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Bifurcation of a square plate twisted by corner forces Miya, Raymon 1994-02-25

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BIFURCATION OF A SQUARE PLATETWISTED BY CORNER FORCESByRaymon MiyaB.A.Sc. (Mechanical Engineering), University of Waterloo, 1992A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF MECHANICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1994©Raymon Miya, 1994In presenting this thesis inpartial fulfilment of the requirementsfor an advanceddegree at the University ofBritish Columbia, I agree thatthe Library shall make itfreely available for reference andstudy. I further agreethat permission for extensivecopying of this thesisfor scholarly purposes may begranted by the head ofmydepartment or by hisor her representatives. Itis understood that copyingorpublication of this thesis forfinancial gain shall not beallowed without my writtenpermission.(Signature)Department ofCCCP1The University of British ColumbiaVancouver, CanadaDateJDE-6 (2/88)AbstractA square plate twisted by corner forcesis described by classical linear theoryas a saddlesurface. In an experiment, as the platedeforms to any noticeable deflection,it appearsnot as a saddle surface, but as a cylindricalsurface. The transformationin mode shapespresents problems in determining materialbehaviour by shear in a platetwisting experiment. The two mode shapes can be describedby either displacement or curvatureof thesurface. The purpose of this workis to investigate the bucklingof a square plate twistedby corner forces by determiningthe bifurcation point andcomparing the presentFEAwork with the experimental resultsof Howell and other resultsfound in literature. Theproblem is examined using nonlinearfinite element buckling analysis.The bifurcationpoint is determined by load-displacementplots. The critical value of Gaussiancurvatureat the centre of the plate is determinedby the Southwell plotmethod. The critical valueof Gaussian curvature is foundto occur before the bifurcationpoint. Gaussian curvatureis found to vary by an orderof magnitude over the plate atbifurcation.11Table of ContentsAbstract iiTable of Contents iiiList of TablesList of Figures viList of Symbols viiiAcknowledgement x1 Introduction 11.1 Purpose 11.2 Literature Review 22 Theory 32.1 Physics of the Problem 32.2 Finite Element Theory 62.3 Plate Theory 82.4 Southwell Plot 93 Analysis 103.1 Preprocessing 103.2 Solution 111113.3 Postprocessing.6 ConclusionsBibliography1313192024273034353T373939404142444512Curvature4 Results4.1 Deflection4.2 Southwell Plot4.3 Gaussian Curvature and Mean4.4 Curvature4.5 Midsurface Strain4.6 Fixed Plate Centre4.7 Alternate Finite Element .4.8 Non-convergence5 Discussions5.1 FEA Comparison ofPs,. and FK,.5.2 Comparisons with Bifurcation Points in Literature .5.2.1 Howell5.2.2 Ramsey5.2.3 Miyagawa, Hirata, and Shibuya5.2.4 Lee and HsuivList of Tables3.1 Material Constants103.2 Plate Geometry114.1 Critical Values from load-deflection plot164.2 Critical Values from load-deflection plot174.3 Critical Values from Southwell plot204.4 Critical Values from Gaussian-mean curvature plot234.5 Critical Values from Gaussian-mean curvature plot244.6 SHELL43 Element Critical Values from Southwell plot354.7 SHELL43 Element Critical Values from Southwell plot355.1 Comparison ofPKand P375.2 Comparison of Coefficient with Howell405.3 Modified Coefficient40VList of Figures2.1 Square Plate Twisted by Corner Forces 32.2 Saddle Surface 42.3 Cylindrical Surface 42.4 Mohr’s Circle for Curvature 53.1 Constraints 114.1 Deflection Contour Plot for 3 pinned corners at Pa, 144.2 Load-Deflection Plot for 3 pinned corners 144.3 Deflection Contour Plot for 3 pinned corners (rotated) at P 154.4 Load-Deflection Plot for 3 pinned corners (rotated) with varying a/h ratios 164.5 Load-Deflection Plot for 3 pinned corners (rotated) with varying meshdensity 174.6 Deflection Contour P1st for 3 pinned corners (rotated) atF,./2 184.7 Deflection Contour Plot for 3 pinned corners (rotated) at2P, 184.8 Southwell Plot 194.9 Critical Value of Gaussian Curvature from Southwell plot 214.10 Coefficient from Southwell plot 214.11 Gaussian-Mean Curvature Plot 224.12 Load-Gaussian Curvature Plot224.13 Load-Mean Curvature Plot234.14 Gaussian Curvature Contour Plot atPKc,254.15 Gaussian Curvature Contour Plot at Pa, 25vi4.16 Mean Curvature Contour Plot atPK 264.17 Mean Curvature Contour Plot at P . 264.18 Load-Curvature Plot 274.19 Curvature ic Contour Plot at PK.284.20 Curvature i Contour Plot at F284.21 Twist ic, Contour Plot atPKc,’ 294.22 Twist ic., Contour Plot at Ps,. .. 294.23 Midsurface Strain Plot304.24 Midsurface Strain € Contour Plot atPKc,’ 314.25 Midsurface Strain e Contour Plot atPa, 314.26 Midsurface Strain y Contour Plot atFKc,’ . . . 324.27 Midsurface Strain y Contour Plot at P,324.28 Load-Deflection Plot for fixed plate centre334.29 Load-Gaussian Curvature Plot for fixed plate centre345.1 Load-Gaussian Curvature Plot .385.2 Load-Deflection Plot385.3 Load-Deflection Plot41viiList of Symbolsa plate lengthC coefficient for the critical value of twistCa, Cb, C, Cdplate cornersD plate flexural rigidityE Young’s modulus of elasticityh plate thicknessP corner forceP nondimensionalised corner forcePa,. corner force at bifurcationFKcrcorner force at critical value of Gaussian curvature (from Gaussian-mean curvature plot)corner force at critical value of Gaussian curvature (from Southwellplot)u, v, w displacementdeflection at cornerCof platedeflection at centre of plateviiinondimensionalised deflection,es,,7strainic, ic,,, curvature of the surfaceIC, twist of the surfaceK Gaussian curvatureK,. critical value of Gaussian curvaturepmean curvatureii Poisson’s ratioixAcknowledgementThe author would