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# FINITE ELEM METHODS CEE 504

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This 39 page Class Notes was uploaded by Mason Hackett on Wednesday September 9, 2015. The Class Notes belongs to CEE 504 at University of Washington taught by Staff in Fall. Since its upload, it has received 55 views. For similar materials see /class/191986/cee-504-university-of-washington in Civil and Environmental Engineering at University of Washington.

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Date Created: 09/09/15

xi 8697 MWquot y Mg x 3 WWmquotHung f ii g A z w r 2v mm 2 a a M n z 4 Q 2 c F x r LJ39t iv 1 66 4 Exact mi Approx a Exact new loads Approx new loads m 7 a 12253 Gm 71453st i 1 EL r a w m w v49 55 2 i xs 5m n w x Q M m A K r MM LLLLLL x11 33 i Jbe mm W 9 E Q wwwwwmyow39zwwywmW wm nwwm w nwxmquotyym wwymW4WM University of Washington Department of Civil Engineering Fall 2008 Instructor Laura N Lowes CEE457 Using SAP 1 Start the program Create a new model Do this from the pulldown menu for File 7gt Newmodel a Define the units for the model you can always change these in the bottom right corner but setting the default units to your preferred unit system is best b Create a new model with using grid only template i The default grid has the zaxis in the vertical direction ii The grid is generated with the number of spacings that you specify in the x y and 2 directions 3 Change the vertical axis from z to x or y as is necessary to help define your structure Do this from the pull down menus for Options 7gt 3D View Up Direction N 4 Edit the grid to fit your structure a Pulldown menu Delinegt Coordinate S ystems Grids gt Mod l Show System i Add and Subtract grid lines ii Set Snap to grid 5 Drawthe structure a Draw nodes Select Draw Special Jointbutton on left or Add Special Joint from Draw pulldown menu b Draw elements Select Draw FrameCable Elementbutton on left or Draw Frame Element from Drawpulldown menu i Note that you can use the popup Properties of Object Menu to assign member section member moment releases and other properties We ll ignore this for the time being and assign member properties using the pulldown menus 6 Define properties of the elements a Define materials Define 7gt Materials i Add new material ii Modify existing material b Define frame sections Define 7gt FrameCable Sections i Choose type of frame section rectangle Wide flange etc ii Note that if the section you want is not loaded you can import sections from databases that are stored in the SAP2000 directory iii For a truss 1 Set xdimension and ydimension so that area is correct 2 Note that you can use section modification factors to reduce increase stiffness in shear flexure torsion axial etc a To define a rigid response in shear assign a shear stiffness modification of zero iv Specify material 7 Attach frame section to elements a Highlight element gt Assgngt Frame 7gt Sections b For 3D Assign local axes as necessary Highlight element gt Assign gt Frame gt Local axes i Type in a value for Angle in Degrees This is an angle by which local axis 2 of the Element will be rotated around local axis 1 Local axis 1 is along the length of the Element By default local axis 2 is always in the 12 plane except if the Element is vertical and then it is parallel to the global X axis The definition of the local axes followthe right hand rule The angle is measured anticlockwise as positive if local axis 1 is pointing towards you Local xdirection Red Arrow Local ydirection While Arrow weak bending axis Local zdirection Blue Arrow strong bending axis 8 Assign frame member releases a To model truss elements we need to release moments at both ends of the element b Highlight the frame element using the mouse c To model truss elements assign frame releases using the Assign 7gt FrameCable 7gt Releases Pania Fixy pulldown menu Set the moment releases at each end of the member and set the spring stiffness to be zero 9 Assign frame member rigidend offsets a Highlight element gt Assgngt FrameCable 7gt End Length Offsets b In the Frame End Offsets dialog box i If you want to use User Specified End Offsets then check Define Lengths and type in the values for the Offsets at End l and End J If you want the program to calculate the End Offsets from the connectivity of the model then check Update lengths from current connectivity The program will automatically calculate the End Offsets from the Depth Major and Width Minor specified for the Frame Element properties Specify a Rigid Zone Factor in the edit box This is a factor used to define the percentage ofthe Zone specified through End Offsets to be taken as fully rigid 0 means no portion of the zone is considered to be rigid and 1 means that the entire zone is taken as rigid A value between 0 and 1 indicates the percentage of the zone that is considered to be rigid 10 Assign section modification factors a Approach 1 for individual elements i Highlight element gt Assgngt FrameCable 7gt Frame Property Modi er 1 Define one or more modification factors for the original member axial shear flexural or torsional stiffness This approach may be used to stiffness a section or make it more flexible b Approach 2 for all elements with a specific section 39 