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EDI CI ONS
UPC
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EDI CI ONS
UPC
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MECNICA DE ESTRUCTURAS I, RESISTENCIA DE MATERIALESPublished byATARAXIAINC111 III World StreetHoboken, NJ 07030-5774
Copyright 2006 by Ataraxiainc, Bogota, ChibchombiaPublished by Ataraxiainc, Bogota, ChibchombiaPublished simultaneously in the Earth planetAll parts of this publication may be reproduced, stored in a retrieval system or transmitted in any form orby any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as nopermitted under Sections of the Copyright Act, without either the prior written permission of thePublisher, or authorization through payment of the appropriate per-copy fee to the Copyright ClearanceCenter, 2*2 Rosewood Drive, Danvers, MA 0+-23, (978) 75.-84/*00, fax (9%) 646-/600. Requests tothe Publisher for permission should be addressed to the Legal Department,Ataraxiainc, Bogota, Chibchombia e-mail: [email protected]: ATARAXIAINC
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POLITEXT 111
Mecnica de estructurasLibro 1
Resistencia de materiales
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POLITEXT
EDICIONS UPC
Miguel Cervera Ruiz
Elena Blanco Daz
Mecnica de estructurasLibro 1
Resistencia de materiales
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La presente obra fue galardonada en el octavo concurso
"Ajuts a l'elaboraci de material docent" convocado por la UPC.
Primera edicin: septiembre de 2001
Segunda edicin: septiembre de 2002Reimpresin: septiembre de 2003
Diseo de la cubierta: Manuel Andreu
Los autores, 2001
Edicions UPC, 2001
Edicions de la Universitat Politcnica de Catalunya, SL
Jordi Girona Salgado 31, 08034 BarcelonaTel.: 934 016 883 Fax: 934 015 885
Edicions Virtuals: www.edicionsupc.es
E-mail: [email protected]
Produccin: CPET (Centre de Publicacions del Campus Nord)
La Cup. Gran Capit s/n, 08034 Barcelona
Depsito legal: B-28576-2002
ISBN: 84-8301-622-2
ISBN Obra completa: 84-8301-623-0
Quedan rigurosamente prohibidas, sin la autorizacin escrita de los titulares del copyright, bajo las san-
ciones establecidas en las leyes, la reproduccin total o parcial de esta obra por cualquier medio o pro-
cedimiento, comprendidos la reprografa y el tratamiento informtico, y la distribucin de ejemplares de
ella mediante alquiler o prstamo pblicos.
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odv fxdohv od whqvlq wdqjhqfldo vh dqxod> glfkdv gluhfflrqhv vrq odv gluhfflrqhv sulqflsd0ohv gh whqvlq | odv whqvlrqhv qrupdohv fruuhvsrqglhqwhv vrq odv whqvlrqhv sulqflsdohv1
Vl vh frqrfhq odv frpsrqhqwhv gho whqvru gh whqvlrqhv fruuhvsrqglhqwhv d grv sodqrv
shushqglfxoduhv hqwuh v +7
/ C
/ 7 C
, hv srvleoh ghwhuplqdu odv whqvlrqhv sulqflsdohv
phgldqwh xqd frqvwuxfflq jufd oodpdgd fufxor gh Prku/ txh vh pxhvwud hq od Iljxud
41 7 C
, |
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+7
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, > 3 ,= Odv lqwhuvhfflrqhv gho fufxor frq ho hmh
ghwhuplqdq orv ydoruhv gh odv whqvlrqhv sulqflsdohv/ |d txh sdud hoorv @ 3=
D sduwlu gh od frqvwuxfflq jufd gho fufxor vh sxhghq fdofxodu orv ydoruhv gh odv
whqvlrqhv sulqflsdohv/
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Los autores, 2003; Edicions UPC, 2003.
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45 F D S W X O R 4 1 I X Q G D P H Q W R V G H H O D V W L F L G D G
Ilj1 41 |,1 Sru rwur odgr/ od whqvlq wdqjhqfldo p{lpd hq ho sxqwr hv
ljxdo do udglr gho fufxor/ hv ghflu=
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jhqhudo gh xq hvwdgr wuld{ldo gh whqvlrqhv/ dxqtxh gh irupd dojr pv frpsohmd d od dtx
frphqwdgd1
419 Frqfhswr gh ghirupdflq
Frqvlghuhprv xq fxhusr fxdotxlhud vrphwlgr d od dfflq gh ixhu}dv dsolfdgdv/ | frq
yqfxorv vxflhqwhv frpr sdud lpshgluoh prylplhqwrv gh volgr ujlgr1 Gdgr txh qr
h{lvwh pdwhuldo dojxqr txh vhd lqqlwdphqwh ujlgr/ od dfflq gh odv ixhu}dv vh wudgxfh
hq txh ho fxhusr vh ghirupd1 Vxsrqguhprv hq or txh vljxh txh ho fxhusr vh frpsruwd
gh irupd vxflhqwhphqwh ujlgd frpr sdud txh orv prylplhqwrv txh vh surgxfhq hq
ho surfhvr gh ghirupdflq vhdq shtxhrv frpsdudgrv frq odv glphqvlrqhv gho fxhusr
+sulqflslr gh uljlgh},1
Od ghirupdflq gh xq hohphqwr glihuhqfldo gh yroxphq gh glphqvlrqhv +g{>g|>g},/
wdo frpr ho txh vh pxhvwud hq od Iljxud 4143/ sxhgh ghvfrpsrqhuvh hq wuhv sduwhv=
urwdflq | wudvodflq +prylplhqwrv gh volgr ujlgr, | ghirupdflq sxud +fdpelr gh
irupd,1 Qrv rfxsduhprv hq hvwh dsduwdgr gh od ghirupdflq sxud qlfdphqwh1
Los autores, 2003; Edicions UPC, 2003.
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4 1 9 1 F R Q F H S W R G H G H I R U P D F L Q 46
Ilj1 4143= Surfhvr gh ghirupdflq gh xq hohphqwr gh yroxphq
Orv odgrv gho sdudohohsshgr hohphqwdo prglfdq vxv orqjlwxghv lqlfldohv g{/ g|/ g}
gh pdqhud txh odv sur|hfflrqhv gh odv qxhydv orqjlwxghv vreuh orv wuhv hmhv gh uhihuhqfld
sdvdq d ydohu +4 . 7 , g{> +4 . C , g|> +4 . S , g}/ uhvshfwlydphqwh1 Dvlplvpr/ odvsur|hfflrqhv gh orv qjxorv uhfwrv txh irupdq hqwuh v odv fdudv gho hohphqwr dqwhv gh
od ghirupdflq ydudq sdud sdvdu d ydohu/ h{suhvdgdv hq udgldqhv/ @5 7 C
> @5
7 S
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1 Vh oodpdq dodujdplhqwrv xqlwdulrv r ghirupdflrqhv olqhdohv d orv ydoruhv
7
> C
> S
| glvwruvlrqhv r ghirupdflrqhv dqjxoduhv d orv ydoruhv 7 C
> 7 S
> C S
1 Ghelgr do
sulqflslr gh uljlgh}/ wdqwr odv ghirupdflrqhv olqhdohv frpr odv dqjxoduhv vrq shtxhdv/
frpsdudgdv frq od xqlgdg1
Vh frqvlghudq srvlwlydv odv ghirupdflrqhv olqhdohv gh dodujdplhqwr | qhjdwlydv odv
gh dfruwdplhqwr1 Dvlplvpr/ vh frqvlghudq srvlwlydv odv ghirupdflrqhv dqjxoduhv txh
ruljlqdq glvplqxflq gh orv qjxorv uhfwrv irupdgrv sru orv sodqrv fruuhvsrqglhqwhv d
orv hmhv gh uhihuhqfld/ | qhjdwlydv odv txh ruljlqdq dxphqwr gh glfkrv qjxorv1H{shulphqwdophqwh/ vh revhuyd txh/ hq orv pdwhuldohv phfqlfdphqwh lvwursrv/ odv
whqvlrqhv qrupdohv gdq oxjdu d dodujdplhqwrv/ plhqwudv txh odv whqvlrqhv wdqjhqfldohv
suryrfdq glvwruvlrqhv +yhu Vhfflq 4143,1
Ilj1 4144= Dodujdplhqwr | glvwruvlq hq xq pdwhuldo lvwursr
Los autores, 2003; Edicions UPC, 2003.
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47 F D S W X O R 4 1 I X Q G D P H Q W R V G H H O D V W L F L G D G
41: Whqvru gh ghirupdflrqhv
Frqvlghuhprv xq hohphqwr xqlwdulr$RG @ $q > gh frvhqrv gluhfwruhv +o >p>q,1 Vh ghqh
frpr ghirupdflq gho hohphqwr $RG do ghvsod}dplhqwr $GG
gho sxqwr G/ ruljlqdgrsru od ghirupdflq gho sdudohohsshgr hohphqwdo txh whqjd d RG sru gldjrqdo/ suhvflq0
glhqgr gh urwdflrqhv | wudvodflrqhv gh volgr ujlgr +yhu Iljxud 4145d,1
Fdofxodprv d frqwlqxdflq odv frpsrqhqwhv fduwhvldqdv gho yhfwru / r vhd +7
> C
S
,1
Sdud hoor/ vxpdprv odv frpsrqhqwhv gho ghvsod}dplhqwr gho sxqwr G ghelgdv d orv dodu0
jdplhqwrv xqlwdulrv 7
> C
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| d odv glvwruvlrqhv 7 C
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+Iljxudv 4145e | f,=
7
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Ilj1 4145= +d, Frqfhswr gh yhfwru ghirupdflq +e, Ghvsod}dplhqwrv ghelgrv d orv
dodujdplhqwrv +f, Ghvsod}dplhqwrv ghelgrv d odv glvwruvlrqhv
Los autores, 2003; Edicions UPC, 2003.
