi-body dynamics are accessible via two input formats. Finite element solutions via Nastran-type Bulk Data Format include: o Linear static analysis o Non-linear implicit quasi-static contact analysis o Linear buckling analysis o Normal modes analysis o Frequency response analysis o Random response analysis o Linear transient response analysis o Linear fluid-structure coupled (acoustic) analysis o Linear steady-state heat transfer analysis coupled with static analysis o Inertia relief analysis with static, non-linear contact, modal frequency response, and modal transient response analyses o Component Mode Synthesis (CMS) for the generation of flexible bodies for multi-body dynamics analysis o Reduced matrix generation o One-step (inverse) sheet metal stamping analysis A typical set of finite elements including shell, solid, bar, scalar, and rigid elements as well as loads and materials is available for modeling complex events. Finite element solutions via RADIOSS Block format include: o Explicit dynamic analysis o Non-linear implicit static analysis o Transient heat transfer and thermo-mechanical coupling o Explicit Arbitrary Euler-Lagrangian (ALE) formulation o Explicit Computational Fluid Dynamics (CFD) o Smooth Particle Hydrodynamics (SPH) HyperWorks 10.0 RADIOSS for Linear 5 Proprietary Information of Altair Engineering, Inc. Chapter 1: Introduction o Incremental sheet metal stamping analysis with mesh adaptivity o Linear static analysis o Normal modes analysis o Linear and non-linear buckling analysis A typical set of finite elements including shell, solid, bar, and spring elements, rigid bodies as well as loads, a number of materials, contact interfaces are available for modeling complex events. Multi-body dynamics solution integrated via Nastran-type Bulk Data format for rigid and flexible bodies includes: o Kinematics analysis o Dynamics analysis o Static and quasi-static analysis o Linearization All typical types of constraints like joints, gears, couplers, user defined constraints, and high-pair joints can be defined. High pair joints include point-to-curve, point-to-surface, curve-to-curve, curve-to-surface, and surface-to-surface constraints. They can connect rigid bodies, flexible bodies, or rigid and flexible bodies. For this multi-body dynamics solution, the power of Altair MotionSolve has been integrated with RADIOSS. 2.1 – RADIOSS process RADIOSS for Linear 6 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 1: Introduction 3 – Guide lines On section 2 we described RADIOSS as a very powerful solver that can solve almost all type of problems in solid mechanics. Then before starting a FEA model the analyst should ask the following questions: 1. What are the loads on this system or component? 2. Are the loads static or dynamics? 3. Which is the load path? 4. What are the design criteria? (Stress, displacement, Strain, Life, etc…) 5. Where do we expect the high stresses and what limit can I accept? 6. Is it a linear or non-linear problem? 7. How can I verify the FEA results? Depending on the answers to these questions the analyst will define the type of FEA solution that is necessary to capture the responses necessary to solve the problem. This is a basic training where the focus is on deformable bodies with LINEAR behavior. The diagram below show in bold the areas that will be discussed during this course: Solid mechanics diagram HyperWorks 10.0 RADIOSS for Linear 7 Proprietary Information of Altair Engineering, Inc. Chapter 1: Introduction RADIOSS for Linear 8 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Chapter 2 Linear Static Analysis 1 – What is a static analysis? In mechanics we can define static state as the state of a system that is in equilibrium under an action of balanced forces and torques so that they remain at rest (V=0). A static load is one which does not vary. If it changes slowly, the structure's response may be determined with static analysis, but if it varies quickly (relative to the structure's ability to respond), the response must be determined with a dynamic analysis. When solving a static problem, all finite element solvers will solve the following equation: Kx = f Where: o K : It is the global stiffness o x : It is the displacement vector response to be determined. o f : It is the external forces vector applied to the structure. matrix We will proposal now a small static example that will be used to better understand what a static analysis is. The example is showed on the following image: A1= 314.16 mm2 L1= 100 mm A2= 78.54 mm2 L2= 200 mm Material Steel: F = 10 KN E =210 GPa Simple rod example This is a column that is formed by two different regions with uniform section. The methodology that will be described here can be easily extended to any kind of problem. First we need to represent the structure as nodes and elements, for this case is clear that we need at least 2 elements, one for each distinct section. HyperWorks 10.0 RADIOSS for Linear 9 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis First we need a finite element model to solve the problem, on our case we will use rod elements as showed on the following image: The model is composed then by 3 nodes {1, 2, and 3} and 2 elements {1 and 2}, a material that is associated to 2 different properties A1 and A2. Every node has only one DOF (x) and the finite element matrix for this one-dimensional rod element can be written as:  AE  K= L AE −  L AE  L  AE   L  − Then we can evaluate the matrix for each element based on the input data:  314.16 * 210  100 K1 =  314.16 * 210 − 100   78.54 * 210  200 K2 =  78.54 * 210 − 200  314.16 * 210    659.74 − 659.74 100 = 314.16 * 210   − 659.74 659.74   100  − 78.54 * 210    82.47 − 82.47 200 = 78.54 * 210  − 82.47 82.47   200  − Now the next step is to assembly this element to form the global stiffness matrix: KG = 1 1 2 3 659.74 − 659.74 0 2 − 659.74 659.74 + 82.47 − 82.47 3 0 − 82.47 82.47 Now we need to write the force and the displacement vector:  0    f= 0  − 10     x1    x =  x2  x   3 Then we can finally write the global system: RADIOSS for Linear 10 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis 0   x1   0   659.74 − 659.74 − 659.74 742.21 − 82.47  x  =  0     2       0 − 82.47 82.47   x3  − 10  We have a prescribed null displacement for node 1, then we can eliminate the first line and the first column of our system:  742.21 − 82.47  x2   0  − 82.47 82.47   x  = − 10     3   To solve this problem we just need to invert the Global stiffness matrix and multiply both sides:  x 2  − 0.0152   =   x3  − 0.1364  With the displacement vector defined is possible to determine the element strain, stresses and forces: STRAIN ε1 = ε2 = ∆L x2 − x1 − 0.0152 − 0 = = = −1.52 * 10 −4 mm / mm L L1 100 ∆L x3 − x 2 − 0.1364 − 0.0152 = = = −6.06 * 10 −4 mm / mm L L2 200 STRESS σ 1 = Eε 1 = −1.52 * 10 −4 * 210 = −0.032GPa σ 2 = Eε 2 = −6.06 * 10 −4 * 210 = −0.127GPa FORCES f 1 = σ 1 A1 = −0.032 * 314.16 = −10 KN f 2 = σ 2 A2 = −0.127 * 78.54 = −10 KN This is a very simple example, but it is very efficient in summarize the finite element methodology, all these calculus demonstrated here are made automatically per the solver. If the user needs more detail about finite element method it can be found on the online documentation or at the referenced books. HyperWorks 10.0 RADIOSS for Linear 11 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis 2 – Model Definition Structure The input deck is formed per 3 different sections as shoed on the following image: 2.1 – Input/output section The I/O Section is the first part of a OptiStruct input file, it controls the overall running of the analysis or optimization. It controls for example the type, format, and frequency of the output, the type of run (analysis, check, or restart), and the location and names of input, output, and scratch files. This is not a required section, if the user doesn’t specify any I/O control this section will not be on the input deck, but OptiStruct has a default I/O setup that will generate these outputs: 1- ANALYSIS o ASCII output o .out This file is always created. It contains a report with comments on the solution process. o .stat This file is always created. This file provides details on CPU and elapsed time for each solver module. RADIOSS for Linear 12 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis o o o HTML Reports This file is always created. This file contains a problem summary and results summary of the run. o .html o _frames.html This file is output when the H3D FORMAT is chosen. The file contains two frames. The top frame opens one of the .h3d files using the HyperView Player browser plug-in. The bottom frame opens the _menu.html file, which facilitates the selection of results to be displayed. o _menu.html This file is output when the H3D FORMAT is chosen. This file facilitates the selection of the appropriate .h3d file, for the HyperView Player browser plug-in in the top frame of the _frames.html file, based on chosen results Model results o .res The .res file is a HyperMesh binary results file. o .h3d The .h3d file is a compressed binary file, containing both model and result data. HV session file o .mvw The .mvw file is a HyperView session file that is linked with the h3d result file and can be open directly from HyperMesh using the HyperView button on OptiStruct or RADIOSS panel. 