Analysis of composite beams

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FaultyofArhiteture

ANALYSIS OF COMPOSITE BEAMS

Aniko Pluzsik

Supervisor:

Laszlo P. Kollar

Budapest, January, 2003

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I would liketo thankProfessor LaszloP. Kollarfor allof his help and ooperation. He is an

extraordinary teaher not only in the lassroom, but in life's lessons as well. His approah to

people, his endless patiene,and his outlookon theworld havemadealasting impression. Iam

privilegedtohavehadtheopportunitytoworkwithhim.

ThankstothehairmenofthedepartmentsforprovidingmyPh.Dstudies:

ProfessorGyorgyFarkas,ChairmanoftheDepartmentofStruturalEngineering,

ProfessorTamasMatussakand ProfessorGabor Domokos,ChairmenoftheDepartment

ofMehanisandStrutures.

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1 Introdution 9

2 Review ofompositebeamtheories 9

2.1 Constitutiveequations . . . 10

2.2 Classialbeam-theories(norestrainedwarping,nosheardeformation) . . . 10

2.2.1 Basiassumptions . . . 10

2.2.2 Isotropibeam . . . 11

2.2.3 Orthotropibeam . . . 11

2.2.4 Generallyanisotropibeam . . . 12

2.3 Vlasovtypebeamtheories(restrainedwarping,nosheardeformations). . . 13

2.3.1 Basiassumptions . . . 13

2.3.3 Beamswithsymmetriallaminates . . . 14

2.4 Timoshenkotypebeamtheories(sheardeformation,norestrainedwarping) . . . . 14

2.4.1 Basiassumption . . . 14

2.5 Theoriestakingintoaountrestrainedwarpingand sheardeformations . . . 15

2.5.2 Isotropiandorthotropibeam . . . 15

2.5.3 Symmetrialbeam . . . 16

2.6 Theories taking into aount restrained warping, shear deformationsand warping induedshear . . . 16

2.7 Generalizedbeamtheories . . . 18

2.8 Transverselyloadedbeams. . . 18

2.8.1 Classialbeam-theories(norestrainedwarping,nosheardeformation) . . . 18

2.8.2 Theories taking into aount restrained warping indued shear (restrained warping,sheardeformation) . . . 19

3 Theoryofopen andlosedsetion,generallyanisotropithin-walledbeams-no restrainedwarping 19 3.1 Problemstatement . . . 19

3.2 Opensetionbeams . . . 20

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3.2.2 Step2. Foresandmomentsin thewallsegments. . . 23

3.2.3 Step3. Foresin thebeam . . . 26

3.2.4 Step4. Stinessmatrix . . . 26

3.3 Closed setionbeams. . . 28

3.3.1 Step1. Strainsin thewallsegments . . . 30

3.3.2 Step2. Foresandmomentsin thewallsegments. . . 30

3.3.3 Step3. Foresin thebeam . . . 31

3.3.4 Step4. ForesX 1 andX 2 . . . 32

3.3.5 Step5. Stinessmatrix . . . 32

3.4 Centroid. . . 33

3.4.1 Stinessandomplianematriesintheentroidoordinatesystem . . . . 34

3.4.2 Stinessmatrixof orthotropibeams. . . 36

3.5 Stressesandstrains. . . 36

3.5.1 Opensetionbeams . . . 36

3.5.2 Closedsetionbeams. . . 37

3.6 Veriation . . . 38

3.6.1 I-beamwithunsymmetriallayup[0 6 =45 6 ℄. . . 41

3.6.2 I-beamwithdierentlayups. . . 45

3.6.3 Box-beamwithunsymmetriallayup ([ 0 6 =45 6 ℄) . . . 45

3.6.4 Box-beamwithdierentlayups . . . 49

3.7 Disussion . . . 50

3.7.1 Eetofanisotropy . . . 51

3.7.2 Eetofloalstiness . . . 51

4 Theory ofopen setion,orthotropi,thin-walled beams-with restrainedwarping 54 5 Theoryoflosedsetion,orthotropi,thin-walledbeams-withrestrainedwarp- ing 56 5.1 Literature . . . 56

5.2 Problemstatement . . . 59

5.3 Assumptions . . . 59

5.4 Governingequations . . . 60

5.5 \Exat"solutionforsinusoidalloads . . . 63

5.6 SolutionbytheRitzmethod . . . 64

5.7 Beamtheory . . . 66

5.7.1 Assumptions . . . 66

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5.7.3 Replaementstinesses . . . 68

5.7.4 Unsymmetrialbeams . . . 75

5.7.5 Veriation . . . 76

6 Eetof sheardeformation and restrainedwarping 80 6.1 Problemstatement . . . 81

6.2 Eetofsheardeformation . . . 81

6.3 Eetofrestrainedwarping . . . 86

6.3.1 Opensetionbeams . . . 86

6.3.2 Closedsetionbeam . . . 91

6.4 Numerialexample . . . 92

7 Conlusions 96 8 Referenes 97 A Stiness matrix ofa laminatedompositeplate 100 B Solution ofthe dierential equation of beamsin torsion 102 B.1 Simply supportedbeam . . . 102

B.2 Cantileverbeamsubjetedto aonentratedtorqueattheend . . . 103

C Buklingof ompositeolumnssubjetedto onentrated loads 105

D The \neutral" surfaes ofa laminate 105

E Solution ofa simplysupported beam 107

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A area

[A℄;[B℄;[D℄ stinessmatriesofaplate

[e℄; h

e

i

; h

e

Æ i

omplianematriesofaplate

A

ij

;B

ij

;D

ij

elementsofthestinessmatries

EA tensilestinessofanorthotropibeam

EI

yy

;EI

yz

;EI

zz

bendingstinessesofanorthotropibeam

EI

!