like to thank Professor Hilton Ramsey for his guidance during theresearching and writing of this thesis.The author would also like to thank the Natural Sciences and Engineering ResearchCouncil of Canada for their financial support.xChapter 1Introduction1.1 PurposeThe purpose of this work is to investigate the buckling of a linear elastic, isotropicsquareplate twisted by corner forces, by determining the bifurcation point.The problem isexamined by a nonlinear finite element buckling analysis using thecommercially available software package ANSYS Revision 5.0. A square plate twistedby corner forces isdescribed by classical linear theory as a saddle surface. In an experiment,as the plate deforms to any noticeable deflection, it appears not as a saddle surface,but as a cylindricalsurface. The transformation in mode shapes presents problems indetermining materialbehaviour in shear by experiment. The findings of the present FEAwork can be used infuture study to develop a nonlinear relationship to account for thistransformation and/oran upper bound to the application of a plate twist experiment.The two mode shapescan be described by either displacement or curvature of the surface.The critical value ofcorner force at bifurcation is determined from load-displacement plots.The critical valueof Gaussian curvature at the centre of the plate is determinedfrom the Southwell plotmethod. There is a discrepancy in the results found in literatureusing different methodsof analysis and assumptions. The present FEA workis compared to the experimentalresults of Howell and other results in literature.1Chapter 1. Introduction21.2 Literature ReviewIn 1890, Kelvin and Tait noted a transition in deformation surfaces of asquare platetwisted by corner forces but did not attempt to find the point of instability.In 1971, Lee and Hsu[2j investigated the buckling problem numerically,using finitedifference methods and the nonlinear von K.rmán equations for plates. Thecritical valueof corner force at bifurcation was determined by displacement-load plots.In 1975, Miyagawa, Hirata, and Shibuya[3] investigated the buckling problemexperimentally and numerically, using deflection measurements in the experimentalapproach,and using a polynomial deformed configuration, von Kármán theory,and stress functions in the numerical approach. The critical value of corner force atbifurcation wasdetermined by load-deflection plots.In 1985, Ramsey[5] investigated the buckling problem analytically, using thekinematicresults of Green and Naghcli for small deformations superposed on a largedeformationof an elastic Cosserat surface, and the restricted form of the generalnonlinear theory ofshells and plates of Naghdi. The critical value of twist at bifurcation wasdeterminedfrom a Rayleigh quotient.In 1991, Howell[1] investigated the buckling problem experimentally, usingstrain measurements and Kirkhhoff theory to determine curvatures. The critical valueof Gaussiancurvature at bifurcation was determined by the Southwellplot.Chapter 2Theory2.1 Physics of the ProblemClassical linear theory of flat plates describes deflection w of a square plate twisted bycorner forces P (figure 2.1):U= 2(1 —v)D(2.1)in terms of the surface coordinates of the plate x,y. Flexural rigidity of the plate D:D= 12(1_v2)(2.2)is a function of Young’s modulus of elasticity E, Poisson’s ratio z’, and plate thicknessh.Deflection can also be expressed in terms of twist ic of the surface:w = ixy(2.3)FFigure 2.1: Square Plate Twisted by Corner Forces3Chapter 2. Theory 4...... .. .. .. . ..2 2 Saddle SurfaceFigureFigure 2.3: Cylindrical SurfaceThese results (equations 2.1, 2.3) are well known in fundamental classical linearplatetheory. The plate appears as a saddle surface (figure 2.2).However, in an experiment, as the plate deforms to any noticeable deflection,itappears not as a saddle surface, but as a cylindrical surface with generatorsparallelto a plate diagonal (figure 2.3).The mode of the plate can be determined by the surface characteristics with eitherdisplacement or curvature attributes. The saddle surface has equal magnitude deflectionsin the four corners relative to a fixed centre. The cylindrical surface has equal magnitudedeflections in two opposite corners and zero deflection in the other two cornersrelativeto a fixed centre.Curvature can be viewed on a Mohr’s circle for curvature (figure 2.4). The abscissaChapter 2. Theory5i2().,a. Anticlastic Curvature b. Synclastic CurvatureFigure 2.4: Mohr’s Circle for Curvaturerepresents curvature ic and the ordinate represents twist iiof the surface. Principalcurvatures are ic, ic2. The saddle surface is anticlastic—the two principalcurvatureshave opposite signs (figure 2.4a). Principal directions are parallel tothe plate diagonals.Principal curvatures are equal and opposite, resulting in zero mean curvatureand negativeGaussian curvature. The cylindrical surface is synclastic—curvatures in allorientationshave like signs (figure 2.4b). Principal directions are parallel to the platediagonals. Oneprincipal curvature is zero and the other non-zero, resulting in a non-zero meancurvatureand a zero Gaussian curvature.Classical linear theory of flat plates neglects all quadratic termsin the Green-Lagrangestrain:= (2.4)=vY+(uY+vY+wY) (2.5)= (2.6)= v+ u, + (uu + vv + ww) (2.7)= w,+v+ (uu + + ww) (2.8)7zx= U + W + (uu + vv + WZW) (2.9)The approximation of neglecting the nonlinear terms failsto account for the deformation in the middle plane of the plate due to bending. Midsurfacestrains can onlyChapter 2. Theory6be neglected if the defiections of the plate are small in comparisonwith its thickness innon-developable surfaces (non-zero Gaussian curvature, such as saddle shapes,spheres)or the defiections are of the order of its