2.2 – Subcase information section The Subcase or Case Control Section contains information for specific subcases. It identifies which loads and boundary conditions are to be used in a subcase. It can control output type and frequency, and may contain objective and constraint information for optimization problems. For more information on solution sequences, please see the table included on the Solution Sequences page of the help. HyperWorks 10.0 RADIOSS for Linear 13 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Descriptions for individual Subcase Control entries can be accessed on the online documentation. 2.3 – Bulk data section The Bulk Data Section contains all finite element data for the finite element model, such as grids, elements, properties, materials, loads and boundary conditions, and coordinates systems. For optimization, it contains the design variables, responses, and constraint definitions. The bulk data section begins with the BEGIN BULK statement. 3 – Linear Static Analysis Setup There are some basic steps that the user needs to follow to setup a static analysis for RADIOSS, we will describe these steps using the HyperMesh to setup the example described on section 1. Step 1 – Define the material. From the collectors tool bar the user can select the Material icon button, as showed below: Then the user should input the material name and select the material type and click on create/edit: The material card now needs to be filled with the Young Modulus [E] and Poisson [Nu] values and click return: *If there are only 1D elements the Poisson value can be ignored. After this step the user should see at the model browser the Material group with the instance steel. RADIOSS for Linear 14 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Step 2 – Define the properties and associate it with the appropriate material. From the collectors tool bar the user can select the Properties icon button, as showed below: Then the user should input the property name, pick the card image, pick a material and click on create/edit: This will open the PROD panel where the user should enter with the Area [A]: Repeat the process for the second property: After create these 2 properties the user should see a new group called Property with 2 instances called A1 and A2: HyperWorks 10.0 RADIOSS for Linear 15 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Step 3 – Define the components and associate it with their relative property. From the collectors tool bar the user can select the Components icon button, as showed below: Then the user should input the Component name, pick the respective property and click on create: Repeat the process for the second component: After create these 2 components the user should see 2 new groups called Component and Assembly Hierarchy with 2 instances called A1 and A2 on both: RADIOSS for Linear 16 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Step 4 – Create the Finite element mesh with the appropriated properties associated with the elements: This step is very model dependent, it can start with a CAD geometry that is imported or from other FEA Model, or either can be a combination of existent models and imported geometries. On our case this is a very simple model that we can start creating the nodes and the respective elements. To create the nodes the user can use the shortcut key F8 that will open the Create Nodes panel: The user should create a node at this coordinates: (0, 0, 0), (100, 0, 0) and (300, 0, 0) HyperWorks 10.0 RADIOSS for Linear 17 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis On our case it is necessary to create crod elements then the user should use the Element Types panel to setup the rod to crod as below: The Create rod panel can be accessed by the Mesh > Create > 1D Elements > Rods pull down menu: This will open the rod elements create panel where the user should first select the right property for each element and then pick the nodes to create it: RADIOSS for Linear 18 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Step 5 – Define the constraint load collector and apply the model constraint. With a right click on the model browse the user can chose Create > LoadCollector to access the load collector pop-up window to create the load collector. After create this load collector the user should see a new group called LoadCollector with an instance called SPC: HyperWorks 10.0 RADIOSS for Linear 19 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Now the user can use the BCs > Create > Constraints to fix the DOF that are not allowed to move: On RADIOSS the rod element is a spatial element and has 3 DOFs (Ux, Uy and Uz), to reproduce the simple configuration we had on the first section we need to remove all Uy and Uz DOFs and the Ux at node 1 as we did on the first section: Now there are only 2 DOFs in this model Ux2 and Ux3. Step 6 – Define the force load collector and apply the loads. The Create Load Collector panel can be accessed from the Collectors > Create > Load Collectors pull down menu: RADIOSS for Linear 20 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis On this panel the user should enter with the name and a card image if necessary and click on create: After create this load collector the user should see a new instance called Force on the LoadCollector group: Now the user should create the force on -X direction at the node3, it can be done from BCs > Create > Forces: HyperWorks 10.0 RADIOSS for Linear 21 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis That access the create forces panel where the user needs to select the node, magnitude and direction to create the force: Step 7 – Define the load step. Now the user should create the loadstep, it can be done from Setup > Create > LoadSteps: This will access the LoadSteps panel where the user should select the SPC and the load for a static load case and click on create to create the load step, This will add a new group to the model browser tree called Loadstep with an instance called Force: RADIOSS for Linear 22 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Step 8 – Define the extra parameters to your analysis. Optional On this step the user can select for example the results that would be necessary, some analysis control card that define the configuration that will be used by the solver. Here are some examples of these settings, for more examples please go to the online documentation: Control cards From Setup > Create > Control cards the control cards panel can be accessed: o Request displacement to be written on H3D Result file: o Request H3D Result file and suppress the html and status output: HyperWorks 10.0 RADIOSS for Linear 23 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis o Auto SPC off: Step 9 – Run the analysis. From Application > RADIOSS the launch panel can be accessed: Now the user should define a file name and submit the job: RADIOSS for Linear 24 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Step 11 – Post-process the results. After run is complete the easiest way to access the results is suing the HyperView green button that is on the right hand side bellow the RADIOSS button used to call the solver. On HyperView on the toolbar click on contour Apply this will generate the contour showed below: and select displacement and click The measure button can be used to compare this results with the one solved on the first section, just click on Add and change the measure type for Nodal contour and select the nodes 2 and 3: As we can see on the measures notes the values match with the results evaluated in our first section, there are many other post-processing functions that will be discussed later and for more specific details the user should use the online documentation. HyperWorks 10.0 RADIOSS for Linear 25 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Exercise 2.1: Stress and displacement analysis in a Simple supported beam. In this exercise, a structural analysis is performed on a simple supported beam. The structural model with loads and constraints applied are shown in the figure below. The objective is to create a finite element model that is good enough to predict the theoretical solution for this model. 