warpingstinessofanorthotropibeam

GI

t

torsionalstiness ofanorthotropibeam

S

ij

shearstinessesofanorthotropibeam

s

ij

shearomplianesofanorthotropibeam

h platethikness

b

k

widthofawallsegmentofthebeam

K numberofwallsegments

L lengthofthebeam

l eetivelengthofthebeam

b

N

x

axialforeatingonabeam

M

y

;

M

z

bendingmomentsatingonabeam

M

!

bimomentatingonabeam

b

T

x

torqueatingonabeam

b

T

sv

SaintVenanttorqueatingonabeam

b

T

!

restrainedwarpinginduedtorqueatingonabeam

b

V

y

; b

V

z

shearforesatingonabeam

X

1 ,X

2

foreandmomentperunit lengthapplied attheutofalosedsetionbeam

N

x ,N

y

;N

zy

in-planeforesatingonaplate

M

y

;M

z

;M

zy

bendingandtwistmomentsatingonaplate

M

!

bimomentatingonabeam

[P℄ stinessmatrixofabeamreferstotheentroid

P

stinessmatrixofabeamreferstoanarbitrarilyhoosenpoint

[W℄ omplianematrixofabeamreferstotheentroid

W

omplianematrixofabeamreferstoanarbitrarilyhoosen point

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u;v,w displaementsinthex,y andzdiretions

w

B ,w

S

displaementsdueto bendingandshear deformations

y

,z

oordinatesoftheentroid

o

x

axialstrainofthebeam

xy

strainsoftherefernesurfaeof aplate

1

y

; 1

z

urvaturesofthebeam

# twistperunit length

# B

;# B

twistperunit lengthdueto bending andsheardeformations

seondderivativeofthetwistofthebeam

y

;

z

rotationoftheross-setionofthebeaminthex zandy z planes

twistofthebeam(rotationoftheross-setionaboutthexaxis)

loationofthe\tension neutral"surfae

loationofthe\torqueneutral"surfae

parameterforestimatingtheerroroftheshear deformation

q shearow

C

k

onstansintheRitzmethod

k

basifuntions intheRitzmethod

[F℄; f o-eÆtientmatrixandvetorintheRitzmethod

[A

1

℄;[A

1

℄;[B

1

℄;[B

2

℄;[B

3

℄ submatriesof[F℄

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Figures

Figure1 page20 Figure22 page59 Figure43 page84

Figure20 page57 Figure41 page82

Figure21 page58 Figure42 page84

Tables

Table1 page27 Table7 page52 Table13 page88

Table6 page51 Table12 page88

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Instruturalmodelingofbeamstherearetwoontraditoryrequirements: themodelingofabeam

must be simple and aurate at the same time. In the literature several beam theories an be

foundforanisotropithin-walledbeams. Setion1summarizesthese theories,from thesimplest,

tothemoreadvanedones.

Beammodelsthat anbefoundin theliteraturedonotoverallthepratialases. Theaim

ofthisthesisisto fullthelakonthis eld.

First we develop beam models for generally anisotropi beams with open and losed ross-

setionsusingtheassumptionsof lassialbeamtheories(Setion 3). (Solutionthat anbeused

for pratial purposes is available only for losed setion beams [16℄. However Manseld [16℄

negletstheloal bending stinessesof thewallsegmentsof thebeam. Thus, his solutionis not

appliableforallases.)

Thenweonsiderorthotropi,losedsetionbeamstakingtheeetofrestrainedwarpinginto

aount(Setion 5). (Abeammodel isavailable onlyforpure torsionof symmetriross-setion

beams in the literature[20℄, however,Urban's solution [20℄ results in an unaeptable error for

shortbeams.) Ouraimisto developabeammodelwhih anbeused forallpratialases.

Inaddition,ouraimistodeterminetheaurayandappliabilityoftheexistingbeammodels

withtheaidofnumerialomparisons(Setion6).

2 Review of omposite beam theories

This setion summarizes the dierent beam theories available in the literature. However, the

questionremains: how aurate are these models, and under whih irumstanes they an be

used.

Thetheoriesdierfromeahotherin

the alloweddeformations ofthe beam's elements (e.g. the ross-setionsremain planeand

perpendiulartotheaxisortheymaydeform),

theallowedmaterialtypesandlayupsofthewallsofthebeam(e.g. orthotropi,symmetrial

layup,et.),

theallowedgeometryof theross-setion(openorlosed setionbeams,symmetrialross-

setions,et.),

loadingonditions(endloadsortransverseloads),

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Setion2.2.4).

2.1 Constitutive equations

Weonsideropenorlosedsetionthin-walledbeams.

Thegeneralizedstressesofawallofthebeamaretheintegralsofthenormalandshearstresses

(

x

;

y

;

xy

)alongthethikness. ThegeneralizedstressesofthebeamfNgaretheintegralofthese

stressesofthewallsalongtheirumferenes.Thegeneralizedstrainsf"garealulatedfrom the

displaementsof thebeambythegeometrialequations. Theonstitutiveequationsrelatethese

stressestothestrains. Itan bewrittenas follows:

fNg=

P

f"g (2.1)

where

P

isthestinessmatrixofthebeam.

Wesummarizethebeam-theoriesbelowaordingtothefollowingtwoquestions

Howarethegeneralizedstresses( fNg)andstrains( f"g)aredened?

Whihelementsofthestinessmatrix(

P

)arenon-zero?

2.2 Classial beam-theories (no restrained warping, no shear deforma-

tion)

Inthefollowingwesummarizethetheorieswhereaxialonstraints(restrainedwarping)andshear

deformationsinbendingarenottakeninto aount.

