The influence of shear deformations on the stability of thin-walled beams under non-conservative loading

THE INFLUENCE OF SHEAR DEFORMATIONS ON THE STABILITY OF THIN-WALLED BEAMS UNDER NON-CONSERVATIVE LOADING F. LAUDIERO. M. S.AVOIA and D. ZACCARIA lnstltutodi Scienza delle Costruzioni. Universitri di Bologna. Facolt$

dl Ingegncria.

40136 Bologna. Italy

(Rereiwi

27 January I990

: in rcri.tcti

form

I2 Mu! 1990)

Abstract-This paper analyses the influenceof shear deformations on the stability of thin-walled beams of open section subjected to follower-type loads. The warping of the cross-sections is assumed to depend on the shear forces as well as on both primary and secondary torsion. The internal stresses are consistently deduced from the kinematic field. An algorithm is presented which works in the framework of the least squares method making use of a vector iteration technique. An extensive comparison is performed with the results offered by the classical models.

I. INTRODUCTION

Starting

from

a torque

vector

the pioneering

(Ptliigcr.

1950; Beck, 1952)

acting

analysis

by Nikolai

(1928) of a cantilever

at the free end. subsequent

systems. A comprehensive

works

were devoted to single problems of non-conservative

mechanical came from

of the basic concepts

(1953. 1956). who otfercd a classitication

and of the rclatcd

methods

Holotin

Hcrrmann

of bending-torsional

elastic

(1963).

continuum

and Jong.

1966~). In thiscontext. be associated placements solutions

only. virtual

Convergence

value problem single-valued

Hence classical

variational

obtained

work

by making

principle

by Barsoum investigations

(1971)

type loads

and

principles

do not

of the Galerkin

recently method

method

or of the

for an F.E. analysis and

can bc found

dis-

true and numerical

I98 I). Further by Attard

in general.

on generalized

hold

foundation

and Symeonidis,

conditions

forces cannot.

dcpcndcnt

by 196-I;

and Hcrrmann.

USC’of the Galcrkin-Pctrov

more

(Lcipholz.

boundary

since circulatory functional

(Iluttcr). well ;Is that

was tirst formulated

to those studies

1966: Ncmat-Nasser

when a convenient

CI ~1.. 1981 ; Argyris

(Argyris

were given

boundary

cquilihrium

bcilm x

with tllccorrcsponding

a stationary

were often

incremental rsquired

with

1966; Augusti,

of static amplitude

elastic thin-walled

to follower

theequationsofmotion

dc:fne a non-self-adjoint

of an

systems

that in the prcscncc of

of increasing

gave a new impulse

1965; Coma.

both of the mechanical

conligurations

motion

tluttcr

subjected

whose monograph

of elastic stability

It was, then. more ovidcnt

may consist of a vibrational

The problem of iI linearly

of analysis.

forces, the system may lack adjacent

non-consorvativc

to

schools

clarification

the work of Zicglcr

and the instability

beam subjected

of Russian and German

was

F.E. formulations Somrrvaillc

in Lcipholz

(1987).

(1962,

1963.

1983) and Levinson ( 1966). An alternative procedure has been followed by several authors (Ncmat-Nasser and Hcrrmann. 1966b; Prasad and Herrmann, 1969, 1972: Dubcy and Lcipholz. 1975) in the attempt

to extend the variational

of the linearized

theory ofelastic

formulations stability.

to non-conservative

All thcsc formulations.

problems

in the conte.\t

which make use ofadjoint

variational methods. can be seen as particular casts of a more general Telcga (1979). An iterative method for the solution of non-conservative

approach stability

was recently proposed by Xiong et al. (1989) by solving a set of related problems. In the same framework a least squares quadratic functional (Altman de Oliveira. was proposed

1984). rather by Leipholz

with an appropriate conservative ones.

than the adjoint (1980).

“semi-scalar”

who product,

equation, introduced where

was employed. a suitable

conservative and Marmo

An original function

non-conservative

given b) problems

approach

space supplied

systems

behave

like

F

I;52 Sufficient BcnLcnuto

conditions

Sutlicient loads.

for

and Corsanego conditions

for

abstract

formulation.

extended detinite

non-conservative

(1951)

of linear

formulated

(1981.

obtainin

operators.

;L general rather

1956)

g cigenixlue

giLen

by Caprir systems

suitable

problems

Lvith

of live

and Podio by

functionals.

deduced

in

in order to obtain

procedure

than

were method.

in the presence

non-self-adjoint

of operators

and Tralli

Liapunov

were determined

non-linear.

variations

systems

use of the second

displacements.

almost

procedure

<'I J/

adopting

Starting F.E.

from

models

symmetric

and

but

not

of

the

matrices. approach

correction

must

for

Alliney

the quoted

,A diKcrent load

Tonti

formulations

spaces involving

Tonti’s

of

self-adjointnrss

incremental

(I 98 I ). Recently.

variational

st;Lhilit!

( 197-I) by making

in a class of small

Guidupli

the

L>\\r LIILKI)

be

matrix

remarked

N’rigps.

that

of

;t direct

b>

hlang

consisting

~\:Is proposed such

;L procedure

IOS?) and physically

(I pr~l~iori

and

only

( 1983).

cumbersome

it; of

mwningful

symmetriration

Gallagher

for

Nonctheles5.

application

divergence-type

(Klee

sqstcms

it

and

(Argkris

1’1 r/l.. 19s I ). Recent

dcvelopmcnts

in the field of space mechanics.

and aeroelasticity. mech-

hydro

anical and structural engineering have underlined the increasing importance of the structures subjected to non-conservative l&is.

pressure

on cooling

loads acting

towers

forces and the need for proper

on ofTshore

are typical

or submarine

cxaniplcs

methods

structures.

of IiOIi-conscrv;ltive

on

of analysis.

