Download - AME2503
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AME2503
P li i iPreliminaries
Session delivered by:Session delivered by:
Dr Vinod K BanthiaDr Vinod K BanthiaDr. Vinod K. BanthiaDr. Vinod K. Banthia
M.S. Ramaiah School of Advanced Studies, Bengaluru 1
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AME2503
• Session Objectives– At the end of this session the delegate would
have understood the• Need for analysis• Approach to solving complex problems• Finite Element Method
B i t i l d d fi iti• Basic terminology and definitions• Strength of material approach to design• Variables of interest in structural problemVariables of interest in structural problem• Application of “analysis” approach to structures
M.S. Ramaiah School of Advanced Studies, Bengaluru 2
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Session TopicsSession Topics1. Design process and need for analysis tools
2. Different methods and need for numerical methods
3. Analysis approach to solving complex problems
4. Finite element analysis approach, its steps and key concepts
5. Basics of Mechanics of Materials
6. Stress, Strain, their relation and material failure
7 Component level analysis in SOM7. Component level analysis in SOM
8. Typical structural problem
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9. Structure as a combination of springs
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AME2503
Why?The Need
What does it help one accomplish?
Design
What is design?What is design?
1: to create fashion execute or construct according to plan : devise contrive1: to create, fashion, execute, or construct according to plan : devise, contrive2 a: to conceive and plan out in the mind b: to have as a purpose : intend c: to devise for a specific function or end3archaic : to indicate with a distinctive mark, sign, or name, g ,4 a: to make a drawing, pattern, or sketch of b: to draw the plans forintransitive verb1: to conceive or execute a plan
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2: to draw, lay out, or prepare a design http://www.m-w.com/dictionary/design
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Design
Checking the behaviour under operating conditionsChecking the behaviour under operating conditions
Must know/understand the behaviour before one can proceed to designbefore one can proceed to design
Must know/understand what the product has to accomplish
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what the product has to accomplish
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“So many things (to design) – So little time”
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In the beginning, there was product development
ConceptObservations Trials
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ConceptObservations Trials
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What is involved in product development?• Understanding of functional requirements
• Understanding of operational requirements
• Development of conceptual design
• Development of geometrical design
• Assessment of the design • Hand calculations • Testing • Simulation
“A simulation is an imitation of some real thing, state of affairs, or process”http://en.wikipedia.org/wiki/Simulation
Imitation of Physics
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How to assess? How to make a decision? Quantification
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Some Definitions
“Mechanics is the branch of physics concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effect of the bodies on their environment.” http://en.wikipedia.org/wiki/Mechanics
“Engineering is the applied science of acquiring and applying knowledge to design, analysis, and/or construction of works for practical purposes.”
http://en.wikipedia.org/wiki/Engineering
Quantification
Quantification “covers all those acts which quantify observations and experiences by converting them into numbers through counting and measuring. It is thus the basis for mathematics and for science.” http://en.wikipedia.org/wiki/Quantification
Mathematical modelling
A mathematical model is an abstract model that uses mathematical language
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A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system http://en.wikipedia.org/wiki/Mathematical_model
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Mathematical Models
dtvmdF )(
dxdTkq uc
tu 222
2
)2l (
)2ln(2/1 t )( BvEqF
fTpvv
tv
rtoePtP )(
Malthusian growth modeltxxCxxptx )()(),(
niPVFV )1( Compound InterestxxxxUtilityOptimise )(
txxCxxptx )()(),(Profit maximisation
Compound Interest
n
iii
n
Budgetquantityprice
xxxxUtilityOptimise
1
321
)()( subject to
),....,,,(
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i 1
Rational behaviour of consumer
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Closed Form Solutions
mk
m
xxx , ,
k
1 kx xm
(Restoring Force) (Inertia Force)
0 xmkx 0 kxxm 0xmkx 0kxxm
tBtAtxAssume sincos)( ps rmk
n )(
)cossin()( tBtAtx
)()sincos()( 22 txtBtAtx
m
Hzkfn 21
)()sincos()( txtBtAtx
0)( 2 xkmkxxm sec2
n
mn 2
A closed-form solution (or closed form expression) is any formula that can
M.S. Ramaiah School of Advanced Studies, Bengaluru 11
nn
( p ) ybe evaluated in a finite number of standard operations.
