8. PLATE BENING ELEMENTS Plate Bendng s a Smple Etenson of Beam Theor 8. INTROUCTION { XE "Plate Bendng Elements" }Before 960, plates and slabs were modeled usng a grd of beam elements for man cvl engneerng structures. Onl a small number of closed form solutons ested for plates of smple geometr and sotropc materals. Even at the present tme man slab desgns are based on grd models. Ths classcal appromate approach, n general, produces conservatve results because t satsfes statcs and volates compatblt. However, the nternal moment and shear dstrbuton ma be ncorrect. The use of a converged fnte element soluton wll produce a more consstent desgn. The fundamental dfference between a grd of beam elements and a plate-bendng fnte element soluton s that a twstng moment ests n the fnte element model; whereas, the grd model can onl produce one-dmensonal torsonal moments and wll not converge to the theoretcal soluton as the mesh s refned. { XE "Plate Bendng Elements:Thn Plates" }The followng appromatons are used to reduce the three-dmensonal theor of elastct to govern the behavor of thn plates and beams:. { XE "Plate Bendng Elements:Reference Surface" }It s assumed that a lne normal to the reference surface (neutral a of the plate (beam) remans straght n the loaded poston. Ths dsplacement constrant s the same as statng that the n-plane strans are a lnear functon n the thckness drecton. Ths assumpton does not requre that the rotaton of the normal lne to be equal to the rotaton of the reference surface; hence, transverse shear deformatons are possble.. In addton, the normal stress n the thckness drecton, whch s normall ver small compared to the bendng stresses, s assumed to be zero for both beams and plates. Ths s accomplshed b usng plane stress materal propertes n-plane as defned n Chapter. Note
8- STATIC AN YNAMIC ANALYSIS that ths appromaton allows Posson s rato strans to est n the thckness drecton. 3. { XE "Krchhoff Appromaton" }If the transverse shearng strans are assumed to be zero, an addtonal dsplacement constrant s ntroduced that states that lnes normal to the reference surface reman normal to the reference surface after loadng. Ths appromaton s attrbuted to Krchhoff and bears hs name. { XE "Plate Bendng Elements:Shearng eformatons" }{ XE "Shearng eformatons" }Classcal thn plate theor s based on all three appromatons and leads to the development of a fourth order partal dfferental equaton n terms of the normal dsplacement of the plate. Ths approach s onl possble for plates of constant thckness. Man books and papers, usng complcated mathematcs, have been wrtten based on ths approach. However, the Krchhoff appromaton s not requred to develop plate bendng fnte elements that are accurate, robust and eas to program. At the present tme, t s possble to nclude transverse shearng deformatons for thck plates wthout a loss of accurac for thn plates. In ths chapter, plate bendng theor s presented as an etenson of beam theor (see Append F) and the equatons of three-dmensonal elastct. Hence, no prevous background n plate theor s requred b the engneer to full understand the appromatons used. Several hundred plate-bendng fnte elements have been proposed durng the past 30 ears. However, onl one element wll be presented here. The element s a three-node trangle or a fournode quadrlateral and s formulated wth and wthout transverse shearng deformatons. The formulaton s restrcted to small dsplacements and elastc materals. Numercal eamples are presented to llustrate the accurac of the element. The theor presented here s an epanded verson of the plate bendng element frst presented n reference [] usng a varatonal formulaton. 8. THE QUARILATERAL ELEMENT { XE "Quadrlateral Element" }Frst, the formulaton for the quadrlateral element wll be consdered. The same approach apples to the trangular element. A quadrlateral of arbtrar geometr, n a local - plane, s shown n Fgure 8.. Note that the parent four-node element, Fgure 8.a, has 6 rotatons at the four node ponts and at the md-pont of each sde. The md-sde rotatons are then rotated to be normal and tangental to each sde. The tangental rotatons are then set to zero, reducng the number of degrees-of-freedom to, Fgure 8.b. The sdes of the element are constraned to be a cubc functon n u and four z
