Professor Terje Haukaas Unversty of Brtsh Columba, ancouver www.nrsk.ubc.ca The Q Element Ths document consders fnte elements that carry load only n ther plane. These elements are sometmes referred to as D contnuum elements, but the term plane elements s preferred n these documents. The prncple of vrtual dsplacements s utlzed to derve expressons for the stffness matrx and load vector, commencng wth the generc prncple of vrtual dsplacements, whch reads where, n ths document, p represents the dstrbuted element loads: The prncple of vrtual dsplacements does by tself represent an average satsfacton of equlbrum. Conversely, materal law and knematcs must be added. To ths end, the materal law for plane problems s wrtten σ=dε and s ether plane stress: or plane stran: σ x σ y τ xy σ x σ y τ xy E = ν ν ν ν ε x ε y γ xy ν ν E = (+ν)( ν) ν ν ν Substtuton of the materal law nto Eq. () yelds Next, the knematc relatonshp δ ε T σ d δ u T p d = ε x ε y γ xy δ ε T Dε d δ u T pd = (5) p = p x (x, y) p y (x, y) () () () () ε = ε x ε y γ xy = u(x, y) v(x, y) = u (6) The Q Element Updated December, Page
Professor Terje Haukaas Unversty of Brtsh Columba, ancouver www.nrsk.ubc.ca s combned wth the fundamental fnte element assumpton to read u = Nu (7) ε = u = Nu Bu (8) where the B-matrx s defned as B = N. It s understood that the B-matrx contans the dervatve of the shape functons. Substtuton of Eq. (8) nto (5) wth the same dscretzaton of the vrtual dsplacement feld as the real one yelds ( Bδu) T D( Bu)d ( Nδu) T p d = (9) Because the transpose of a matrx product s the same as transposng each matrx or vector and swtchng the multplcaton order, rearrangng yelds δu T B T DBd u NT p d = () The vrtual dsplacement pattern s arbtrary, whch means that the large parenthess must be zero, and the result s: B T DBd u = N T p d K where the stffness matrx and load vector are dentfed. So far n ths dervaton the shape functon relatonshp u = Nu have remaned abstract. The specfc formulaton of these shape functons for an arbtrary element s dffcult because each element has a dfferent shape and dfferent dmensons. Ths problem s addressed next, by the soparametrc element formulaton, whch has two crucal steps:. The shape functons are formulated n a normalzed coordnate system (ξ,η) nstead of the orgnal (x,y) coordnate system, and. The orgnal shape of the element s descrbed by means of the same shape functons that descrbe the deformaton of the element. Ths lkeness gves rse to the word so, whch means equal. F () The Q Element Updated December, Page
Professor Terje Haukaas Unversty of Brtsh Columba, ancouver www.nrsk.ubc.ca y 8 η 7 6 5 ξ 8 η 7 (-,) (,) (-,-) (,-) 6 5 ξ x Fgure : Transformaton of the Quad element nto the normalzed doman. Fgure shows an arbtrary element on the left-hand sde, whch s transformed nto the normalzed (ξ,η)-doman. All elements, regardless of shape, are equal n the (ξ,η)- doman. Hence, the problem of formulatng shape functons s reduced to that of formulatng shape functons for the rght-most element n Fgure. Each shape functon must equal unty at the node that contans the DOF t s assocated wth. Furthermore, t must be zero at all other nodes. It s also natural to employ the same nterpolaton of the ξ-drecton dsplacement as the η-drecton dsplacement. In other words, t s natural to select the shape functon of u equal to that of u. Smlarly, the shape functons of u and u, u 5 and u 6, u 7 and u 8 are equal. Consequently, the dscretzaton s wrtten u(ξ,η) N (ξ,η) N (ξ,η) N (ξ,η) N (ξ,η) = v(ξ,η) N (ξ,η) N (ξ,η) N (ξ,η) N (ξ,η) where the shape functons N, N, N, and N are assocated wth the four nodes of the element. Specfcally, the followng shape functons are derved N (ξ,η) = ( ξ)( η) u u u u u 5 u 6 u 7 u 8 () N (ξ,η) = (+ ξ)( η) N (ξ,η) = ( + ξ)( + η) () N (ξ,η) = ( ξ)( + η) The Q Element Updated December, Page
Professor Terje Haukaas Unversty of Brtsh Columba, ancouver www.nrsk.ubc.ca Substtuton of these shape functon nto Eq. () to calculate the stffness matrx, K, and the load vector, F, poses a new problem: The ntegrals n Eq. Eq. () are formulated n the orgnal (x,y)-doman, whle the shape functons are formulated n the normalzed (ξ,η)-doman. Coordnate transformaton s necessary to remedy ths problem. For pedagogcal purposes, consder frst the coordnate transformaton of a sngle-fold ntegral: x f (x)dx = f (x(ξ)) dx dξ dξ () ξ Notce that the factor dx/dξ appears n the ntegrand when the ntegral s calculated n the ξ-doman. Next, consder the transformaton of a two-fold ntegral: f (x, y)dx dy = f x(ξ,η), y(ξ,η) (5) y x η ξ ( ) J dξ dη where J, sometmes denoted smply by J, s the determnant of the Jacoban matrx, that s establshed shortly wth the followng terms: J = ξ ξ In summary, the ntegral for the stffness matrx from Eq. () reads K = h (6) B T DB J dξ dη (7) where the element thckness h s consdered constant and pulled outsde the ntegral. The load vector s addressed n a subsecton below. The B-matrx must be computed for the evaluaton of Eq. (7). Accordng to Eq. (8) t contans dervatves of the shape functons. Specfcally, the dervatve operator n Eq. (6) appled to the matrx of shape functons n Eq. () gves B = N N N N N N N N However, whle the dfferentaton s wth respect to the (x,y)-coordnates, the shape functons, N, are defned n terms of the (ξ,η)-coordnates. Suppose a relatonshp between the two coordnate systems s avalable, such that the coordnates of one system (8) The Q Element Updated December, Page