2.2.1 Basi assumptions

1. Theeet of hange in geometry oftheross-setion isnottakenintoaountin theequi-

libriumequations.

2. TheKirhho-Lovehypotesisisvalidforeahplateelement.

3. Thenormalstressesintheontourdiretions issmallomparedtotheaxialstresses.

4. Thestrains(

x

)varylinearlyalongtheyand zaxes:

x

= du

dx y

d

y

dx z

d

z

dx

(2.2)

where uis the axial displaement, and the rotation of the ross-setion

y and

z in the

x y and x z planesanbealulatedfrom thedisplaementsof theaxisin they and z

diretions(v andw):

y

= dv

dx

(2.3)

z

= dw

dx

(11)

y z

gleted.

2.2.2 Isotropi beam

Abeamisisotropiwhenitismadeofisotropimaterial. IntheSaintVenanttheoryfourgeneral-

izedstressesaredened: normalfore

b

N

x

= R

N

ds

, bendingmomentsaboutthey andzaxes

M

y

= R

(N

y+M

os)dsand

M

z

= R

( N

z+M

sin)ds

andtorque

b

T

x

= 2

R

M ds

.

Thegeneralizedstrainsare: theaxialstrainofthebeam'saxis o

x

= u

x

, urvaturesofthex

axisaboutthe y and z axes

1

y

= d

y

dx and

1

z

= d

z

dx

; and the rateof the twist

#= d

dx

.

Beauseofisotropytorqueloadsdonotauseeitheraxialstrainorurvatures. Thestinessmatrix

(

P

)an bediagonalizedbytheproperhoieoftheoordinatesystem: Whentheoriginof the

oordinatesystemisattahedtotheenterofgravityoftheross-setion,andtheaxesarein the

prinipaldiretions,Equation(2.1)beomes

8

>

<

>

: b

N

x

M

y

M

z

b

T

x 9

>

=

>

;

= 2

6

4 EA

EI

yy

EI

zz

GI

t 3

7

5 8

>

<

>

:

o

x

1

y

1

z

# 9

>

=

>

;

(2.4)

Calulation of the tensile (EA) and bending stinesses ( EI

yy

andEI

zz

) are similar to open

andlosed setion beams, thetorsional stiness(GI

t

)is alulateddierently, for losedsetion

beams. The torsional stiness (Bredt-Batho formula) is signiantly higher than the torsional

stinessoftheequivalentopensetionbeam.

(Asimplerbeamtheory forisotropibeamswas derivedbyBernoulliand Euler. Inthis ase

thelast assumptioninSetion2.2.1beomes:

1. Inbendingtheross-setionsofthebeamremainplaneduringthedeformation.

2. Inbendingtheross-setionsofthebeamremainperpendiularto theaxis.

Thesearethe\Bernoulli-Navier"hypoteses.)

2.2.3 Orthotropi beam

Athin-walledbeamis orthotropiwhen eahwallsegmentis madeof orthotropi laminates. (A

laminate is orthotropi when the 1,6 and 2,6 terms in its stiness matries ([ A℄; [B℄; [D℄ see

Appendix A) are zero [9℄.) The Saint Venant theory an be easily generalized for orthotropi

beams. Thestiness matrixremainsdiagonal. Analogouslytothe\enter ofgravity"thetheory

denestheentroidsuhthatanormalforeatingattheentroiddoesnotausetheurvatures

ofthe beam's axispassingthroughthe entroid. The alulationof theelements of thestiness

matrixandtheoordinatesoftheentroidisstraightforward[13℄,[4℄. Formalistiallywewrite

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>

<

>

: b

N

x

M

y

M

z

b

T

x

>

=

>

;

= 6

6

4 P

11

P

22

P

33

P

44 7

7

5

>

<

>

:

o

x

1

y

1

z

#

>

=

>

;

(2.5)

2.2.4 Generally anisotropi beam

Athin-walledbeamisalled\generallyanisotropi"whenatleastoneofitswallisnonorthotropi.

For generallyanisotropi beamstension, bending and twisting modes of displaements interat,

whihrepresentsasigniantdierenebetweenthetheoryofberreinforedbeamsandoflassial

beams. Theonstitutiveequationsbeomes

8

>

<

>

: b

N

x

M

y

M

z

b

T

x 9

>

=

>

;

= 2

6

4 P

11 P

12 P

13 P

14

P

12 P

22 P

23 P

24

P

13 P

23 P

33 P

34

P

14 P

24 P

34 P

44 3

7

5 8

>

<

>

:

o

x

1

y

1

z

# 9

>

=

>

;

(2.6)

Theinverseofthestinessmatrixistheomplianematrix:

W

=

P

1

(2.7)

By the proper hoie of the entroid of the oordinate system and alulation of the priniple

diretions3ofthese elementsoftheompliane matrixmaybezero:

8

>

<

>

:

o

x

1

y

1

z

# 9

>

=

>

;

= 2

6

4 W

11

0 0 W

14

0 W

22

0 W

24

0 0 W

33 W

34

W

14 W

24 W

34 W

44 3

7

5 8

>

<

>

: b

N

M

y

M

z

b

T

x 9

>

=

>

;

(2.8)

Manseld and Sobey [16℄ derived the ompliane matrix ( W) for anisotropi, losed setion

beams. The wallsof thebeamsaremade oflaminated omposite. Ineah layerthere areunidi-

retionalbers,thediretion ofthebersisoptional. Inadditionaltotheabovefourassumption

(Setion2.2.1),theystate

1. The[B℄and[D℄matriesofthelaminates(seeAppendixA)arenegligible. Aordinglythey

takeinto aountonlythein-plane stinessesof thelaminate.