thickness in developable surfaces(zero Gaussiancurvature, such as cylinders, cones)[9]. Because of this approximation, classicallinearplate theory cannot predict buckling.2.2 Finite Element TheoryThe finite element used in the analysis is an 8 node isoparametricquadrilateral shellelement. It is labeled SHELL93 in the ANSYS Revision5.0 element library. There are5 degrees of freedom per node: 3 translations and 2 rotations. This element includesfeatures of Green-Lagrange strains and Mindlin plate theory. Green-Lagrangestrains(equations 2.4—2.9) take into account midsurface strains of the plate.Mindlin platetheory allows for transverse shear deformation. This means that a linethat is straightand normal to the midsurface before loading, is assumed toremain straight but notnecessarily normal to the midsurface after loading. Displacementsu, v of a point in theplate a distance z from the midsurface are:u = ii—zc (2.10)v =(2.11)where a, /3 are small angles of rotation of a line that was normal to the midsurfacebeforeloading and i, i are the displacements at the plate midsurface. Strainsc,e.g, and shearstrains7, 7yz,are:= (2.12)= (2.13)Yxy = +,+ (ii, + + ww) — z(/3 + a) (2.14)Chapter 2. Theory77yz= Wy+ + (ii + iJYiYZ + ww) — (2.15)7zx= z+wx+(z11x+t1zUx+wzwx)—a (2.16)Strains,and shear strain y, are assumed to vary linearly through the platethickness.Transverse shear strains y,7zaare assumed to be constant through the plate thickness.In the stress-strain relationship:{a} [D]{€}(2.17)the stress vector {o-}, the strain vector {e}, and the materialproperty matrix for theelement [D] are defined as:{a}LXa T T2 Tzx (2.18){e}= [xEy7xy 7yz 7zxj(2.19)lu 0 0 0ui 0 0 0E[D] —0 0 0 0(2.20)0 0 0 0o o 0 02wherefis the shear factor:11.2, A/h2 <25f=— (2.21)(1.0+0.2, A/h2 > 25where A is the area of the element and h is theplate thickness. The shear factor isdesigned to avoid shear locking. As the element becomes thin,the A/h2 ratio becomeslarge. The shear factorfis thus increased and the stiffness associatedwith the transverse shears is reduced. The correct method toavoid shear locking is through selectiveintegration, but ANSYS does not accommodate this. The SHELL93element uses a 2 x 2reduced quadrature rule.Chapter 2. Theory82.3 Plate TheoryKirkhhoff theory is used to calculate curvatures from strain output of the finite elementsoftware. This is to be consistent with Howell’s experimental analysis so results canbecompared. Kirkhhoff theory neglects transverse shear deformations. Thismeans that aline that is straight and normal to the midsurface before loading, is assumed toremainstraight and normal to the midsurface after loading.Extensional strain e at an arbitrary point a distance z from the platemidsurface is:= m + (2.22)where the membrane strain6mappears along the plate midsurface, and the curvatureicis associated with bending strain.Solving the above equation for the top and bottom of the plate and equatingmidsurface strains gives the curvaturesicr, ic1, and the twist t of the midsurface:=Ii(2.23)—El,=_______where ê, € are the top and bottom surface strains of theplate respectively, and h is theplate thickness.Principal curvatures i, K2 from Mohr’s circle of curvatures are:IC1,1C2 = ±;C)2+, (2.26)Mean curvature p. is the average of the two principal curvatures:p.=+i)=(i+‘2)(2.27)Chapter 2. Theory9Gaussian curvature K is product of the two principal curvatures:K=— 1Sry(2.28)2.4 Southwell PlotThe Southwell plot is a common method to determine the elastic buckling loadof astructural system. In experiments, there exists some imperfection in the undeformedshape and/or applied loading. As the compressive load increases, the lowest critical loadbuckling mode dominates. A linear function can be expressed in terms of applied loadand deflection by neglecting contributions from higher modes.In 1932, Southwell considered a simply supported column with an initial imperfectionsubjected to a compressive load P[6]. He expressed a linear relationship:S 1 1= —S+ —a (2.29)in terms of the incremental deflection 5, the Euler loadFe,.,and coefficient a. TheSouthwell plot of S/P versus S gives a straight line whose slope is equal to the inverseofthe buckling load.The Southwell plot method claims accuracy only as P —*Ps,..Spencer[7] states thatconstructing Southwell plots using K.rmn’s strut data with loads up to O.91P,., toO.88P,., and to O.82P,. (Pa,. being defined as the critical load which Southwell obtainedby plotting Kármán’s data to O.98F,.) gives errors of 3, 5, and 25 percent respectively.The critical load P,. is a theoretical concept and should be independent of initialdeflection. Spencer[7] showed that in buckling of a uniaxially compressed simply supportedplate, the Southwell plot begins to underestimate the critical load when:w0/h > 0.5(2.30)where w0 is the initial deflection at the plate centre and h is the plate thickness.Chapter 3AnalysisThe analysis was performed on a SUN SPARC workstation. The preprocessing and thesolution utilized ANSYS Revision 5.0, and the postprocessing utilized FORTRAN77 andTECPLOT Revision 5.0.3.1 PreprocessingThe plate is modelled with the SHELL93 8 node isoparametric shell element.The plate material is modelled as T6061-T6 Aluminium (table 3.1) for comparisonwith the experimental results of Howell{1j. Material nonlinearity, such as plasticity,isnot considered in the analysis.The plate geometry is square with plate length to thickness ratios a/h (table3.2) forcomparison with the experimental results of Howell[1].The plate is meshed with square elements N per side, where N is even to provide anode at the centre of the plate to take displacement and strain measurements—the samelocation as Howell’s strain gauges [1]. There are a total of N2 elements and (3N2+4N +1)nodes for the