Numerical Modelling in Geo engineering

8/19/2019 Numerical Modelling in Geo engineering

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CIE4366 Numerical Modelling in GEO-ENGINEERING

Coursework 3:

2‐Dimensional Finite Element Method Programming

Name: CX. Azua-Gonzalez

Student Nr: 449!94


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INE!"# $%S&R$C&............................................................................................................................... "

'# IN&RO(C&ION...................................................................................................................... "

3# &)EORE&IC$* %$C+GRO(N............................................................................................... 4

a# Sk,line Storage.................................................................................................................. 4

# C.olesk, decom/osition...................................................................................................4

c# Iso/arametric elements.................................................................................................... #

d# %ending o0 Euler-%ernoulli eams...................................................................................#

e# E1ui2alent nodal loading s,stems...................................................................................$

4# M$&)EM$&IC$* ORM(*$&IONS.........................................................................................!

a# Gauss-*egendre 1uadrature.............................................................................................!

# S.a/e 0unctions and local deri2ati2es..........................................................................%c# acoian matrices............................................................................................................. %"

d# Gloal deri2ati2es............................................................................................................%"

e# Constituti2e relations......................................................................................................%4

0# 5% matri7.......................................................................................................................... %&

g# Stresses............................................................................................................................. %&

8# GENER$* &9O-IMENSION$* EM ORM(*$&ION........................................................%#

6# $*I$&ION O RES(*&S.................................................................................................... 2"

a# $7ial com/ression o0 a column;laterall, restrained


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"A# $BBENI!.........................................................................................................................."

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"# $%S&R$C&

A t'o general FEM (rogram 'as de)elo(ed in Fortran 9& as (art o* the assignment "A

and "+ *or the ,ourse CE 4"## CE umeri,al Modelling in Geo Engineering. Di/erent

t0(es o* elements 'ere in,or(orated in,luding linear1 a uadrati, elements. 3his ,ode

'as used to sol)e sim(le t'o- dimensional stru,tural me,hani,s (rolems under (lain

strain *ormulation1 and one- dimensional stru,tural elements under ,om(ression and

ending. A standard FEM (ro,edure 'as (ro(osed *or the ,ode es(e,iall0 *or the

,al,ulation o* the element sti/ness matri,es1 and ,al,ulation o* stresses and interna

*or,es. 3he standardized (ro,edure in ,omination 'ith the elaoration o* suroutines

'as (ro)en to assist in 'riting a ,learer ,ode to the reader. 3he main (rogram 'as

modi5ed to assist the user in generating automati, node numering element numering

Furthermore1 )alidation o* the results 'as done 0 means o* sim(li5ed anal0ti,a

solutions. Finall01 the in6uen,e o* the )i,init0 to restraints1 t0(e o* element1 and numer

o* elements 'as studied.

+e,words: FEM1 Gauss-7engendre uadrature1 s80line storage1 solidme,hani,s

'# IN&RO(C&ION

3'o dimensional stru,tural anal0ses o* solid odies ,an e (er*ormed 0 means o* the

Finite Element Method FEM:1 and its ,om(le;it0 ,an e ,ontrolled 0 the %?.

3he ,urrent re(ort aims to e;(lore the use o* a Fortran 9& Code to (er*orm a stru,tura

anal0sis o* t'o-dimensional solid odies using Gauss-7egendre uadrature and ,lassi,a

*ormulations o* stresses1 and strains in me,hani,s. 3he (rogram sol)es the (rolems indis(la,ements as (rimar0 )ariales@ as a result1 oundar0 ,onditions are regarded as

5;ed nodes. odal *or,es are a((lied in the *rame'or8 o* eui)alent e;ternal 'or81

'hi,h means all the *or,es are a((lied on the nodes in su,h a manner that the e;terna

'or8 'ould e eui)alent as that o* the original loading s0stem.

Additionall01 one- dimensional solid odies are sol)ed to ma8e a ,om(arison among the

results a,hie)ed 0 a t'o- dimensional anal0sis and anal0ti,al solutions i* a)ailale.

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3he goals o* this assignment are

%. E;(lain ho' 5nite element (rograms 'or8.2. Bse asi, s,ienti5, (rogramming te,hniues1 in,luding limited modi5,ation o* FEM

(rograms.

3o a,hie)e these goals1 a general FEM ,ode 'as 'ritten (artiall0 and deli)ered to the

students o* the ,ourse CE 4"## CE umeri,al Modelling in Geo Engineering. 3he tas8 o*

the students 'as to ,om(lete the missing (arts o* the ,ode using a)ailale suroutines

Additional suroutines 'ere as8ed to e elaorated 'here needed.

Finall01 it should e mentioned that the stru,ture o* the (rogram deli)ered in this re(ort

aims to 5nd an a,,e(tale solution. n *a,t1 the ,ode does not re(resent the most

e,ient manner to sol)e a stru,tural anal0sis o* solid odies in terms o* ,om(utational

resour,es. e)ertheless1 the im(lementation o* s80line storage (ro,edures o* element

sti/ness matri,es into a gloal sti/ness matri; 'as reuired to ma8e the (rolems

resol)ale 0 means o* a standard ,om(uter.

3# &)EORE&IC$* %$C+GRO(Na# Sk,line Storage

Bnder the situation o* e;tremel0 s(arse matri,es the (ro,edure o* s80line storage is

ne,essar0. 3his (ro,ess im(lies the storage o* the )alues 'hi,h are ta8en into a,,ount in

the resolution o* the matri; linear s0stem. 3he (ro,edure aims to e an alternati)e to old

)ersions o* anded storage >2?. 3he (ro,edure reuires the storage o* the )alues in a

s80line matri; and the (osition o* the main diagonal terms. An elu,idation o* the s80linestorage (ro,edure is de(i,ted lo'

Figure %. Elucidation o0 sk,line storage /rocedure

Furthermore1 it ,an e seen that sometimes it is use*ul to start the (osition )e,tor at zero

de(ending on the *urther ,al,ulation (ro,edures ado(ted.:

# C.olesk, decom/osition

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Choles80 de,om(ositions use the s0mmetr0 and (ositi)e de5niteness o* the sti/ness

matri; as a starting (oint o* the anal0sis. n *a,t1 i* the (re)ious ,ondition is true1 this

(ro,edure reuires one hal* o* the 6oating (oint o(erations than Gauss elimination >"?

3here are t'o )arians o* ,holes80 de,om(osition

%. K ¿BT

B

2. K ¿B

T DB

n general1 the se,ond )ariant o* Choles80 de,om(osition is (re*erred to a)oid the

,al,ulation o* suare roots. For that reason1 this (ro,edure is e;(lained elo'1 a*ter

de5ning matri,es as de(i,ted

E1uation "# De5nition o* matri,es in Choles80 de,om(osition

te(%. De,om(ose K ¿BT DB

A. For ea,h i%1 -% in in,rements o* %a. 8i. +ihHdi,. For ea,h 8iI%1 in in,rements o* % 88 is re(la,ed 0 88-hi8

+. da

te(2. For'ard sustitution f ¿ BT

z , Db= z *or ea,h %1n in in,rements o*

A. JKFK+. For ea,h i%1-% in in,rements o* % zz-iziC. zHd

te(". +a,8 sustitution Bu=b *or ea,h n1% in in,rements o* -%

A. u+. *or ea,h iI%1n in in,rements o* % uu-iui

3he determinant o* L is ,al,ulated 'ith

| K |=|BT || D||B|=| D|= (d1 d2… . dn )

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igure '# Elu,idation o* the se,ond Choles80 de,om(osition )ariant >"?

c# Iso/arametric elements

3he *ormulation o* inter(olation *un,tions de(ends on the a((li,ation o* the element

degrees o* *reedom:. e)ertheless1 in t'o dimensional stru,tural anal0sis o* solid odies

the use o* the sha(e *un,tions1 o*ten ,alled inter(olation *un,tions1 ,an e e;tended to

inter(olate ,oordinates *rom a lo,al s0stem to a gloal s0stem. 3he elements 'hi,h(osses this 8ind o* *ormulation are ,alled iso(arametri, elements. 3he ,on)enien,e o*

using iso(arametri, elements lie in the uniue *ormulation o* the sha(e *un,tion and

deri)ati)es to elaorate the train-dis(la,ement matri; >+?.

* the ,hain rule is a((lied to the (artial deri)ati)es o* the sha(e *un,tions1 a relationshi(

et'een deri)ati)e in the lo,al s0stem and in the gloal s0stem ,an e *ound >%?. 3he

term that relates these deri)ati)es is ,alled the a,oian. 3he mathemati,al relationshi(

is de(i,ted elo'

E1uation '. Ka,oian matri; *ormulation

d# %ending o0

Euler-%ernoulli eams

+ending is (rodu,ed 0 trans)erse loading in a eam. 3he relationshi( o* the ,ur)ature

81 the modulus o* elasti,it0 and the moment o* inertia sho'n in 5gure elo'1 ,an e

used to a,8 ,al,ulate internal ending moments in a FEM *ormulation1 a*ter the

resolution o* the (rolem in terms o* the (rimar0 )ariales trans)erse dis(la,ement and

rotational slo(e theta:. tresses ,an e ,al,ulated as the multi(li,ation o* the ,ur)ature

the modulus o* elasti, and the distan,e 'ith res(e,t to the neutral a;is.

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N#%: 3he *ortran ,ode de)elo(ed in this re(ort ,al,ulates the multi(li,ation E8

automati,all01 ho'e)er the user must (er*orm the ,al,ulation o* EL0 to ,al,ulate the

stress at a desired distan,e.

igure 3# Elu,idation o* stresses1 ,ur)ature and ending moments in an Euler-+ernoull

+eam >4?.

e# E1ui2alent nodal loading s,stems

3he *ormulation o* element sti/ness matri,es1 and loading matri,es is ased u(on the

sim(li5,ation o* loading s0stems. n *a,t1 distriuted loads in a sur*a,e or de5ned as

)olume loads loads (er unit area1 or od0 loads a((lied 0 gra)it0:. Di/erent (ro,edures

,an e a((lied to a,hie)e this goal1 )ariational a((roa,hes are used to 5nd eui)alent

nodal loads 'hi,h ensure the same e;ternal 'or8 as the original s0stem.

e)ertheless1 this (ro,edure might not e the most ,ommon due to its mathemati,al

*ormulation. n ,ontrast1 stru,tural engineers in their desire o* getting good estimation o*

results tend to use the stri( method. 3his (ro,edure ,onsists in di)iding distriuted loads

o)er its area o* in6uen,e *or ea,h node as it is sho'n elo'. 3he (ro,edure tends to e

,alled a node 0 node lum(ing >2?.

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igure 4# Elu,idation o* the node 0 node dire,t lum(ing >2?.

n that sense1 'hen the internal *or,es are ,al,ulated in a s0stem resol)ed 0 means o* an

eui)alent s0stem the real *or,es must e ,om(uted sutra,ting the eui)alent *or,es (rodu,ed

o)er ea,h element.

{ f internal}=[k m ] {u }−{f equivalent }

E1uation 3# Com(utation o* real internal *or,es

N#%: 3he (rogram ,ode de)elo(ed in the ,urrent re(ort ,al,ulates the (rodu,t o* >8m? and uN

and the user is in ,harge o* sutra,ting the *eui)alentN in ,ase that the nodal nodes 'ere

deri)ed *rom another loading s0stem. 3his s,heme gi)es the users o* the ,ode the *reedom o

de5ning the loads )ia )ariational a((roa,h or other (ro,edures 0 themsel)es. 3he e;am(les

sho'n in this re(ort1 are ased on sim(li5ed dire,t nodal ,onditions in 'hi,h *eui)alentN is a

zero )e,tor. en,e1 internal *or,es sho'n in the a((endi; are 5nal ans'ers and do not need ane;tra mani(ulation.

