A hybrid formulation for determining torsion- and warpingconstantsCitation for published version (APA):Menken, C. M. (1987). A hybrid formulation for determining torsion- and warping constants. (DCT rapporten; Vol.1987.079). Technische Universiteit Eindhoven.

Document status and date:Published: 01/01/1987

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A HYBRID FORMULATION FOR DETERMINING

TORSION- AND WARPING CONSTANTS

(I.M. Menken

Eindhoven Unj.versj.t.y of Technology I I?. O. Box 5 13 I

5600 #B Eindhoven, The Netherlands

Abstract- Theoretically, i.he t.orsion const.ant. of a bar subjecteed t.o a torque can he calculated irrespective of i t s cross-sectional shape; eiL1ie.r w i t h a potent.ia.2 energy (displacement 1 or wj.t.h a complement.ary energy ( sixes5 func t ion) formu lat5.m. The potential energy formulation, however , provides the so-called warping constani., t.oo, which may be important. i n the case of thin-wal l ed sections , but the complementary energy formulation does not:. on the ot.her hand , t.he complementary energy formula t.ion leads t.o simple ana1yt:ica.l expressions forr the torsion constant of thin-walled sections , while a f i n i t e . element cal cu.lai.j on based on pot.ent.j.al energy requires a re la t ive ly Large number: o f elements to obtain comparable results.

A general fini.t.e element formu1at.j on, however, should provide all torsi.ona1. constants w i t l i one algoritlm and a small number of elements for a l l k i n d s of cross-section, a1.bei.t. solid or thin-walled, open or closed. Since i1ybr:j.d formulations often give more accurate results and combine some potential energy a.; wel 1 as complementary energy feat.ures, a hybrid variat:j.onat formulation was derived for the torsion problem. T h i s was done i.n siich a way that both the torsion- and warpjng conritmts could he calc!~la!:ed consi:akeni:ly w i t h t:his formulation and coiild be followed by the usua l one dimensional response calculation for a bar subjected t o a torque. Tiie fj.,:j t: fi.ni.t:e elemttni. calciil.ai:j.on for rectangular cross-sections resul Led i n an improved warping constant.

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NOTATION

a

a c

A

Ae

b C

c . J j k l

G

- H It: k

- k I.

L lul

n n y' z N - 8 N 0

.. -

Ni 'i

'i -

- P Lq -1

Q L

S

SC? SC?

- T P

xy ' t.xz t: * * xy ' tXz t.

half w i d t h of rectangle

column representing cross sectional coordinat.es in t.he functional, see eq. (3.14) cross-sectional area of tiie bar: cross sectional area of t.he e-t:h element. half height of rectangle constanl. , pol a r moment. of inertia component:: of tiie elas t i c compliance tensor nonzero components af t.he st.rain t.ensor shear modulus f l e x i b i l i t y matrix, sec? eq.(3.6) torsion constant constant i n J s t i f fness matrix length of bar shape function.; for 41 on the element- boundary prescribed t.orqiie components of outer normal column w i t h shape funct.jons for fl column w i t h shape funct:jons for 11 shape function for t.he i-t.h coordinat.e i -t.h componenl. of boundary t.ract.icm

t

prescri bed boundary t.raction matrix, see eqs. (3 .2) and (3.4) Yamada displacement. vector [ i ó j column w i t h nodal values of IJ coliimn w i t h generalized loads coordinate along element. boundary bounda ry of AC? porticin of the boundary of f where t.he t.ract.j.ons are prescribed ma t.rix , see eq. ( 3 . 1 2 ) nonzero components of t.he stress tensor

Lagranye multi pl iers

-4-

U I vr w displacements U.

V"

ave W . nodal a x i a l displacement X I Y r s p a t i a l coordj.nates

i - t h component of the displacement.

volume of the e-th element. boundary of the e-t.h element

3.

