Plate heores for Classcal and Lamnated plates Weak Formulaton and Element Calculatons PM Mohte Department of Aerospace Engneerng Indan Insttute of echnolog Kanpur EQIP School on Computatonal Methods n Engneerng Applcaton Knowledge Incubaton for EQIP
Plate lke Elements n Structural Applcatons
Classcal Plate heor
Classcal Plate heor: Plate Geometr and notatons, n-plane coordnates coordnate plane md-plane or md-surface h q(,) z coordnate n thckness drecton h thckness of the plate u, v and w deformatons/dsplacements along, and z aes, respectvel z - doman of the plate, - boundng surface
Classcal Plate heor: Basc Features: Deformaton feld s based on the of Krchhoff-Love assumptons Applcable to thn plates Plane stress condton 1. Normal to the md-plane (.e., transverse normals) before deformaton reman straght and normal after deformaton. ransverse normals to the md-plane are netensble 3. Materal s homogeneous and sotropc
Classcal Plate heor: Knematcs of Deformaton Geometr of Deformaton Approach Undeformed Geometr: a b Or z Deformaton Stretchng acton due to load the edges of plate and parallel to the md-surface Bendng acton due to loads the top and bottom surface
Classcal Plate heor: Knematcs of Deformaton Deformaton due to Stretchng: An normal to the mdplane s translated horzontall as a rgd lne n the plane Undeformed Geometr a X Deformed Geometr z b u (,) a X z Vew n drecton b
Classcal Plate heor: Knematcs of Deformaton Deformaton due to Bendng: An normal to the mdplane s translated vertcall as rgd lne mantanng, coordnates then rotates as a rgd element Undeformed Geometr a b Deformed Geometr z a w (,) z Vew n drecton b
Classcal Plate heor: Knematcs of Deformaton Deformaton due to Stretchng and Bendng: Undeformed Geometr a z b u (,) w (,) Deformed Geometr a z Vew n drecton b
Classcal Plate heor: Knematcs of Deformaton Deformaton of a Generc Pont on the Normal: z α w tan For small deformatons, w tan w z a a p p In-plane deformaton due to rgd rotaton of the normal z b α w b z w Vew n drecton
Classcal Plate heor: Knematcs of Deformaton Deformaton of a Generc Pont on the Normal due to both Stretchng and Bendng:,,, u z u z w Smlarl, the deformaton vewed n drecton can be gven as,,, v z v z w Stretchng part Bendng part And the deformaton n the transverse drecton,,, w z w Bendng part
Classcal Plate heor: Knematcs of Deformaton Alternate Approach: Small deformatons, Strans are nfntesmal he frst assumpton leads to the condton that transverse shear strans are zero. z and z Inetensblt of a transverse normal mples the transverse normal stran s zero. hat s, zz Integratng ths wth respect to z, w z, z w w,, - functon of and onl
Classcal Plate heor: Knematcs of Deformaton Deformaton feld: From the frst assumpton z u w z u z w Integratng wth respect to z w,,z z u, u Smlarl, z v w z v z w
Classcal Plate heor: Deformaton feld: Integratng wth respect to z w,,z z v, v Snce,, v, and w, are ndependent of z, u, ther varaton along thckness s zero or these are constants along z. herefore, u,, v, and w, are taken as md-plane or md-surface dsplacements.