Cable

bridges

suspe~~&xi

loads considcrcd

or

by slructur;ll

cn~inccrinc. c c

III tllc more spccitic licld of stability hcrcd that the bcnm model by including

the clYccts of

Ncmat-Nasser

ofwarping

the cll&l

ol’xxial

the Timc~shcnko dynaniic

distrihuti<~n ofthc

paper.

rcsponsc

dcform;ltions

More

conscrvativc

an open

a~i;il tlisplaccmcnt

ofa

( l909)

two-b;Ir

(1977)

theory

or

by including

Sophi~lrlopoulc’s

and

arc unilicd

hq the

( IOSh) an;~ly~d

frame

by the authors

thin-walled

in ;L symmetry

xcording

to

given

hy the theory

of

scctorial

ricpcncl on the local beam resultants functions

yield

by C’apurso authors

through

by making

IO ciiIfcrcnti~~l

the hcam pursued

through

homogcncous (I%3).

The

to :lnaly~

to any

given

use of ;I description

matrix. trpriori

the subspaco

problem

ruled

trigonometric

cigcnvaluc

All

dclincd

by ;I full

problem.

IWO)

makes

1975)

For

problem functions

non-symmetric purpose.

to obtain

use. in the framework

functions

resulting

solution

constraints.

through The

is a hard

method.

of linear

and strongl)

a projection

which

Lit-c

tcchnicluc

of an cigenvaluu

especially

has been dcvclopcd

of the Icast cquarcs

since

by LcipholL

conditions

solution task.

the from

is then

in a system of Rit/”

by a non-symmetric

arc enforced

obtained paper.

is obtained

numerical

to

Thcsc

cqu;ltion

decoupling

The

and homogeneous)

an algorithm

model.

In the present

cxtencicd equations

matrix

are assumcti

;1 variationul

Partial

is ruled

(linear

by the mentioned this

warping

expansion.

of symmetry.

and

functions

of the de St. Vcnant

series expansion

the bounci:lry

and

\tarping

are dcduccd.

one plane

by the coordinate

conditioned lvhich

of motion

:ind S;lvoia

of the inlinite series

called “gcner:lli/cd

corresponding

ill-conditiorlctl not satislicd

the classical

The

in the context

law (I3ailcy.

to cvhibit

equations

arcas.

only.

representation

equations

is assumed

by l.audicro

an eigcnfunction

make use of Hamilton’s

ivhich

onto

:tn approximate

(196-I)

proposed

plane.

in order

hcam subjcctcti

( IWO~~.h).Accorciing to this model. for ;I beam subjected to non-uniform bcnding anti torsion, the local shear &formations ofthe middle surface of the beam arc taken into account ; hence. with reference lo torsion . :I “secondary” warping is consicicrcti which is superimposed to the “primary” warping

ticld rcccntly

Ts;ti

hc rcmcm-

Kounadis

lo;~ti~ only).

symmetric

Loomis acting

it must

and

to Timoshcnko anti

cigcnfrcclucncics

all thcsc aspects

of

of I’ollowcr

(for

Kounadis

hc;lms.

( 1967)

according

Ncmat-Nxscr

rcccntly,

on the hcnding

hcam model

of thin-\vallcd

by Ncmat-Nasser

( IW~k~) anti

rigidity. incrlia

In the prcscnt lhc

shear

and llcrrmann

inllucnce

analysis

has been relined

for

(,Illincy

an ill(‘I (II..

of a tc‘ctor itcrati<)n

tcchniquc. A variety

of example

various

cross-sections

mntions

on (possibly

comp;Irisons

arc ma&

problems

subjected coupled) with

illustrate

to different flcxural

the classical

the response load conditions.

and torsional models.

modes

of a cantilever The

influence

is investigated

i-mm.

having

of shear dcforand extensive

Thin-wllrJ

1353

beams under non-conservative loadiq 2. KlNEMATICS

Figure I shows a typical open thin-walled

prismatic beam. A right-handed

orthogonal

coordinate system C,,.VK is adopted. where .\: and ~-are the centroidal principal axes. displacement

components

are illustrated

cenrre St.\-,$._I.,). in the local reference

in the same figure. The coordinates

system Pnp. shown in Fig. I). are given by :

11= (x-sy)

where s is a curvilinear coordinate

Positive

of the shear

if -I-(_I’-_vS) d$,

(1)

lying on the middle line and the axesp and II are

tangent

and normal to the middle line respectively. It is assumed that the cross-sections are rigid in their own planes whereas the shearing deform:ltion

on the middle surf&x

the displacement assumed

gradient

is not constrained

components

corresponding

to zero (assumption

a). Moreover.

to the loaded configuration

to t-x ncgligiblc with rcspcct to unity (assumption b).

are

If wc consiclcr assumption of the cross-section

;I. Ict x’ hc the plant which dclincs the avcragc displaccmcnt

contour c (I:ig. 3). C’ and S’ dcnotc the location of the ccntroiri and

the shear ccntrc of (.’ on the plant IT’. By,/” WC will dcnotc the intcrscction bctwccn x’ and the plane containing C” parallcl to the original plane n,, of the cross-section, whcrcas./‘will dcnotc the straight linc/‘is

lint

parallel to,/” and containing the original

oricntcd so that the s axis rotutcs countcrclockwisc

coincitlc with/: 3. and rotuts

Let us dcfinc the angle 0 like 0 = /I’--/{, 1.’ by an angle -0

counterclockwise.

ccntroid. C. The straight

by the angle 3. less than x, to

whcrc /I’ and /I arc shown in Fig.

