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

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BIFURCATION OF A SQUARE PLATE TWISTED BY CORNER FORCES By Raymon Miya B.A.Sc. (Mechanical Engineering), University of Waterloo, 1992  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MECHANICAL ENGINEERING  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  April 1994  ©  Raymon Miya, 1994  In presenting this thesis in partial fulfilment of the requirements for degree at the University of British Columbia, I agree that the Library freely available for reference and study. I further agree that permission copying of this thesis for scholarly purposes may be granted by the department  or  by  his  or  her  representatives.  It  is  understood  that  an advanced shall make it for extensive head of my copying  or  publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of  CCC  P1 J  The University of British Columbia Vancouver, Canada Date  DE-6 (2/88)  Abstract  A square plate twisted by corner forces is described 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 cylindrical surface. The transformation in mode shapes presents problems in determining material behaviour by shear in a plate twisting experi ment. The two mode shapes can be described by either displacement or curvature of the surface. The purpose of this work is to investigate the buckling of a square plate twisted by corner forces by determining the bifurcation point and comparing the present FEA work with the experimental results of Howell and other results found in literature. The problem is examined using nonlinear finite element buckling analysis. The bifurcation point is determined by load-displacement plots. The critical value of Gaussian curvature at the centre of the plate is determined by the Southwell plot method. The critical value of Gaussian curvature is found to occur before the bifurcation point. Gaussian curvature is found to vary by an order of magnitude over the plate at bifurcation.  11  Table of Contents  Abstract  ii  Table of Contents  iii  List of Tables List of Figures  vi  List of Symbols  viii  Acknowledgement  x  Introduction  1  1  2  3  1.1  Purpose  1  1.2  Literature Review  2  Theory  3  2.1  Physics of the Problem  3  2.2  Finite Element Theory  6  2.3  Plate Theory  8  2.4  Southwell Plot  9  Analysis  10  3.1  Preprocessing  10  3.2  Solution  11  111  3.3 4  5  6  Postprocessing  .  Results  12 13  4.1  Deflection  13  4.2  Southwell Plot  19  4.3  Gaussian Curvature and Mean Curvature  20  4.4  Curvature  24  4.5  Midsurface Strain  27  4.6  Fixed Plate Centre  30  4.7  Alternate Finite Element  4.8  Non-convergence  34  .  35  Discussions  3T  5.1  FEA Comparison of Ps,. and FK,.  5.2  Comparisons with Bifurcation Points in Literature  37 .  39  5.2.1  Howell  39  5.2.2  Ramsey  40  5.2.3  Miyagawa, Hirata, and Shibuya  41  5.2.4  Lee and Hsu  42  Conclusions  44  Bibliography  45  iv  List of Tables  3.1  Material Constants  10  3.2  Plate Geometry  11  4.1  Critical Values from load-deflection plot  16  4.2  Critical Values from load-deflection plot  17  4.3  Critical Values from Southwell plot  20  4.4  Critical Values from Gaussian-mean curvature plot  23  4.5  Critical Values from Gaussian-mean curvature plot  24  4.6  SHELL43 Element Critical Values from Southwell plot  35  4.7  SHELL43 Element Critical Values from Southwell plot  35  5.1  Comparison of PK and P  37  5.2  Comparison of Coefficient with Howell  40  5.3  Modified Coefficient  40  V  List of Figures  2.1  Square Plate Twisted by Corner Forces  3  2.2  Saddle Surface  4  2.3  Cylindrical Surface  4  2.4  Mohr’s Circle for Curvature  5  3.1  Constraints  11  4.1  Deflection Contour Plot for 3 pinned corners at Pa,  14  4.2  Load-Deflection Plot for 3 pinned corners  14  4.3  Deflection Contour Plot for 3 pinned corners (rotated) at P  15  4.4  Load-Deflection Plot for 3 pinned corners (rotated) with varying a/h ratios 16  4.5  Load-Deflection Plot for 3 pinned corners (rotated) with varying mesh density  17  4.6  Deflection Contour P1st for 3 pinned corners (rotated) at F,./2  18  4.7  Deflection Contour Plot for 3 pinned corners (rotated) at 2P,  18  4.8  Southwell Plot  19  4.9  Critical Value of Gaussian Curvature from Southwell plot  21  4.10 Coefficient from Southwell plot  21  4.11 Gaussian-Mean Curvature Plot  22  4.12 Load-Gaussian Curvature Plot  22  4.13 Load-Mean Curvature Plot  23  4.14 Gaussian Curvature Contour Plot at PKc,  25  4.15 Gaussian Curvature Contour Plot at Pa,  25  vi  4.16 Mean Curvature Contour Plot at PK 4.17 Mean Curvature Contour Plot at P  26 26  .  4.18 Load-Curvature Plot  27  4.19 Curvature  ic  Contour Plot at PK  4.20 Curvature  i  Contour Plot at F  4.21 Twist  ic,  Contour Plot at  4.22 Twist  ic.,  Contour Plot at Ps,.  28  .  28 29  PKc,’ .  29  .  4.23 Midsurface Strain Plot  30  4.24 Midsurface Strain  31  €  Contour Plot at PKc,’  4.25 Midsurface Strain e Contour Plot at Pa, 4.26 Midsurface Strain y Contour Plot at FKc,’  31 .  .  .  32  4.27 Midsurface Strain y Contour Plot at P,  32  4.28 Load-Deflection Plot for fixed plate centre  33  4.29 Load-Gaussian Curvature Plot for fixed plate centre  34  5.1  Load-Gaussian Curvature Plot  5.2  Load-Deflection Plot  38  5.3  Load-Deflection Plot  41  38  .  vii  List of Symbols  a  plate length  C  coefficient for the critical value of twist  Ca, Cb, C, Cd  plate corners  D  plate flexural rigidity  E  Young’s modulus of elasticity  h  plate thickness  P  corner force  P  nondimensionalised corner force  Pa,.  corner force at bifurcation  FKcr  corner force at critical value of Gaussian curvature (from Gaussianmean curvature plot) corner force at critical value of Gaussian curvature (from Southwell plot)  u, v, w  displacement deflection at corner C of plate deflection at centre of plate  viii  nondimensionalised deflection ,  es,, 7  strain  ic, ic,,,  curvature of the surface  IC,  twist of the surface  K  Gaussian curvature  K,.  