12345 23 FEA model Model Information o Force = 1000 N (Applied in a segment equivalent to 2mm) o Beam properties: L = 1000, B = 10 and H = 20 mm o Material Steel: E =210000 MPa and Nu=0.3 o UNITS: N, mm, ton, s Theoretical Results: σ max = U max = − M max c = I F *L H 4 2 3 B* H 12 * = 3FL = 375MPa 2 BH 2 FL3 FL3 FL3 =− = 14.881mm 3 = − 48EI 4 EBH 3 48E BH 12 Problem Setup You should copy this File: Beam_shell_geometry.hm; RADIOSS for Linear 26 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Step 1: Launch HyperMesh and Set the User Profile 1. Launch HyperMesh through the start menu. The User Profiles dialog will appear by default. 2. Choose RADIOSS as the user profile by selecting the radio button beside it. 3. Chose BulkData as format and Click OK. Step 2: Open the HyperMesh data base model This HM database only contains geometry information. 1. From the pull down menu chose File > Open…. An Open File popup window appears to select the HyperMesh database. 2. Browse on the training directory for a file named Beam_shell_geometry.hm and click Open. Step 3: Define the Material 1. Right click on the Model Browse tab and chose Create > Material. 2. On the popup window enter Name: Steel and MAT1 for Card image. HyperWorks 10.0 RADIOSS for Linear 27 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis We will use here MAT1 that is a Linear isotropic material that can represent well the steel behavior, for more details about this material or other material formulations please go to the online documentation. 3. Click Create/Edit and fill the values as showed on the following image. IMPORTANT: CONSISTENT UNITS!!! 4. Click return to exit the panel. Step 4: Create Model Properties 1. From the pull down menu click on Properties > Create. 2. Enter as prop name = Beam. 3. Change the color to match with the component. 4. Change type = to 2D. 5. Click on card image = and pick pshell. 6. Click on material = and pick Steel. 8. Click create/edit. 9. Fill out the card as showed on the image: The thickness represents the base valued of our Beam section. 14. Click return twice to exit the panel. RADIOSS for Linear 28 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Step 5: Assign the property to the component 1. From the pull down menu click on Collectors > Assign > Component Property. 2. Click on comps and select the beam component. 3. Click on Property = and pick the property Beam. 4. Click assign. This will make that all elements from this component to use this property. If an element from this component has another property associated with itself directly this prop will be preserved, i.e. HM will ignore the component property for this element. Step 6: Create the finite element mesh 1. From the pull down menu click on Mesh > create > 2D Automesh. 2. Click on surfs and chose all to select all surfaces. 3. Click mesh. The element size is 10mm 4. Click return twice to accept the mesh and exit the panel. Step 7: Apply the constraint to your model 1. Right click on the Model browse tab and click on Create > LoadCollector. 2. On the Create LoadCollector popup window enter Name: SPC and change the color to green and click Create. HyperWorks 10.0 RADIOSS for Linear 29 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis 3. From the pull down menu click on BCs > Create > Constraints 4. Change the entity selection from nodes to points 5. Select the lower left-hand side point and fix 1, 2, 3, 4 and 5 DOFs. 12345 6. Select now the lower right-hand side point and fix 2 and 3 DOFs. 23 12345 It is always important to setup the right BCs, the user should never over constrain the model because it in general drives to wrong results. Be careful before add any constraint to the model. Step 8: Apply the forces to your model 1. Right click on LoadCollector group at Model browse and click on Create. 2. On the Create LoadCollector popup window enter Name: Force and change the color to red and click Create. 3. From the pull down menu click on BCs > Create > Forces 4. Change the entity selection from nodes to points 5. Select the middle point at the upper side of the beam. 6. Enter for magnitude = 1000 and change the direction selector to y-axis. 7. Click create. RADIOSS for Linear 30 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis 8. Click return to exit the panel. On our case the force should be applied uniformly in a 2 mm segment in the middle of the beam, as we have only one node on this region the total load is applied to it, but it is very important because it can generate some singularities that can lead to a very high stress that are not Physical, and appear only in the mathematic model. Step 9: Define the load step 1. From the pull down menu click on Setup > Create > LoadSteps. 2. On the LoadStep panel enter Name: Force and change the type: to linear static select the SPC load collector as SPC and the Force as LOAD and click Create. The static analysis is already setup and ready to be solved! Step 10: Define the Analysis parameters (Optional) 1. From the pull down menu click on Setup > Create > Control Cards. 2. Look for the DISPLACEMENT card and fill out as showed below: With this card the displacement result will be wrote at the H3D result file. 3. Click return to exit the DISPLACEMENT panel. 4. Repeat the item 2 and 3 for STRESS and fill out the card as showed below: HyperWorks 10.0 RADIOSS for Linear 31 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis 5. Look for the card SCREEN and fill out as showed below: This will make OS show at the screen what it is writing on the out file. 6. Look for the card PARAM panel and setup AUTOSPC NO as showed below: RADIOSS as default uses AUTOSPC, ON it helps to prevent undesired stops or failure runs. For example if the model has an element unattached to the structure with no constraint applied to it the run would stop complaining about a rigid body movement, with AUTOSPC ON, RADIOSS would automatically fix this element and run the analysis. But the user should be aware of any DOF fixed by the AUTOSPC, as we discussed before it can lead to a wrong behavior. Then don’t forget in the end if he run is made with “AUTOSPC ON” to verify which DOF was fixed and if this will not change the solution you are looking for. Step 11: Run the analysis 1. From Application > RADIOSS the launch panel can be accessed: 2. Click on Radioss to start the solution. 3. Wait until the message Process completed successfully appears on the prompt window. RADIOSS for Linear 32 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis This message means that the process had ran without error and the result files are available for post-processing. Step 12: Post-process the first results 1. From RADIOSS launch panel click on HyperView to launch the application. 2. Close the message window. 3. Change the animation mode from transient to linear static . 4. Click on contour toolbar button and select as result Displacement click on Edit Legend… and change the properties as showed below: 5. Click on scale toolbar button edges and click Apply. set the Value: to 10 change the undeformed shape: to 6. Click on Top button at the lower right-hand side of the window at the permanent menu. 7. Click on Page Layout toolbar button and select the 3 window layout . 8. Click on lick on Note toolbar button and change the actual text on the Description: to BEAM MODEL and click Apply from the pull down menu click Edit > Copy Window and click on the second window and click Edit > Past Window repeat it for the third window. HyperWorks 10.0 RADIOSS for Linear 33 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis In the end the page should look as below: Total displacement 9. Click the contour toolbar button change the contour type to Von Mises Stress and using the ctrl + middle mouse button apply a zoom to the maximum stress on the window 3 as showed bellow: Total displacement (mm) and Von Mises (MPa) [ELEMENT SIZE 10 mm] As we can see the displacement results is very good with an error ~0.5% but the stress results are not good with and error superior to 50%. Here if the user plots the XX stress on the global system it will be easy to understand why the model can’t represent the right solution, the first element on the top is in compression and the bottom element is tension that means that there is a BIG STEP between it that is not captured for this course mesh. To improve it the user will need to refine the mesh. RADIOSS for Linear 34 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Let’s automate the post-processing of this model saving this page as a report template to reproduce these contours for different models. 