4# M$&)EM$&IC$* ORM(*$&IONS

3he summar0 o* mathemati,al *ormulations1 and euations to e sol)ed are de(i,ted in

this se,tion o* the re(ort. t is assumed that the reader has a asi, a,8ground o* linear

algera1 and solid me,hani,s.

a# Gauss-*egendre 1uadrature

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An integral ,an e re(resented 0 its a((ro;imate summation o* the *un,tion e)aluated

in ,ertain ,oordinates i* its domain goes *rom -% to %. 3his is an ad)antage *or FEM

anal0sis1 due to the situation that dimensionless lo,al ,oordinates are de5ned in that

domain. 3he 'eightings ,om *rom a mathemati,al a((ro;imation o* 7egendre

(ol0nomials o* the same order into the e;(ression sho'n elo'. 3he deri)ation o* these

,oordinates and ,oe,ients is not ,onsidered under the s,o(e o* this re(ort

e)ertheless1 it is im(ortant to mention that there is a minimum numer o* terms

reuired in the summation1 this )alues a((l0 *or a uni-)ariale (ol0nomial and ,an ee;tra(olated to multi-)ariale (ol0nomials as a the one *or ilinear re,tangular elements.

3he ,oordinates and 'eighting ,oe,ients are sho'n in the ,hart elo'.

E1uation 4# Gauss-7egendre Ouadrature

c.art "# Gauss-7egendre Ouadrature ,oordinates and 'eighting ,oe,ients >"?

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igure 8# Degree o* the (ol0nomial *or t0(i,al uadrilateral elements >"?For the integration o* the >+? matri;1 the ,orre,t numer o* (oints to e in,or(orated in

the Gauss-7egendre Ouadrature is essential. For that reason1 a (reliminar0 anal0sis o*

the nature o* the sha(e *un,tions must e (er*ormed. E)en 'ithout 8no'ing the sha(e

*un,tion *ormulas1 ut its general de5nition in (o'ers the ma;imum order o* the

(ol0nomial ,an e *oreseen.

3he + matri; ,olle,ts (artial deri)ati)es and de,reases the order o* the (ol0nomial in

e)er0 ,ase1 e;,e(t *or the uadrati, uadrilateral element. irtuall01 the (rodu,t

[B

]

T

[B

] rises this e;(onentials to the suare as it ,an e oser)ed in the 5gure sho'n.

igure 6# Degree o* the (ol0nomials in the (ro,ess o* integration >"?

# S.a/e 0unctions and local deri2ati2es

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A summar0 o* the sha(e *un,tions and (artial deri)ati)es *or the elements used in the

Fortran ,ode are sho'n elo'. Ouadrilateral elements are regarded 'ith t'o degrees o*

*reedom (er node1 'hile the rod elements are treated 'ith onl0 one degree o* *reedom.

E;,e(tionall0 one dimensional eams ha)e degrees o* *reedom (er node.

igure =# Distriution o* lo,al numering o* nodes in a +ilinear Ouadrilateral element

[ N

1

N 2

N 3

N 4

]=[0.25 (1−ξ ) (1−η )0.25 (1−ξ ) (1+η )

0.25 (1+ξ ) (1+η )0.25 (1+ξ ) (1−η )

] .E1uation 8# ha(e *un,tions *or a +ilinear Ouadrilateral element

[

∂ N 1

∂ ξ

∂ N 1

∂ η

∂ N 2

∂ ξ

∂ N 2

∂ η

∂ N 3

∂ ξ

∂ N 3

∂ η

∂ N 4

∂ξ

∂ N 4

∂ η

]=[

−0.25( 1−η ) −0.25 (1−ξ )−0.25(1+η ) 0.25 (1−ξ )

0.25 (1+η ) 0.25 (1+ξ )

0.25 (1−η ) −0.25 (1+ξ ) ] .

E1uation 6# 7o,al deri)ati)es *or a +ilinear Ouadrilateral element

igure ?# Distriution o* lo,al numering o* nodes in a Ouadrati, Ouadrilateral element

%2

%

"2

4

4

!

&

#

$

"

2

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[

N 1

N 2

N 3

N 4

N 5

N 6

N 7 N 8

]=

[

0.25 (1−ξ ) (1−η ) (−ξ−η−1 )

0.5 (1−ξ ) (1−η2)0.25 (1−ξ ) (1+η ) (−ξ+η−1 )

0.5 (1−ξ2) (1+η )0.25 (1+ξ ) (1+η ) (ξ+η−1 )

0.5 (1+ξ ) (1−η2 )0.25 (1+ξ ) (1−η ) (ξ−η−1 )

0.5 (1+ξ ) (1−η2 )

].

E1uation =# ha(e *un,tions *or a Ouadrati, Ouadrilateral element

∂ N 1

∂ ξ

∂ N 1

∂ η

∂ N 2

∂ ξ

∂ N 2

∂ η∂ N 3

∂ ξ

∂ N 3

∂ η

∂ N 4

∂ξ

∂ N 4

∂ η

∂ N 5

∂ ξ

∂ N 5

∂ η

∂ N 6

∂ ξ

∂ N 6

∂ η

∂ N 7

∂ ξ

∂ N 7

∂ η

∂ N 8

∂ ξ

∂ N 8

∂ η

=[0.25 (1−η ) (2 ξ+η ) 0.25 (1−ξ ) (2 η+ξ )

−0.5( 1−η2 ) −η (1−ξ )0.25 ( 2ξ−η ) (1+η ) 0.25 (1−ξ ) (2η−ξ )

−ξ (1+η ) 0.5 (1−ξ2 )

0.25 ( 1+η ) (2 ξ+η ) 0.25 (2 η+ξ ) (1+ξ )

0.5 (1−η2 ) −η (1+ξ )0.25 ( 2ξ−η ) (1−η ) 0.25 (1+ξ ) (2η−ξ )

−ξ ( 1−η ) −0.5 (1−ξ2 )

] .E1uation ?# 7o,al deri)ati)es *or a Ouadrati)e Ouadrilateral element

igure @# Distriution o* lo,al numering *or a linear rod element and a one dimensionaleam

[ N 1 N 2]=[0.5 (1− ξ )0.5 (1+ξ ) ] .

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E1uation @# ha(e *un,tions *or a linear rod or t'o-node eam element s(atiall0:

[∂ N

1

∂ ξ∂ N

2

∂ ξ ]=[−0.50.5 ] .

E1uation "A# 7o,al deri)ati)es *or a linear rod or a t'o-node eam element s(atiall0:

igure "A# Distriution o* lo,al numering *or a uadrati, rod

[ N

1

N 2

N 3

]=[−0.5 ξ (1−ξ )( 1−ξ ) (1+ξ )0.5 ξ (1+ξ ) ] .

E1uation "". ha(e *un,tions *or a uadrati, rod

[∂ N

1

∂ ξ

∂ N 2

∂ ξ

∂ N 3

∂ ξ

]=[−0.5 (1−2 ξ )−2 ξ0.5(1+2 ξ ) ] .E1uation "'# 7o,al deri)ati)es *or a uadrati, rod

N#%#: Qne dimensional eams in the ,urrent Fortran ,ode are treated as non iso(arametri,elements. 3hus1 the sha(e *un,tions used *or a rod element1 'ere used onl0 *or the s(atia

inter(olation o* the one-dimensional eam element. 3he sha(e *un,tions to inter(olate the

trans)erse dis(la,ement o* a eam are de(i,ted elo'. ote that there are t'o degrees o*

*reedom in ea,h node des(ite the one dimensional *ormulation

igure ""# Distriution o* lo,al numering and degrees o* *reedom in a one dimensionaleam element

[ N 1v N

1θ

N 2v

N 2θ

]=[ 0.25 (1−ξ )2 (2+ξ )

0.125 l (1−ξ )2 (1+ξ )

0.25 (1+ξ )2 (2−ξ )

−0.125 l (1+ξ )2 (1−ξ )] .

E1uation "3# ha(e *un,tions *or a one dimensional eaminter(olation o* BN:

%4

% 2

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[

∂2 N

1v

∂ ξ2

∂2 N

1θ

∂ ξ2

∂2 N

2v

∂ ξ2

∂2 N 2θ

∂ ξ2

]=[

1.5ξ

0.25 l (3ξ−1 )−1.5ξ

−0.25 l (3ξ+1 )] .

E1uation "4# 7o,al se,ond order deri)ati)es

N#%#: 3he ,ur)ature-dis(la,ement matri; >+? *or a +eam is elaorated 'ith gloal se,ond orde

sha(e *un,tion deri)ati)es 'hi,h are deri)ed 0 means o* the Ka,oian matri; *rom the lo,a

se,ond order deri)ati)es as sho'n in the ust (re)ious euation.

c# acoian matrices

A summar0 o* the Ka,oian matri,es *or the elements used in the Fortran ,ode are

sho'n

J =[∂ N

1

∂ ξ

∂ N 2

∂ ξ

∂ N 3

∂ ξ

∂ N 4

∂ ξ

∂ N 1

∂ η

∂ N 2

∂ η

∂ N 3

∂ η

∂ N 4

∂η] [

x1

y1

x2 y2

x3

x4

y3

y4

] .E1uation "8# Ka,oian matri; o* a ilinear uadrilateral element

J =[∂ N

1

∂ ξ

∂ N 2

∂ ξ

∂ N 3

∂ ξ

∂ N 4

∂ ξ

∂ N 5

∂ ξ

∂ N 6

∂ ξ

∂ N 7

∂ ξ

∂ N 8

∂ ξ

∂ N 1

∂ η

∂ N 2

∂η

∂ N 3

∂ η

∂ N 4

∂ η

∂ N 5

∂ η

∂ N 6

∂ η

∂ N 7

∂ η

∂ N 8

∂ η] [

x1

y1

x2

y2

x3

x4 x

5

x6

x7

x8

y3

y4 y

5

y6

y7

y8

] .E1uation "6# Ka,oian matri; o* a uadrati, uadrilateral element

J =[ ∂ N 1∂ ξ ∂ N 2∂ ξ ][ x1 x2] .

E1uation "=# Ka,oian matri; o* a linear rod or one-dimensional eam

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J =[ ∂ N 1∂ ξ∂ N

2

∂ ξ

∂ N 3

∂ ξ ][ x

1

x2

x3

] . E1uation "?# Ka,oian matri; o* a uadrati, rod

d# Gloal deri2ati2es

3he gloal deri)ati)es are otained 0 means o* the in)erse o* the a,oian matri;. A

es(e,ial ,ase is (resented *or the elaoration o* gloal se,ond order deri)ati)es *or one-

dimensional *ormulation o* an Euler-+ernoulli eam. n the later ,ase1 the ,hain in)ersion

(ro,edure is used t'i,e. t ,an e seen straight *or'ard that e,ause the Ka,oian o* a

t'o node element is 5lled 'ith ,onstants1 the in)erse o* the Ka,oian is 5lled 'ith

,onstants and gets out o* the deri)ati)e in the se,ond in)ersion ste(.

[∂ N

1

∂ x

∂ N 2

∂ x

∂ N 3

∂ x

∂ N 4

∂ x

∂ N 1

∂ y

∂ N 2

∂ y

∂ N 3

∂ y

∂ N 4

∂ y ]=J

−1

[∂ N

1

∂ ξ

∂ N 2

∂ ξ

∂ N 3

∂ ξ

∂ N 4

∂ ξ

∂ N 1

∂ η

∂ N 2

∂ η

∂ N 3

∂ η

∂ N 4

∂η ].