critical value of Gaussian curvature  p  mean curvature  ii  Poisson’s ratio  ix  Acknowledgement  The author would like to thank Professor Hilton Ramsey for his guidance during the researching and writing of this thesis. The author would also like to thank the Natural Sciences and Engineering Research Council of Canada for their financial support.  x  Chapter 1  Introduction  1.1  Purpose  The purpose of this work is to investigate the buckling of a linear elastic, isotropic square plate twisted by corner forces, by determining the bifurcation point. The problem is examined by a nonlinear finite element buckling analysis using the commercially avail able software package ANSYS Revision 5.0. A square plate twisted by corner forces is described by classical linear theory as a saddle surface. In an experiment, as the plate de forms to any noticeable deflection, it appears not as a saddle surface, but as a cylindrical surface. The transformation in mode shapes presents problems in determining material behaviour in shear by experiment. The findings of the present FEA work can be used in future study to develop a nonlinear relationship to account for this transformation and/or an upper bound to the application of a plate twist experiment. The two mode shapes can be described by either displacement or curvature of the surface. The critical value of corner force at bifurcation is determined from load-displacement plots. The critical value of Gaussian curvature at the centre of the plate is determined from the Southwell plot method. There is a discrepancy in the results found in literature using different methods of analysis and assumptions. The present FEA work is compared to the experimental results of Howell and other results in literature.  1  Chapter 1. Introduction  1.2  2  Literature Review  In 1890, Kelvin and Tait noted a transition in deformation surfaces of a square plate twisted 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 finite difference methods and the nonlinear von K.rmán equations for plates. The critical value of corner force at bifurcation was determined by displacement-load plots. In 1975, Miyagawa, Hirata, and Shibuya[3] investigated the buckling problem exper imentally and numerically, using deflection measurements in the experimental approach, and using a polynomial deformed configuration, von Kármán theory, and stress func tions in the numerical approach. The critical value of corner force at bifurcation was determined by load-deflection plots. In 1985, Ramsey[5] investigated the buckling problem analytically, using the kinematic results of Green and Naghcli for small deformations superposed on a large deformation of an elastic Cosserat surface, and the restricted form of the general nonlinear theory of shells and plates of Naghdi. The critical value of twist at bifurcation was determined from a Rayleigh quotient. In 1991, Howell[1] investigated the buckling problem experimentally, using strain mea surements and Kirkhhoff theory to determine curvatures. The critical value of Gaussian curvature at bifurcation was determined by the Southwell plot.  Chapter 2  Theory  2.1  Physics of the Problem  Classical linear theory of flat plates describes deflection  w  of a square plate twisted by  corner forces P (figure 2.1): U =  (2.1)  2(1 —v)D  in terms of the surface coordinates of the plate x, y. Flexural rigidity of the plate D: D  =  (2.2)  12(1_v2)  is a function of Young’s modulus of elasticity E, Poisson’s ratio z’, and plate thickness h. Deflection can also be expressed in terms of twist  ic  of the surface:  w = ixy  F  Figure 2.1: Square Plate Twisted by Corner Forces  3  (2.3)  Chapter 2. Theory  4  ......  .. .. ..  .  ..  Figure 2 2 Saddle Surface  Figure 2.3: Cylindrical Surface These results (equations 2.1, 2.3) are well known in fundamental classical linear plate theory. The plate appears as a saddle surface (figure 2.2). However, in an experiment, as the plate deforms to any noticeable deflection, it appears not as a saddle surface, but as a cylindrical surface with generators parallel to a plate diagonal (figure 2.3). The mode of the plate can be determined by the surface characteristics with either displacement or curvature attributes. The saddle surface has equal magnitude deflections in the four corners relative to a fixed centre. The cylindrical surface has equal magnitude deflections in two opposite corners and zero deflection in the other two corners relative to a fixed centre. Curvature can be viewed on a Mohr’s circle for curvature (figure 2.4). The abscissa  Chapter 2. Theory  ( 2 i  5  ).  ,  a. Anticlastic Curvature  b. Synclastic Curvature  Figure 2.4: Mohr’s Circle for Curvature represents curvature ic and the ordinate represents twist curvatures are ic,  . 2 ic  ii  of the surface. Principal  The saddle surface is anticlastic—the two principal curvatures  have opposite signs (figure 2.4a). Principal directions are parallel to the plate diagonals. Principal curvatures are equal and opposite, resulting in zero mean curvature and negative Gaussian curvature. The cylindrical surface is synclastic—curvatures in all orientations have like signs (figure 2.4b). Principal directions are parallel to the plate diagonals. One principal curvature is zero and the other non-zero, resulting in a non-zero mean curvature and a zero Gaussian curvature. Classical linear theory of flat plates neglects all quadratic terms in the Green-Lagrange strain:  == == = 7zx =  (2.4) (2.5)  vY+(uY+vY+wY)  (2.6) v + u, + (uu + vv + ww) w,  + v + (uu +  U  +  W  (2.7)  + ww)  (2.8)  + (uu + vv + WZW)  (2.9)  The approximation of neglecting the nonlinear terms fails to account for the defor mation in the middle plane of the plate due to bending. Midsurface strains can only  Chapter 2. Theory  6  be neglected if the defiections of the plate are small in comparison with its thickness in non-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 Gaussian curvature, such as cylinders, cones)[9]. Because of this approximation, classical linear plate theory cannot predict buckling.  