around S’ where the positive rotation

is assumed to bc

As ;I conscqucncc, C’ and (.’ move to C” and L”’ rcspcctivcly and the

displaccmcnt functions <, rl and ; arc dctincd as the components of CC” along .v. J and : rcspcctivcly. By ./“’ WC will dcnotc the intcrscction bctwcen z’ and thu horizontal plane n;, containing C”. Let P bc ;t typical point belonging to (‘. !’ its linal position and P’ the projection of I’ on ths plant 71’. Rccausc of the rotation -0 around S’. the point P’ arrives at P” which. with rcspcct to the swung axes .v’ and _I”, keeps the same coordinates s and J’ of the point [’ in the unstrained configuration (Fig. 4~1). Analogously, S’ keeps the same coordinates x S. Hcncc. the coordinates of P ‘, with rcspcct to .Y’ and J,‘. take the form : .Y’ = .\---(,~~--~,~)sin O-(.\---.~,~)(l

-cos

0)

js’ =?‘+(.\:-.~,s)sinO-(?,-!,,)(I

-cosO).

(2)

With reference to Fig. 4,. WC have : P’A =

--s’ sin r+~,’ cos z

C”A = -x’

cos r--J”

sin z.

(3)

Thin-walled beams under non-conservative loading

1355

Plane 7r’ (a) Fig. 4. In-plane section displacements.

Let cp be the angle by which the plane KG rotates around J”’ to coincide Moreover,

by c we will denote the out-of-plane

P’p.

displacement

with n’.

If P; and P,, are the

projections of P’ and P on R$. we obtain (Fig. 4b) : P,,A = P’A cos cp+[sin

cp

P,,F’ = - P’A sin cp+[cos

cp.

(4)

Finally. considering that the coordinates of I’ in the rcfcrcncc C”.VJ*Zarc: I,,

=

-

I’,,,4 sin

2 - C”A cos z

?‘I1= P,,A cos x - C”A sin r =o = P,,I’. itnd keeping in mind that the coordinates

(5) of C” in the reference CU.r_~~:are given by <. q,

the displrrcement components of P along the axes s, _Pand z are given by:

z+<.

II, = <-.\-+.r’(cos

cpsin’~+cos~iw)+~‘(l

llY = q--)‘+y’(cos

cpcos’r+sin’r)+.r’(

u: = C+.r’sin

cpsin Z-J.’

sin

-cos I -cos

fp cos r+[cos

fp) sin crcos r- .esin cp)sin r cos z+[sin

cpsin 2 cpcos r

fp,

(6)

where s’ ilnd _I” arc given by eqns (2). If we consider assumption and [(z,s)

in a neighborhood

linear and quadratic

b. for an arbitrary

variation

of the static equilibrium

of <(I),

configuration

r,~(:). i(z).

(p(z). O(z)

and for any z(z). the

terms of the displacement components are:

(7’) and

(7”) where

cp, = are the first order rotations definition

(8) implies

lp co5

cp\= -_c? sin 2

2.

of the cross-section

that the corresponding

negative .Y and positive _r axes respectively. displacement Jis cast in the form (Laudiero

(3)

around the axes .Y and A’ respecti\,ely. The positive

Finall!.

rotation

vectors are oriented like

the linear part of the out-of-plane

and Sa\oia. IY9Oa) :

where w(s) is assumed to coincide with the sectorial areas. The functions Q, and cp, art able to reproduce the Reissner-Mindlin

formulation

(klindlin.

coincident with q’ and 5’ respective1y.t the Euler-Bernoulli

1951) : when they are ma& model is obtained. Analogously.

when the function cp,,,is made coincident with O’, the Vlasov formulation terms x,(:)$,(s)

and x,.(:)$,(s)

is recovered. The

take into account the shear strains due to the beam forces

T, and T,, whereas the term x_,(z)II/,.,(.s) (secondary warping) plays the analogous role with respect to the shear stresses arising from non-uniform

torsion.

By making use of cqns (7’)

and (9) the linearized displacement components of the cross-section

contour in the local

reference system Plrp: are given by :

The warping functions II/,. I,!I$. I//,,,arc ohtaincd under the assumption only on lhc local hcanl resultants.

Analogou4)

that they dcpcntl

Lvith lhc tic St. Vcnanl niodcl,

as\urning

bending and twisting momcnls linearly varying along the beam axis and making u5c ofcqr13 (IO), the warping functions can bc obtained by means of the li>llowing and Savoia, IWOb) :

Q,(.).)= IG,(.S)-C.j(.s)(.,,--‘,(i.

rclations (Laudicro

j = .Y.~‘.I~J)

(11)

where d D,S, ds 6, = -

J,r

d

’

(I?)

ds

f(s) is the thickness of the cross-section,

d.4.

A is the area of the cross-section

t Superscript

conditions :

( )’ meansdcrivatiw

(13)

and S,. S,. S,,, and J,. J,. J,, arc the first and the second

area and sectorial moments rcspcctivcly. The three constants C, and the nine constants orthogonality

II,.. =

with rcspcct

IO :

c,, can bc dctcrmincd by imposing

the

Thin-wallcdbeams under non-conserwtive

I .I

loading

1357

$,d.-f = 0 (i, j = 1. ?‘. W),

(14)

j$, d.4 = 0

resulting in :

c, = f

I,

4, d/f

s

i&, d.4

L’,,

=

(i. j = x.,v.

(15)

0~).

;_--_

J

j’dil

4

3. STRAIN

DISPLACEMENT

RELATIONS

Adopting 3 Lngnmgian description. the linear and quadratic terms of the Green strain tensor arc given by :

E”) = #7u”‘+u”‘v) E

‘2) =

$“I

Il.

U”‘Vf~(VU’~‘fU’~‘V)

( I6a.b)

whcrc the operator V mc;lns gradient (or its transpose when it is postponctl).