Definegt FrameCable Section ldentify property type to modify Choose Modify Show Property Choose Property Modifiers Define one or more modi cation factor for the original member axial shear flexural or torsional stiffness This approach may be used to stiffness a section or make it more flexible ltE39EE 11 Assign joint restraints a To model fixed boundary conditions we introduce joint restraints and restrain the joint from translating or rotating b Highlight nodes gt Assign 7gt Joint Restraints 12 Define load patterns a These are typically DEAD LIVE EQ etc b If you want to neglect selfweight in your analysis it is easiest to do this by setting the selfweight multiplier to be zero when you define the load pattern c If you have more than one load case it is easiest to define multiple load cases and then solve all at once If you have a forced displacement case as well as nodal load cases you can t solve both at the same time but multiple load cases means it s easy to solve one and then the other d To define load patterns De ne 7gt load Patterns i Modify patterns so that there is no selfweight load 13 Define loads a Highlight a node gt Assign 7gt Jointloads a gt Forces Displacements i Note which load casequot you are you modifying ii Do you want to add load or replace existing loads for the load case iii If you are applying displacements you will need to restrain those nodes 14 Analyze a Set analysis options for frames or truss basically how many DOF at each node Do this using the popup menu on Analyze 7gt SetAnalysis Options b Set load cases to run You probably want to run all of the load cases that you have defined However you may want to turn off some defaults Do this from the popup menu on Analyze 7gt Set Analysis Cases to Run n c 15 Results a Use Displaypulldown menu to view anything you want to see b Right clicking the mouse will give you data for nodes and elements in a tabular format if you are in the tabular display mode Displayegt Analysis Results Table or in a graphical format Results can be saved to text files Display 7gt Analysis Results Table Results can be printed File 7gt Print Tables 90 University of Washington Department of Civil Engineering Fall 2008 Instructor Laura N Lowes CEE457 Using SAP to Model 2D Continua Start the program Create a new model Do this from the pulldown menu for File gt lVewmode a Define the units for the model you can always change these in the bottom right corner but setting the default units to your preferred unit system is best b Create a new model with using grid only template i The default grid has the zaxis in the vertical direction ii The origin of the grid is 0 0 0 iii The grid is generated with the number of spacings that you specify in the x y and 2 directions 3 Edit the grid to fit your structure a Change the vertical axis from z to x or y as is necessary to help define your structure Do this from the pulldown menus for Options 7gt 3D View Up Direction b Pulldown menu Definegt Coordinate S ystems Grids gt Mod l Show System i Add and Subtract grid lines ii Set Snap to grid Define properties of the elements a Define materials Define 7gt Materials i Add new material ii Modify existing material b Define area sections Define 7gt Area Sections i Name the section ii Choose type of material for the section If the material is orthotropic wood for example has different material properties in direction parallel to the grain versus perpendicular to the grain you will need to define the material angle to use iii Choose the area element type 1 Shell element these elements represent 2D stressstrain response in the plane of the element referred to as membrane action as well as out of plane bending 2 Plane element these elements represent only 20 stressstrain response in the plane of the element These include the 3node triangular elements that we ve developed in class as well as 4node quadrilateras which are similar 3 Axisymmetric Solids these elements can be used to model a threedimensional object that is axisymmertic for example a barrelshaped water tank One models only a 2D slice of the object using these elements iv Define the thickness 1 For shell elements you can define different thickness for the inplane stress strain membrane response and out of plane bending 2 For plane elements you define a single thickness equal to the outofplane thickness of the element 3 For axisymmetric elements you define the thickness in terms of the angle out of 211 radians that the element represents gt F V Further define the element type 1 For shell elements your options are to model i both the inplane stressstrain membrane response as well as the outofplane bending response ii only the inplane action or iii only the out of plane action 2 For plane elements you have the choice of a plane stress or a plane strain formulation You also have the choice to include incompatible modesquot this is an improved element formulation that actually allows for discontinuities in the displacement field and results in a more exible model 3 For Axisymmetric elements you have only the choice of including incompatible modesquot this is an improved element formulation that actually allows for discontinuities in the displacement field and results in a more flexible model c Set Modifiers This menu allows