2.2  Finite Element Theory  The finite element used in the analysis is an 8 node isoparametric quadrilateral shell element. It is labeled SHELL93 in the ANSYS Revision 5.0 element library. There are 5 degrees of freedom per node: 3 translations and 2 rotations. This element includes features of Green-Lagrange strains and Mindlin plate theory. Green-Lagrange strains (equations 2.4—2.9) take into account midsurface strains of the plate. Mindlin plate theory allows for transverse shear deformation. This means that a line that is straight  and normal to the midsurface before loading, is assumed to remain straight but not necessarily normal to the midsurface after loading. Displacements u, v of a point in the plate 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 midsurface before loading and i, i are the displacements at the plate midsurface. Strains c, e.g, and shear strains 7, 7yz,  are:  Yxy  =  (2.12)  =  (2.13)  =  + , + (ii, +  + ww)  —  z(/3 + a)  (2.14)  Chapter 2. Theory  Strains  ,  7  + (ii +  +  + ww)  (2.15)  7yz  =  Wy  7zx  =  z+wx+(z11x+t1zUx+wzwx)—a  iJYiYZ  —  (2.16)  and shear strain y, are assumed to vary linearly through the plate thickness.  Transverse shear strains y,  are assumed to be constant through the plate thickness.  7za  In the stress-strain relationship: {a}  [D]  (2.17)  {€}  the stress vector {o-}, the strain vector {e}, and the material property matrix for the element [D] are defined as:  L [  {a} {e}  =  f is  x  E  [D]  where  X  —  a Ey  (2.18)  2 T  Tzx  7 x y  7yz  7zx  lu  0  0  0  ui  0  0  0  0  0  T  0  0  0 0  0  o o  0  j  (2.19)  (2.20)  0 0  2  the shear factor:  11.2,  f=  2 <25 A/h —  (2.21)  (1.0 + 0.2, A/h 2 > 25  where A is the area of the element and h is the plate thickness. The shear factor is designed to avoid shear locking. As the element becomes thin, the A/h 2 ratio becomes large. The shear factor  f  is thus increased and the stiffness associated with the trans  verse shears is reduced. The correct method to avoid shear locking is through selective integration, but ANSYS does not accommodate this. The SHELL93 element uses a 2 x 2 reduced quadrature rule.  Chapter 2. Theory  2.3  8  Plate Theory  Kirkhhoff theory is used to calculate curvatures from strain output of the finite element software. This is to be consistent with Howell’s experimental analysis so results can be compared. Kirkhhoff theory neglects transverse shear deformations. This means that a line that is straight and normal to the midsurface before loading, is assumed to remain straight and normal to the midsurface after loading. Extensional strain e at an arbitrary point a distance z from the plate midsurface is: = m  where the membrane strain  (2.22)  +  appears along the plate midsurface, and the curvature ic  m 6  is associated with bending strain. Solving the above equation for the top and bottom of the plate and equating midsur face strains gives the curvatures  , 1 icr, ic  and the twist  t  of the midsurface:  =  (2.23)  Ii  = where ê,  €  —  El,  are the top and bottom surface strains of the plate respectively, and h is the  plate thickness. Principal curvatures  i, K 2  from Mohr’s circle of curvatures are: ±  , 1C2 = 1 IC  ;  2 C)  +  ,  (2.26)  Mean curvature p. is the average of the two principal curvatures: p. =  +  i) = (i  +  ‘2)  (2.27)  Chapter 2. Theory  9  Gaussian curvature K is product of the two principal curvatures: K  2.4  =  —  (2.28)  Sry 1  Southwell Plot  The Southwell plot is a common method to determine the elastic buckling load of a structural system. In experiments, there exists some imperfection in the undeformed shape and/or applied loading. As the compressive load increases, the lowest critical load buckling mode dominates. A linear function can be expressed in terms of applied load and deflection by neglecting contributions from higher modes. In 1932, Southwell considered a simply supported column with an initial imperfection subjected 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 load Fe,., and coefficient a.  The  Southwell plot of S/P versus S gives a straight line whose slope is equal to the inverse of the buckling load. The Southwell plot method claims accuracy only as P  —*  Ps,.. Spencer[7] states that  constructing Southwell plots using K.rmn’s strut data with loads up to O.91P,., to O.88P,., and to O.82P,. (Pa,. being defined as the critical load which Southwell obtained by 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 initial de flection. Spencer[7] showed that in buckling of a uniaxially compressed simply supported plate, the Southwell plot begins to underestimate the critical load when: w / 0 h > 0.5 where w 0 is the initial deflection at the plate centre and h is the plate thickness.  (2.30)  Chapter 3  Analysis  The analysis was performed on a SUN SPARC workstation. The preprocessing and the solution utilized ANSYS Revision 5.0, and the postprocessing utilized FORTRAN77 and TECPLOT Revision 5.0.  