Making use

or the lincarizcd displaccmsnt field (I 0), cqn (I 6a) yields : r’*,1 ” =

,:!,!,’= ()

2; = <’ -fI,:..r-cp:?'-(P;,,(fJ+X:'l/,+X:.~,

Moreover,

+X;,,$w

recalling that the cross-section does not distort

and assuming that any

straight line remains normal to the middle surface of the beam. wc have: ‘1) = c./’

&” Analogously.

Moreover,

=

( 17”)

0

assuming the lincarizcd displaccmcnt field. cqn (l6b) becomes :

in accordance with the kinematic assumptions.

the following

is adopted :

Since 11i.7is expected to be very small compared with unity, the terms ofeqns (18’) containing such a derivative will be neglected in the following.

4 THE

The loading condition

COSDITIOS

of the beam will be defined

load q and in t\vo distributions and : components.

LOAD

in terms of a distribution of a line

of loads acting at the end sections. f(0) and f(l).

having J

Both the load q and the loads fare assumed to act in a symmetry plane

(say J,--z). By making use of the linear part of the displacement

field (7’). the corresponding

variations of the applied loads arc obtuincd : 9

III = (R,-I)q.

f”’ = (R,-I)f.

where I is the unit tensor and the tirst order rotation

(19)

tensors R, and Rr may assume either

of the follobving forms :

r

- 0 <’ -

1

R,; =

I

11'

- I]’

I

0 IL. C’ + (.I’ -j.s)w

(,r-_l~,s)(l’

I

(70)

R, =

The knsor

R,; tlcri\cs

Urom the assumption

that. during

the deformation.

follo\vs the .I’ I plant keeping the same angle with the transformed other hand. R, rcllccts the assumption plant

the load

IocnI gcncratrix ; on the

that fhc load is rigidly conncctcd with the middle

of’thc crabs-section. In the ti)ilo\ring,

the cases R, 5 R,; and R, 2 Rs will be developed

in clctail tvhcrca ;I numerical application will be given for the C;ISC’R, E Rc. With rct’erencc to ;I thin-walled

beam element, the non-vanishing

Cauchy stress tensor u. corresponding

to the static equilibrium

components of the

configuration

and referred

to the local reference system. may bc written as follows:

(22) whcrc .I/,,, =

. I &Il/,

d.4.

s,,, =

*> i l/Q, ds.

Jo

I

J,, = i (tl/,)'dA Jc

(‘3)

of .\/,, with respect to 1; tinally the positive components ol the beam forces arc sho\vn in f’ig. I.

and 7;. is Ihc lirst dcrivati\,c

5. ST:\BILITY The basic method of investigating

CRITERION

the stability

of non-conservative

problems consists

of the analysis ol’rhc oscillations ofthc system close to its equilibrium position. An integral formulation govcrnin, *pthe motion of ;1 bodv, U can be expressed through Hamilton’s 1awt (Bnilcy.

I975) :

Thin-walled

beams under non-conservative

loading

1359

(24)

for any admissible the variation strain

variation

energy.

Moreover,

the conservative

components

f

”

components

to zero

compatible

by imposing

the requirement

a non-self-adjoint

from a potential.

problem

The residual

II”’ and the further

the virtual that

work

&“‘(I,)

work

of impulses

= 3u”‘(fl)

since the external

terms appearing

whcrc IC is the elastic tensor at the origin

of small

untlcr the assumption

gradients,

part of the

the work

done by done by

may be made

= 0 which

work

6L cannot,

is fully equation

generally.

be

in eqn (24) may be given the compact

:

expressions

which.

collecting

of free oscillations of the system. The variational

with the assumption

(24) represents derived

In eqn (24) ST is

functions.

of the second order

and q”’ of the external loads for the displacement

II’ I’. The last term of eqn (24) containing

equal

displacement

a second order work

SL represents

loads for the displacement

the non-conservative field

of the generalized

energy and SW is the variation

of the kinetic

rcrluccs to the Cauchy

displaccnicnt cquilibriiim

ilnd

the loads acting

p’ ” cxprcsscs

KirchhofTstrcss

of both displaccmcnts

stress tensor u. Morcovcr

lick! (7”). p rcproscnts contiguriltiou

and S is the second Pi&

components

Ou“I is the quadratic

on the body

the variation

of

tensor

and displuccmcnt

surface

the cxtcrnal

pilrt of the in the static loads

cor-

rcspontling to the lincarizotl displaccmcnt field u’ I’. Making

use ofcqns

(7”), (IO). (l7),

loacl p in cqns (25) symbolizcs

(18). (19) and (22) and keeping

in mind

that the

the loads q and f, the cqns (3s) may be rcwrittcn as follows :t

+J,.(~~s(~,.+J,,~,oS~.,+J,,;i.,S;ir+J,,;i,S;i,.+J,,,,;i,,,SR,,)d=

(26)

I 6 LC’=

i I,

(E[Ai’S;‘+

J,cp;Scp;. + J,.cp;.Scp;.+ J,,cp:Scp:, + J&,Sx:

- T,.(j(i”(l)+r~/,S(,“n’)+~(c,~I,+c,,,r/,,)n’so’-(c,r,.+c,~ r,,)s(W); d=

(27)

t In the expression nf$ltl as usual in (hc beam theory. the Poisson effect has ban accounted for by assuming 6,. = 8:“”= -WC;, whcrcas in cqn (IS”), c,, and E, were assumed to vanish.

where E and G are the normal and tangential elastic moduli. components in the j’and

: directions.

ql. and (1: are the line load

F,, F, and f/, are the beam resultants of the external

forces f. J, is the torsion constant. J, is the polar moment of inertia about the shear center and u here :+

iid.