you to reduce the stiffness of the element for any mode of response as well as modify the mass and weight of the element 5 Draw the structure a Create nodes that define the perimeter of your structure Select Draw Special Jointbutton on left or Add Special Jointfrom Drawpulldown menu b Draw a few area elements that define the basic geometry of your structure For rectangular areas you can Draw Rectangular Area Elements by either selecting the Draw Rectangular Area Element button on the left or Draw Rectangular Area Element from the Draw pulldown menu For nonrectangular areas either select the Draw Poly Area Element button on the left or Draw Poly Area Element from the Draw pulldown menu Whey you choose to draw an area element a windows pops up that allows you to choose the area element that you want to use c Once you have drawn the few area elements that define the basic geometry of your structure you will likely want to remesh these Select the area that you want to remesh and Use the Editmenu Editgt llleshAreas Use the Assign menu ASSgn 7gt Area 7gt AutomaticArea lllesh 1 Note that the new mesh will not be visible until you run your analysis d Note that once you have remeshed an area you can use the undo feature to go back to the original coarser mesh but you cannot remesh an area to increase the size of the elements 6 Assignjoint restraints a To model fixed boundary conditions we introduce joint restraints and restrain the joint from translating or rotating b Highlight nodes gt ASSgn 7gt Joint Restraints 7 Define load patterns a lfyou have more than one load pattern it is easiest to assign multiple load patterns and then solve all at once You may want to modify the default load pattern so that gravity selfweight loading is not included b To define load patterns Define 7gt load Pattern 8 Define loads Modify patterns so that there is no selfweight load a Highlight a node gt ASSgn 7gt Joint loads a gt Forces Displacements Note which load patternquot you are you modifying Do you want to add load or replace existing loads for the load case 9 Analyze a Set analysis options basically how many DOF at each node Do this using the popup menu on Analyze 7gt SetAnalysls Options b Set load cases to run You probably want to run all of the load cases that you have defined However you may want to turn off some defaults Do this from the popup menu on Analyze 7gt Set Analysis Cases 0 Run c Run 10 Results a Use Displaypulldown menu to view anything you want to see b To display stress fields view element forces stresses II You can choose the stress fields that you want to see S11 822 S12 refer to 011 022 012 where axes 1 amp 2 are the local xaxis and yaxis equal to the global if you have not redefined the local axes for the element SMax and SMin will give you principal stresses SVMax will give you maximum shear stress ll You can choose to i not smooth the stress field ii smooth the stress field at joints or iii smooth the stress field within groups 7 Typically you will want to smooth the stress field at the joints This gives you a better representation of the actual stress eld as is smooths out the jumps in the stress eld that occur because of the element formulation 2 You don t want to smooth the stress field if there is a discontinuity in the structure that will result in a discontinuity in the stress field such as a change in the thickness of the material Smoothing stresses within groups takes care of this issue c The labes Display gt Show Tables provides a good way of determining maximum stress values with more accuracy than is provided by reading values off of the contour plots CEE5o4 Finite Element Methods in Structural Mechanics 0 Introduction to the Finite Element Method What is the nite element method Why do we use nite element models to analyze the response of structural systems What are the steps in an solution 0 Course Objectives see handout 0 Course Outline see handout 0 Course Conduct see handout 0 Reading Assignment 1 Chapters 1 amp 2 CEE504 University ofWashington Winter 2009 D epartment of civil and Environmental Engineering What is the FEM 0 Two perspectives mathematical View FEM is a numerical method for solving a boundary value problem that consists of developing a discrete model of the continuous eld engineering FEM is a procedure for determining the deformation andor stress eld in a structural system The global structural response is determined by modeling the relatively simple response of pieces of the system and combining the response of these individual components to determine the global response University of Washington Winter 2009 D epanment of civil and Environmental Engineering Mathematical View of the FEM o It is a numerical method that provides an approximate solution to a boundary value problem that is defined by a partial or ordinary differential equation that consists of developing a discrete model of the continuous field 0 x ordinary differential equation AEdiXux Tx o d boundary conditions uw 0 AE d F at 1 ex it NOTE analytical solution 0 E t Fquot 3e 3e University ofWashington Department of civil and Environmental Engineering CEES 04 Vifmter 2009 Engineering View of the FEM o It is a method of determining the global response of a complex structure by modeling the relatively