3.

a twist.

coliimn wi t .h nodal values o f l.he stress function warping constant dimensionless coordina t.es angle o f t.wist potential energy functional Hii-Washizu func lkm1 Hellj nger-Reissner funct.iona1 h y h r j d f unctimna 1 coliimn with non-zero shear stxesses sixer;.; function nodal value of + ( y , z ) waxpi.ng functj on

see eg. ( 3 . 3 9 )

Subscripts

i t j t k , l

X 1 Y r Z components w i t h respect tn the x,y or z-tli coordinate components w i t h respect t.cl 1.he i, j k or 1-t.h coordinat.e

Superscripts e number: a t eiiement

(i 1 i - t h smuoth side of element. T tran:ipo:id

1 . INTRODUCTION

The general linear theory o f t.he iinj.furm t,orsion o f prjsmatic bars was

well-íormulated by the phneers of mechanical science, de Saint-Venank [ 11 and Prandt.1 [ % I , according t o the t.hen main approaches t.o solid mechanics; an ultimate focmu1ai.j on j.n ed t.lier displacements or stresses. A common featiire was i.hat a t.riily three-dimensional problem was s p l i t . into a t w o - rìimensi.ona1 problem and a one-dimensional problem w i t h the a i d of assumptions siich as r i g i d i t y of the cross-sectkm i n i t s plane. The t w o - dimensional problem includes the determination of geometric cross-sectional propert.ie., such as the de Saint-Venanl. t.orsi.on constant and t.he so-called warping cun:j tant., while tilie one-dimensional problem is the response problem of the remaining one-dimensional model of t.he har. Analyi.j.ca1 solutions were restr icted t:o bat:: w j t:h simple cross-sections. The introduction o f the computer i t«gether wi th the advent. of t.he finj.t.e clemeni. rnet.hod enabled the geomet,rj.c con:i{.ani.:i 1.0 be ca1 culated for cross-sections w i t h complicated shape.;, parti ciilarl y 2.hoi;e of extruded rods [RI. Alt.hough i n analytical calcu1at.i on.5 both Lhe di :jplacement approach and the stress approach were iised (with preference for t.he stress approach i n t.he case of t-hin-walled 5 e c t i o n : j ) , i n compukr calcuht:j.on:j, the displacement approach prevails.

Fireqiientl y the tursion problem was treated i n pre-computer t,exthnoks [ 4 ] , Es], [ G I I rare1.y do piibIj.cat.iuns mention f i n i t e element calculations.

I’erhifpc, this is diie t o the facl. t.hat. such ca lcula thns require the soltit i on o f t.he plane Ttaplace equat,i.on, a beloved subject of introductory books about f in i t e element niethods.

From one of tiie earliest. publications [ T I , tiowever, it: appears that an accwate dekermi n2t.i on of tiie torsion constant. requires a relat.ively large number of elemen1,:j. The same applies t o tfie warping constant;. More striking is t h e observatj o n that, for some simp1 e 1.hj.n-walled cross sections ( e . g . the narrow rectangl e and tiie slotted tube) where tiie analytical stress approach yrwi de:; ai,mpl e ayproximat.ionS for t.he de Saint.-Venant. t-orsion cons bant. I the ti isplacement based f in i t e element approach, even w i t h 8-node isoparametrlc elements, requires a re la t ive ly large number of elements [ 31.

T h i s prompted us to develop a hybrid torsion formulation, since liyhrid fc)rmulal.ions oft.en perform more accurat.ely [ 8 ] . For a consi st.ent. formuiat-ion

of tiie tor:iion- and warping constank, i t s development; cannot be based on a

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simple transformation of the plane Laplace problem, as suggested by Martin

[9], hut i n j t i a l ].y, recourse miist be. made t o t.he f u l l three-dimensional problem.

As far as we know, only a few hybrid t:orsj.on formulations have been published :

Yaniada et. al Cl01 djvided t.he bar under tmrsion i n t o longitmlinal prisms w i 2 l h a triangular cross section. Their element displacement vector;

p j - €I w w w 1 1 2. 3 j contains, apart. from the t.hree ax ia l displacements w. which tiie whole cross-section rotates; consequently, i t is common to a l l the elements. Since t h e angle of rotat.jon 0 and t.he nodal displacements w calculated simultaneously, the procedure does not comply with beam- and bar theory where geometric constants have i-.o he det.ermined separa t.ely and subsequently are used t o calculate t.he angle of rot.at.i.on 8 . Moreover, an older hybrid formiilat;.i.on was used that requireti a priori fulfilment of t he dynamj c boundary condit..i.ons.