Theexibilitymatrixisompletelylled,onlyW

12

;W

13

;W

23

anberesettozerowithmoving

theoriginto theentroidandrotating theaxes tothe prinipal axesof bending. Manseldand

Sobeygavethese transformations.

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tions)

In the following we summarize the theories where the axial onstraints are taken into aount,

howeverthetransversesheardeformationisnegleted.

2.3.1 Basi assumptions

The rst three assumptions in Setion 2.2.1 are valid. Instead of Assumption 4 these theories

assumethattheaxialstrainvariesasfollows:

o

x

= du

dx y

d

y

dx z

d

z

dx

! d#

dx

(2.9)

wherethealulationof

y and

z

isthesameas inSetion 2.2.1:

y

= dv

dx ,

z

= dw

dx ,#=

d

dx is

therateoftwist,and!= R

rdsisasetionpropertyalledthesetorialarea.

The last term in Equation (2.9) allowsto the points of the ross-setion anadditional axial

displaement,alledwarping,proportionalto therateoftwist[17℄.

2.3.2 Isotropi beam

Vlasov[21℄ gaveabeamtheory withthe aboveassumptionsforisotropi open setion beams. If

thewarpingisrestrained,normalandshear stressesarise inthebeaminaddition tothoseof the

lassialtheory. (Iftheross-setionislosed,theseadditionalstressesdeayinashortdistane.)

Thetorqueonsistsoftwoparts : theSaintVenanttorqueandthewarpinginduedtorque:

b

T = b

T

sv +

b

T

!

(2.10)

Vlasovdenesanadditionalgeneralizedstress

M

!

(alledbimoment)

M

!

= Z

N

!ds (2.11)

M

!

is relatedto the generalizedstrain ( = d#

dx

) bythe warpingstiness (EI

!

), and hene

thestress-strainrelationshipbeomes:

8

>

<

>

: b

N

x

M

y

M

z

M

!

b

T

sv 9

>

=

>

;

= 2

6

4 EA

EI

yy

EI

zz

EI

!

GI

t 3

7

5 8

>

<

>

:

o

x

1

y

1

z

# 9

>

=

>

;

(2.12)

The bimoment is related to b

T

! as

b

T

!

= d

M

!

=dx, whih yields the well known equation of

restrainedwarping

b

T = b

T

sv +

b

T

!

=GI

t

#+EI

! d

2

#

dx 2

(2.13)

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BauldandTzeng[6℄improvedtheabovetheoryforlaminatedompositebeams. Theross-setion

isopen andthelayupisrestritedtomidplanesymmetriberreinforedones([B℄=0). Beause

ofthisonditionforthelayupthereisnoouplingbetweentensionandtorsion,andthe1,5and5,1

elementsarezerointhe

P

stinessmatrix. Thewhole

P

matrixislled,exeptP

15

=P

51

=0.

Hene weobtain therelationshipbetween thestrainsand the stressesof thebeam: thestiness

matrix:

8

>

<

>

: b

N

x

M

y

M

z

M

!

b

T

sv 9

>

=

>

;

= 2

6

4 P

11 P

12 P

13 P

14 0

P

21 P

22 P

23 P

24 P

25

P

31 P

32 P

33 P

34 P

35

P

41 P

42 P

43 P

44 P

45

0 P

52 P

53 P

54 P

55 3

7

5 8

>

<

>

:

o

x

1

y

1

z

# 9

>

=

>

;

(2.14)

2.4 Timoshenko type beam theories (shear deformation, no restrained

warping)

Inthefollowingwesummarizethetheorieswheretheaxialonstraintsarenegleted,howeverthe

transverseshear deformationistakenintoaount.

2.4.1 Basi assumption

Therst three assumptionsin Setion 2.2.1is valid. The form of theaxial strainis idential to

Assumption4:

o

x

= du

dx y

d

y

dx z

d

z

dx

(2.15)

howeverinthisexpression

y and

z

aredened as

y

= dv

dx

y

z

= dw

dx

z

(2.16)

where

y and

z

aretheshearstrainin thex yandx zplanes,respetively. Theshearstrain

issupposedtobeonstantintheross-setionwhih isreferredtoas therstordersheartheory.

2.4.2 Isotropi beam

Theeetofthesheardeformationisinludedinthistheory. Henethetransverseforesandthe

shear strains are inluded in the ompatibility equations (Equation2.17). These are related to

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8

>

<

>

: b

N

x

M

y

M

z

b

V

y

b

V

z

b

T

sv 9

>

=

>

;

= 2

6

4 EA

EI

yy

EI

zz

S

y

S

z

GI

t 3

7

5 8

>

<

>

:

o

x

1

y

1

z

y

z

# 9

>

=

>

;

(2.17)

Bank[2℄,[3℄investigatedtheombinedbendingandtransverseloadingofanisotropi,opensetion

beams(torsionalandaxialloads arenotonsidered). Helaimsthattheirmodelan beused for

hand alulationpreliminary designstudies only. The geometry ofthe ross-setionis restrited

toopen,singly anddoubly symmetrialones. From

M

z and

b

V

y

hedetermines o

x

;1=

y

;1=

z and

#forthelayers,andgivesanaveragevalueofthese strainsfor thebeam. Thestinessmatrixis

notpresented.

2.5 Theoriestakingintoaount restrainedwarping andshear deforma-

tions

ThismodelusesAssumptions1to3inSetion2.2.1,andfortheaxialstrainthefollowingassump-

tion:

o

x

= du

dx y

d

y

dx z

d

z

dx

! d#

dx

(2.18)

where

y

= dv

dx

y

z

= dw

dx

z

# =

d

dx

(2.19)

Henetheeet ofwarpingand sheardeformationsareinluded, howeverwarpinginduedshear

isnotinluded. Thetheoryassumesnoouplingbetweennormalandshearingeets.