model.Table 3.1: Material ConstantsE 69x109 Pai’ 0.3310Chapter 3. Analysis11Table 3.2: Plate Geometrya/h ratio a (m) h (m)49.2 0.1524 0.00309963.2 0.2032 0.00321680.3 0.2540 0.00316296.0 0.3048 0.003175196.7 0.6096 0.003099w0Cd Ccy,vL,Ca Cbuvw0 vw0Figure 3.1: Constraints3.2 SolutionThe plate is constrained at cornersCa, Cb, Cd (figure 3.1) to zero displacement in the zdirection, to simulate the self equilibrating corner forces associated with theapplied forceat corner C. These are the same constraints in the experiment by Howell[1].To preventrigid body motion, additional DOF constraints are specified. The plate is constrainedat cornerCato zero displacement in the x andy directions to prevent translation, andconstrained at cornerCb to zero displacement in the y direction to prevent rotation.These constraints satisfy the kinematic, but not the static boundary conditionsof aplate with free edges.When applying the Southwell plot method to find critical values, an initial hydrostaticChapter 3. Analysis12pressure is applied in the positive z direction. A nominal value of hydrostaticloading isused which produces deflections small compared to the plate thickness.The applied force P at corner C is in the positive z direction.These boundary conditions provide a stable post buckling response. The appliedcorner force P can exceed the value at the bifurcation pointPawithout the instabilityof ill conditioned matrices, such as a negative main diagonal in the stiffness matrix.Body forces, such as gravity loads, are not included in the analysis.3.3 PostprocessingDisplacements 5,, 6 are calculated at the centre of theplate and at corner C. Thecritical value of corner force at bifurcation is determined from load-deflectionplot.•Strains c3,, e,7xyare calculated at the top and bottom surfaces at the node at thecentre of the plate using nodal point averaging in ANSYS. Thesevalues are exported toa FORTRAN code which calculates curvatures using Kirkhhoff platetheory. The criticalvalue of Gaussian curvature Kr,. is determined from the Southwell plot.The Southwellplot uses p. as the abscissa and p./K as the ordinate. The asymptoticbehaviour of thecurve determines K asK/KCI.—*1.Chapter 4Results4.1 DeflectionThe deflection for the plate with 3 pinned corners is zero at the pinned cornersCa, Cb,Cd and a maximum at the corner with the applied force C, (figure 4.1).The load-deflection curve of corner with the applied force C is smooth and showsnoindication of buckling (figure 4.2). The load-deflection curve of the centre of theplatehas an abrupt change in the slope at the bifurcation pointPa,..The finite element analysis deflection of the plate centre agrees well withlinear theory(equation 2.1) for deflections less than a plate thickness (figure 4.2). The FEAdeflectionof the plate cornerC agrees well with linear theory for deflections less than 4 ,platethicknesses.The finite element results of the plate with 3 pinned corners can berotated to showthe characteristic surface. The plate can be rotated so the deflections of cornersCa andC are equal, and translated so the deflection of the centre of the plate is zero. Thedeflections becomeS/2 —SatCaand C, and atCband Cd (figure 4.3), where SandSare the deflections of the unrotated results for the centre of theplate and cornerCrespectively.The bifurcation point is where the magnitude of rotated cornersCa, C and rotatedcornersCb, Cdsignificantly diverge. The critical value of corner load variesslightly withHowell’s[l] a/h ratios (figure 4.4 and table 4.1). The mesh densityof 144 elements is13Chapter 4. ResultsDeflection Contour Plot3 pInned corners at P96 h rallo, 144 elementslevel öAi15.945913.910711.87569.840417.805255.770093.734921.6997614Figure 4.1: Deflection Contour Plot for 3 pinned corners at F,.25201510500load Pa2I2Dh30Load-Deflection Plot3 pinned corners96 ajh rallo, 144 elements5 10 15 20deflection jhFigure 4.2: Load-Deflection Plot for 3 pinned cornersChapter 4. Results15Deflection Contour Plot3 pinned corners (rolaled) at P,96 h ratio, 144 elementsöih3.442622.443671.444720.44577-0.55318-1.55213-2.55108-3.55003levelI7654Figure 4.3: Deflection Contour Plot for 3 pinned corners (rotated) atPc.sufficient to provide displacement convergence (figure 4.5 andtable 4.2).Below the bifurcation point, the magnitudes of deflection for rotatedcornersCa, Ccand rotated corners Cb,Cdare almost equal (figure 4.4). The plate is bendingto a saddlesurface (figure 4.6).Above the bifurcation point, the magnitude of deflection for rotated cornersCa, Cis decreasing, and the magnitude of deflection for rotatedcornersCb, Cd is increasing.(figure 4.4). The plate is bending to a cylindrical surface(figure 4.7).Nondimensionalized corner force P is defined as [3]:—p2Pj (4.1)Nondimensionalized deflection is defined as [3]:—Sh(4.2)Chapter 4. Results16load Pa2I2Dh3025201510500.0Figure 4.4: Load-Deflection Plot for 3 pinned corners (rotated) with varying a/hratiosTable 4.1: Critical Values from load-deflection plot144 elementsa/h ratio P (N) P, &,49.2 1380 26.9 4.17 4.4863.2 870 26.0 4.32 4.5280.3 510 25.5 4.41 4.5396.0 355 25.1 4.45 4.55196.7 80 24.9 4.54 4.58Load-Deflection Plot3 pinned corners (rotated)144 elements49.2a/hratio --63.2-.- —- ... —--•_%_.rotated COrnS C1,C, \\- ----:_-—80.3V..