Module 3: Element Properties Lecture 1: Natural Coordinates


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1 Module 3: Element Propertes Lecture : Natural Coordnates Natural coordnate system s bascally a local coordnate system whch allows the specfcaton of a pont wthn the element by a set of dmensonless numbers whose magntude never exceeds unty. hs coordnate system s found to be very effectve n formulatng the element propertes n fnte element formulaton. hs system s defned n such that the magntude at nodal ponts wll have unty or zero or a convenent set of fractons. It also facltates the ntegraton to calculate element stffness. 3.. One Dmensonal Lne Elements he lne elements are used to represent sprng, truss, beam lke members for the fnte element analyss purpose. Such elements are qute useful n analyzng truss, cable and frame structures. Such structures tend to be well defned n terms of the number and type of elements used. For example, to represent a truss member, a two node lnear element s suffcent to get accurate results. However, three node lne elements wll be more sutable n case of analyss of cable structure to capture the nonlnear effects. he natural coordnate system for one dmensonal lne element wth two nodes s shown n Fg Here, the natural coordnates of any pont P can be defned as follows. N =  x and N x l = l (3..) Where, x s represented n Cartesan coordnate system. Smlarly, x/lcan be represented as ξ n natural coordnate system. hus the above expresson can be rewrtten n the form of natural coordnate system as gven below. N =  x and N = x (3..) Now, the relatonshp between natural and Cartesan coordnates can be expressed from eq. (3..) as ìü é ùìn ü í = í x ê0 lú N î ë ûî (3..3) Here, N and N s termed as shape functon as well. he varaton of the magntude of two lnear shape functons (N and N ) over the length of bar element are shown n Fg hs example dsplays the smplest form of nterpolaton functon. he lnear nterpolaton used for feld varable can be wrtten as ( ) = N+ N fx f f (3..4)
2 Fg. 3.. wo node lne element Fg. 3.. Lnear nterpolaton functon for two node lne element Smlarly, for three node lne element, the shape functon can be derved wth the help of natural coordnate system whch may be expressed as follows:
3 3 { N} ì 3x x ü  + l l ì N ü ì 3x+ x ü 4x 4x N í í 4 4 í l l N 3 x + x = =  = x x (3..5) î x x î  + l l î he detaled dervaton of the nterpolaton functon wll be dscussed n subsequent lecture. he varaton of the shape functons over the length of the three node element are shown n Fg Fg Varaton of nterpolaton functon for three node lne element Now, f s consdered to be a functon of L and L, the dfferentaton of wth respect to xfor two node lne element can be expressed by the chan rule formula as df f L f L =. +. (3..6) dx L x L x hus, eq.(3..4) can be wrtten as L L = and = (3..7) x l x l herefore,
4 4 d æ ö = dx l ç  ç L L è ø he ntegraton over the length ln natural coordnate system can be expressed by pq!! (3..7) p q ò L L dl = l (3..9) ( p + q +! ) l Here, p! s the factoral product p(p)(p).() and 0! s defned as equal to unty. 3.. wo Dmensonal rangular Elements he natural coordnate system for a trangular element s generally called as trangular coordnate system. he coordnate of any pont Pnsde the trangle s x,y n Cartesan coordnate system. Here, three coordnates, L, L and L 3 can be used to defne the locaton of the pont n terms of natural coordnate system. he pont P can be defned by the followng set of area coordnates: ; ; (3..0) Where, = Area of the trangle P3 = Area of the trangle P3 = Area of the trangle P A=Area of the trangle 3 hus, and (3..) herefore, the natural coordnate of three nodes wll be: node (,0,0); node (0,,0); and node 3 (0,0,).
5 5 Fg rangular coordnate system he area of the trangles can be wrtten usng Cartesan coordnates consderng x, y as coordnates of an arbtrary pont P nsde or on the boundares of the element: A = A = A = A 3 = he relaton between two coordnate systems to defne pont P can be establshed by ther nodal coordnates as (3..) Where,
6 6 he nverse between natural and Cartesan coordnates from eq.(3..) may be expressed as (3..3) he dervatves wth respect to global coordnates are necessary to determne the propertes of an element. he relatonshp between two coordnate systems may be computed by usng the chan rule of partal dfferentaton as L L L3 = x L x L x L3 x b b b3 = A L A L A L3 (3..4) 3 b = å. A L = Where, b = y y 3 ; b = y 3 y and b 3 = y y. Smlarly, followng relaton can be obtaned. 3 c = å. y A L = (3..5) Where, c = x 3 x ; c = x x 3 and c 3 = x x. he above expressons are looked However, the man advantage s the ease wth whch polynomal terms can be cumbersome. ntegrated usng followng area ntegral expresson. p!q!r! p q r ò L L L3 da= A (3..6) ( p q r )! A Where 0! s defned as unty Shape Functon usng Area Coordnates he nterpolaton functons for the trangular element are algebracally complex f expressed n Cartesan coordnates. Moreover, the ntegraton requred to obtan the element stffness matrx s qute cumbersome. hs wll be dscussed n detals n next lecture. he nterpolaton functon and subsequently the requred ntegraton can be obtaned n a smplfed manner by the concept of area
7 7 coordnates. Consderng a lnear dsplacement varaton of a trangular element as shown n Fg. 3..5, the dsplacement at any pont can be wrtten n terms of ts area coordnates. u=a L+a L+a 3L3 Or, u = { f} { a } (3..7) where, { f } = [ L L L ] and { a } = { a a a } 3 3 And ; ; (3..8) Here, A s the total area of the trangle. It s mportant to note that the area coordnates are dependent as. It may be seen from fgure that at node, L = whle L = L 3 = 0. Smlarly for other two nodes: at node, L = whle L = L 3 = 0, andl 3 = whle L = L = 0. Now, substtutng the area coordnates for node, and 3, the dsplacement components at nodes can be wrtten as ìu ü é 0 0ù í u 3 ê0 0 ú î ë û { u } = u = 0 0 { a} hus, from the above expresson, one can obtan the unknown coeffcenta : é 0 0ùìu ü a = í { } ú 0 0ú u ú ê0 0 ú u ë ûî 3 (3..9) (3..0) Fg Area coordnates for trangular element
8 8 Now, eq.(3..7) can be wrtten as: é 0 0ùì u ü é 0 0ù ê ú ê ú í ê0 0 ú u ê0 0 ú { u} = { f } ê0 0ú u = { f} ê0 0ú{ u } ë ûî 3 ë û he above expresson can be wrtten n terms of nterpolaton functon as = { } { } Where, é 0 0ù ê ú ê0 0 ú ë û { N} = [ L L L ] ê0 0ú= [ L L L ] 3 3 u N u Smlarly, the dsplacement varaton v n Y drecton can be expressed as follows. { } { } (3..) (3..) v= N v (3..3) hus, for two dsplacement components u and v of any pont nsde the element can be wrtten as: ìü u é{ N} { 0} ù u d = = ú ì ü í v î ê ë{} 0 { N} ú ûî { } úí v hus, the shape functon of the element wll become [ N] él ê L L ù ú ë û 3 = ê L L L3 ú (3..4) (3..5) It s mportant to note that the shape functon N become unty at node and zero at other nodes of the element. he dsplacement at any pont of the element can be expressed n terms of ts nodal dsplacement and the nterpolaton functon as gven below. u=n u +N u +N u 3 3 v=n v +N v +N v 3 3 (3..6)
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