---- http://www.riskglossary.com/link/closed_form_solution.htm
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Solutions with Simplifying Assumptions
X u(x y)
Y,v(x)
P
q(x) X,u(x,y) Mq( )
2
yvyxxvyxu
)(),(
yvyxvy
xu
2
2
v2
EyI
M
EyxvEyE
2
EIdxyEdxyM 2
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EIdxyEdxyM
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Complexities in Engineering Problems
P Transient Harmonic Random
t
P
t
P
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t t
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Boundary Value Problems
)(12
2
21
1
1 xhyadxdya
dxyda
dxyda
dxyd
nnn
n
n
n
n
n
dxdxdxdx
General solution contains ‘n’ arbitrary constantsRequires ‘n’ specified conditions for unique solution
Initial Value Problems Boundary Value Problems• Statement of differential equation • Statement of differential equation
q p q
Statement of differential equation
• Specified value of the unknown
))(,()( tytfty • Statement of differential equation
02
2
2
2
yT
xT
pfunction at a given point in the domain
0000 )( )( ytyyty
• Specified value of the unknown and/or its derivative all around the boundary
0
TbaT
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. 0
n
baTNewton’s law: mFtx /)(
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Solution Techniques for Boundary Value ProblemsShooting Method:
),,()( yytfty )( )( byay
At ‘a’, guess a value
a b
, gof )(ty
It may not lead to 2
2.51 0.8 0.6 0.4 0.2
correct value of y at b
T f1
1.5
Try sequence of increasingly accurate guesses until specified 0
0.5
0 0.2 0.4 0.6 0.8 1 1.2
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value of y at b is matched -0.5
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Fi i Diff M h dSolution Techniques for Boundary Value Problems
Finite Difference Method:),,()( yytfty )( )( byay
iyy 1iy
2iy
2iy1iy
3iy
3iy 2iy
i 1i 2i 3i1i2i3i
h
i 1i 2i 3i1i2i3i
hyyty ii
i 2)( 11
211 2)(
hyyyty iii
i
),,(22
112
11
hyyytf
hyyy ii
iiiii
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Solve for yi, i=1,2,3,….,n
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Solution Techniques for Boundary Value ProblemsWeighted Residual Method
)())(( xpxuDT l )())(( xpxuD To solve
n
iiauxu ~)( Assume solution to be
i
iiauxu1
)( Assume solution to be
0))())(~(( xpxuDResidual R(x)
Goal is to force this residual to zero in some average sense over the domain.
n.,1,2,3,....i 0)(
dxWxR i
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Leads to a set of n algebraic equations for the unknowns ai
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Solution Techniques for Boundary Value Problems
Collocation method:
Forcing residual to be zero at finite number of pointsg p
0 1
)(i
iii xfor x
xfor xxxW
At the points, Wi is non-zero, forcing R(xi)=0
Sub-domain method:
Forcing residual to be zero over various sub-sections of the domain by setting weighting functions to unity over the sub-domains
n1 2 3i0)()( dxxRdxWxR
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n.,1,2,3,....i 0)()( i
i
i
dxxRdxWxR
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Solution Techniques for Boundary Value ProblemsLeast Square method:
Summation of all the squared residuals is minimised
0)()()( 2
dxxRdxxRxRS
For minimisation of S
)( )(20
dxWxRdx
aRxR
aS
iii
Galerkin method:
Similar to Least Square methodDerivative of approximating function is used as the weighting function
)(~uW
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)( xa
W ii
i
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Solution Techniques for Boundary Value Problems
Ritz Method
P i i l f Vi t l W k
Close Form Solutions
Finite Element MethodPrinciple of Virtual Work:
Principle of Min. Potential Energy:
Finite Element Method
Finite Difference Method
Variational Principle: Boundary Element Method
Finite Volume Method
Spectral MethodSpectral Method
Mesh-Free Method
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How to deal with Complexity?
Divide each difficulty into as many parts i f ibl d t l itas is feasible and necessary to resolve it.
- Rene Descartes
“Divide and Conquer”AnalysisAnalysis The separation of an intellectual or material whole i i i f i di id linto its constituent parts for individual
The study of such constituent parts and their
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interrelationships in making up a whole.
