THE EQUATIONS OF THE SECOND ORDER LINEAR MODEL OF SURFACE GRIDS

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1 MECHANKA TEORETYCZNA STOSOWANA 2/3 2 (983) THE EQUATIONS OF THE SECOND ORDER LINEAR MODEL OF SURFACE GRIDS PIOTR W I Ś N I A K O W S KI Poltechnka Warszawska. Introducton The paper deals wth the method of fndng the governng equatons for a surface structure havng a form of a dense and regular grd made of bars. The lateral deformaton of elements of the system are taken nto account. It s assumed that the materal of the structure s elastc homogeneous and sotropc. The problem of statcs s analysed wthn the lnear theory. The numercal methods employed to solve the problems related to the consdered systems were based on dscret representaton of the structure (see among others [ 2 3]) and lead to a system of algebrac equatons wth a large number of unknowns. The dmensons of nodes ther deformablty and the lateral deformablty of structure's bars were not taken nto account. The applcaton of a contnuum model of a structure conssts n an approxmaton of the mult connected geometry of the system by a certan smply connected and contnuous model (see among others [4 5 6]). The advantage of the dscussed approach over the prevous one les n the fact that the analytcal methods can be employed. The negatve aspects are: a) consderable naccuracy of results for not suffcently dense grds b) the requred geometrcal symmetry of the structure. An nterestng dea of a contnuum model of such structures based on the concept of a contnuum wth nternal mcrostructure and hgher order nternal reactons s presented n [6]. In the present paper Cz. Woźnak's model wll be appled to obtan equatons of the second order theory. An energetc approach dfferent from the prevously consdered one whch wll be employed makes t possble to descrbe n the explct from all propertes of the contnuum model. As a specal case (n whch the hgher order effects are neglected) equatons of the frst approxmaton wll be obtaned. 2. Basc assumptons It s assumed that the structure consst of (homogeneously) deformable cubcod nodes connected by means of the prsmatc lnks of rectangular cross sectons (and subject to homogeneous deformaton n ther plane) and consttute the regular and orthogonal

2 22 P. WlŚ NIAKOWSKI surface grd made of bars (Fg. ). The lengths of the elements of structure are small as compared wth the lengths of the surface and ts curvature rad. A system of JC x 2 coordnates on the л surface on whch the structure s shaped and a z coordnate n the drecton normal to surface л were chosen n such a way that x' x 2 z axes represent a rght hand system of coordnates. It was assumed that the geometrc centers of the nodes le at ntersectons of parametrc lnes.y' = const x 2 = const z = 0 and that the axes of the lnks concde wth drectons of these parametrc lnes. A typcal segment of such a structure s shown n Fg.. x (x'=const) D JC T~ A t ; г f Щ Щ I lx I к I 4 Д у Д / a v 72 H ' I (xr=const) Fg. We shall ntroduce twelve contnuous suffcently smooth functons defned on the surface л of the structure. These functons represent translatons rotatons deformatons along the coordnate axes and the shape deformatons. The forementoned functons consttute unknown quanttes of the model and have a physcal sense only at the node centres. In every net mesh they can be treated as lnear nature. 3. The analyss of the structure components Node. When a structure s loaded a typcal node s subjected to a homogeneous deformaton havng 2 degrees of freedom. Let (u x u y u : ) be the dsplacements (d x #>. the components of an ndependent vector of rotaton (o) x w y co : ) and (ш х у co X2 oo yz ) the lnear and devatorc components of a homogeneous deformaton respectvely. Denotng by w x w u> dsplacements wthn the node area n drectons X y z respectvely

