course book mechanics of materials and fea aa

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Course BookMechanics of Materials and Finite

Element Method

Lecturer: Assist .Prof.Dr.Basim M. Fadhel

Coordinator:

3rd

Stage

Petroleum Engineering Department

School Of Chemical and Petroleum Engineering

Engineering Faculty

Koya University


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Course overviewMechanics of materials is a basic engineering subject that must be understood

by anyone concerned with the strength and physical performance of structures,

whether those structures are man-made or natural. The subject matter includes

such fundamental concepts as stresses and strains, deformations anddisplacements, elasticity and inelasticity, strain energy, and load-carrying

capacity. These concepts underlie the design and analysis of a huge variety of

mechanical and structural systems.

Finite element method (FEM) is a numerical approach by which partial

differential equations can be solved approximately. From an engineering

standpoint, the FEM is a method for solving engineering problems such as

stress analysis, heat transfer, fluid flow and electromagnetics by computer

simulation.

It is intended that this course provide the student with a clear and through

presentation of the theory and applications of the principles of mechanics of

materials and finite element method.

1-Mechanics of materials,R.C.Hibbeler2-A First Course in Finite Elements ,Jacob Fish


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Lecture SchedulesWeeks Contents

Mechanics of Materials (1st

Course)

1Stress, Introduction, Equilibrium Of Deformable Body ,Stress,

Normal Stress, Shear Stress, Allowable Stress

2 Strain, Deformation, Strain.

3Mechanical Properties Of Materials, Tension And Compression

Test, The StressStrain Diagram.

4

StressStrain Behavior Of Ductile And Brittle Materials,

Hookes Law, Strain Energy, Poissons Ratio. The Shear Stress

Strain Diagram.

5Axial Load, Saint-Venants Principle, Elastic Deformation Of An

Axially Loaded Member, Principle Of Superposition.

6Thermal Stress, Stress Concentration, Inelastic Axial

Deformation, Residual Stresses.

7Torsion,Torsional Deformation Of A Circular Shaft,Torsion

Formula,Power Transmission .Angle Of Twist

8 Stress Concentration, Inelastic Torsion,Residual Stress

9Bending, Shear And Moment Diagram, Flexure

Formula,Composite Beams,

10Combined Loading, ThinWalled Pressure Vessels, State Of

Stress Caused By Combined Loading11

Plane-Stress Transformation, Principal Stress ,Mohrs Circle-

Plane Stress

12Buckling Of Columns, Critical Load Ideal Column With Pin

Supports.

13Columns Having Various Types Of Supports, Secant Formula

,Inelastic Buckling

14Energy Method, External Work And Strain Energy, Conservation

Of Energy

15 Impact Loading, Principle Of Virtual Work

FEM( 2n

Course)

16Introduction,\basic concepts, why FEM, applications of FEM in

Eng.,FEM in Structural analysis, Objective of This course.

17Review of matrix algebra, spring Element ,one spring

element,spring system,

18,19Bar and beam element (linear static analysis, bar element

,stiffness matrix ,direct method,formal approach

20,21Distributed load, bar element in 2D and 3D,stiffness matrix in the

2D space

22,23 Beam element, direct method ,formal approach,3D beam element

24 FE analysis of frame structure,

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2D Problems,plane 2D problems,stress-strain

temp.relations,strain and displacement relations,boundary

conditions

27,28

FE for 2D problems,general formula for the stiffness matrix,linear

strain triangle ,linear quadrilateral element,quadratic quadrilateralelement.

29Equation solving,gauss elimination ,iterative method, nature of

FE solutions


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Exams:There will be two exams (each at the end of first and second semester) and one final.

General instructions and commandments:

1-Not eligible for the student to enter the lecture after the professor.

2-The student is responsible for any oral or written notes that mention

inside the lecture hall.

3- It is not allowed to the student to borrow a pen or calculator or

anything during the exam.

4-It is not allowed to re-exam, just by an official excuse.


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Topic No: 1

A

P

A

Fave

A

0

lim


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The corresponding average shear stress is,

A

Pave


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Topic No: 2

MECHANICAL PROPERTIES OF MATERIALS

Stress-strain diagram for a typical structural steel in tension


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Hookes Law

Topic No: 3

Axial Load

strainnormal

stress

L

A

P

L

A

P

A

P

2

2

LL

A

P

2

2


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A temperature change results in a change in

length or thermal strain. There is no stress

associated with the thermal strain unless theelongation is restrained by the supports.

i ii

ii

EA

LP


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Treat the additional support as

redundant and apply the principle of

superposition.

The thermal deformation and thedeformation from the redundant

support must be compatible

coef.expansionthermal

AE

PLLT PT

0 PT

TEA

P

TAEP

AE

PLLT

0


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Topic No: 4Torsion

Torsion of a screwdriver due to a torque T applied to the handle

Torsional Deformation Of A Circular Shaft,Torsion

Where T =torque

=angular velocity


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Topic No: 5

Bending


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Topic No: 6

Combined Loading, ThinWalled Pressure Vessels, State Of Stress Caused

By Combined Loading


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Topic No: 7Plane-Stress Transformation, Principal Stress ,Mohrs Circle-Plane Stress


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Topic No: 8Buckling of Columns

Critical LoadThe transition between the stable and unstable conditions occurs at a special value of the

axial force known as the critical load (denoted by the symbol Pcr).


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Topic No: 9

Energy Method


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Where Ucis the external work and U

iis the internal work

Impact loading

The max. force


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2

nd

CourseFINITE ELEMENT METHOD


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Can interpret and evaluate the quality of the results (know the physics of the

problems)

Be aware of the limitations of the FEM (dont misuse the FEM - a numerical

tool)

Topic No: 2Review of matrix algebra; spring Element, one spring element,

spring system


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Types of Finite Elements

III. Spring Element


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Everything important is simple.One Spring Element

Topic No: 2

Bar and Beam Elements.

Linear Static AnalysisBar Element

Stiffness Matrix --- Direct Method


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Stiffness Matrix --- A Formal Approach

Topic No: 3Distributed load, bar element in 2D and 3D, stiffness

matrix in the 2D space


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Distributed Load


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Topic No: 4Beam element, direct method, formal approach, 3D beam

element

Beam Element


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Direct MethodUsing the results from elementary beam theory to compute each column of the stiffness

matrix.

Formal ApproachApply the formula,

Stiffness matrix of a general 2-D beam element,


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Topic No: 4FE analysis of frame structure

Members in a frame are considered to be rigidly connected. Both forces and

moments can be transmitted through their joints. We need the general beamelement (combinations of bar and simple beam elements) to model frames.

Two-Dimensional Problems


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Topic No: 4

FE for 2D problems

A General Formula for the Stiffness Matrix


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Constant Strain Triangle (CST or T3)This is the simplest 2-D element, which is also called

linear triangular element.

Applying formula (13), we obtain the element stiffness matrix for the

CST element,

Example


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course book mechanics of materials and fea aa

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