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ApplicationFinding the perimeter and area of a circle using a straight edge measureFinding the perimeter and area of a circle using a straight edge measure
n Perimeter/2R Area0 0 01 1.22515E-16 -1.22515E-162 2 1.22515E-164 2 828427125 2
R4 2.828427125 28 3.061467459 2.828427125
16 3.121445152 3.061467459
)/sin(RP ii 22 32 3.136548491 3.12144515264 3.140331157 3.136548491
128 3 141277251 3 140331157
ii sinRA 2
21
128 3.141277251 3.140331157256 3.141513801 3.141277251512 3.14157294 3.141513801
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2 1024 3.141587725 3.14157294
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AME2503Steps in Analysis
• Understand the nature (underlying Physics) of the problem(Representation of area and “perimeter” of the small part)
• Try to mimic the behaviour of a small entity
(Representation of area and “perimeter” of the small part)
• Try to mimic the behaviour of a small entity(Calculation of area and “perimeter” of small part)
• Through a collection of small entity, mimic the behaviour of the whole quantityof the whole quantity
(Calculation of area and perimeter of full part)
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Extension to Finite Element Method• Write down the equation • Understand the nature• Write down the equation for calculation of quantities of interest
• Understand the nature (underlying Physics) of the problem
• Try to mimic the behaviour to be modelled over small
• Use the formula to calculate the quantities of
entities
• Through collection of ll i i i i h
interest over small regions
• Add up the value of the small entities, mimic the behaviour of the whole quantity
Add up the value of the quantities to be calculated over all the small elements q y
• Solve the simplified representation of the
• Find the value of the quantities of interest directly
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problemor after some calculations
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AME2503How to apply this to structural ProblemsProblems
On what basis to break it up? What parameter should be looked at?p p
How to apply the methodology here?Is this a new methodology?Is this a new methodology?
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AME2503Finite Element Analysis -- Process
SYSTEMS
CONTINOUSResponse is described by
DISCRETEResponse is described byResponse is described by
variables at an INFINITEnumber of points.
Response is described by variables at a FINITE
number of points.
Set of differential equations Set of algebraic equations
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Computational Continuum MechanicsStatics Dynamics
Study of phenomena with negligible inertia component
Study of phenomena with i ifi i inegligible inertia component
• Static
significant inertia component(time dependence is explicitly
considered) •Quasi-static
)
Linear Non-LinearResponse of the structure is proportional to the applied
loads
Response of the structure is not proportional to the applied
loads & is load path dependent
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loads loads & is load path dependent
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Computer Aided EngineeringAnalyse the functional and performance characteristics
Structural: Fluids Flow:Structural:StrengthStiffnessDurability
PressureVelocityTemperatureDurability
(under given loads)
TemperatureMixingThermal
Kinematics Characteristics(under given motion input)
Other application areas:Acoustics, electro-magnetics, multibody dynamics
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g y yBio-medial applications, weapons, weather
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AME2503Mathematical Model Approach
Starting Point: gDE/ PDE in space/time
Modelling: UsingUsing Variational or Weak form
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AME2503Physical Model Approach
Starting Point: gPhysical System
Modelling:Modelling: Idealisation and Discretization
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PEMP
AME2503Finite Element Analysis – A brief history
1943 Richard Courant proposes breaking a continuum problem into triangular regions with piecewise approximation of field variable
history
1945 Electronic digital computer gains usage
1952 B. Langefors describes global behaviour as assimilation of l l b h i P bl ith 100 d f b i l dlocal behaviour. Problems with 100 dof are being solved
1953 N.J.Turner develops triangular plane stress modelTurner Clough Martin and Topp publish paper describing
1959 Development of irregularly shaped cells
1956 Turner, Clough, Martin and Topp publish paper describing determination of FE properties using direct stiffness method
1960 The new method is christened Finite ElementStructural Analysis by Digital Simulation of Analog Methods
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1963Structural Analysis by Digital Simulation of Analog Methods (SADSAM) developed in NASA
http://www.asme.org/Communities/History/Resources/Interactive_Timeline.cfm
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Finite Element Analysis – A brief history1963
Structural Analysis by Digital Simulation of Analog Methods (SADSAM) developed in NASA
y y
NASA Structural Analysis Program (NASTRAN) developedFirst conference on Matrix Methods in Structural MechanicsPortable desktops and mocrocomputers debut
1965Portable desktops and mocrocomputers debutVariational form developed opening FEM to problems other than structural mechanics (Zienkiewicz and Cheung)
First public release of NASTRAN ANSYS grows out of
1976 CRAY supercomputer developed
1971First public release of NASTRAN, ANSYS grows out of Westinghouse, Interactive pre and post processor programs appear,use of FEM in automotive industry begins
