S6 l04 analytical and numerical methods of structural analysis
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Transcript of S6 l04 analytical and numerical methods of structural analysis
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Lecture #4Analytical and Numerical Methodsof Structural Analysis
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FLOWCHART OF STRUCTURAL ANALYSIS
2
Real object
Structurallayout
Design model
Only load-carrying structureis kept
Assumptions and simplificationsare applied, loads are calculated
according to the problem
Results ofanalysis
Implementationon real object
This step is beyond the scopeof structural analysis
Structural analysis
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FLOWCHART OF STRUCTURAL ANALYSIS
3
Real object
Structurallayout
Design model
Results ofanalysis
Implementationon real object
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FLOWCHART OF STRUCTURAL ANALYSIS
4
Real object
Structurallayout
Design model
Results ofanalysis
Implementationon real object
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FLOWCHART OF STRUCTURAL ANALYSIS
5
Real object
Structurallayout
Design model
Results ofanalysis
Implementationon real object
Depending on the kind of problem which is solved, the design model could be either as detailed as structural layout, or as generalized as below:
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METHODS OF STRUCTURAL ANALYSIS
Analytical methods Numerical methods
Best for designing calculations, suit for
checking calculations with certain limitations
Best for checking calculations, practically
effete for designing calculations
Solutions exist for partial cases (specific objects)
Versatile and flexible
Need much work to be developed, but only simple software for
application
Need expensive and complex software and
hardware6
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ANALYTICAL METHODS OF STRUCTURAL ANALYSIS
7
Widely used nowadays:• methods of Mechanics of Materials;• methods for statically indeterminate structures:
• method of forces;• method of displacements;
• beam theory for thin-walled structures;• method of reduction coefficients.
Rarely used nowadays (mostly due to the progress of numerical methods):• methods based on Calculus of Variations;• methods of Theory of Elasticity.
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EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
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Aim:Design the steel bracket to minimize the self-weight displacement of the load. Given data:D2=80 mm;L2=500 mm;L1=200 mm;mload=0.5 kg.
Objectives:D1, 1, 2.
loadL1, D1, 1
L2, D2, 2
base
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BASIC EQUATIONS OF SOLID MECHANICS
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Equilibriumequations
Constitutiveequations
Compatibilityequations
This is not only the sum offorces or moments, but appliesfor elementary volume as well
Physical law, expressesthe relation between
stress and strain
Solid body shouldremain continuous
while being deformed
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BASIC EQUATIONS OF SOLID MECHANICS
10
Equilibriumequations
Constitutiveequations
Compatibilityequations
This is not only the sum of forcesor moments, but applies forelementary volume as well.
0
0
0
xyx xz
xy y yz
yzxz z
X ,x y z
Y ,x y z
Z ,x y z
where X ,Y ,Z external forces.
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BASIC EQUATIONS OF SOLID MECHANICS
11
Equilibriumequations
Constitutiveequations
Compatibilityequations
Physical law, expresses the relation between stress and strain.
1
1
1
x x y z
y y x z
z z x y
xy yzxzxy xz yz
,E
,E
,E
, ,G G G
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BASIC EQUATIONS OF SOLID MECHANICS
12
Equilibriumequations
Constitutiveequations
Compatibilityequations
Solid body should remaincontinuous while being deformed.
2 22 2 22
2 2 2 2
2 22
2 2
2
2
2
2
2
2
y xyx x xzz
y yzz
yz xyxz x
yz xy yxz
yz xyxz z
, ,y x x y z x x z
,z y y z
,x x y z y z
,y x y z x z
.z x y z x y
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BASIC EQUATIONS OF SOLID MECHANICS
13
Equilibriumequations
Constitutiveequations
Compatibilityequations
Boundaryconditions:• geometrical
features; • supports and other means of
fixation;• applied
displacements and forces.
Designmodel
From analytical point of view, every design model can be expressed as a set of three basic equations with corresponding boundary conditions.
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WAYS TO SOLVE A SOLID MECHANICS PROBLEM
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Displacements are set as unknowns
Strains are derived
Stresses are derived
Compatibility equations
Constitutive equations
Equilibrium equations are
solved
Equilibrium equations
Stresses are set as
unknowns
Strains are derived
Constitutive equations
Equilibrium equations
Compatibility equations are
solved
Compatibility equations
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EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
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Analytical methods used:• Methods of Mechanics of Materials;• Mohr’s integral to calculate displacements.
Software used:MathCAD V14.0
loadL1, D1, 1
L2, D2, 2
base
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EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
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EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
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EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
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EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
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EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
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EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
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EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
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EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
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EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
24
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EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
25
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METHODS OF STRUCTURAL ANALYSIS
Analytical methods Numerical methods
Best for designing calculations, suit for
checking calculations with certain limitations
Best for checking calculations, practically
effete for designing calculations
Solutions exist for partial cases (specific objects)
Versatile and flexible
Need much work to be developed, but only simple software for
application
Need expensive and complex software and
hardware26
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NUMERICAL METHODS OF STRUCTURAL ANALYSIS
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Most widely used nowadays is Finite Element Method (FEM).