t.he angle 0 over 1 '

are i

Xn a more recent paper by Tra l l j . [ 11 J khat, i s rest.ri.ct.ed t:o t h i n - walled cectj.ctns, the Hellinger-Reissner principle [ 121 was used, The secondary warping of t.he jndjvidual components was taken into account: by the a priori known torsional ri .y. idity of the plate-like components. Tiie predominant primary torsional r j yj.di1.y due t.o eit.her closed p a r t s ( c e l l s ) or restrained warphg j.;; 1ij.tiden i n the assembled stif fness matrix. As in tiie aforementioned paper, the angle of xoi.at.i.on of t.he cross-sect.ion as a whole appear:$ j.n t:lie dj.spJ.acement vector of each individual (har-) element. Again, a x i a l displacements and l.he angle of rot.atj.on are calculated simult-aneously and t;Jie txxsion- anü warping constants are not obtained exp l i c i t l y (!i merit of Tra l l j ' s work, however, is that. i t . can deal w j t.h varjations of the cross- section along the length of tiie beam).

I n the following section a hybrid formulat.j.on for arbitmry cross- :sections i.s presented which yields tiie torsional constants e x p l i c i t l y .

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2 . DERIVATION OF THE HYDRXD FUNCTIONAL

There are several o p t i o n s for derivj.ng t.he h y b r i d functional of uniform torsion.

One option i.s 1.0 introduce the c lassical assumpt.i ons of t.orsion theory into tiie general iiybr id f unctionaf that was originally derived by Pian [ 131 , 1751;

( 2 . 1 ) 1 “mc [ i i j k 1 i-J kl 1 1. - - c -1.. . t av - Jepiuias t Je p . u . a s V av S

P

Where: ave i s t.he ent.i.re hoiindary of the e-th element having volume V”. i n this functional, tiie displacements u only appear i n the boundary integral i and c a n be viewed as Lagrange multj.plj.ers tiiat have been introduced exclusi vely on the element. boundary i n order t.o relax the conti nu3 t.y requirements for the inter-element tractions p i ’ element models, such as plate elements, this formulation has proved to he quj t.e advant.ageous, because for the boundary displacements simple functions of the boiindaxy coordinates can be chosen independently of tiie imp1j.ci.t internal displacements.

I n a number of f in i t e

One objective here, however , was t.o calculate t.he warpj ng const.ant. r using tiie c lassical formula:

Where: A i.5 t.he cross-sectional area of t.he bar under t.orsj.on, while the so- called warping $ ( y , z ) represents t.he ax ia l cfi.splacemen1.s per u n i t of twist. T k coordhates y and z are located i n the cross-section. T h u s , tiie dj spiacements have 1.0 l-,e defined over the whole cross-sect.j.on and not. only along the element boiindaries .

Thai was why another way of constructjng t.he hybrid formulai-ion had t o b e cho:jen. It: started w i t h the potential energy formulation, adapted to tiie problem of uni farm t.orsj on, then, proceeded t:o the hybr id formulation along tiie line o f the so-called Friedrich transformation. Apart from making some feat.ures of the ultimate formulation clearer, it. shows that . t.he (warping-)

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displacements occuring exclusively i n the boundary integrals , represent the dj splacements original.ly defined w i t h i n l.he ent.j.re element.

e

Fig. 1 . Twisked bar and coordinates

Consider a homogeneous and isotropic prismat.ic bar of 1engt.h 1 (Fig. 1 ) and introduce Cartesian coordinates i n such a way that the x-axis passes through l.he centroids of the cross-sections. Since det.erminat.ic)n of bchh t.he Saint-Venant torsion constant and the warping constant are based on the situation of uniform torsion, orient.at.ion of t.he y- and z-axis witeh respect to e.q. principal iìXe:j is irrelevant. Moreover, any longitudinal f iber can he chosen a s t.he axi.s of rotation, i.n t.his case, t.he x-axis was chosen. The displacements i n the x, y and z directions are denoted by u, v and w respectively, and the twist per u n i t length j.s « ( x ) . When scilving the torsion problem, de Saint-Venant assumed that the ax ia l displacement u was prcrportjonal t.o t.he consimt twist o: and dependent. on t.he coordinates y and 2:

( 2 . 3 )

-9-

where : IJ i s t.he unknown warpj.ng f unct.j.on. The potentia 1 energy formula t3 on requires J1 t o he continuous. Diie t o the assumption t.hat. a cross-sect.ion does not deform i n its plane, the other displacements become:

v = - u x z

( 2 . 4 ) w = u x y

lmplicjt. far these relatj.on:; i s the fact. that. rotat.jan ai: x = O has heen

suppressed. As a consequence to ( 2 . 3 ) and (3,. 4 ) , the non-vanishiny strains are :

A subscript preceded by a comma denot.es pari.j.al d i f ferent.iat.ion wi t.h respect t o the independent: variable indicated by the subscript. The per1.j nent shear siresses are:

.[- = 2 G e X Y XY

( 2 . 6 )

= 2 Ge t:xz xz

where: CI i s the shear modulus of t.he mater ia l . The stresses should s a t i s f y the local equilibrium equation :

i i. = o xy,y X % , % t

For a har which i:; free t o warp when loaded wj.t.h a t-wj.st.ing moment

M at. x = 1, i.he potential energy i s : -

( 2 . 7 )

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.If t he bar i s subdivided i n t o l o n g i t u d i n a l p r i sma t i c prisms over i t s whole

l e n g t h , i.he e1as t j . c energy becomes t h e sum o f t.he e l a s t . i c energy o f t.he

indivi.dua1 pr ism:: :

e where : V con ta ins the element: e . The summatj.on include:: a l l tlie e lements . Zn uniform t .ors ion, t.he t.w.ist. u i.s constant. ; t h e r e f o r e , t.he cont.i.nuj.t.y requirement. f o r t.he axja3 displacement ii(y, z ) (2.3) j.mplies t.hat. t.he warping f u n c t i o n $ ( y , z ) must: he cont inuous over i n t e r - e l emen t boundaries . Comyati.bj.lity i n t h e plane

of t.he c r o s s - s e c t i o n is giiarant.eed by equat ions (2.4). The s t r e s s - h y b r i d formula t ion r equ i r ed can be der ived by us ing the gene ra l t r ans fo rma ik in scheme desc r ibed by Washj mi [: 141 and Pi an [ 151.

is t.he volume of t.he 1ongi.t.udj n a l prism whose c ros s - sec t ion

I n the f i r s t s t e p , subsidiary cond i t ions ( 2 . 5 ) are in t roduced i n the * * f u n c t i o n a l ( 2 . 9 ) by mean:; o f Lagrange miil.ti.pliers t and txz:

XY

t t.* {e - - 1 u($,, t y)}]dV] - ti xz xz 2

has t:o be st .al . ionary ’m1 Jn t.he second s t e p , strains, t h i s l e a d s t o the relati .on:j :

* i. = 2G e

X Z Xi!

(2.10)

wj.t.h respect . t.o t.he

(2 .11 )

Comparing (2 .11) wi th ( 2 . 6 ) shows that . t.he m u l t i p l i e r s can be iden1, i f ied

w i . t h the shea r ing stresses. Replac.hg Lhe m u l t i p l i e r s by t h e s e st.cesses and adopt ing Hooke‘s law ( 2 . 6 ) i n order t.