———rotated corners C,,, C11.0 2.0 3.0 4.0 5.0 6.0deflection hChapter 4. Results17Load-Deflection Plot3 pinned Corners (rotated)96 h ratio256 elementsrotated corners C,, C1Cd25201510500.0 1.0 2.0 3.0 4.0 5.0 6.0deflection hFigure 4.5: Load-Deflection Plot for 3 pinned corners (rotated) with varying meshdensityTable 4.2: Critical Values from load-deflection plot96 a/h ratioelements Pc,. (N)Sa,S&, 8d16 352 25.0 4.45 4.5564 355 25.1 4.45 4.55144 355 25.1 4.45 4.55256 355 25.1 4.45 4.558/h2.538511.812771.087030.361294-0.364444-1.09018-1.81592-2.541668/h1.705020.081854-1.5413 1-3.16448-4.78764-6.41081-8.03397-9.65714Chapter 4. Results 18Deflection Contour Plot3 pInned corners (rotated) at P,,,/ 296 h ratio, 144 elementslevel7654Figure 4.6: Deflection Contour Plot for 3 pinned corners (rotated) at Pc,./2Deflection Contour Plot3 pInned corners (rotated) at 2 P,,96 h ratio, 144 elementslevel87654321Figure 4.7: Deflection Contour Plot for 3 pinned corners (rotated) at 2Pc,Chapter 4. Re8ults 19Southwell Plotii 1K Im]at centre of plate-0.020096 alh ratio, 144 elements-0.0150-0.010010.0 Pa.—..—...... // 1.OPa-0.0050 •..,‘0.lPa-0.0000 I I0.0000 0.0005 0.0010 0.0015 0.0020Im1Figure 4.8: Southwell Plot4.2 Southwell PlotHowell[1] determined the critical value of Gaussian curvature using theSouthwell plotmethod. The Southwell plot method requires an initial curvature in the structure.The FEA Southwell plot is constructed from strains at the centreof the plate with3 pinned corners and initial hydrostatic pressure. The Southwell plot producesparallellines for varying intensity of initial hydrostatic pressure (figure 4.8).The initial hydrostatic pressure creates an initial deflection of the centreof the plateS. The critical value of Gaussian curvature and corner force is determinedby the Southwell plot method are not affected by deflections S less than onetenth of aplate thickness (table 4.3).The coefficient C is defined as [5]:C = (4.3)Chapter 4. Results20Table 4.3: Critical Values from Southwell plot96 a/h ratio, 144 elements-o —pressure (Pa) S,PScr(N)Ps,.C0.1 0.0000961 284 20.1 9.041 0.000961 284 20.1 9.0410 0.00961 284 20.19.03100 0.0961 274 19.4 9.01500 0.478 226 16.0 8.65for the critical value of twist ic at the centre of the plate.For load levels less than Ps,., the Southwell plot method overor under predictsKcj.(figure 4.9) and C (figure 4.10) depending on the magnitude of the initialdeflection.4.3 Gaussian Curvature and Mean CurvatureThe Southwell plot determines the critical value of Gaussian curvaturewhere the slope onthe Gaussian-mean curvature plot (figure 4.11) is zero[7]. The Gaussian-meancurvatureplot is constructed from strain calculations at the centre of theplate with 3 pinnedcorners and no initial hydrostatic pressure.Gaussian curvature is zero for the undeformed plate (no initial curvature),increases inmagnitude as the plate deforms to a saddle surface,reaches a maximum value, beginstodecrease in magnitude, and after bifurcation decreasesin magnitude as the plate deformsto a cylindrical surface (figure 4.12).Mean curvature is zero for the undeformed plate (no initial curvature)remains zeroas the plate deforms to a saddle surface, and after bifurcationincreases in magnitude asthe plate deforms to a cylindrical surface (figure4.13).The corner load at the critical value of Gaussian curvaturePK,. (tables 4.4—4.5) isless than the corner load at bifurcationPa,. (figure 4.12).Ksrm2J-0.2000.000 —0.0000CoefficientCMfrom Southwell Plot20atcentre of plate96 h ratio, 144 elements1510.0 Pai.opa0.1 PaChapter 4. Results21Critical Value of Gaussian Curvaturefrom Southwell Plotat centre of plate96 h ratio, 144 elements1.0 Pa,loçPa•—0.1 Pa-0.150-0.100-0.0500.0010 0.0015 0.0020p(m1]0.0005Figure 4.9: Critical Value of Gaussian Curvature fromSouthwell plot1050 —0.0000 0.0005 0.0010p jmt]0.0015 0.0020Figure 4.10: Coefficient from Southwell plotChapter 4. Results 22Gaussian-Mean Curvature Plotat centre of plate96 h ratio, 144 elements-50Gaussian curvature Ka41h20.0050 0.0100 0.0150 0.0200mean curvature ImiGaussiancurvature Ka4/h2-75-50-2500.0000Figure 4.11: Gaussian-Mean Curvature Plotload Pa2I2Dh25 -2015 -10 -5.Load-Gaussian Curvature Plotat centre of plate144 elements0 -25 -75Figure 4.12: Load-Gaussian Curvature PlotChapter 4. Results 23Load-Mean Curvature Plotat centre of plate96 alh ratio, 144 elements0.050 0.100 0.150 0.200 0.250 0.300 0.350mean curvature ji 1m1Figure 4.13: Load-Mean Curvature PlotTable 4.4: Critical Values from Gaussian-mean curvature plot144 elementsload Pa2I2Dh302520 -15 -10 -50 —0.000a/h ratioPK7.(N)PKC,C49.2 1112 21.7 9.0163.2 696 20.8 9.0180.3 406 20.3 9.0296.0 284 20.1 9.03196.7 63 19.7 9.06Chapter 4. Results 24Table 4.5: Critical Values from Gaussian-mean curvature plot96 a/h ratioelementsPK,. (N)PKcrC16 278 19.7 9.0964 276 19.6 9.03144 284 20.1 9.03256 286 20.3 9.01Gaussian curvature is a minimum absolute value at the centre of the plate, a maximumabsolute value near the corners of the plate, and varies over the plate by an order ofmagnitude at PK (figure 4.14) and P (figure 4.15).Mean curvature is a zero at the centre of the plate and varies positive and negativevalues over the plate at (figure 4.16) andPa,. (figure 4.17).4.4 CurvatureThe load-curvature plot is constructed from strains at the centre of the plate with3pinned corners and no initial hydrostatic pressure. Curvatures i, ,, are zero for theundeformed plate (no initial curvature), remain zero as the plate deforms to a saddlesurface, and after bifurcation increase in magnitude as the plate deforms to a cylindricalsurface (figure 4.18).Twist ic is zero for the undeformed plate (no initial curvature), increases in magnitude as the plate deforms to a saddle surface, reaches a maximum value, begins todecrease in magnitude, and after bifurcation continues to increase in magnitudeas theplate deforms to a cylindrical surface.The FEA twist ic, agrees well with linear theory (equations 2.1—2.3)for defiectionsless than half the plate thickness, and agrees well with membrane stress theory ([4] andKa41h2-57.4374-114.875-172.3 12-229.75-287.187-344.625-402.062-459.5Ka4/h2-81.7554-163.5 11-245.266-327.022-408.777-490.533-572.289-654.044Chapter 4. Results25Gaussian Curvature Contour Plotat P96 h ratio, 144 