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4 1 : 1 W H Q V R U G H G H I R U P D F L R Q H V 48
r pdwulfldophqwh=
B @ ( ? +415:,
grqgh B | ? vrq orv yhfwruhv gh ghirupdflq | ho hohphqwr xqlwdulr/ ghqlgrv dqwhulru0phqwh/ | ( hv ho whqvru gh ghirupdflrqhv txh hv ljxdo d=
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Ho whqvru gh ghirupdflrqhv hv vlpwulfr/ do kdehu suhvflqglgr hq vx frqvwuxfflq gh
odv urwdflrqhv gh volgr ujlgr1
Vh ghgxfh gh od h{suhvlq +415:, txh/ frqrflgdv odv frpsrqhqwhv gho hvwdgr gh
ghirupdflq gh xq sxqwr +whqvru gh ghirupdflrqhv, | orv frvhqrv gluhfwruhv gh xqhohphqwr olqhdo fxdotxlhud/ vh sxhgh frqrfhu od ghirupdflq gh glfkr hohphqwr1
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ho yhfwru q=
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49 F D S W X O R 4 1 I X Q G D P H Q W R V G H H O D V W L F L G D G
41; Hodvwlflgdg | olqhdolgdg1 Oh| gh Krrnh
Wrgr fxhusr vh ghirupd edmr od dfflq gh odv ixhu}dv dsolfdgdv/ | do fhvdu vwdv/ ho
fxhusr wlhqgh d uhfxshudu vx irupd sulplwlyd1 Hvwd whqghqfld txh/ hq pd|ru r phqrujudgr/ wlhqhq wrgrv orv fxhusrv vh ghqrplqd hodvwlflgdg1
Hq uhdolgdg/ orv fxhusrv qr vrq ql shuihfwdphqwh hovwlfrv ql shuihfwdphqwh lqhov0
wlfrv1 Odv ghirupdflrqhv txh hq hoorv vh surgxfhq frqvwdq gh xqd sduwh gh ghirupdflq
hovwlfd/ txh ghvdsduhfh do fhvdu odv ixhu}dv dsolfdgdv/ | xqd sduwh gh ghirupdflq shu0
pdqhqwh/ txh vh pdqwlhqh srvwhulruphqwh1 Hq xq hohydgr qphur gh fxhusrv/ vl odv
ixhu}dv qr vreuhsdvdq ghwhuplqdgrv ydoruhv/ odv ghirupdflrqhv shupdqhqwhv vrq px|
shtxhdv/ | hq frqvhfxhqfld/ glfkrv fxhusrv sxhghq frqvlghuduvh hovwlfrv1
Frqvlghuhprv xq h{shulphqwr vhqfloor= vh prqwd xqd yljd eldsr|dgd frpr od gh od
Iljxud 4147 | vh plgh ho ghvsod}dplhqwr yhuwlfdo gh xq flhuwr sxqwr D gh od plvpd
surgxflgr sru xqd ixhu}d S dsolfdgd hq rwur sxqwr E gh glfkd yljd1 Od ixhu}d S vhdxphqwd gh irupd judgxdo kdvwd xq flhuwr ydoru | vh glexmd od fxuyd S rewhqlgd1
Vl dkrud ghvfdujdprv od yljd gh irupd judgxdo | glexmdprv wdpelq od fxuyd S hq
ghvfdujd/ revhuyduhprv txh sdud ydoruhv p{lprv gh S phqruhv txh flhuwr ydoru oplwh
Se
/ oodpdgr oplwh hovwlfr/ odv udpdv gh fdujd | ghvfdujd frlqflghq/ qr kd| ghirupdflq
shupdqhqwh/ | sru or wdqwr/ ho frpsruwdplhqwr hv hovwlfr1 Vl od fdujd S h{fhgh ho
oplwh hovwlfr/ od udpd gh ghvfdujd vh vhsdud gh od gh fdujd/ vh surgxfhq ghirupdflrqhv
shupdqhqwhv | vh glfh txh ho frpsruwdplhqwr qr hv shuihfwdphqwh hovwlfr1
Hq ho plvpr h{shulphqwr revhuyduhprv txh h{lvwh rwur ydoru oplwh S
/ phqru txh
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+|/ qrupdophqwh/ su{lpr d vwh,/ oodpdgr oplwh gh sursruflrqdolgdg/ wdo txh sdud
fdujdv phqruhv txh vwh orv ghvsod}dplhqwrv vrq sursruflrqdohv d odv ixhu}dv txh orv
ruljlqdq1 Hvwh hqxqfldgr ghqh ho frpsruwdplhqwr hovwlfr olqhdo | vh frqrfh frpr Oh|
gh Krrnh/ |d txh ixh hvwdeohflgr sru Urehuw Krrnh hq 49:; hq vx wudedmr Gh Srwhqwld
Uhvwlwxwlyd +ri Vsulqjv,1
Ilj1 4147= H{shulphqwr | fxuyd fdujdghvsod}dplhqwr
Los autores, 2003; Edicions UPC, 2003.
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4 1 < 1 S U L Q F L S L R G H V X S H U S R V L F L Q 4:
Pdwhpwlfdphqwh/ od oh| gh Krrnh vh h{suhvd gh od irupd=
@ n S +4164,
grqgh n hv od frqvwdqwh gh sursruflrqdolgdg hqwuh od ixhu}d dsolfdgd S | ho ghvsod}d0
plhqwr txh vwd surgxfh1 Reyldphqwh/ hvwd frqvwdqwh ghshqgh gh od jhrphwud gho
sureohpd | gh odv surslhgdghv phfqlfdv gho pdwhuldo gh od yljd1
41< Sulqflslr gh vxshusrvlflq
Vl vh fxpsoh od oh| gh Krrnh | vh vxsrqh txh orv ghvsod}dplhqwrv surgxflgrv sru odv
ixhu}dv dfwxdqwhv vrq px| shtxhrv hq uhodflq d odv glphqvlrqhv gho fxhusr/ gh wdo
pdqhud txh vh sxhgd frqvlghudu txh vwh pdqwlhqh od irupd | glphqvlrqhv ruljlqdohv/
hqwrqfhv sxhgh dsolfduvh ho Sulqflslr gh Vxshusrvlflq r Sulqflslr gh Olqhdolgdg1 HoSulqflslr gh Vxshusrvlflq hvwdeohfh txh orv hihfwrv txh xq vlvwhpd gh ixhu}dv dsolfdgdv
ruljlqd hq xq fxhusr vrq ljxdohv d od vxpd gh orv hihfwrv txh ruljlqdq hvdv plvpdv
ixhu}dv dfwxdqgr sru vhsdudgr1
Vl vreuh xq fxhusr dfwd xq vlvwhpd irupdgr sru wuhv ixhu}dv S
> SP
| S
> ho Sulqflslr
gh Vxshusrvlflq hvwdeohfh txh od frpsrqhqwh x gho ghvsod}dplhqwr gh xq sxqwr lqwhulru
R> vhjq xqd flhuwd gluhfflq gh uhihuhqfld/ vhu=
x @ x
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grqgh x
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vrq orv ydoruhv txh wrpd ho fruuhvsrqglhqwh ghvsod}dplhqwr fxdqgr
dfwdq S
> SP
| S
vrodv/ uhvshfwlydphqwh1 Vl od oh| gh Krrnh hv dsolfdeoh/ vhu=
x
@ n
S
xP
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x
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x @ n
S
. nP
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. n
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Ilj1 4148= Sulqflslr gh vxshusrvlflq
Los autores, 2003; Edicions UPC, 2003.
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4; F D S W X O R 4 1 I X Q G D P H Q W R V G H H O D V W L F L G D G
Hq yluwxg gh odv surslhgdghv frqpxwdwlyd | dvrfldwlyd gh od vxpd/ uhvxowd fodur
txh ho hqxqfldgr gho Sulqflslr gh Vxshusrvlflq hv htxlydohqwh d hvwdeohfhu txh orv
hihfwrv txh xq vlvwhpd gh ixhu}dv dsolfdgdv ruljlqd hq xq fxhusr vrq lqghshqglhqwhv gho
rughq gh dsolfdflq gh odv ixhu}dv1 Pdwhpwlfdphqwh/ ho Sulqflslr gh Vxshusrvlflq
hvwdeohfh txh od uhodflq dfflq0uhvsxhvwd hv olqhdo/ | wlhqh/ sru wdqwr/ odv surslhgdghv
gh odv ixqflrqhv olqhdohv1
4143 Oh| gh Krrnh jhqhudol}dgd
Dsolfduhprv hq hvwh dsduwdgr od oh| gh Krrnh/ txh hvwdeohfh od sursruflrqdolgdg hqwuh
ixhu}dv +dfflrqhv, | ghvsod}dplhqwrv +hihfwrv,/ d od uhodflq h{lvwhqwh hqwuh whqvlrqhv
| ghirupdflrqhv dfwxdqwhv hq xq sxqwr1 Hqxqfldgd gh hvwd irupd/ vh oh oodpd oh| ghKrrnh jhqhudol}dgd/ | orv frhflhqwhv gh sursruflrqdolgdg txh dsduhfhq vrq frqvwdqwhv
fdudfwhuvwlfdv gho pdwhuldo/ | qr ghshqghq gh od jhrphwud gho fxhusr/ |d txh ho hvwdgr
whqvlrqdo | gh ghirupdflq vrq surslrv gh xq sxqwr1
Dvlplvpr/ dgplwluhprv hq wrgrv qxhvwurv ghvduuroorv txh orv fxhusrv vrq phfql0
fdphqwh lvwursrv/ hv ghflu/ txh vxv surslhgdghv phfqlfdv vrq ljxdohv hq wrgdv odv
gluhfflrqhv1 Hvwd klswhvlv gh lvrwursd phfqlfd qr vh fxpsoh h{dfwdphqwh hq dojxqrv
pdwhuldohv gh frqvwuxfflq wdohv frpr od pdghud/ txh hv eurvd/ ql hq odv urfdv hvwudwl0
fdgdv/ ql hq pdwhuldohv ideulfdgrv phgldqwh xq surfhvr gh odplqdflq/ hwf1 D shvdu gh
hoor/ orv uhvxowdgrv txh vh rewlhqhq frq hvwd klswhvlv vrq vdwlvidfwrulrv hq od pd|rud
gh orv fdvrv1
Ilj1 4149= Hohphqwr glihuhqfldo gh yroxphq | ghirupdflq edmr whqvlq qrupdo
Los autores, 2003; Edicions UPC, 2003.
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4 1 4 3 1 O H \ G H K R R N H J H Q H U D O L ] D G D 4 S
/ sursruflrqdohv d 7
/
Pdwhuldo H ^JSd` Pdwhuldo H ^JSd`
Dfhur 543 Krupljq 53
Freuh 453 Flqf 433
Odwq 433 Gxudoxplqlr :3
Eurqfh 443 Doxplqlr :9
Judqlwr 83 Ixqglflq 4:3
Pdghud 43 Hvwdr 73
Wdeod 414= Pgxor gh \rxqj sdud glihuhqwhv pdwhuldohv
Los autores, 2003; Edicions UPC, 2003.
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53 F D S W X O R 4 1 I X Q G D P H Q W R V G H H O D V W L F L G D G
vlhqgr xq frhflhqwh gh sursruflrqdolgdg dglphqvlrqdo oodpdgr frhflhqwh gh Srlvvrq
+V1 G1 Srlvvrq/ 4;5 6 | hq ho krupljq ydoh hqwuh 3> 48 | 3> 531
Gh irupd dqorjd/ od dfwxdflq gh od frpsrqhqwh qrupdo gh whqvlq C
surgxfh
xqdv ghirupdflrqhv orqjlwxglqdohv | wudqvyhuvdohv gh ydoru=
C
2
@
C
H
S
2
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2
@
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| od dfwxdflq gh od frpsrqhqwh qrupdo gh whqvlq S
surgxfh xqdv ghirupdflrqhv=
S
@
S
H
C
@ 7
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Dsolfdqgr ho sulqflslr gh vxshusrvlflq/ od ghirupdflq olqhdo vhjq ho hmh { ghelgd
d od dfflq frqmxqwd gh odv whqvlrqhv qrupdohv 7
> C
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vhu 7
@ 7
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2
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1
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| S
1 Sru or wdqwr/ vh wlhqh=
7
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7
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Frqvlghuhprv dkrud odv ghirupdflrqhv txh odv frpsrqhqwhv wdqjhqfldohv gh od whq0
vlq surgxfhq hq ho sdudohohsshgr hohphqwdo1 H{shulphqwdophqwh vh frpsuxhed txh/ hq
xq pdwhuldo phfqlfdphqwh lvwursr/ od dfwxdflq gh fdgd xqr gh orv sduhv gh whqvlrqhv
wdqjhqfldohv uhfsurfrv +hvwr hv/ 7 C
| C 7
> C S
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> 7 S
| S 7
, surgxfh xqd ghirupd0
flq dqjxodu +7 C > C S > S 7 > uhvshfwlydphqwh, txh wudqvirupd glfkr sdudohohsshgr uhfwrhq xqr reofxr +yhu Iljxud 414:,1
Ilj1 414:= Ghirupdflq edmr whqvlq wdqjhqfldo
Los autores, 2003; Edicions UPC, 2003.
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4 1 4 3 1 O H \ G H K R R N H J H Q H U D O L ] D G D 54
Vhjq od oh| gh Krrnh/ odv glvwruvlrqhv surgxflgdv vhuq sursruflrqdohv d od whqvlq
wdqjhqfldo dfwxdqwh=
7 C @
7 C
J
C S
@
C S
J+416
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55 F D S W X O R 4 1 I X Q G D P H Q W R V G H H O D V W L F L G D G
Ilj1 414;= +d, Hvwdgr gh fruwdqwh sxur +e, Fufxor gh Prku
Ghvsuhfldqgr lqqlwvlprv gh vhjxqgr rughq/ odv orqjlwxghv DE/ EF/ FG | GD qr
fdpeldq/ shur v or kdfhq orv qjxorv hqwuh orv vhjphqwrv1 Dv/ ho qjxor _RDE fdpeld
do _RD E / fx|d wdqjhqwh ydoh=
wdq
7
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4 . 7
+4176,
|d txh RD @ RE1 Sru od oh| gh Krrnh jhqhudol}dgd/ | uhfrugdqgr txh 7
@ |
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@ / vh wlhqh=
7
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7
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sdud qjxorv @5 shtxhrv1 Ljxdodqgr odv hfxdflrqhv +4176, | +4178, vh wlhqh=
5
@ 4 . H
+4179,
| uhfrugdqgr txh @ @J vh wlhqh txh=
J @H
5 +4 . ,+417:,
Los autores, 2003; Edicions UPC, 2003.