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http://www.asme.org/Communities/History/Resources/Interactive_Timeline.cfm
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Finite Element Analysis – A brief history1980
Use of FE for diverse problems, growth in model generation and result display, development in non-linear arena, PC debuts
y y
1990 Use of FE for CFD and other non-structural applications increases
1994 MSC/NASTRAN for Windows released
1998Fe is being used for the solution of fully coupled problems, applications in biomedical research
1999 Growth in CFD applications
2000+ Application to more complex problems
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http://www.asme.org/Communities/History/Resources/Interactive_Timeline.cfm
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AME2503Finite Element Method -- Idealisation
Idealisation
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Finite Element Method -- Discretisation
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Ch f
Finite Element Analysis -- ProcessPhysical problem Change of
physical problem
I h i lMathematical model Improve mathematical model
FE SolutionRefine mesh, solution
FE solution of mathematical
parameters etc.Assessment of accuracy
model
Interpretation of results Refine analysis
Design
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Design improvements/optimization
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AME2503Finite Element Analysis -- Process
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AME2503Finite Element Analysis -- ProcessMathematical model Governed by by differential
equations
Assumptions on• Geometry
Ki i• Kinematics• Material law• LoadingLoading• Boundary conditions • etc
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Fi it l tTHE PROCESS OF FINITE ELEMENT ANALYSIS
Finite element solution
Choice of • Finite element• Mesh density
S l ti t• Solution parametersRepresentation of• Loadingg• Boundary conditions • etc
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AME2503
Start Stop IMPLEMENTATION OF FINITE ELEMENT
Problem Analysis and design decision
O N NANALYSIS
Preprocessor1 Reads control
Postprocessor1. Prints and plots contours for
1. Reads control parameters.
2. Reads or generates nodal or
state variables.
2. Returns element domain & calculates flux & othercoordinates.
3. Reads or generates element data. Pr
oces
s calculates flux & other variables.
3. Prints and plots contours for flux and other physical
4. Reads material constants.
5. Reads boundary conditions
flux and other physical conditions.
4. Evaluates and prints error
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bounds.
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AME2503
Start Stop IMPLEMENTATION OF
Problem Analysis and
design decision Processor
N ON OFINITE ELEMENT ANALYSIS
Preprocessor
Processor1. Generates element shape functions. Postprocessor2. Calculates master element equations.3. Calculates transformation
p
matrices.4. Maps element equations into global system.5. Assembles element equations.6. Introduces boundary conditions.
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7. Performs solution procedures.
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AME2503Six Steps of FEA Procedure
Creation of Finite Element model (Pre-Processing)Creation of Finite Element model (Pre-Processing)a) Idealize the structure and discretize it into a collection of Elements connected at Nodes.b) S if M t i l ti B d diti db) Specify Material properties, Boundary conditions and Loading conditions.
Analysis with the Finite Element Program (Solver)a) Generate element stiffness equations and assemble them.b) Modify global equations to suit BC and Loads) y g qc) Solve resulting set of equations for primary unknowns.
Finite Element Program Results (Post Processing)Finite Element Program Results (Post-Processing)a) Using the nodal values of primary unknowns
(displacement, temperature etc.) calculate and check the al es of the primar deri ed ariables
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values of the primary derived variables.
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FEM FEM –– IdealisationIdealisationFEM FEM –– IdealisationIdealisation
Idealisation
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FEM FEM –– DiscretisationDiscretisationFEM FEM –– DiscretisationDiscretisation
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FEM FEM –– Assembly & SolutionAssembly & SolutionFEM FEM –– Assembly & SolutionAssembly & Solution
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AME2503Degree of FreedomMinimum number of independent coordinates required to determine
completely the positions of all parts of a system at any instant of time
Discrete and Continuous SystemsSystems
• Systems with a finite number of degrees of freedom are called discrete or lumped parameter systemsdiscrete or lumped parameter systems
• Systems with an infinite number of degrees of freedom are called continuous or distributed systems
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Distributed (Continuous) ( )and Lumped (Discrete) Systems
1
1.2
1.2
0
0.2
0.4
0.6
0.8
0.4
0.6
0.8
1
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http://www.zentech.co.uk/zencrack_example_helicopter.htm
00 0.5 1 1.5 2 2.5 3 3.5
0
0.2
0 0.5 1 1.5 2 2.5 3 3.5
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Features of Finite Element methodology
• The piecewise approximation of the physical field (continuum) on finite elements provides good precision even with simple
gy
on finite elements provides good precision even with simple approximating functions. Simply increasing the number of elements can achieve increasing precision.
