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An experimental study of instability in square plates twisted by corner forces Williams, Gordon Colin 1988

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AN EXPERIMENTAL  STUDY OF I N S T A B I L I T Y IN SQUARE PLATES TWISTED BY CORNER FORCES BY GORDON C. WILLIAMS  B.Sc,  Queen's U n i v e r s i t y , 1984  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE Department  We  accept to  of Mechanical  this  STUDIES Engineering  t h e s i s as c o n f o r m i n g  the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA June  1988  • G o r d o n C. W i l l i a m s , 1988  In presenting this thesis in partial fulfilment  of the  requirements for an advanced  degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department  or  by  his  or  her  representatives.  It  is  understood  that  copying or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  rtee-U.*^!  Ev^^gcr'in^  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  ABSTRACT  An  experimental  twisting of  A  of  study  plates  is  with  self-equilibrating  simple  were  twisted  sides to  The  apparatus  at  the  unwanted  while  free  edges,  corner  loads.  was  designed  measuring  centre.  of  method  between  the  pre-  post-bifurcation  the  experimental  strains  are  and  found  to  midsurface  to  the  various  sizes  was  results, with  of  plates  on  both  encountered  its  own  determine  of  application  strains  under  strains  linearly  non-linear  through  regions  analytical vary  the  surface  plate  used  relationship and  on  d i f f i c u l t y  the  is  and  the  I n i t i a l  deflection  Rayleigh-Ritz  presented  and  weight.  an  analytical  curvatures  twisting. the  due  In  in both  midsurface  the  Gaussian  which  collapse  curvature.  Non-dimensional experimental  and  curvature  groups  non-linear  bifurcation  are  load-strain,  strain-Gaussian dimensional  groups  is  identified  load-curvature  and  relationships.  These  collapse  regions.  The  identified  as  the  results  curvature a  i i  at  function  in  of  midsurface non-  both  the  the  the  point  plate  linear of  dimensional  parameters. surface  The  Also  strain  are  literature.  resolved  The  and  experimental  bifurcation the  shown  for  present  analytical  the  corner  the  expressions  load  at  load-curvature compared w i t h A  large  results,  form and  which  a are  relationships  point  in of  for  important using  the  c r i t i c a l  bifurcation  analytical  basis  for  relationship  discrepancy  theoretical  results  constitutive  are  occurs.  and  point  of  results  found  in  the  literature  is  bifurcation.  verification in  the  twisted  of  future  measurement plate  test.  of  TABLE OF CONTENTS  Abstract Table  of  ii Contents  List  of  Figures  List  of  Tables  iv v i i x  Nomenclature  xi  Acknowledgements  1.  1  INTRODUCTION 1.1  1.2  1.3  2.  x i i i  Background  1  1.1.1  Description  1.1.2  Determination  Literature  of  the of  Instability  Constitutive  1 Coefficients  Review  4  1.2.1  Material  Testing  5  1.2.2  Point  Instability  6  Purpose  and  of  Scope  of  the  Study  7  10  PLATE BENDING THEORY 2.1  Linear 2.1.1  2.2  3  Theory Limitations  Non-Linear 2.2.1  10 the  Linear  Theory  Governing  of  16 17  Equations  Geometric  Theory  Strain iv  18 18  Constitutive  Compatibility  20  Equilibrium  21  Stress  23  2.2.2  Boundary  2.3 A p p r o x i m a t e  19  Function  Conditions  24  Solutions  2.3.1 A n a l y t i c a l Plates  3.  Relations  26  Solution  with  Unequal  for Midsurface  Strain  Curvatures  3.1 A p p a r a t u s  37  3.1.1  Design  37  3.1.2  Instrumentation  42  Strain S t r a i n  3.1.3  Plate  Selection  Indicator  42  and B r i d g e  Configuration  44  Design  45 S i z e Experimental 3.2.1  Initial  3.2.2  Test  EXPERIMENTAL  Gauge  Estimate  4.  27  37  DESCRIPTION OF EXPERIMENT  3.2  in  of  Plate  and M a t e r i a l  Procedure  Procedure  Principal  4.2  Plate  Curvature  49  51 52  55  RESULTS  Plate  46  51  Preparation  4.1  Non-Linearity  Strain  56 61  v  5. C O M P A R I S O N  OF RESULTS  WITH  THEORETICAL  AND NUMERICAL  MODELS 5.1  72  Collapsing  Data  into  Form  72  5.1.1  Principal  Strains  73  5.1.2  Principal  Curvature  79  5.1.3  Midsurface  Strain  5.1.4  Comparison  of  Approximate  81  Load-Curvature  Results  With  Solutions  84  Chandra[5]  86  Reissner[9]  87  Ramberg  88  L e e a n d Hsu[7]  5.2  Plate  5.3  Comparison  5.4  Non-Dimensional  Curvature of  Under  Experimentally  5.3.2  Analytical Load  Measurement  of  i t s  88 Own W e i g h t  the Curvature  5.3.1  Limiting  and Miller[2]  Determined  Bifurcation  and Surface Shear  Bifurcation  Point  Point  During  91 92  Points  Strain  Modulus  89  95 the 96  6.  