simple response of small pieces of the system and combining these pieces to determine the global response Representative loads X L Elements wi h constant area elastic modulus and stress We know the relationship between stress strain displacement and force for these simple elements University of Washington D epartment of civil and Environmental Engineering Vifmter 2009 Why use FEM 1 Because analytical solutions can be found only for very very simple problems CEE504 University ofWashington Winter 2009 D epartment of civil and Environmental Engineering Why use FEM Boundary Value Problem Vo 0 for X E Q o xx7s forXEI39g1 o XXS forxefgz oyy0forxe1quot93 o 0 forxefg4 azzaxzayz0 fOI39XEQ University of Washington Winter 2009 Department of Civil andEnvimnmental Engineenng Why use FEM lt gt lt gt lt gt s lt gt s lt gt lt gt lt gt I 1 14 3 0056 166 13 00526 i rr9 12 3 sin26 I I I CEE504 Univer51ty ofWashington Winter 2009 Department of Civil and Environmental Engineering Why use FEM 2 It provides a tool for use in determining the response of complex structures Wallframe systems Frame elements Plane stress or shell elements HHH JUDDDU CEE5 04 Winter 2009 University of Washington Department of Civil and Environmental Engineering Why use FEM Complex structures in the design office Structural wall With irregular Slab With iIregular 0A eninus subgected to simulated Openings Sub fted t0 earthquake loading graVlty loadlng gt wx 50 psf DD University of Washington CEEs 04 Winter 2009 Department of civil and Environmental Engineenng Why use FEM Complex structures in the design office diaphragms Floor systems in buildings With unusual layouts may behave as exible diaphrang under earthquake loading University of Washington Winter 2009 D epartment of civil and Environmental Engineering Why use FEM 3 It enables highresolution prediction of stress strain and damage elds Cypress Viaduct Oakland CA following Loma Prieta Earthquake in 1989 CEE504 University of Washington Winter 2009 Department of Civil and Environmental Engineering Why use FEM bond Stress Steel tensile stress concrete compression stress compression tension w H Steel compression stress CEE504 University of Washington Winter 2009 Department of Civil and Environmental Engineering Why use FEM 4 The mathematical formulation of the provides Proof of convergence Methods for estimating errors 5 Formulation provides a means of enhancing individual aspects of the model improved material constitutive relationships nite deformation and large strain theory element formulation CEE504 University of Washington Winter 2009 Department of Civil and Environmental Engineering The Finite Element Method Conceptual Road Map from CEE 501 de ne BVP Example 1D system Generate the weak form of the BVP using Principle of Minimum Potential Energy Galerkin s Method Virtual Work d Equilibrium EAUX TX 0 Kinematics 5X dX Constitution 0X EEX 39 I I Approx the displacement eld in the structure using shape functions that de ne the displacement eld as a function of nodal displacements d Use approximate displacement eld to compute strains stresses reactions Formulate stiffness equations for the structure F Kd Solve for nodal displacements CEE504 University of Washington Winter 2009 Department of Civil and Environmental Engineering Steps in the FEM per Logan Chapter 1 o Assumptions We are solving a structural engineering problem The unknowns we seek are nodal displacements that de ne the displacement at points in the body From these we can compute strains and stresses The solution is a displacement eld that satis es equilibrium compatibility and constitutive requirements 0 Step 1 discretize the body Element size delity of the eld accuracy computational effort Type of element behavioral mechanisms that are represented CEE504 University of Washington Winter 2009 Department of Civil and Environmental Engineering Discretization of a wallframe structure Wallframe systems Frame elements Plane stress or shell elements Hull JDDDDD CEE504 University of Washington Winter 2009 Department of Civil and Environmental Engineering Steps in the FEM 0 Step 2 select a displacement function Within each element The displacement function is associated with the element type beam elements are typically assumed to exhibit exural deformation but not shear deformation plane strain elements are assumed to deform only in plane etc Displacement functions are usually very simple polynomials Typically all elements of one type are assumed to have the same displacement function Dis lacement within the element is de ned by the displacement function the nodal displacements CEES 04 Vifmter 2009 University of Washington D epanment of civil and Environmental Engineen Element Formulations amp Displacement Fields 1D Truss Element 1 displacement dof per node in direction of bar axis two total nodal displacements enable de nition of a linear displacement function x3 39 I X2 nodal dof x1 0 2D Plane Stress and Plane Strain Elements 2 orthogonal displacement dof per node displacement is linear in x amp y with one crossterm X2 I X1 I nodal dof University of Washington Winter 2009 D epartment of civil and Environmental Engineering Element Formulations amp Displacement Fields 0 Pseudo 3D Axisymmetric Elements 2 orthogonal disp dof per node