• The locality of the approximation leads to sparse equation systems for a discretized problem. This helps to ease the solution of problems having very large numbers of nodal unknowns. It is not uncommon today to solve systems containing a millionnot uncommon today to solve systems containing a million primary unknowns.
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AME2503
Governing Equation for Solid Mechanics Problems
[K] { } {F } + {Fth} + {F } + {F } + {F l} + {F } + {F } + {Fld}
• Basic equation for a static analysis is as follows:
Governing Equation for Solid Mechanics Problems
•[K] {u} = {Fapp} + {Fth} + {Fpr} + {Fma} + {Fpl} + {Fcr} + {Fsw} + {Fld}•[K] = total stiffness matrix•{u} = nodal displacement•{Fapp} = applied nodal force load vector•{Fth} = applied element thermal load vector•{Fpr} = applied element pressure load vector{Fpr} applied element pressure load vector•{Fma} = applied element body force vector•{Fpl} = element plastic strain load vector{F } l t t i l d t•{Fcr} = element creep strain load vector
•{Fsw} = element swelling strain load vector•{Fld} = element large deflection load vector
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PEMP
AME2503Boundary Conditions
PK
Representation of interaction with the surroundings
Why are boundary constraints required? What will happen to the body otherwise?
K P
= infinite= finite
H H tHomogeneous Heterogeneous
Skew
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Normal Skew
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AME2503Boundary Conditions
• Geometric or Essential Boundary ConditionsKi i C i Di l B d– Kinematic Constraints or Displacement Boundary Conditions
– Ux,; Uy; Uz; Rx; Ry; Rz;Ux,; Uy; Uz; Rx; Ry; Rz;– The order of the derivative in the essential boundary
conditions is in a Cm-1problem at most m-1• Natural or Forced Boundary Conditions
– Correspond to prescribed forcesF F F M M M– Fx; Fy; Fz; Mx; My; Mz;
– The order of the derivative in these boundary conditions are of order m to 2m-1
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conditions are of order m to 2m 1
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AME2503
St th f M t i lStrength of Materials
Session delivered by:Session delivered by:
Dr Vinod K BanthiaDr Vinod K BanthiaDr. Vinod K. BanthiaDr. Vinod K. Banthia
M.S. Ramaiah School of Advanced Studies, Bengaluru 54
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AME2503
Session Topics1 Basics of Mechanics of Materials1. Basics of Mechanics of Materials
2. Stress, Strain, their relation and material failure
3. Component level analysis in SOM
4. Typical structural problem
5. Structure as a combination of springs
6. Structure idealisation, discretisation and assemblyy
7. Element stiffness formulation
8 Displacement approach and shape functions
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8. Displacement approach and shape functions
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AME2503Mechanics
Study of interaction between physical objects and theStudy of interaction between physical objects and the effect of this interaction on the objects involved.
Theoretical Applied Computational21mm
221
rmmGFG Planetary Motion Routing a satellite
Mechanics of MaterialMechanics of Material
What happens to solid bodies when subjected to various types of loading?
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(Strength of Materials, Mechanics of Deformable Bodies, Solid Mechanics, ….)
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AME2503Mechanics of Materials“Forces make the world turn”
Goal of structural designCreate a structure that will safely withstand the operational loadsCreate a structure that will safely withstand the operational loads
LoadsStatic, Sustained, Dynamic, Impact, Concentrated, Distributed ………
Material subjected to loading
Structure Components Material
Strength: Intrinsic property of ability to carry load
Structure subjected to loadingStrength: Of the material (YS, UTS)Stiffness: Geometric details of the componentsDensity: Weight of the structure
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What parameter to compare for design assessment?