elementslevelT:,8654Figure 4.14: Gaussian Curvature Contour Plot atPK,.Gaussian Curvature Contour Plotat P0,96 h ratio, 144 elementsleveli...87654Figure 4.15: Gaussian Curvature Contour Plot at PChapter 4. Results 26p [n11]0.3255760.2325260.1394750.046424-0.046625—0.139676-0.232727-0.325777p[mj0.4604310.3297320.1990320.068332-0.062366-0.193066-0.323766-0.454465Mean Curvature Contour Plotat P96 h ratio, 144 elementslevel7654Figure 4.16: Mean Curvature Contour Plot atPKlevel7654Mean Curvature Contour Plotat Pt,,96 alh ratio, 144 elementsFigure 4.17: Mean Curvature Contour Plot atPa,.Chapter 4. Results27Load-Curvature Plotat centre of plate96 alh ratio 144 elementsFigure 4.18: Load-Curvature Plotequation 2.22) for defiections less than a plate thickness(figure 4.18).Curvature ic., is a zero at the centre of the plate and varies positiveand negativevalues over the plate atPK (figure 4.19) and Pa,. (figure 4.20).Twist tc, is a minimum absolute value at the centre of the plate,a maximum absolutevalue near the corners of the plate, and varies over theplate by an order of magnitudeatPK(figure 4.21) and P (figure 4.22).4.5 Midsurface StrainThe load-midsurface strain plot is constructed from strains at the centreof the plate with3 pinned corners and no initial hydrostatic pressure. Midsurface strains, ,are zerofor the undeformed plate (no initial curvature), are equal and compressiveas the platedeforms to a saddle surface, and after bifurcation decrease in magnitudeas the plateloadFEAx1,FEA x, K72015-10 .5 0curvature ica2lh5 10Chapter 4. Results28Ka2/h8.114095.799913.485741. 17156-1. 14262-3.4568-5.77097-8.08515ica2/h11. 12847.975634.822861.67009-1.48269-4.63546-7.78823-10.941Curvature x Contour Plotat P,96 h ratio, 144 elementslevelj...8:654321Figure 4.19: Curvature ic,.., Contour Plot atPKCurvaturexContour Plotat P0,96 h ratio, 144 elementslevel65432Figure 4.20: Curvature ic, Contour Plot atFcj.Chapter 4. Results29ica2/h-2.57822-5.15645-7.73467-10.3129-12. 89 11-15.4693-18.0476-20.6258-3.09801-6.19602-9.29404-12.392-15.4901-18.5881-21.6861-24.7841Twist Contour Plotat P,96 alh ratio, 144 elementsleveli..87654Figure 4.21: Twist ic Contour Plot at Pjç,.Twist Contour PlotatP,96 h ratio, 144 elementslevel.....8:::::654Figure 4.22: Twist ic, Contour Plot at P.Chapter 4. Results30Load-Midsurface Strain Plotatcenh’eotpteload Pa2!2Dh96 alh ratio, 144 elements30 -25Yxy2015e105 --0.0.00 -0.25 -0.50 -0.75 -1.00 -1.25 -1.50 -1.75strain £a21hFigure 4.23: Midsurface Strain Plotdeforms to a cylindrical surface (figure 4.23).Midsurface shear strainy, is zero for the undeformed plate (no initial curvature),remains zero as the plate deforms to a saddle surface, and afterbifurcation increases inmagnitude as the plate deforms to a cylindrical surface.The midsurface of the plate is in maximum compressionat the centre of the plateand maximum tension at the edge of the plate atPK (figure 4.24) and P (figure 4.25).Midsurface shear strain-y, is zero at the centre of the plate and varies positive andnegative values over the plate atPK (figure 4.26) and P (figure 4.27).4.6 Fixed Plate CentreThe FEA plate buckles without an initial perturbation because ofthe 3 pinned cornerconstraints—the same used by Howell[1j. As the plate deforms onlycorner C is free toChapter 4. Results31ea2)h7.653346.33035.007273.684232.361 191.03815-0.284883-1.60792Ea21h9.602647.942386.282124.621862.96161.30134-0.35892-2.0 1918Midsurface Strain Contour Plotat P,96 h ratio, 144 elementslevelj87654Figure 4.24: Midsurface Strain , Contour Plot atPKlevelI.8III654Midsurtace Strain Contour Plotat P96 a(h ratio, 144 elementsFigure 4.25: Midsurface Strain Contour Plot atPa,.Chapter 4. Results 32ya2/h4.435713.167641.899570.6315-0.63657-1.90464-3.1727 1-4.44078ya2/h5.414963.859862.304770.749669-0.805429-2.36053-3.91562-5.47072Midsurlace Shear Strain y, Contour Plotat P,96 aTh ratio, 144 elementslevelI.V:.:V:87654Figure 4.26: Midsurface Strain -y, Contour Plot atPKc,Midsurface Shear Strain Contour Plotat P0,96 h ratio, 144 elementslevelj....8654Figure 4.27: Midsurface Strain-y,Contour Plot atPs,.Chapter 4. Results 33Load-Deflection Plotcorner deflections relative to centre of plateload Pa2!2Dh96 h ratio, 144 elements30253 pinned corners (rcted) ,‘/‘ fixed plate centre201510500.0 5.0 6.0Figure 4.28: Load-Deflection Plot for fixed plate centredeflect. The corner forces remain parallel to the z axis and are no longer normal to thetangent plane of the centre of the plate.Constraints which do not initiate buckling are created by fixing the centre of the platein 5 degrees of freedom: displacements u, v, w and rotations about the x andy axis. Thedrilling rotation about the z axis is fixed by a constraint onCa in the direction of they = —x diagonal. The applied corners loads are P atCa and C and —P at C, and Cd.The orientation of the corner forces in the fixed plate centre loading case remainnormal to the tangent plane of the centre of the plate. There is no perturbation, and theplate is loaded beyond the bifurcation point without experiencing buckling (figure 4.28).The critical value of Gaussian curvature remains the same for the 3 pinned cornersand the fixed plate centre loading cases (figure 4.29).The 3 pinned corners and the fixed plate centre loading cases create membrane tensionfor large deflections due to the corner forces remaining in their original orientation and1.0 2.0 3.0 4.0deflection hChapter 4. ResultsFigure 4.29: Load-Gaussian Curvature Plot for fixed plate centrestretching the plate. Further study involving “follower forces” which remain normal tothe plate surface is recommended to study the effects of the added membrane tension.3425 -Load-Gaussian Curvature Plotat centre of plate96 alhralio,l 44 elements3 pinned cornersload Pa’I2Dh20 -15 -1050 -25 .50Gaussian curvature Ka41h2.754.7 Alternate Finite ElementThe finite element analysis was also performed modelling the plate with the SHELL434 node shell element. The SHELL43 element is claimed by ANSYS to be well suitedto model nonlinear thin to moderately-thick shell structures[8]. The SHELL43 elementaccommodates rotational degrees of freedom and shear deformations but since the problem under consideration is highly nonlinear, the bilinear SHELL43 element would notbe expected to model the plate as well as the quadratic SHELL93 element. The platemodelled with SHELL43 elements did not buckle for the loading case of 3 pinned cornersand no initial hydrostatic pressure. The SHELL43 element model only buckled withChapter 4. Results35Table 4.6: SHELL43 Element Critical Values from Southwell plot96 a/h ratio, 400 elementspressure (Pa) P5 (N)PScr1 0.00094 did not buckle5 0.0047 did not buckle10 0.0094 did not buckle20 0.018 did not buckle30 0.028 295 20.940 0.037 292 20.750 0.047 285 20.2100 0.094 285 20.2500 0.47 225 15.9Table 4.7: SHELL43 Element Critical Values from Southwell plot96 a/h ratio, 50 Pa hydrostatic pressureelements P5 (N)PScr16 did not buckle64 did not buckle144 285 20.2256 285 20.2400 285 20.2an initial deflection greater than 0.3 plate thicknesses (table4.6) and greater than 144elements (table 4.7).For the reasons of problems in buckling, the SHELL43 elementwas not used in theanalysis.4.8 Non-convergenceThe post buckling response for the finite element analysis of the196.7 a/h ratioplatewith 3 pinned corners does not converge. The plate bendsto a saddle surface up toChapter 4. Results36the bifurcation point without numerical difficulties. Shear lockingdoes not seem to bea factor, since increasing the mesh density—decreasing the elementa/h ratio—does notrectify the problem.Chapter 5Discussions5.1 FEA Comparison of Pc,. andPK,.Gaussian curvature at the centre of the plate reaches a maximum absolute valueK,. atcorner force The absolute value of K then begins to decrease in magnitudebeforethe bifurcation point Pc,. (figure 5.1).Attempting to determine the bifurcation point by a critical Gaussian curvaturecriterion, such as the Southwell plot method, will underestimate the bifurcationpoint (figure 5.2).The critical Gaussian curvature pointPKc,.is 80 percent of bifurcation pointPa,.(table 5.1).The values of K,. andPKc,.remain the same for the 3 pinned corners and the fixedplate centre loading cases (figure 4.29). The present work uses cornerforces which maintain their original vertical direction parallel to z axis. For large defiections,the orientationTable 5.1: Comparison ofPKc,andPc,.a/h ratio Pc,.PKcr PKc,./Pc,.49.2 26.9 21.7 0.8163.2 26.0 20.8 0.8080.3 25.5 20.3 0.8096.0 25.1 20.1 0.80196.7 24.9 19.7 0.7937Chapter 5. Discussions38Figure 5.1: Load-Gaussian Curvature PlotLoad-Deflection Plot3 pinned corners (rotated)96 h ratio, 144 elementsI I.... I.... . . .1....1.0 2.0 3.0 4.0defieclion jhLoad-Gaussian Curvature Plotat centre of plate144 element96-25 .50 .75Gaussian curvature Ka4/h2load Pa2J2Dh25201510 -5-00load Pa2j2Dh30252015100•0.0cr5.0 6.0Figure 5.2: Load-Deflection PlotChapter 5. Discussions39of the forces causes tensile membrane stresses which may havean effect on the value ofPK4.Further investigation using “follower forces” which remain normal tothe platesurface is recommended.5.2 Comparisons with Bifurcation Points in Literature5.2.1 HowellHowell determined the bifurcation point by experiment[1]. The constraintson the platein the experiment were three corners pinned and the loaded corner free todeflect withthe load applied by a constant direction tensile cable. The 3 pinned cornersloading casein the present FEA models this experimental setup. Strain gauges measuredstrains onthe top and bottom surfaces of the plate, and Kirkhhoff theory was used tocalculate thecurvatures from the strains. The critical value of Gaussian curvature atbifurcation wasdetermined using the Southwell plot method.Howell gives the bifurcation point:= 1O.8h/a (5.1)for the critical value of twist , at the centre of the plate.The present FEA work using the Southwell plot method gives:= 9.Oh/a (5.2)The difference between the result of Howell andthe FEA is mainly due to Howell’slimit of applied corner force. Howell limited the maximum appliedcorner forceP0hltoavoid plastic yielding of the material. This corresponds tocorner loads less than half ofFKcr(table 5.2). The Southwell Plot method only claims accuracyas the load approachesthe critical load P — Pa,. (section 2.4).Chapter 5. Discussions40Table 5.2: Comparison of Coefficient with Howella/h ratioCHohlcHoweUpHowell/p49.2 17.85 9.01 4.8 0.1880.3 11.07 9.02 10.0 0.3996.0 10.61 9.03 9.9 0.39196.7 10.25 9.06 10.9 0.43Table 5.3: Modified Coefficient—HowellatFma= 9.996 a/h ratio, 144 elementspressure (N) C’0.1 8.91 9.510 10.0100 9.9300 9.4500 8.9The magnitude of initial deflection affects the Southwell’s plot predictionof the criticalvalue. The initial deflection for the experiment of Howell is unknown(figure 5.3).The Southwell plot method using Gaussian and mean curvaturesonly up topoe11gives modified coefficient C’ values closer to Howell’s results (table5.3).5.2.2 RamseyRamsey determined the bifurcation point by analytical methods[5].The kinematic results of Green and Naghdi for small deformations superposedon a large deformation ofan elastic Cosserat surface, and the restricted form of thegeneral nonlinear theory ofshells and plates of Naghdi were used. The critical value of twist ,at bifurcation wasdetermined from a Rayleigh quotient.Chapter 5. Discussions41Southwell Plotat centre of platepIKImI 96hralio-0.75 - - - -Howell - --0.50 - - - - - - -FEA 500 Pa initial hydrostatic pressure-0.25 -FiOPa..PEA lOOPa5.i.. i... I i... i.....1J.vu0.000 