Professor Terje Haukaas Unversty of Brtsh Columba, ancouver www.nrsk.ubc.ca are expressed n terms of the other coordnates,.e., ξ(x,y), η(x,y), x(ξ,η), and y(ξ,η). Then the sought dervatves are obtaned by the chan rule of dfferentaton: and N ( ξ(x, y),η(x, y) ) = N ξ ξ + N N ( ξ(x, y),η(x, y) ) = N ξ ξ + N In these expressons the dervatves N ξ and N are readly computed by hand. The dervatves ξ, etc. are pcked from the nverse of the Jacoban matrx: J = ξ The remanng queston s how to calculate the Jacoban matrx,.e., the matrx of dervatves of orgnal coordnates wth respect to the normalzed coordnates. Ths s at the heart of the soparametrc element formulaton. In ths approach, the coordnates of any pont n the element, denoted x, s expressed n terms of the nodal coordnates, whch are collected n the vector x: ξ (9) () () x = Nx () Ths equaton has exactly the same form as Eq. () and N are the same shape functons, hence the phrase soparametrc. Eq. () s wrtten as two equatons n ndex notaton as follows: x = N x y = N y () where x and y dentfy the locaton of any pont n the element and summaton over equal ndces s mpled. Gven ths soparametrc element formulaton the dervatves n the Jacoban matrx are ξ = N ξ x ξ = N ξ y = N x = N y Ths concludes the evaluaton of the ntegrand, but not the ntegral, n the stffness matrx expresson n Eq. (7). For arbtrary element shapes t s not possble to evaluate the ntegral analytcally. Rather, numercal ntegraton s necessary. () The Q Element Updated December, Page 5
Professor Terje Haukaas Unversty of Brtsh Columba, ancouver www.nrsk.ubc.ca Quadrature Several numercal ntegraton schemes, called quarature, are avalable to evaluate the ntegral n Eq. (7) n an approxmate manner. The most common quadrature for fournode quadrlateral elements s Gauss ntegraton. Consder frst Gauss quadrature for sngle-fold ntegraton of the arbtrary functon f(ξ) over the doman - to : f (ξ) dξ = w f (ξ ) (5) where N s the number of ntegraton ponts, w are ntegraton weghts, and ξ are ntegraton ponts. More detals about quadrature are wrtten n a math document on the topc. For example, for N= the rule gves w = and ξ =. For N=, Gauss quadrature specfes w =w =, ξ.577, and ξ.577. Gauss quadrature s readly extended to two-fold ntegrals by applyng the same ponts and weghts n both drectons. For example, two-pont Gauss quadrataure n the (ξ,η)-doman evaluates the ntegrand at the four ponts (.577,.577), (-.577,.577), (.577, -.577), and (-.577, -.577) wth unt weght assocated wth each pont. In concluson, the stffness matrx ntegral n Eq. (7) s evaluated by N = K = h B T D B J dξ dη = h w w j B T D B J (6) N N = j = ( ) j It s noted that all the nformaton about the element s geometry s contaned n J. Mechansms Ths element has the followng egenmodes,.e., ndependent dsplacement modes: 5 6 7 8 Fgure : Independent dsplacement modes of the four-node quadrlateral element. The Q Element Updated December, Page 6
Professor Terje Haukaas Unversty of Brtsh Columba, ancouver www.nrsk.ubc.ca Consstent Load ector In fnte element analyss t s not generally possble to lump dstrbuted element loads as concentrated nodal loads accordng to smple trbutary area. Rather, the load vector as defned n Eq. () must be evaluated. Evaluaton of the load vector accordng to that expresson, whch contans the shape functons of the element formulaton, results n what s called the consstent load vector. One useful case s load that acts on the element edge. The load acts n the element plane, and usually has both an x and a y-component. That s, both components of the vector p, as defned n Eq. (), are generally non-zero. In the followng dervatons, let the coordnate Γ run along the loaded edge n the orgnal (x,y)-coordnate system. In the normalzed coordnate system, ether ξ or η runs long the edge, dependng on whch sde of the element s loaded. Wthout loss of generalty, consder an example where the edge ξ= s loaded. The coordnate η run along ths edge. The expresson for the load vector reads N T p d = N T p dγ = N T p dγ dη dη (7) Γ The value of the Jacoban, dγ/dη, depends on the orentaton of the loaded edge. An nfntesmal length dγ along the edge s related to nfntesmal lengths along the coordnate drectons, dx and dy, by dγ = dx + dy (8) Furthermore, because ξ s constant along the edge, the dfferentals dx and dy are related to the dfferentals n the normalzed coordnate system by Combnaton of Eqs. (8) and (9) yelds dx = dx dη dη dy = dy dη dη (9) dγ = dη dx dη + dy dη () where the dervatves from the Jacoban matrx are evaluated at the edge ξ=. The Q Element Updated December, Page 7