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4 1 4 3 1 O H \ G H K R R N H J H Q H U D O L ] D G D 56
Ilj1 414 ,> +H> J, +> ,1
Sru lqwxlflq ivlfd | ud}rqhv whuprglqplfdv/ ho pgxor gh uljlgh} d fruwdqwh gh0
eh vhu srvlwlyr/ sru or wdqwr gheh vhu 4> 31 Vlq hpedujr/ qr vh kdq phglgr
frhflhqwhv gh Srlvvrq qhjdwlyrv hq qlqjq pdwhuldo1 Ho ydoru pqlpr gh > phglgr
h{shulphqwdophqwh/ hv ho gho ehulolr + @ 3> 34 3> 38,1
414315 Pgxor gh hodvwlflgdg yroxpwulfr
Frqvlghuhprv xq sdudohohsshgr glihuhqfldo gh odgrv +g{>g|>g}, vrphwlgr d od dfflq
gh whqvlrqhv qrupdohv 7
> C
> S
dfwxdqgr vreuh vxv fdudv1 Od ghirupdflq olqhdo vhjq
orv hmhv +{>|>}, hv ljxdo d 7
> C
> S
+Iljxud 4153,1 Hq frqvhfxhqfld/ ho yroxphq gho
hohphqwr h{shulphqwd xqd yduldflq xqlwduld ljxdo d=
gY
gY@
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,+4 . S
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grqgh gY @ g{g|g} hv ho yroxphq lqlfldo | gY
@ +4 . 7 ,g{+4 . C ,g|+4 . S ,g} hv hoyroxphq ghvsxv gh od ghirupdflq1
Vl vh ghvsuhfldq lqqlwvlprv gh rughq vxshulru/ hvwd yduldflq hv ljxdo d=
gY
gY@ 4 .
7
. C
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+417
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57 F D S W X O R 4 1 I X Q G D P H Q W R V G H H O D V W L F L G D G
Ilj1 4153= Ghirupdflq edmr whqvlrqhv qrupdohv
Gh od uhodflq dqwhulru vh revhuyd txh od glodwdflq felfd/ hv ghflu/ ho lqfuhphqwr
xqlwdulr gh yroxphq/ hv ljxdo d=
gY gY
gY@
t D
@ 7
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+4183,
Sdud rewhqhu od h{suhvlq gh hvwd glodwdflq yroxpwulfd hq ixqflq gh odv whqvlrqhv
dfwxdqwhv/ vh vxpdq odv ljxdogdghv +416;, gh od oh| gh Krrnh jhqhudol}dgd/ hv ghflu=
t D
@ 7
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7
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Vl ho hvwdgr whqvlrqdo d txh hvw vrphwlgr ho sdudohohsshgr hohphqwdo hv wdo txh vh
fxpsoh txh 7
@ C
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@ s +hvwdgr gh suhvlq klgurvwwlfd,/ hqwrqfhv ho ydoru gh od
glodwdflq yroxpwulfd hv ljxdo d=
t D
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hv xqd frqvwdqwh ivlfd gho pdwhuldo txh vh oodpd pgxor gh hodvwlflgdg yroxpwulfr/ |
wlhqh glphqvlrqhv ^I O3 P `1
Sru lqwxlflq ivlfd | ud}rqhv whuprglqplfdv/ ho pgxor gh hodvwlflgdg yroxpwulfr
gheh vhu srvlwlyr/ or fxdo lpsolfd txh 4@5= Hv ghflu/ txh hq fxdotxlhu pdwhuldo
lvwursr ho frhflhqwh gh Srlvvrq gheh vhu qhfhvduldphqwh lqihulru d 3> 8/ txh fruuhvsrqgh
do oplwh gh frpsruwdplhqwr lqfrpsuhvleoh +N @ 4,1
Los autores, 2003; Edicions UPC, 2003.
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4 1 4 4 1 U H O D F L Q W H Q V L Q 0 G H I R U P D F L Q 1 H V W X G L R H [ S H U L P H Q W D O 58
4144 Uhodflq whqvlq0ghirupdflq1 Hvwxglr h{shulphqwdo
Od uhodflq uhdo hqwuh whqvlq | ghirupdflq gh xq pdwhuldo vh ghwhuplqd gh irupd
h{shulphqwdo phgldqwh hqvd|rv hq oderudwrulr1 Orv hqvd|rv pv vlpsohv gh uhdol}du vrqorv gh wudfflq r gh frpsuhvlq sxud vreuh surehwdv floqgulfdv r sulvpwlfdv qrupdol0
}dgdv1 Hvwrv hqvd|rv vh uhdol}dq dsolfdqgr hq orv h{wuhprv gh od surehwd xqd ixhu}d S
hq gluhfflq gho hmh gh od plvpd1 Oodpdqgr D do uhd lqlfldo gh od vhfflq wudqvyhuvdo
gh od surehwd/ vh ghqh frpr whqvlq qrplqdo +phgld, d od uhodflq=
@S
D+4187,
Sru rwur odgr/ oodpdqgr O d od orqjlwxg fdoleudgd gh od surehwd | O do dodujdplhqwr
gh vwd surgxflgd sru od ixhu}d d{ldo/ vh ghqh frpr ghirupdflq d{ldo qrplqdo d od
uhodflq=
@ OO +4188,
Vl hq ho hqvd|r vh plgh wdpelq od yduldflq g gh xqd gh odv glphqvlrqhv wudqvyhu0
vdohv gh od surehwd/ gh orqjlwxg lqlfldo g> vh sxhgh fdofxodu od ghirupdflq wudqvyhuvdo
qrplqdo frpr=
|
@g
g+4189,
Ho hqvd|r vh uhdol}d dxphqwdqgr surjuhvlydphqwh od ixhu}d S dsolfdgd gh irupd
ohqwd | judgxdo | plglhqgr orv dodujdplhqwrv d{ldohv O | orv dfruwdplhqwrv wudqvyhu0
vdohv g fruuhvsrqglhqwhv d fdgd qlyho gh fdujd1 D sduwlu gh orv frqmxqwrv gh ydoruhv
+S> O> g, rewhqlgrv/ vh fdofxodq orv fruuhvsrqglhqwhv frqmxqwrv gh ydoruhv gh whq0
vlq | ghirupdflq qrplqdohv +> > | ,1 Orv sduhv +> , vh uhsuhvhqwdq hq xqd fxuydtxh vh ghqrplqd fxuyd gh whqvlq0ghirupdflq gho pdwhuldo +Iljxud 4155,1
Od fxuyd whqvlq0ghirupdflq rewhqlgd ghshqgh gh pxowlwxg gh idfwruhv +yhorflgdg
gh dsolfdflq gh od fdujd/ whpshudwxud/ glvsrvlwlyr h{shulphqwdo hpsohdgr/ hwf1, sru
or txh orv hqvd|rv ghehq uhdol}duvh gh irupd qrupdol}dgd/ frq htxlsrv hvwdqgdul}d0
grv | hq frqglflrqhv frqwurodgdv1 Gh hvwd irupd vh frqvljxh txh orv uhvxowdgrv vhdq
uhsuhvhqwdwlyrv gh od hvwuxfwxud lqwhuqd fdudfwhuvwlfd gho pdwhuldo hqvd|dgr1
Ilj1 4154= Hqvd|r gh wudfflq xqld{ldo
Los autores, 2003; Edicions UPC, 2003.
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59 F D S W X O R 4 1 I X Q G D P H Q W R V G H H O D V W L F L G D G
Ilj1 4155= Fxuyd whqvlqghirupdflq xqld{ldo
Dxqtxh ho frpsruwdplhqwr gh fdgd pdwhuldo hv glihuhqwh/ hq od fxuyd whqvlq0
ghirupdflq vh glvwlqjxhq/ hq jhqhudo/ odv }rqdv | whqvlrqhv oplwh vljxlhqwhv=
Wudpr sursruflrqdo +RD,1 Hq hvwh wudpr lqlfldo vh fxpsoh od oh| gh Krrnh/ hv
ghflu/ od uhodflq hqwuh whqvlq | ghirupdflq hv olqhdo sdud whqvlrqhv lqihulruhv
do ydoru
txh vh oodpd oplwh gh sursruflrqdolgdg1 Od shqglhqwh gho wudprghwhuplqd ho ydoru gho pgxor gh hodvwlflgdg r gh \rxqj +H, gho pdwhuldo1 Ho
frhflhqwh gh Srlvvrq sxhgh fdofxoduvh frpr od uhodflq hqwuh od ghirupdflq
wudqvyhuvdo | od orqjlwxglqdo/ @ |
@=
Wudpr hovwlfr +RE,1 Hq hvwh wudpr od ghvfdujd vh surgxfh gh irupd hovwlfd/ hv
ghflu/ vlq txh dsduh}fdq ghirupdflrqhv shupdqhqwhv do glvplqxlu od fdujd kdvwd
dqxoduod1 Vh oodpd oplwh hovwlfr +e
, d od p{lpd whqvlq txh vh sxhgh dofdq}du
vlq txh vh surgx}fdq ghirupdflrqhv shupdqhqwhv/ wdpelq oodpdgdv lqhovwlfdv r
sovwlfdv1 Ho wudpr DE/ gh frpsruwdplhqwr hovwlfr qr olqhdo/ sxhgh vhu pv r
phqrv revhuydeoh ghshqglhqgr gho pdwhuldo1
Wudpr sovwlfr +EG,1 Hq hvwh wudpr vh revhuyd ghirupdflq shupdqhqwh do
ghvfdujdu od surehwd/ sru hmhpsor od pdufdgd RO hq ho gldjudpd1 Vh oodpd oplwh
gh xhqfld +s
, d od whqvlq d sduwlu gh od fxdo ho pdwhuldo vh ghirupd fdvl vlq
dxphqwr gh od whqvlq1
Los autores, 2003; Edicions UPC, 2003.