PLATE BENING ELEMENTS 8-3 dsplacements are ntroduced at the corner nodes of the element, Fgure 8.c. Fnall, the md-sde rotatons are elmnated b statc condensaton, Fgure 8.d, and a OF element s produced. 3 3 8 7 (a) 6 (b) 5 s 3 r 3 θ (d) (c) u z θ Fgure 8. Quadrlateral Plate Bendng Element The basc dsplacement assumpton s that the rotaton of lnes normal to the reference plane of the plate s defned b the followng equatons: θ ( r, θ ( r, 8 N ( r, θ + N ( r, θ + 8 5 8 5 N ( r, θ N ( r, θ The eght-node shape functons are gven b: N ( r)( / N ( + r)( / N ( + r)( + )/ N ( r)( + )/ N 3 s 5 ( r )( s 7 ( r )( s s 6 ( + r)( s 8 ( r)( s (8.) )/ N )/ (8.) N + )/ N )/ { XE "Herarchcal Functons" }Note that the frst four shape functons are the natural blnear shape functons for a four-node quadrlateral. The four shape functons for the md-sde nodes are an addton to the blnear functons and are
8- STATIC AN YNAMIC ANALYSIS often referred to as herarchcal functons. A tpcal element sde s shown n Fgure 8.. θ j θ j L α m θ θ,,3, j,3,, m 5,6,7,8 θ Fgure 8. Tpcal Element Sde The tangental rotatons are set to zero and onl the normal rotatons est. Therefore, the and components of the normal rotaton are gven b: θ sn α θ cosα θ θ (8.3) Hence, Equaton (8.) can be rewrtten as: θ ( r, θ ( r, 8 N ( r, θ + N ( r, θ + 8 5 8 5 M M ( r, θ ( r, θ (8.) { XE "Plate Bendng Elements:Postve splacements" }The number of dsplacement degrees-of-freedom has now been reduced from 6 to, as ndcated n Fgure 8.b. The three-dmensonal dsplacements, as defned n Fgure 8.3 wth respect to the - reference plane, are:
PLATE BENING ELEMENTS 8-5,u z,u z h θ θ,u Fgure 8.3 Postve splacements n Plate Bendng Element u ( r, z θ ( r, u ( r, z θ ( r, (8.5) Note that the normal dsplacement of the reference plane u z ( r, has not been defned as a functon of space. Now, t s assumed that the normal dsplacement along each sde s a cubc functon. From Append F, the transverse shear stran along the sde s gven b: γ (uzj - uz) - ( θ + θ j) - θ (8.6) L 3 From Fgure 8., the normal rotatons at nodes and j are epressed n terms of the and rotatons. Or, Equaton (8.6) can be wrtten as: sn α α ( θ + θ j) + cos ( θ + θ j) - (8.7) γ (uzj - uz) - θ L 3 Ths equaton can be wrtten for all four sdes of the element.
8-6 STATIC AN YNAMIC ANALYSIS It s now possble to epress the node shears n terms of the sde shears. A tpcal node s shown n Fgure 8.. θ k α k k γ θ j u z θ,,3, j,3,, k,,,3 γ z γ γ k z α θ Fgure 8. Node Pont Transverse Shears The two md-sde shears are related to the shears at node b the followng stran transformaton: γ cosα sn α γ z (8.8) γ k cosα k sn α k γ z Or, n nverse form: γ z sn α k cosα k γ (8.9) γ z det sn α cosα γ k where det cosα sn α cosα sn α. k k The fnal step n determnng the transverse shears s to use the standard four-node blnear functons to evaluate the shears at the ntegraton pont. 8.3 STRAIN-ISPLACEMENT EQUATIONS { XE "Plate Bendng Elements:Stran-splacement Equatons" }{ XE "Stran splacement Equatons:Plate Bendng" }Usng the three-dmensonal strandsplacement equatons, the strans wthn the plate can be epressed n terms of the node rotatons. Or:
PLATE BENING ELEMENTS 8-7 u ε z θ ( r,, u ε z θ ( r,, u u γ + z [ θ ( r,, θ ( r, ] (8.0) Therefore, at each ntegraton pont the fve components of stran can be epressed n terms of the 6 dsplacements, shown n Fgure 8.c, b an equaton of the followng form: ε z 0 0 0 0 0 z 0 0 0 θ X ε θ Y γ 0 0 z 0 0 b or d Bu a( z) b( r, u (8.) uz γ z 0 0 0 0 z 0 0 0 0 θ γ Hence, the stran-dsplacement transformaton matr s a product of two matrces n whch one s a functon of z onl. 8. THE QUARILATERAL ELEMENT STIFFNESS { XE "Plate Bendng Elements:Propertes" }From Equaton (8.), the element stffness matr can be wrtten as: where T B EBdV k b b da T (8.) T a Ea dz (8.3) After ntegraton n the z-drecton, the 5 b 5 force-deformaton relatonshp for orthotropc materals s of the followng form: M M M V V z z 3 5 3 5 3 3 33 3 53 3 5 5 5 35 5 55 ψ ψ ψ γ z γ z (8.)