Classcal Plate heor: Stran Feld All other stran components are zero. hs also mples that the transverse shear stresses are zero w z v u v u w z v v w z u u 1 1 z z
Classcal Plate heor: Consttutve Relatons Stran Feld: where, z u v u v Md-plane strans w w w Md-plane Curvatures
Classcal Plate heor: Consttutve Relatons Isotropc Materals: Strans n terms of stresses ν Posson s rato, E- Young s Modulus Stresses n terms of strans 1 1 1 E 1 1 E 1 1 1
Classcal Plate heor: Stress Resultants z Defne shear force resultants as z z z Q h h z dz
Classcal Plate heor: Stress Resultants z z z Q h h z dz
Classcal Plate heor: Stress Resultants Resultant Forces: h h h N dz, N dz, N dz h h h Resultant Moments: h h h M z dz, M z dz, M z dz h h h
Classcal Plate heor: Stress Resultants Resultant Forces: N 1 N A 1 N 1 where, A Eh 1 Resultant Moments: M 1 M D 1 M 1 where, D Eh Bendng Rgdt 3 1 1
Classcal Plate heor: Equlbrum Equatons Equaton n -drecton wthout an bod force: z z Multpl wth z and ntegrate over plate thckness h h h z dz z dz z dz z z h h h M M z z dz z h h
Classcal Plate heor: Equlbrum Equatons Smplfng, h h h z z dz z z h z z dz Q z h h No shear stress on the top and bottom of the plate. M M Q a Smlarl, equlbrum equaton n drecton s M M Q b
Classcal Plate heor: Equlbrum Equatons Integratng the equlbrum equaton n z drecton over the plate thckness In terms of resultants h h h z z h h h ransverse load s appled on top face onl. zz dz dz dz z Q Q zz z z h h z zz z zz z h and z h q, z zz z
Classcal Plate heor: Equlbrum Equatons hs gves Q Q q, c Combnng Eqs. (a), (b) and (c) nto a sngle equaton and elmnatng Q and Q M M M q, Fnall, epressng the resultant moments n terms of dsplacement w 4 q w Nonhomogeneous Bharmonc Equaton frst developed b Sophe German n 1815. D
Fnte Element Formulaton of Classcal Plate heor
Classcal Plate heor: Fnte Element Formulaton Let the doman be dvded nto (3 noded) trangular or (4 noded) quadrlateral elements e k e l j j k Each node has three nodal dsplacements or 3 degrees of freedom (DOF) or prmar varables w w θ w θ e w a w z
Classcal Plate heor: Fnte Element Formulaton Each node has three nodal forces (secondar varables) q z (q θ ) (q θ ) z e q q q q z
Classcal Plate heor: Fnte Element Formulaton otal Potental Energ of an element π: D w w w w 1 d d a q he frst varaton of π w w w 1 D w w w w d d a q 1 w w
Classcal Plate heor: Fnte Element Formulaton he frst varaton of otal Potental Energ of an element usng resultant moments and curvatures w w w M M M d d a q Moments and Curvatures n matr forms usng assumed dsplacement over an element M d d a q
Classcal Plate heor: Fnte Element Formulaton Re-arrangement of Stran Feld: s a lnear operator For eample, for the element shown the DOFS are a L w L w w w L j k e w 3 9 8 3 7 6 5 4 3 1, c c c c c c c c c w
Classcal Plate heor: Fnte Element Formulaton Nodal dsplacement for th node: Or can be found as ([A] s nvertble) c w 3 1 3 1 1 3 3 e e c A a e e a A c 1 e c
Classcal Plate heor: Fnte Element Formulaton Md-plane curvatures for e th element: Or where, 9 8 7 6 5 4 3 1 4 6 6 c c c c c c c c c w w w e e e c H 4 6 6 H
Classcal Plate heor: Fnte Element Formulaton Md-plane curvatures for e th element n terms of nodal varables: e 1 H A a e Let B H A 1 Now the resultant moments are epressed as M 1 M D 1 M 1 or where, M D* D* 1 D 1 1
Classcal Plate heor: Fnte Element Formulaton he frst varaton of otal Potental Energ of an element usng resultant moments and curvatures a B D Ba d d a q Usng the relatons developed for resultant moments and md-plane curvatures a B D* B a dd a q or a B D * Bdd a q
Classcal Plate heor: Fnte Element Formulaton hs can be further wrtten as where, e a K a q e k B D* Bdd as a s arbtrar. Fnall, for an element e we wrte K e a e q e
Classcal Plate heor: Fnte Element Formulaton Element load vector calculaton: From Prncpal of vrtual work Now, re-wrte the dsplacement over the element,,,,, a q p w dd P w r w ds M w ds n 3 9 8 3 7 6 5 4 3 1, c c c c c c c c c w j k P P 1 M r(,) p(,)
Classcal Plate heor: Fnte Element Formulaton Dsplacement n an element or 3 3, 1,,,,,,,, w c 3 3 1 e, 1,,,,,,,, * 1 w A a C A a e e Let herefore, and N C * A 1 e w N a a N e w a N
Classcal Plate heor: Fnte Element Formulaton Puttng these n vrtual work of eternal forces a q a p, N dd a P N, N, a r, N, ds a M ds n Snce a s arbtrar e q p, N dd P N, N, r, N, ds M ds n
Classcal Lamnated Plate heor
Classcal Lamnated Plate heor: Nomenclature H otal lamnate thckness 13 Prncpal materal drectons z Global materal drectons θ k Pl orentaton of k th pl z k bottom z-coordnate of k th pl Abbrevated as CL or Classcal Lamnated Plate heor - CLP
Classcal Lamnated Plate heor: Assumptons hs s a drect etenson of classcal plate theor to lamnates Same assumptons as n classcal plate theor wth Laers are perfectl bonded. here s no slp between adjacent laers. he dsplacements are contnuous through the thckness. Each laer s consdered to be homogeneous and can be sotropc, orthotropc or transversel sotropc wth known propertes.