2.5.2 Isotropi and orthotropi beam

Barbero[4℄,[5℄givestheompatibilityequationsalsoforopen andlosedross-setion. Thevetors

ofthegeneralizedstressesandstrainsarethesumofthesestressesandstrainsin Setion2.3and

Setion2.4. Thenondiagonalelementsofthestinessmatrixarezerobeauseoforthotropy,and

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beausetheouplingbetweenthenormalandshearingeetsisnegleted. Henethestress-strain

relationshipisasfollows

8

>

<

>

: b

N

x

M

y

M

z

b

V

y

b

V

z

M

!

b

T

sv 9

>

=

>

;

= 2

6

4 EA

EI

yy

EI

zz

S

y

S

z

EI

!

GI

t 3

7

5 8

>

<

>

:

o

x

1

y

1

z

y

z

# 9

>

=

>

;

(2.20)

2.5.3 Symmetrialbeam

Kabirand Sherbourne [10℄ improved the beam theory of Bauld and Tzeng [6℄. The transverse

shear deformationof the beam ross-setionis inluded in the expression of the stiness matrix

(seeEquation2.21).

8

>

<

>

: b

N

x

M

y

M

z

M

!

b

T

sv

b

F

y

b

F

z

b

V

y

b

V

z 9

>

=

>

;

= 2

6

4 P

11

P

18 P

19

P

22

P

25 P

26

P

33

P

35

P

37

P

44 P

45

P

52 P

53 P

54 P

55 P

56 P

57

P

62

P

65 P

66

P

73

P

75

P

77

P

81

P

88

P

91

P

99 3

7

5 8

>

<

>

:

o

x

1

y

1

z

#

0

z

0

y

z 9

>

=

>

;

(2.21)

Thetheoryisvalidforopensetion beamswithsymmetriallaminates.

2.6 Theoriestakingintoaountrestrainedwarping,sheardeformations

and warping indued shear

Therst three assumptionsin Setion 2.2.1is valid. The form of theaxial strainis idential to

thatofinSetion 2.5.1:

o

x

= du

dx y

d

y

dx z

d

z

dx

! d#

B

dx

(2.22)

where

(17)

y

=

dx

y

(2.23)

z

= dw

dx

z

howeverinthisexpression#

B

isdenedas

#

B

= d

dx

#

S

(2.24)

The warpingis assumed to be proportional to the rst derivative of the restrained warping

induedtwist. Couplingsbetweennormalandshearingeets arenegleted.

Kollar[12℄investigatedtransverselyloaded,opensetion,orthotropibeams. Thenormalstresses

induedbytherestrainedwarpingresultadditionalshear stresses. Thesestressesause anaddi-

tionaltwistof thebeam, and henethewarpingis notproportional tothe rstderivativeof the

twist. The twist has two parts: one from bending (whih auses warping, like in the previous

setion)andanotherpartfromtherestrainedwarpinginduedshearstress:

#=#

B +#

S

(2.25)

isdened as = d#

B

dx

. TheonstitutiveequationsaordingKollar:

8

>

<

>

:

M

y

M

z

M

!

b

T

sv

b

V

y

b

V

z

b

T

! 9

>

=

>

;

= 2

6

4 EI

yy EI

yz

EI

yz EI

zz

EI

!

GI

t

S

yy S

yz S

y!

S

yz S

zz S

z!

S

y!

S

z!

S

!!

3

7

5 8

>

<

>

: 1

y

1

z

#

y

z

#

s 9

>

=

>

;

(2.26)

Roberts and Ubaidi [18℄ proposed takeinto aount the shear deformationin torsion by the

alulationofthewarpingstinessforopen setion,orthotropibeams.

Urban [20℄ and Kristek [7℄ investigated the torsion of doubly symmetrial, losed setion,

isotropi beams. Using his basi equations we an derive the same equations as for open se-

tionbeams (Setion 2.6.2). The stinesses in the expression of the onstitutive equations dier

fromthoseoftheopensetions.

8

>

<

>

:

M

!

b

T

sv

b

T

! 9

>

=

>

;

= 2

6

4 EI

!

GI

t

S

!!

3

7

5 8

>

<

>

:

#

s 9

>

=

>

;

(2.27)

Inadditionaltotheassumptionsof[12℄, theystate

(18)

TheirresultstogetherwithothermoregeneralapproaheswillbedisussedinSetion5.1.

WuandSun[23℄usethesamegeneralizedstrainsasKollar[12℄. Adierentialequationsystemis

statedfortheproblem. Thereisnorestritionforthelayersoftheplatesonsistingthebeam. The

ross-setionan beeither open or losed. Thestiness matrixisnotexpressed, andthesolution

beomesveryompliated.

2.7 Generalized beam theories

IntheTimoshenkotypebeamtheories(rstordershear theories,Setion2.4) thedistributionof

the shear strain arossthe ross-setion is uniform. There are higherorder shear theories with

moregeneralassumptionsforthedistributionof theshearstrain.

The Vlasovtype beam theories (Setion 2.3) allowsthe ross-setionto warpaording to a

determinedfuntion. InBahautheorynoassumptionismadeonaxialdisplaementdistribution.

Inthis beam theoryaset oforthonormal eigenwarpings(!

i

(s)) is derived. With these theaxial

strain( o

x

)ofthelassialtheories(inludingsheardeformationsornot) anbeimproved:

o

x

= o

xb +

X

W

i

(s)F 0

i

(x) (2.28)

whereF

i

(x)andW

i

( s)areunknownfuntionsofxands,whihanbedeterminedbyminimizing

thepotentialenergy.