10. From pull down menu click on File > Save Session File as: 11. On the Save Session File Window type Beam_Report.tpl and change the type to Report definition (*.tpl) and click save. Step 13: Refinement study (Optional Elem = 5 mm) The next steps are used to determine a good mesh to solve this problem and it can be let aside if the user has a good background in FEA analysis. 1. Now coming back to HyperMesh the user should click return to close the RADIOSS launch panel. 2. To refine the mesh the user should use the automesh panel with uniform size 5 mm. (refer to Step 6 for more detail) Refined mesh (5mm) 3. Save this model as Beam_5mm.hm. 4. Rerun the model using the RADIOSS panel. (Refer to Step 11 for reference) HyperWorks 10.0 RADIOSS for Linear 35 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Step 14: Post-processing the refined model (Optional 5 mm) 1. Going back to HyperView click on Reports toolbar button and change the GRAPHIC_FILE_1 and the RESULT_FILE_1 to Beam_5mm.h3d and click apply. As we can see on the image below the stress results looks much better, now the error is ~ 26% that is a lot better on what we had for 10mm. Total displacement (mm) and Von Mises (MPa) [ELEMENT SIZE 5 mm] Now with 4 elements on the height it is possible to represent better the bending behavior. If the user plot the XX stress again it will be clear that there is some step yet but the transition now is a lot better. Step 15: Refinement study (Optional Elem = 2.5 mm) 1. Now coming back to HyperMesh the user should click return to close the RADIOSS launch panel. 2. To refine the mesh the user should use the automesh panel with uniform size 2.5 mm. (refer to Step 6 for more detail) Refined mesh (2.5mm) RADIOSS for Linear 36 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis 3. Save this model as Beam_2.5mm.hm. 4. Rerun the model using the RADIOSS panel. (Refer to Step 11 for reference) Step 16: Post-processing the refined model (Optional 2.5 mm) 1. Going back to HyperView click on Reports toolbar button and change the GRAPHIC_FILE_1 and the RESULT_FILE_1 to Beam_2.5mm.h3d and click apply. As we can see on the image below the stress results looks better again, now the error is ~ 13% that is a lot better on what we had for 5mm. Total displacement (mm) and Von Mises (MPa) [ELEMENT SIZE 2.5 mm] Now with 8 elements on the height it is possible to represent better the bending behavior. If the user plot the XX stress again it will be clear that there is some step yet but the transition now is a lot better. Global normal stress on X direction. HyperWorks 10.0 RADIOSS for Linear 37 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Step 17: NON-uniform refinement study (Optional Elem = 1 mm) It is easy to notice that if we refine the whole model the results will get better, but refine the whole model is very inefficient procedure, mainly because the solution time is a DOF exponential function that can easily arrive in unfeasible time solutions. Looking on the models we had simulated it will be easy to notice that there is no important change in stress or displacement at the end of the beam, then we can conclude that the model with 5mm it was good for this 2 regions, but looking o the center of the beam we can easily see that the last model is much better. To solve this problem the best approach is to refine the mesh only where it is necessary. 1. Now coming back to HyperMesh the user should click return to close the RADIOSS launch panel. 2. To refine the mesh where it is necessary the user should look at the stress results and define regions based on how much the stress gradient, to divide the component the user should use from the pull down menu Geometry > Edit > Surface Surface divided in 14 segments 1:5 linear. This is just a suggestion size the number of segments and progression is dependent of the problem, but a good reference is that the mesh transition should not exceed 25% in size. Other important point here is that we want to have element with 0.5 mm at the force region, this means now that region where the force will be applied will have more then one node, then it should be distributed among them to don’t create a mathematical singularity. 3. To don’t loose time on creating this model this case is already prepared, just open the file BEAM_REF.hm. 4. Run the model using the RADIOSS panel. (Refer to Step 11 for reference) Step 16: Post-processing the final model (Optional NON-UNIFORM size) 1. Going back to HyperView click on Reports toolbar button and change the GRAPHIC_FILE_1 and the RESULT_FILE_1 to BEAM_REF.h3d and click apply. As we can see on the image below the stress results looks better again, now the error is ~ 1% that is a lot better on what we had for 2.5mm, and the most important the number of DOFs is 55939 that compared with 21647 for 2.5mm is a very good improvement without increase a lot the model. RADIOSS for Linear 38 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Total displacement (mm) and Von Mises (MPa) [NON-UNIFORM ELEMENT SIZE] Now with 50 elements on the height it is easy to see that the bending behavior is well represented. Normal XX stress. Normal XX stress variation on Y axis As we can see the on the XY plot above the stress distribution evaluated by our model is in accordance with the analytical solution that we trying to reproduce. That is good, we now know how to improve the model to match a know solution, but this is not a case for real world models where the analytical solution doesn’t exist and where the finite element method can really show its advantage. The next exercise will cover this application. HyperWorks 10.0 RADIOSS for Linear 39 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Exercise 2.2: Static analysis of a solid bracket. In this exercise, a structural analysis is performed on a bracket modeled with solid elements. The structural model with loads and constraints applied are shown in the figure below. The objective is to create a finite element model that is good enough to predict an accurate solution for this problem with a reasonable model size. 123456 13456 FEA model Model Information o Force = (12000,12000, -20000) N o Material Aluminium: o • E =70000 MPa • Nu = 0.33 • S0 = 240 Mpa • SADM = 0.7*S0 UNITS: N, mm, ton, s Problem Setup You should copy these files: nafems1.hm; RADIOSS for Linear 40 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Step 1: Launch HyperMesh with RADIOSS bulk profile and open the nafems1.hm model HyperMesh with the bracket model loaded Step 2: Create the Aluminum material with the properties showed on the image below. Material data HyperWorks 10.0 RADIOSS for Linear 41 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Step 3: Create the solid property and assign it to the solid component (BRACKET) Step 4: Apply the constraint to the model. (SPC) Step 5: Apply the Force (12000, 12000, -20000). (Force) RADIOSS for Linear 42 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Step 6: Create the load step. (Linear static => Force) Step 7: Define the control cards: FORMAT H3D DISPLACEMENT (H3D) = ALL PARAM, AUTOSPC, NO STRESS (H3D, ALL, CENTER) = YES STRAIN (H3D, ALL) = ALL TITLE = NAFEMS BRACKET OUTPUT, H3D, ALL OUTPUT, HTML, , NO OUTPUT, STAT, , NO SCREEN OUT SPCFORCE (H3D, ALL) = ALL HyperWorks 10.0 RADIOSS for Linear 43 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Step 8: Run the analysis and Post-process the results. ? Von Mises Stress (Elem. Size = 10 mm) Total displacement (Elem. Size = 10 mm) o It is easy to notice that the stress results are not good. (Discontinuities) o The next step is to rerun this model with a refined mesh Model 1 Element Size (mm) 10 Von Mises (MPa) 60.2 Displacement (mm) 1.06 RADIOSS for Linear 44 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Step 9: Repeat the whole process for the model with 6.5 mm. 6.5 mm Model with the refinement regions Von Mises Stress and total displacement (Elem. Size = 6.5 mm) Model 1 2 Element Size (mm) 10 6.5 Von Mises (MPa) 60.2 63.3 Displacement (mm) 1.06 1.08 HyperWorks 10.0 RADIOSS for Linear 45 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Step 10: To save time the next models are already solved, the user should only use the report template to confirm the values. Von Mises Stress and total displacement (Elem. Size = 5 mm) Von Mises Stress and total displacement (Element Size = 3 mm) RADIOSS for Linear 46 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Von Mises Stress and total displacement (Element Size = 2 mm) Von Mises Stress and total displacement (Element Size = 0.55 mm) HyperWorks 10.0 RADIOSS for Linear 47 Proprietary Information of Altair Engineering, Inc. Chapter 2: Linear