The structural analysis using FEM is called finite element analysis (FEA).
Generally, FEM is just a method to solve differential equations.The physical background of FEM is the same as in analytical methods.
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FINITE ELEMENT METHOD
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The smooth field of stresses is calculated using the combination of basic functions, which are constructed on a mesh.
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FINITE ELEMENT METHOD
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Software work on principle GIGO: “garbage in – garbage out”…
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FINITE ELEMENT METHOD
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Software work on principle GIGO: “garbage in – garbage out”…
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FINITE ELEMENT METHOD
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Software work on principle GIGO: “garbage in – garbage out”…
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FINITE ELEMENT METHOD
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Even if the result is realistic,you should always check it.
It’s not a big problemto create a “nice picture” using FEA.
The problem is to get the reliable and verified result.That’s a difference between professional and
beginner.
FEA is a way to find a stress state,but not the way to design or to check the strength!
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EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
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base
Object FEM model
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EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
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Assignment of parameters
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EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
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Deformed shape
35 finite elements in model
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EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
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Optimization took 7
minutes on average
office PC.990 points
were studied.
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Analytical method Finite element analysis
Displacement is 7.66 m (or 8.07 m in FEA)
Displacement is 7.81 m (or 8.22 m in FEA)
D1=67 m, 1=2 m,
2= 11.23 mm
D1=58 m, 1=2 m,
2= 13 mm
Need only simple software for application
Need expensive and complex software
37
EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
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METHODS OF STRUCTURAL ANALYSIS
Analytical methods Numerical methods
Best for designing calculations, suit for
checking calculations with certain limitations
Best for checking calculations, practically
effete for designing calculations
Solutions exist for partial cases (specific objects)
Versatile and flexible
Need much work to be developed, but only simple software for
application
Need expensive and complex software and
hardware38
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EXAMPLE ILLUSTRATING THE DESIGN OF FINITE ELEMENT MODEL
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Shell model.12K elements in model, around 5 minutes to solve on poor office PC
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EXAMPLE ILLUSTRATING THE DESIGN OF FINITE ELEMENT MODEL
40
Solid model.250K elements, around 20
minutes to solve on powerful PC(4x4.2GHz processor,
16 GB memory).
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EXAMPLE ILLUSTRATING THE DESIGN OF FINITE ELEMENT MODEL
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Difference?Certain there is a difference, but for some results it is negligible.
Thus, the model should be built according to problem,as in analytical methods
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FINITE ELEMENT SOFTWARE
42
Widely used nowadays:• ANSYS;• Patran/Nastran;• ABAQUS;• Solidworks Simulation (former COSMOS);And many-many others, freeware and proprietary…
Any difference? Generally, no difference. FEA packages are often customized to be more convenient for some specific problems. But for most cases, differences between packages and limitations have marketing but not the technical reason.
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1940Theoretical aspects were developed for the
first time
1950,1960
Theoretical background was developed, the rise of computer power allowed to make first
specific software
1970Large general-purpose packages appeared
(ANSYS, NASTRAN)
1980 Graphical pre- and post-processing
1990Rise of the role of automatic meshing, and
thus the model scale
2000 Multi-disciplinary and large-scale problems43
FINITE ELEMENT METHOD HISTORICAL PROGRESS
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PRINCIPAL COMPONENTS OF FEA SOTWARE
44
Pre-processor
Solver
Post-precessor
Construction of the model, setting the properties, loads, constraints
etc.
Solution itself. Two main types of solvers are iterative or sparse .
Looking on results, processing of results, generating “nice figures”
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FINITE ELEMENT METHOD – OLD INTERFACE
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FINITE ELEMENT METHOD – MODERN INTERFACE
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CONCLUSIONS• “Analytical method” does not mean “by hand” or “on paper”, modern software can give large benefits.
• Numerical methods are very exacting in terms of software and hardware.
• FEA is not a magic key that opens every door. It’s just a versatile and effective tool to calculate stresses.
• Deep insight is required for successful structural analysis, no matter is it analytics of FEA.
• Generally, analytical methods are good for designing, FEA – for checking. However, in many cases both of methods can be used for both phases of analysis. 47
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CONCLUSIONS
What does it mean to be a professional in structural analysis?
Knowledge of complex formulas? Usage of modern software?
Neither of it is an answer. Answer is a deep understanding of how the structure works, and
application of proper methods according to the given problem.
Analytical and numerical methods loss their disadvantages when they are combined together.
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WHERE TO FIND MORE INFORMATION?BOOKS AND USEFUL LINKS
49
Good book about structural analysis and Ansys is E. Madenci, I.Guven. The Finite Element Method and Applications in Engineering Using ANSYS, 2006
Very useful links are:
Official website resource library ansys.com
Resourse for Ansys users ansys.net
Perfect forum (not for beginners, though) xansys.org
Youtube (it’s not a joke, search “Ansys” for amazing training videos) youtube.com
… Internet is boundless …
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TOPIC OF THE NEXT LECTURE
50
Statically determinate trusses
All materials of our course are availableat department website k102.khai.edu
1. Go to the page “Библиотека”2. Press “Structural Mechanics (lecturer Vakulenko S.V.)”