ame2503

PEMP

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

PEMP

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


PEMP

AME2503

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


9. Structure as a combination of springs

PEMP

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


2: to draw, lay out, or prepare a design http://www.m-w.com/dictionary/design

PEMP

AME2503

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


what the product has to accomplish

PEMP

AME2503

“So many things (to design) – So little time”


PEMP

AME2503

In the beginning, there was product development

ConceptObservations Trials


ConceptObservations Trials

PEMP

AME2503

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

k d i i Q ifi i M.S. Ramaiah School of Advanced Studies, Bengaluru 8

How to assess? How to make a decision? Quantification

PEMP

AME2503

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


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

PEMP

AME2503

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

),....,,,(


i 1

Rational behaviour of consumer

PEMP

AME2503

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


nn

( p ) ybe evaluated in a finite number of standard operations.

---- http://www.riskglossary.com/link/closed_form_solution.htm

PEMP

AME2503

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


EIdxyEdxyM

PEMP

AME2503

Complexities in Engineering Problems

P Transient Harmonic Random

t

P

t

P


t t

PEMP

AME2503

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


. 0

n

baTNewton’s law: mFtx /)(

PEMP

AME2503

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


value of y at b is matched -0.5

PEMP

AME2503

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


Solve for yi, i=1,2,3,….,n

PEMP

AME2503

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


Leads to a set of n algebraic equations for the unknowns ai

PEMP

AME2503

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


n.,1,2,3,....i 0)()( i

i

i

dxxRdxWxR

PEMP

AME2503

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


)( xa

W ii

i

PEMP

AME2503

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


PEMP

AME2503

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


interrelationships in making up a whole.

PEMP

AME2503

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


2 1024 3.141587725 3.14157294

PEMP

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)


PEMP

AME2503

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


problemor after some calculations

PEMP

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?


PEMP

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


PEMP

AME2503


PEMP

AME2503

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


loads loads & is load path dependent

PEMP

AME2503

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


g y yBio-medial applications, weapons, weather

PEMP

AME2503Mathematical Model Approach

Starting Point: gDE/ PDE in space/time

Modelling: UsingUsing Variational or Weak form


PEMP

AME2503Physical Model Approach

Starting Point: gPhysical System

Modelling:Modelling: Idealisation and Discretization


PEMP

AME2503


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


1963Structural Analysis by Digital Simulation of Analog Methods (SADSAM) developed in NASA

http://www.asme.org/Communities/History/Resources/Interactive_Timeline.cfm

PEMP

AME2503

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



PEMP

AME2503

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



PEMP

AME2503Finite Element Method -- Idealisation

Idealisation


PEMP

AME2503

Finite Element Method -- Discretisation


PEMP

AME2503

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


Design improvements/optimization

PEMP

AME2503Finite Element Analysis -- Process


PEMP

AME2503Finite Element Analysis -- ProcessMathematical model Governed by by differential

equations

Assumptions on• Geometry

Ki i• Kinematics• Material law• LoadingLoading• Boundary conditions • etc


PEMP

AME2503

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


PEMP

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


bounds.

PEMP

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.


7. Performs solution procedures.

PEMP

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


values of the primary derived variables.

PEMP

AME2503

FEM FEM –– IdealisationIdealisationFEM FEM –– IdealisationIdealisation

Idealisation


PEMP

AME2503

FEM FEM –– DiscretisationDiscretisationFEM FEM –– DiscretisationDiscretisation


PEMP

AME2503

FEM FEM –– Assembly & SolutionAssembly & SolutionFEM FEM –– Assembly & SolutionAssembly & Solution


PEMP

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


PEMP

AME2503

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


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

PEMP

AME2503

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.


PEMP

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


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


Normal Skew

PEMP

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


conditions are of order m to 2m 1

PEMP

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


PEMP

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


8. Displacement approach and shape functions

PEMP

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?


(Strength of Materials, Mechanics of Deformable Bodies, Solid Mechanics, ….)

PEMP

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


What parameter to compare for design assessment?

PEMP

AME2503StressStressAs internal reaction/resistance to applied load

PP P= P/A

PP

M M

P

M

P


PEMP

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


Stress at a point or Stress on an area?

PEMP

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

PEMP

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


2 y

x

PEMP

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


yxy

PEMP

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


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


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


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


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


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


llAP / /

PEMP

AME2503Material Behaviour Material Behaviour –– Theories of FailureTheories of Failure

y

x xy

xy

Fn

Mn?

y

xxyxy

n?


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/


PEMP

AME2503

Material Behaviour Material Behaviour –– Theories of FailureTheories of Failure

(Rankine Theory)

(Coulomb’s Theory)

(St Vnenamt’s Theory)(St. Vnenamt s Theory)



PEMP

AME2503

Material Behaviour Material Behaviour –– Theories of FailureTheories of Failure



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


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 ``


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 ''


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


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


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)


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


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


)( 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)


• Differential cooling rates (locked in stresses)

PEMP

AME2503

Structural Problem and FE


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


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?


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

“Breaking up” of the structure


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


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


ame2503

Documents