-11-

( 2 . 1 2 )

The t .h i rd s t ep requj res p a r t i a l i .n teyra l . i on i n order t o generat.e t.he l o c a l e q u i l i b r i u m equakion:

2 x y , y + X Z , Z

- -(t2 -C t - a( i : 'mR1

-12-

The last; requj.rement; c a n be met by i n t r o d u c i n g the scalar stress f u n c t i o n

+ ( y , z ) together w i t h t h e prescr j .p t . ion:

and t. = - + ' Z XZ

t = XY

( 2 . 1 6 )

The re la t . i .ons which make llmc s t a t i o n a r y wj.1:h respect. t.o all. t h e admissi h l e v a r i a t i o n s are:

1 . compat . jhi l i1 .y withj .n each prkm, 2 . a cont inuous stress component p between the prisms, 3 , p

X = O on t;he stress-fïee 1ongi tudi .na l face of tiie bar as a whole,

X

4 . t.he n a t u r a l boundary condii-.j.cm at. x = 1 :

tx,y) dA ( 2 . 1 7 )

Apart. from it.s s i g n , f i inct . iona1 ( 2 . 1 5 ) represent .s a modified somplement.ary energy func t j c ina l w j t.h c o n t i n u i t y o f p on t.he l o n y i t u d j n a i faces of t.he prisms and, on t.he t . r a n s v e r s e e n d f a c e , t h e n a t u r a l boundary c o n d i t i o n ( 2 . 1 7 ) relaxed by rot .at . ion u l .

relaxed by a x i a l d i s p l a c e m e n t s cx$ ( s ) X

it is worth n o t i n g t h a t , i n some hybrid formula t ions tiie stress cont . inu i t y i s r e l a x e d by j n i r o d u c j ng boundary di sp1acement.s t.hal. are o n l y deíined on tiie e lement boundar ies b u t are independent of tiie i n t e r n a l d i s p l a c e m e n t s [ 1 5 ] . However, t.he warping f u n c t i o n J ) ( s ) I alt.hoiigh it. o n l y appears i.n the contour i - n t e g r a l i.n (2 .13) r e p r e s e n t s a cont inuous warping f u n c t i o n $ ( y , z ) t h a t i s d e f i n e d over t h e whole area A, a s shown by the Ùerj va Li on cif ii with l.he classical formiila :

7 4 c m thus he used !.o c a l c i l a t e t.he warging cons tan t . r mC *

( 2 . 1 8 )

Alt.hough t . h i s formula complies wit.h t h e pot.ent.ia1 energy formulat . ian, it* may also be used w i t h tiie hybrid formulat ixm s i n c e , i€ a term a c c o u n t i n g for tiie s t r a i n e n e r g y due t.o non-uni form t . o r s i o n i s added t.o t.he pot .ent . ia1 e n e r g y ( 2 . 9 1 , this term w i l l remain u n a f f e c t e d by tiie t r a n s f o r m a t i o n to kfie tiybrid

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formulation. As this term conkaim ( 2 . 1 8 1 , tiie c lass ica l formula w i l l a lso

remaj n unaffected - i n l.he following section, t.he hybrid formulat.j.on w i l l be used to

construct a stress-hybrid fini.te element: model. After assembling the elemeni.s, i t w i l l become clear t h a t determining t.he t.orsion- and warping constanLs can he performed separakly and w i l l precede calculating the twist: a a s a funct.j.an of t.orqiie h , 1j.ke f o r t.he c lassical djsplacement. formulation.

3 . DISCRETISATION OF THE FUNCTIONAL

The f in i t e element discretisation proceeds along the general lines

given by P i a n [!I]. Since t.he aim was t o generate a formulat.jon where t.he determination of torsj.ona1 constants could be d i s t.hgiiished from tiie one- djmensional C.orsion problem; t.he end rnt.al.j.an al and t.he shear modulus G w i l l be l e f t out of the brackets, when formulating matrices.

We start. by assuming a stress function dist.rihut.ion:

(3.1)

Where column f.l contains dj.screte valiies of l.he element. st.ress funct.ion. Cont inui t y of CP between elements is not. a requirement. Due tm the boundary jntegral appearing i n ( 2 . 1 3 1 , nodal valiies of CP may be advant.ageous; i n which case tiie so-called shape functions could be taken as coiuponents of N The pertinent; stresses can now be expressed as:

4

UB*

where: u contains t.he relevant stresses: 5

( 3 . 2 )

m ( 3 . 3 )

while matrix is defined as:

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- P = ( 3 . 4 )

The f a c t t.hai. l.he st.resses are related t.o the derjva1:j.ves of t.he stress function w i l l affect: ( 3 . 1 ) :

Xf a polynomjal shape i s chosen for ( 3 . 1 ) I the constant. value of O can be omitted.