3.1  Preprocessing  The plate is modelled with the SHELL93 8 node isoparametric shell element. The plate material is modelled as T6061-T6 Aluminium (table 3.1) for comparison with the experimental results of Howell{1j. Material nonlinearity, such as plasticity, is not considered in the analysis. The plate geometry is square with plate length to thickness ratios a/h (table 3.2) for comparison with the experimental results of Howell[1]. The plate is meshed with square elements N per side, where N is even to provide a node at the centre of the plate to take displacement and strain measurements—the same location as Howell’s strain gauges [1]. There are a total of N 2 elements and 2 (3N + 4N +1) nodes for the model. Table 3.1: Material Constants E i’  69x10 9 0.33  10  Pa  Chapter 3. Analysis  11  Table 3.2: Plate Geometry a/h ratio 49.2 63.2 80.3 96.0 196.7  a (m) 0.1524 0.2032 0.2540 0.3048 0.6096  h (m) 0.003099 0.003216 0.003162 0.003175 0.003099  w0  Cc  Cd y,v  L Ca uvw0  ,  Cb vw0  Figure 3.1: Constraints 3.2  Solution  The plate is constrained at corners Ca, Cb, Cd (figure 3.1) to zero displacement in the z direction, to simulate the self equilibrating corner forces associated with the applied force at corner C. These are the same constraints in the experiment by Howell[1]. To prevent rigid body motion, additional DOF constraints are specified. The plate is constrained at corner Ca to zero displacement in the x and y directions to prevent translation, and constrained at corner Cb to zero displacement in the y direction to prevent rotation. These constraints satisfy the kinematic, but not the static boundary conditions of a plate with free edges. When applying the Southwell plot method to find critical values, an initial hydrostatic  Chapter 3. Analysis  12  pressure is applied in the positive z direction. A nominal value of hydrostatic loading is used 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 applied corner force P can exceed the value at the bifurcation point Pa without the instability of 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  Postprocessing  Displacements 5,, 6 are calculated at the centre of the plate and at corner C. The critical value of corner force at bifurcation is determined from load-deflection plot.• Strains 3 ,c, e, 7xy are calculated at the top and bottom surfaces at the node at the centre of the plate using nodal point averaging in ANSYS. These values are exported to a FORTRAN code which calculates curvatures using Kirkhhoff plate theory. The critical value of Gaussian curvature Kr,. is determined from the Southwell plot. The Southwell plot uses p. as the abscissa and p./K as the ordinate. The asymptotic behaviour of the curve determines K as K/KCI.  —*  1.  Chapter 4  Results  4.1  Deflection  The deflection for the plate with 3 pinned corners is zero at the pinned corners Ca, 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 shows no indication of buckling (figure 4.2). The load-deflection curve of the centre of the plate  has an abrupt change in the slope at the bifurcation point Pa,.. The finite element analysis deflection of the plate centre agrees well with linear theory (equation 2.1) for deflections less than a plate thickness (figure 4.2). The FEA deflection of the plate corner C agrees well with linear theory for deflections less than 4 ,plate thicknesses. The finite element results of the plate with 3 pinned corners can be rotated to show the characteristic surface. The plate can be rotated so the deflections of corners Ca and  C are equal, and translated so the deflection of the centre of the plate is zero. The deflections become S/2  —  S at Ca and C, and  at Cb and Cd (figure 4.3), where S  and S are the deflections of the unrotated results for the centre of the plate and corner C respectively. The bifurcation point is where the magnitude of rotated corners Ca, C and rotated corners Cb, Cd significantly diverge. The critical value of corner load varies slightly with Howell’s[l] a/h ratios (figure 4.4 and table 4.1). The mesh density of 144 elements is  13  Chapter 4. Results  14  Deflection Contour Plot 3 pInned corners at P 96 h rallo, 144 elements  level  öAi 15.9459 13.9107 11.8756 9.84041 7.80525 5.77009 3.73492 1.69976  Figure 4.1: Deflection Contour Plot for 3 pinned corners at F,.  Load-Deflection Plot 3 pinned corners 96 ajh rallo, 144 elements  load Pa I2Dh 2 30  25  20  15  10  5  0 0  5  10 deflection jh  15  20  Figure 4.2: Load-Deflection Plot for 3 pinned corners  Chapter 4. Results  15  Deflection Contour Plot 3 pinned corners (rolaled)  at P,  96 h ratio, 144 elements  level  I  öih 3.44262  7  2.44367  6 5  1.44472 0.44577  4  -0.55318 -1.55213 -2.55108 -3.55003  Figure 4.3: Deflection Contour Plot for 3 pinned corners (rotated) at Pc. sufficient to provide displacement convergence (figure 4.5 and table 4.2). Below the bifurcation point, the magnitudes of deflection for rotated corners Ca, Cc and rotated corners Cb, Cd are almost equal (figure 4.4). The plate is bending to a saddle surface (figure 4.6). Above the bifurcation point, the magnitude of deflection for rotated corners Ca, C is decreasing, and the magnitude of deflection for rotated corners Cb, 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]: 2 p Pj  —  Nondimensionalized deflection  (4.1)  is defined as [3]: —  S h  (4.2)  Chapter 4. Results  16  Load-Deflection Plot 3 pinned corners (rotated) load Pa I2Dh 2  144 elements  49.2a/hratio  --  63.2-.-  —-  30 rotated COrnS C , C, \\ 1  --  ...  —  •_%_.  ----:_-—  80.3  V..———  25 rotated corners C,,, C 1 20  15  10  5  0 0.0  1.0  2.0  4.0 3.0 deflection h  5.0  6.0  Figure 4.4: Load-Deflection Plot for 3 pinned corners (rotated) with varying a/h ratios  Table 4.1: Critical Values from load-deflection plot  a/h ratio 49.2 63.2 80.3 96.0 196.7  144 elements P (N) P 26.9 1380 26.0 870 510 25.5 25.1 355 24.9 80  ,  4.17 4.32 4.41 4.45 4.54  &,  4.48 4.52 4.53 4.55 4.58  Chapter 4. Results  17  Load-Deflection Plot 3 pinned Corners (rotated) 96 h ratio  256 elements  rotated corners C,, C 1  25  Cd 20  15  10  5  0 0.0  1.0  2.0  4.0 3.0 deflection h  5.0  6.0  Figure 4.5: Load-Deflection Plot for 3 pinned corners (rotated) with varying mesh density  Table 4.2: Critical Values from load-deflection plot  elements 16 64 144 256  96 a/h ratio Pc,. (N) 352 25.0 25.1 355 25.1 355 25.1 355  S 4.45 4.45 4.45 4.45  Sa,  d &, 8  4.55 4.55 4.55 4.55  Chapter 4. Results  18  Deflection Contour Plot 3 pInned corners (rotated) at P,,,/ 2 96 h ratio, 144 elements  level  8/h 2.53851  7  1.81277  6  1.08703  5  0.361294  4  -0.364444 -1.09018 -1.81592 -2.54166  Figure 4.6: Deflection Contour Plot for 3 pinned corners (rotated) at Pc,./2  Deflection Contour Plot 3 pInned corners (rotated) at 2 P,, 96 h ratio, 144 elements  level  8 7 6 5  8/h 1.70502 0.081854 -1.5413 1  -3.16448  4  -4.78764  3  -6.41081 -8.03397 -9.65714  2 1  Figure 4.7: Deflection Contour Plot for 3 pinned corners (rotated) at 2Pc,  Chapter 4. Re8ults  19  Southwell Plot at centre of plate  ii 1K Im] -0.0200  96 alh ratio, 144 elements  -0.0150  -0.0100  10.0 Pa .