UG FEM - Lec 1 Introduction
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Transcript of UG FEM - Lec 1 Introduction
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Finite Element Method
ME438
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Introduction
Dr Aamir Mubashar
PhD Mechanical Engineering Loughborough University, Loughborough, United Kingdom
MSC Advanced Manufacturing Technology and Systems Management The University of Manchester, Manchester, United Kingdom
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Course Contents
Introduction
Stiffness (Displacement) Method
Development of Truss Equations
Development of Beam Equations
Plane Stress / Strain Stiffness Equations
Practical Considerations in Modelling
Usage of Commercial Finite Element Software (Abaqus)
Term Project (4 weeks)
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Marks Distribution
Quizzes 15%
Projects / Assignments 10%
OHT-1 15%
OHT-2 15%
End Semester Exam 45%
5% Class Quizzes20% Term Project
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INTRODUCTION TO FEM
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Finite Element Method
FEM is a numerical method for solving problems or engineering and mathematical physics
Typical classes of problems
Structural analysis
Heat transfer
Fluid flow
Mass transport
Electromagnetic potential
etc.
Can solve problems involving complicated geometries, loadings and material properties for which analytical solutions are not possible
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GENERAL STEPS OF FEM
Introduction to FEM
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General Steps of FEM
For simplicity, we will consider the structural problem for now
Engineers seeks to determine displacements and stresses throughout a structure, which is in equilibrium and subjected to applied loads
Two general approaches traditionally associated with FEM
Force or flexibility method: uses internal forces as unknowns, result is a set of algebraic equations for determining unknown forces
Displacement or stiffness method: uses displacements of nodes as unknowns
For computational purposes, displacement or stiffness method is simpler to formulate for most of the problems and will be discussed further
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How FEM Works?
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How FEM Works?
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General Steps of FEM
Step 1
Discretise and select the element types
Step 2
Select a displacement function
Step 3
Define the strain / displacement and stress / strain relationships
Step 4
Derive the element stiffness matrix and equations
Step 5
Assemble to obtain global equations and apply boundary conditions
Step 6
Solve for unknown degree of freedom (displacements)
Step 7
Solve for element strains and stresses
Step 8
Interpret the results
Step 1 & 8 are generally decided by AnalystSteps 2-7 are carried out automatically by computer program
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General Steps of FEM
Step 1
Discretise and select the element types
Step 2
Select a displacement function
Step 3
Define the strain / displacement and stress / strain relationships
Step 4
Derive the element stiffness matrix and equations
Step 5
Assemble to obtain global equations and apply boundary conditions
Step 6
Solve for unknown degree of freedom (displacements)
Step 7
Solve for element strains and stresses
Step 8
Interpret the results
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Step 1: Discretise and Select the Element Types
Dividing the body into an equivalent system of finite elements with associated nodes
Choose most appropriate element type to model most closely the actual physical behaviour
Primary Engineering Judgements
Total number of elements used
Variation in size and type of elements within a given body
Elements
Small enough to give useable results
Large enough to reduce computational effort
Small element (and possibly higher order elements) are generally desirable where the results are changing rapidly such as where changes in geometry occurs
Large elements can be used where results are relatively constant
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General Steps of FEM
Step 1
Discretise and select the element types
Step 2
Select a displacement function
Step 3
Define the strain / displacement and stress / strain relationships
Step 4
Derive the element stiffness matrix and equations
Step 5
Assemble to obtain global equations and apply boundary conditions
Step 6
Solve for unknown degree of freedom (displacements)
Step 7
Solve for element strains and stresses
Step 8
Interpret the results
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Step 2: Select a Displacement Function
Involves choosing a displacement function within each element
Defined using nodal values of the element
Linear, quadratic and cubic polynomials are frequently used functions as they are simple to use in FEM formulation
Trigonometric series can also be used
For a two dimensional element, displacement function is a function of the coordinates in its plane (e.g. x-y plane)
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Step 2: Select a Displacement Function
The functions are expressed in terms of nodal unknowns (in two dimensional problem, in terms of x-y components)
Same general displacement function can be used repeatedly for each element
Hence, in FEM a continuous quantity such as displacementthroughout the body is approximated by a discrete modelcomposed of a set of piece-wise continuous functions defined within each finite domain or finite element
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General Steps of FEM
Step 1
Discretise and select the element types
Step 2
Select a displacement function
Step 3
Define the strain / displacement and stress / strain relationships
Step 4
Derive the element stiffness matrix and equations
Step 5
Assemble to obtain global equations and apply boundary conditions
Step 6
Solve for unknown degree of freedom (displacements)
Step 7
Solve for element strains and stresses