If a descrj.pi.jon j.s chosen j.n nodal values of 41, column $ should - contain the appropriate d i f fecences between these nodal values ; whj.le N( y r z ) should be adapted accordingly. SubsI;ituting ( 3 - 2 ) into the f i r s t kerm of (2.13) gives:

c

where

w i t h

(3.5)

(3.6)

(3.7)

The second term of (2 . 13 ) represents t h e cont.ribut.ion of t.he prismatic faces 1.0

jnciividiial face. Wit.h the a i d of ( 2 . 1 4 ) ' px can be expressed along each smooth s i d e i as:

p . u . d S of ( 2 . 1 ) . lt. contains t.he a x i a l shear component p on each 1 1 1 X

( 3 . 8 )

According t c ) the aim, the warping $ ( y , % ) w i l l be defined over t.he whole elemental area as:

Thj.s warping di st.ribution appears on the i-t,h element. boundary as :

(3.9)

(3.10)

-15-

( 3 . 1 1 )

where (3 . 12)

The t h i r d term of (2. 13) represents l.he conl-rj.hut.ion t.o t.he term p. u. clC of ( 2 . 1 ) oí tiie cross-section a t x = i wìiicfi ~ i a s a surface A". Diie t o tiie assumption of t.he x i g j ù i t y of t.he cross-secticin i n i ts plane, t.he in-plane displacements v and w are restrained by ( 2 . 4 ) . Together wi.t:Ii the stress dist.rj.biit.ion ( 3 . 2 ) , L h i s gives:

I I L

where : A"

(3 .13)

(3 - 14)

The i n d i v i d u a l terms o f functional ( 2 . 1 5 ) have now heen dkcret-esized SD that the tokal matrix representation becomes :

(3.15)

For l.he traction y elemental boundaries, nor a r e t.here reqi1iremeni.s at. t.he t.ractj on-free 1ongjt.iid.inal facea o f the bar. Thiir;, a i l t h e stress parameters of fl can be varied ai: element 1.eveI.. T f kiie fiinct:i.on ( 3 . 1 5 ) is reqiii.red t o be stationary w i t h respect t.o i.he variations of p , t.liis yi.elds for each element.:

P.here is neither a coni3.nuj.t.y requj.rement. at- the inter- x i

.. c

and since i s not: singular:

( 3 . 1 6 )

( 3 . 1 7 )

- 16-

T h k express j .on js used t o e1imj.nat.e 1.he stress parameters of fl from (3.15) : 5

where :

T Q = T a .. 5 1 T - 1 C = , a H a L.. c

( 3 . 1 8 )

( 3 . 1 9 )

(3.20)

(3 .21)

Expressi .on ( 3 . 1 8 ) has t h e same appearance a s the common d i s p l a c e m e n t f o r m u l a t i o n . I n this special ca:je, the disp lacement vector q o n l y c o n t a i n s unknown n o d a l warping val lies. It. does not: c o n t a i n t.he t.wist. u ! Cont.ribut,ions

c

to the s t r a i n energy o f d i s p l a c e m e n t s i n tiie c r o s s - s e c t i o n a l p l a n e are T c o n t a i n e d i.n C; whereas, q Q r e p r e s e n t s t.he c o u p l h g between t.he warping-

and in -p lane displacements. Again a p l a n e problem has t.o be solved i n order - -

t o f i n d tiie unknown warping valiles . Comparing ( 3 . 1 8 ) w i ti1 the po t en t i a l energy e x p r e s s i o n for a iinj formly t.wj.st.ed b a r ;

t h i s shows that. l.he t o r s i o n c o n s i m t I c a n he c a l c u l a t e d wi.1:h: t

( 3 . 2 2 )

(3.23)

Moreover, t.he warpj ng cons tan l . f' ( 2 . 1 8 ) c a n now he c a î c u l a t . e d , because t h e noda l warpi.ng v a l u e s r e p r e s e n t a cont inuous warping d i s t ; r i h u t i o n ( 3 . 9 ) over t h e e n t i . r e c rc t ss - sec t ion .