—..—......  -0.0050  /  /  1.OPa  •..  ,‘  0.lPa  -0.0000 0.0000  I  I  0.0010  0.0005  0.0015  0.0020  Im1  Figure 4.8: Southwell Plot 4.2  Southwell Plot  Howell[1] determined the critical value of Gaussian curvature using the Southwell plot method. The Southwell plot method requires an initial curvature in the structure. The FEA Southwell plot is constructed from strains at the centre of the plate with 3 pinned corners and initial hydrostatic pressure. The Southwell plot produces parallel lines for varying intensity of initial hydrostatic pressure (figure 4.8). The initial hydrostatic pressure creates an initial deflection of the centre of the plate S. The critical value of Gaussian curvature  and corner force is  determined  by the Southwell plot method are not affected by deflections S less than one tenth of a plate thickness (table 4.3). The coefficient C is defined as [5]: C  =  (4.3)  Chapter 4. Results  20  Table 4.3: Critical Values from Southwell plot  pressure 0.1 1 10 100 500  96 a/h ratio, 144 elements -o (Pa) S, PScr (N) Ps,. 0.0000961 20.1 284 0.000961 284 20.1 0.00961 20.1 284 0.0961 274 19.4 0.478 226 16.0  for the critical value of twist  —  ic  C 9.04 9.04 9.03 9.01 8.65  at the centre of the plate.  For load levels less than Ps,., the Southwell plot method over or under predicts Kcj. (figure 4.9) and C (figure 4.10) depending on the magnitude of the initial deflection. 4.3  Gaussian Curvature and Mean Curvature  The Southwell plot determines the critical value of Gaussian curvature where the slope on the Gaussian-mean curvature plot (figure 4.11) is zero[7]. The Gaussian-mean curvature plot is constructed from strain calculations at the centre of the plate with 3 pinned corners and no initial hydrostatic pressure. Gaussian curvature 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 to decrease in magnitude, and after bifurcation decreases in magnitude as the plate deforms to a cylindrical surface (figure 4.12). Mean curvature is zero for the undeformed plate (no initial curvature) remains zero as the plate deforms to a saddle surface, and after bifurcation increases in magnitude as the plate deforms to a cylindrical surface (figure 4.13). The corner load at the critical value of Gaussian curvature PK,. (tables 4.4—4.5) is less than the corner load at bifurcation Pa,. (figure 4.12).  Chapter 4. Results  Ksr -0.200  21  m2J  Critical Value of Gaussian Curvature from Southwell Plot at centre of plate 96 h ratio, 144 elements  -0.150 1.0 Pa ,loçPa•  -0.100  —  0.1 Pa -0.050  0.000 — 0.0000  0.0005  0.0010 ] 1 p(m  0.0015  0.0020  Figure 4.9: Critical Value of Gaussian Curvature from Southwell plot  Coefficient from Southwell Plot  CM  atcentre of plate 96 h ratio, 144 elements  20  15 10.0 Pa  i.opa 10  0.1 Pa 5  0 — 0.0000  0.0005  0.0010  0.0015  p jm ] t  Figure 4.10: Coefficient from Southwell plot  0.0020  Chapter 4. Results  22  Gaussian-Mean Curvature Plot Gaussian  at centre of plate 96 h ratio, 144 elements  /h 4 Ka curvature 2 -75  -50  -25  0 0.0000  0.0050  0.0100  mean curvature  0.0150  0.0200  Imi  Figure 4.11: Gaussian-Mean Curvature Plot  Load-Gaussian Curvature Plot at centre of plate 144 elements  load Pa I2Dh 2 25  -  20  15  -  10  -  5.  0  -25  -50  -75  Gaussian curvature 2 1h 4 Ka  Figure 4.12: Load-Gaussian Curvature Plot  Chapter 4. Results  23  Load-Mean Curvature Plot at centre of plate ratio, 144 elements  I2Dh 2 load Pa 30  96 alh  25  20  15  10  -  -  -  5  0 — 0.000  0.050  0.100  0.150  0.200  mean curvature  0.250 ji  0.300  0.350  1 1m  Figure 4.13: Load-Mean Curvature Plot  Table 4.4: Critical Values from Gaussian-mean curvature plot 144 elements a/h ratio PK . (N) PKC, 7 49.2 1112 21.7 63.2 696 20.8 80.3 20.3 406 284 20.1 96.0 196.7 19.7 63  C 9.01 9.01 9.02 9.03 9.06  Chapter 4. Results  24  Table 4.5: Critical Values from Gaussian-mean curvature plot 96 a/h ratio elements PK,. (N) PKcr 16 278 19.7 64 276 19.6 144 284 20.1 256 286 20.3  C 9.09 9.03 9.03 9.01  Gaussian curvature is a minimum absolute value at the centre of the plate, a maximum absolute value near the corners of the plate, and varies over the plate by an order of magnitude 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 negative values over the plate at  4.4  (figure 4.16) and Pa,. (figure 4.17).  Curvature  The load-curvature plot is constructed from strains at the centre of the plate with 3 pinned corners and no initial hydrostatic pressure. Curvatures  i, ,,  are zero for the  undeformed plate (no initial curvature), remain zero as the plate deforms to a saddle surface, and after bifurcation increase in magnitude as the plate deforms to a cylindrical surface (figure 4.18). Twist  ic  is zero for the undeformed plate (no initial curvature), increases in mag  nitude as the plate deforms to a saddle surface, reaches a maximum value, begins to decrease in magnitude, and after bifurcation continues to increase in magnitude as the plate deforms to a cylindrical surface. The FEA twist  ic,  agrees well with linear theory (equations 2.1—2.3) for defiections  less than half the plate thickness, and agrees well with membrane stress theory ([4] and  Chapter 4. Results  25  Gaussian Curvature Contour Plot at  P 96 h ratio, 144 elements  level  1h 4 Ka 2 -57.4374 -114.875  6 5 4  T:,8  -172.3 12 -229.75 -287.187 -344.625 -402.062 -459.5  Figure 4.14: Gaussian Curvature Contour Plot at PK,.  