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4 1 4 4 1 U H O D F L Q W H Q V L Q 0 G H I R U P D F L Q 1 H V W X G L R H [ S H U L P H Q W D O 5:
Wudpr gh xhqfld +FG,1 Hq hvwh wudpr od ghirupdflq dxphqwd vlq txh vh sur0
gx}fd xq dxphqwr dsuhfldeoh gh od whqvlq1 Kdvwd oohjdu do sxqwr G od ghirupdflq
orqjlwxglqdo | odv ghirupdflrqhv wudqvyhuvdohv vrq vhqvleohphqwh xqliruphv d or
odujr gh od orqjlwxg wlo gh od surehwd1
Wudpr gh hqgxuhflplhqwr sru ghirupdflq +GH,1 D sduwlu gh xq ghwhuplqdgr
ydoru gh od ghirupdflq/ od whqvlq qhfhvduld sdud vhjxlu dxphqwdqgr od ghiru0
pdflq sovwlfd vh lqfuhphqwd1 D hvwh ihqphqr vh oh oodpd hqgxuhflplhqwr sru
ghirupdflq | vxhoh yhqlu dfrpsddgr sru od suglgd gho hvwdgr xqliruph gh gh0
irupdflrqhv | ho lqlflr gh od orfdol}dflq gh vwdv hq od }rqd phgld gh od surehwd1
Wudpr gh hvwulfflq +HI,1 Hq hvwh wudpr vh revhuyd txh od vhfflq gh xqd
sduwh gh od surehwd frplhq}d d glvplqxlu gh irupd dsuhfldeoh1 Hvwh ihqphqr gh
orfdol}dflq gh odv ghirupdflrqhv orqjlwxglqdohv | wudqvyhuvdohv/ frq suglgd gho
hvwdgr xqliruph dqwhulru/ vh ghqrplqd hvwulfflq1 Od dsduhqwh suglgd gh whqvlq
hq hvwh wudpr vh gheh d od ghqlflq qrplqdo gh od whqvlq/ txh hvw fdofxodgd
vreuh ho uhd lqlfldo gh od surehwd/ hq oxjdu gh fdofxoduod vreuh ho uhd uhdo hq fdgd
prphqwr/ | txh hv phqru ghelgr d od hvwulfflq1 Vh ghqrplqd whqvlq gh urwxud
+o
, d od p{lpd whqvlq phglgd hq ho hqvd|r1
Ho wlsr gh fxuyd whqvlq0ghirupdflq ghvfulwr fruuhvsrqgh d orv pdwhuldohv oodpdgrv
gfwlohv/ frpr sru hmhpsor ho dfhur r ho freuh/ txh suhvhqwdq xq pdufdgr frpsruwd0
plhqwr sovwlfr | dofdq}dq od urwxud frq xq qlyho hohydgr gh ghirupdflq1 Dv/ sdud
hvwrv pdwhuldohv/ @
* 53
| V
* 533
/ | orv wudprv ghvfulwrv vrq iflophqwh
lghqwlfdeohv1
Ilj1 4156= Fxuyd whqvlqghirupdflq sdud pdwhuldohv iujlohv
Los autores, 2003; Edicions UPC, 2003.
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5; F D S W X O R 4 1 I X Q G D P H Q W R V G H H O D V W L F L G D G
Hq rwurv pdwhuldohv frpr/ sru hmhpsor/ od ixqglflq/ ho krupljq r ho ylgulr/ ood0
pdgrv iujlohv/ qr vh suhvhqwd xqd }rqd gh xhqfld sovwlfd elhq ghqlgd1 Hq hvwrv
pdwhuldohv vh wrpd frqyhqflrqdophqwh frpr whqvlq gh xhqfld/ s
/ od fruuhvsrqglhqwh
d xqd ghirupdflq shupdqhqwh gh 3/5 ( +Iljxud 4156,1 Hvwrv pdwhuldohv urpshq frq
srfd ghirupdflq +V
* 8
,/ euxvfdphqwh/ flufxqvwdqfld sholjurvd sdud od vhjxulgdg
gh odv hvwuxfwxudv1
Qr revwdqwh or glfkr/ qr hv gho wrgr uljxurvr fdolfdu d xq pdwhuldo frqfuhwr frpr
gwlo r iujlo/ |d txh ho frpsruwdplhqwr phfqlfr ghshqgh hq judq phglgd gh odv
frqglflrqhv gh fdujd1 Dv/ xq dfhur gfwlo hq frqglflrqhv qrupdohv sxhgh prvwudu
xq frpsruwdplhqwr pdufdgdphqwh iujlo d edmdv whpshudwxudv r d dowdv yhorflgdghv gh
fdujd1 Sru rwud sduwh/ ho klhuur ixqglgr r ho ylgulr pxhvwudq xq frpsruwdplhqwr gfwlo
d dowdv whpshudwxudv1
Gheh vhdoduvh wdpelq txh pxfkrv pdwhuldohv iujlohv h{klehq uhvlvwhqfldv px| gl0ihuhqwhv vhjq hvwq vrolflwdgrv d wudfflq r d frpsuhvlq1 Dv sru hmhpsor/ ho krupljq
| od pd|rud gh odv urfdv qdwxudohv wlhqhq xq frpsruwdplhqwr px| iujlo d wudfflq frq
xqd uhvlvwhqfld uhodwlydphqwh edmd1 Sru rwur odgr/ vx frpsruwdplhqwr d frpsuhvlq hv
pxfkr pv gfwlo/ dxqtxh vlq prvwudu xqd }rqd fodud gh xhqfld/ | vx uhvlvwhqfld d
frpsuhvlq hv pxfkr pd|ru txh d wudfflq/ gho rughq gh 43 yhfhv pd|ru/ wdo frpr vh
pxhvwud hq od Iljxud 41571
Orv dfhurv gh frqvwuxfflq/ hq fdpelr/ wlhqhq xq frpsruwdplhqwr sduhflgr hq wudf0
flq | hq frpsuhvlq +sdud slh}dv srfr hvehowdv,/ | frq pdufdgrv wudprv gh xhqfld hq
Ilj1 4157= Fxuyd whqvlqghirupdflq sdud ho krupljq
Los autores, 2003; Edicions UPC, 2003.
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4 1 4 5 1 W H Q V L Q O P L W H / W H Q V L Q D G P L V L E O H \ F R H I L F L H Q W H G H V H J X U L G D G 5 hv ghflu/ vl p d {
hv ho ydoru p{lpr gh od whqvlq hq od
hvwuxfwxud/ gheh fxpsoluvh txh
p d {
d g p
o l p
+418;,
Od ghwhuplqdflq gho frhflhqwh gh vhjxulgdg ghshqgh gh xq judq qphur gh idfwr0
uhv/ shur vwh gheh vhu pd|ru fxdqwr pd|ru vhd ho judgr gh lqfhuwlgxpeuh hq od idvh gh
sur|hfwr r fxdqwr pd|ru vhd ho lpsdfwr/ hq wuplqrv vrflrhfrqplfrv/ txh dfduuhdud
xq hyhqwxdo idoor hvwuxfwxudo1 Ho frhflhqwh gh vhjxulgdg sxhgh uhgxfluvh hq hvwuxfwx0
udv gh srfd frpsohmlgdg r gh ylgd wlo uhgxflgd1 Hq fxdotxlhu fdvr/ hvwh frhflhqwh
vh md vhjq od h{shulhqfld dgtxlulgd hq hvwuxfwxudv dqorjdv | hq ho qlyho whfqrojlfr
Los autores, 2003; Edicions UPC, 2003.
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4 1 4 6 1 W H Q V L Q H T X L Y D O H Q W H \ F U L W H U L R V G H U H V L V W H Q F L D 64
dsolfdeoh1 Hq hglfdflq h lqjhqlhud gh od frqvwuxfflq vh xvdq frhflhqwhv gh vhjxul0
gdg hqwuh 5 | 8/ dgrswqgrvh ydoruhv pv dowrv sdud od frqvwuxfflq lq vlwx txh sdud
od suhideulfdflq1 Hq lqjhqlhud dhurqdwlfd vh xvdq/ qhfhvduldphqwh/ frhflhqwhv gh
vhjxulgdg phqruhv/ hqwuh 4/8 | 5/ shur dvrfldgrv d pwrgrv gh dqolvlv/ frqvwuxfflq/
frqvhuydflq | frqwuro pxfkr pv frpsohmrv1
4146 Whqvlq htxlydohqwh | fulwhulrv gh uhvlvwhqfld
Hq od Vhfflq dqwhulru vh kd lqwurgxflgr ho frqfhswr gh whqvlq dgplvleoh |/ wdpelq/
ho gh frhflhqwh gh vhjxulgdg/ hq uhodflq do gh whqvlq oplwh1 Hvwh owlpr vh ghqh
xvxdophqwh sdud orv glyhuvrv pdwhuldohv hq ixqflq gh hqvd|rv gh uhvlvwhqfld hq txh ho
hvwdgr whqvlrqdo hv ixqgdphqwdophqwh xqld{ldo1 Vlq hpedujr/ hv kdelwxdo hq orv fdvrv gh
lqwhuv hq Phfqlfd gh Volgrv | hq Fofxor gh Hvwuxfwxudv txh orv pdwhuldohv wudedmhq
hq hvwdgrv elglphqvlrqdohv r wulglphqvlrqdohv gh whqvlq1 Hvwr lpsolfd od qhfhvlgdg gh
uhghqlu ho frqfhswr gh whqvlq oplwh gh irupd txh vhd xwlol}deoh edmr fdujd pxowld{ldo1
Od glfxowdg hvwuled hq od glfxowdg sufwlfd gh uhdol}du hqvd|rv edmr hvwdgrv gh fdujd
frpsohmrv/ sru xq odgr/ | od pxowlsolflgdg gh frpelqdflrqhv gh whqvlrqhv d frqvlghudu/
sru rwur1 Ghelgr d hoor/ hv kdelwxdo phglu ho qlyho gh vrolflwdflq gh xq pdwhuldo d wudyv
gho frqfhswr gh whqvlq +xqld{ldo, htxlydohqwh/ h t
1
Vh ghqh frpr whqvlq htxlydohqwh +r gh frpsdudflq, gh xq hvwdgr pxowld{ldo gh
whqvlrqhv d xq ydoru hvfdodu/ ixqflq gh odv frpsrqhqwhv gho whqvru gh whqvlrqhv +r
gh orv ydoruhv gh odv whqvlrqhv sulqflsdohv gh vwh,/ txh sxhgh fruuhodflrqduvh frq orv
ydoruhv gh whqvlq hq xq hqvd|r xqld{ldo1 Dv/ grv hvwdgrv pxowld{ldohv gh whqvlq vrqhtxlydohqwhvd hihfwrv gh uhvlvwhqfld vl ohv fruuhvsrqghq lgqwlfdv whqvlrqhv htxlydohqwhv1
Hq sduwlfxodu/ xq hvwdgr pxowld{ldo hv htxlydohqwh d xq hvwdgr xqld{ldo hq ho txh dfwd
xqd whqvlq ljxdo d od whqvlq htxlydohqwh gho sulphur1
Ho frqfhswr gh whqvlq htxlydohqwh shuplwh dsolfdu d orv hvwdgrv pxowld{ldohv orv
plvprv fulwhulrv gh vhjxulgdg txh hq hvwdgrv xqld{ldohv1 Dv/ vh h{ljh txh od p{lpd
whqvlq htxlydohqwh hq od hvwuxfwxud qr vxshuh ho ydoru gh od whqvlq dgplvleoh/ ghqlgd
dsolfdqgr xq ghwhuplqdgr frhflhqwh gh vhjxulgdg d od whqvlq oplwh xqld{ldo=
+h t
,p d {
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q+418
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65 F D S W X O R 4 1 I X Q G D P H Q W R V G H H O D V W L F L G D G
D hvwdv ghqlflrqhv vh ohv ghqrplqd fulwhulrv gh uhvlvwhqfld1 Orv pv kdelwxdohv vrq orv
vljxlhqwhv=
+d, Fulwhulr gh Udqnlqh
Vhjq hvwh fulwhulr/ ho pdwhuldo idood fxdqgr ho p{lpr ydoru devroxwr gh xqd gh odv
whqvlrqhv sulqflsdohv dofdq}d xq ydoru ljxdo d od whqvlq oplwh1 Hv ghflu/ vl odv whqvlrqhv
sulqflsdohv vrq
P
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h t
@ +m
m,p d {
l @ 4> 5> 6 +4193,
Hvwh fulwhulr vxhoh hpsohduvh sdud pdwhuldohv frpr ho krupljq/ hq dtxhoodv vlwxdflrqhv
hq txh ho idoor gho pdwhuldo vh surgxfh suhgrplqdqwhphqwh d wudfflq/ vhjq ho oodpdgr
prgr L gh iudfwxud1
+e, Fulwhulr gh Wuhvfd
Vhjq hvwh fulwhulr/ ho pdwhuldo idood fxdqgr od whqvlq wdqjhqfldo p{lpd dofdq}d ho
ydoru fruuhvsrqglhqwh d od whqvlq wdqjhqfldo p{lpd hq ho hqvd|r xqld{ldo1 Hv ghflu/
vl odv whqvlrqhv sulqflsdohv vrq
P
/ hqwrqfhv od whqvlq wdqjhqfldo p{lpd
gheh vhu
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Los autores, 2003; Edicions UPC, 2003.