8-8 STATIC AN YNAMIC ANALYSIS The moments M and shears resultant V are forces per unt length. As n the case of beam elements, the deformatons assocated wth the moment are the curvature ψ. For sotropc plane stress materals, the non-zero terms are gven b: 3 Eh ( ν ) 3 νeh (8.5) ( ν ) 5Eh 55 ( +ν ) 8.5 SATISFYING THE PATCH TEST { XE "Patch Test" }{ XE "Plate Bendng Elements:Patch Test" }For the element to satsf the patch test, t s necessar that constant curvatures be produced f the node dsplacements assocated wth constant curvature are appled. Equaton (8.) can be wrtten n the followng form: ψ θ ψ b b θ ψ (8.6) b b w γ z θ γ z { XE "Plate Bendng Elements:Constant Moment" }where, for a quadrlateral element, b s a 3 b matr assocated wth the node dsplacements ( θ,θ, w ) and b s a 3 b matr assocated wth the ncompatble normal sde rotatons ( θ ). In order that the element satsfes the constant moment patch test, the followng modfcaton to b must be made: b b b da (8.7) A The development of ths equaton s presented n the chapter on ncompatble elements, Equaton (6.). 8.6 STATIC CONENSATION { XE "Statc Condensaton" }The element 6 b 6 stffness matr for the plate bendng element wth shearng deformatons s obtaned b numercal ntegraton. Or:
PLATE BENING ELEMENTS 8-9 K K T K B B da (8.8) K K where K s the b matr assocated wth the ncompatble normal rotatons. The element equlbrum equatons are of the followng form: K K u F (8.9) K K θ 0 where F s the node forces. Because the forces assocated wth θ must be zero, those deformaton degrees-of-freedom can be elmnated, b statc condensaton, before assembl of the global stffness matr. Therefore, the b element stffness matr s not ncreased n sze f shearng deformatons are ncluded. Ths quadrlateral (or trangular) plate bendng element, ncludng shear deformatons, s defned n ths book as the screte Shear Element, or SE. 8.7 TRIANGULAR PLATE BENING ELEMENT { XE "Plate Bendng Elements:Trangular Element" }The same appromatons used to develop the quadrlateral element are appled to the trangular plate bendng element wth three md-sde nodes. The resultng stffness matr s 9 b 9. Appromatel 90 percent of the computer program for the quadrlateral element s the same as for the trangular element. Onl dfferent shape functons are used and the constrant assocated wth the fourth sde s skpped. In general, the trangle s stffer than the quadrlateral. 8.8 OTHER PLATE BENING ELEMENTS The fundamental equaton for the dscrete shear along the sdes of an element s gven b Equaton (8.6). Or: γ (uzj - uz) - ( θ + θ j) - θ (8.0) L 3 { XE "Plate Bendng Elements:PQ" }If θ s set to zero at the md-pont of each sde, shearng deformatons are stll ncluded n the element. However, the nternal moments wthn the element are constraned to a constant value for a thn plate. Ths s the same as the PQ element gven n reference [], whch s based on a second order polnomal appromaton of the normal dsplacement. The dsplacements produced b ths element tend to have a small error; however, the nternal moments for a coarse mesh tend to have a sgnfcant error. Therefore, ths author does not recommend the use of ths element.