Gaussian Curvature Contour Plot , 0 at P  96 h ratio, 144 elements  level  /h 4 Ka 2 -81.7554  7  -163.5 11  6 5  -245.266 -327.022  i. .8  -408.777  4  -490.533 -572.289 -654.044  Figure 4.15: Gaussian Curvature Contour Plot at P  Chapter 4. Results  26  Mean Curvature Contour Plot at P 96 h ratio, 144 elements  level  p [n1 ] 1 0.325576  7 6 5 4  0.232526 0.139475 0.046424 -0.046625 —0.139676 -0.232727  -0.325777  Figure 4.16: Mean Curvature Contour Plot at PK  Mean Curvature Contour Plot at Pt,, 96  alh ratio, 144 elements  level  p [mj 0.460431  7  0.329732  6  0.199032  5  0.068332  4  -0.062366 -0.193066 -0.323766 -0.454465  Figure 4.17: Mean Curvature Contour Plot at Pa,.  Chapter 4. Results  27  Load-Curvature Plot at centre of plate  96 alh ratio 144 elements  load  FEA  x, K 7  , 1 FEAx 20  15  -10  .5  0 curvature  5  lh 2 ica  10  Figure 4.18: Load-Curvature Plot equation 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 positive and negative values over the plate at PK (figure 4.19) and Pa,. (figure 4.20). Twist  tc,  is a minimum absolute value at the centre of the plate, a maximum absolute  value near the corners of the plate, and varies over the plate by an order of magnitude at PK (figure 4.21) and P (figure 4.22). 4.5  Midsurface Strain  The load-midsurface strain plot is constructed from strains at the centre of the plate with 3 pinned corners and no initial hydrostatic pressure. Midsurface strains  ,  ,  are zero  for the undeformed plate (no initial curvature), are equal and compressive as the plate deforms to a saddle surface, and after bifurcation decrease in magnitude as the plate  Chapter 4. Results  28  Curvature x Contour Plot at P, 96 h ratio, 144 elements  level  /h 2 Ka 8.11409 5.79991  :  6  Figure 4.19: Curvature  ic,..,  3.48574  5  1. 17156  4 3 2 j...81  -1. 14262 -3.4568 -5.77097 -8.08515  Contour Plot at PK  Curvature x Contour Plot , 0 at P 96 h ratio, 144 elements  level  /h 2 ica 11. 1284 7.97563  6 5  4.82286 1.67009  4  -1.48269  3 2  -4.63546 -7.78823 -10.941  Figure 4.20: Curvature  ic,  Contour Plot at Fcj.  Chapter 4. Results  29  Twist  Contour Plot  at P,  96 alh ratio, 144 elements  level  7 6 5  i. 8 4  /h 2 ica -2.57822 -5.15645 -7.73467 -10.3129 -12. 89 11 -15.4693 -18.0476 -20.6258  Figure 4.21: Twist ic Contour Plot at Pjç,.  Twist  Contour Plot  at P, 96 h ratio, 144 elements  level  :::::  6 5  4  .....8  Figure 4.22: Twist  ic,  Contour Plot at P.  -3.09801 -6.19602 -9.29404 -12.392 -15.4901 -18.5881 -21.6861 -24.7841  Chapter 4. Results  30  Load-Midsurface Strain Plot atcenh’eotpte load Pa !2Dh 2 30  96 alh ratio, 144 elements  -  25  Yxy 20  e  15  10  5  0 .0.00  --  -0.25  -0.50  -0.75  -1.25 -1.00 1h £a strain 2  -1.50  -1.75  Figure 4.23: Midsurface Strain Plot deforms to a cylindrical surface (figure 4.23). Midsurface shear strain y, is zero for the undeformed plate (no initial curvature), remains zero as the plate deforms to a saddle surface, and after bifurcation increases in magnitude as the plate deforms to a cylindrical surface. The midsurface of the plate is in maximum compression at the centre of the plate and maximum tension at the edge of the plate at PK (figure 4.24) and P (figure 4.25). Midsurface shear strain -y, is zero at the centre of the plate and varies positive and negative values over the plate at PK (figure 4.26) and P (figure 4.27). 4.6  Fixed Plate Centre  The FEA plate buckles without an initial perturbation because of the 3 pinned corner constraints—the same used by Howell[1j. As the plate deforms only corner C is free to  Chapter 4. Results  31  Midsurface Strain  Contour Plot  at P, 96 h ratio, 144 elements  level  )h ea 2 7.65334  7  6.3303  6 5  5.00727 3.68423  4  2.361 19 1.03815 -0.284883  j8  Figure 4.24: Midsurface Strain  Midsurtace Strain  ,  -1.60792  Contour Plot at PK  Contour Plot  at P 96 a(h ratio, 144 elements  1h Ea 2  level  III 6 5  I.8 4  Figure 4.25: Midsurface Strain  9.60264 7.94238 6.28212 4.62186 2.9616 1.30134 -0.35892 -2.0 1918  Contour Plot at Pa,.  Chapter 4. Results  32  Midsurlace Shear Strain y, Contour Plot 96 aTh  at P, ratio, 144 elements  level  /h ya 2  7  4.43571 3.16764  6 5 4  I.V:. V:8  1.89957  0.6315 -0.63657 -1.90464 -3.1727 1 -4.44078  Figure 4.26: Midsurface Strain -y, Contour Plot at PKc,  Midsurface Shear Strain  at P , 0 96 h ratio, 144 elements  Contour Plot  level  /h ya 2  5.41496 3.85986  6 5 4  j....8  2.30477  0.749669 -0.805429 -2.36053  -3.91562 -5.47072  Figure 4.27: Midsurface Strain -y, Contour Plot at Ps,.  Chapter 4. Results  33  Load-Deflection Plot corner deflections relative to centre of plate load Pa !2Dh 2 30  96 h ratio, 144 elements  25  3 pinned corners (rcted)  ,‘/ ‘  fixed plate centre  20  15  10  5  0 0.0  1.0  2.0  3.0  4.0  5.0  6.0  deflection h  Figure 4.28: Load-Deflection Plot for fixed plate centre deflect. The corner forces remain parallel to the z axis and are no longer normal to the tangent plane of the centre of the plate. Constraints which do not initiate buckling are created by fixing the centre of the plate in 5 degrees of freedom: displacements u, v, w and rotations about the x and y axis. The drilling rotation about the z axis is fixed by a constraint on Ca in the direction of the y  =  —x diagonal. The applied corners loads are P at Ca and C and —P at C, and Cd. The orientation of the corner forces in the fixed plate centre loading case remain  normal to the tangent plane of the centre of the plate. There is no perturbation, and the plate 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 corners and the fixed plate centre loading cases (figure 4.29). The 3 pinned corners and the fixed plate centre loading cases create membrane tension for large deflections due to the corner forces remaining in their original orientation and  Chapter 4. Results  34  Load-Gaussian Curvature Plot at centre of plate 96 alhralio,l 44 elements  load Pa’I2Dh 25  -  3 pinned corners  20  -  15  -  10  5  0  -25  .50 Gaussian curvature 2 1h 4 Ka  .75  Figure 4.29: Load-Gaussian Curvature Plot for fixed plate centre stretching the plate. Further study involving “follower forces” which remain normal to the plate surface is recommended to study the effects of the added membrane tension. 