E1uation "@# Gloal deri)ati)es *or a ilinear uadrilateral element

[∂ N

1

∂ x

∂ N 2

∂ x

∂ N 3

∂ x

∂ N 4

∂ x

∂ N 5

∂ x

∂ N 6

∂ x

∂ N 7

∂ x

∂ N 8

∂ x

∂ N 1

∂ y

∂ N 2

∂ y

∂ N 3

∂ y

∂ N 4

∂ y

∂ N 5

∂ y

∂ N 6

∂ y

∂ N 7

∂ y

∂ N 8

∂ y]=J −1[

∂ N 1

∂ ξ

∂ N 2

∂ ξ

∂ N 3

∂ ξ

∂ N 4

∂ ξ

∂ N 5

∂ ξ

∂ N 6

∂ ξ

∂ N 7

∂ ξ

∂ N 1

∂ η

∂ N 2

∂ η

∂ N 3

∂ η

∂ N 4

∂η

∂ N 5

∂ η

∂ N 6

∂ η

∂ N 7

∂ η

#

E1uation 'A# Gloal deri)ati)es *or a uadrati, uadrilateral element

[ ∂ N 1∂ x ∂ N 2∂ x ]=J −1[ ∂ N 1∂ ξ ∂ N 2∂ ξ ] . E1uation '"# Gloal deri)ati)es *or a linear rod

[ ∂ N 1∂ x∂ N

2

∂ x

∂ N 3

∂ x ]=J −1[∂ N

1

∂ ξ

∂ N 2

∂ ξ

∂ N 3

∂ ξ ] . E1uation ''# Gloal deri)ati)es *or a uadrati, rod

[ ∂2 N

1v

∂ x2

∂2 N

1θ

∂ x2

∂2 N

2v

∂ x2

∂2 N

2θ

∂ x2 ]= ( J −1 )2[ ∂

2 N

1v

∂ ξ2

∂2 N

1θ

∂ ξ2

∂2 N

2v

∂ ξ2

∂2 N

2θ

∂ ξ2 ] .

E1uation '3. Gloal se,ond order deri)ati)es *or a one-dimensional eam

%#


17/82

N#%: 3he suare o* the in)erse o* the Ka,oian is ,on,e(tuall0 )alid *or this (arti,ular ,ase gi)en

the *a,t that the Ka,oian *or a linear rod or a t'o node one dimensional eam element is a

s,alar.

e# Constituti2e relations

n general1 the ,onstituti)e relation matri; )aries 'ith its a((li,ation. For the ,urrent general FEM

(rogram the *ormulation o* .,/erelastic materials 'as ,onsidered. n that1 sense (lasti,it0 isnot ta8en into a,,ount nor *ailure en)elo(es to lo,8 the stress state o* in5nitesimall0 smal

(arti,les in the ,ontinuum.

Due to the addition o* the terms o* Area1 and moment o* nertia during the integration o* the

element sti/ness matri,es1 the ,onstituti)e relation >D? in the Fortran ,ode ,an e seen to e

merged 'ith the Area1 or moment o* nertia. 3his a((ears to e ,on)enient in the standardization

o* the Finite element (ro,edure. e)ertheless1 the original *ormulation o* >D? as in me,hani,s is

,onsidered *or the ,om(utation o* stresses.

[ D ]= (1−v )

(1+v ) (1−2 v ) [ 1

v

1−v 0

v

1−v 1 0

0 0 1−2 v

2 (1−v ) ] #E1uation '4# Constituti)e Elasti,it0 matri; *or (lain strain *ormulation o* t'o-dimensional solid odies

[ D ]= #

E1uation '8# Constituti)e relation *or one-dimensional rod and eam elements

0# 5% matri7

3he deri)ation o* the >+? matri; *or di/erent solid me,hani,s (rolems is not ,onsidered

as (art o* the s,o(e o* this re(ort. 3he >+? matri; ,alled strain-dis(la,ement matri; *or

me,hani,al (rolems o* t'o- dimensional elements *or 4!:-node uadrilateral elements1

and one dimensional rod elements is (resented. Additionall0 the ,ur)ature-dis(la,ementmatri; >+? *or one dimensional ending (rolems is sho'n as 'ell.

%$


18/82

[ B ]=

∂ N 1

∂ x 0

∂ N 2

∂ x 0

∂ N 3

∂ x 0

∂ N 4

∂ x 0

0∂ N

1

∂ y 0

∂ N 2

∂ y 0

∂ N 3

∂ y 0

∂ N 4

∂ y

∂ N 1

∂ y

∂ N 1

∂ x

∂ N 2

∂ y

∂ N 2

∂ x

∂ N 3

∂ y

∂ N 3

∂ x

∂ N 4

∂ y

∂ N 4

∂ x

#

E1uation '6# train-dis(la,ement matri; *or a +ilinear Ouadrilateral element.

[ B ]=

[

∂ N 1

∂ x 0

∂ N 2

∂ x 0

∂ N 3

∂ x … .0

∂ N 8

∂ x 0

0∂ N

1

∂ y 0

∂ N 2

∂ y 0 … .

∂ N 7

∂ y 0

∂ N 8

∂ y

∂ N 1

∂ y

∂ N 1

∂ x

∂ N 2

∂ y

∂ N 2

∂ x

∂ N 3

∂ y … .

∂ N 7

∂ x

∂ N 8

∂ y

∂ N 8

∂ x

]#

E1uation '=# train Rdis(la,ement matri; *or a Ouadrati, Ouadrilateral element

[ B ]=[ ∂ N 1∂ x ∂ N 2∂ x ] .

E1uation '?. train-dis(la,ement matri; *or a linear rod element

[ B ]=[ ∂ N

1

∂ x

∂ N 2

∂ x

∂ N 3

∂ x ] .E1uation '@# train-dis(la,ement matri; *or a uadrati, rod element

[ B ]=[ ∂2 N 1v

∂ x2

∂2 N 1θ

∂ x2

∂2 N 2

∂ x2

∂2 N θ

∂ x2 ] .

E1uation 3A# Cur)ature dis(la,ement matri; *or one-dimensional eam element

g# Stressestresses 'ithin this Fortran ,ode are ,al,ulated at the integration (oints. Con,e(tuall01 i*

the dis(la,ements are ,al,ulated 0 means o* the resolution o* the s0stem o* euations

the strain 5eld ,an e ,al,ulated at an0 (oint inside the domain o* the elements

e)ertheless1 it ,an e *oreseen that the mathemati,al *ormulation o* the sha(e

*un,tions in6uen,e greatl0 in the estimation o* stresses sin,e some sha(e *un,tions are

%!


19/82

more D? and ,ur)ature k}. 3his result must e multi(lied 0 the distan,e A? and the sha(e *un,tions in >? lies in the

generation o* the >+? matri;. 3his matri; is ,alled the strain-dis(la,ement matri; *or solid

(rolems1 and ,ur)ature-dis(la,ement matri; *or ending o* eams.

3he general e;(ression sho'n ao)e is modi5ed *or the ,ase o* one-dimensional

(rismati, rod elements as de(i,ted elo'

∫ $ [ % ]T [ $ ]T [ D ] [ $ ] [ % ] d& {# }=∫ $ [ % ]T d& {f } . E1uation 38

n the ,ase that a (rismati, eam element is anal0zed the e;(ression lies as de(i,ted elo'

∫ 'ner [ % ]T [ $ ]T [ D ] [ $ ] [ % ] d& {# }=∫ 'ner [ % ]T d& {f } . Euation "#

%9


20/82

3he e;(ressions sho' ,learl0 the standardized (ro,edure to e ta8en. n *a,t1 the maor

di/eren,e in the (ro,edure *or di/erent (rolems is the de,ision in ho' the right hand side o* the

euations is ,onsidered1 under traditional )ariational a((roa,h or sim(li5ed methodologies to

5nd the eui)alent nodal loads. Bsing the Gauss-7egendre Ouadrature the element sti/ness

matri,es lie as de(i,ted elo'

[k m ]=∑i=1

ni(

[ B ]T

[ D ] [B ] det ( J ) ) i .

Euation "$. Element sti/ness matri; *or a general t'o dimensional FEM (rolem

[k m ]=∑i=1

ni(

$ [ B ]T

[ D ] [ B ] det (J ) ) i .

Euation "!. Element sti/ness matri; *or one-dimensional (rismati, FEM (rolem

[k m

]=

∑i=1

ni(

'ner [ B ]T

[ D ] [B ] det ( J ) ) i.

Euation "9. Element sti/ness matri; *or one-dimensional (rismati, ending (rolem

a# Conce/tual low c.art o0 t.e

resolution in ortran

BROGR$M E

-Use library

-Initialization

3he 5rst ste( lies in the inititalization o* )ariales. D0nami, arra0s are (re*erred to

set u( the ,orre,t size and sa)e ,om(utational resour,es. ere the )ariales are

read *rom the in(ut data 5le1 and more im(ortant the numer o* euations and size

o* the >loads? )e,tor is determined in ,omination 'ith the (osition o* diagonal

elements in 8diag.

-Calculate Stifness matrix

3he general (ro,edure to ,al,ulate the >8)? sti/ness matri; in s80line storage is

sho'n in a des,ri(ti)e manner

Ste/"# et >8)a,um? to zero. Thi,h is a tem(orar0 )ariale *or a,,umulation o*

>8)? ,onstriutions isn s80line storage

*oo/ t.e elements 0rom iel"Dnels2


21/82

Ste/'# et the >8m? matri; to zero

Ste/3# 3he ,oordinates1 length in the ,ase o* a eam: o* the elements are

otained

Suroutine: *,oord1 6el

Ste/4# 3he ,onstituti)e relation matri; >D? is ,al,ulated. torage *or *urther

use.

Suroutine: *,onst

*oo/ t.e integration /oints O ini/"Dni/

Ste/8# 3he lo,al deri)ati)es o* the sha(e *un,tions are otained. * a

one dimensional eam (rolem is eing sol)ed1 the sha(e *un,tions o*

a linear rod 'or8 *or the a,oian o* the eam. e)ertheless1 the sha(e

*un,tions *or inter(olating BN is di/erent. * the reader reuires the

re)ision o* the *ormulas.Suroutine: 6o,deri)e1 and 6o,se,deri) in the ,ase o* a

ending (rolem.

Ste/6# 3he multi(li,ation o* the lo,al deri)ati)es and the ,oordinates

is made to ,om(ute the Ka,oian.

Suroutine: *a,

N#%#: 3he Ka,oian matri; o* a linear rod ,an e used *or a one-

dimensional Euler +ernoulli eam. 3his ,omes *rom the *a,t thatoth elements are de5ned geometri,all0 0 t'o nodes

e)ertheless1 the sha(e *un,tions to inter(olate the trans)erse

are di/erent1 thus1 the eam element ,an no longer e regarded

as an iso(arametri, element.

Ste/=# 3he in)erse o* the Ka,oian matri; is ,al,ulated to ,hange the

re*eren,e s0stem1 *rom the lo,al to the gloal s0stem. * a ending

(rolem is ta8en into a,,ount1 the doule in)ersion (ro,edure must e

(er*ormed. ere the *un,tion o* in)ersematri;: *rom the lirar0 is

used.

Suroutine: *gloderi)1 *glose,deri) in the ,ase o* a ending

(rolem is a,ti)ated automati,all0.

Ste/?# 3he determinant o* the Ka,oian matri; is determined. 3he

*un,tion determinantmatri;: is used inside the suroutine.

Suroutine: *det

2%


22/82

Ste/@.3he >+? matri; is ,al,ulated. 3he in(ut )alues into this

suroutine must e the des,ri(tion o* the element and the gloa

deri)ati)es se,ond order in ,ase o* ending (rolem:.

Suroutine: *strainsdis(

Ste/"A# 3he ,ontriution o* the ini(-th integration (oint into the

element sti/ness is a,,umulated in >8m?. 3he suroutine retri)es the

(reious )alue o* >8m?1 and the ,om(onents *or integration >+?