within a plane displacement is linear in z amp r with one crossterm nodal dof 0 2D Frame Element 1 displacement and 1 rotational dof per node in the direction perpendicular to the axis of the element Four nodal displacements enable definition of a cubic displacement field x3 K nodal dof X CEE504 University of Washington Winter 2009 Department of Civil and Environmental Engineering Element Formulations 0 Plate Elements 2 rotations and 1 outofplane disp dof per node the outof plane displacement field is cubic in x amp y with crossterms nodal dof a 0 Shell Elements 3 displacement and 2 rotation dof per node 5 39 39 39 i 39 quot quot 39 395 Out of plane bending Inplane membrane action CEE504 University of Washington Winter 2009 Department of Civil and Environmental Engineering 10 Steps in the FEM 0 Step 3 de ne the straindisplacement kinematics and stressstrain relationships constitution For CEE 504 we will assume that structures exhibit small deformations and small displacements rotations and remain elas c oz small deformations and small displacements rotations Small in nimsimal strains these con gurations are essentially tine same same 39239 Linear elastic materials o E5 o 05 The nite element method can be extended to consider 0 Finite deformations Green strain logarithmic strain in CEE 506 do I 39 39 39 39 39 CEE 506 do Inelastic materials plasticity visco7elasticity etc in CEE 503 CEE504 University ofWasliington Winter 2009 D epartment of Civil and Environmental Engineering Steps in the FEM 0 Step 4 derive the element stiffness matrix and equilibrium equations using Direct equilibrium method 7 This is probably what you have done so far Work amp Energy Methods 7 The principle ofniininiuni potential energy Castigliano s threorem amp the principle ofvirtual work Methods of weighted residuals 7 Galerl 39n s method 0 Step 5 Assemble the element equilibrium equations to obtain the global equilibrium equations and introduce boundary conditions i F kd University ofWashington Winter 2009 Department of civil and Environmental Engineering Steps in the FEM 0 Step 6 Solve for unknown nodal displacements 0 Step 7 Solve for element strains and stresses or resultants moments for beams plates shells Element strains determined by displacement eld Element stresses resultants determined by constitutive relationship FEM theory provides a basis for improving the calculation of the stress resultant eld within the structure 0 Step 8 Interpret and evaluate the results FEM theory provides a basis for interpreting results evaluating the error associated with the nite element solution and determining if the finite element solution represents a converged solution CEES 04 University of Washington Winter 2009 39 Department of Civil and Environmental Engineenng Course Objectives Upon completion of CEE504 students will have the knowledge and skills to use the nite element method to predict stress and strain elds is elastic structural subassemblages subjected to a variety of static load conditions 1 Understand the theory of the finite element method and demonstrate this understanding by formulating the finite element problem 2 Solve a global structural analysis problem for a structure and solution 3 Use a typical finite element analysis software package to analyze structures and interpret the results of these analyses University ofWashington Mnter 2009 Department of Civil and Environmental Engineering Course Outline Week 1 Introduction to the FEM Weeks 1 amp 2 Development and application of the FEM for analysis of trusses Weeks 3 5 2D continua solving plane stress amp strain problems using triangular amp quadrilaterial elements Week 6 Practical modeling issues Weeks 7 amp 8 Development and application of the FEM for analysis of exural elements Weeks 9 amp 10 Plates and shells Umversty ufWashmglun Wilmer znna Depanmmt nf ciml and Envlmnmmtal Engineenng CEE 504 Course Outline Quick Glance at Topics gigging mi gigging in rat elim39ixwel mYllm Course Conduct Prerequisites CEE 501 Required textbook A First Course in the Finite Element Method 4 1 Ed by Logan Grading homework 30 midterm exam 30 nal exam 40 Computer programs To solve FEM problems for small systems Matlab To facilitate solution of theoretical problems 39239 Matlab symbolic toolbox 39239 Mathematica 39239 MathCad To solve FEM problems for large systems 39239 COMSOL Multiphysics 33 39239 SAP 2000 V10 39239 MSCMarc Mentat CEE504 University ofWashington Winter 2009 D epartment of Civil and Environmental Engineering Additional Reference Texts I Texts that are comparable to Logan Concepm and applications offinite element analysisH by Cook Malkus and Plesha and Witt 2002 Introduction to the Finite Element MethodH by Ottosen and Peterson Introduction to FE in EngineeringH by Chandrupatla and Belegundu I Comprehensive references The Finite Element Method Volume lH by Zienkiewicz and Taylor 2000 The Finite Element Method Linear Static and Dynamic Finite ElementAnalysisH by Hughes 1987 Finite Element ProceduresH by Bathe 1995 I Other references Fundamentals of Structural MechanicsH by Hjelmstad University of Washington Winter 2009 D epanment of Civil and Environmental Engineering Cy 8m f3 52 c M f k g 3 quotbe fi z g O 139

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