8-0 STATIC AN YNAMIC ANALYSIS If the shear s set to zero along each sde of the element, the followng equaton s obtaned: 3 3 θ (w j - w ) - ( θ + θ j) (8.) L Hence, t s possble to drectl elmnate the md-sde relatve rotatons drectl wthout usng statc condensaton. Ths appromaton produces the screte Krchhoff Element, KE, n whch transverse shearng deformatons are set to zero. It should be noted that the SE and the KE for thn plates converge at appromatel the same rate for both dsplacements and moments. For man problems, the SE and the KE tend to be more fleble than the eact soluton. 8.9 NUMERICAL EXAMPLES { XE "Plate Bendng Elements:Convergence" }{ XE "Plate Bendng Elements:Eamples" }Several eamples are presented to demonstrate the accurac and convergence propertes of quadrlateral and trangular plate bendng elements wth and wthout transverse shear deformatons. A four-pont numercal ntegraton formula s used for the quadrlateral element. A three-pont ntegraton formula s used for the trangular element. 8.9. One Element Beam To llustrate that the plate element reduces to the same behavor as classcal beam theor, the cantlever beam shown n Fgure 8.5 s modeled as one element that s nches thck. The narrow element s 6 nches b 0. nch n plan. E0,000 ks G3,86 ks.0 k 6.0.0 Fgure 8.5 Cantlever Beam Modeled usng One Plate Element 0.
PLATE BENING ELEMENTS 8- { XE "Plate Bendng Elements:KE" }{ XE "Plate Bendng Elements:SE" }The end dsplacements and base moments are summarzed n Table 8. for varous theores. Table 8. splacement and Moment for Cantlever Beam THEORY and ELEMENT Tp splacement (nche Mamum Moment (kp-n.) Beam Theor 0.000050 6.00 Beam Theor wth Shear eformaton 0.0000587 6.00 SE Plate Element 0.0000587 6.00 KE Plate Element 0.000050 6.00 PK Plate Element Ref. [] 0.00005 3.00 Ths eample clearl ndcates that one plate element can model a onedmensonal beam wthout the loss of accurac. It s worth notng that man plate elements wth shear deformatons, whch are currentl used wthn computer programs, have the same accurac as the PQ element. Hence, the user must verf the theor and accurac of all elements wthn a computer program b checkng the results wth smple eamples. 8.9. Pont Load On Smpl Supported Square Plate { XE "Plate Bendng Elements:Pont Load" }To compare the accurac of the SE and KE as the elements become ver thn, a b mesh, as shown n Fgure 8.6, models one quadrant of a square plate. Note that the normal rotaton along the pnned edge s set to zero. Ths hard boundar condton s requred for the SE. The KE elds the same results for both hard and soft boundar condtons at the pnned edge.
8- STATIC AN YNAMIC ANALYSIS 5.0 5.0 θ 0 θ 0 5.0 5.0 E 0.9 ν 0.3 h., P.0 u z 0 0., at and 0.0, center θ 0 n 0.00, at 0.000 sdes Fgure 8.6 Pont Load at Center of Smpl Supported Square Plate The mamum dsplacement and moment at the center of the plate are summarzed n Table 8.. For a thn plate wthout shear dsplacements, the dsplacement s proportonal to /h 3. Therefore, to compare results, the dsplacement s normalzed b the factor h 3. The mamum moment s not a functon of thckness for a thn plate. For ths eample, shearng deformatons are onl sgnfcant for a thckness of.0. The eact thn-plate dsplacement for ths problem s.60, whch s ver close to the average of the KE and the SE results. Hence, one can conclude that SE converges to an appromate thn plate soluton as the plate becomes thn. However, SE does not converge for a coarse mesh to the same appromate value as the KE. Table 8. Convergence of Plate Elements b Mesh Pont Load splacement tmes h 3 Mamum Moment Thckness, h KE SE KE SE.95.383 0.355 0.73 0..95.9 0.355 0.69 0.0.95.8 0.355 0.69 0.00.95.8 0.355 0.69 0.000.95.8 0.355 0.69 To demonstrate that the two appromatons converge for a fne mesh, a 6 b 6 mesh s used for one quadrant of the plate. The results obtaned are summarzed n Table 8.3.