>D?1detK:1Ti and A or ner i* reuired:. Ea,h >+? is stored *or *urther use.

Ste/ ""# Qut(ut (artial results

EN O ;0or integration /oints<

Ste/ "'# Cal,ulate steering )e,tor o* the element1 and store *or *urther use.

tore >8m? *or *urther use.

Ste/"3# et to Jero >8)? to a)oid undesirale (re)ious )alues. Com(ute ne'

,ontriution to >8)?.

Suroutine: *s(ar) ,olle,ts g1 8m and 8diag ,om(uted in the

intitialization stage:

Ste/ "4# A,,umulate ne' >8)? in >8)a,um?tem(orar0 name *or the )ariale

>8)? to a)oid (rolems 'ith the use o* the suroutine s(ar) *rom the lirar0:

EN O ;0or elements<

Ste/ "8# tore >8a,um? in >8)? *or *urther usage in the ,ode.

-Loading conditions

Ste/ "6# Uetrie)e the loads *rom the in(ut data 5le. As soon as (ossile1 ,lose the

s(a,e %: in 'hi,h the data 5le 'as eing read. odes 'ithout *or,e ,om(onents

are le*t 'ith zeros.

-Equation solution

Ste/ "=# 0stem o* euations is sol)ed and solutions are stored in >loads?.

Suroutines: (arin and s(aa,

-Output results

Ste/ "?# Dis(la,ements are re-stored in a gloal dis(la,ement matri; in,luding zeros *or

5;ed *reedoms

22


23/82

Suroutine *glodis(

*oo/ t.e elements O iel"Dnels

Ste/ "@# Call a,8 >8m? and retrie)e the ,oordinates o* the element.

A*ter'ards the ,oordinates are used to ,om(ute the lo,ation o* the

integration (oints in the gloal s0stem o* re*eren,e

Suroutine: *,oord

Ste/ 'A# * a one dimensional rod or ending (rolem is eing anal0zed >D?

matri; is restored as in me,hani,s that is >D?AEHA or >D?Ener:Hner:. 3his

is done due to the *a,t that >D? and the Area or moment o* nertia 'ere

merged in >D? to gain a more standard (ro,edure *or an0 8ind o* (rolem.

*oo/ t.e integration /oints O ini/"Dni/

Ste/ '"# Call a,8 >+? to e used *or ,al,ulation o* stresses at

Gaussian (oints

Ste/ ''. Uetrie)e the dis(la,ements at the nodes o* the element *rom

gloal arra0

Suroutine: *eldis(

Ste/ '3# Cal,ulate the stresses as >D?>+?BN1 in the ,ase o* ending

the results 'ill e normalized 0 the distan,e to the neutral a;is. 3he

user o* the (rogram must e a'are o* e)aluating this e;(ression at the

neutral a;is. t should e re,alled that at 0 *rom the neutral a;is o* a

eam no stresses are de)elo(ed.

Ste/ '4# 3he sha(e *un,tions are e)aluated at the lo,al ,oordinates

,orres(onding to the lo,ation o* the ini(-th integration (oint.

Ste/ '8# 3he gloal ,oordinates ate 'hi,h the stresses 'ere

,al,ulated are ,om(uted as the matri; multi(li,ation o* the sha(e

*un,tions and the gloal ,oordinates.

Suroutine: *,on)

Ste/ '6# Qut(ut o* the stresses1 and lo,ation at 'hi,h the ini(-thintegration (oint at 'hi,h ,al,ulations 'ere (er*ormed

EN O ;0or t.e integration /oints<

Ste/ '=# Cal,ulate internal *or,es 'ithout sutra,tion o* eui)alent noda

*or,es. 3he user is in ,harge o* identi*0ing i* eui)alent nodal *or,es 'ere2"


24/82

used. * the later is true1 the real internal *or,es are otained 0 sutra,ting

the eui)alent nodal *or,es *rom the result o* the (rogram.

3he internal *or,es are otained as the matri; multi(li,ation o* >Lm? and BN.

N#+ * onl0 nodal *or,es 'ere ,onsidered in the original loading s0stem1 the

result o* the (rogram is de5niti)e. Ue)ise euation " *or more details.

Ste/ '?# Qut(ut o* the internal *or,es

EN O ;0or elements<

-Visualization o results

Ste/ '@# Generate the mesh using the gloal ,oordinates arra0 and the gloal

numering arra0. 3he 5le generated 'ill e named as


25/82

igure "'# Elu,idation o* *or,ed meshing te,hniues 0 means o* Delaune0triangulation

-Initialization

Ste/"# 3he 5rst ste( lies in the inititalization o* )ariales. 3he numer o* ,olumns

and ro's o* elements n;1n0: are read *orm the in(ut data 5le. 3he numer o* node

,olumns and node ro's are ,al,ulated nn;1 nn0:. Additionall01 the )e,tors o*

guiding ,oordinates ;,oord and 0,oord are allo,ated the ,orre,t size.

nnx=nx+1 # E1uation 4A

nny=ny+1 # E1uation 4"

nn=nnx∗nny # E1uation 4'

Ste/ '# Che,8 the a;is in 'hi,h the smallest numer o* nodes is reuired. * eual

numer o* nodes is reuired1 numering 'ill e (er*ormed along the ;- dire,tion1

'hi,h means the ro' is 8e(t ,onstant until another ro' is in)ol)ed.

N#%. A tri(le loo( is (ro(osed ha)ing a dumm0 )ariale 8 that in,reases the

node numering 'hile the loo(s 8ee( going. A similar (ro,edure is a((lied to

numer the ,orres(onding elements. ere the re(etiti)e (attern along the

mo)eale a;is is ta8en as an ad)antage to relate the (osition o* the lo,a

nodes %121"14: to the gloal numering as sho'n elo'.

if (nx>ny) then

!------------Numbering along y axis----------

!---------Numbering of nodes----------

k=1

DO WHILE (k


26/82

g_coord(1,k)=xcoord(1,i)

g_coord(2,k)=ycoord(1,j)

k=k+1

END DO

END DO

END DO

!--------Numbering of elements---------

k=1

DO WHILE (k


27/82

Ste/ 3# 3he (re)ious Fortran ,ode ,ontinues 'ithout ,hanges.

6# $*I$&ION O RES(*&Sa# $7ial com/ression o0 a

column;laterall, restrained<

igure "3# Gra(hi,al re(resentation o* the sam(le (rolem1 5;ed asement andlaterall0 restrained

3he sam(le (rolem gi)en to sol)e 'as related to solid od01 laterall0 restrained in the

nodes and in the ase. 3he (rolem in the sam(le out(ut 'as related to a uniue

element. 3he same tas8 'as (er*ormed 'ith the Fortran ,ode atta,hed to this re(ort

3he results mat,hed until the 4th digit a*ter the de,imal (oint.

e)ertheless1 the solution (ro)ided must ha)e een (ro)ed to e right. As result1 and

anal0ti,al solution 'as deri)ed under the theor0 o* elasti,it0. 3his (rolem 'as treatedas a +ia;ial test in 'hi,h the lateral dis(la,ement is im(osed to e zero. 3he anal0ti,al

solution *or the stresses is de(i,ted elo'.

For the (rolem )." and* ! y=¿ FHA-.&V2H%.-%.. As a result the lateral stress

must e

* ! x= v

1−v * ! y=

0.3

1−0.3(−1.0)=−0.4286

Uesults gi)en 0 the Program mat,h 'ith this anal0ti,al solution 'ith a minor error

Solutio

n

* ! x Error

* ! y Error

$nal,ti

cal

-

.42!#

- -%. -

Brogra - .%W -.$4 2#W

2$

2 "

% 4


28/82

m .42!&

$c.art '# Com(arison o* solutions *or the sam(le (rolem

3he internal *or,es at the ase nodes 'as another (ro)ed that 'as made to this (rolem. n *a,t1

the internal *or,es o* the element at the asement must e eual to the total load on to(. 3he

linear distriuted *or,es on the asement are

+ 2= + 4=0.5

As it ,an e seen in the in(ut data 5le1 the internal *or,es o* this uniue element are in *a,t the

rea,tion *or,es that euilirate the s0stem.

# $7ial com/ression o0 a

column;laterall, unrestrained<

igure "4# Column under a;ial ,om(ression1 +ilinear uadrilateral elements more thanone element:

3he same (rolem 'as sol)ed 0 means o* a 4 element ,olumn. e)ertheless1 the oundar0

,onditions 'ere ,hanged to e;(lore the stailit0 o* the ,al,ulation s,hemes as sho'n in the in(ut

data 5le sam(le. 3he ,hosen element t0(e 'as a ilinear uadrilateral 4 nodes (er element:.

3he (rolem ,an e sol)ed 0 means o* a sim(li5ed s0stem in 'hi,h onl0 a;ial ,om(ression is

ta8en into a,,ount1 *or that anal0ti,al solution the dis(la,ement on to( the ,olumn 'as

,om(uted

* y=* ! ∗ y

=

−1.0∗1

1.0E6=−1.0E-6

Solutio

n

! y Error

* y Error

$nal,ti

cal

-%. - -%.E-

#

-

Brogra

m

-%. .W -.!$9 %2W

2!


29/82

c.art 3# Com(arison o* solutions *or a 4 ilinear element ,olumn laterall0 unrestrained

3he results are at least *rom the same order o* magnitude regarding dis(la,ements. o'e)er1 it

should e mentioned that (erha(s the assum(tion o* onl0 a;ial intera,tion 'as a it iased and

so the results o* the (rogram are still o8. * the )isualization tool o* the (rogram is used1 the

dis(la,ement )e,tors in *a,t a((ear to e slightl0 tilted to the outer (art o* the ,olumn. 3his

should ha)e een the 5rst assum(tion *or the anal0ti,al solution e,ause o* the t'o-dimensiona

solution o* the FEM s,heme. +elo' the mesh and dis(la,ement )e,tors generated 0 the

(rogram are sho'n.

igure "8# Bnde*ormed and de*ormed mesh1 dis(la,ement )e,tors *or ilinear element,olumn laterall0 unrestrained

igure "6# Eui)alent nodal *or,es im(osed *or uadrati, uadrilateral elements morethan one element:

A ro' o* 4 uadrati, elements 'as used to anal0ze the ,olumn (rolem unrestrained laterall0:.

3he eui)alent nodal *or,es 'ere ,al,ulated ased on the ode 0 ode lum(ing. As a result1 the

middle node gets hal* o* the total load on to( o* the ,olumn1 'hile the nodes on the ,orners

re,ei)e one *ourth o* the total *or,e ea,h. 3he situation e;(lained is elu,idated in the Figure %#.

-.%2"2#E-&

Solution* y Error * ! y Error

$nal,ti

cal

-%.E-# - -%. -

Brogra

m

-%.2"E-

#

2"W -%.4 4W

c.art 4# Com(arison et'een a((ro;imate anal0ti,al solution and (rogram results *or 4uadrilateral elements unrestrained laterall0.

29

"

2

%

&

#

$


30/82

igure "=# Bnde*ormed1 de*ormed mesh1 dis(la,ement )e,tors *or uadrati,re,tangular elements laterall0 unrestrained more than one element:.

Another im(ortant as(e,t to e ta8en into a,,ount is that the restrained asement must e

(la0ing a great in6uen,e in the dis(la,ement 5eld o* the ,olumn (arti,les. 3o e;(lore that

matter1 the modi5ed )ersion o* the Program 'as used to ,reate a denser mesh.

3he results sho' that the dis(la,ement 5eld starts to redu,e as the lo,ation a((roa,hes to the

restrained asement. n that sense1 the anal0ti,al assum(tion ,ould e )alid onl0 *or the to((arti,les o* the od0. 3he (reliminar0 ,on,lusion that the restraints a/e,t the distriution o*

stresses arises so *ar.

