Module 6: FEM for Plates and Shells Lecture 6: Finite Element Analysis of Shell

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1 Module 6: FEM for Plates and Shells Lecture 6: Fnte Element Analyss of Shell Introducton A shell s a curved surface, whch by vrtue of ther shape can wthstand both membrane and bendng forces. A shell structure can take hgher loads f, membrane stresses are predomnant, whch s prmarly caused due to n-plane forces (plane stress condton). However, localed bendng stresses wll appear near load concentratons or geometrc dscontnutes. he shells are analogous to cable or arch structure dependng on whether the shell ressts tensle or, compressve stresses respectvely. Few advantages usng shell elements are gven below.. Hgher load carryng capacty. Lesser thckness and hence lesser dead load 3. Lesser support requrement 4. Larger useful space 5. Hgher aesthetc value. he example of shell structures ncludes large-span roof, coolng towers, ppng system, pressure vessel, arcraft fuselage, rockets, water tank, arch dams, and many more. Even n the feld of bomechancs, shell elements are used for analyss of skull, Crustaceans shape, red blood cells, etc Classfcaton of Shells Shell may be classfed wth several alternatves. Dependng upon deflecton n transverse drecton due to transverse shear force per unt length, the shell can be classfed nto structurally thn or thck shell. Further, dependng upon the thckness of the shell n comparson to the rad of curvature of the md surface, the shell s referred to as geometrcally thn or thck shell. ypcally, f thckness to rad of curvature s less than 0.05, then the shell can be assumed as a thn shell. For most of the engneerng applcaton the thckness of shell remans wthn 0.00 to 0.05 and treated as thn shell Assumptons for hn Shell heory hn shell theores are bascally based on Love-Krchoff assumptons as follows.. As the shell deforms, the normal to the un-deformed mddle surface reman straght and normal to the deformed mddle surface undergo no extenson..e., all stran components n the drecton of the normal to the mddle surface s ero.. he transverse normal stress s neglected. hus, above assumptons reduce the three dmensonal problems nto two dmensonal Overvew of Shell Fnte Elements Many approaches exst for dervng shell fnte elements, such as, flat shell element, curved shell element, sold shell element and degenerated shell element. hese are dscussed brefly bellow.

2 39 (a) Flat shell element he geometry of these types of elements s assumed as flat. he curved geometry of shell s obtaned by assemblng number of flat elements. hese elements are based on combnaton of membrane element and bendng element that enforced Krchoff s hypothess. It s mportant to note that the couplng of membrane and bendng effects due to curvature of the shell s absent n the nteror of the ndvdual elements. (b) Curved shell element Curved shell elements are symmetrcal about an axs of rotaton. As n case of axsymmetrc plate elements, membrane forces for these elements are represented wth respected to merdan drecton as,, and n crcumferental drectons as,,. However, the dffcultes assocated wth these elements ncludes, dffculty n descrbng geometry and achevng nter-elemental compatblty. Also, the satsfacton of rgd body modes of behavour s acute n curved shell elements. (c) Sold shell element hough, use of 3D sold element s another opton for analyss of shell structure, dealng wth too many degrees of freedom makes t uneconomc n terms of computaton tme. Further, due to small thckness of shell element, the stran normal to the md surface s assocated wth very large stffness coeffcents and thus makes the equatons ll condtoned. (d) Degenerated shell elements Here, elements are derved by degeneratng a 3D sold element nto a shell surface element, by deletng the ntermedate nodes n the thckness drecton and then by projectng the nodes on each surface to the md surface as shown n Fg (a) 3D sold element (b) Degenerated Shell element 6.6. Degeneraton of 3D element

3 40 hs approach has the advantage of beng ndependent of any partcular shell theory. hs approach can be used to formulate a general shell element for geometrc and materal nonlnear analyss. Such element has been employed very successfully when used wth 9 or, n partcular, 6 nodes. However, the 6- node element s qute expensve n computaton. In a degenerated shell model, the numbers of unknowns present are fve per node (three md-surface dsplacements and two drector rotatons). Moderately thck shells can be analysed usng such elements. However, selectve and reduced ntegraton technques are necessary to use due to shear lockng effects n case of thn shells. he assumptons for degenerated shell are smlar to the Ressner-Mndln assumptons Fnte Element Formulaton of a Degenerated Shell Let consder a degenerated shell element, obtaned by degeneratng 3D sold element. he degenerated shell element as shown n Fg 6.6.(b) has eght nodes, for whch the analyss s carred out. Let, are the natural coordnates n the md surface. And ς s the natural coordnate along thckness drecton. he shape functons of a two dmensonal eght node soparametrc element are: ( -x)( -h)( -x-h- ) ( + x)( -x)( -h) N N5 4 ( + x)( -h)( x-h- ) ( + x)( + h)( -h) N N6 4 (6.6.) ( + x)( + h)( x+ h- ) ( + x)( - x)( + h) N3 N7 4 ( - x)( + h)( - x+ h-) ( - x)( + h)( -h) N4 N 4 he poston of any pont nsde the shell element can be wrtten n terms of nodal coordnates as ì xü ì ìx ü ì x ü ü y N( xh, ) + - y y + å (6.6.) î î top î î bottom Snce, ς s assumed to be normal to the md surface, the above expresson can be rewrtten n terms of a vector connectng the upper and lower ponts of shell as Or, ìxü ì ì (, ) ì x ü ì x ü ü ì ì x ü ì x ü üü y N xh y y + + y - y å top bottom î î î î top î î î î bottom

