finite element method (FEM) & Structural modelling and fem analysis

1. NAST-621: COMPUTATIONAL METHODS FOR MODELING, DESIGN
AND SIMULATIONS
TITLE: FINITE ELEMENT METHOD (FEM)
& STRUCTURAL MODELLING AND FEM ANALYSIS
COURSE INSTRUCTOR:
DR.P.THANGADURAI
ASSISTANT PROFESSOR
Centre for Nano Sciences & Technology
Madanjeet School of Green Energy Technologies
PRESENTED BY,
MUGILAN N
M.TECH NAST 1ST year
Reg no:16305012

2. BASIC CONCEPT
• THE FINITE ELEMENT METHOD (FEM) IS ALSO KNOWN AS THE FINITE ELEMENT
ANALYSIS (FEA).
• IT IS BASED ON THE IDEA OF BUILDING A COMPLICATED OBJECT WITH SIMPLE
BLOCKS OR DIVIDING A COMPLICATED OBJECT INTO SMALL AND MANAGEABLE
PIECES.
• APPLICATION OF THIS SIMPLE IDEA CAN BE FOUND EVERYWHERE IN EVERYDAY
LIFE, AS WELLAS IN ENGINEERING.
EXAMPLES:LEGO (KIDS’ PLAY),BUILDINGS,APPROXIMATION OF THE AREA OF A
CIRCLE.
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3. ELEMENT SiAREA OF ONE TRIANGLE: Si=1
2R2 SIN 𝜃
AREA OF THE CIRCLE: Sn=
• WHERE N = TOTAL NUMBER OF TRIANGLES (ELEMENTS)
• OBSERVATION: COMPLICATED OR SMOOTH OBJECTS CAN BE REPRESENTED BY
GEOMETRICALLY SIMPLE PIECES (ELEMENTS).
𝑖=1
𝑁
𝑠𝑖 = 1
2R2 N sin
2𝜋
𝑁
Basic equation
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4. WHY FINITE ELEMENT METHOD?
• DESIGN ANALYSIS: HAND CALCULATIONS, EXPERIMENTS, AND COMPUTER
SIMULATIONS
• FEM/FEA IS THE MOST WIDELY APPLIED COMPUTER SIMULATION METHOD IN
ENGINEERING
• CLOSELY INTEGRATED WITH CAD/CAM APPLICATIONS
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5. APPLICATIONS OF FEM IN ENGINEERING
• Mechanical/Aerospace/Civil/Automobile Engineering
• Structure analysis (static/dynamic, linear/nonlinear)
• thermal/fluid flows
• Electromagnetics
• Geomechanics
• Biomechanics
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6. FEM IN STRUCTURAL ANALYSIS (THE PROCEDURE)
• DIVIDE STRUCTURE INTO PIECES (ELEMENTS WITH NODES)
• DESCRIBE THE BEHAVIOR OF THE PHYSICAL QUANTITIES ON EACH ELEMENT
• CONNECT (ASSEMBLE) THE ELEMENTS AT THE NODES TO FORM AN
APPROXIMATE SYSTEM OF EQUATIONS FOR THE WHOLE STRUCTURE
• SOLVE THE SYSTEM OF EQUATIONS INVOLVING UNKNOWN QUANTITIES AT
THE NODES (E.G., DISPLACEMENTS)
• CALCULATE DESIRED QUANTITIES (E.G., STRAINS AND STRESSES) AT
SELECTED ELEMENTS
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7. FEM MODEL FOR A GEAR TOOTH
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8. • COMPUTER IMPLEMENTATIONS:
• PREPROCESSING (BUILD FE MODEL, LOADS AND CONSTRAINTS)
• FEA SOLVER (ASSEMBLE AND SOLVE THE SYSTEM OF EQUATIONS)
• POSTPROCESSING (SORT AND DISPLAY THE RESULTS)
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9. TYPES OF FINITE ELEMENTS
1-D (Line) Element
(Spring, truss, beam, pipe, etc.)
2-D (Plane) Element
(Membrane, plate, shell, etc.)
3-D (Solid) Element
(3-D fields - temperature, displacement, stress, flow velocity)
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10. SPRING ELEMENT
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12. CONSIDER THE EQUILIBRIUM OF FORCES FOR THE SPRING. AT NODE I,
WE HAVE
• Fi= - F = - K (Uj-Ui ) =KUi – KUj
AND AT NODE J,
Fj= F = K (Uj-Ui ) =-KUi + KUj
In matrix form:
𝑘 −𝑘
−𝑘 𝑘 𝑢𝑖
𝑢𝑗
=
𝑓𝑖
𝑓𝑗
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13. HENCE,
KU =F
• K = (ELEMENT) STIFFNESS MATRIX
• U = (ELEMENT NODAL) DISPLACEMENT VECTOR
• F = (ELEMENT NODAL) FORCE VECTOR
NOTE THAT K IS SYMMETRIC
SPRING ELEMENTS
• SUITABLE FOR STIFFNESS ANALYSIS
• NOT SUITABLE FOR STRESS ANALYSIS OF THE SPRING ITSELF
• CAN HAVE SPRING ELEMENTS WITH STIFFNESS IN THE LATERAL
DIRECTION, SPRING ELEMENTS FOR TORSION, ETC. 4/4/2017(FEM) 13

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18. PROBLEM 1:
GIVEN, FOR THE SPRING SYSTEM SHOWN ABOVE,
K1 = 100 N / MM,
K2 = 200 N / MM,
K3 = 100 N / MM
P = 500 N, U1 = U4 = 0
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19. TO FIND: .THE GLOBAL STIFFNESS MATRIX
• DISPLACEMENTS OF NODES 2 AND 3
• THE REACTION FORCES AT NODES 1 AND 4
• THE FORCE IN THE SPRING 2
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25. THANK YOU
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