Solutio

n

* y Error

$nal,ti

cal

-%.E-# -

Brogra

m

-.$4E-

#

2#W

-.!$E-#

%"W

-.2E-# !W.EI -

c.art 8# Com(arison o* solutions *or an irregular element arra0 ,olumn laterall0unrestrained more than one element:

igure "?# Mesh and dis(la,ement )e,tors *or an irregular element arra0 ,olumnlaterall0 unrestrained

"


31/82

As a 5nal (ro,edure to test the sensiilit0 o* the (rolem additionall0 to the )i,init0 to restrainedoundaries and the distriution o* elements1 the t0(e o* element 'as ,hanged. 3he same(rolem 'as anal0zed 0 means o* one dimensional rod elements to oser)e ho' mu,h itde)iates *rom the


32/82

igure "@# Cantile)er eam under a ,on,entrated load at the *ree e;treme

3he (rolem o* a eam (er*e,tl0 emedded in one e;treme1 under a ,on,entrated load on the

*ree e;treme 'as anal0zed 0 means o* an eui)alent (lain strain *ormulation1 'hi,h means that

an in5nite sla is treated as ada,ent eam disregarding the *ri,tion et'een the (ortions o* the

sla. E;a,t solutions *or the ma;imum slo(e and trans)erse dis(la,ement o* the eam at the

*ree e;treme are a)ailale and 'ere used to )alidate the results o* the (rogram. 3he eui)alentdistriuted load o* -2. o)er a 'idth o* .& is eui)alent to -%. in an Euler-+ernoulli eam in

a one-dimensional *ormulation

3he anal0ti,al solution ,onsidering the stati, euilirium sho's that the ending moment should

e eual to M2. n that sense the stress at e;treme atta,hed to the 'all might e ,al,ulated

and ,om(ared to the result gi)en in the (rogram. Qne stress (oint Gauss (oint: 'as (i,8ed

randoml0 to ,he,8 its )alue against an anal0ti,al solution as de(i,ted elo'

! = ,y

'ner=

200∗(2−0.0528)2

∗(0.25−0.0132 )=8850.23

Solution ; 0 ! y Error

$nal,tical .&2

!

.%"2 !!&.2" -

Brogram Fuadrilateral

element

.&2

!

.%"2 !$"". %.4W

c.art =# Com(arison o* a ,antile)er t'o dimensional solid eam and anal0ti,al solutions.

igure 'A# De*ormed mesh1 and dis(la,ement )e,tors o* a ,antile)er eam.

"2


33/82

N#%: 3he eam 'as modeled as a t'o dimensional solid od01 in that sense the *reedom o*

rotation is someho' in)ol)ed. Furthermore1 oth solutions mat,h in good a,,ura,0. Finall01 to

e;(lore the e/e,t o* the sha(e *un,tions1 one dimensional eam elements 'ere used instead o* a

t'o dimensional element. 3he ma;imum slo(e and trans)erse dis(la,ement 'as ,om(uted *rom

e;isting solutions and ,om(ared 'ith the results gi)en 0 the (rogram.

θmax= − - l2

2 'ner=

−100 (22 )2 (200e6 ) (5.21e-3 )

=−0.19194e-3

vmax= − - l

3

3 'ner=

−100 (23 )3 (200e6 ) (5.21e-3 )

=−2.5592e-4

Solution θmax vmax Error

$nal,tical −0.19194e-3 −2.5592e-4 -

Brogram Fuadrilateral

element

−0.19194e-3 −2.5592e-4 .W

,hart !. Com(arison o* a ,antile)er one dimensional solid eam and anal0ti,al solutions.

As it ,an e seen1 results *rom a one dimensional a((roa,h mat,h 'ith the anal0ti,al solution.

e)ertheless1 the rigidit0 in terms o* the gradient (rodu,ed 0 the sha(e *un,tions might di/er

*rom those gradients (rodu,ed in realit0.

=# $N$*>SIS O RES(*&S

n general uadrati, sha(e *un,tions tend to gi)e smoother results in terms o* the stressgradients (rodu,ed 0 the sha(e *un,tions inside the domain o* the elements. e)ertheless1 the

use o* larger numer o* nodes might e ,umersome 'hile (re(aring the in(ut data 5le. 3his

issue 'as o)er,ome 'ith the generation o* automati, node and element numering inside the

,ode *or uadrilateral elements.

Additionall01 ,he,8ing internal *or,es at e;treme elements 'as (ro)ed to e a good tool o*

)alidation o* the results. From the grou( o* sam(le in(ut 5les anal0zed 0 means o* the Fortran

,ode1 it ,ould e seen that at least the order o* magnitude o* stresses1 and dis(la,ements must

mat,h 'ith anal0ti,al solutions1 no matter ho' sim(li5ed this anal0ti,al solutions are assumed as

long as engineering udgment is used to test the assum(tions.

Furthermore1 the e/e,t o* the numer o* nodes ,ould e oser)ed. 3he higher the numer o

nodes1 the etter the )isualization out(uts tend to e. 3hese gra(hi,al out(uts are a good tool to

,he,8 i* the oundar0 ,onditions are ,orre,tl0 a((lied es(e,iall0 i* s0mmetr0 is dominant.

3he node 0 node lum(ing a((eared to e e/e,ti)e in assisting in the load matri; ,om(onents o

the loading matri;. o'e)er1 its e/e,ti)eness 'as onl0 tested in loading s0stems 'ith noda

*or,es1 and ,al,ulation o* rea,tion *or,es might get more di,ult *or the user due to the *a,t that

the (rogram does not distinguish et'een eui)alent and nodal *or,es. 3his issue might e""


34/82

o)er,ome 'ith a slight modi5,ation o* the ,ode1 ma8ing it to ,al,ulate eui)alent nodal *or,es 0

itsel*.

Moreo)er1 the use o* s80line storage s,hemes 'as (ro)ed to e hel(*ul in ma8ing a standard

,om(uter sol)e 3'o-dimensional (rolems 'ithout the ris8 o* e;,eeding its a)ailale memor0.

3he ,om(utational resour,es reuired *or (rodu,ing more nodes is de5nitel0 im(er,e(tile *o

the 8ind o* (rolems that 'ere sol)ed in this re(ort. n that sense1 it is highl0 )aluale to retrie)e

a more realisti, solution under the solution o* t'o dimensional FEM euations.

Finall01 it ,an e mentioned that the


35/82

@# REERENCES

5" Moa2eni S# ;"@@@


36/82

"A#

"A# $BBENI!

$# INB(& S$MB*ES: PAU3 %A-D:1 PAU3 2A-+:1 PAU3 "A-+:

%# RES(*&S: PAU3 %A-D:1 PAU3 2A-+:1 PAU3 "A-+:

C# OR&R$N COEPAU3 %1 PAU3 2

"#


37/82

INB(& S$MB*ES:PART 1-“LINEAR ELEMENTS INPUT DATA FILES”:

B$R& "$:

-% OBADU7A3EUA7 E7EME3 BDEU AXA7 CQMPUEQ

7A3EUA77 UE3UAED:

B$R& "%:

-4 OBADU7A3EUA7 E7EME3 BDEU AXA7 CQMPUEQ 7A3EUA77

BUE3UAED:

B$R& "C:

-4 UQD E7EME3 A CQ7BM BDEU AXA7 CQMPUEQ Q 3Q

B$R& ":-

PART 2-“QUADRATIC ELEMENTS INPUT DATA FILES”:

B$R& '$:

-4 E7EME3 CQ7BM 2D CBADUA3C OBADU7A3EUA7 E7EME3:

B$R& '%:-4 E7EME3 CQ7BM %D CBADUA3C UQD E7EME3:

PART 3-“AUTOMATIC MESHING INPUT DATA FILES”:

B$R& 3$:

- UUEGB7AU E7EME3 CQ7BM 2D +7EAU OBADU7A3EUA7

E7EME3:

B$R& 3%:

- UUEGB7AU E7EME3 CA37EEU +EAM 2D +7EAU

BADU7A3EUA7 E7EME3

"$


38/82

PART 1-LINEAR ELEMENTS INPUT DATA FILES.

N.B.: Bold !""#$!" "%#&d 'o( !)*l#%+o& #%%!(" o&l,.

B$R& "$:

" F($RI*$&ER$* E*EMEN& (NER $!I$* COMBRESSION ;*$&ER$**>

RES&R$INE

(NRES&R$INE


39/82

B$R& "C:

4 RO E*EMEN&S IN $ CO*(MN (NER $!I$* COMBRESSION ON &OB:

!l!!&% %,*! &o. &od!"'r%d' 2

&o. !l!!&%" &o. &od!" %o%#l/ &o. +*" *!( !l!!&%/ &o o''(!!do" *!( &od! &o. "%(!"" %!(" &o o' d+!&"+o&"4 ! 2 1 1 1

&*0%,*!"1

P(o* E A/1.0e 1.0

Glo#l &od#l oo(d+%!"0.0 -0.2! -0.! -0."! -1.0

Glo#l !l!!&% &od! &4!(+&$ $lo#l &od! &4!(" #" lo#l &od!"/2 1 3 2 4 3 ! 4

Glo#l '(!!do" &' &od! &4!( 15'(!! 65'+)!d/1

! 0

Lo#d!d &od!" && &od! &4!( lo#d ,/

11 -1.0

B$R& ":

4 %E$M E*EMEN&S IN $ C$N&I*EER %E$M (NER BOIN& *O$ $& &)E

E!&REME:

!l!!&% %,*! &o. &od!"'&ea' 2

&o. !l!!&%" &o. &od!" %o%#l/ &o. +*" *!( !l!!&%/ &o o''(!!do" *!( &od! &o. "%(!"" %!(" &o o' d+!&"+o&"4 ! 2 2 1 1

&*0%,*!"1

P(o* E I&!(/200.0e !2.1e-4

Glo#l &od#l oo(d+%!"0.0 0.!0 1.0 1.! 2.0

Glo#l !l!!&% &od! &4!(+&$ $lo#l &od! &4!(" #" lo#l &od!"/1 2 2 3 3 4 4 !

Glo#l '(!!do" &' &od! &4!( 15'(!! 65'+)!d/

11 0 0

Lo#d!d &od!" && &od! &4!( lo#d ,/1

"9


40/82

PART 2-(UADRATI) ELEMENTS INPUT DATA FILES.

N.B.: Bold !""#$!" "%#&d 'o( !)*l#%+o& #%%!(" o&l,.

B$R& '$:

4 F($RI*$&ER$* E*EMEN&S (NER $!I$* COMBRESSION ;*$&ER$**>

(NRES&R$INE 11 ; < > 9 16 13 12 1< 1 11 12 13 1= 19 1>21 1; 1< 1> 19 26 23 22Glo#l '(!!do" &' &od! &4!( 15'(!! 65'+)!d/321 6 6 22 6 6 23 6 6Lo#d!d &od!" && &od! &4!( lo#d ,/3

B$R& '%:4 RO E*EMEN&S IN $ CO*(MN (NER $!I$* COMBRESSION ON &OB:

!l!!&% %,*! &o. &od!"'r%d' 3

&o. !l!!&%" &o. &od!" %o%#l/ &o. +*" *!( !l!!&%/ &o o''(!!do" *!( &od! &o. "%(!"" %!(" &o o' d+!&"+o&"4 $ 2 1 1 1

&*0%,*!"1

P(o* E A/1.0e 1.0

Glo#l &od#l oo(d+%!"0.0 -0.12! -0.2! -0.3"! -0.! -0.2! -0."! -0.#"! -1.0Glo#l !l!!&% &od! &4!(+&$ $lo#l &od! &4!(" #" lo#l &od!"/3 2 1 ! 4 3 " ! $ # "

Glo#l '(!!do" &' &od! &4!( 15'(!! 65'+)!d/1

$ 0

L%aded *%de+ ,** *%de *u&er l%ad /

1

1 -1.0

4


41/82

PART 3- AUTMATI) MESIN INPUT DATA FILES

N.B.: Bold !""#$!" "%#&d 'o( !)*l#%+o& #%%!(" o&l,.

