Scarica Modeling of Space Structures - Summary e più Schemi e mappe concettuali in PDF di Meccanica Computazionale delle Strutture solo su Docsity!
FORMULATION OFTHE LINEAR ELASTIC PROBLEM
STRESSTENSOR E CAUCHY RELATION EL E u E C a
symmetrical tensor
STRAINTENSOR (^) E (^) defines the (^) deformation process GREENLAGRANGE (^) E (^) Io grande graditi gradygradì
non linear term
INFINITESIMAL displacements Linearization
condition
equilibrio e è^ grade gradi Ein (^) ECuintuni for the^ initial configuration compatibility EQUATION (^) E (^) I grande gradi e
CONSTITUTIVE LAW band between
strains
linear (^) elastic materials (^) E E E^ E^ E^
E
La (^) no (^) permanent (^) deformation
3D isotropic material Eaa
È tifi
Poisson's (^) ratio
fa
shearing modulus
pane stress^ che
figli ma^ iI
Axial stress^ tre (^) 53 93 D E
la
g È
fin 22in
ROBLEM FORMULATION
divide the^ surface
Placements
su (^) se s Sf forces
compatibility equations^ E constitutive con^ E equations E^ AV^ I^ ds^ o
s
IdV (^) III d si incorona stava Lance ar^ É
Il dice^ LV^
O (^) di (^) E (^) E 0
boundary conditions^ a^ L^ in^ Su I (^) IL in^ Sf
in V
Gi
Ce (^) I E (^) E grande grande e^ in^ V
constitutive law in V
in (^) sa
Unknowns
I in Su
E (^) e a
multi field (^) problemi (^) Simplification to^ a^ SINGLE FIELD differential (^) problem polis placements^ evaluated^ in^ every^ paint strong form^ solution is obtained (^) by solving exactly approxima
WEAK FORM (^) equations are (^) multiplied (^) far a test (^) function when I (^) have (^) integrals average displacement
ii
Displacement based^ approaches a are^ the^ unknowns
I
PUN minimum (^) potential (^) energy method Force (^) based (^) approaches (^) forces are tue unknowns I
PCVw
minimum potential ENERGY (^) HYPER elastic materials
stress o^
GLI
strain energy density
potential function
Main (^) energy sue (^) sfide
Pun (^) swi sue (^) se (^) IdV e^ tu (^) IdV tu (^) I (^) dsa iii se di^81 I^ due (^) se (^) E (^) da
su due (^) se I due^ se (^) Idsf
S UTU O
St O^ t (^) UN (^) total (^) potential energy equilibrium compatibility is^
imposed satisfied a^ priori
HAMILTON PRINCIPLE INERTIAL FORCES used to define the equations
of motion
virtual (^) displacements sale t 0
SI Ha ta sucxa.to o
in S in V perturbed configuration
is equal to the actual
displacement
pun swi sue 8W^ e Swat
integration over^ time^ swi^ sve^ sve^ at^ o
f su sv^ Lsu (^) pè di^ at o Corneto condition
Ilsa due^ sufi sienavate integration by (^9)
parts
E sipudvat^ fisk da
outsutsk at o
8 ut^ K^ at^0 Isl de^0
Le lagrangiani
function
BEAM KINEMATIC^ MODELS
3D structure^ ad^ structure
M e
g
e l
È BEAM Axis materia
mie that contain
info related^ to the (^2) alimenti as
that we ami tt
To (^) simplify from 3D (^) to ad we must (^) consider
MMÈ aime (^) of the analysis