4.7  Alternate Finite Element  The finite element analysis was also performed modelling the plate with the SHELL43 4 node shell element. The SHELL43 element is claimed by ANSYS to be well suited to model nonlinear thin to moderately-thick shell structures[8]. The SHELL43 element accommodates rotational degrees of freedom and shear deformations but since the prob lem under consideration is highly nonlinear, the bilinear SHELL43 element would not be expected to model the plate as well as the quadratic SHELL93 element. The plate modelled with SHELL43 elements did not buckle for the loading case of 3 pinned corners and no initial hydrostatic pressure. The SHELL43 element model only buckled with  Chapter 4. Results  35  Table 4.6: SHELL43 Element Critical Values from Southwell plot 96 a/h ratio, 400 elements pressure (Pa) 1 5 10 20 30 40 50 100 500  0.00094 0.0047 0.0094 0.018 0.028 0.037 0.047 0.094 0.47  P (N) PScr 5 did not buckle did not buckle did not buckle did not buckle 295 20.9 292 20.7 285 20.2 285 20.2 225 15.9  Table 4.7: SHELL43 Element Critical Values from Southwell plot 96 a/h ratio, 50 Pa hydrostatic pressure elements P 5 (N) PScr 16 did not buckle 64 did not buckle 144 285 20.2 256 285 20.2 400 285 20.2  an initial deflection greater than 0.3 plate thicknesses (table 4.6) and greater than 144 elements (table 4.7). For the reasons of problems in buckling, the SHELL43 element was not used in the analysis.  4.8  Non-convergence  The post buckling response for the finite element analysis of the 196.7 a/h ratioplate with 3 pinned corners does not converge. The plate bends to a saddle surface up to  Chapter 4. Results  36  the bifurcation point without numerical difficulties. Shear locking does not seem to be a factor, since increasing the mesh density—decreasing the element a/h ratio—does not rectify the problem.  Chapter 5  Discussions  5.1  FEA Comparison of Pc,. and PK,.  Gaussian curvature at the centre of the plate reaches a maximum absolute value K,. at corner force  The absolute value of K then begins to decrease in magnitude before  the bifurcation point Pc,. (figure 5.1). Attempting to determine the bifurcation point by a critical Gaussian curvature cri terion, such as the Southwell plot method, will underestimate the bifurcation point (fig ure 5.2). The critical Gaussian curvature point PKc,. is 80 percent of bifurcation point Pa,. (table 5.1). The values of K,. and PKc,. remain the same for the 3 pinned corners and the fixed plate centre loading cases (figure 4.29). The present work uses corner forces which main tain their original vertical direction parallel to z axis. For large defiections, the orientation Table 5.1: Comparison of PKc, and Pc,. ratio 49.2 63.2 80.3 96.0 196.7  a/h  Pc,. 26.9 26.0 25.5 25.1 24.9  37  PKcr  PKc,./Pc,.  21.7 20.8 20.3 20.1 19.7  0.81 0.80 0.80 0.80 0.79  Chapter 5. Discussions  38  Load-Gaussian Curvature Plot at centre of plate load Pa J2Dh 2  96  144 element  25  20  15  10  -  5-  0 0  -25  .50  .75  Gaussian curvature 2 /h 4 Ka  Figure 5.1: Load-Gaussian Curvature Plot  Load-Deflection Plot 3 pinned corners (rotated) 96 h ratio, 144 elements  load Pa j2Dh 2 30  25 cr 20  15  10  0• 0.0  I  1.0  I....  2.0  I....  3.0 4.0 defieclion jh  .  .  .1....  5.0  Figure 5.2: Load-Deflection Plot  6.0  Chapter 5. Discussions  39  of the forces causes tensile membrane stresses which may have an effect on the value of PK4.  Further investigation using “follower forces” which remain normal to the plate  surface is recommended.  Comparisons with Bifurcation Points in Literature  5.2 5.2.1  Howell  Howell determined the bifurcation point by experiment[1]. The constraints on the plate in the experiment were three corners pinned and the loaded corner free to deflect with the load applied by a constant direction tensile cable. The 3 pinned corners loading case in the present FEA models this experimental setup. Strain gauges measured strains on the top and bottom surfaces of the plate, and Kirkhhoff theory was used to calculate the curvatures from the strains. The critical value of Gaussian curvature at bifurcation was determined using the Southwell plot method. Howell gives the bifurcation point: = 1O.8h/a  for the critical value of twist  ,  (5.1)  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 and the FEA is mainly due to Howell’s limit of applied corner force. Howell limited the maximum applied corner force  P0hl  to  avoid plastic yielding of the material. This corresponds to corner loads less than half of FKcr (table 5.2). The Southwell Plot method only claims accuracy as the load approaches  the critical load P  —  Pa,. (section 2.4).  Chapter 5. Discussions  40  Table 5.2: Comparison of Coefficient with Howell  a/h ratio 49.2 80.3 96.0 196.7  CHohl  c  HoweU  17.85 11.07 10.61 10.25  9.01 9.02 9.03 9.06  4.8 10.0 9.9 10.9  pHowell/p  0.18 0.39 0.39 0.43  Table 5.3: Modified Coefficient —Howell  at Fma = 9.9 96 a/h ratio, 144 elements pressure (N) C’ 0.1 8.9 1 9.5 10 10.0 100 9.9 9.4 300 500 8.9  The magnitude of initial deflection affects the Southwell’s plot prediction of the critical value. The initial deflection for the experiment of Howell is unknown (figure 5.3). The Southwell plot method using Gaussian and mean curvatures only up to  poe11  gives modified coefficient C’ values closer to Howell’s results (table 5.3). 