FEA Assignment - Lucas Dos Santos Almeida p13175018

 

Faculty of Technology Department of Mechanical Engineering DE MONTFORT UNIVERSITY

Module: ENGD 3016 Student: Lucas dos Santos Almeida P-number: 131701! "ourse: Mec#anical En$ineerin$ Lecturer: %arren Mannin$

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"ontents #ontent$%%%%%%%%%%%%%%%%%%%%%%%% #ontent$% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%2 Matla& #ODE%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #ODE%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%' %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%' Re$ult$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Re$ult$%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%( %%%%%%%%%%%%%%%%%%%%%%%%%%%(

S&uare cross-sectional area results''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''10 "ircular cross-sectional area results''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''1( Di$cu$$ion an) conclu$ion%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% conclu$ion%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%!" %%%%%%%%%%%%%%%%%!" *ppen)i+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%!,

Matri) anal*sis o+ a truss'''''''''''''''''''''''''''''''''''''''''''''' truss''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''1 ''''''''''''''''''''''''1 Reference$%%%%%%%%%%%%%% Refere nce$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%!!-

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Matlab ",DE #e +ollo.in$ code .as de/eloed in order to sol/e t#e truss roblem' #e mode #o. t#e data is entered is .ell e)lained in aendi)' #e aendi) artiall* resented2 and t#e ro$ram .ere de/eloed b* me as a course monitor in ra4il to aid a ro+essor in t#e lecture o+ 5EA in Solid Mec#anics ' t .as onl* necessar* to modi+* t#e enterin$ data as t#e ro$ram calculates +or e/er* truss i+ t#e data is correctl* entered'

clear all clc   num_nos = 3; coord = [0 0;0 .3;.4 0]; num_elem = 3; incid = [1 2;1 3;2 3]; mat = [200E9]; num_load = 1; load = [3 2 -5000]; prop_geo = [1.11E-4]; num_restr = 3; restr = [1 1; 2 1; 2 2]; counter = 1;   glo!al = "eros#num_nos$2%; &or n=1'num_elem  n=1'num_elem &or   incid" = incid#n('%; ) *ets t+e incidence o& element n   ) *ets t+e coordinate o& t+e element nodes 1   ,o1 = coord#incid"#1%('%;   ,o2 = coord#incid"#2%('%; coord#incid"# 2%('%;   ) ,alculate t+e lengt+ o& t+e !ar ! t+e t+agorean relations+ip. relations+ip.   / = srt##,o2#1%-,o1#1%%2#, srt##,o2#1%- ,o1#1%%2#,o2#2%-,o1#2%% o2#2%-,o1#2%%2%; 2%;   ) calculates t+e cosine and sine !ars.   s = #,o2#2% - ,o1#2%%/;   c = #,o2#1% - ,o1#1%%/;   c2 = c2;   s2 = s2;   cs = c$s;   ) ,alculate t+e element sti&&ness matri n ) mat #1% re&ers to t+e oung6s modulus( prop_geo #1% t+e crosssectional area.   e = mat#1%$prop_geo/$[c2 mat#1%$prop_geo/$[c2 cs -c2 -cs; cs s2 -cs -s2; -c2 -cs c2 cs; -cs -s2 cs s2];   ) *ets t+e 7egrees o& 8reedom &or t+e element. o &ind t+e degree o& 3

 

) &reedom &or eac+ element :as used t+e node logic $ 2  and node $ 21 )    7(00GPa' ?

 

S.uare cro$$/$ectional area re$ult$

Figure: Displacement in X 

Figure: Displacement in Y 

10

 

Figure: Reactions in X 

Figure: Reactions in Y 

11

 

Figure: Stress results

#ircular cro$$/$ectional area re$ult$

Figure: Displacement in X 

1(

 

Figure: Displacement in Y 

Figure: Reactions in X 

13

 