3 SECOND ORDER SURFACE GRIDS 23 the followng formula hold w x (xyz) = u x co x x (w xy e z )y (& y (o x: )z (3.) w y (xyz) = u y (&. co xy )x co y y (co yz '& x )z w z (xyz) = u z (oj xz & y )x (& x co yz )y(o z z. After applyng the prncple of deal constrants we can arrve at 2 equatons descrbng the node equlbrum wth 6 generalzed nternal forces and 2 generalzed external forces. From the equatons of the lnear theory of elastcty the general consttutve relatons can be obtaned together wth a formula for the stran energy of a node. Lnk. Let us take nto account a typcal element connectng the th and the j th nodes stuated on the x 2 = const parametrc lne (see Fg. ). Let w x w y w. represent dsplacements of the lnk area n drectons of a local coordnates x y z (see Fg. 2). It s assumed that the lateral cross sectons of a lnk are subjected to homogeneous deformatons n ther planes as well as to the rgd dsplacements (9 degrees of freedom). Hence: w x (xyz) = v x (x) ycp z (x)z(p y (x) (3.2) w y (xyz) = v y (x)yy y (x)z y y (x) 9> x (x)j w : (xyz) = v z (x)y Јy yw(»)c)(*)j2y«(») where v x v y v z are dslocatons <p x q> y q> z rotatons y y y. y yz deformatons of the cross secton of an element along the.v coordnate. 'ytvy f P 4 Fg. 2 The state of lnk area dsplacements s descrbed by 9 functons of the varable x beng the Lagrange's generalzed dsplacements. The assumpton (3.2) can be called the hypothess of a flat homogeneously deformable cross secton wth ndependent rotatons. Ths s a generalzaton of the well known hypothess of Bernoull and Tmoshenko for the classcal model of a bar. The constrants for stresses are assumed n the form (3.3) = o 0. Ths assumpton smplfes consderably the formulae gven below. After applyng the prncple of deal constrants of the statc and knematc types we shall obtan 9 equatons 7 Mech. Teorct Stos. 2 3/83

4 24 P. WlŚ NIAKOWSKl descrbng the equlbrum of the lnk wth 0 generalzed nternal forces and 9 generalzed external forces. Takng nto account the known equatons of the lnear elastcty the generalzed consttutve equatons and the formulae defnng the stran energy of a lnk can be found. Node lnk node system. A system consstng of the / th node the j th node and the ( j) th lnk connectng these two nodes s presented below (see Fg. 2). From the equlbrum and consttutve equatons descrbng the lnk the dfferental equatons for the generalzed dsplacements can be obtaned. The knematc boundary condtons result from the assumpton that the dsplacements of the approprate boundares of the / th and v' th nodes have to be competble wth the dsplacements of the sutable boundares of the lnk stuated between them. In ths way we obtan functons v x v y ) (p x <p y q> z y y y z y y whch are expressed as the functons of the / th node and the A operator defned as follows: (3.4) zj() =()' ()' These functons can be understood as certan shape functons of the bar treated as a three dmensonal body. The total elastc energy of the (/ j) th lnk s a functon of parameters attrbuted to the / th node and the A operator defned above. The analogous procedure can be appled to lnk stuated on the x = const parametrc lne and connectng the / th and the A: th nodes. Instead of the A operator we deal now wth the A operator defned as follows: (3.5) ~A{ )=()* ( )' 4. Bovcrnng equatons Accordng to the forementoned assumptons the parameters descrbng the dsplacement deformaton strans and stresses as well as the elastc modul are descrbed by the contnuous suffcently regular functons of arguments x x 2. These functons have a physcal sense only n certan ponts of the surface. The dsplacement state of a structure s defned by 2 parameters for each node. A contnuous suffcently regular extenson of these dscrete functons leads to the relatons: U(x} X 2 ) = u x Ux x 2 ) = Uy u(x\ x 2 ) = u z ^(.x x 2 ) = 9 X 2 {x\ x 2 ) = y {x\ x 2 ) = a> lt (x l x 2 ) = co x ш 22(х x 2 ) = co y w(x l x 2 ) = w 2 (o 2 (x\ x 2 ) = co 2 (x x 2 ) = co xy co^x x 2 ) = co xz a> 2 (x l x 2 ) = co yz. The stran energy of a typcal structure segment (.e. the energy of the ( )) th and the (/ /<) th lnks energy of the th node Д уf energy of the 7 th node of energy of the Ar th node) s related to ABCD surface segment wth / l 2 dmensons (see Fg. ). Ths energy s a functon of parameters assgned to the th node and nvolve A and A operators.