Kinematic (^) formulation description of the^ displacement (^) field using GENERALIZED^ variables
evaluate e in ANY^ point of the^ bean do not
repreactualsent
displacement
Principal centraidal^ axis^ de^ couple^ axial^ and^ bending
behaviours
extention to^ full
linear problems
plates smells
TIMOSHENKO (^) MODEL
AXIAL Behaviour
è e
pp P^ sax
beamaxis
AA no
Az (^) a Bending behaviour p^ sax
Wo (^) in n
g
smeardeforma
q sgg
hits we^ y
complete (^) formulation u^ x^ X^ not^ Zyx Xa
W XI XI We Cx
E UN NON Z Lxx
generalized strains Xx UN twix Image
Xx NON
Suo (^) QIdV Sex (^) Adv SuoNix 84 Mix Owo (^) la du integration (^) a byparts Suon (^) SAM Suo^ I SexAdv (^) suonateMix tw^ a di
Suo o^ N o Suo e N e^89 a Mio Sex e M e owo o^ o^ tonde e
onlyvolume^ one^ I^ daIdV^ concentrated^ forces^84 Fx^ SwFa^ di^ Suoni
contributes
84 M^ 8W^ É suo^ Z 54 Fx Suofa du Suo^ a^ No
suo e^ Nè^ sax o io 8 ex e mi (^) suo a^ è Suo e^ ai
Suoni Sexin^ twovàda^ du^ a^ No^ suo^ e^ Nè^ a^ Sly o^ mi^ sale^ mi
Ono o^ Èo suo e^ QI
Ja saxa_su Nix duxMix^ 8W^ Qix^ tu^ ni^ Sex^ in^ swore du
suolo N^ o^ tuo C nell^ Sex^ o^ m^ o^ saxein e^8 no e^ a^ a^ suo^ e^ e t
Suo o^ Ni_suo e^ Ni Sex^ o^ mi _sax leimi 5 no^ o^ ai_suoe^ ai 0
suo (^) Nix ni Jlx^ Q (^) Mix in^ two^ an nà dV^ Suo^ o^ Enco
Ni (^) tu e nce^ Ni sax o^ Mio^ mi^ sale nce mi
Sw o a a ai tw e (^) ale ai o
Equilibrium equations (^) fa
tono No^ MI^ O
a ma ni o Ax né^ o
Bes values^ of^ displacements at^ boundary
mancano V89x o^ M^ a^ mò O^ V84 e M.cc (^) né 0 (^8) wo e ale (^) è e
NixTUE^ O Q (^) Mix m O
Qix the O
NCO No
nce (^) ne
suo suo e 0 (^8) Ex a 0 MÈInè a^ sacer 0
è
naturalBes (^) essenziale
constitutive con
I (^) b D (^) ftp.tuoixtZ lGTxZIasZE
È (^) ast (^) EEEE as.IE
4 asGUx
yea
yea
s
of mass^ is
www.st (^) E i IE
e G (^) wax (^) Ids principal and centroidae
EAnoix reference from
E UNA arcane i^
e a E
4
E bending stiffness
SHEARFACTOR
GENTEtwitter.mx^ ui o^ BCS (^) seta ode GA (^) LENTWAIN UE O
inconsistency (^) of the^ model
Kinematics Zzz Wiz O
Constitutive con azz o^ e o
how to^ fix this
Let's (^) consider the (^) energetic contribution
SWI (^) SE E AV^ ETA (^) EYYTyytfzztz.tv
in both^ cases
Zzz 0
Ez O
EULER BERNOULLI^ MODEL
AXIAL Behaviour
è e
a pp P^ Saxa
beamaxis
AB no same as
Timoshenko
Setwo (^) x (^) MixAV (^) Swann (^) è SesuoNixdx Suo e Nce
integrationby 9 parts Suo O^ N^ O^ le8WMixxdu^ OwoMix^ Owen^ e^ M e^ t
(^8) Won o (^) MIO (^) lsuoNo DX Suo e N^ e (^) Suo a^ Nco
SetwoMixxdu^ Suo^ e^ Mix l^ Suo^ e^ Mix o^ Swan e^ Mie^ t
(^8) Won O (^) M O
Gaff
only theEwe^ su^ IdV^ concentrated^ loads^ su^ ZSA^ Fxx volume
(^8) non e^ MI 8 non e (^) MI (^) lasuoni di (^) SesuoEzdx
suo a^ No^ Suo e^ Né Suo^ a^ ai^ Suo e (^) Qi Suo Lo^ Mi
no e^ Me^ the rotation^ is^ won
I 8 no N^ ui^ dx^ finof Mixx Az^ da^ Suo^ a^ Ncd^ Ni
suo e^ nce Ni tono^ o^ Mix o^ ai^5 no^ e^ Mix e^ ai
