Analytical and Numerical Methods of Structural Analysis (1)

7/23/2019 Analytical and Numerical Methods of Structural Analysis (1)

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Lecture #4Analytical and Numerical Methodsof Structural Analysis



FLOWCHART OF STRCTRALANAL!S"S

Real object

Structural

layout

Design model

Only load-carrying structureis kept

Assumptions and simplifications

are applied, loads are calculatedaccording to the problem

Results of analysis

Implementation

on real object

This step is beyond the scope

of structural analysis

Structural analysis




$

Real object

Structural

layout

Design model

Results of analysis

Implementation

on real object




4

Real object

Structural

layout

Design model

Results of analysis

Implementation

on real object




%

Real object

Structural

layout

Design model

Results of analysis

Implementation

on real object

Depending on the kind of

problem which is solved, thedesign model could be either as

detailed as structural layout, or

as generalized as below



M&THO'S OF STRCTRAL ANAL!S"S

Analytical methods Numerical methods

!est for designingcalculations, suit for

checking calculations with

certain limitations

!est for checkingcalculations, practically

effete for designing

calculations

Solutions e"ist for partialcases #specific ob$ects%

&ersatile and fle"ible

'eed much work to bedeveloped, but onlysimple software for

application

'eed e"pensive andcomple" software and

hardware

(



ANAL!T"CAL M&THO'S OFSTRCTRAL ANAL!S"S

)

Widely used nowadays:( methods of )echanics of )aterials*( 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 progressof numerical methods%( methods based on alculus of &ariations*( methods of Theory of lasticity+



&*AM+L& TO COM+AR& ANAL!T"CALAN' NM&R"CAL M&THO'S

,

Aim:

Design the steelbracket to minimize

the self-weight

displacement of the

load+

Gien data:

D./01 mm*

2./311 mm*

24/.11 mm*

mload/1+3 kg+

!bjecties:D4, δ4, δ.+ load

"#$ D#$ #

"%$ D%$ %

base



-AS"C &.AT"ONS OF SOL"'M&CHAN"CS

/

&'uilibrium

e'uations

(onstitutie

e'uations

(ompatibility

e'uations

This is not only the sum of

forces or moments, but appliesfor elementary volume as well

5hysical law, e"pressesthe relation between

stress and strain

Solid body should

remain continuous

while being deformed




01

&'uilibrium

e'uations

(onstitutie

e'uations

(ompatibility

e'uations

This is not only the sum of forces

or moments, but applies for elementary volume as well+

0

0

0

xy x xz

xy y yz

yz xz z

X ,

x y z

Y , x y z

Z , x y z

where X ,Y ,Z external forces.

τ σ τ

τ σ τ

τ τ σ

∂∂ ∂+ + + =

∂ ∂ ∂∂ ∂ ∂

+ + + =∂ ∂ ∂

∂∂ ∂+ + + =

∂ ∂ ∂

−




00

&'uilibrium

e'uations

(onstitutie

e'uations

(ompatibility

e'uations

5hysical law, e"presses the

relation between stress and strain+

( )( )

( )( )

( )( )

1

1

1

x x y z

y y x z

z z x y

xy yz xz xy xz yz

, E

, E

, E

, ,G G G

ε σ ν σ σ

ε σ ν σ σ

ε σ ν σ σ

τ τ τ γ γ γ

= − +

= − +

= − +

= = =



WA!S TO SOL2& A SOL"' M&CHAN"CS

+RO-L&M

04

Displacements

are set asun*nowns

Strains are

deried

Stresses are

deried

ompatibility

e6uations

onstitutive

e6uations

&'uilibrium

e'uations are

soled

6uilibriume6uations

Stresses are

set asun*nowns

Strains arederied

onstitutive

e6uations

6uilibrium

e6uations

(ompatibility

e'uations are

soled

ompatibilitye6uations



&*AM+L& TO COM+AR& ANAL!T"CALAN' NM&R"CAL M&THO'S

0%

Analytical methods

used:( )ethods of

)echanics of

)aterials*

( )ohr8s integral tocalculate

displacements+

Software used:

)athAD &49+1

load

"#$ D#$ #

"%$ D%$ %

base



&*AM+L& TO COM+AR& ANAL!T"CAL AN'NM&R"CAL M&THO'S

0(




0)




0,




0/




1




0




$




4




%







certain limitations



calculations




application


hardware

(



NM&R"CAL M&THO'S OFSTRCTRAL ANAL!S"S

)

)ost widely used nowadays is+inite &lement ,ethod -+&,./

The structural analysis using 7) is called finite

element analysis #7A%+

Generally$ +&, is just a method to sole

differential e'uations/The physical background of 7) is the same as in

analytical methods+



F"N"T& &L&M&NT M&THO'

,

The smooth field of stresses is calculated using the combination

of basic functions, which are constructed on a mesh+




/

Software work on principle :;:O <garbage in = garbage out>?