Static Analysis Von Mises Stress and total displacement (Element Size = 0.17 mm) Convergence table Model 1 2 3 4 5 6 7 Element Size (mm) 10 6.5 5 3 2 0.55 0.17 Von Mises (MPa) 60.2 63.3 69.5 73.0 80.0 84.4 89.3 Displacement (mm) 1.06 1.08 1.09 1.10 1.10 1.09 1.09 Von Mises Stress convergence RADIOSS for Linear 48 HyperWorks 10.0 Proprietary Information of Altair Engineering, Inc. Chapter 3: Modal Analysis Chapter 3 MODAL ANALYSIS 1 – Definitions A modal analysis calculates the frequency modes or natural frequencies of a given system, but not necessarily its full time history response to a given input. The natural frequency of a system is dependent only on the stiffness of the structure, and the mass which participates with the structure (including self-weight) and the boundary conditions. Consider the motion equation, where the damp and external forces are null, this leave the equation on the reduced form kwon as Free vibration equation: M&x& + Kx = 0 (1) The solution for this equation can be evaluated if we proposal a general harmonic solution with the form: x = Φ sin (ωt ) (2) Where: Φ → Mode shape or Eigenvector ω → Circular natural frequency The harmonic hypothesis helps on find the equation solution, but it has a physical importance that we will discuss further, this solution shows that all DOFs of the structure when submitted to a free vibration will move synchrony with each other. If we substitute the equation 2 into 1 performing the differentiation on the first term: − ω 2 MΦ sin (ωt ) + KΦ sin (ωt ) = 0 (K − ω 2M )Φ = 0 (3) This is the equilibrium equation for a structure performing free vibration, which can be rewrite in terms of the eigenvalues λ =ω2: HyperWorks 10.0 RADIOSS for Linear Analysis 49 Proprietary Information of Altair Engineering, Inc. Chapter 3: Modal Analysis [K − λM] Φ = 0 Where: • K is the stiffness matrix of the structure • M is the mass matrix. • The solution of the eigenvalue problem yields n eigenvalues λ, where n is the number of degrees of freedom. • The vector Φ is the eigenvector corresponding to the eigenvalue λ. The eigenvalue problem on RADIOSS is solved using a matrix method called the Lanczos Method. This method is very efficient when not all eigenvalues are required that is the case for structural problems where only a small number of the lowest eigenvalues are normally important. * It requires that the mass matrix be positive semidefinite and the stiffness be symmetric. 1.1 - Natural frequency (Eigenvalue) The natural frequency of a structure is the frequency value at the structure naturally tends to vibrate if it is subjected to pulse. For example, the strings of a guitar are made to vibrate at a specific frequency. A system with N DOFs will have N natural frequencies. The natural frequencies can be evaluated from solution of the Eigenvalues as showed below: fi = ωi 2π (4) Where: f i → i-th natural frequency ωi → i-th circular frequency 1.2 - Mode shape (Eigenvector) It is the deformed shape that the structure will vibrate when excited at a specific natural frequency, this is called too normal mode or Eigen vector. Each mode shape is associated with a specific natural frequency or Eigen value. The natural frequencies and normal modes of a structure can be function of the load and the damping present on the system, this kind of analysis is defined as Pre-stressed and damped modal analysis respectively, these analyses will not be covered here. RADIOSS for Linear Analysis 50 Proprietary Information of Altair Engineering, Inc. HyperWorks 10.0 Chapter 3: Modal Analysis 2 - Why Modal Analysis Is Important On any kind of structural simulation a modal analysis will help the Engineer to understand the global behavior of the system, doing a modal analysis first it is possible to: • Identify the natural frequencies and modal shapes of the system. • Verify if there are rigid modes on the system, and the link between components. • Understand if the BCs applied to the system are correct. • With the strain energy density for example, the Engineer can determine where the part should be reworked to improve the performance. • It helps on predict the dynamic responses that this system will have, then all the other dynamic simulations should be done only after a MODAL Analysis. It is useful to know the modal frequencies of a structure as it allows you to ensure that the frequency of any applied periodic loading will not coincide with a modal frequency and hence cause resonance, which could leads to large responses and consequently fails. To define the subsequent dynamic analyses (i.e., transient, frequency response, PSD, etc.) should be based on Modal results. With a previous knowledge about the important modes the analyst can chose the appropriate time or frequency step to solve the problem. If the analyst needs to work with a big model then the modal analysis results can be used to solve the FRF or Transient simulation, this is called a modal FRF or Modal Transient, where the equations are solved using a method called Modal superposition, this make the dynamic solution much less expansive then the direct integration. A modal analysis plays a key role when the analyst needs to compare the dynamic analyses with physical test, it helps to define the right equipment that have to be used and the right location for accelerometers and strain gages. It helps during the test too to understand the test results and correlate the virtual model with the prototype. It is possible sometimes only with a modal analysis find out if a design change will improve the dynamic performance of the system. In summary the modal analysis is used to determine the normal modes and normal shapes, but it helps on understand the whole system and helps on understand all other dynamic analysis. All output quantities for a modal analysis are based on the relative displacements of a mode shape, and then the output quantities can be compared for a certain mode, but not necessarily between different modes. HyperWorks 10.0 RADIOSS for Linear Analysis 51 Proprietary Information of Altair Engineering, Inc. Chapter 3: Modal Analysis 3 – Eigenvalue Solution Methods 1. Vector Iteration Methods o Inverse Iteration o Forward Iteration o Shifting in Vector Iteration o Rayleigh Quotient Iteration o Matrix Deflation and Gram-Schmidt Orthogonalization 2. Transformation Methods o Jacobi Method o Generalized Jacobi Method o Householder-QR-Inverse Iteration Solution 3. Polynomial Iterations Methods o Explicit polynomial Iteration o Implicit Polynomial Iteration o Iteration Based on the Sturm Sequence Property 4. Lanczos Iteration Method 5. Subspace Iteration Method For linear elastic problems that are properly setup (no rigid body rotation or translation), the stiffness and mass matrices and the system in general are positive definite. These are the easiest matrices to deal with because the numerical methods commonly applied are guaranteed to converge to a solution. When all the qualities of the system are considered: 1) Only the smallest eigenvalues and eigenvectors of the lowest modes are desired 2) The mass and stiffness matrices are sparse and highly banded 3) The system is positive definite The Lanczos algorithm is an iterative algorithm invented by Cornelius Lanczos that is an adaptation of power methods to find eigenvalues and eigenvectors of a square matrix or the singular value decomposition of a rectangular matrix. It is particularly useful for finding decompositions of very large sparse matrices as the FEA ones. A typical solution is first to tridiagonalize the system using the Lanczos algorithm. Next, use the QR algorithm to find the eigenvectors and eigenvalues of this tridiagonal system. RADIOSS for Linear Analysis 52 Proprietary Information of Altair Engineering, Inc. HyperWorks 10.0 Chapter 3: Modal Analysis 4 - Modal Analysis using FEM The goal of modal analysis in structural mechanics is to determine the natural mode shapes and frequencies of an object or structure during free vibration. It is common to use the finite element method (FEM) to perform this analysis because, like other calculations using the FEM, the object being analyzed can have arbitrary shape. Sometimes, the only desired modes are the lowest frequencies because they can be the most prominent modes at which the object will vibrate, dominating all the higher frequency modes. It is also possible to test a physical object to determine it's natural frequencies and mode shapes. This is called an Experimental Modal Analysis. The results of the physical test can be used to calibrate a finite element model to determine if the underlying assumptions made were correct (for example, material properties, boundary conditions, etc.). 