s6 l04 analytical and numerical methods of structural analysis

Lecture #4Analytical and Numerical Methodsof Structural Analysis

FLOWCHART OF STRUCTURAL ANALYSIS

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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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Real object

Structurallayout

Design model

Results ofanalysis



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Real object

Structurallayout

Design model

Results ofanalysis



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Real object

Structurallayout

Design model

Results ofanalysis


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:

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

ANALYTICAL METHODS OF STRUCTURAL ANALYSIS

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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.

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

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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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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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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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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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.

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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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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17


18


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21


22


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24


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application


hardware26

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.

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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Software work on principle GIGO: “garbage in – garbage out”…


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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!


33load

base

Object FEM model


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Assignment of parameters


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Deformed shape

35 finite elements in model


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Optimization took 7

minutes on average

office PC.990 points

were studied.

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

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application


hardware38

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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Solid model.250K elements, around 20

minutes to solve on powerful PC(4x4.2GHz processor,

16 GB memory).


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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

FINITE ELEMENT SOFTWARE

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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.

http://en.wikipedia.org/wiki/List_of_finite_element_software_packages

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

PRINCIPAL COMPONENTS OF FEA SOTWARE

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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

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

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

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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 …

http://ansys.com/Resource+Library

http://www.ansys.net/

http://www.xansys.org/

http://www.youtube.com/results?search_query=ansys&oq=ansys&gs_l=youtube-reduced.3..0l4.1165.1659.0.1883.5.5.0.0.0.0.139.519.2j3.5.0...0.0...1ac.1.Sa8fpI1ujBc

TOPIC OF THE NEXT LECTURE

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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.)”

http://k102.khai.edu/

http://k102.khai.edu/ru/site/library.html

http://k102.khai.edu/ru/site/structural-mechanics.html