Classcal Lamnated Plate heor: Lamna Consttutve Relatons Orthotropc Materals, Prncpal Materal Drectons Strans n k th lamna n terms of stress Stresses n k th lamna n terms of strans 1 11 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 Q G E E E E 1 1 1 E E and 1 11 1 1 1 1 1 1 11 1 1 1 G E E E E
Classcal Lamnated Plate heor: Lamna Consttutve Relatons Global Drectons ransformaton of stresses ransformaton of strans n m mn mn mn m n mn n m 1 11 and cos m where, sn n n m mn mn mn m n mn n m 1 11
Classcal Lamnated Plate heor: Lamna Consttutve Relatons Stresses n Global Drectons 11 1 11 Q Q 1 1 Q Reduced transformed stffness matr: 1 Q Q Q Q Q 11 1 16 Q Q Q 1 6 Q Q Q 16 6 66 Q
Classcal Lamnated Plate heor: Lamna Consttutve Relatons Strans n Global Drectons S Reduced transformed stffness matr: 1 1 1 S Q Q S S S 11 1 16 S S S 1 6 S S S 16 6 66 Q, Q, S are smmetrc matrces
Classcal Lamnated Plate heor: Lamna Consttutve Relatons Stresses n k th lamna k Q k k Usng defnton of strans k k k k Q Q z
Classcal Lamnated Plate heor: Lamnate Consttutve Relatons Resultant Forces: H H H N dz, N dz, N dz Usng defnton of stresses H H H where N N N N H k k k k N Q dz Q Q z dz H H H Stresses var pecewse lnearl n each laer.
Classcal Lamnated Plate heor: Lamnate Consttutve Relatons k are constant n each laer Q, and Integraton over lamnate thckness s evaluated as sum of ntegratons over laer thcknesses Nla k k k k N Q dz Q Q z dz z k Nla k1 z k1 k1 k1 z z k where, Nla N A B k 1 A Q z k zk1, B Q zk zk1 k1 k1 Nla k
Classcal Lamnated Plate heor: Lamnate Consttutve Relatons Resultant Moments: H H H M zdz, M zdz, M zdz H H H where M M M M Usng defnton of stresses H H k k k k M Q zdz Q Q z zdz H H Stresses var pecewse lnearl n each laer.
Classcal Lamnated Plate heor: Lamnate Consttutve Relatons Integraton over lamnate thckness s evaluated as sum of ntegratons over laer thcknesses Nla k k k k M Q zdz Q Q z zdz z k Nla k1 z k1 k1 k1 M B D z z k where, Nla Nla 1 k 1 3 3 3 B Q z k zk1, D Q zk zk1 k1 k1 k A, B and D are smmetrc.