8/14/2019 Ataraxiainc - 2006 - Edicions UPC - 2003 - Mecanica de Estructuras I, Resistencia de Materiales
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4 1 4 6 1 W H Q V L Q H T X L Y D O H Q W H \ F U L W H U L R V G H U H V L V W H Q F L D 66
+g, Fulwhulr gh PrkuFrxorpe
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dsolfdu d orv jhrpdwhuldohv r pdwhuldohv iulfflrqdohv/ wdohv frpr ho krupljq r orv vxhorv/
txh suhvhqwdq xqd pdufdgd glihuhqfld hqwuh odv uhvlvwhqfld d wudfflq | frpsuhvlq1
Hq od Iljxud 4159 vh pxhvwud jufdphqwh od glihuhqfld hqwuh orv glyhuvrv fulwhulrv
gh idoor lqwurgxflgrv1 Sdud xq hvwdgr sodqr gh whqvlrqhv/
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ghwhuplqdq xq ydoru oplwh gh od fruuhvsrqglhqwh whqvlq htxlydohqwh1 Sxhgh ghfluvh
txh ho lqwhulru gh odv fxuydv dv ghqlgdv fruuhvsrqgh d vlwxdflrqhv hovwlfdv/ hv ghflu/
vlq ulhvjr gh idoor1 Ho h{whulru gh glfkdv fxuydv fruuhvsrqgh d hvwdgrv gh whqvlq
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+e, fulwhulrv gh Wuhvfd | PrkuFrxorpe
Los autores, 2003; Edicions UPC, 2003.
8/14/2019 Ataraxiainc - 2006 - Edicions UPC - 2003 - Mecanica de Estructuras I, Resistencia de Materiales
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5 Ixqgdphqwrv gh Uhvlvwhqfld gh Pdwhuldohv
514 Lqwurgxfflq
Od Uhvlvwhqfld gh Pdwhuldohv |/ sru h{whqvlq/ od Phfqlfd gh Hvwuxfwxudv sxhghqfrqvlghuduvh frpr dtxhood sduwh gh od Phfqlfd gh Volgrv txh uhvxowd gh dsolfdu odWhrud gh od Hodvwlflgdg d xq wlsr uhvwulqjlgr gh sureohpdv txh vh sodqwhdq hq ho gd dgd gh od Lqjhqlhud Hvwuxfwxudo sdud srvlelolwdu vx uhvroxflq gh irupd dqdowlfd1 Hvwdfdwhjrud gh sureohpdv ylhqh ghqlgd/ ixqgdphqwdophqwh/ sru=
Od uhvwulfflq hq od ghqlflq gho sureohpd hovwlfr1 Hvwd uhvwulfflq dihfwd dod jhrphwud gh orv hohphqwrv hvwuxfwxudohv/ d vxv frqglflrqhv gh vxvwhqwdflq |ylqfxodflq | do wlsr gh dfflrqhv frqvlghudgdv1 Dv/ od Uhvlvwhqfld gh Pdwhuldohvvh dsolfd d slh}dv sulvpwlfdv +yljdv/ froxpqdv/ eduudv/ hmhv/ hwf1, txh hvwq dsr0|dgdv/ duwlfxodgdv r hpsrwudgdv | vrphwlgdv d od dfflq gh ixhu}dv sxqwxdohv r
uhsduwlgdv/ ghvfhqvrv gh dsr|rv/ fdujdv wuplfdv/ hwf1
Od dgrsflq gh flhuwdv klswhvlv ixqgdphqwdohv1 Hvwdv klswhvlv shuplwhq/ sruxq odgr/ ghqlu xq sureohpd olqhdo |/ sru rwur/ derugdu xq sureohpd lqlfldophqwhwulglphqvlrqdo +odv slh}dv wlhqhq orqjlwxg/ dowxud | dqfkxud, frpr xq hqvdpeod0 mh gh sureohpdv evlfdphqwh xqlglphqvlrqdohv +odv slh}dv vh uhsuhvhqwdq frproqhdv,1
Vh suhvhqwdq hq hvwh Fdswxor/ hq irupd euhyh/ orv ixqgdphqwrv gh odv grv glvfl0solqdv phqflrqdgdv1 Vh lqwurgxfhq/ sulphur/ frqfhswrv hohphqwdohv frpr ho gh slh}d/hvwuxfwxud/ dsr|r/ hqodfh/ hwf1/ | ghvsxv rwurv gh lpsruwdqfld ixqgdphqwdo frpr orvgh hvixhu}rv vreuh xqd vhfflq/ hvwuxfwxudv lvrvwwlfdv h klshuhvwwlfdv/ oh|hv gh hvixhu0}rv/ hwf1 Hq od owlpd Vhfflq vh lqwurgxfh od sureohpwlfd dvrfldgd d od uhvroxflq ghhvwuxfwxudv klshuhvwwlfdv> sdud hoor/ vh glvfxwhq orv uhtxlvlwrv gh htxloleulr | frpsdwl0elolgdg gh od vroxflq | vh sodqwhdq/ hq ixqflq gh vwrv/ odv edvhv gh orv pwrgrv ghvroxflq gh hvwuxfwxudv klshuhvwwlfdv1
68
Los autores, 2003; Edicions UPC, 2003.
8/14/2019 Ataraxiainc - 2006 - Edicions UPC - 2003 - Mecanica de Estructuras I, Resistencia de Materiales
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69 F D S W X O R 5 1 I X Q G D P H Q W R V G H U H V L V W H Q F L D G H P D W H U L D O H V
515 Remhwlyr gh od Uhvlvwhqfld gh Pdwhuldohv
Od Uhvlvwhqfld gh Pdwhuldohv/ frpr sduwh lqwhjudqwh gh od Phfqlfd gh Hvwuxfwxudv/ wlh0
qh frpr remhwlyr ixqgdphqwdo ghwhuplqdu od uhvsxhvwd gh odv hvwuxfwxudv fxdqgr vwdvvh yhq vrphwlgdv d odv glihuhqwhv dfflrqhv txh ghehq vrsruwdu gxudqwh vx frqvwuxfflq| ylgd wlo1
Sru uhvsxhvwd hvwuxfwxudo vh hqwlhqgh/ evlfdphqwh/ od ghwhuplqdflq gh orv hv0wdgrv gh whqvlq | ghirupdflq d orv txh od hvwuxfwxud yd d hvwdu vrphwlgd sru hihfwrgh orv glihuhqwhv hvwdgrv gh fdujd txh vh frqvlghudq1 Od ghwhuplqdflq gh orv hvwdgrvgh whqvlq hv qhfhvduld sdud frpsuredu od vdwlvidfflq gh orv fulwhulrv gh uhvlvwhqfldtxh hvwdeohfhq odv fruuhvsrqglhqwhv qrupdwlydv | orv xvrv gh exhqd sufwlfd/ gh fdud djdudqwl}du od vhjxulgdg gh odv hvwuxfwxudv1 Sru vx sduwh/ od ghwhuplqdflq gh orv hvwdgrvgh ghirupdflq vxhoh vhu qhfhvduld sdud vdwlvidfhu orv fulwhulrv gh uljlgh}/ txh hvwq dphqxgr oljdgrv d uhtxlvlwrv gh ixqflrqdolgdg gh odv hvwuxfwxudv1
Sru wdqwr/ od Uhvlvwhqfld gh Pdwhuldohv suhwhqgh hvwdeohfhu odv frqglflrqhv gh uh0vlvwhqfld | uljlgh} gh odv hvwuxfwxudv dqdol}dgdv1 Dperv dvshfwrv vh derugdq hq orvgrv wlsrv gh sureohpdv ixqgdphqwdohv txh vh sodqwhdq hq od sufwlfd lqjhqlhulo= hoglphqvlrqdplhqwr | od yhulfdflq gh hvwuxfwxudv1
Ho surfhvr gh frqvwuxfflq gh xqd hvwuxfwxud hpslh}d sru od frqfhsflq gh vwd1 Hqhvwd idvh vh ghwhuplqd vx wlsrorjd | vh lghqwlfdq odv dfflrqhv txh dfwxduq vreuh hood1Hvwd lghqwlfdflq suhyld vh kdfh gh irupd dsur{lpdgd/ |d txh orv ydoruhv uhdohv ghpxowlwxg gh dfflrqhv ghshqghq gh od irupd | glphqvlrqhv ghqlwlydv gh od hvwuxfwxud1
Hq hvh prphqwr hqwud hq mxhjr od Uhvlvwhqfld gh Pdwhuldohv sdud hydoxdu od uhvsxhvwdhvwuxfwxudo gho fruuhvsrqglhqwh suhglvhr hvwuxfwxudo1 Hq ixqflq gh orv uhvxowdgrvrewhqlgrv/ vh surfhgh do glphqvlrqdplhqwrgh orv hohphqwrv hvwuxfwxudohv/ hv ghflu/ d gduydoruhv frqfuhwrv d odv glphqvlrqhv gh vwrv1 Ghvsxv/ vh sdvd d od idvh gh ghqlflq ghosur|hfwr/ hq od fxdo vh shuvljxh vdwlvidfhu orv uhtxlvlwrv suhylvwrv1 Hq hvwd idvh vh sxhghqprglfdu r udwlfdu odv glyhuvdv ghflvlrqhv wrpdgdv hq od idvh suhyld1 Hq fxdotxlhu fdvr/vh ghqhq | frqfuhwdq od wlsrorjd hvwuxfwxudo | odv dfflrqhv d frqvlghudu +shvr surslr/
Ilj1 514= Od Uhvlvwhqfld gh Pdwhuldohv hq ho Dqolvlv gh Hvwuxfwxudv
Los autores, 2003; Edicions UPC, 2003.
8/14/2019 Ataraxiainc - 2006 - Edicions UPC - 2003 - Mecanica de Estructuras I, Resistencia de Materiales
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5 1 6 1 F R Q F H S W R G H S L H ] D \ H V W U X F W X U D 6:
fdujdv ylydv/ hwf1,1 Vh sdvd hqwrqfhv d od yhulfdflq gho prghor hvwuxfwxudo uhvxowdqwh1Vl hvwd frpsuredflq d srvwhulrul gho glvhr hvwuxfwxudo hv vdwlvidfwruld/ vh hoderud ho
sur|hfwr ghqlwlyr> vl qr or hv/ hv qhfhvdulr yroyhu d od idvh gh ghqlflq1 Vh surfhgh/sxhv/ gh irupd lwhudwlyd kdvwd txh vh vdwlvidfhq dghfxdgdphqwh wrgrv orv uhtxlvlwrvqhfhvdulrv1
Ho surfhvr ghvfulwr vh pxhvwud gh irupd hvtxhpwlfd hq od Iljxud 514/ grqgh vhlqglfd ho sdsho txh od Uhvlvwhqfld gh Pdwhuldohv ghvhpshd hq o1
516 Frqfhswr gh slh}d | hvwuxfwxud
Vh oodpd slh}d sulvpwlfd/ yljd r eduud do fxhusr volgr hqjhqgudgr sru xq uhd sodqd
V txh vh pxhyh hq ho hvsdflr/ gh pdqhud txh vx fhqwur gh judyhgdg J uhfruuh xqd oqhdgdgd o | vx sodqr vh pdqwlhqh frqvwdqwhphqwh qrupdo d glfkd oqhd1 D od oqhd o vh ohghqrplqd gluhfwul}/ oqhd phgld r hmh gh od slh}d | d odv glyhuvdv srvlflrqhv gho uhd Vvh ohv oodpd vhfflrqhv uhfwdv r vhfflrqhv qrupdohv gh od slh}d +yhu Iljxud 515d,1
Od gluhfwul} gh od slh}d sxhgh vhu xqd fxuyd dodehdgd/ sodqd r uhfwd/ gdqgr oxjdud slh}dv dodehdgdv +yljd hq kolfh,/ slh}dv sodqdv +dufrv/ yljdv edofq, r slh}dv uhfwdv+pqvxodv/ froxpqdv/ hwf1,1 Od vhfflq gh od slh}d sxhgh vhu frqvwdqwh r yduldeoh d orodujr gh od slh}d/ gdqgr oxjdu d slh}dv gh vhfflq frqvwdqwh r yduldeoh/ uhvshfwlydphqwh1
Ilj1 515= Frqfhswr gh= +d, slh}d sulvpwlfd/ +e, uhedqdgd | +f, eud
Los autores, 2003; Edicions UPC, 2003.