0.010 0.020 0.030 0.040 0.050 0.0601m1iFigure 5.3: Load-Deflection PlotRamsey gives the bifurcation point:= 3.29h/a(5.3)for the critical value of twist ic at bifurcation atthe centre of the plate.The present FEA work using the Southwell plot method givesresults in equation 5.2.The difference between the result of Ramsey andthe FEA is mainly due to Ramsey’s assumption of the Gaussian curvature behaviour.Ramsey assumed the Gaussiancurvature at bifurcation to be uniform over the plate.The present FEA work shows theGaussian curvature varies by an order of magnitude overthe plate (figure 4.14—4.15).5.2.3 Miyagawa, Hirata, and ShibuyaMiyagawa, Hirata, and Shibuya determined the bifurcation pointby experimental andnumerical methods [3].Chapter 5. Discussions 42In the experiment of Miyagawa et al., the critical value of corner force at bifurcationwas determined from the load-deflection plot. The plate dimension ratio varied 40<a/h < 120. Bifurcation occurred only at ratios a/h> 80.Miya.gawa et al. give the bifurcation point experimentally:= []= 21 (5.4)for the dimensionless corner force .In the numerical work of Miyagawa et al., the deformed configuration of the plate wasapproximated as a polynomial. Stresses in the middle of the plate were approximatedby combining von Kármn theory, an assumed stress function, and experimental results.The relation between load and deflection was determined by minimizing the total energyof: strain energy due to bending and twisting, strain energy in the middle of the platedue to membrane stretching, and work done by the loads.Miyagawa et al. give the bifurcation point numerically:= 22.8 (5.5)The present FEA work using load-deflection plot gives:Pc.2. = 25 (5.6)In the experiment of Miyagawa et al., the four loading points were applied by flat rollerbearings which simulated “follower loads” to reduce the stretching forces along the plate.The plate material experienced plastic yielding resulting in the experimental bifurcationpoint of Miyagawa et al. lower than the numerical bifurcation point of Miyagawa etal. [3].5.2.4 Lee and HsuLee and Hsu determined the bifurcation point by finite difference methods[2]. The criticalvalue of corner force at bifurcation was determined by the displacement-load plot.Chapter 5. Discussions43Lee and Hsu give the bifurcation point:= [(12(1_v2))p] = 21 (5.7)for the dimensionless corner force M.The present FEA work using load-deflection plot gives:Mc,. = 61 (5.8)The difference between the result of Lee and Hsu and the FEA is mainlydue to thelimited model of Lee and Hsu. The mesh used by Lee and Hsu inthe finite differencescheme was not dense enough to provide convergence of M. No attemptwas made tocalculate Me,. more precisely.Chapter 6ConclusionsDescribing the surface of a square plate twisted by corner forces basedon either displacement or curvature values gives different results for the critical point.The load-displacement plot determines the bifurcation pointFe,.. The present FEA work gives= 25. The Southwell plot based on curvature determines the criticalGaussian curvature point PK. The present FEA work gives Pjç,. = 20.The present FEA work gives the coefficient for the critical value oftwist at thecentre of the plate C = 9.0 from the Southwell plot. Thisresult compares well with theexperiment of Howell taking into account the low load levels Howell used toavoid plasticyielding of the material. Southwell plots constructed from curvaturedata of load levelsless thanPKc,,will overpredict the calculated value ofPKc,’for initial defiections of theplate centre between 0.001 < S,/h< 0.5.The result of the present FEA work does not compare well with theanalytical work ofRamsey. Ramsey assumed Gaussian curvature to be uniform overthe plate at bifurcation.The present FEA work shows that the problem is highly nonlinear andGaussian curvaturevaries over the plate by an order of magnitude at and Pa,..The applied forces in the present FEA work maintain their original orientationevenfor large deflections. This will create significant tensile membrane stressesin the platefor defiections much larger than the plate thickness. Further FEA investigationinvolving“follower forces” which remain normal to the plate surface, and inclusionof nonlinearmaterial properties is recommended.44Bibliography[1] R. A. Howell. An experimental investigation of the bifurcation in twisted squareplates. Master’s thesis, University of British Columbia, 1991.[2] S. S. Lee and C. S. Hsu. “Stability of Saddle-like Deformed Configurations of Plateand Shallow Shells”. International Journal of Non-linear Mechanics, 6:221—236, 1971.[3] M. Miyagawa, T. Hirata, and S. Shibuya. “Deformation of Square Plates underContrary Transverse Load”. Memoirs of Faculty of Technology: Tokyo MetropolitanUniversity, 25, 1975.[4] W. Ramberg and J. A. Miller. “Twisted Square Plate Method and Other Methodsfor Determining the Shear Stress-Strain Relation of Flat Sheet”. Journal of Researchof the National Bureau of Standards, 5O(2):111—123, 1953.[5] H. Ramsey. “A Rayleigh Quotient for the Instability of a Rectangular Plate with FreeEdges Twisted by Corner Forces”. Journal de Mdchanique tMorique et appliqueé,4(2):243—256, 1985.[6] R. V. Southwell. “On the Analysis of Experimental Observations in Problems ofElastic Stability”. In Proceedings of the Royal Society, volume 135 of A, pages601—616, London, 1935.[7] H. H. Spencer and A. C. Walker. “Critique of Southwell Plots with Proposals forAlternative Methods”. Experimental Mechanics, 15(8): 303—310, 1975.[8] Swanson Analysis Systems Inc. ANSYS User’s Manual for Revision 5.0, 1992.[9] S. Timoshenko and S. Woinowsky-Krieger. Theory of Plates and Shells. McGraw-HillBook Company, 1959.45

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