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AME2503StressStressAs internal reaction/resistance to applied load
PP P= P/A
PP
M M
P
M
P
M.S. Ramaiah School of Advanced Studies, Bengaluru 58
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AME2503StressStress: A measure of internal force distribution (reaction of one
part of the body on the other) in a loaded bodyT2 P3
Z P
P
P1Y
Z
nP
Aplane specified aon force internal on theIntesity
T1T
P2X
A
APStress
A
0lim
li 1T3
St t i t St ?
?lim0
AA
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Stress at a point or Stress on an area?
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AME2503
StressZ
nP Z nP
Y
Zn+
X
Yn
X
Yn
X
Yn
tP+=
Normal Shear
PZ
nP Z
nnP
Zn
nP
= =Y
XX
Yn
tP X
Y
PtyP
= = M.S. Ramaiah School of Advanced Studies, Bengaluru 60
ttxP
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AME2503
Transformation of Stress2
2
2,1 22 xyyxyx
yxy 22
tan2 1
xx
xy
xy
2
tan2yx
y
xy
2cos22
2121,
yx
x
y
xyyxy
2sin2
21 xy
x
y
xy
yx
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2 y
x
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AME2503
Combined Stress1. Find stress state for each loading2. Sum up “like” stresses3 Use Mohr’s circle3. Use Mohr s circle
Like Stresses: Normal (bending, axial)Shear (torsion, shear)
T T
McPn IA
n
JTr
APs yxy
xx
xy
xy
M.S. Ramaiah School of Advanced Studies, Bengaluru 62
yxy
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AME2503
lStrain
l lll
dx
dxudx
xuu /)(
udu
u
y
dy v
xu
x
dxx
u
xu
xdx
yy
y
dudy
yu
dyv
Shear Strain = =- xv
yu
xy
vu dx
xvv
dxu
dyy Normal Strain: No distortion, Volume change
Shear Strain: Distortion, no volume change
M.S. Ramaiah School of Advanced Studies, Bengaluru 63
xdx
x, g
PEMP
AME2503Stress-Strain RelationshipLoad Deformation?
Uniqueness for structure?
Load Stress Strain Deformation
q
Stress Strain?Unique for material
Robert Hooke (1676)
iii tt
Simeon Poissony yxlt
Auxetic MaterialsAuxetic MaterialsP
ceiiinossssttuu
Ut tensio, sic vis (1678)
Auxetic MaterialsAuxetic Materials
E1
M.S. Ramaiah School of Advanced Studies, Bengaluru 64
As the extension, so the forcex http://home.um.edu.mt/auxetic/www/properties.htm
PEMP
AME2503
StressStress--Strain RelationshipStrain Relationship E (Hooke’s Law) E
Poisson’s Effect:
(Hooke s Law)
lt x x E
xx
x
EEEzzyyxx
321 ; ;x
yz
zz
xx
Ex
zy
EEE 321 ;;
xx
yyyy
zz
xx
321 xx
zzyyxxxx E
1 xxzzyyyy E
1 yyxxzzzz E
1
1 1 1xyxy G
1 yzyz G
1 zxzx G
1
EG
M.S. Ramaiah School of Advanced Studies, Bengaluru 65
12
G
PEMP
AME2503
Stress-Strain relations 0001
yy
xx
yy
xx
E
21000100010001
yz
xy
zz
yz
xy
zz E
21
02210000
00221000
)21)(1(
zxzx 22100000
Stiffness Representation
zz
yy
xx
zz
yy
xx
000100010001
1
zx
yz
xy
zx
yz
xy E
)1(2000000)1(2000000)1(2000
M.S. Ramaiah School of Advanced Studies, Bengaluru 66
Flexibility Representation
PEMP
AME2503
Different Kinds of Material BehavioursDifferent Kinds of Material BehavioursI t i• Isotropic
Same properties in all directions2 material constants
• OrthotropicProperties have two orthagonal planes of symmetry9 material constants
Composites, rolled sheets
• Transversely OrthotropicSame property in one plane and different normal to it5 material constants
• AnisotropicProperties have no planes of symmetry
5 material constants
M.S. Ramaiah School of Advanced Studies, Bengaluru 67
p p y y21 material constants
PEMP
AME2503
Material BehaviourMaterial Behaviour
u
fy
Elastic limit
iona
lt
E
enin
g
y
Prop
orti
limi
last
ic
yiel
ding
train
har
de
neck
ing
llAP //
e y st n
M.S. Ramaiah School of Advanced Studies, Bengaluru 68
llAP / /
PEMP
AME2503Material Behaviour Material Behaviour –– Theories of FailureTheories of Failure
y
x xy
xy
Fn
Mn?
y
xxyxy
n?