8/14/2019 Ataraxiainc - 2006 - Edicions UPC - 2003 - Mecanica de Estructuras I, Resistencia de Materiales
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6; F D S W X O R 5 1 I X Q G D P H Q W R V G H U H V L V W H Q F L D G H P D W H U L D O H V
Vh oodpd uhedqdgd do hohphqwr glihuhqfldo gh slh}d frqwhqlgr hqwuh grv vhfflrqhvuhfwdv lqqlwdphqwh su{lpdv +yhu Iljxud 515e,1 Dv/ vh sxhgh frqvlghudu d od yljdfrpr xq volgr jhqhudgr sru xqd vxfhvlq gh uhedqdgdv glihuhqfldohv1 Vh oodpd eud dohohphqwr glihuhqfldo gh yroxphq jhqhudgr sru xq hohphqwr glihuhqfldo gho uhd sodqd Vhq ho prylplhqwr txh jhqhud od slh}d sulvpwlfd +yhu Iljxud 515f,1 Sru wdqwr/ vh sxhghfrqvlghudu d od yljd frpr xq volgr irupdgr sru xq kd} gh eudv glihuhqfldohv1 Oduhedqdgd | od eud vrq orv hohphqwrv glihuhqfldohv evlfrv gh hvwxglr hq od Uhvlvwhqfldgh Pdwhuldohv1
Vh oodpd slh}d gh sodqr phglr d dtxhood slh}d sodqd hq od txh od vhfflq uhfwd hvvlpwulfd uhvshfwr d xq sodqr txh frqwlhqh d od gluhfwul} gh od slh}d | txh hvw vrphwlgdd fdujdv frqwhqlgdv hq glfkr sodqr phglr1 Vh oodpd slh}d gh flolqgur phglr d dtxhoodslh}d sodqd hq od txh od vhfflq uhfwd hv vlpwulfd uhvshfwr d xq sodqr shushqglfxodu dotxh frqwlhqh d od gluhfwul} gh od slh}d | txh hvw vrphwlgd d fdujdv shushqglfxoduhv dosodqr gh od gluhfwul}1
Vh oodpd hvwuxfwxud gh eduudv do vlvwhpd phfqlfr irupdgr sru ho hqvdpeodmh ghslh}dv sulvpwlfdv1 Vh kdeod gh hvwuxfwxud sodqd fxdqgr odv gluhfwulfhv gh wrgdv odvslh}dv txh irupdq od hvwuxfwxud hvwq frqwhqlgdv hq xq plvpr sodqr/ | gh hvwuxfwxudhvsdfldo hq fdvr frqwudulr1 Vh oodpd hvwuxfwxud gh sodqr phglr d od hvwuxfwxud sodqdirupdgd sru slh}dv gh sodqr phglr/ | txh hvwq vrphwlgdv d fdujdv frqwhqlgdv hq glfkrsodqr phglr1 Vh oodpd hpsduuloodgr sodqr d od hvwuxfwxud sodqd irupdgd sru slh}dv ghflolqgur phglr/ | txh hvwq vrphwlgdv d fdujdv shushqglfxoduhv d vx sodqr phglr1
Hq hvwh Oleur vh wudwduq vlhpsuh hvwuxfwxudv gh eduudv |/ ixqgdphqwdophqwh/ hv0wuxfwxudv gh eduudv gh sodqr phglr1
517 Sulqflslrv gh od Uhvlvwhqfld gh Pdwhuldohv
Od Uhvlvwhqfld gh Pdwhuldohv hv xqd glvflsolqd txh vh edvd hq wuhv sulqflslrv r klswhvlvixqgdphqwdohv= ho Sulqflslr gh uljlgh}/ ho Sulqflslr gh vxshusrvlflq | ho Sulqflslr ghVdlqw0Yhqdqw1 Od ydolgh} gh orv uhvxowdgrv rewhqlgrv phgldqwh hvwd whrud ghshqgh/evlfdphqwh/ gho judgr gh vdwlvidfflq gh hvwrv sulqflslrv hq ho fdvr frqfuhwr hq txh vhdsoltxh1
51714 Sulqflslr gh uljlgh}
Ho Sulqflslr gh uljlgh} hv xqd frqvhfxhqfld gluhfwd gh od klswhvlv gh shtxhrv pryl0plhqwrv dsolfdgd do dqolvlv gh hvwuxfwxudv | hvwdeohfh txh
odv hfxdflrqhv gh htxloleulr vh sxhghq irupxodu vreuh od jhrphwud lqghirupdgd/
hv ghflu/ vlq frqvlghudu orv prylplhqwrv suryrfdgrv sru ho vlvwhpd gh fdujdv1
Los autores, 2003; Edicions UPC, 2003.
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5 1 7 1 S U L Q F L S L R V G H O D U H V L V W H Q F L D G H P D W H U L D O H V 6 vlq hood +ghuhfkd,/ odv uhdfflrqhv ghshqghq gh odghirupdflq gh od hvwuxfwxud/ | vwd hv ghvfrqrflgd d sulrul1
51715 Sulqflslr gh vxshusrvlflq
Frpr |d vh lqglf hq ho Fdswxor dqwhulru/ vl hv yolgd od klswhvlv gh shtxhrv ghv0sod}dplhqwrv | vh fxpsoh/ dghpv/ od oh| gh Krrnh/ hqwrqfhv ho sureohpd hv olqhdo1 Oduhvroxflq gh xq sureohpd hvwuxfwxudo olqhdo hv pxfkr pv vhqflood txh od gh xqr qrolqhdo> dsduwh gh rwudv glfxowdghv gh wlsr sufwlfr/ hq xq sureohpd qr olqhdo qr h{lvwh/hq jhqhudo/ jdudqwd gh txh od vroxflq h{lvwd | vhd qlfd1
Dghpv/ vl ho sureohpd hv olqhdo sxhgh dsolfduvh ho Sulqflslr gh vxshusrvlflq1 vwhhvwdeohfh txh
orv hihfwrv txh xq vlvwhpd gh ixhu}dv ruljlqd vreuh xqd hvwuxfwxud vrq ljxdohv d
od vxpd gh orv hihfwrv txh ruljlqdq fdgd xqd gh odv ixhu}dv gho vlvwhpd dfwxdqgr sru
vhsdudgr1Xqd frqvhfxhqfld lqphgldwd gho flwdgr sulqflslr hv txh ho hvwdgr qdo +whqvlrqdo
| ghirupdflrqdo, gh xqd hvwuxfwxud vrphwlgd d xq vlvwhpd gh ixhu}dv qr ghshqgh ghorughq gh dsolfdflq gh glfkdv ixhu}dv1 Hvwr shuplwh/ sru hmhpsor/ dqdol}du ho fdvr gh xqdvrolflwdflq frpsohmd ghvfrpsrqlqgrod hq fdvrv pv vlpsohv gh hvwxgldu1 Hv srvleoh/
Los autores, 2003; Edicions UPC, 2003.
8/14/2019 Ataraxiainc - 2006 - Edicions UPC - 2003 - Mecanica de Estructuras I, Resistencia de Materiales
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73 F D S W X O R 5 1 I X Q G D P H Q W R V G H U H V L V W H Q F L D G H P D W H U L D O H V
Ilj1 517= Lpsolfdflq gho Sulqflslr gh vxshusrvlflq
dvlplvpr/ od ghqlflq | ho dqolvlv gh hvwdgrv gh fdujd gh lqwhuv frpr frpelqdflq ghhvwdgrv gh fdujd suhyldphqwh ghqlgrv | dqdol}dgrv1 Dpedv srvlelolgdghv vrq xwlol}dgdviuhfxhqwhphqwh hq Uhvlvwhqfld gh Pdwhuldohv1
Dv/ sru hmhpsor/ hq od Iljxud 517 vh pxhvwud xqd yljd eldsr|dgd vreuh od txh dfwdxq vlvwhpd gh fdujd frqvlvwhqwh hq grv ixhu}dv sxqwxdohv> hvwh vlvwhpd sxhgh ghvfrp0srqhuvh hq orv grv vlvwhpdv pv vlpsohv gh od sduwh lqihulru gh od Iljxud1 Ho sulqflslrgh vxshusrvlflq lpsolfd txh odv uhdfflrqhv/ prylplhqwrv/ whqvlrqhv | ghirupdflrqhvtxh suryrfd vreuh od hvwuxfwxud ho hvwdgr gh fdujdv ruljlqdo hv ljxdo d od vxpd gh odvuhdfflrqhv/ prylplhqwrv/ whqvlrqhv | ghirupdflrqhv txh suryrfdq orv hvwdgrv hq orv txhvwh vh ghvfrpsrqh1
51716 Sulqflslr gh Vdlqw0Yhqdqw
Ho Sulqflslr gh Vdlqw0Yhqdqw sxhgh frqvlghuduvh od yhugdghud slhgud dqjxodu vreuh odtxh vh hglfd od Uhvlvwhqfld gh Pdwhuldohv/ |d txh hvwdeohfh txh
hq xqd slh}d sulvpwlfd/ odv whqvlrqhv txh dfwdq vreuh xqd vhfflq uhfwd/ dohmdgd
gh orv sxqwrv gh dsolfdflq gh xq vlvwhpd gh fdujdv/ vor ghshqghq gh od ixhu}d | gho
prphqwr uhvxowdqwhv gh odv ixhu}dv vlwxdgdv d xq odgr gh od vhfflq frqvlghudgd1
Hvwd klswhvlv lpsolfd txh orv hihfwrv orfdohv +ixhu}dv frqfhqwudgdv/ dsr|rv/ yduld0flrqhv gh vhfflq, vor dihfwdq d xqd }rqd orfdol}dgd d xqr | rwur odgr gh odv vhfflrqhvhq txh vh surgxfhq1 Od irupd suhflvd hq txh vh dsolfdq odv fdujdv r orv glvsrvlwlyrvhpsohdgrv sdud dsolfduodv/ dv frpr od glvsrvlflq frqvwuxfwlyd frqfuhwd gh orv dsdud0wrv gh dsr|r/ vor dihfwdq do hvwdgr whqvlrqdo gh hvwd }rqd orfdol}dgd/ gh xqd orqjlwxgdsur{lpdgdphqwh ljxdo do fdqwr gh od slh}d1 Ho hvwxglr whqvlrqdo gh odv }rqdv dihfwdgdvsru orv hihfwrv orfdohv hv frpsohmr | gheh kdfhuvh phgldqwh od Whrud gh od Hodvwlflgdg1
Los autores, 2003; Edicions UPC, 2003.