Equilibrium equations^ le
IEIE.IE
Boundary conditions
IIII
macos at^0 tono^ e^ male^ ai 0
NextMI^ O
Mixx MI^0 as^ expected
NCO Né
suo e 0 aw o^ o
a game
Mix o^ Qi È
suo e^0
Swan o^0
Swan e^0
I È^ base^ EI
as
fece
won Ez (^) non zio
ds
E (^) non Ids Eeas ÉTÉ
LEI (^) I FA (^) e E
E A^ NOIXX UI O
E WOMAN TUE o^
Bs
Shear (^) force a does^ not (^) appear in the (^) equilibrium equation
but Q Mix EJ NON O
it (^) appears as a (^) Lagrange multiplier 9 is the Reaction Force needed to^ keep the
section STRAIGHT
DYNAMIC EQUATIONS^ SWI^8 We^8 Wet
Ewe supiù sn qui di^
f
suo zona^ p ciò zio
Twopuò AV^ suopuò_suopzuioix (^) zdwoxf.no
E (^) Swoopmia twogia due Sesuoni S^ dx
suonò (^) Il dsdxtfswoixliolszpdsdx fowoxwix.frpdsa Ia Se 8 wow Spatsdx^ fuori Io suo (^) mix 7 suo (^) cio (^) In t
GWONWON 72 suo io Io du
E A^ NOIxx UI IO IO NÉ Ia
ET WOMAN TUE io^ Io^ ciò^7 mio^ I
B
Az (^) a Bending behaviour^ p^ sexe
Won (^) in U^ Y^ G^ Zay^ x
smear deformation
sgg
complete formulation x^ y vo^ a Ze^ x
x XD Wo ta
compete formulation^ pu^ xe xe a^ volta^ a^ ze^ Ha^ a
μ
Le (^) 9,2 (^) B 9,
Ua Xx^ N^ no^ Xa^ Zla^ x
WCM A^ we^ Xa
E voi Zayn
www ms
fini
rxy uythx U.ly voix^ Zla^ t^ lu
generalized membrane^ matrix^ farm strains generalized bending^ Exa^ Mox^ Y strains (^) Ery (^) non e
generalized transversal È^ a^ È
Ii
strains (^) tiè tx
la way ty
Coop (^) I Chop tuopa Gap definition^ of E
K (^) B (^) E Gap EB a Xxy is^ inside^ Eap y
G E
B Ekar
non
GENERALIZED STRESSES (^) SWI (^) GSE IdV GENE TSENG (^) Stay Gt
STA Azt (^) SKATE AV^ SExp Tap di^ t
Grazia AV^ f 89 78kg (^) Gp a^ t
Sla Swan Taz di^ Slap
thickness
Te de^ ds
8kg Zap de^ ds^ Slats^ won^ fradza
Map (^) E Zap dz
GENERALIZED EXTERNALLOADS SWe (^) I 81 di^ contribution at the boundaries CAB su (^) Fx Su (^) fy Sw (^) F di Cab
Sua Fa^ Sn^ Fz^ AV^ CAB
suo Zola fa AV Sonofed Vacab
I 8 nonffàde^ ds^ Sla zfadzdsttIsnoffzdzdStCA
ha Fa^ da^ LsgCsu.aNatSlaMI Suod (^) dry mia (^) f z fa da
p fadz
EQUILIBRIUMCONDITIONS
Pun s (^) 8Wi SWe
SWi (^) TE (^) E AV (^) f EN ON (^) SEM Ty (^) SKYON (^) STATE Stata du
ExpTap Stazcaz AV^ I 8 Gap^ Zika^ IB^ AV^ t
Sla Swan (^) razaV^89 Nap 8kgMap Glatsword a ds
E Ono^ Suora^ Nap^ ds^ E Slap^ sera^ Map^
ds
slat 8^ won (^) Qazds
E (^) symmetric Gp Opa (^) Nap Nba
suoxp Nap 8 lapMap Sla Swan Aaa ds integration by parts^ suoa Nap (^) p as^ suo^ Napnedif (^) f 8laMappd^ S^ t normal component
Ideas
STRAIN (^) ENERGY SU (^) ESWI (^) E 89 Nap SKapMAB VA Qaz DS
constitute E^
E N^ 84K M^548 a^ as
Con (^) E 5193 A (^93 8) KID K^ 8285A 8 ds
BOUNDARY CONDITIONS associated to
bending behaviour
free side^ My^ Muy^ May^ o
clamped side^ Wo^ Ex^ Ey^ O
simply (^) supported side^ no^ la^ Max Ooca^ o
IRCHNOFF MODEL extention of Euler Bernoulli
Good assumption for Thin^ Plates^ E I noooo
we have to pay attention to composite structures because we