4 4 ì xü ì ìx ü ü y N y + î îî Where, å ( xh, ) (6.6.3) 3 ì x ü ì x x ü ì ü ì ü ì x ü ì x ü y y + y and, 3 y - y î î top î î bottom î top î bottom (6.6.4) Fg Local and global coordnates For small thckness, the vector 3 can be represented as a unt vector t v 3 : ì xü ì ìx ü ü y N( xh, ) y + tv 3 å (6.6.5) î î î Where, t s the thckness of shell at th node. In a smlar way, the dsplacement at any pont of the shell element can be expressed n terms of three dsplacements and two rotaton components about two orthogonal drectons normal to nodal load vector 3 as, ì uü ì ìu ü ü t a ì ü v N( xh, ) v + [ v -v] å (6.6.6) b w w î î î î Where,, are the rotatons of two unt vectors v & v about two orthogonal drectons normal to nodal load vector 3.he values of v and v can be calculated n followng way: he coordnate vector of the pont to whch a normal drecton s to be constructed may be defned as x xˆ+ yj ˆ+ kˆ (6.6.7)

5 4 In whch, ˆ, ˆj, k ˆ are three (orthogonal) base vectors. hen, s the cross product of î & 3 as shown below. ˆ & 3 (6.6.) and, v 3 & v (6.6.9) Jacoban matrx he Jacoban matrx for eght node shell element can be expressed as, é * N * N * N ( x + tx) ( y + ty) ( + t å å å ) x x x * N * N * N [ J] å( x + tx) å( y + ty) å( + t) h h h * * * ånx åny ån ù úû (6.6.0) Stran dsplacement matrx he relatonshp between stran and dsplacement s descrbed by { } [ B]{ d} e (6.6.) Where, the dsplacement vector wll become: { d} { u v w v v u v w v v } (6.6.) And the stran components wll be ì u ü x v y u v [ e] + (6.6.3) y x v w + y w u + î x Usng eq. (6.6.6) n eq. (6.6.3) and then dfferentatng w.r.t.,, the stran dsplacement matrx wll be obtaned as

6 43 é u v wù ì N ü ì N ü ì N ü x x x x x x b é ù é a ù u v wú N tv N [ u v w ] ú - tv N b h h h å h å + a h å h êb ú ê 3 u v w 0 N a ú ë û 3 N ë û ê ú ë û î î î (6.6.4) Stress stran relaton he stress stran relatonshp s gven by s D e (6.6.5) { } [ ]{ } Usng eq. (6.6.) n eq. (6.6.5) one can fnd the followng relaton. { } [ D][ B]{ d} s (6.6.6) Where, the stress stran relatonshp matrx s represented by é m ù m m E [ D] (6.6.7) -m a( -m) a( -m) úû he value of shear correcton factor a s consdered generally as 5/6. he above consttutve matrx can be splt nto two parts ([D b ] and [D s ] )for adopton of dfferent numercal ntegraton schemes for bendng and shear contrbutons to the stffness matrx. é[ Db ] [ 0] ù [ D] (6.6.) [ 0] [ Ds ] úû hus, é ù m 0 E [ Db ] 0 m (6.6.9) -m -m 0 0 úû and Ea é 0ù [ Ds ] (6.6.0) ( + m) ê ë 0 ú û

7 44 It may be mportant to note that the consttutve relaton expressed n eq. (6.6.9) s same as for the case of plane stress formulaton. Also, eq. (6.6.0) wth a multplcaton of thckness h s smlar to the terms corresponds to shear force n case of plate bendng problem Element stffness matrx Fnally, the stffness matrx for the shell element can be computed from the expresson k òòò B D B dw (6.6.) [ ] [ ] [ ][ ] However, t s convenent to dvde the elemental stffness matrx nto two parts: () bendng and membrane effect and () transverse shear effects. hs wll facltate the use of approprate order of numercal ntegraton of each part. hus, k k + k (6.6.) [ ] [ ] [ ] b s Where, contrbuton due to bendng and membrane effects to stffness s denoted as [k] b and transverse shear contrbuton to stffness s denoted as [k] s and expressed n the followng form. k B D B dw and k B D B dw òòò òòò (6.6.3) [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] b b b b s s s s Numercal procedure wll be used to evaluate the stffness matrx. A Gauss Quadrature can be used to evaluate the ntegral of [k] b and one pont Gauss Quadrature may be used to ntegrate [k] s to avod shear lockng effect.

2x x l. Module 3: Element Properties Lecture 4: Lagrange and Serendipity Elements

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