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5 1 7 1 S U L Q F L S L R V G H O D U H V L V W H Q F L D G H P D W H U L D O H V 74
Ilj1 518= Loxvwudflq gho Sulqflslr gh Vdlqw0Yhqdqw
Hq od Vhfflq 518 vh ghqh ho frqfhswr gh hvixhu}rv vreuh xqd vhfflq/ suhflvdphqwh/frpr odv frpsrqhqwhv gh od ixhu}d | gho prphqwr uhvxowdqwhv gh odv ixhu}dv vlwxdgdv dxq odgr gh od vhfflq1 Ho fofxor gh hvixhu}rv mxhjd xq sdsho ixqgdphqwdo hq od Whrudgh Uhvlvwhqfld gh Pdwhuldohv | Fofxor gh Hvwuxfwxudv/ |d txh/ vhjq ho Sulqflslr ghVdlqw0Yhqdqw/ odv whqvlrqhv hq xqd vhfflq vor ghshqghq gho ydoru gh vwrv1
Hq od Iljxud 518 vh pxhvwudq grv hmhpsorv gh yrodgl}rv frq prphqwr dsolfdgr hqvxv h{wuhprv oleuhv1 Hq ho sulphu fdvr/ ho hpsrwudplhqwr vh frqvljxh dsr|dqgr od yljdfrqwud ho vrsruwh | frduwdqgr ho jlur phgldqwh vhqgrv dqjxoduhv vrogdgrv d dperv> hoprphqwr hq ho h{wuhpr hv od uhvxowdqwh gh xq sdu ixhu}dv krul}rqwdohv1 Hq ho vhjxqgrfdvr/ ho hpsrwudplhqwr vh frqvljxh hqfdvwudqgr dghfxdgdphqwh od yljd hq ho vrsruwh>ho prphqwr hq ho h{wuhpr hv od uhvxowdqwh gh xq sdu ixhu}dv lqfolqdgdv uhvshfwr do hmhgh od yljd1 Hq dpedv vlwxdflrqhv/ ho prphqwr uhvxowdqwh gh odv ixhu}dv dfwxdqwhv hvlgqwlfr/ plhqwudv txh od uhvxowdqwh gh ixhu}dv hv qxod1
D shvdu gh txh odv pdwhuldol}dflrqhv gh odv frqglflrqhv gh hpsrwudplhqwr | gh orvprphqwrv dsolfdgrv vrq glihuhqwhv/ vx lghdol}dflq d hihfwrv gh fofxor hv lgqwlfd/ wdofrpr vh pxhvwud hq od Iljxud 5181 Vhjq ho Sulqflslr gh Vdlqw0Yhqdqw/ odv whqvlrqhvtxh dsduhfhq hq odv }rqdv fhqwudohv gh odv slh}dv ghshqghq/ h{foxvlydphqwh/ gho pr0phqwr uhvxowdqwh gh odv ixhu}dv dfwxdqwhv> odv }rqdv h{wuhpdv/ sru ho frqwudulr/ vh yhqdihfwdgdv sru orv hihfwrv orfdohv/ | od ghwhuplqdflq gh vx hvwdgr whqvlrqdo uhtxhulufrqvlghudflrqhv pv fxlgdgrvdv1
Los autores, 2003; Edicions UPC, 2003.
8/14/2019 Ataraxiainc - 2006 - Edicions UPC - 2003 - Mecanica de Estructuras I, Resistencia de Materiales
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75 F D S W X O R 5 1 I X Q G D P H Q W R V G H U H V L V W H Q F L D G H P D W H U L D O H V
Hq od sufwlfd lqjhqlhulo/ ho hvwxglr gho hvwdgr whqvlrqdo gh xq hohphqwr hvwuxfwxudovxhoh kdfhuvh vxsrqlhqgr txh ho Sulqflslr gh Vdlqw0Yhqdqw hv dsolfdeoh d od wrwdolgdggh odv vhfflrqhv dqdol}dgdv1 Ho glphqvlrqdplhqwr gh odv slh}dv vxhoh kdfhuvh hq ixqflqgh orv uhvxowdgrv dv rewhqlgrv1 D srvwhulrul/ vh dgrswdq odv glvsrvlflrqhv frqvwuxfwlydvqhfhvduldv sdud uhgxflu r vrsruwdu gh irupdu dghfxdgd orv hihfwrv orfdohv hq odv }rqdvhq odv txh orv uhvxowdgrv fdofxodgrv hq hvwd irupd qr vrq dsolfdeohv1
51717 Uhvwulfflrqhv jhrpwulfdv
Sdud txh orv sulqflslrv ixqgdphqwdohv hqxqfldgrv vh fxpsodq | orv uhvxowdgrv gh odUhvlvwhqfld gh Pdwhuldohv vhdq yolgrv/ odv slh}dv ghehq fxpsolu flhuwdv frqglflrqhv=
Jhrphwud gh od gluhfwul}1 Hq slh}dv gh gluhfwul} fxuyd/ orv udglrv gh fxuydwxud
gh vwd ghehq vhu judqghv hq uhodflq do fdqwr gh odv slh}dv1 Vl hvwd uhodflq hvvxixflhqwhphqwh shtxhd/ orv hihfwrv gh od fxuydwxud sxhghq oohjdu d ghvsuhfldu0vh hq ho dqolvlv gho frpsruwdplhqwr gh odv uhedqdgdv | xwlol}du odv h{suhvlrqhvrewhqlgdv sdud slh}dv gh gluhfwul} uhfwd1
Jhrphwud gh odv vhfflrqhv=
Glphqvlrqhv1 Odv glphqvlrqhv wudqvyhuvdohv gh odv slh}dv/ fdqwr | dqfkr/ghehq vhu shtxhdv hq uhodflq d vx orqjlwxg1 Hvwr hv qhfhvdulr sdud txh vhfxpsodq ho sulqflslr gh Vdlqw0Yhqdqw | odv klswhvlv gh ghirupdflq gh odvvhfflrqhv txh vh xwlol}dq kdelwxdophqwh1 Sru rwur odgr/ odv fdudfwhuvwlfdv
jhrpwulfdv gh odv vhfflrqhv uhfwdv ghehq dvhjxudu txh odv slh}dv whqjdq oduljlgh} qhfhvduld sdud txh orv prylplhqwrv vhdq shtxhrv1 Hq jhqhudo/ vrqdgplvleohv odv uhodflrqhv fdqwr2ox} +k@o, vljxlhqwhv=
k
o@
4
43d
4
45sdud slh}dv uhfwdv gh krupljq dupdgr
k
o@
4
48d
4
58sdud slh}dv uhfwdv gh krupljq suhwhqvdgr
k
o@
4
53d
4
68sdud slh}dv uhfwdv gh dfhur odplqdgr
k
o@
4
73
d4
433
sdud dufrv
Yduldflq1 Hq slh}dv gh vhfflq yduldeoh/ od yduldflq gh odv glphqvlrqhvwudqvyhuvdohv gheh vhu judgxdo | ohqwd1 Odv yduldflrqhv euxvfdv hq odv fdudf0whuvwlfdv jhrpwulfdv gh odv vhfflrqhv surgxfhq hihfwrv orfdohv txh lqydolgdqho sulqflslr gh Vdlqw0Yhqdqw1
Los autores, 2003; Edicions UPC, 2003.
8/14/2019 Ataraxiainc - 2006 - Edicions UPC - 2003 - Mecanica de Estructuras I, Resistencia de Materiales
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5 1 8 1 G H I L Q L F L Q G H H V I X H U ] R V H Q X Q D V H F F L Q 76
518 Ghqlflq gh hvixhu}rv hq xqd vhfflq
Gdgd xqd yljd vrphwlgd d od dfflq gh xq vlvwhpd gh fdujdv h{whulruhv/ vxsrqguhprv
txh xqd vhfflq uhfwd fxdotxlhud/ V/ glylgh od slh}d hq grv sduwhv/ +D, | +E,/ vlwxdgdvfdgd xqd d xq odgr gh od vhfflq frqvlghudgd/ wdo frpr vh pxhvwud hq od Iljxud 5191
Ho vlvwhpd gh ixhu}dv +dfflrqhv | uhdfflrqhv, txh dfwdq vreuh od yljd gheh hvwdu hqhtxloleulr hvwwlfr1 Vl oodpdprv I
do vlvwhpd gh ixhu}dv txh dfwd vreuh od sduwh +D,gh od slh}d | I
do vlvwhpd gh ixhu}dv txh dfwd vreuh od sduwh +E,/ sru htxloleulr ghod yljd gheh vhu=
I
. I
@ 3 +514,
Dvlplvpr/ wrpdqgr prphqwrv uhvshfwr do fhqwur gh judyhgdg J gh od vhfflq V/ ghehfxpsoluvh=
Pg
V
. Pg
V
@ 3 +515,
Frqvlghuhprv dkrud/ sru vhsdudgr/ ho htxloleulr gh odv grv sduwhv gh od slh}d +D, |+E,= Sdud txh wdo htxloleulr h{lvwd/ hv qhfhvdulr vxsrqhu txh h{lvwhq xqdv ixhu}dv ghlqwhudfflq txh dpedv sduwhv vh hmhufhq hqwuh v/ d wudyv gh od vxshuflh V= Dv/ od sduwh+E, hmhufhu vreuh od sduwh +D, xqdv ixhu}dv txh srghprv uhgxflu d xqd uhvxowdqwh U
| xq prphqwr uhvxowdqwh/ uhvshfwr d J/ Pg
1 Gh irupd dqorjd/ od sduwh +D, hmhufhvreuh od sduwh +E, xqd ixhu}d uhvxowdqwh U
| xq prphqwr uhvxowdqwh Pg
1
Ilj1 519= Ixhu}d | prphqwr uhvxowdqwhv hq xqd vhfflq
Los autores, 2003; Edicions UPC, 2003.
8/14/2019 Ataraxiainc - 2006 - Edicions UPC - 2003 - Mecanica de Estructuras I, Resistencia de Materiales
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77 F D S W X O R 5 1 I X Q G D P H Q W R V G H U H V L V W H Q F L D G H P D W H U L D O H V
Sru htxloleulr gh od sduwh +D, gheh vhu=
I
. U
@ 3Pg
V
. Pg
@ 3 +516,
Sru htxloleulr gh od sduwh +E, gheh vhu=
I
. U
@ 3Pg
V
. Pg
@ 3+517,
Vxpdqgr odv hfxdflrqhv +516, | +517, | whqlhqgr hq fxhqwd ho htxloleulr joredo +hfxdflr0qhv +514, | +515,,/ vh wlhqh=
+ I
. I
, . U
. U
@ 3 , U
@ U +518d,
+ PgV
. PgV
, . Pg
. Pg
@ 3 , Pg
@ Pg
+518e,
Sru or wdqwr/ odv dfflrqhv txh xqd sduwh hmhufh vreuh od rwud vrq ljxdohv | gh vhqwlgrfrqwudulr1
Oodpdqgr sru vlpsolflgdg= U @ U
@ U
| P @ Pg
@ Pg
/ vh whqguwdpelq txh=
U @ I
@ I
P @ PgV
@ PgV
,+519,
Fxdqgr od ixhu}d uhvxowdqwh U | ho prphqwr uhvxowdqwh P vh fdofxodq hq od irupdU @ I
| P @ PgV
/ hv ghflu/ whqlhqgr hq fxhqwd odv ixhu}dv txh dfwdq sru ghodqwh
gh V +hq od sduwh E,/ ho fofxor vh oodpd iurqwdo1 Fxdqgr U | P vh fdofxodq hq odirupd U @ I
| P @ PgV
/ hv ghflu/ whqlhqgr hq fxhqwd odv ixhu}dv txh dfwdqsru ghwuv gh V +hq od sduwh D,/ ho fofxor vh oodpd gruvdo1 Hv hylghqwh txh dpervsurfhglplhqwrv frqgxfhq d uhvxowdgrv lgqwlfrv1
Ghqlprv ho wulhgur fduwhvldqr gh{wujlur {|}/ frq ruljhq hq ho fhqwur gh judyhgdgJ gh od vhfflq V/ hq ho txh ho hmh { hv qrupdo d od vhfflq V/ rulhqwdgd gh +D, d +E,| wdqjhqwh d od gluhfwul} gh od yljd/ | orv hmhv +|> }, vrq orv hmhv sulqflsdohv gh lqhufld
gh od vhfflq V +yhu Iljxud 51:,1 Sur|hfwdqgr od ixhu}d uhvxowdqwhU vreuh hvwrv hmhv vhghqh frpr=
Hvixhu}r D{lo QHvixhu}r Fruwdqwh W
C
Hvixhu}r Fruwdqwh WS
d od sur|hfflq gh U vhjq ho hmh
;A?A=
{
|
}
Los autores, 2003; Edicions UPC, 2003.