-17-

Exam P 1 e :j

I n order t.c) gj.ve a f j . rst . i.mpression of i.he hybrid formulat-ion i n

operaLion, a simple rectangular element was derived w i t h four nodes.

F i g . 2 . Rectangular element

A hi-l ineal: distcubution was chosen f o r both the skress- and warping

f unct.j.cins :

Unlj.ke t:he usiial element fo.rmulat.ions, t.he g l o b a l mordinat.es y and z

undeformable in i t s plane. They were t.reat.ed i n t.he usual way:

-18-

13 cv

_t

where: y . and z , are t h e c o o x d i n a k s of the i - t h node. In o r d e r t o avoj.d

singiilari.1.y of ma t r ix H ( 3 . 6 ) , the column p ( 3 . 2 ) was d e f i n e d as: 3 3

..

- m

-Tñ A= va --

The results obt.ained for t.he t.or:;j.on int .egra1 I factor: k, def ined i n :

could he expressed wi th a i

a

of

These rexul2.s were compared t.hen wi th l.he r e s u l t s ob ta ined from t h e classical compatj.ble Eormulat;i.on based on a bi-lj.neax warping f o r each

element.; a l s o , wi1.h Llie exact va lues of k i n Table 1 .

-19-

2x2

4x4

6x6

2x10

5x10

5x1 5

4

16

36

20 50

65

nxm

.27 3 -

.2?2 -

number of

- 272

I eimenix k

compa - t i b l e I h y b r i d .148

.142

. 1 4 1

-315

.314 -

- .139

. 1 4 0

. 2 9 4 -

.309

exact

. 1 4 1

-312

The resulis obtaj.ned for t h e warping c o n s t a n t are gj.ven i n Table 11.

nxm

2x3.

4x4

6x6

8x8

12x12

12x18

2x10

5x10

5x15

4x3s

niim h e r Of

ei ement.s 4

16

36

64

4 4

196

20

50

75

140

-764 -

. 630

.616 * 553 -

-313

.275 -

.27 2

1 Ö h r

.551

-20-

Concltis ions

A h y b r i d formulai.ion for calculat.ing l.he t-orsion- and warping constants fo r prismatic bar:j by means of f in i t e elements, can be derived. i n the c lass j cal formulation, t.hj.s requires a f in i t e element solut.ion of t.he t w o dimensional Laplace equati.on. Wlien deriving the hybrid formula tion , however, t h e f u l l three dimensional model must. be considered. Separating t.he expression f o r tiie torsion constant from the response of the bar to torsion j s bui. possihl e j n the discretesi zed formul a1.j.m. A procedure for l.he f i n i t e element calculation of the torsion constant, that is consistent w i t h tiie hybrid formiilation, could he deve1.ope.d. The. c lassjxal expression for the warping constant, which is relevant i n cases of non-uniform torsion, i s st.i 11 valid i n the hybrid formiilat.i.on. Calculat-ing t.his constant: requires the warping to be known al: every point: on the cross-section. I n the funct j onal pertaining t.0 Lhe hybrj d formu1at.j on (2 .15) t.he warpj ng only appears on tfie element; boundaries. Tiie der.i.vakion of this functlonal, however, shows k h a t this warping of 1.he element. boundaries represents a warping that was defined originally at; every poink of the cross-section and 1.hus can be used f c ~ calculating l.he warping const.an1. consist.ent.ly. A few simple examples showed that tiie torsion constant: calculated with tiie hybrid method was l i t t l e better than that calcula1.ed by the c lass ica l method . The warping cons k a n t , however , was more accurate.

Acknowledgements : The author i s gra t.ef u1 t.o W - Groot , H. Ceeverens , J - v . Vroonlioven and I,. J . W . Hulsen for t1iej.r conkribution t o this papel:. T h i s work was part. of a research prcigram on t.he latexal-t.orsj onal hcklj.ng of ext.ruded L J C ~ S S financed by The Net!ieJ:€ands Technology Foudation (STW) . 1- .- -

-21-

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-22-

T. H , H. P i a n , F i n i t e e lement methods by v a r i a t i o n a l p r i n c i p l e s w i t h

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