D,, =

I!clu;ltion

I

li/,.~L,.$~.,)

(29)

(i.i=.\-.~,.(,).~,.rl/~.~~,,,)

(30)

(2X) has hccn derived ncglccting all the warping terms of the displacement licld

consistently

with the assumption

terms. All the unknown of the vari:ilion;\l To

di di ,d,,d:sd.-l

ti.j=

functions

formulation

sunImarkc,

extensional intcrmll

(74)

that the li)llowcr

forces arc insensitive

have to belong to I/‘((). I) xcording

to the warping

to the requirement

(3).

the present formulation

neglects flexural

c.utcnsional and torsional

energy terms as well 3s the tlcxural- tlc?cur;il terms given by the geometric

cl~ects of an applied torque. These contributions

were cv;~Iu;~ted in La udicro and %accari;l

( 19XYa.b). 6. TIIE

With

rcfercncc Lo Hamilton’s

and integrating

I:yuATroNs

01’ \lOTION

law cxprcsscd by cqn (2-I). making use ofeqns (26)

by parts yield the following

(2X)

cqu~~lions of motion :

has txcn disrcprdcd in eqns (27) and (3) t The coupling term J,,, an.~logous coupling terms. Rcpx~tcd subscripts arc reported oncc only.

SIIICC’11 is ncghg~hlc Hith rcspcct to

Thin-walled beams undtm

loading

n0n-c0t~~e~~311w

1361

=0

PJ,.~~~,-EJ,~~;~-GD.~(~‘-~~,.)-GD,,.,(B’-~~,,,)-GD,,~X,-GD,,,,,X,,-~,(~-)‘~)~ ~Jwcii,,- EJwcp::-GDw(@-v<,)

-GD,,,,(5’-

v,) -GD,tv,\~.r - GD,o+

=0

PJ,,~.,-EJ,~X.~+GD,~X,+GD,,~(~‘-~,)+GD,.,,,(O’-~,.,)+GD,~,,,,X,~, +GD,,,(rl’-9,)

PJ&--EJ,,x:+GD,,x, PJ,,,,~,,--J~,,,x::+GD,,,,x~“+GD,~*,.,x.,

+GD,,,,,(:‘-9,.)+GD

= 0

= 0 ,.,w,, (0’-9w)

= 0.

(33)

with the relevant boundary conditions :

[(GD,(~‘-~~)+GD,,,(~‘-~~,)+GD,,~X,+GD,,,,,X,,,+R’~’ i-N_v#‘-

T,e+icl,@‘+F,e-F,cp,)Sr]b

[(GD,.(~‘-9,)+GD,.,,X~+iV~‘-F,9,)3rl]:, [(Et!;‘+

= 0

=O

F,.cp,)d;]b = 0

[(GJ,~)‘+GD,,,(~‘-~,,,)+GD,,,,(~’-~,.)+GD,,,,,X.,

+GD,.,,,,x,,,

+ Nr,y:‘+ N(J,/A)(I’+il/,;‘+~(C,:l/,

+ C,/U,,)V’-

[(EJ,cp;hhp,j:,

(C, TV+ C,, T,,)O)M];,

= 0

= 0

[(EJ,qv - I;:(?,-?.,~)n)acll,]I, = 0 [(~J,,.(~l,)~~CP,.,lI, = (1 [(I~J+~~:)h~,j~, = 0 [( kY~,~;,,s~~]:, = 0 (34)

[(~~J,,,,XS,)~~X,,,l~, = 0.

The solution

to the problem (34) is obtnincd by making USC of trigonometric

expansions so as to adhere strictly makes the comparison Bernoulli All

to the problem of the beam vibrations.

with the aniilogous

results

given by the Timoshcnko

series

Such 11choice and Euler-

models more direct. the boundary conditions

which are not sibtisficd by the coordinate functions

will

be enforced by ;L projection onto iI proper subspocc. Assuming

the displacements to bc functions of time and space independently, each of

the unknown functions and the corresponding

variation can be given the compact form :

y(z, I) = a&:)

C””

Sy(z. I) = sa‘;E(z) c””

(35)

whcrc the vectors e ilnd a,, contain it suitable number of coordinate functions corresponding Substituting

and the

Lagrangian coordinates rcspcctivcly. eqns (35) into (26)

(28). the variations

of the kinetic and strain energy

and that of the work of the external loads can bc written in the f’orm :

where the vector a represents a global vector of unknown generalized coordinates collecting

F. LALXJIERO ef (11.

IX’

all the vectors a,. the matrix loads and the matrix Hence. setting

K,_ is non-symmetric

since 6L contains

the work

of the follower

%I is the mass matrix.

K = Ku+KL,

the variational

equation

(24) turns out to be discretized

:

as follows

‘2

I

GaT(Ka-p’ibla)

e”!“d(

(37)

= 0

11

to :

subjected

Ca e2’p’ = 0,

where

eqn (38) collects

functions

the boundary

S. By decomposing

conditions

the matrix

and setting x = LTa. the problem

C(ja

e:‘p’ = 0 which

M according

(35)

are not satisfied

to a Cholesky

a priori

by the

scheme, i.e. hl = LL’

(37). (38) becomes: ‘2

I

Sx~'(Ax-p~x)~~'~'df

0

=

(39)

‘I

to :

subjcctcd

Bx = 0,

BSx = 0

(4Oa,h)

whcrc A = I, In order

to satisfy

‘KL

the constraints

‘,

(40).

B = CL

‘.

a projection

(JI) operator

.30 is dcfincd

(Aoki.

1971): 9’ =

9” = I-B(BB”)

“B,

which projects any given vector x E R” onto the subspace defined equation reduced

(40a).

Hence,

making

use of the condition

2%’= 2”,

(41, by the linear homogeneous the problem

(39). (40) is

to the form :

I

‘2

6xTi4'(9'A9“x

e”“’ dl = 0.