Figure: Reactions in Y 

Figure: Stress results

Discussion and conclusion A+ter all t#e calculation done b* t#e ro$rams MALA and "8E,= it is no time time +or +or disc discus ussi sion on'' # #ee re resu sult ltss $ene $enera rate ted d b* bo bot# t# r ro$ o$ra ram ms in indi dica cate te a $o $ood od con/er$ence' #e di++erence is mainl* e)lained b* t#e considerations ta@en b* t#e met#ods o+ calculations' MALA ro$ram uses t#e matri) anal*sis o+ a truss usin$ onl* bars' #is means t#at t#is ro$ram does not considerate bendin$' ,n t#e ot#er  1

 

#and= "8E, uses beams as elements to simulate' t is ossible to obser/e t#at "8E, simulation roduces di++erent results .#en usin$ di++erent cross-section areas' Note t#at in t# t#ee MA MALA LA al$o al$orit rit#m #m t#e t#e s#a s#aee o+ crosscross-sec sectio tion n area area isnt isnt e/ e/en en ta ta@e @en n in consideration t onl* $ets t#e /alue o+ t#e area to do t#e calculation' #e simulation o+  t#e same truss .as carried .it# ALG,8 simulate and roduced EBA"LC t#e same results obtained in t#e calculation b* t#e MALA al$orit#m' #is is e)lained b* t#e +act t#at ALG,8 uses t#e same calculation rocess as MALA' MALA'

Aendi) Matri+ analy$i$ of a tru$$

#e model o+ t#e +ollo.in$ inut /ariables is E( > E3 > (00)103 MPa= P > 1000N #e areas o+ #ori4ontal and /ertical bars is 1mm( area .#ile t#e inclined 1=1 mm(' #e len$t#s in t#e +i$ure are in millimeters

1

 

  Figure 1: Truss

%e can de+ine t#e be$innin$= t#e number o+ nodes o+ t#e truss and number o+ elements' 5or t#is case:

  Figure : !odes and defned elements"

#e red numbers indicate t#e element and blac@ indicate t#e node'  Numnodes >  #en de+ine t#e coordinates o+ us and ut in a matri)' coord > 16

 

  100   ( 0 (1   3 (1 0    (1 (1 #e +irst column reresents t#e node= t#e second and t#ird reresent t#e coordinates ) and *'

%e no. de+ine t#e elements' n t#is e)amle are 6 bars'  Numelem > 6 Cou must set t#e cost o+ eac# element= t#at is= tell t#e ro$ram .#ere t#e* are situated' 5or e)amle= t#e element 1 is connected at node 1 and node (= .#ile t#e element 3 is connected node 3 and node ' #e +irst column reresents t#e number o+ t#e element' N"D >  

11(

 

1(3

 

33

 

(

 

(3

 

61

  t is necessar* to de+ine t#e $eometric roerties o+ t#e materials and t#e bars' As t#e 6  bars #a/e t#e same material= .ill be a simle /ector' n ro$eo ro$eo /ector laced t#e areas o+ bars' Mat > 17

 

  (00E3 Pro$eo >  

11

 

(1

 

31

 

1

 

 1=1

 

6 1=1

%e see t#at mo/ement is restricted in t#e lattice  de$rees o+ +reedom' Node 1 is restricted in ) and *= t#e node ( in ) and * as .ell'  Numrestr >  %e must create a matri) matr i) .it# restrictions' n t#e +irst column ut .#at is t#e node .#ere t#e restriction is' n t#e second lace .#ic# is t#e de$ree o+ +reedom #as been restricted' #e number 1 reresents t#e de$ree o+ +reedom in ) and number ( reresents t#e de$ree o+ +reedom in *' + t#is trellis= node 1 and node ( are restricted in D,5 1 and ( ) and *2' s t#en: restr >  

11

 

1(

 

(1

 

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#en remains remains to de+ine t#e s#iments s#iments o+ t#e trellis' t creates a /ector .it# t#e number  number  o+ s#iments' n t#is case= t#ere is a +orce' 1!

 

 Numloads > 1 ,ne s#ould tell t#e ro$ram .#ere t#is load is located' + t#is lattice o.er is bein$ alied to t#e node = t#e D,5 (' n t#e matri)= t#e +irst column indicates t#e node= t#e second t#e D,5 and t#e t#ird ma$nitude' Loads >

 

 ( -1000

8e+erences ittencourt = Marco - Analise "omutacional de Estruturas= com alicaFo do mtodo de elementos +initos'

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