5 SECOND ORDER SURFACE GRIDS 25 Assumng that: (4 2) _ = _ L ( ) = _ L = ( b and that the densty of elastc energy a s equal to the densty of energy a () n the th node the basc relaton of the contnuum model of the consdered structure n ts explct form was found: ao = ^ C^ >»y.kly.mn ~ A^y K y L G^ 'x K L y M F^ y K L r M N D K y K co L H KL»x KL co M R«I M y K r L M l G KLMFL T K Lr MJV I G*4 K r L у F KL o> K to L у Am 2 C KLM co K r LM А к т к(о у A KLMN y KL y MN (4.3) ~ C KL y.k x E KLM K K y LM B K N y KL co MN D««x K r LMN pklm.. JKLMNP _ pklmn. ' r KLMN. у Zl*' LMPJS T KI A T P?s 'y4 KLMP/S T KLM T^jjs C^^^W^LTj^pn f tf"c XL a> D KLM OJ KL т м B K L M T K L M M H K L M N r K L M r N. KL MN P R S = 2. The densty of work of the external forces can be olso defned. The relaton (4.3) was orgnally expressed n the Cartesan coordnate system and then generalzed to a curvlnear orthogonal system of coordnates n terms of whch the surface system s descrbed. The parameters: y KL y K x KL x K r KLM = r KML r KL r K a) KL = co LK co K OJ consttute generalzed components of the state of deformaton wth the geometrc relatons takng the form YKL = L\K b LK 4 e LK & y K = u\ K b I { l L e KL <& L (4.4) T KLM U>LM\K~Ь щ К () b K m M *KL = b>l\c bk(o N L b LK m r K Ы (o\ K 2b%co N where b KL e KL ( ) K represent the components of the second metrc tensor of the surface Rcc's bvector and the symbol of covarant dfferentaton on the surface respectvely. The functons A K L M P R S... C K L M... A K A stand for the tensor of elastc modul of the structure and descrbe ts geometrc and physcal propertes. The components of the stress state of the structure are gven by the formulae

6 26 P. WlŚ NIAKOWSKl (4.5) [cont.] v J s klm 8a... 8а. 8а = = s = s* = 8r KL 8r KL dv 8a K 8a 8a. dft) KŁ ' 8co K ' dco We see that 2 from 30 ntroduced aqove components of the stress state s of the force type (p KL p K r KL r K r) and the remanng 8 s of the couple type (m KL m K s KLM s KL s K ). Wth the ad of the prncple of vrtual work the equlbrum equatons and the boundary condtons for the contnuum model of the structure can be obtaned n the form p KL \K b L Kp K q L = 0 p K \ K b LK p KL q = 0 m KL \ K bkm K e' I tp K h L = 0 m K \ K e KL p KL b LK m KL h = 0 (4.6) s KLM \ K ~ j(b L Ks KM b^s KL ) r LM f LM = 0 s KL \ K 2b MKS KLM 2b L Ks K r L f L = 0 s*\ K b LK s KL rf = 0. (4.7) p KL n K p L 0 o r "_*=«_; Р к п к р = 0 or u = u * * * m KL n K m L = 0 or # L = L ; m K n K m = 0 or SKL U K_*LM Ш o o r W ł m = 4 A (; sr^n F s Ł = 0 or s K «K * = 0 or w = *J where «K represents the components of a unt normal vector to the boundary 8Q of the structure q K q h K hj' KL f K f are the denstes of the surface type external stress; p pm K tn s KL s K s are the denstes of boundary stresses u L u # Ł co LS OJ LOJ are the gven values of generalzed dsplacements wthn the 8Q. The equlbrum equatons (4.6) and the boundary condtons (4.7) together wth the consttutve (4.5) and geometrc (4.4) relatons form the basc system of equatons descrbng the contnuous model of the structure. Ths system enables us to calculate the dsplacement dstrbuton n the lnk and node areas as well as the stress dstrbuton. It must be stressed that parameters y K y.kl r KL K (ol [see (4.4)] defne the components of the plate lke deformaton state whle y K L co KL x K r KLM the components of the plane lke deformaton state. The components yxl. w *. *к» _ L m a r e defned exactly as n the rst order model (see [6]) however the parameters whch do not appear n that model.e. T K L M T K L T K <X>kl cok ш result from the deformablty of a node and the deformablty of lnk's lateral cross sectons. The analyss of the nfluence of the second order parameters on the nternal forces together wth the sutable numercal calculaton wll be the subject of separate papers.