Classcal Lamnated Plate heor: Lamnate Consttutve Relatons Stress Resultants Moment Resultants
Classcal Lamnated Plate heor: Lamnate Consttutve Relatons Lamnate Consttutve Relatons: B D N A B M Equatons of moton: N N M N N M M q,
Classcal Lamnated Plate heor: Weak Formulaton From the frst equaton of moton: e u u v u v w w w A11 A1 A16 B11 B 1 B 16 dd u u v u v w w w A16 A6 A66 B16 B 6 B 66 e N u ds n n
Classcal Lamnated Plate heor: Weak Formulaton From the second equaton of moton: e e v u v u v w w w A1 A A6 B1 B B 6 v u v u v w w w N A16 A6 A66 B16 B 6 B 66 s u ds s dd
Classcal Lamnated Plate heor: Weak Formulaton From the thrd equaton of moton: e w u v u v w w w A11 A1 A16 B11 B 1 B 16 w w u v u v w w w A16 A6 A66 B16 B 6 B 66 dd w u v u v w w w A1 A A6 B1 B B 6 w w u v u v w w w A16 A6 A66 B16 B 6 B 66
Classcal Lamnated Plate heor: Weak Formulaton From the thrd equaton of moton: w u v u v w w w B11 B1 B16 D11 D 1 D 16 w u v u v w w w B1 B B6 D1 D D 6 dd e w u v u v w w w B16 6 66 16 6 B B D D D 66 wq w Vn w M n ds n e
Classcal Lamnated Plate heor: Weak Formulaton where, N N n N n, N N n N n n s M ns Vn Qn P s M M n M n, M M n M n n s w w w w P N N n N N n (n, n ) denote the drecton cosnes of the unt normal on the element boundar Γ e
Classcal Plate heor: Fnte Element Formulaton Let the doman be dvded nto (3 noded) trangular or (4 noded) quadrlateral elements e k e l j j k Each node has fve nodal dsplacements or 5 degrees of freedom (DOF) or prmar varables v u θ w z θ a e u v w w w
Classcal Plate heor: Fnte Element Formulaton Each node has fve nodal forces (secondar varables) q q q z (q θ ) (q θ ) z Nodal force vector: e q q q q q q z
Classcal Lamnated Plate heor: Fnte Element Formulaton Appromaton of soluton n an element: u Node, 1, u N Smlarl, for the other varables v, w, w and w hat s, appromaton of soluton vector: Node Number of nodes n an element e N (,) Interpolaton functon assocated wth node a a value of correspondng to node Node a N, a 1
Classcal Lamnated Plate heor: Weak Formulaton otal Potental Energ Nelem Number of elements Nelem e1 d e e e d U W e U e - Stran energ of an element W e - work done b eternal forces n an element
Classcal Lamnated Plate heor: Fnte Element Formulaton Md-plane strans: or where, E stands for etenson. E L a w v u
Classcal Lamnated Plate heor: Fnte Element Formulaton Md-plane curvatures: or where, B stands for bendng. w v u a L B
Classcal Lamnated Plate heor: Fnte Element Formulaton Generalzed Md-plane strans: where, and BE LE N Node B E B E 1 L a L N a Node 1 B E E E Node 1 B E d a and d a a a a 1 3 Node
Classcal Lamnated Plate heor: Fnte Element Formulaton Generalzed Md-plane curvatures: where, and BB LB N Node B B B B 1 L a L N a Node B 1 B B B Node 1 B B d a and d a a a a 1 3 Node
Classcal Lamnated Plate heor: Fnte Element Formulaton Stran Energ of an element: Stran energ due to etenson Stran energ due to bendng Usng the notatons, B E e U U U e A e da M N U 1 e A B B B E E B E E e da d B D B d d B B B d d B B B d d B A B d U 1
Classcal Lamnated Plate heor: Fnte Element Formulaton Stran Energ of an element n a concse form e 1 e U d K d where, K e s the element stffness matr 1 K B A B B B B B B B B D B da e E E B E E B B B e A
Classcal Lamnated Plate heor: Fnte Element Formulaton Eternal work done: e N W d N q, da d N r, ds d M ds n A e e e Also can be wrtten concsel as e W d q e where, q e s the element load vector N q N q da N r ds M ds n e,, A e e e
Classcal Lamnated Plate heor: Fnte Element Formulaton Numercal Integraton of Stffness Matr (for quadrlateral element): 11 e * j j K B D B J dd 11 NIN NIN e * j J j l m l1 m1 K B D B W W NIN - s the number of Gauss quadrature ponts nξ and η drectons W weght ponts n drecton J - Determnant of the Jacoban matr
Classcal Lamnated Plate heor: Fnte Element Formulaton where, B BE B B and D * A B B D