2.8 Transversely loaded beams

2.8.1 Classialbeam-theories (norestrained warping, no sheardeformation)

Fromthetransverseloadswean alulatetheresultantnormalforeand thebending moments,

andwiththeseforeandmomentswean examinethebeamsimilarlytotheanalysisdisussedin

theprevioussetion. Toalulatetheresultanttorqueresultingfromthetransverseloadswehave

tointroduethe notionofthe shearenter, aboutwhih thetorque due to ashearfore haveto

bemeasured.

For isotropibeamsthedenition oftheshear enteris asfollows: transverseforesatingat

theshearenter donotausetwist. Theshearowandhenetheloationoftheshearenterare

dierentforopenandforlosedross-setionbeams.

For orthotropi beams the denition of theshear enter and the proedure of alulating its

loationare thesameas for isotropibeams. Barbero [5℄ givesthe expressions ofoordinatesof

theshearenter'sforopen andlosedsetion beams.

(19)

bendingmoments. Consequently,thetransverseforesatingatanypointoftheross-setionthe

beammaytwist. ManseldandSobey[16℄gavethedenition oftheshearenter foranisotropi,

losedsetion beams: theshearenterisatthepointwherethetwistissolelyduetotheresultant

bendingmoments. ManseldandSobeygavetheexpressionoftheloationoftheshearenter for

losedross-setions. Beauseoftheomplexityofthisexpressionasimplerapproximationisalso

given.

2.8.2 Theories taking into aount restrained warping indued shear (restrained

warping,shear deformation)

When warping indued shear is taken into aount, even for orthotropi beams, as a rule, the

ross-setionofthebeamwill rotate. Inthisasethere isno loationfortheexternaltransverse

loads,forwhih,therotationofthebeamiszero. Fororthotropi,opensetion beamsKollar[12℄

dened two speied points on the ross-setion: the shear deformation shear enter (when the

shearforesareappliedatthatpoint,thereisnoshearinduedtwist)andthebendingdeformation

shearenter(whentheshearforesareappliedatthat point,thereisnobendinginduedtwist).

3 Theory of open and losed setion, generally anisotropi

thin-walled beams - no restrained warping

Inthissetion wewill derivethestiness matrixofthin walledopenor losed setionomposite

beams[24℄ (see

P

in Setion2.2.4 (Equation2.6)). There isno restritiononthe layup of the

wallsegmentsandtheloalbendingstinessesaretakenintoaount. Wewill neglettheeets

ofrestrainedwarpingandsheardeformation. (Theeetofthisapproximationwillbeinvestigated

inSetion6.)

3.1 Problem statement

Weonsider thin-walled open and losed setion prismati beams. Thewallsof the beams may

onsistofasinglelayeror ofseverallayers,eah layermaybemadeof ompositematerials. The

beam's wall onsists of at segments (Figure 1) designated bythe subsript k (k = 1;2;:::;K,

whereK isthetotalnumberofwallsegments). Theross-setionmaybesymmetrialorunsym-

metrialandthelayupof thebeamis arbitrary(i.e. thelayersarenotneessarilyorthotropior

symmetrial).

Thebeamissubjetedtoanaxialfore b

N,bendingmoments

M

y ,

M

z

,andtorque b

T atingat

theentroid(theentroidisdened suhthatthefore b

N doesnotresultin theurvaturesofthe

axisofthebeam).

Weemploythefollowingoordinatesystems(Figure2).

(20)

y-z oordinate system with the origin at an arbitrarily hosen point. For the k-th segment we

employthe

k -

k

oordinatesystemwiththeoriginattheenter oftherefereneplaneof the

k-thsegment. isparallel tothe xoordinate, isalong theirumfereneof thewall, and is

perpendiulartotheirumferene.

Thedisplaementsofthelongitudinalaxis,passingthroughtheentroid,areu, v, w, (Fig-

ure3),whereuistheaxialdisplaement,vandwarethetransversedisplaementsinthey andz

diretions,and istherotationoftheross-setion(twist).

Therelationshipsbetweenthesedisplaements,andtheaxialstrain o

x

andtheurvatures1=

y ,

1=

z

ofthexaxis,andtherateoftwist#are

u

x

= o

x

2

v

x 2

= 1

z

2

w

x 2

= 1

y

x

=# (3.1)

Inthex-y-zoordinatesystemthese relationshipsbeome

u

x

= o

x

2

v

x 2

= 1

z

2

w

x 2

= 1

y

x

=# (3.2)

where o

x , 1=

y , 1=

z

, are the axial strain and the urvatures of the longitudinal axis passing

throughtheoriginofthex-y-z oordinatesystem. u, v,w, and arethedisplaementsofthex

axis.

Theross-setionsofbeamswitharbitrarylayupdonotremainplaneandtheBernoulli-Navier

hypothesis is inappliable. Nonetheless, in a long body subjeted to the loads given in Figure

1,suh as a beam, the strainsmay be onsidered to be onstant in the axial diretion, and the

plane-strainondition (the stresses and strainsvary onlyin planes perpendiular to the x axis)

maybeapplied intheanalysis[15℄.

3.2 Open setion beams

Theanalysis ofthin-walledopensetionbeamsisperformedinfoursteps.

1 2

k

K

b _k z

x

M _y y M _z

T

1 2

k

K

b _k z

x

M _y y M _z

T

N N

Figure 1: Foresinopenandlosedsetionthin-walledbeamswithatwallsegments

(21)

beam'saxis.