7 SECOND ORDER SURFACE GRIDS 27 References. W. NOWACKI Problems related to the theory of flat grds. Arch. Mech. Stos (n Polsh) 2. W. GUTKOWSKI Grd surface structures. Mech. Teor. Stos (n Polsh) 3. H. FRĄ CKIEWICZ Deformaton of a dscrete set of ponts. Arch. Mech. Stos M.T. HUBER Probleme der Statk technsch wehlger orthotroper Platten Zurch C. WOŹ NIAK Introducton to mechancs of fbrous meda Arch. Mech. Stos С. WOŹ NIAK Lattce type surface structures PWN Warsaw 970. (n Polsh). Р е з ю ме У Р А В Н Е НЯ И Л И Н Е Й Н Й О Т Е О Р И В Т О Р О О Г Р Я ДА У П Р У Г Х И П О В Е Р Х Н О С Т Х Н Ы Р О С Т В О РК О В р а б ое т п р е д с т а в о л ел ни н е й не ы у р а в н е я н ис т а т ии к у п р у гх и п о в е р х н о с х т нр ыа с т в о к р ои м е ю щ их п л о т ню у и р е г у л я рю н ус е т ку э л е м с о е д и н е н е н мы е ж ду с о б й о р пи п о м о щи п р и з м а т и ч ех с кс ти е р ж ч е н и. е е н т окво т о р х ы д е ф о р м и р о в а н нп ыр яе м о у г о л е ь нуыз лы нй е и м е ю щ х и п р я м о у г о л е ь сн ое П р и н и мя а и с х о д не ы д а н н ы: еу р а в н е я н ил и н е й нй о т е о р и и у п р у г о с т а и т а к же п о д х о д я е щ и к и н е м а т и ч е сгк ии п о т ы е зп о л у ч о е нв а р и а ц и о нм н мы е т о дм о у р а в н е я н ис п л о ш Р а б оа т с о д е р жт ио б о б щ в о л я ю ще иу ч и т ы е е н ит е о р и и В р о в а не и п о п е р е ч но осге ч е ня и с т е р ж н о о мг о д е я л п р о г о н. а о з н я к ав ы х о д я ще из а п р е д еы л т е о р и и г о р я д а п о з в ь а эт ф ф е кы т в ы с ш х и р я д о " в (р а з м еы р у з л о в и х д е ф о р м и р о в а нд ие еф о р м и нй ес о е д и н я ю х щ уи з л ы. ) Streszczene RÓWNANIA LINIOWEJ TEORII DRUGIEGO RZĘ DU SPRĘ Ż YSTYC H POWIERZCHNIOWYCH RUSZTÓW W pracy wyprowadzono lnowe równana statyk sprę ż ystyc h rusztów powerzchnowych o gę stej regularnej satce elementów których odksztalcalne prostopadloś cenne wę zły połą czone są za pomocą pryzmatycznych prę tów o przekroju prostoką tnym. Przyjmując za punkt wyjś ca równana lnowej teor sprę ż ystoś coraz zakładając odpowedne hpotezy knematyczne otrzymano na drodze waracyjnej równana cą głego modelu dź wgara. Praca zawera uogólnene teor Woź naka wykraczają ce poza teorę I go rzę du zezwalają ce na uwzglę dnene efektów wyż szych rzę dów" (wymary wę złów ch odkształcalnoś ć odksztalcalność przekrojów poprzecznych prę tów łą czą cyc h wę zły). Praca została złoż ona w Redakcj dna 3 marca 983 roku