-p%i)

(43)

‘I

Since in the quotient immediately

obtained

space R”/.U

the variation

.YA.Px Equation

66x

is arbitrary,

from

eqn (43) it is

:

(44) will be rewritten

-p’.+‘x

= 0.

(44)

in the following: dy

- iy

= 0.

(4%

and y = .YXE R,‘/.U. as required by the original problem. where i. = p’, ,rJ = YAP Equation (45) represents a standard cigenvaluc problem which satisfies all the boundary conditions, including the non-essential etficient since the structural response will will be seen in the following.

ones. This technique turns out to be particularly be extremely sensitive to the loading features as

Thin-walled beams under nonconservative loading

Since the matrix non-symmetric,

hereafter.

with the higher

We will restrict the analysis

will

algorithm

ill-conditioned

(Alliney

under

et al.. 1990) which

of the ill-conditioning

frequencies

(square roots of the corresponding

the assumption

In the framework

of the matrix

to the problem

that the eigenvalue

of the least squares method.

(45) ; hence

problem

admits

the solutions

min J(y. i.)(J(y.

L) =

1).dy - i.y III (46)

IIYIIi

that the problem

iff the pair (y.i)

is a solution

(46) can be given the equivalent.

min *(y)@(y)

where Q. the Householder

will

:

the condition

where J 2 0 vanishes

and

counterpart

to the search of the real solutions

be developed

from

are strongly

modes of vibration.

ourselves

one or more real solutions.

&.

a suitable

The mechanical

the rapid increase of the circular

eigenvalues)

be derived

the matrix

it was necessary to develop

is briefly outlined K involves

K. and consequently

I363

reflector

(45). It can be shown form :

yTdTQ(yLdy IIY II 5

=

(Stewart,

of the problem Rayleigh-type,

(47)

1973). is given by the expression

:t

Q(z) = I-ii.

The minimization rrth iteration giving

of tip(y) is obtained

the Houscholdcr

rctlcctor

Q,, = Q(y,,) ; then the gradient

is dctcrmined.

at the point

by means of il vector iteration is dctcrmined

tcchniquc.

by mcuns of the 11th trial

At the

vector

yn

of the form :

.% (y)

expression

(48)

y'l'.drQ,.dy (49)

= ----j($--

y,,, resulting

of a scalar

(to the accuracy

multiplier)

in the

:

r, = .c/“Q,,.c!?,

Hence. a step is performed vector as follows

towards

- (j,f.~/*Q,,.c/j)~,.

the minimization

(50)

of R(y) by correcting

the current

:

Yn+l =

where r may bc determined resulting in :

through

_

the condition

a, = ~{[9n~jfn)-9P.(~,)l+

.

yn--ar,

(51) that .~#~(i,)--.ti,,(jm+

[~~~~~~-~,~~,~lz+~ll~~ll~~.

t The symbol (“) indicates normalizarion of the vector acted upon.

,) is maximum,

(52)

I?64

F

L.at I>IIHO ;‘f 01

Y. ~UILlERICAL

All

RESULTS

the examples solved are constituted

corresponding

by cantilever

beams having a cross-section

to one of the two shown in Fig. 5. with the exception of the first example

which does not require such a specification. obtained by making use of relations

(12)

The warping functions

11,. rl/, and I/,,, were

( 15) and are reported in the Appendix together

with the series expansions adopted for the displacement functions. The first e.uample refers to the classical Beck problem which was solved by adopting the Euler-Bernoulli

model. The latcrul displacement was interpolated

with an increasing

number of terms of the series expansion and the rate of convergence is shown in Table The Hutter loads Lvere obtained by checking the trajectories

of the tirst

t\to circular

I. fre-

quencies while increasing the applied load. The starting trial vectors to be used at each load level were assumed to coincide with the optimal vectors of the preceding load lecel. Subscqucntly.

the same problem was sol\ed by adopting I6 terms of the series espan-

sion for each unknown function in order to compare the results of the proposed model with those given by the Euler-Bernoulli

and Timoshenko

models (Table

NUMBER OF TERMS

8

16

32

48

PC, e.‘/EJ

20.0068

20.04 19

20.0496

20.0505

2). When the beam

EULER-BERNOULLI P P’/E J = 20.0509

‘l’;~hlc 2. I~tllucncc 01 ~I~IIcI~~II~~\ 01, Ilcxural Iltrllcr I~utl (Ikch e/h

2.50

5.00

10.00

prohlcm)

15.00 I/

EULER-

P

BERNOULLI

20.04

20.04

20.04

20.04

TIMOSHENKO

2.065

4 661

10.55

14 23

;/e+e i_-----*

ANALYSIS

P,ez -

ISectIon

PfWSENT 2.207

I------b 4

4 777

10 72

p--

14.37

b=h

2h

E

I

J.

Thm-walled

beams under non-comcrwt1cc

loading

Table 3. lntluence of slenderness on Herural flutter load (Leipholz

P/h

2.50

5.00

10.00

40.05

40.05

15’oo

CLASSICAL MODEL

problem).

40.05

+q

40.05

A

27.26

I Section b= h

%C PRESENT ANALYSIS

3.914

7.265

Tuhlc 4. Intluenceof~l~ndernessun

Q/h

q

torsional flutter Iwd (Semal-Naswrand pmhlem).