2. Theforesineahwallsegmentaredeterminedfromthestrainsinthewallsegment.

3. Theresultantaxialfore,moments,andtorqueatingattheaxisofthebeamaredetermined

fromthewallsegmentfores.

4. Thestinessmatrixisestablishedbyrelatingtheresultantaxialfore,moments,andtorque

totheaxialstrain,urvaturesandtwistofthebeam'saxis.

3.2.1 Step1. Strains in the wallsegments

Atapointontherefereneplaneofeah wallsegment theaxial strainisalulatedbytheplane

strainondition

o

k

= o

x +z

1

y +y

1

z

(3.3)

where z and y are the oordinates of anarbitrary point on thek-th segment'sreferene surfae

and o

k

istheaxialstrainatthis point.

Thestrainsoftheaxisofthek-thwallsegmentan beexpressedas

8

>

<

>

:

w

k

w

k

w

mk

# w

k 9

>

=

>

;

= 2

6

4

1 z

k

y

k 0

0 os

k

sin

k 0

0 sin

k

os

k 0

0 0 0 1

3

7

5

| {z }

[R

k

℄

8

>

<

>

:

o

x

1

y

1

z

# 9

>

=

>

;

(3.4)

where y

k and z

k

are the oordinates of the

k -

k

oordinate system's origin, whih is at the

midpointof therefereneplane(Figure 2), and

k

istheangle betweenthe

k

andy oordinate

axes. Thesupersriptwreferstothesegment'slongitudinalaxiswhihpassesthroughthemidpoint

oftherefereneplane(=0; =0). TherstoftheseequationsiswrittenbyvirtueofEquation

z

x

y

® _k

» _k

´ _k

³ _k z

y x

z _k y _k

k-th wall segment

Figure2: Coordinatesystemsemployedintheanalysisofthin-walledbeamswitharbitrarylayup

(22)

(3.3).

w

k and

w

mk

arethe urvaturesof thek-thwallsegment's axisin the- and -planes,

respetively(Figure4). Thelastequation(#

w

k

=#)iswrittenbyobservingthatthetwistofevery

pointin thewallsegmentisequaltothetwistofthebeam.

Theaxialstraininthek-thsegmentvarieslinearlywith(seeEquation3.3)andweanwrite

o

k

= w

k

+

w

mk

(3.5)

Theurvature

k

isuniformin eahatsegment

k

= 1

y os

k 1

z sin

k

(3.6)

Theratiooftwistanbewrittenas

#=

x

=

w 0

y

x

=

2

w 0

xy

(3.7)

whih,byusingthedenition of

(

= 2

2

w

xy

)an bewrittenas

#= 1

2

(3.8)

henewehave

k

= w

k and

k

= 2#

w

k

= 2#. These relationshipstogetherwith Equation

(3.5)maybewrittenin matrixform as

(23)

o

² x

k

k x

1 z

r y

1 y

r z

1

o

² x

w

k mk

1 z

² _»

y

» _k

³ _k » _k

´ _k

Figure4: Deformationsofanopensetionthin-walledbeam

8

>

<

>

:

o

k

k 9

>

=

>

;

= 8

>

<

>

:

o

9

>

=

>

;

k

= 2

6

4

1 0 0

0 1 0 0

0 0 0 2

3

7

5

| {z }

[R

℄

8

>

<

>

:

w

k

w

k

w

mk

# w

k 9

>

=

>

;

(3.9)

3.2.2 Step2. Fores and moments in the wallsegments

Thestrain-forerelationshipin eahwallsegmentisgivenby:

8

>

<

>

:

o

0

9

>

=

>

;

k

= 2

6

4

11

12

16

11

12

16

12

22

26

21

22

26

16

26

66

61

62

66

11

21

61 Æ

11 Æ

12 Æ

16

12

22

62 Æ

12 Æ

22 Æ

26

16

26

66 Æ

16 Æ

26 Æ

66 3

7

5

k 8

>

<

>

: N

N

M

M 9

>

=

>

;

k

(3.10)

wherethe[℄,[℄,[Æ℄matriesaretheinverseofthe[A℄,[B℄, [D℄matriesofthewallsegment[9℄

2

4

[℄ [℄

[℄

T

[Æ℄

3

5

= 2

4

[A℄ [B℄

[B℄

T

[D℄

3

5 1

(3.11)

N

;N

;M

are zeroalong the freelongitudinal edges(see Figure 5) and, onsequently, are ap-

proximatelyzeroeverywhere(N

=N

=M

=0).

Withtheseapproximationsthestrain-forerelationshipsforthek-thwallsegmentbeome(see

Equation3.10)

(24)

N _´ = 0

N _» _´ = 0 M _´ = 0

Figure5: Theforesandmomentsalongthelongitudinaledgeofanopensetionthin-walledbeam

8

>

<

>

:

o

9

>

=

>

;

k

= 2

6

4

11

16

11 Æ

16

16 Æ

66 3

7

5

k

| {z }

[e

k

℄

8

>

<

>

: N

M

9

>

=

>

;

k

=[e

k

℄ 8

>

<

>

: N

M

9

>

=

>

;

k

(3.12)

Weintroduethefollowingstressresultantsinthesegment's

k -

k

oordinatesystem(Figure

6,left)

b

N w

k

= Z

(b

k )

N

k d=b

k N

k ;at=0

(3.13)

M w

k

= Z

(bk) M

k d=b

k M

k ;at=0

(3.14)

M w

mk

= Z

(b

k )

N

k

d (3.15)

b

k

isthewidthofthewallsegment(Figure1). Thetorqueis[5℄

b

T w

k

= 2

Z

(b

k )

M

k

d= 2b

k M

k ;at=0

(3.16)

Notethat b

N w

,

M w

,and b

T w

representforesandmoments,andthesupersriptwreferstothe

wallsegment. Thesamesymbolswithouthatrepresentforesandmomentsperunit length.