M.S. Ramaiah School of Advanced Studies, Bengaluru 69
Seely, F.B. and Smith, J.O, “Advanced Mechanics of Materials,” Second edition, John Wiley & Sons, Inc., New York
PEMP
AME2503
http://web.utk.edu/~prack/mse201/Chapter%208%20Failure.pdf http://web.utk.edu/~prack/mse201/Chapter%208%20Failure.pdf
http://www-outreach.phy.cam.ac.uk/physics_at_work/2005/exhibit/matsci.php
http://www.aloha.net/~icarus/
M.S. Ramaiah School of Advanced Studies, Bengaluru 70
PEMP
AME2503
Material Behaviour Material Behaviour –– Theories of FailureTheories of Failure
(Rankine Theory)
(Coulomb’s Theory)
(St Vnenamt’s Theory)(St. Vnenamt s Theory)
M.S. Ramaiah School of Advanced Studies, Bengaluru 71
Seely, F.B. and Smith, J.O, “Advanced Mechanics of Materials,” Second edition, John Wiley & Sons, Inc., New York
PEMP
AME2503
Material Behaviour Material Behaviour –– Theories of FailureTheories of Failure
M.S. Ramaiah School of Advanced Studies, Bengaluru 72
Seely, F.B. and Smith, J.O, “Advanced Mechanics of Materials,” Second edition, John Wiley & Sons, Inc., New York
PEMP
AME2503
Components Subjected to Axial LoadingComponents Subjected to Axial LoadingP Assume
AEP
E 1
A, L E
=
P LL
P i ti M b (U if C ti )
• Loads/Weight act only along the axis of the element
P
PAEPLL
• Prismatic Member (Uniform Cross-section)
• Loads applied at the centroid of the cross-section
• Applied Loads are constant
• Cross-section remains plane
PPP
Deformation is uniformStresses and Strains are constant
• Applied Loads are constant
• Material is homogeneous and isotropic
• Poisson’s effect is neglected
• Buckling effect is not consideredBuckling effect is not considered
M.S. Ramaiah School of Advanced Studies, Bengaluru 73
PEMP
AME2503
Components Subjected to Bending LoadingComponents Subjected to Bending LoadingP
E, L, I
( ) P
X,u(x,y)
Y,v(x)
M
P
q(x) ( y)M
yvyvyu
2
2
yvyxvyxu
)(),( yyx
yx 2
yyx
y
)(
EyxvEyE
2
2
EIdxyEdxyM 2
=Mc/I
=Mc/2I=P/2a ``
M.S. Ramaiah School of Advanced Studies, Bengaluru 74
Mc/2I
PEMP
AME2503
Components Subjected to Bending LoadingComponents Subjected to Bending LoadingPP
E, L, IShearing Bending
M M
+ ++
– –
dVVdMMddw qdx
Vdx
EIdx
dx
w
EMyVadMya ''
M.S. Ramaiah School of Advanced Studies, Bengaluru 75
yI
Ity
dxIty
PEMP
AME2503
Components Subjected to Bending LoadingComponents Subjected to Bending LoadingSt ti l d• Static loads
• Stress is purely a result of external loads• Bending is the predominant deformation pattern• Bending is the predominant deformation pattern• Bending is not accompanied by twisting of the cross-section• Beam is subjected to pure bendingBeam is subjected to pure bending• Initial configuration of the beam is straight• Cross-section does not change abruptlyg p y• Material follows Hooke’s law• Material is homogeneous • Point of interest is far from the point of loading and constraint
M.S. Ramaiah School of Advanced Studies, Bengaluru 76
PEMP
AME2503
Components Subjected to Torsional LoadingComponents Subjected to Torsional Loadingl
T
l
T
r
A l f i ( )• Prismatic member
Angle of twist = (x)l/ Pure TorsionlG / Stress-Strain
• Torque applied at the end• Small angle of twist
N h i “ ”GStress StraindAlGT )/(* Equilibrium
2
2r
• No change in “r”• Cross-section twists as rigid body
0 0
2 )/()/( p lGIdAlGT
GT
M.S. Ramaiah School of Advanced Studies, Bengaluru 77
lrJ
PEMP
AME2503
Components Subjected to Torsional LoadingComponents Subjected to Torsional Loading
Beams – Torsionx
yz
T
T LG
JT
r
L
xy
zT b )/(3/ 3GbtTL 3/3btJ
Tt)/(6 3btTy )/(3 2
max btT
3)(/3/ tbGTL i is shape coefficientC(1 1) Z(1 17)
M.S. Ramaiah School of Advanced Studies, Bengaluru 78
3)/(6 tbTy i C(1.1), Z(1.17),
Angle(0.83), T(1.00)2max )/(3 tbT i
PEMP