2.50

5.00

10.00

15.00

CLASSICAL MODEL

18.56

18.86

20.70

22.20

PRESENT ANALYSIS

10.44

15.42

19.09

21.75

rather short.

hccomcs

19.17

Herrmann

it can bc seen that prcvcnting the built-in section from warping makes

the hcam stiircr with rcspcct to the Timoshcnko Iwant ~nodcl. ~~hcthird cxa~ilplc rcfcrs to ;I cnntilcvcr kiln suh,jcctodlo ;I uniformly :ixi;ll load which follows the dcformcd ccntroid;~l axis (Lciph& results rcportcd in Tahlc 3 conlirm thcdctcrmining in

the higher modes of the Jlcxur;II vihrations

distributed

prohlcm). The numerical

role that the shear strains play. cspccially

( I6 tams wcrc adoptctl for each unknown

funclion).

eJ‘hc

cxamplc analy~s

liwrth

the inllucncc of the shear strains

OII tlt~’

Lorsionnl Jluttsr

lo;~l for ;I c;lntilcvcr l-mm (having an J cross-suction) subjcctcd to two axial forces acting at the intcrscctions bctwecn the web ilnd the Ililngcs of the end cross-section (Ncmat-Nasser and t tcrrtnilnn gcnerutricos. tion)

problem).

The

The

two forces arc assumed to follow the local transformed

numerical results rcportcd in Table 3 (I6 terms for ciich unknown

func-

show that the shear strains still ploy ;I signilicant role even if not such iI dramatic one

as in the cast of the IlcxuraI Iluttcr. In the Jifth example, ;I cantilever beam having an J cross-section is subjected to an end il.\i;ll force with throc diJJ’crcnt eccentricities. The case .I*,,= 0 corresponds to at the ccntroid uncoupled

of the cross-section

rwdcs

of’

vibration.

and. conscqucntly. the first column

The

cast

,I; =

0.5/r corresponds

II

force acting

of Table

5 refers to

to iI force acting at the

intersection of the web with ;t tlangc. Hence. the second and third columns refer to coupled

II~OCICS of vibrations

and

the symbols T (torsional)

component of the vibration

and F (tlcxural) denote the dominant

mode. The numerical results were obtained by adopting eight

terms for each unknown function. The cilses corresponding to the first three ro\vs of Table 5 kvcrc an;~ly~cd by Ncm;lt-Nasser

and Tsui (1960).

In the subscqucnt examples ths trajsctorics thcmsclvcs of the non-dimensional

circular

frcqucncics are shown in order to emphasize iI phcnomcnon :llrciidy revc:tIcd by the previous cxamplc. i.c. how the critical rcsponsc may bc characterized by buckling or lluttcr dcpcnding whsthcr the load follows the middle plant of the cross-section or the local gcncratrix. The structural clcmcnt considcrcd wits ~1cilntilcvcr bcilm with U section subjected to an end axial force: thercforc. the flcxural-torsional coupling was an inherent consequence of the structural gcomctry. The analogous problem (with no shear strain influcncc) has been recently studied by Aida (1986).

Figure

6 shows how the critical response additionally

depends on the slcndcrncss ratio: in fact. it tends to the buckling mode for both the loadin_c fcntures as the beam length is reduced. In this example eight terms for each unknown function were adopted and the shear strain influence was fully considered.

I266

F.

LALVIEHO t’l ui.

T.~hlr 5. Cnt~cal values for a canr~levrr barn wnh an I arIaI follower force. Yp = 0.

PC,EZ EJ”

Finally,

XCIIOIIsubpxed

yp = 0.25 h

Mode

yp = 0.50 h

Mode

Mode

T-ON

2.407

T-Div

3.330

F-Flu

9 516

T-DIV

19.82

F-Flu

10.61

T-Div

38.73

F-Flu

20.01

T-DIV

2.353

T-Div

3.269

F-Flu

7.285

F-Flu

15.15

F-Flu

8.912

T-Div

39.18

T-Div

16.16

T-Div

2.407

T-Div

2.548

T-Div

3.430

F-Flu

10.02

F-Flu

7.566

T-Div

19.82

F-Flu

20.01

T-Div

2.353

T-Div

2.464

T-Div

3.339

F-Flu

15.12

F-Flu

8.253

F-Flu

6.357

T-Div

16.16

the cast of a cantilever beam. with an I cross-section.

lateral force was considered. Again (out-of-plane) be coupled. For the sake example, for which tight

of simplicity,

to an rc‘ctxtrlc

subjcctcd to an end

Hcxural illltl torsional

modes turn out lo

the shear strain inllucncc was not considered in this

terms for each unknown

function

were retained. This

problem

does not stem to have been solved in the literature since only thin rectangular cross-sections wcrc consitlcrcd (Conio. the non-dimensional

1966: Harsoum.

I97 I ; A tlarrl and Soniervaillc.

(Uexural and torsional)

frequencies of the unloaded structure.

two frequencies was obtained by assuming dilfcrent height. Figure 7 shows how the flutter to unity. This

I W7).

In f:ig. 7

values of the flutter load were plotted against the ratio of the tirst t\vo

load approaches zero as the frequency ratio tends

paradoxical result was already underlined

by Bolotin

that, when the natural frequencies of the unloaded structure slight damping may have a stabilizing

The relative variation of the

ratios of the Ilange width over the web

( 1963) who noticccl

approach each other. even

elTect. In these casts, howcvcr. a non-linear analysis (Sethna and Shapiro, 1977) seems

which goes beyond the limits of a linearized formulation to be the required development.

9. CONCLUSIONS

A model has been presented for thin-walled account the middle surface shear deformations

beams of open section which takes into

associated with shear forces as well as with

secondary torsion. All the loads have been assumed to act in a symmetry plant of the beam according to two distinct features. i.e. the loads may follow either the middle plane of the cross-section

or the local deformed generatrix.

The

internal

stresses have been directly

derived from the kinematic field by making use of the constitutive relations. Following the usual separation of space and time variables. the variational equation expressing Hamilton’s law has been discretized by making use of trigonometric series expansions. This choice

Thin-walled

beams under non-conservative

1367

loading

+PP’

G

4

3.