Classcal Lamnated Plate heor: Fnte Element Formulaton Numercal ntegraton of Load vector: wo parts: Area ntegral and lne ntegral N q N q da N r ds M ds n e,, where, area ntegral part of the load vector: And lne ntegral part of the load vector: A e e e e q e q e q 1 e q N q, 1 e A e q N r, e ds e da N n M ds
Classcal Lamnated Plate heor: Fnte Element Formulaton Numercal ntegraton of Load vector: 11 e q N, q J d d 1, 1 1 NIN NIN e q N q,,, J l m l m l, 1 l1 m1 m W l W m All other terms have same meanng as n stffness matr numercal ntegraton
Classcal Lamnated Plate heor: Fnte Element Formulaton Numercal ntegraton of Load vector: Lne ntegral 1D Numercal ntegraton q 1 e N 1D N r M J d n 1 NIN1D N l l l M l l J l1 n e q N r NIN1D - s the number of Gauss quadrature ponts nξ drecton 1D 1D W D W 1 l 1D J weght ponts n drecton - Jacoban for one dmensonal ntegratons (=length/)
Hgher Order Shear Deformable Plate heores
Hgher Order Plate heor: Dsplacement feld: 3 1 3 4 u,,z u, zu, z u, z u, 3 1 3 4 v,,z v, zv, z v, z v, 3 1 3 4 w,,z w, zw, z w, z w, Hgher order representaton of strans n the thckness drecton ransverse strans are non-zero. Stran components: zz z z
Frst Order Shear Deformaton heor: Dsplacement feld: 1 1 u,,z u, zu, v,,z v, zv, w,,z w, 1 Hgher order representaton of strans n the thckness drecton ransverse shear strans are non-zero but transverse normal stran s zero. Stran components: z z
hrd Order Plate heor: Dsplacement feld: 3 1 3 4 u,,z u, zu, z u, z u, 3 1 3 4 v,,z v, zv, z v, z v, w,,z w, 1 No transverse stresses on top and bottom face of plate. z z v w w v zv z v z 1 3 3 4 u w w u zu z u z 1 3 3 4
hrd Order Plate heor: Dsplacement feld: u v : s used smplf the feld 3 3 3 w1 3 w1 v4 4 : also used to smplf the feld v, u u 4 h 4h For the (earler) dsplacement feld the resultant moments and shear force H H H * 3 * 3 * 3,, M z dz M z dz M z dz H H H H H H H * * z, z, z, z Q dz Q dz Q z dz Q z dz H H H H
hrd Order Plate heor: Fnte Element Formulaton In the FE formulaton the defntons of are changed: Now there are seven varables per node! Dsplacement vector changed. L E, L B d s also L S s defned for shear part. Stran energ s d B AB d d B BB d E E B E e 1 U d BE BBB d d BB DBB d da A e d BS DS BS d
hrd Order Plate heor: Fnte Element Formulaton Stran Energ of an element n a concse form e 1 e U d K d where, K e s the element stffness matr K e 1 B E A BE BB B BE BE B BB BB D B B da A e BS DS BS
Hgher Order Plate heor: Most popular hgher order theores have constant or lnear varaton of transverse dsplacement. hat s, the transverse normal stran s zero or constant! he development of these theores was domnated b bendng loadng. he stretchng of aal part s not gven mportance. In case of unsmmetrc lamnates there s stretchng bendng couplng. hese theores ma not be sutable.
Hgher Order Plate heor: Dsplacement feld: u v w p, p, p p u z 1, u z 1 u,,z v pz 1 v,,z v, z w,,z 1 w pz 1 w, z 1 z z z - order of polnomal drector functons n z-drecton for u, v and w hese orders can be dfferent.
Hgher Order Plate heor: In general, p u z For bendng domnated problems p v z, v,,z u,,z w,,z - antsmmetrc wth respect to z (odd powers) - smmetrc wth respect to z (even powers) For frst order shear deformable theor hs s Denoted as: 1,1, u v w z z z p p 1, p For thrd order shear deformable theor hs s Denoted as: 3,3,
Hgher Order Plate heor: Natural Herarch of models for bendng domnated problems: 1,1,, 1,1,, 3,3,, 3,3,4, For membrane domnated problems: u,,z, v,,z - smmetrc wth respect to z (even powers) w,,z - antsmmetrc wth respect to z (odd powers) Natural Herarch of models for membrane domnated problems:,,1,,,1,,,3, 4,4,3,
Hgher Order Plate heor: For both bendng and membrane domnated problems: 1,1,1,,,, 3,3,3, 4,4,4, he sequences of the models were based on the convergence of ther solutons to the three dmensonal soluton n a sutable norm.
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