B$R& 3$:

IRREG(*$R E*EMEN& CO*(MN ;' %I*INE$R F($RI*$&ER$* E*EMEN&<

6


42/82

RES(*&S:PART 1-“LINEAR ELEMENTS OUTPUT DATA FILES:

B$R& "$:

-% OBADU7A3EUA7 E7EME3 BDEU AXA7 CQMPUEQ

7A3EUA77 UE3UAED:

B$R& "%:

-4 OBADU7A3EUA7 E7EME3 BDEU AXA7 CQMPUEQ 7A3EUA77

BUE3UAED:

B$R& "C:

-4 UQD E7EME3 A CQ7BM BDEU AXA7 CQMPUEQ Q 3Q

B$R& ":

-

PART 2-“QUADRATIC ELEMENTS OUTPUT DATA FILES”:

B$R& '$:

-4 E7EME3 CQ7BM 2D CBADUA3C OBADU7A3EUA7 E7EME3:

B$R& '%:

-4 E7EME3 CQ7BM %D CBADUA3C UQD E7EME3:

PART 3-“AUTOMATIC MESHING OUTPUT DATA FILES”:

B$R& 3$:

- UUEGB7AU E7EME3 CQ7BM 2D +7EAU OBADU7A3EUA7E7EME3:

B$R& 3%:

- UUEGB7AU E7EME3 CA37EEU +EAM 2D +7EAU

BADU7A3EUA7 E7EME3

42


43/82

RES(*&S:

B$R& "$

" F($RI*$&ER$* E*EMEN& (NER $!I$* COMBRESSION

;*$&ER$**> RES&R$INE<

4"


44/82

" F($RI*$&ER$* E*EMEN& (NER $!I$* COMBRESSION ;*$&ER$**>

RES&R$INE


45/82

integration /oint '

s.a/e 0unction deri2ati2es wit. res/ect to local coordinates:

-.%#24"&29"%9$#" -."94""$$#!2"$ -."94""$$#!2"$

."94""$$#!2"$ ."94""$$#!2"$ .%#24"&29"%9$#"

.%#24"&29"%9$#" -.%#24"&29"%9$#"

acoian matri7: .& . . .&

determinant o0 t.e acoian matri7:

.2&

s.a/e 0unction deri2ati2es wit. res/ect to gloal coordinates:

-.2%%"24!$&!#"9&2# -.$!!#$&%294%"#4$ -.$!!#$&%294%"#4$

.$!!#$&%294%"#4$ .$!!#$&%294%"#4$ .2%%"24!$&!#"9&2# .2%%"24!$&!#"9&2#

-.2%%"24!$&!#"9&2#

strain dis/lacement matri7D %D i/: ' -.2%%"24!$&!#"9&2# . -.$!!#$&%294%"#4$ . -.$!!#$&%294%"#4$

-.2%%"24!$&!#"9&2# -.$!!#$&%294%"#4$ . .$!!#$&%294%"#4$ .

.$!!#$&%294%"#4$ -.$!!#$&%294%"#4$ .$!!#$&%294%"#4$ .

cumulati2e km matri7D km

"4"9$&.9!"!"$&9 %!9&!&."#$#4$&%%2 -$4"$.&2229!&444$ -"$9%$.$"&29&22

-%442".$$%!%94* -%#"%".#9&$%2#!9! -%92"$.#!9$%9&! -4&"&4.&9!4%#$"%94

%!9&!&."#$#4$&%%2 4!2$#2.%"$4$"92!" "$9%$.$"&29&22 -"!##!.2&$&9"&4#$

-%"4$.9%9#!""4#"! -%442".$$%!%94* -9"4"%.&2%49"#&49$ 4!$#.92#%2!

-$4"$.&2229!&444$ "$9%$.$"&29&22 "4"9$&.9!"!"$# -%!9&!&."#$#4$&%%2

-%92"$.#!9$%9&! 4&"&4.&9!4%#$"%94 -%442".$$%!%94* %#"%".#9&$%2#!9!

-"$9%$.$"&29&22 -"!##!.2&$&9"&4#$ -%!9&!&."#$#4$&%%2 4!2$#2.%"$4$"929

9"4"%.&2%49"#&49$ 4!$#.92#%2! %"4$.9%9#!""4#"! -%442".$$%!%94*

-%442".$$%!%94* -%"4$.9%9#!""4#"! -%92"$.#!9$%9&! 9"4"%.&2%49"#&49$

2"294$.!$9!#9#4$ &$99.24$$"$%%424 %"&9%."$"#29$#&" -%%&9.!49&4$422!&

-%#"%".#9&$%2#!9! -%442".$$%!%94* 4&"&4.&9!4%#$"%94 4!$#.92#%2!

&$99.24$$"$%%424 94%#.9#$99!"9&! %%&9.!49&4$422!4! %992.!$!%&&&"#&&"&

-%92"$.#!9$%9&! -9"4"%.&2%49"#&49$ -%442".$$%!%94* %"4$.9%9#!""4#"!%"&9%."$"#29$#&" %%&9.!49&4$422!4! 2"294$.!$9!#9#& -&$99.24$$"$%%424

4&


46/82

integration /oint 3


-.%#24"&29"%9$#" -.%#24"&29"%9$#" -."94""$$#!2"$

.%#24"&29"%9$#" ."94""$$#!2"$ ."94""$$#!2"$

.%#24"&29"%9$#" -."94""$$#!2"$



.2&


-.2%%"24!$&!#"9&2# -.2%%"24!$&!#"9&2# -.$!!#$&%294%"#4$

.2%%"24!$&!#"9&2# .$!!#$&%294%"#4$ .$!!#$&%294%"#4$ .2%%"24!$&!#"9&2#

-.$!!#$&%294%"#4$ strain dis/lacement matri7D %D i/: 3

-.2%%"24!$&!#"9&2# . -.2%%"24!$&!#"9&2# . -.2%%"24!$&!#"9&2#

-.2%%"24!$&!#"9&2# -.$!!#$&%294%"#4$ . .2%%"24!$&!#"9&2# .

.2%%"24!$&!#"9&2# -.$!!#$&%294%"#4$ .$!!#$&%294%"#4$ .

.$!!#$&%294%"#4$ . .$!!#$&%294%"#4$ .$!!#$&%294%"#4$ .2%%"24!$&!#"9&2#


"#"299.24"!&4!"" 2"2.&%2%%4""4 44"&!.%#4&%#""!4& -2!""2.&%!!!!&!#42#

-2%#"4#.%&$$29%!4#" -%4#"$$.$9!9!44&%"4 -%9%"%%.2&#4%2"#! -2%.%9422!"9&2#2"2.&%2%%4""4 &2!&."#"$#44$2 &$##%.4$$$%$!"#!$ -"!%%.!%!&%&$$!4

-%$4%"&.229##&2!$2 -2%#"4#.%&$$29%!4#" -!"!4#.9##!&2$4%%! 99!$2.#%24!492!

44"&!.%#4&%#""!4& &$##%.4$$$%$!"#!$ &&$&99.!%%$2!#2&% -229#49.4$9"#!"4#

-"!%%.!%!&%&$$!4# -2!""2.&%!!!!&!#42# -2%#"4#.%&$$29%!4#" 2"2.&%2%%4""4

-2!""2.&%!!!!&!#42# -"!%%.!%!&%&$$!4 -229#49.4$9"#!"4# &&$&99.!%%$2!#2&

&$##%.4$$$%$!"#!$ 44"&!.%#4&%#""!4& 2"2.&%2%%4""4 -2%#"4#.%&$$29%!4#"

-2%#"4#.%&$$29%!4#" -%$4%"&.229##&2!$2 -"!%%.!%!&%&$$!4# &$##%.4$$$%$!"#!$

&2!&."#"$#44$2$ 2"2.&%2%%4"" 99!$2.#%24!492! -!"!4#.9##!&2$4%%!

-%4#"$$.$9!9!44&%"4 -2%#"4#.%&$$29%!4#" -2!""2.&%!!!!&!#42# 44"&!.%#4&%#""!4&

2"2.&%2%%4"" "#"299.24"!&4!"4 -2%.%9422!"9&2# -%9%"%%.2&#4%2"#!

-%9%"%%.2&#4%2"#! -!"!4#.9##!&2$4%%! -2%#"4#.%&$$29%!4#" 2"2.&%2%%4""4

99!$2.#%24!492! -2%.%9422!"9&2# "$$!4.$9&!!992!#& -9!#"."&%229#&!

-2%.%9422!"9&2# 99!$2.#%24!492! 2"2.&%2%%4""4 -2%#"4#.%&$$29%!4#"

4#


47/82

integration /oint 4


-."94""$$#!2"$ -.%#24"&29"%9$#" -.%#24"&29"%9$#"

.%#24"&29"%9$#" .%#24"&29"%9$#" ."94""$$#!2"$

."94""$$#!2"$ -."94""$$#!2"$



.2&


-.$!!#$&%294%"#4$ -.2%%"24!$&!#"9&2# -.2%%"24!$&!#"9&2#

.2%%"24!$&!#"9&2# .2%%"24!$&!#"9&2# .$!!#$&%294%"#4$ .$!!#$&%294%"#4$

-.$!!#$&%294%"#4$

strain dis/lacement matri7D %D i/: 4 -.$!!#$&%294%"#4$ . -.2%%"24!$&!#"9&2# . -.2%%"24!$&!#"9&2#

-.$!!#$&%294%"#4$ -.2%%"24!$&!#"9&2# . .2%%"24!$&!#"9&2# .

.2%%"24!$&!#"9&2# -.2%%"24!$&!#"9&2# .2%%"24!$&!#"9&2# .

.$!!#$&%294%"#4$ . .$!!#$&%294%"#4$ .2%%"24!$&!#"9&2# .$!!#$&%294%"#4$


&$#92".$%$4&$24 24"!4.#%&"!4#%&"! 9#%&".!&%""%22 -4!$#.92"$#92"$

-2!!4#%.&4"#"!9%2!4 -24"!4.#%&"!4#%&"! -"!4#%&."$94"!%%# 4!$#.92"$#92"$

24"!4.#%&"!4#%&"! &$#92".$%$4&$2" 4!$#.92"$#92"$ -"!4#%&."$94"!%%

-24"!4.#%&"!4#%&"! -2!!4#%.&4"#"!9%2!4 -4!$#.92"$#92"! 9#%&".!&%""%22

9#%&".!&%""%22 4!$#.92"$#92"$ &$#92".$%$4&$2& -24"!4.#%&"!4#%&"!

-"!4#%&."$94"!%%# -4!$#.92"$#92"$ -2!!4#%.&4"#"!9%2!4 24"!4.#%&"!4#%&"!

-4!$#.92"$#92"$ -"!4#%&."$94"!%% -24"!4.#%&"!4#%&"! &$#92".$%$4&$24

4!$#.92"$#92"$ 9#%&".!&%""%22 24"!4.#%&"!4#%&"! -2!!4#%.&4"#"!9%2!4

-2!!4#%.&4"#"!9%2!4 -24"!4.#%&"!4#%&"! -"!4#%&."$94"!%%# 4!$#.92"$#92"$

&$#92".$%$4&$24 24"!4.#%&"!4#%&"& 9#%&".!&%""%22 -4!$#.92"$#92"!-24"!4.#%&"!4#%&"! -2!!4#%.&4"#"!9%2!4 -4!$#.92"$#92"$ 9#%&".!&%""%22

24"!4.#%&"!4#%&"& &$#92".$%$4&$2& 4!$#.92"$#92"$ -"!4#%&."$94"!%%#

-"!4#%&."$94"!%%# -4!$#.92"$#92"! -2!!4#%.&4"#"!9%2!4 24"!4.#%&"!4#%&"!