PLATE BENING ELEMENTS 8-3 Table 8.3 Convergence of Plate Element 6 b 6 Mesh Pont Load splacement tmes h 3 Mamum Moment Thckness h KE SE KE SE.63.393 0.587 0.570 0.0.63.6 0.587 0.595 0.000.63.6 0.587 0.595 One notes that the KE and SE dsplacements converge to the appromatel same value for a pont load at the center of the plate. However, because of stress sngulart, the mamum moments are not equal, whch s to be epected. 8.9.3 Unform Load On Smpl Supported Square Plate To elmnate the problem assocated wth the pont load, the same plate s subjected to a unform load of.0 per unt area. The results are summarzed n Table 8.. For thn plates, the quadrlateral KE and SE dsplacements and moments agree to three sgnfcant fgures. Table 8. Convergence of Quad Plate Elements 6 b 6 Mesh - Unform Load splacement tmes h 3 Mamum Moment Thckness h KE SE KE SE 9.807 0.3.. 0.0 9.807 9.85.. 0.000 9.807 9.85.. 8.9. Evaluaton of Trangular Plate Bendng Elements The accurac of the trangular plate bendng element can be demonstrated b analzng the same square plate subjected to a unform load. The plate s modeled usng 5 trangular elements, whch produces a 6 b 6 mesh, wth each quadrlateral dvded nto two trangles. The results are summarzed n Table 8.5. For thn plates, the quadrlateral KE and SE dsplacements and moments agree to four sgnfcant fgures. The fact that both moments and dsplacements converge to the same value for thn plates ndcates that the trangular elements ma be more accurate than the quadrlateral elements for both thn and thck plates. However, f the trangular mesh s changed b dvdng the quadrlateral on the other dagonal the results are not as mpressve.
8- STATIC AN YNAMIC ANALYSIS Table 8.5 Convergence of Trangular Plate Elements - Unform Load Thckness h splacement tmes h 3 Mamum Moment KE SE KE SE 9.807 0.308.5.5 0.0 9.807 9.807.5.5 0.000 9.807 9.807.5.5 0.000* 9.800 9.807..5 * Quadrlateral dvded on other dagonal It should be noted, however, that f the trangular element s used n shell analss, the membrane behavor of the trangular shell element s ver poor and naccurate results wll be obtaned for man problems. 8.9.5 Use of Plate Element to Model Torson n Beams { XE "Plate Bendng Elements:Torson" }For one-dmensonal beam elements, the plate element can be used to model the shear and bendng behavor. However, plate elements should not be used to model the torsonal behavor of beams. To llustrate the errors ntroduced b ths appromaton, consder the cantlever beam structure shown n Fgure 8.7 subjected to a unt end torque. FIXE EN 0. 6.0 0 τ E0,000,000 ν 0.30 0 τ z z γ z 0 0. T.0 Fgure 8.7 Beam Subjected to Torson Modeled b Plate Elements The results for the rotaton at the end of the beam are shown n Table 8.6. Table 8.6 Rotaton at End of Beam Modeled usng Plate Elements Y-ROTATION KE SE 6 9 9 6 9 9
PLATE BENING ELEMENTS 8-5 free 0.08 0.033 0.368 0.9 fed 0.07 0.08 0.089 0.0756 The eact soluton, based on an elastct theor that ncludes warpage of the rectangular cross secton, s 0.03 radans. Note that the shear stress and stran boundar condtons shown n Fgure 8.6 cannot be satsfed eactl b plate elements regardless of the fneness of the mesh. Also, t s not apparent f the - rotaton boundar condton should be free or set to zero For ths eample, the KE element does gve a rotaton that s appromatel 68 percent of the elastct soluton; however, as the mesh s refned, the results are not mproved sgnfcantl. The SE element s ver fleble for the coarse mesh. The results for the fne mesh are stffer. Because nether element s capable of convergng to the eact results, the torson of the beam should not be used as a test problem to verf the accurac of plate bendng elements. Trangular elements produce almost the same results as the quadrlateral elements. 8.0 SUMMARY { XE "FLOOR Program" }{ XE "SAFE Program" }A relatvel new and robust plate bendng element has been summarzed n ths chapter. The element can be used for both thn and thck plates, wth or wthout shearng deformatons. It has been etended to trangular elements and orthotropc materals. The plate bendng theor was presented as an etenson of beam theor and three-dmensonal elastct theor. The KE and SE are currentl used n the SAFE, FLOOR and SAP000 programs. In the net chapter, a membrane element wll be presented wth three OF per node, two translatons and one rotaton normal to the plane. Based on the bendng element presented n ths chapter and membrane element presented n the net chapter, a general thn or thck shell element s presented n the followng chapter. 8. REFERENCES. { XE "Ibrahmbegovc, Adnan" }Ibrahmbegovc, Adnan. 993. Quadrlateral Elements for Analss of Thck and Thn Plates, Computer Methods n Appled Mechancs and Engneerng. Vol. 0 (993). 95-09.
8-6 STATIC AN YNAMIC ANALYSIS