characterized by weak^ properties with^ respect to^ shear
small G^ implies shear^ deformation
not included in Kirchhoff model
O (^) G x^ plane
e
AXIAL BEHAVIOUR se geomaxis
Ulas no^ Xa
Az (^) A bending behaviour l u (^) x (^) g zwoix a
jf.fr
noshear
W Xa XD we x
deformation
be
complete formulation (^) ulxa.es no^ x^ Zwan x W (^) Xa X No (^) Xa
A x^ plane
a
a
AXIAL Behaviour^
È e
l fanta^
Usa vo^ a
Az (^) A Bending behaviour^ p^ su x
W E XD No x
Ing
shear
VK.st zwo.ca
deformation
complete formulation if t.fi Zwe4duCXa
xa.X3 noCxa xa^ ZwoxCxa xa
Complete formulation xe^ xe Xa vo^ Xa Xa^ z^ wo^ y ta ta
W (^) Xa A X wo (^) ta Xa
e 9,2 p a^2
U X^ I Noa X^ ZNO^ XD
N Ca we Xa
E (^) y zwoiyy.werauzeasu.ws
fi
IIII IIII III
ZWxytvox zw.mx
matrix (^) forme
1 11 at
von NON
Gap (^) E Moab no^ pa strains
generalized Mht
generalized bending Kap WOMB
strains
Boundary conditions
annotavano complex Suo^ should^ be^ integra
normal u^ and^ tone
N (^) app ha o^ genti al s (^) directions
Mappa p^ o
min
or Sua o
ii tie'mia
Constitutive con (^) from midline
È b^ E^
e (^1) S
casa
b (^) il D^ E
cosa
1 S sa^4
laser
Let's consider (^) only the^ bending contribution
MaBsap P^ O
D k xxxx wkyyxxtukxxxxtkyyxtkxxyytvkyyyytwkxxyytkyyyyJ.in
D WONXxx UWOMYXXTOWOIxxxxtwolyyxxtwoxxyytowoiyyyytuw.MY t
woryyyy p
D wocxxxxtwocyyyytz.noxxyytO woIxxxxtwosyyyytZwoixxyy p
DA No (^) P
A o^ o^ Xx G my
STRAIN (^) ENERGY U (^) ESWI E SE IdV^ ES^59 ZKap^ CapaV
E (^) I SapNap^8 KapMap DS
E 195 LA^93 K^ DI^ K^ as
RITZ METHOD
make (^) complex problems solvable (^) using small^ member (^) of daf used (^) during PRELIMINARY^ Analysis DIRECT METHOD (^) uses (^) functionals variational (^) principles to
obtain an approximate^ solution
ORMULATION Total (^) potential (^) energy it^ e^ p Iii Info L (^) Potential OFEXTERNAL
general unknown^ LOADS
Salve the problem Sit^ u o^ s^ set of differential equations
a f
differential operator
The basic idea is to transform Pde into Algebraic equations
approximation unknown^ function
ne (^) è (^) E ci di do (^) di TRIAL Functions
do introduced^ when^ we^ nome
non (^) homogeneous BCS
Ci unknown^ amplitudes SCALAR
terms
specify non^ to^ combine
the trial functions
PROCEDURE MINIMUMPOTENTIAL (^) ENERGY METNOD a (^) calculate (^2) Substitute TIÈ u (^) TCE (^3) Impose (^) equilibrium sit ù o I
Sci o^ voci
n (^) equations (^4) salve the (^) equation
PRINCIPLEOF (^) VIRTUALWORK
a Calculate sui and Sue
2 Substitue è into sui and Sue
(^3) Impose equilibrium swi sue (^) separate in^ N^ equations
one equation associated
to each^ Ci
4 Salve the equation
CRITERIA FOR Choosing TRIAL FUNCTIONS
fulfill essential^ Bcs^ natural^ one^ are^ embedded^ in^ the
variational principle
if (^) di satisfy both^ essential^ and^ natural^ nata
requirement are^ called^ COMPARISON^ FUNCTIONS