Theseond equalities in Equations(3.13), (3.14)and (3.16) are written byobserving that in

eah segment n

o

;

o

varies linearly along the width ( diretion, see Equation 3.9), and,

onsequently, N

, M

also vary linearly along the width (see Equation 3.12). Equations

(3.13),(3.14),(3.16),(3.12), and(3.9)resultin

8

>

<

>

:

w

k

w

k

# w

k 9

>

=

>

;

= 1

b

k 2

6

4

11

11 1

2

16

11 Æ

11 1

2 Æ

16

1

2

16 1

2 Æ

16 1

4 Æ

66 3

7

5 8

>

<

>

: b

N w

k

M w

k

b

T w

k 9

>

=

>

;

(3.17)

(25)

»

´

w

M mk

w

M _x k w

N _x k w

T _x k

k-th wall segment

Figure6: Thefore resultantsin thek-thwallsegment

k k

w

k mk

1

Figure7: Curvature w

mk

ofthewallsegment

ToevaluatetheintegralinEquation(3.15)weonsideronlytheurvature w

mk

ofthesegment,

andassumethat theatsegmentremainsat(

=0,

=0,Figure7).

With thisapproximationtheinverseofEquation(3.12)yields

8

>

<

>

: N

M

9

>

=

>

;

k

= 2

6

4 e

A

11 e

A

12 e

A

13

e

A

12 e

A

22 e

A

23

e

A

13 e

A

23 e

A

33 3

7

5

k 8

>

<

>

:

o

k

0

0 9

>

=

>

;

(3.18)

Thematrix h

e

A i

isdenedas

2

6

4 e

A

11 e

A

12 e

A

13

e

A

12 e

A

22 e

A

23

e

A

13 e

A

23 e

A

33 3

7

5

k

= 2

6

4

11

12

16

12

22

26

16

26

66 3

7

5 1

k

(3.19)

Withthisdenition N

k isN

k

=

e

A

11

k

o

k

. BysubstitutingthisexpressionandEquation(3.5)

intoEquation(3.15)and byperformingtheintegration,weobtain

M w

mk

=

e

A

11

k b

3

k

12

w

mk

(3.20)

(26)

k k k

natesystem

8

>

<

>

:

w

k

w

k

w

mk

# w

k 9

>

=

>

;

= 1

b

k 2

6

4

11

0

1

2

16

11 Æ

11

0

1

2 Æ

16

0 0

12

( e

A11)

k b

2

k 0

1

2

16 1

2 Æ

16 0

1

4 Æ

66 3

7

5

| {z }

[!

k

℄

8

>

<

>

: b

N w

k

M w

k

M w

mk

b

T w

k 9

>

=

>

;

(3.21)

3.2.3 Step3. Fores in thebeam

Inthebaroordinatesystem(x -y-z)theforesin thebeamarethesumofthefores inthewall

segments(Figures6and2)

8

>

<

>

: b

N

x

M

y

M

z

b

T

x 9

>

=

>

;

= K

X

k =1 2

6

4

1 0 0 0

z

k

os

k

sin

k 0

y

k

sin

k os

k 0

0 0 0 1

3

7

5

| {z }

[R

k

℄ T

8

>

<

>

: b

N w

k

M w

k

M w

mk

b

T w

k 9

>

=

>

;

(3.22)

Wenotethat[R

k

℄ T

isthetransposeof thematrix[R

k

℄(seeEquation3.4).

3.2.4 Step4. Stiness matrix

Equations(3.22),(3.21), and(3.4)yield

8

>

<

>

: b

N

x

M

y

M

z

b

T

x 9

>

=

>

;

= K

X

k =1 [ R

k

℄ T

8

>

<

>

: b

N w

k

M w

k

M w

mk

b

T w

k 9

>

=

>

;

= K

X

k =1 [R

k

℄ T

[!

k

℄ 1

8

>

<

>

:

w

k

w

k

w

mk

# w

k 9

>

=

>

;

= (3.23)

= K

X

k =1

[R

k

℄ T

[!

k

℄ 1

[R

k

℄

| {z }

P

8

>

<

>

:

o

x

1

y

1

z

# 9

>

=

>

;

(3.24)

P

isthestiness matrixin thebaroordinatesystem(Table1).

Thedisplaementsmustbedeterminedbysolvingthegoverningequations(i.e. theequilibrium

equations, the strain displaement relationships (Equation 3.1), and the onstitutive equations

(Equation 2.6)) for the beam under onsideration. These equations are given in the x, y, z

oordinate system. Hene thestiness (andompliane) matriesmustalso be expressedin this

Analysis of composite beams

1 2

k

K

b k z

x

M y y M z

T

1 2

k

K

b k z

x

M y y M z

T

N N

z

x

y

® k

» k

´ k

³ k z

y x

z k y k

k-th wall segment

o

² x

k

k

k x

1

z

r y

1

y

r z

1

o

² x

w

k mk

1 z

² »

² »

y

» k

³ k » k

´ k

N ´ = 0

N » ´ = 0 M ´ = 0

»

´

w

M mk

w

M x k w

N x k w

T x k

k-th wall segment

k k

w

k mk

1

b _k z

M _y y M _z

b _k z

M _y y M _z

® _k

» _k

´ _k

³ _k z

z _k y _k

² _»

² _»

» _k

³ _k » _k

´ _k

N _´ = 0

N _» _´ = 0 M _´ = 0

M _x k w

N _x k w

T _x k