5.2.2  Ramsey  Ramsey determined the bifurcation point by analytical methods[5]. The kinematic re sults of Green and Naghdi for small deformations superposed on a large deformation of an elastic Cosserat surface, and the restricted form of the general nonlinear theory of shells and plates of Naghdi were used. The critical value of twist  determined from a Rayleigh quotient.  ,  at bifurcation was  Chapter 5. Discussions  41  Southwell Plot at centre of plate pIKImI  -0.75  -0.50  96hralio  -  -  Howell  -  -  -  - -  - -  - - - -  FEA 500 Pa initial hydrostatic pressure  -0.25  -  FiOPa.. PEA lOOPa 5. 1J.vu  0.000  i..  0.010  i...  0.020  I  i...  0.030  0.040  i.....  0.050  0.060  i 1 1m  Figure 5.3: Load-Deflection Plot Ramsey gives the bifurcation point: =  for the critical value of twist  ic  3.29h/a  (5.3)  at bifurcation at the centre of the plate.  The present FEA work using the Southwell plot method gives results in equation 5.2. The difference between the result of Ramsey and the FEA is mainly due to Ram sey’s assumption of the Gaussian curvature behaviour. Ramsey assumed the Gaussian curvature at bifurcation to be uniform over the plate. The present FEA work shows the Gaussian curvature varies by an order of magnitude over the plate (figure 4.14—4.15). 5.2.3  Miyagawa, Hirata, and Shibuya  Miyagawa, Hirata, and Shibuya determined the bifurcation point by experimental and numerical methods [3].  Chapter 5. Discussions  42  In the experiment of Miyagawa et al., the critical value of corner force at bifurcation was 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:  = for the dimensionless corner force  []  = 21  (5.4)  .  In the numerical work of Miyagawa et al., the deformed configuration of the plate was approximated as a polynomial. Stresses in the middle of the plate were approximated by combining von Kármn theory, an assumed stress function, and experimental results. The relation between load and deflection was determined by minimizing the total energy of: strain energy due to bending and twisting, strain energy in the middle of the plate due 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:  . = 25 2 Pc.  (5.6)  In the experiment of Miyagawa et al., the four loading points were applied by flat roller bearings which simulated “follower loads” to reduce the stretching forces along the plate. The plate material experienced plastic yielding resulting in the experimental bifurcation point of Miyagawa et al. lower than the numerical bifurcation point of Miyagawa et al. [3]. 5.2.4  Lee and Hsu  Lee and Hsu determined the bifurcation point by finite difference methods[2]. The critical value of corner force at bifurcation was determined by the displacement-load plot.  Chapter 5. Discussions  43  Lee 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 mainly due to the limited model of Lee and Hsu. The mesh used by Lee and Hsu in the finite difference scheme was not dense enough to provide convergence of M. No attempt was made to calculate Me,. more precisely.  Chapter 6  Conclusions  Describing the surface of a square plate twisted by corner forces based on either dis placement or curvature values gives different results for the critical point. The loaddisplacement plot determines the bifurcation point Fe,.. The present FEA work gives = 25. The Southwell plot based on curvature determines the critical Gaussian cur  vature point PK. The present FEA work gives Pjç,. = 20. The present FEA work gives the coefficient for the critical value of twist at the centre of the plate C = 9.0 from the Southwell plot. This result compares well with the experiment of Howell taking into account the low load levels Howell used to avoid plastic yielding of the material. Southwell plots constructed from curvature data of load levels less than PKc,, will overpredict the calculated value of PKc,’ for initial defiections of the plate centre between 0.001 < S,/h < 0.5. The result of the present FEA work does not compare well with the analytical work of Ramsey. Ramsey assumed Gaussian curvature to be uniform over the plate at bifurcation. The present FEA work shows that the problem is highly nonlinear and Gaussian curvature varies over the plate by an order of magnitude at  and Pa,..  The applied forces in the present FEA work maintain their original orientation even for large deflections. This will create significant tensile membrane stresses in the plate for defiections much larger than the plate thickness. Further FEA investigation involving “follower forces” which remain normal to the plate surface, and inclusion of nonlinear material properties is recommended.  44  Bibliography  [1] R. A. Howell. An experimental investigation of the bifurcation in twisted square plates. Master’s thesis, University of British Columbia, 1991. [2] S. S. Lee and C. S. Hsu. “Stability of Saddle-like Deformed Configurations of Plate and Shallow Shells”. International Journal of Non-linear Mechanics, 6:221—236, 1971. [3] M. Miyagawa, T. Hirata, and S. Shibuya. “Deformation of Square Plates under Contrary Transverse Load”. Memoirs of Faculty of Technology: Tokyo Metropolitan University, 25, 1975. [4] W. Ramberg and J. A. Miller. “Twisted Square Plate Method and Other Methods for Determining the Shear Stress-Strain Relation of Flat Sheet”. Journal of Research of the National Bureau of Standards, 5O(2):111—123, 1953. [5] H. Ramsey. “A Rayleigh Quotient for the Instability of a Rectangular Plate with Free Edges 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 of Elastic Stability”. In Proceedings of the Royal Society, volume 135 of A, pages 601— 616, London, 1935. [7] H. H. Spencer and A. C. Walker. “Critique of Southwell Plots with Proposals for Alternative 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-Hill Book Company, 1959.  45  

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