$1





$0





$

ven if the result is realistic,

you should always check it+

;t8s not a big problem

to create a <nice picture> using 7A+

The problem is to get the reliable and verified result+

That8s a difference between professional and

beginner+

7A is a way to find a stress state,

but not the way to design or to check the strength@




$$load

base

!bject +&, model




$4

Assignment

of parameters

O CO C




$%

Deformed

shape

3 finite elements

in model



&*AM+L& TO COM+AR& ANAL!T"CAL AN'



Analytical method +inite element analysis

Displacement is B+CC µm

#or 0+1B µm in 7A%

Displacement is B+04 µm

#or 0+.. µm in 7A%

D4/CB µm, δ4/. µm,

δ./ 44+. mm

D4/30 µm, δ4/. µm,

δ./ 4 mm

'eed only simplesoftware for application

'eed e"pensive andcomple" software

$)








certain limitations



calculations




application


hardware

$,

&*AM+L& "LLSTRAT"N3 TH& '&S"3N OF F"N"T&



&*AM+L& "LLSTRAT"N3 TH& '&S"3N OF F"N"T&&L&M&NT MO'&L

$/

Shell model+

4. elements inmodel, around 3

minutes to solve

on poor office 5





41

Solid model+

.31 elements, around .1

minutes to solve on powerful 5#9"9+.:Ez processor,

4C :! memory%+





40

Difference5ertain there is a difference, but for some results it is negligible+

Thus, the model should be built according to problem,as in analytical methods

F"N"T& &L&M&NT SOFTWAR&



F"N"T& &L&M&NT SOFTWAR&

4

Widely used nowadays:

( A'SFS*( 5atranG'astran*( A!AHIS*( Solidworks Simulation #former OS)OS%*

And many-many others, freeware and proprietary?

Any difference5 :enerally, no difference+ 7A

packages are often customized to be more convenientfor some specific problems+ !ut for most cases,

differences between packages and limitations have

marketing but not the technical reason+

F"N"T& &L&M&NT M&THO' H"STOR"CAL



4J91Theoretical aspects were developed for the

first time

4J31,4JC1

Theoretical background was developed, therise of computer power allowed to make first

specific software

4JB12arge general-purpose packages appeared

#A'SFS, 'ASTKA'%

4J01 :raphical pre- and post-processing

4JJ1Kise of the role of automatic meshing, and

thus the model scale

.111 )ulti-disciplinary and large-scale problems 4$

F"N"T& &L&M&NT M&THO' H"STOR"CAL+RO3R&SS

+R"NC"+AL COM+ON&NTS OF F&A



+R"NC"+AL COM+ON&NTS OF F&ASOTWAR&

44

2re6processor

Soler

2ost6precessor

onstruction of the model, setting

the properties, loads, constraintsetc+

Solution itself+ Two main types of

solvers are iterative or sparse +

2ooking on results, processing of

results, generating <nice figures>

F"N"T& &L&M&NT M&THO' OL'



4%

F"N"T& &L&M&NT M&THO' OL'"NT&RFAC&

F"N"T& &L&M&NT M&THO' MO'&RN



4(

F"N"T& &L&M&NT M&THO' MO'&RN"NT&RFAC&

CONCLS"ONS



CONCLS"ONS

( <Analytical method> does not mean <by hand> or <on

paper>, modern software can give large benefits+

( 'umerical methods are very e"acting in terms of

software and hardware+

( 7A is not a magic key that opens every door+ ;t8s $ust a versatile and effective tool to calculate stresses+

( Deep insight is re6uired for successful structural

analysis, no matter is it analytics of 7A+

( :enerally, analytical methods are good for

designing, 7A = for checking+ Eowever, in many

cases both of methods can be used for both phases of

analysis+ 4)

CONCLS"ONS



CONCLS"ONS

Lhat does it mean to be a professional in structuralanalysisM

nowledge of comple" formulasM Isage of modern

softwareM

'either of it is an answer+ Answer is a deep

understanding of how the structure wor*s, and

application of proper methods according to the given

problem+

Analytical and numerical methods loss their

disadvantages when they are combined together+

4,

WH&R& TO F"N' MOR& "NFORMAT"ON5



WH&R& TO F"N' MOR& "NFORMAT"ON5-OO6S AN' S&FL L"N6S

4/

:ood book about structural analysis and Ansys isE. Madenci, I.Guven. The Finite Element Method and Applications inEngineering Using ANSYS, !!"

&ery useful links are

Official website resource library ansys+com

Kesourse for Ansys users ansys+net

5erfect forum #not for beginners, though% "ansys+org

Foutube #it8s not a $oke, search <Ansys> for amazing training

videos% youtube+com

? ;nternet is boundless ?



TO+"C OF TH& N&*T L&CTR&

Statically determinate trusses

All materials of our course are available

at department website k41.+khai+edu

4+ :o to the page <NPQRUV>

.+ 5ress <Structural )echanics #lecturer &akulenko S+&+%>

Analytical and Numerical Methods of Structural Analysis (1)

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