Step 8
Interpret the results
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Step 3: Define the Strain / Displacement and
Stress / Strain Relationships
Necessary for deriving the equations for each finite element
In case of one dimensional deformation, say x-direction
=
Stresses must be related to strains by a constitutive law. Simplest relationship is given by Hookes law
=
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General Steps of FEM
Step 1
Discretise and select the element types
Step 2
Select a displacement function
Step 3
Define the strain / displacement and stress / strain relationships
Step 4
Derive the element stiffness matrix and equations
Step 5
Assemble to obtain global equations and apply boundary conditions
Step 6
Solve for unknown degree of freedom (displacements)
Step 7
Solve for element strains and stresses
Step 8
Interpret the results
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Step 4: Derive the Element Stiffness Matrix
and Equations
Stiffness matrix and element equations relating nodal forces to nodal displacements are obtained using force equilibrium conditions
Most easily adaptable to line or one dimensional elements
Can be used to illustrate development for spring, bar, and beam elements
Several methods are used for determining the element stiffness matrix
Direct Equilibrium or Stiffness Method
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Step 4: Derive the Element Stiffness Matrix
and Equations
Using any of the above methods, we get the equations to describe the behaviour of an element
123
=
11 12 13 121 22 23 231 32 33 3 1
123
Vector of element
nodal forces
Element Stiffness Matrix
Vector of unknown element
nodal dofs
= In compact form
Generalised displacements may include such quantities as actual displacements, slopes or even curvatures
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General Steps of FEM
Step 1
Discretise and select the element types
Step 2
Select a displacement function
Step 3
Define the strain / displacement and stress / strain relationships
Step 4
Derive the element stiffness matrix and equations
Step 5
Assemble to obtain global equations and apply boundary conditions
Step 6
Solve for unknown degree of freedom (displacements)
Step 7
Solve for element strains and stresses
Step 8
Interpret the results
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Step 5: Assemble to Obtain the Global
Equations and Introduce Boundary Conditions
The individual element nodal equilibrium equations generated in Step 4 are assembled into the global nodal equilibrium equations
The direct method of superposition (called the direct stiffness method), based on nodal force equilibrium, can be used
Implicit in the direct stiffness method is the concept of continuity or compatibility
Compatibility means that the structure remains together and no tears occur anywhere inside the structure
The final assembled equation can be written as
=
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Step 5: Assemble to Obtain the Global
Equations and Introduce Boundary Conditions
For most problems, global stiffness matrix is square and symmetric
It can be shown that global stiffness matrix is singular i.e. its determinant is equal to zero
To remove singularity, boundary conditions (constraints or supports) are applied so structure remains in place and no rigid body motion occurs
Applied known loads are accounted for in the global force matrix
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General Steps of FEM
Step 1
Discretise and select the element types
Step 2
Select a displacement function
Step 3
Define the strain / displacement and stress / strain relationships
Step 4
Derive the element stiffness matrix and equations
Step 5
Assemble to obtain global equations and apply boundary conditions
Step 6
Solve for unknown degree of freedom (displacements)
Step 7
Solve for element strains and stresses
Step 8
Interpret the results
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Step 6: Solve for Unknown Degrees of
Freedom (or Generalised Displacements)
After modification to account for boundary conditions, a set of simultaneous algebraic equations is obtained
Where is the structure total number of unknown nodal degrees of freedom
Can be solved for s using elimination method (such as Gausss method) or an iterative method (such as Gauss-Seidel method)
123
=
11 12 13 121 22 23 231 32 33 3
1
123
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General Steps of FEM
Step 1
Discretise and select the element types
Step 2
Select a displacement function
Step 3
Define the strain / displacement and stress / strain relationships
Step 4
Derive the element stiffness matrix and equations
Step 5
Assemble to obtain global equations and apply boundary conditions
Step 6
Solve for unknown degree of freedom (displacements)
Step 7
Solve for element strains and stresses
Step 8
Interpret the results
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Step 7: Solve for the Element Strains and Stresses
For structural stress analysis, important secondary quantities of strain and stress can be obtained from displacements
Typical relationships between strain and displacements and between stress and strain can be used
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General Steps of FEM
Step 1
Discretise and select the element types
Step 2
Select a displacement function
Step 3
Define the strain / displacement and stress / strain relationships
Step 4
Derive the element stiffness matrix and equations
Step 5
Assemble to obtain global equations and apply boundary conditions
Step 6
Solve for unknown degree of freedom (displacements)
Step 7
Solve for element strains and stresses
Step 8
Interpret the results
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Step 8: Interpret the Results
Final goal is to interpret and analyse the results for use in the design / analysis process
Determination of locations in structures where large deformationsand large stresses occur is generally important
Post processor computer programs help the user to interpret the results by displaying them in graphical form