4.1 - FEA eigensystem For the most basic problem involving a linear elastic material which obeys Hooke's Law, the matrix equations take the form of a dynamic three dimensional spring mass system. The generalized equation of motion is given as: && + Cu& + Ku = F Mu Where M is the mass matrix, ü is the 2nd time derivative of the displacement u (i.e. the acceleration), u& is the velocity, C is a damping matrix, K is the stiffness matrix, and F is the force vector. The only terms kept are the 1st and 3rd terms on the left hand side which give the following system: && + Ku = 0 Mu This is the general form of the eigensystem encountered in structural engineering using the FEA. Further, harmonic motion is typically assumed for the structure so that ü is taken to equal -λu, where λ is an eigenvalue, and the equation reduces to: (K − λM )u = 0 where the solution of the eigenvalue problem yields n eigenvalues λ, where n is the number of degrees of freedom. The vector u is the eigenvector corresponding to the eigenvalue. HyperWorks 10.0 RADIOSS for Linear Analysis 53 Proprietary Information of Altair Engineering, Inc. Chapter 3: Modal Analysis 5 – How to Setup a Modal Analysis In order to run a normal modes analysis, an EIGRL bulk data entry needs to be given because it defines the number of modes to be extracted. The EIGRL card needs to be referenced by a METHOD statement in a SUBCASE in the subcase information section. Step 1 – Generate a FEM model with the material and properties already setup (Chapter 1 section 3) FEM model Step 2 – Define the constraint LoadCollector and the constraint for this simulation. SPC LoadCollector definition Step 3 – Define the EIGRL LoadCollector with the desired number of frequencies. EIGRL LoadCollector definition RADIOSS for Linear Analysis 54 Proprietary Information of Altair Engineering, Inc. HyperWorks 10.0 Chapter 3: Modal Analysis EIGRL LoadCollector input data Step 4 – Define the modal load case. Modal loadcase setup Step 5 – Run and post process the results. First Eigen mode contour The Lanczos eigensolver implemented on RADIOSS and OptiStruct provides two different ways of solving the problems. If the eigenvalue range is defined on EIGRL has no upper bound and less than 50 modes the faster method is automatic applied. It is not necessary to define boundary conditions using an SPC statement. If no boundary conditions are applied, a zero eigenvalue is computed for each rigid body degree of freedom of the model. It is possible to request the computation of residual vectors in conjunction with a normal modes analysis. Residual vectors are static displacements ortho-normalized with the HyperWorks 10.0 RADIOSS for Linear Analysis 55 Proprietary Information of Altair Engineering, Inc. Chapter 3: Modal Analysis eigenvectors to be used in an external FRF - Frequency Response Analysis. In order to get this output, users have to define degrees of freedom using USET, USET1. The degrees of freedom are then used to define loads in the unit load method to compute the residual vectors. RESVEC = YES needs to be defined in the normal modes subcase. Boundary conditions can be defined using SPC or inertia relief. A Modal loadstep definition for RADIOSS looks like the following lines: SUBCASE 1 SPC=1 METHOD(STRUCTURE)=2 This defines: 1. The Normal modes subcase 1 2. The Constrain are defined on the LoadCollector 1 3. The number of modes and other parameters are defined on the LoadCollector 2 that have to be an Eigrl type. RADIOSS for Linear Analysis 56 Proprietary Information of Altair Engineering, Inc. HyperWorks 10.0 Chapter 3: Modal Analysis Exercise 3.1: Shell Clamped BEAM model This exercise runs a modal analysis on a very simple problem where the Eigen values and eigenvectors are well known and can be found using analytical formulas. The problem intends to describe all the cards involved in a modal analysis and the procedure of setup a modal loadstep. In this exercise, you will learn how to: Define a modal analysis on RADIOSS • Pos-process and understand the modal results h • b L Physical model description Model Information • Geometry: o • One load case: Normal Modes o • (L = 1000, h = 10, b = 10 mm) 3 First modes Material STEEL: o o o ρ = 7.8e-9 E = 210000 ν = 0.3 T/mm3 MPa - [RHO] Density [E] Young’s modulus [nu] Poisson’s ratio Problem Setup You should copy this File: BEAM_SHELL_MODAL.hm; HyperWorks 10.0 RADIOSS for Linear Analysis 57 Proprietary Information of Altair Engineering, Inc. Chapter 3: Modal Analysis Step 1: Open the Start model on HyperMesh 10.0. 1. Launch HyperMesh 10.0. with User Profile > RADIOSS > BulkData. 2. Open the HyperMesh database BEAM_SHELL_MODAL.hm. Step 2: Model Setup 1. 2. 3. 4. Mesh the surface with element size = 500 mm. Create the beam material (Mat1 [E = 210000, nu = 0.3 and rho = 7.9e-9]) Create a pshell property T=10 mm, assign the material created above. Assign the property created above to the comp beam. Step 3: Create modal subcase 1. Create a LoadCollector and call it SPC. 2. Apply the constrain as follow: • Surface Uz = 0 • Clamped line All dof = 0 (Our first model has only one element, then we need to fix the points to enforce HM to apply this BC to the corner nodes, on HM10.0 when a line has only 2 nodes the BCs are not transferred to them) 3. Create/Edit a LoadCollector with card image EIGRL and call it EIGRL. 4. Create a loadstep and call it Normal Modes as follow. Step 4: Run and study the results 1. Save this model BEAM_SHELL_BASELINE_MODAL.HM. 2. Run this model and call it BEAM_ELEM500.FEM 3. Run the same model with different mesh size, see the table below for reference and fill the values you get for all models. RADIOSS for Linear Analysis 58 Proprietary Information of Altair Engineering, Inc. HyperWorks 10.0 Chapter 3: Modal Analysis • Do it until you think the model can represent well the 3 first modes of the Beam. • Remember you can take advantage of the BCs on the geometry on your Baseline model. Just redo the mesh and run with another name… Result Table SIZE DOF 1ST MODE 2ND MODE 3RD MODE 500 250 100 50 20 10 5 1 0.5 Analytical Solution: f 1 = 0.748 2 HyperWorks 10.0 EI mL4 f 2 = 1.8732 EI mL4 f 3 = 3.134 2 EI mL4 RADIOSS for Linear Analysis 59 Proprietary Information of Altair Engineering, Inc. Chapter 3: Modal Analysis Exercise 3.2: Compressor Bracket Modal Analysis This exercise runs a modal analysis on a compressor system. This is very common problem for an engine designer, who needs to find the best way to link the compressor with the engine. To make this system viable the vibration produced by the engine can’t have resonance with the compressor system, and then the key to the project is to develop a bracket that makes the frequencies higher than excitations. Suppose that our 4-cycle engine can work up to 8000 RPM, and then the excitations from the second order (2 explosions per cycle) are up to ~266 Hz. Then the objective of this project is to have a Bracket with the first frequency higher than 350 Hz. In this exercise, you will learn how to: • Determine if a FEA model is well defined • Understand how good are the modal results 5 Bolt link Problem description Model Information • Determine if the Bracket Baseline geometry pass the Dynamic criteria: o Natural frequencies > 350 Hz. • Compressor: (Mass = 3 kg and CG = (-5.2, -14.5, 65.2) • Material STEEL: o ρ = 7.8e-9 T/mm3 [RHO] Density o E = 210000 MPa [E] Young’s modulus o ν = 0.3 [nu] Poisson’s ratio Problem Setup: You should copy these Files: BRACKET_COMPRESSOR_FEA_2nd.hm; RADIOSS for Linear Analysis 60 Proprietary Information of Altair Engineering, Inc. HyperWorks 10.0 Chapter 3: Modal Analysis Step 1: Open the Start model on HyperMesh 10.0. 1. Launch HyperMesh 10.0. with User Profile > RADIOSS > BulkData. 2. Open the HyperMesh database BRACKET_COMPRESSOR_FEA_2nd.hm. Step 2: Model Review 1. As the user can see this FEA model is already idealized. FEA model with tetra10, RB2 and RB3. a) The model is made with tetra10 ~5mm. Is it ok? What would you change on this model? __________________________________________________________________ b) It is used RB3 to link the mass element. Why we don’t use RB2 (rigid) for it? __________________________________________________________________ c) There is no representation for the Bolts and the compressor. How much it can change? What is needed to make this hypothesis? __________________________________________________________________ d) The engine wall is considered rigid. When this is important? __________________________________________________________________ Step 3: Create MAT with these properties HyperWorks 10.0 RADIOSS for Linear Analysis 61 Proprietary Information of Altair Engineering, Inc. Chapter 3: Modal Analysis Step 4: Create a PSOLID property and assign the Bracket component with it. Step 5: Create a Mass element at the dependent node of the RB3. Step 6: Create modal subcase 3. Create a LoadCollector and call it SPC. 4. Add a displacement constrain to all RB2 independent nodes (All DOFs = 0). 