9#%&".!&%""%22 4!$#.92"$#92"$ &$#92".$%$4&$24 -24"!4.#%&"!4#%&"!

4!$#.92"$#92"$ 9#%&".!&%""%22 24"!4.#%&"!4#%&"! -2!!4#%.&4"#"!9%2!4

4$


48/82

Cumulati2e gloal stiHness matri7 in sk,line storage:

&$#92".$%$4&$24 9#%&".!&%""%22 &$#92".$%$4&$2&

kdiag: > % " ?3

Results:

Node 7-dis/ ,-dis/

% .EI -.$42!#E-#

2 .EI -.$42!#E-#

" .EI .EI

4 .EI .EI

Stresses at Gaussian /oints and Internal orces:

Element no# "

&.e integration /oint;ni/ 4 < stresses are:

Element 7-coord ,-coord sig7 sig, tau7,

% .2%%" -.$!!$ -.42!&$EI -.%EI% -.924#2E-%$

% .2%%" -.2%%" -.42!&$EI -.%EI% -.%99$#E-%#

% .$!!$ -.2%%" -.42!&$EI -.%EI% -.%99$#E-%#

% .$!!$ -.$!!$ -.42!&$EI -.%EI% -.924#2E-%$

&.e node ;nod 4 < internal 0orces are:

Element *ocal numer Gloal numer Int# )or# orce Int# ert# orce

% % " .2%429EI .&EI % 2 % .2%429EI -.&EI

% " 2 -.2%429EI -.&EI

% 4 4 -.2%429EI .&EI

4!


49/82

RES(*&S:

B$R& "%

4 F($RI*$&ER$* E*EMEN&S (NER $!I$* COMBRESSION

;*$&ER$**> (NRES&R$INE<

49


50/82

4 F($RI*$&ER$* E*EMEN&S (NER $!I$* COMBRESSION ;*$&ER$**>

(NRES&R$INE


51/82


Element no# "


Element 7-coord ,-coord sig7 sig,

tau7,

% .2%%" -.%9$2 -.""!%2E-2 -.%"EI%

-.9%#4"E-"

% .2%%" -.&2! -.%$$$&E-2 -.999##EI -.9%#4"E-"

% .$!!$ -.&2! -.%$$$&E-2 -.999##EI .9%#4"E-"

% .$!!$ -.%9$2 -.""!%2E-2 -.%"EI% .9%#4"E-"


Element *ocal numer Gloal numer Int# )or# orce Int# ert# orce % % " .#44!4E-" .&EI

% 2 % -.2!2$E-%# -.&EI

% " 2 .22$#&E-%# -.&EI

Element no# '



2 .2%%" -.44$2 -.%&&9"E-% -.%2%EI% -.ôE-2

2 .2%%" -."2! -.&$&2E-2 -.99$!9EI -.ôE-2

2 .$!!$ -."2! -.&$&2E-2 -.99$!9EI .ôE-2

2 .$!!$ -.44$2 -.%&&9"E-% -.%2%EI% .ôE-2


Element 7o,al numer Gloal numer nt. or. For,e nt. ert. For,e 2 % & .""%2$E-2 .&EI

2 2 " -.#44!4E-" -.&EI

2 " 4 .#44!4E-" -.&EI

2 4 # -.""%2$E-2 .&EI

&%


52/82

Element no# 3



tau7,

" .2%%" -.#9$2 -.$#$2"E-% -.%%&EI% -.2$9$$E-%

" .2%%" -.&&2! -.2$$#"E-% -.9!9&%EI -.2$9$$E-%

" .$!!$ -.&&2! -.2$$#"E-% -.9!9&%EI

.2$9$$E-%

" .$!!$ -.#9$2 -.$#$2"E-% -.%%&EI%

.2$9$$E-%



" % $ .%#"$"E-% .&EI

" 2 & -.""%2$E-2 -.&EI

- -

Element no# 4



tau7,

4 .2%%" -.94$2 -."$!&&EI -.%&%!EI%

-.%"!%EI

4 .2%%" -.!2! -.%"#!$EI -.94!2%EI -.%"!%EI

4 .$!!$ -.!2! -.%"#!$EI -.94!2%EI .%"!%EI

4 .$!!$ -.94$2 -."$!&&EI -.%&%!EI% .%"!%EI



4 % 9 .!!2E-% .&EI

4 2 $ -.%#"$"E-% -.&EI

4 " ! .%#"$"E-% -.&EI

&2


53/82

RES(*&S:

B$R& "C

4 RO E*EMEN&S IN $ CO*(MN (NER $!I$*

COMBRESSION ON &OB

&"


54/82


55/82


4. -4. -4. 4.

Cumulati2e gloal stiHness matri7 in sk,line storage:

4. -4. !. -4. !. -4. !.

kdiag: > % " & $ ?3

Results:

Nodal is/lacements:

Node dis/lacement

% -.%E-&

2 -.$&E-#

" -.&E-#

4 -.2&E-#

& .EI


Element no# " &.e integration /oint;ni/ ' < stresses are:

Element coord sig

% -.%9$2 -%.

% -.&2! -%.

&.e node ;nod ' < internal 0orces are:

Element *ocal numer Gloal numer Int# orce

% % 2 .%EI%

% 2 % -.%EI%

&&


56/82

Element no# '

&.e integration /oint;ni/ ' < stresses are:

Element coord sig

2 -.44$2 -%.

2 -."2! -%.



Element no# 3


Element coord sig

" -.#9$2 -%.

" -.&&2! -%.



" % 4 .%EI%

" 2 " -.%EI%

Element no# 4


Element coord sig

4 -.94$2 -%.

4 -.!2! -%.



4 % & .%EI%

4 2 4 -.%EI%


57/82

RES(*&S:

B$R& "

4 %E$M E*EMEN&S IN $ C$N&I*EER %E$M (NER

BOIN& *O$ $& &)E E!&REME

&$


58/82

4 %E$M E*EMEN&S IN $ C$N&I*EER %E$M (NER BOIN& *O$ $& &)E E!&REME:

3here are ! euations and the s80line storage is 24

Element stiHness matri7 and assemlage o0 gloal stiHness matri7:

Cumulati2e gloal stiHness matri7 in sk,line storage: 2.#"992!%!4%%EI! . %.##$%999&&%%&#"%EI$ -%."%99#492&&EI!

-2.&$999%2"%2#2EI$ 2.#"992!%!4%%EI! 2.&$999%2"%2#2EI$

4%#$999.$$&&$&"%& . %.##$%999&&%%&#"%EI$ -%."%99#492&&EI!

-2.&$999%2"%2#2EI$ 2.#"992!%!4%%EI! 2.&$999%2"%2#2EI$

4%#$999.$$&&$&"%& . %.##$%999&&%%&#"%EI$ -%."%99#492&&EI!

-2.&$999%2"%2#2EI$ %."%99#492&&EI! 2.&$999%2"%2#2EI$

4%#$999.$$&&$&"%& -2.&$999%2"%2#2EI$ !""&999.$$&&$&"%#

kdiag: > % " # % %" %$ 2 24 ?3

Results:

Nodal is/lacements and rotation:

Node dis/lacement t.eta

% .EI .EI

2 -.2%99"E-4 -.!"9$"E-4

" -.$99$4E-4 -.%4"9&E-"

4 -.%#%9&E-" -.%$994E-"

& -.2&&92E-" -.%9%94E-"

&!


59/82


Element no# "


sig;sigJ,


60/82

Element no# 3


sig;sigJ,


61/82

RES(*&S:

B$R& '$

4 E*EMEN& CO*(MN ;' C($R$&IC F($RI*$&ER$*

E*EMEN&


62/82

4 E*EMEN& CO*(MN ;' C($R$&IC F($RI*$&ER$* E*EMEN&


63/82


Element no# "

&.e integration /oint;ni/ @ < stresses are:


% .%%2$ -.22%! -.%!$$!EI -.%4&%EI% .2%&29EI

% .%%2$ -.%2& -.#"!%2E-% -.%442!EI% .%494"EI

% .%%2$ -.2!2 .2"#4"EI -.%4&%EI% .!"&$E-%

% .& -.2!2 .#4"2EI -.4&&9%EI -.""$2%E-%&

% .!!$" -.2!2 .2"#4"EI -.%4&%EI% -.!"&$E-%

% .!!$" -.%2& -.#"!%2E-% -.%442!EI% -.%494"EI

% .!!$" -.22%! -.%!$$!EI -.%4&%EI% -.2%&29EI

% .& -.22%! .2%!99EI -.4&&9%EI -.2!#"E-%&

% .& -.%2& ."429#EI -.49"#9EI -.22$&&E-%&

&.e node ;nod ? < internal 0orces are:


% % # -.4%4!&E-% .2%$!&EI

% 2 4 .%&9&%E-%# ."2"24E-%4

% " % .22%9E-%& -.2&EI

% 4 2 -."44$E-%& -.&EI

% & " .""$"2E-%# -.2&EI % # & -.#%%49E-%# .%#"9&E-%4

% $ ! .4%4!&E-% .2%$!&EI

% ! $ .2$"#!E-%& ."EI

#"


64/82

Element no# '



2 .%%2$ -.4$%! -.%$%9EI -.%%9"4EI% .%2EI

2 .%%2$ -."$& -.%#$#EI -.%%99%EI% .%&!%!EI

2 .%%2$ -.2$!2 -.%24$EI -.%%9"4EI% .%9$4EI

2 .& -.2$!2 .#4242E-% -.$&2&4EI -.&%!#E-%&

2 .!!$" -.2$!2 -.%24$EI -.%%9"4EI% -.%9$4EI

2 .!!$" -."$& -.%#$#EI -.%%99%EI% -.%&!%!EI

2 .!!$" -.4$%! -.%$%9EI -.%%9"4EI% -.%2EI

2 .& -.4$%! .%!$&%E-% -.$&2&4EI -.#&"&%E-%&

2 .& -."$& .2!%!#E-% -.$&!24EI -.499&$E-%&

&.e node ;nod ? < internal 0orces are: Element *ocal numer Gloal numer Int# )or# orce Int# ert# orce

2 % %% -.24%"!E-% .%!"!%EI

2 2 9 .!%!&$E-%$ -.22!29E-%&

2 " # .4%4!&E-% -.2%$!&EI

2 4 $ -."2!2E-%& -."EI

2 & ! -.4%4!&E-% -.2%$!&EI

2 # % -.4#24E-%# -.2#"$2E-%&

2 $ %" .24%"!E-% .%!"!%EI

2 ! %2 ."$%4$E-%& .#"2"$EI

#4


65/82

Element no# 3



" .%%2$ -.$2%! -.%&!#EI -.%&&"EI% .%"%$&E-%

" .%%2$ -.#2& -.%%!%!EI -.%&%EI% .&$949E-%

" .%%2$ -.&2!2 -.%9$$EI -.%&&"EI% .%2$2EI

" .& -.&2!2 -.&!#"2E-% -.9"#2EI -.4!!$"E-%&

" .!!$" -.&2!2 -.%9$$EI -.%&&"EI% -.%2$2EI

" .!!$" -.#2& -.%%!%!EI -.%&%EI% -.&$949E-%

" .!!$" -.$2%! -.%&!#EI -.%&&"EI% -.%"%$&E-%

" .& -.$2%! -.99$%$E-% -.9"#2EI -.&4!!E-%&

" .& -.#2& -.#$44E-% -.9"!2EI -.&#%"E-%&

&.e node ;nod ? < internal 0orces are: Element *ocal numer Gloal numer Int# )or# orce Int# ert# orce