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5 1 9 1 U H O D F L Q H Q W U H H V I X H U ] R V \ W H Q V L R Q H V 78
Ilj1 51:= Ghqlflq gh hvixhu}rv hq xqd vhfflq
Dqorjdphqwh/ sur|hfwdqgr ho prphqwr uhvxowdqwh P vreuh orv hmhv/ vh ghqh frpr=
Prphqwr Wruvru P|
Prphqwr Iohfwru PC
Prphqwr Iohfwru PS
d od sur|hfflq gh P vhjq ho hmh
;A?A=
{
|
}
519 Uhodflq hqwuh hvixhu}rv | whqvlrqhv
Odv uhvxowdqwhv U | P/ | vxv uhvshfwlydv frpsrqhqwhv/ hv ghflu/ orv hvixhu}rv hq od
vhfflq/ vrq odv uhvxowdqwhv hvwwlfdv gh odv whqvlrqhv txh dfwdq vreuh od vhfflq V1 Wdofrpr vh ghql hq ho Fdswxor dqwhulru/ glfkdv whqvlrqhv vrq odv ixhu}dv gh lqwhudfflqtxh xqd sduwh gh od yljd hmhufh vreuh od rwud/ ghqlgdv sru xqlgdg gh vxshuflh gh odvhfflq V1
Dv/ wdo frpr vh pxhvwud hq od Iljxud 51;/ vreuh xq glihuhqfldo gh uhd gV/ gh odvhfflq V/ dfwxdu xqd whqvlq w/ txh sxhgh ghvfrpsrqhuvh hq xqd whqvlq qrupdo
7
/vhjq ho hmh {/ | xqd whqvlq wdqjhqfldo vreuh od vhfflq V1 Od whqvlq wdqjhqfldosxhgh ghvfrpsrqhuvh/ d vx yh}/ hq vxv frpsrqhqwhv fduwhvldqdv vhjq orv hmhv +|> },/gdqgr odv whqvlrqhv
7 C
> 7 S
/ uhvshfwlydphqwh1Sdud txh U | P iruphq xq vlvwhpd htxlydohqwh do vlvwhpd gh ixhu}dv lqwhuqdv
glihuhqfldohv w gV/ ghehq fxpsoluvh odv vljxlhqwhv uhodflrqhv yhfwruldohv=U @
]U
w gV > P @
]U
u w gV +51:,
grqgh u hv ho yhfwru gh srvlflq gh gV uhvshfwr do fhqwur gh judyhgdg gh od vhfflq/ frqfrpsrqhqwhv u +3> | > },1
Los autores, 2003; Edicions UPC, 2003.
8/14/2019 Ataraxiainc - 2006 - Edicions UPC - 2003 - Mecanica de Estructuras I, Resistencia de Materiales
59/329
79 F D S W X O R 5 1 I X Q G D P H Q W R V G H U H V L V W H Q F L D G H P D W H U L D O H V
Ilj1 51;= Uhodflq hqwuh hvixhu}rv | whqvlrqhv
Odv dqwhulruhv uhodflrqhv yhfwruldohv vh sxhghq ghvfrpsrqhu hq odv fruuhvsrqglhqwhv
hfxdflrqhv hvfdoduhv1 Dv/ od sulphud gh odv hfxdflrqhv +51:, vh ghvfrpsrqh hq=
Q @
]U
7
gV > WC
@
]U
7 C
gV > WS
@
]U
7 S
gV +51;,
Dqorjdphqwh/ od vhjxqgd gh odv hfxdflrqhv +51:, vh ghvfrpsrqh hq=
P|
@
]U
+7 S
| 7 C
}, gV > PC
@
]U
7
} gV > PS
@ ]
U
7
| gV +51
8/14/2019 Ataraxiainc - 2006 - Edicions UPC - 2003 - Mecanica de Estructuras I, Resistencia de Materiales
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5 1 : 1 H V I X H U ] R V H Q S L H ] D V G H S O D Q R P H G L R 7:
Ilj1 51
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7; F D S W X O R 5 1 I X Q G D P H Q W R V G H U H V L V W H Q F L D G H P D W H U L D O H V
od vhfflq J/ | rulhqwdgr vhjq ho vhqwlgr srvlwlyr gh uhfruulgr gh od gluhfwul}1 Ho hmh| hvw wdpelq hq ho sodqr phglr txh/ sru vlphwud/ hv sodqr sulqflsdo gh lqhufld/ | hvqrupdo do hmh {> irupdqgr frq vwh xq qjxor gh @5 udgldqhv +dqwlkrudulr,1 Ho hmh }hv qrupdo d orv hmhv +{> |, | gh vhqwlgr wdo txh ho wulhgur +{>|>}, hv gh{wujlur1 Dv/ hosodqr [\ frlqflgh frq ho sodqr {|/ | ho hmh } hv shushqglfxodu d vwh1
Do dfwxdu odv ixhu}dv h{whulruhv vor hq ho sodqr [\/ od uhvxowdqwh U gh odv ixhu}dvtxh dfwdq d xqr x rwur odgr gh od vhfflq hvw frqwhqlgd hq glfkr sodqr/ | vh ghvfrp0srqh hq xq hvixhu}r d{lo Q +vhjq ho hmh {, | xq hvixhu}r fruwdqwh W +vhjq ho hmh |,1Sru vlphwud uhvshfwr do sodqr phglr/ W
S
hv qxor1 Sru ho plvpr prwlyr/ ho prphqwruhvxowdqwh P gh odv ixhu}dv txh dfwdq d xqr x rwur odgr gh od vhfflq hv shushqglfxodudo sodqr [\ | vh oodpd prphqwr hfwru P d vx qlfd frpsrqhqwh vhjq ho hmh }1 Odvrwudv grv frpsrqhqwhv P
|
| PC
vrq qxodv sru vlphwud uhvshfwr do sodqr phglr1Do uhihuluqrv d orv hvixhu}rv hq slh}dv gh sodqr phglr dgrswduhprv ho vljxlhqwh
frqyhqlr gh vljqrv/ txh vh pxhvwud hq od Iljxud 5144=
Ho hvixhu}r d{lo Q hv srvlwlyr vl hv gh wudfflq +vhjq ho vhqwlgr srvlwlyr gho hmh{ hq od fdud iurqwdo, | qhjdwlyr vl hv gh frpsuhvlq1
Ho hvixhu}r fruwdqwh W hv srvlwlyr vl kdfh jludu od uhedqdgd hq vhqwlgr dqwlkrudulr+vhjq ho vhqwlgr srvlwlyr gho hmh | hq od fdud iurqwdo,/ | qhjdwlyr hq fdvr frqwudulr1
Ho prphqwr hfwru P hv srvlwlyr vl hv dqwlkrudulr vreuh od fdud iurqwdo +vhjq hovhqwlgr srvlwlyr gho hmh } hq od fdud iurqwdo,/ | qhjdwlyr hq fdvr frqwudulr1
Ilj1 5144= Frqyhqlr gh vljqrv sdud hvixhu}rv hq slh}dv gh sodqr phglr
Los autores, 2003; Edicions UPC, 2003.
8/14/2019 Ataraxiainc - 2006 - Edicions UPC - 2003 - Mecanica de Estructuras I, Resistencia de Materiales
62/329
5 1 ; 1 H F X D F L R Q H V G H H T X L O L E U L R H Q S L H ] D V U H F W D V 7 sC
, vhjq orv hmhv frrughqdgrv/ dsolfdgdv hq od gluhfwul} gh glfkd uhedqdgd1Wdo frpr vh pxhvwud hq od Iljxud 5145/ vh wlhqh=
d, sru htxloleulr gh ixhu}dv vhjq ho hmh {=
Q.s7
g{ . +Q . gQ, @ 3 , s7
@ gQg{
+5143,
e, sru htxloleulr gh ixhu}dv vhjq ho hmh |=
WC
.sC
g{ . +WC
. gWC
, @ 3 , sC
@ gWCg{
+5144,
f, sru htxloleulr gh prphqwrv uhvshfwr do fhqwur gh judyhgdg gh od fdud iurqwdo=
PS
. WC
g{ . +PS
. gPS
,sC
g{P
5@ 3 , W
C
@ gPSg{
+5145,
grqgh vh kd ghvsuhfldgr ho wuplqr gh vhjxqgr rughq ghelgr d odv ixhu}dv uhsduwlgdv1Hvwdv hfxdflrqhv hvwdeohfhq uhodflrqhv glihuhqfldohv hqwuh od oh| gh fdujdv | odv oh|hv
gh hvixhu}rv gh xqd yljd uhfwd gh sodqr phglr1 Hq fdvrv vhqfloorv/ hvwdv uhodflrqhvsxhghq lqwhjuduvh gluhfwdphqwh sdud fdofxodu odv oh|hv gh hvixhu}rv1 Hq fdvrv pvfrpsohmrv/ hvwdv uhodflrqhv vluyhq gh d|xgd sdud fdofxodu r glexmdu glfkdv oh|hv1
Ilj1 5145= Dfflrqhv vreuh xqd uhedqdgd gh slh}d uhfwd
Los autores, 2003; Edicions UPC, 2003.
8/14/2019 Ataraxiainc - 2006 - Edicions UPC - 2003 - Mecanica de Estructuras I, Resistencia de Materiales
63/329
83 F D S W X O R 5 1 I X Q G D P H Q W R V G H U H V L V W H Q F L D G H P D W H U L D O H V
Od jhqhudol}dflq gh odv hfxdflrqhv glihuhqfldohv gh htxloleulr gh od uhedqdgd sdudxqd yljd gh gluhfwul} uhfwd jhqhudo +qr gh sodqr phglr,/ vrphwlgd d xqdv ixhu}dv uh0sduwlgdv sru xqlgdg gh orqjlwxg gh frpsrqhqwhv +s
7
> sC
> sS
, vhjq ho wulhgur orfdo ghuhihuhqfld vh sxhghq rewhqhu gh irupd dqorjd/ | vrq=
s7
@ gQg{
sC
@ gWCg{
sS
@ gWSg{
WC
@ gPSg{
WS
@ .gP
C
g{
+5146,
51< Dsr|rv | hqodfhv hq hvwuxfwxudv gh sodqr phglr
51
8/14/2019 Ataraxiainc - 2006 - Edicions UPC - 2003 - Mecanica de Estructuras I, Resistencia de Materiales
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5 1 < 1 D S R \ R V \ H Q O D F H V H Q H V W U X F W X U D V G H S O D Q R P H G L R 84
Ilj1 5146= Dsr|r vlpsoh r pylo
Ilj1 5147= Dsr|r mr r duwlfxodflq
Ilj1 5148= Hpsrwudplhqwr
Los autores, 2003; Edicions UPC, 2003.
8/14/2019 Ataraxiainc - 2006 - Edicions UPC - 2003 - Mecanica de Estructuras I, Resistencia de Materiales
65/329
8/14/2019 Ataraxiainc - 2006 - Edicions UPC - 2003 - Mecanica de Estructuras I, Resistencia de Materiales
66/329
5 1 < 1 D S R \ R V \ H Q O D F H V H Q H V W U X F W X U D V G H S O D Q R P H G L R 86
Ilj1 5149= Hqodfh ghvol}dqwh
Ilj1 514:= Duwlfxodflq lqwhuqd r qxgr duwlfxodgr
Ilj1 514;= Hpsrwudplhqwr r qxgr ujlgr
Los autores, 2003; Edicions UPC, 2003.
8/14/2019 Ataraxiainc - 2006 - Edicions UPC - 2003 - Mecanica de Estructuras I, Resistencia de Materiales
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87 F D S W X O R 5 1 I X Q G D P H Q W R V G H U H V L V W H Q F L D G H P D W H U L D O H V
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8/14/2019 Ataraxiainc - 2006 - Edicions UPC - 2003 - Mecanica de Estructuras I, Resistencia de Materiales
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Los autores, 2003; Edicions UPC, 2003.
8/14/2019 Ataraxiainc - 2006 - Edicions UPC - 2003 - Mecanica de Estructuras I, Resistencia de Materiales
69/329
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