1. Create/Edit a LoadCollector with card image EIGRL and call it EIGRL. 2. Create a loadstep and call it Normal Modes as follow. Step 7: Run and study the results 4. Save this model BRACKET_COMPRESSOR_FEA_2nd_FINAL.hm. 5. Include strain energy results by adding the card ESE. 6. Run the model. Q1: How much you trust on the first mode you have got on this analysis? ______________________________________________________________________ Q2: Is there any result that you can look to identify if your model is good? ______________________________________________________________________ RADIOSS for Linear Analysis 62 Proprietary Information of Altair Engineering, Inc. HyperWorks 10.0 Chapter 3: Modal Analysis Q3: Should you improve the model? ______________________________________________________________________ Q4: How can you determine where the mesh needs to be refined? ______________________________________________________________________ Expected result: 1st mode (Hz) Time (s) N. DOFS *FEA ERROR 475 35 134769 ~2.2% * Based on a very refined model (~2M DOFs) Eigen Vector Contour (First Mode) Strain Energy Contour (First Mode) HyperWorks 10.0 RADIOSS for Linear Analysis 63 Proprietary Information of Altair Engineering, Inc. Chapter 3: Modal Analysis a) The model is made with tetra10 ~5mm. Is it ok? What would you change on this model? It is not a easy determine if a model is good enough, to do it the analyst needs to start with a simple model and refine it until achieve a converged result, and in general the analyst needs to do trade off (time vs. accuracy). On a modal analysis the user should see which part of the model has the highest STRAIN ENERGY to refine it up to achieve the convergence on the frequency value. ERROR FIRST MODE 100.0% ERROR 10.0% 1st 2nd 1.0% 0.1% 0.1 1 10 100 1000 10000 100000 TIME b) It is used RB3 to link the mass element. Why we don’t use RB2 (rigid) for it? An RB2 would include a rigid condition between the compressors links that doesn’t exist. For this model for example it would show that the first mode would be higher than 700 Hz. c) There is no representation for the Bolts and the compressor. How much it can change? What is needed to make this hypothesis? To do this kind of simplification the analyst needs to have know-how about the system behavior, in general we can assume that the bolt is strong enough (SIZE/MAT) to not change the modal result. But the compressor geometry needs to be studied before any simplification. RADIOSS for Linear Analysis 64 Proprietary Information of Altair Engineering, Inc. HyperWorks 10.0 Chapter 3: Modal Analysis d) The engine wall is considered rigid. When this is important? This is very important, some times the engine wall is thin on the region where the bracket is fixed, and it can be very important on the modal behavior. Again here the analyst needs to study the region to make the right assumption. Answer 1: How much you trust on the first mode you have got on this analysis? To answer this question the analyst should verify: • The first mode is like was expected. (shape and value) • The mesh is refined enough (Mode shape, strain energy convergence). • Are there any tests, analytical or past results to calibrate the model. Answer 2: Is there any result that you can look to identify if your model is good? Strain energy can give to the analyst a very good indication if the mode is well refined. It works like the stress for a static analysis. Answer 3: Should you improve the model? Based on the error plotting the answer should be no. But in general the analyst doesn’t know the FEA error, then the measure needs to be made based on the response variance with the mesh discretization, if it is less than a certain amount considered admissible to the problem then the model is considered ok. Answer 4: How can you determine where the mesh needs to be refined? Again the highest strain energy shows the places where the mesh needs to be refined. HyperWorks 10.0 RADIOSS for Linear Analysis 65 Proprietary Information of Altair Engineering, Inc. Chapter 3: Modal Analysis RADIOSS for Linear Analysis 66 Proprietary Information of Altair Engineering, Inc. HyperWorks 10.0 Chapter 4: Buckling Analysis Chapter 4 BUCKLING ANALYSIS Thin structures subject to compression loads that haven’t achieved the material strength limits can show a failure mode called BUCKLING. This failure can be analyzed using a technique well known as “linear buckling analysis”. 1 – Definitions Linear buckling is a mathematic tool used to predict the theoretical buckling strength of an ideal elastic structure. It is solved by first applying a reference level of loading, Pref, to the structure. A standard linear static analysis is then carried out to obtain stresses which are needed to form the geometric stiffness matrix KG. This new matrix is evaluated using the initial stiffness matrix augmented by the initial stress matrix corresponding to the load specified in the static load step, multiplied with a factor that is determined such that the resulting matrix has zero as its lowest Eigen frequency. The buckling loads are then calculated by solving an eigenvalue problem: [K − λK G ]x = 0 Κ is the stiffness matrix of the structure and λ is the multiplier to the reference load. The solution of the eigenvalue problem generally yields n eigenvalues λi, where n is the number of degrees of freedom (in practice, only a subset of eigenvalues is usually calculated). The vector x is the eigenvector corresponding to the eigenvalue. The eigenvalue problem is solved using a matrix method called the Lanczos method. Not all eigenvalues are required. Only a small number of the lowest eigenvalues are normally calculated for buckling analysis. The lowest eigenvalue λCr is associated with buckling and the critical or buckling load is: PCr = λCr PRef In order to run a linear buckling analysis, an EIGRL bulk data entry needs to be given because it defines the number of modes to be extracted. The EIGRL card needs to be referenced by a METHOD statement in a SUBCASE in the subcase information section. 67HyperWorks 10.0 RADIOSS for Linear Analysis 67 Proprietary Information of Altair Engineering, Inc. Chapter 4: Buckling Analysis In addition, it is necessary to use a STATSUB card to reference the appropriate referential static loading, PRef ,SUBCASE. STATSUB cannot refer to a subcase that uses inertia relief. In such cases, the stiffness matrix is positive semi-definite and the buckling eigenvalue solution ends in singularity. The buckling analysis will ignore zero-dimensional elements, MPC, RBE3, and CBUSH elements. These elements can be used in buckling analysis, but they do not contribute to the geometric stiffness matrix, KG. By default, the contribution from the rigid elements to the geometric stiffness matrix is not included. Users have to add PARAM, KGRGD, YES to the bulk data section to include the contribution of rigid elements to the geometric stiffness matrix. In addition, through the EXCLUDE subcase information entry, users may decide to omit the contribution of other elements to the geometric stiffness matrix, effectively allowing users to control which parts of the structure are analyzed for buckling. The excluded properties are only removed from the geometric stiffness matrix, resulting in a buckling analysis with elastic boundary conditions. This means that the excluded properties may still be showing movement in the buckling mode. 1.1 – Linear buckling and offset elements Some one-dimensional and shell elements can use offset to “shift” the element stiffness relative to the location determined by element’s nodes. For example, shell elements can be offset from the plane defined by element nodes by means of ZOFFS. In this case all other information, such as material matrices or fiber locations for the calculation of stresses, are given relative to the offset reference plane. Similarly, shell results, such as shell element forces, are output on the offset reference plane. Offset is applied to all element matrices (stiffness, mass, and geometric stiffness), and to respective element loads (such as gravity). Hence, in principle offset can be used in all types of analysis and optimization, including linear buckling. However, caution is advised when interpreting the results. Without offset, a typical simple structure will bifurcate and loose stability “instantly” at the critical load. With offset, though, the loss of stability is gradual and asymptotically reaches a limit load, as shown below in figure (b): Buckling failure limits RADIOSS for Linear Analysis 68 Proprietary Information of Altair Engineering, Inc. HyperWorks 10.0 Chapter 4: Buckling Analysis In practice then, the structure with offset can reach excessive deformation before the limit load is reached. (Note that more complex structures, such as frames or structures experiencing bending moments buckle via limit load, even in absence of ZOFFS on the element card). Furthermore, in a fully nonlinear approach, additional instability points may be present on the limit load path. 2 – How to Setup a Linear Buckling Analysis STEP 1 – Define the static load step. (Ref. Chapter 1 Section 3) L = 1000 mm D = 20 mm (Circular) E = 210000 MPa (Steel) Maximum axial load that a long, slender, ideal column can carry without buckling: PCr = π 2 EI (KL)2 Where: F = Critical force (vertical load on column) E = Young’s Modulus I = Area moment of inertia L = Column Length K = Column effective length:
Both ends pinned (K=1)
Both ends fixed (K=0.5)
One end fixed and other pinned (K=0.699…)
One end Fixed and the other Free (2.0) STEP 2 – Create an EIGRL LoadCollector. 69HyperWorks 10.0 RADIOSS for Linear Analysis 69 Proprietary Information of Altair Engineering, Inc. Chapter 4: Buckling Analysis STEP 3 – Create a linear buckling loadstep. STEP 4 – Run and post-process the buckling results. First Buckling mode of a column PCr (Theoretical) = 4069 N ~ 4100 N RADIOSS for Linear Analysis 70 Proprietary Information of Altair Engineering, Inc. HyperWorks 10.0 Chapter 4: Buckling Analysis Exercise 4.1: Wing Linear Buckling Analysis This exercise runs a linear buckling analysis on a simple Aircraft wing. This is typical problem in aerospace structures that needs to be very light and consequently became slender. Then because the structure has a high slenderness ratio the buckling failure verification became necessary. The objective of this project is to verify if the static 3 load cases applied to the wing will not make it fail. In this exercise, you will learn how to: • Verify a wing baseline design for buckling criteria: Problem description Model Information • Design Criteria: o Buckling: FIRST MODE > (1.5 x). o Static: U < 20 mm and Von Mises < 70 MPa. • Material Aluminum: o ρ = 2.1e-9 T/mm3 [RHO] Density o E = 70000 MPa [E] Young’s modulus o ν = 0.33 [nu] Poisson’s ratio Problem Setup: You should copy this File: Wing.hm; 71HyperWorks 10.0 RADIOSS for Linear Analysis 71 Proprietary Information of Altair Engineering, Inc. Chapter 4: Buckling Analysis Step 1: Open the start model on HyperMesh 10.0. 1. Launch HyperMesh 10.0. with User Profile > RADIOSS > BulkData. 2. Open the HyperMesh database WING.hm. Step 2: Run a static analysis and verify the design for static failure. Von Mises stress (MPa) and total displacement (mm). o σ ADM = 70MPa o U ADM = 20mm RADIOSS for Linear Analysis 72 Proprietary Information of Altair Engineering, Inc. HyperWorks 10.0 Chapter 4: Buckling Analysis Step 3: Create the linear buckling load cases 1. Create/Edit a LoadCollector with card image EIGRL and call it EIGRL. 2. Create the buckling loadsteps as follow. Step 7: Run and study the results First Buckling mode (Criteria Mode 1 > 1.5x) 73HyperWorks 10.0 RADIOSS for Linear Analysis 73 Proprietary Information of Altair Engineering, Inc. Chapter 4: Buckling Analysis RADIOSS for Linear Analysis 74 Proprietary Information of Altair Engineering, Inc. HyperWorks 10.0 Chapter 5: Inertia Relief Analysis Chapter 5 INERTIA RELIEF ANALYSIS Inertia relief allows the simulation of unconstrained structures. Typical applications are an airplane in flight, suspension parts of a car, or a satellite in space. 1 – Definitions With inertia relief, the applied loads are balanced by a set of translational and rotational accelerations. These accelerations provide body forces, distributed over the structure in such a way that the sum total of the applied forces on the structure is zero. This provides the steady-state stress and deformed shape in the structure as if it were freely accelerating due to the applied loads. Boundary conditions are applied only to restrain rigid body motion. Because the external loads are balanced by the accelerations, the reaction forces corresponding to these boundary conditions are zero. This calculation is automated on RADIOSS. Inertia relief boundary conditions may be defined in the bulk data section of the input deck or they may be determined automatically by the solver. o o The SUPORT and SUPORT1 bulk data entries are used to define up to six reaction degrees of freedom of the free body.
SUPORT entries will be used in all relevant subcases and therefore do not need to be referenced in the Subcase Information section.
SUPORT1 entries need to be referenced by a SUPORT1 data selector statement for use within a subcase. Inertia relief boundary conditions may be generated automatically by using PARAM, INREL, -2. In RADIOSS, inertia relief can be applied to linear static, nonlinear gap, modal frequency response (with residual vectors), and transient response (with residual vectors) analyses. A static case with inertia relief cannot be referenced in a linear buckling analysis. Inertia relief is meaningless in normal modes analysis. HyperWorks 10.0 RADIOSS for Linear Analysis 75 Proprietary Information of Altair Engineering, Inc. Chapter 5: Inertia Relief Analysis 2 – How to Setup an Inertia Relief Analysis STEP 1 – Prepare the FEM model as it was for any other static analysis. FEM model without BCs and Loads STEP 2 – Define fictitious support (suport or suport1) Fictitious support definition example. o It is important to notice that the fictitious supports should just remove the rigid body motion and not add an improper constraint. RADIOSS for Linear Analysis 76 Proprietary Information of Altair Engineering, Inc. HyperWorks 10.0 Chapter 5: Inertia Relief Analysis Definition of SUPORT1 in HyperMesh STEP 3 – Define the equivalent static force. Definition of a FORCE in HyperMesh STEP 4 – Define appropriate INREL parameter. -2: Without suport or suport1 -1: With suport or suport1 0: Constrained analysis (fictitious supports are treated as SPCs) Control Card: Param > INREL in HyperMesh STEP 5 – Create the static load case with fictitious supports. STEP 6 – Run and post-process the inertia Relief results. Inertia Relief results (Total displacement and Von Mises Stress) HyperWorks 10.0 RADIOSS for Linear Analysis 77 Proprietary Information of Altair Engineering, Inc. Chapter 5: Inertia Relief Analysis Exercise 5.1: Satellite Inertia Load Test This exercise runs an inertia relief load case on a simple Satellite, this is a test made with aerospace structures that will need to support inertia loads. The objective of this kind of test is to verify if the structure is strong enough to support these loads without a static failure. In this exercise, you will learn how to: • Setup a static analysis with inertia loads. 4 load cases o 2 Gs on Z o 3 Gs on Y o 3 Gs on X o 4.7 Gs SUM Problem description Model Information • • • Design Criteria: o Max Rel. disp. < 500 mm. o Von Mises < 70 MPa. (Aluminum) Total Mass: 3.09 ton. Material: Material [E] [RHO] MPa Ton/mm3 Aluminum 70000 2.1 x 10-9 Solar_panel 20000 1 x 10-11 System 1000 1 x 10-13 Antenna 20000 1 x 10-11 Nu 0.33 0.4 0.3 0.4 Problem Setup: You should copy this file: Satellite.hm; RADIOSS for Linear Analysis 78 Proprietary Information of Altair Engineering, Inc. HyperWorks 10.0 Chapter 5: Inertia Relief Analysis Step 1: Open the Start model on HyperMesh 10.0. 1. Launch HyperMesh 10.0. with User Profile > RADIOSS > BulkData. 2. Open the HyperMesh database Satellite.hm. STEP 2 – Define INREL parameter as -1. -2: Without suport or suport1 -1: With suport or suport1 0: Constrained analysis (fictitious supports are treated as SPCs) Control Card: Param > INREL in HyperMesh Step 3: Create the four linear load cases with the name and details listed below: All load steps will have the same fictitious support: Suport: i. Node 2: Uy = 0 ii. Node 3: Ux, Uy, Uz = 0 iii. Node 4: Ux, Uy, = 0 1. 2. 3. 4. LOADSTEP: 2 Gz LOADSTEP: 2 Gy LOADSTEP: 2 Gx LOADSTEP: SUM 4.7 G HyperWorks 10.0 F(node 1) =(0,0, 61800) N Equivalent to 2 Gs F(node 1) =(0,0, 92700) N Equivalent to 3 Gs F(node 1) =(0,0, 92700) N Equivalent to 3 Gs F(node 1) =(0,0, 144933.8) N Equivalent to 4.7 Gs RADIOSS for Linear Analysis 79 Proprietary Information of Altair Engineering, Inc. Chapter 5: Inertia Relief Analysis Step 3: Run and study the results Total displacement (Criteria Umax < 500 mm) Von Mises (Criteria σmax < 70 MPa) RADIOSS for Linear Analysis 80 Proprietary Information of Altair Engineering, Inc. HyperWorks 10.0