" % %# .%4%$E-2 .%$%"4EI

" 2 %4 -."%%9%E-%# .%#!!%E-%&

" " %% .24%"!E-% -.%!"!%EI

" 4 %2 -.2929"E-%& -.#"2"$EI

" & %" -.24%"!E-% -.%!"!%EI

" # %& ."!!42E-%# .&!$E-%&

" $ %! -.%4%$E-2 .%$%"4EI

" ! %$ ."##$E-%& .#&$"%EI

#&


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Element no# 4



4 .%%2$ -.9$%! -.4%24%EI -.%!&EI% -.2""44EI

4 .%%2$ -.!$& -.2#"$4EI -.%#92EI% -.%242$EI

4 .%%2$ -.$$!2 -.%!!##EI -.%!&EI% -.%&%&E-%

4 .& -.$$!2 -.%%"49EI -.99EI .$%!!E-%&

4 .!!$" -.$$!2 -.%!!##EI -.%!&EI% .%&%&E-%

4 .!!$" -.!$& -.2#"$4EI -.%#92EI% .%242$EI

4 .!!$" -.9$%! -.4%24%EI -.%!&EI% .2""44EI

4 .& -.9$%! -.""$2"EI -.99EI .#!!E-%&

4 .& -.!$& -.%!!EI -.!9"$9EI .#"%$%E-%&

&.e node ;nod ? < internal 0orces are:


4 % 2% .#4%"E-% .%9!!EI

4 2 %9 -."&"!"E-%# -.2%##E-%&

4 " %# -.%4%$E-2 -.%$%"4EI

4 4 %$ -."2!#&E-%& -.#&$"%EI

4 & %! .%4%$E-2 -.%$%"4EI

4 # 2 -."%"#9E-%# -."9#9"E-%&

4 $ 2" -.#4%"E-% .%9!!EI

4 ! 22 -.!!#E-%& .#"!"EI

##


67/82

RES(*&S:

B$R& '%

4 E*EMEN& CO*(MN ;" C($R$&IC RO E*EMEN&S


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4 E*EMEN& CO*(MN ;" C($R$&IC RO E*EMEN&S


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Element no# '


Element coord sig

2 -.44$2 -%.

2 -."2! -%.



2 % & .%EI%

2 " " -.%EI%

Element no# 3


Element coord sig

" -.#9$2 -%.

" -.&&2! -%.



" % $ .%EI%

" " & -.%EI%

#9


70/82

Element no# 4


Element coord sig

4 -.94$2 -%.

4 -.!2! -%.



4 % 9 .%EI%

4 2 ! .%99$%E-%&

4 " $ -.%EI%

$


71/82

RES(*&S:

B$R& 3$

IRREG(*$R E*EMEN& CO*(MN ;' %I*INE$R

F($RI*$&ER$* E*EMEN&<

$%


72/82

IRREG(*$R E*EMEN& CO*(MN

;' %I*INE$R F($RI*$&ER$*

E*EMEN&<

&.ere are '?? e1uations and

t.e sk,line storage is 8?A4

Results:

Node 7-dis/ ,-dis/

" -A#'A'A'E-A6

-A#?@'?AE-A6

' -A#"83""E-A6

-A#??@="E-A6

3 -A#"A33'E-A6

-A#??6A"E-A6

4 -A#8'"3?E-A=-A#??'@'E-A6

8 -A#846"3E-'A

-A#??"=3E-A6

6 A#8'"3?E-A=

-A#??'@'E-A6

= A#"A33'E-A6

-A#??6A"E-A6

? A#"83""E-A6

-A#??@="E-A6

@ A#'A'A'E-A6

-A#?@'?AE-A6

"A -A#'AA6"E-A6

-A#?38@3E-A6

"" -A#"8"64E-A6

-A#?3'=6E-A6

"' -A#"A"@3E-A6

-A#?'?=@E-A6

"3 -A#8"'=3E-A=

-A#?'848E-A6

"4 -A#4@A'8E-'A

-A#?'4"6E-A6

"8 A#8"'=3E-A=

-A#?'848E-A6

$2


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Element no# "



" A#A'64 -A#A4@3 A#'A83"E-A' -A#@@@3@ELAAA#@"6?'E-A3

" A#A'64 -A#A"3' A#"=8?4E-A' -A#@@@8'ELAA

A#?AA3=E-A3

" A#A@?6 -A#A"3' A#"4A@"E-A' -A#"AAA3ELA"

A#63"@=E-A3

" A#A@?6 -A#A4@3 A#"=A3?E-A' -A#"AAA'ELA"

A#=4?4'E-A3



" " "A -A#"A?"@E-A3 A#6'48'E-A"

" ' " A#=336AE-"6 -A#6'8AAE-A"

Element no# '



' A#"8"4 -A#A4@3 A#@36=AE-A' -A#@@?'AELAA

A#3"38'E-A'

' A#"8"4 -A#A"3' A#@?"@3E-A' -A#@@?A"ELAA

A#'64'"E-A'

' A#''36 -A#A"3' A#?34A"E-A' -A#"AA"8ELA"

A#'@AA6E-A'

' A#''36 -A#A4@3 A#=??=?E-A' -A#"AA"=ELA"

A#33@3=E-A'



' " "" -A#48688E-A3 A#6'3'?E-A"

- - - -

$"


74/82

RES(*&S:

B$R& 3%

IRREG(*$R E*EMEN&S C$N&I*EER %E$M ;' %I*INE$R

F($RI*$&ER$* E*EMEN&S<

$4


75/82

IRREG(*$R E*EMEN&

C$N&I*EER %E$M ;' %I*INE$R

F($RI*$&ER$* E*EMEN&S<

&.ere are "44 e1uations and

t.e sk,line storage is '8"6

Results:

Node 7-dis/ ,-dis/

% .EI

.EI

2 .%!49E-2 -.%4&&"E-

2

" ."4%&%E-2 -.42"!2E-

2

4 .4$9E-2 -.!&$4"E-2

& .&92%4E-2 -.%4%&$E-

%

# .#$9$2E-2 -.2$4%E-

%

$ .$422E-2 -.2!$#E-

%

! .$$994E-2 -."&9#E-

%

9 .$9"!$E-2 -.44"%E-

%

% .EI

.EI

%% .%2!"9E-2 -.%2&"9E-

2

%2 .2&"$"E-2 -.4%&%E-

2

%" ."E-2 -.!4&&"E-

2

%4 .44!E-2 -.%4#"E-

%

2! .EI

.EI

29 .4!!2E-" -.%4%4E-

2

" .!"%2#E-" -."94%&E-

2

"% .%%$&9E-2 -.!"2"E-2

"2 .%4"E-2 -.%"9E-

%

"" .%#$E-2 -.2&9E-

%

"4 .%!"2"E-2 -.2$9$&E-

%

"& .%92"%E-2 -."&!#4E-

%

"# .%9&$&E-2 -.4"99%E-

%

"$ .EI

.EI

"! -.%#9#E-%# -.%%#%E-

2

"9 -.444"E-%# -."924E-

2

4 -.#"9!"E-%# -.!"#&E-

2

4% -.!#$"E-%# -.%"94"E-

%

42 -.%949E-%& -.2&!E-

%

4" -.%"%%&E-%& -.2$9#!E-

%

44 -.%"#!E-%& -."&!#%E-%

4& -.%"%2&E-%& -.4"9!#E-

%

$&


76/82

&$ -.%#$42E-2 -.44E-

2

&! -.2"#"E-2 -.!"$2&E-

2

&9 -.29224E-2 -.%"99#E-

%

# -.""##E-2 -.2#2%E-%

#% -."#$4E-2 -.2$99&E-

%

#2 -."!&$4E-2 -."&!$%E-

%

#" -."92&!E-2 -.442E-

%

#4 .EI

.EI

$ -.&&"42E-2 -.2!29E-

%

$% -.&!%"2E-2 -."&!!&E-

%

$2 -.&9%E-2 -.44%!E-

%

$" .EI.EI

$4 -.%!49E-2 -.%4&&"E-

2

$& -."4%&%E-2 -.42"!2E-

2

$# -.4$9E-2 -.!&$4"E-

2

$$ -.&92%4E-2 -.%4%&$E-

%


Element no# "



% .&2! -.49" .$%%"2EI4 .2"4EI4 -.%"%$EI4

% .&2! -.%"2 .!$""%EI4 .2994#EI4 -.%49&!EI4

% .%9$2 -.%"2 .$##EI4 .49##EI" ."&&4&EI"

% .%9$2 -.49" .#4%EI4 -.2"&$EI" .&"4"EI"



% % % -.!9"2EI2 -.2&%"!EI"

% 2 % -."$2"4EI" .2!%4"EI"

% " 2 .2&2%&EI" .#$44!EI2

% 4 %% .29&%EI" -.9$49#EI2

$#


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Element no# '



2 ."2! -.49" .&429#EI4 -.9&9$%EI" -.#9"!2EI"

2 ."2! -.%"2 .#&"!!EI4 -.4!4"$EI" -.#""%!EI" 2 .44$2 -.%"2 .#92#EI4 ."#4&$EI" .#"4"!EI"

2 .44$2 -.49" .&$9"&EI4 -.%%$$EI" .&$"$&EI"



2 % %% -.%""2"EI" .#9"#EI2

2 2 2 -.2&2%&EI" -.#$44!EI2

2 " " .244$2EI" -.#944&EI%

2 4 %2 .%4##EI" .&!$%EI%

Element no# 63



#" %.&&2! -.4!#! -.%&"42EI4 .$$"9%EI2 -."%%&EI"

#" %.&&2! -.4&$ -.%22!%EI4 .2!EI" -."""2&EI"

#" %.#9$2 -.4&$ -.%"##$EI4 -.%%4!#EI" .%#&4#EI2

#" %.#9$2 -.4!#! -.%#$2!EI4 -.24#"EI" ."9#4$EI2



#" % $9 .$9#%EI2 -.4949#EI%

#" 2 $ .%%&94EI2 .%4%2&EI2

#" " $% -.4!294EI2 -.%!!!EI2

#" 4 ! -.42"#EI2 .9#""%EI%

$$


78/82

Element no# 64



#4 %.!2! -.4!#! -.&2!!EI" .%%#&2EI" -.2"%2&EI"

#4 %.!2! -.4&$ -.4%""&EI" .%#EI" -.2"!$#EI" #4 %.94$2 -.4&$ -.4&!4%EI" .#&4!EI2 -.%$#4EI"

#4 %.94$2 -.4!#! -.&$"%4EI" .%%"$!EI2 -.%%"EI"



#4 % ! .42"#EI2 -.9#""%EI%

#4 2 $% -.%%&""EI2 .222"EI2

#4 " $2 -."!2!EI2 .%999EI%

#4 4 !% -.%9"2$E-%% -.%2&EI2

$!


79/82

OR&R$N COE:

“LINEAR ELEMENTS AND QUADRATIC ELEMENTS”:

B$R& ":

-7EAU AD OBADUA3C 3TQ- AD QE- DMEQA7 E7EME3

GEEUA7 FEM

MABA7 QDE AD E7EME3 BM+EUG

B$R& ':

-7EAU AD OBADUA3C 3TQ- AD QE- DMEQA7 E7EME3GEEUA7 FEM

$9


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OR&R$N COE:

B$R& ":

-*INE$R $N F($R$&IC &9O- $N ONE- IMENSION$*

E*EMEN&S GENER$* EMD M$N($* NOE $N E*EMEN&

N(M%ERING

!


81/82

OR&R$N COE:

B$R& ':

-*INE$R $N F($R$&IC &9O- $N ONE- IMENSION$*

E*EMEN&S GENER$* EMD $(&OM$&E NOE $N E*EMEN&

N(M%ERING

!%


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Numerical Modelling in Geo engineering

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Transcript of Numerical Modelling in Geo engineering