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7/31/2019 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,
25,26
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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