AME2503
Components Subjected to Internal PressureComponents Subjected to Internal PressureThin walled (r>5t)Thin walled (r>5t)
22 rprtl
prpr
dxrpdxtt 22
prprprlt
tpr
tpr
ll
2
2
tpr
tpr
tt
22t
t42max
2 rprtt
tpr
t 2
M.S. Ramaiah School of Advanced Studies, Bengaluru 79
t2
http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/pressure_vessel.cfm
PEMP
AME2503
Components Subjected to Internal PressureComponents Subjected to Internal PressureThick walledThick walled
22
22222 /)
io
iooiooiir rr
rpprrrprp
22
22222 /)
io
iooiooiit rr
rpprrrprp
22
22
io
ooiil rr
rprp
)(1lrtt E
1 )(1lttrt E
)(1
ltv 2
M.S. Ramaiah School of Advanced Studies, Bengaluru 80
)( trll E
PEMP
AME2503
Components Subjected to Thermal LoadingComponents Subjected to Thermal Loading
,lT
lTA,E
P Pl
P P
PlTl TEP
AETl TE
AT
• Temperature increase under constraintsp• Thermal gradient in a component• Phase change during solidification (locked in stresses)
M.S. Ramaiah School of Advanced Studies, Bengaluru 81
• Differential cooling rates (locked in stresses)
PEMP
AME2503
Extension to Finite Element Method• Write down the equation for calculation of quantities of interest
• Understand the nature (underlying Physics) of the problemof interest
• Try to mimic the behaviour to be modelled over small
problem
• Use the formula to calculate the quantities of to be modelled over small
entities
• Through collection of
calculate the quantities of interest over small regions
Add th l f th gsmall entities, mimic the behaviour of the whole quantity
• Add up the value of the quantities to be calculated over all the small elements quantity
• Solve the simplified representation of the
• Find the value of the quantities of interest directly
M.S. Ramaiah School of Advanced Studies, Bengaluru 83
representation of the problem
quantities of interest directly or after some calculations
PEMP
AME2503
How to apply this to structural Problems
On what basis to break it up? What parameter should be looked at?
How to apply the methodology here?Is this a new methodology?
M.S. Ramaiah School of Advanced Studies, Bengaluru 84
PEMP
AME2503
Problems in Structural Mechanic
To determine the Deformation () in a structure of Stiffness (K)
bj t d t L d (P) M.S. Ramaiah School of Advanced Studies, Bengaluru 85
subjected to Load (P)
PEMP
AME2503
Typical Problem in Solid Mechanics
PP
To determine the Deformation () in a structure of stiffness (K) subjected to Load (P)
K = P
K=? M.S. Ramaiah School of Advanced Studies, Bengaluru 86
PEMP
AME2503
Summary• Needs and methods for quantification in design have been explained• Mathematical modals and their solution have been explained• Analysis method for complicated problems have been explained• Finite element approach and steps have been described
B i d fi iti d t i l f fi it l t d h b• Basic definitions and terminology for finite element used have been introduced• Basic parameters stress and strains used in strength of material haveBasic parameters, stress and strains, used in strength of material have been defined• Types of material behaviour under load and at failure have been ypdescribed•Behaviour of typical structural members has been revised
M.S. Ramaiah School of Advanced Studies, Bengaluru 88
PEMP
AME2503
SummarySummary• Basic parameters, stress and strains, used in strength of material have been defined• Types of material behaviour under load and at failure have been describedbeen described• Behaviour of typical structural members has been revised• Implementation of FE approach has been explainedImplementation of FE approach has been explained• Variables of consequence have been identified
M.S. Ramaiah School of Advanced Studies, Bengaluru 89