1

2

3

4

6

5

7

6

9

Fig. 6. Circular frequencies of flexural torsional vibration modes for a cantilever bum suhjcctcd IO an axial force. The dashed line indicata that the force remains orthogonal IO the cross-scclion. The solid line indicates that the force follows the local transformed gcneratrix.

appears to be the most strictly applications which

a suitable

overcomes

related to the features ofa vibrating

algorithm

the difficulties

has been developed, arising

from

beam. For the numerical

based on a vector

an eigenvalue

problem

iteration

technique,,

ruled

by a non-

symmetric and strongly ill-conditioned matrix. A variety of examples clarify the significant influence of the shear deformations on the critical loads involving flexural as well as torsional modes.

F.

LAUDIERO

CI ul

i

F I

jh

em

0

-5

EIG - 2.6

+A?-

b/h = 0.30

03

Fig. 7. Flcxural

05

torwn~tl

1.0

1.5

20

tluttcr loxl versus the ratio txtwccn the first llcxural and torsional ctrculur frcqucnclcs ol’thc unloaded qtructurc.

I%&um. R. S. (I971 ).‘f?mtr clcmcnt method applied to ;hcprohlcm ol’stahility ol’ a non-conscrvativc system. Iltl. /. xlcnrcr. .\/Cf/I. ElIglty,Iy3. 63. ijsck. Xl. ( 1952). Die Knicklast dcs cinscitig cigcnspanntcn. tangential gcdriicktcn Stabcs. Z.4.\IP3. 215. Iknvcnuto. E. and Corsancgo. A. ( 1974). Nonconscrvativc stahllity evaluation by means of non ;tssociatcd vari;ltional cipcnvaluc problem. Euromccir112. fsolotm, V. V. ( 1963). ,Vf~r~cnn.~rrlclri~‘, Proh/crtr.s o/’ I/W 7%eor~ of Elrrsrrc~ .‘dtrhr/ir,r. Pcrgamon Press. New \I’ork. Capriz. G. ;Ind i’o&o Guidupli. I’. (1981). Duality and stability questions for the lincarizcd tructlon prohicm with hvc loads in elasticity. III .S~ohrlir,r iu I/W ,~lcdumic.r of C’onrirtuu (Edi~cd by F. H. Schrocdcr). p. 70. Springer. Bcrlln. C;lpurso.

M.

(IOCJ).$11c:tkolo dcllc

travi di parctc sotlilc in prcscnr;, di forx

c distorsioni.

1-u Rxc*rccr Srxn/r/rcxr

i.213;7.5. Coma. M. ( 1966). .l

Lateral buckling ofa cantilcvcr subjcctcd to a transvcrsc follower

lorcc. /III. J Solids

S~rrcrrtrc.s

CIC

Dubcy. R. N. and Lciphol7. H. t1. E. (1975). On variational methods for nonconscrvativc problems. ,\I&. RCS. C0n1,tntrt. 2, 55. Hcrrmann. G. and Jong. I.-C. (1965). On the dcstahtlizing etfcct of damping in nonconscrvatl\c ciastlc systems. J. .4ppl. ,~lcdr..AS.\IE32. 591. Klec. K. D. and Wriggcrs. P. (1983). An updated Lagrangian formulalion for Timo$henko hwmc mcluding non concerwtive loads for dtverpcncu-type systems. z\(w/I. RCY. CMI~~IWI. 10, 23%

Thin-walled Kounadls.

A. N.

( lY77).

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Stability of elastically restrained fimoshenko

loading

I369

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kams

of

with open

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Nonlinear behavior of llutlcr unstable dyn~unis;il systems w&h gyroscupic rlSME44. 755. Stcu;trt. G. W. ( I ‘J73). Ilrrrr&c~/lr~rr 10 hltrrri~ C’onrprtrcrrWns. Academic Press, New York. Tclq;t. J. J. ( 1979). 011 v;lri;ition;rl formulations for non-potential operators. J. Iusr. Blur/~. Applic. 23, 175. TOIILI. E. ( I%+t), Vtiri;ltional lbrmul;~tion for cvcry non-linear problem. In!. J. Enqm~ Sci. 22. 1.143. ,X~ong. \‘., W;~ng. T. K, ;lnd Tah;lrrok. 13. (lY8Y). A note on the solutions of general nonconscrvativc syslcms. .l/r?h. KCS. c;~,~u~I,oI. 16. x3. Zicglcr. t I. ( 1953). ILinc;tr elastic stability. %.4.\/PJ, HY. 167. Zqlcr. t I. ( 1956). On the concept of&stic st;thility. In rlclrtrnc~ in App/id Mrclurnic:~, Vol. IV. p. 35 I. Academic I’rw. Scu Yorh.

;IIIJ circul;ltory forccs. J. ;lpp/. Mdt.

APPENDIX With rcfcrcncc to thccross-sections adopted in the numerical cxamplcs, Figs Al and A? show the (normal&d) w:lrping functions IL,. $, and IL,,,dctcrmincd for thr I section and JI, and $,,, dctcrmincd for the U scclion. Fln;tlly. in 111snumerical cxamplcs the unknown functions were rcprcscntcd as follows:

i(l). tJ(-‘). IN:) =, ~,u”(cos(2~ICj)-I).

VP.(-)* ~P,(-).%w

X,(--j. x,(-I.

Thccc rcprcscnt;ltions

= ~$,qsin(‘~n~).

x,,(z) =. i

“_I

2n-I --

u-sin (

2

arc ahlc to salisfy all the essential boundary

71:

I)

conditions

of rhc cxamplcs considered.

+

+

+

-i

C . \

1 t

i +

+ Y*

*.