CONCLUSIONS  7.  BIBLIOGRAPHY  105  8.  APPENDICES  107  99  A.  Instrumentation  B.  Finite  Difference  107 Analysis  vi  113  LIST  Figure  1.1,  A n t i c l a s t i c  curvatures Figure  1.2,  shape  along  Deflection  cylindrical  OF  the  of  FIGURES  with  equal  and  opposite  diagonals.  the  surface,  plate a)  2  after  bifurcation  Concave up.  b)  into  a  Concave  down. Figure  2.1, M  2  Corner  loads  forming  couples  of  magnitude  x ' y  1  Figure  2.2,  Constant  twisting  Figure  2.3,  Midsurface  moment  forces  N  x  N  M ,  11  x  N  applied  to  a  small  element.  22  Figure.3.1,  Schematic  Figure  Details  3.2,  Support 3.3,  Quarter  Figure  3.4,  Design  Figure  3.5,  Graph  Figure  4.1a,  Figure  4.1b,  Plate  weight. Figure  4.1c,  of  6"  8"  Plate  weight.  10"  the  the b)  plate  two  twisting  support  Rolling  bridge of  Plate  of  of  Post,  Figure  weight.  1  apparatus  posts,  Support  a)  Pinned  Post.  40  configuration.  plate  and  non-linear  principal  location ratio  strains,  vs.  45 of  strain  plate  gauges.45  size.  uncorrected  for  plate. principal  49 plate 57  strains,  uncorrected  for  plate. principal  38  plate 58  strains,  plate.  uncorrected  for  plate 59  v i i  Figure  4.1d,  Plate  weight. Figure  4.2a,  Plate  weight. Figure  4.2b,  4.2c,  4.2d,  4.3a, 6"  Figure  Figure  4.3c,  4.3d, 12"  Figure  Plate  uncorrected  for  plate  plate. principal  60 strains,  corrected  for  plate  plate. principal  62 strains,  corrected  for  plate  plate. principal  63 strains,  corrected  for  plate  plate. principal  64 strains,  corrected  for  plate  plate. curvature,  65 corrected  for  plate  weight.  Plate  67 curvature,  corrected  for  plate  weight.  plate.  10" Figure  12"  strains,  plate.  4.3b, 8"  10"  Plate  weight. Figure  8"  Plate  weight. Figure  6"  Plate  weight. Figure  12"  principal  5.1a,  Plate  68 curvature,  corrected  for  plate  weight.  plate. Plate  69 curvature,  corrected  for  plate  weight.  plate. Non-dimensional  70 principal  strain.  Top  Gl  direction. Figure  5.1b,  Non-dimensional  74 principal  strain.  Top  G3  direction. Figure  5.1c,  Non-dimensional  75 principal  strain.  Bottom  Gl  direction. Figure  5.1d,  Non-dimensional  76 principal  direction. Figure  5.2,  Non-dimensional  strain.  Bottom  G3 77  curvature. v i i i  80  Figure  5.3,  Non-dimensional  Figure  5.4,  Comparison  of  midsurface  approximate  strains. and  82  experimental  results.  85  Figure  5.5,  Curvature  Figure  5.6,  Bifurcation  Figure  B . l ,  Deflection  supported Figure  B.2, mesh  Figure  B.3,  by  due  varying  of  of  between  Convergence mesh  plate  weight.  90  point.  of  sizes  93  plate  opposite  Convergence sizes  to  under  load  and  corners.  centre 2  uniform  and  deflection  for  varying  10.  centre between  ix  115  116  bending 2  and  moment 10.  for 117  LIST OF  Table  3.1,  6061-T6  Table  3.2,  Plate  size  Table  3.3,  Plate  design  Table  A . l a ,  Strain  gauge  data  sheet,  Lot  1.  109  Table  A . l b ,  Strain  Gauge  Data  Sheet,  Lot  2.  110  Table  A.2,  Vishay  Material  TABLES  and  properties.  non-linearity  50 ratio.  parameters.  P-350A  Digital  51  Strain  Indicator  specifications. Table  A.3,  Vishay  SB-1  50  Ill Switch  and  specifications.  Balance  Unit 112  x  NOMENCLATURE  a  -simplifying  variable  e  x  £y  Y y  -bending  e  x  e  r y  -midsurface  e*  e  y  Y  strain  x'  a  a  relative  s  t  r  i  a  n  n  T  x ' y '  _  edge  ^ij  C  s  t  i  D  j  flexural  e  -plate  dimension  E  - e l a s t i c  f  -plate  F  -Airy  stress  instability axes  axes  r  e  s  s  a n <  ^  strain  relative  to  x'-y'  axes  position  coefficients  r i g i d i t y  function  h  thickness  -plate k y  x-y  axes  dimension  modulus  K  of  axes  modulus  G -shear  x  x-y  to  x-y  x-y  curvature  "displacement  -plate  K  point  relative to  to  to  length  D  j  relative  relative at  ratio  ratio  -non-dimensional  i j  i  s  Poisson's  surface  diagonal  xy  y'  -plate  c  along  strains  ~  at  strains  Py - n o n - d i m e n s i o n a l  x  £  strains  Y.^y - s u r f a c e  -Poisson's  P CT  e*  x  v  X  -surface  e* e  X  containing  K  1  -indices  xy  "lending y  curvatures  axes  xi  and  twist  relative  to  the  x-  K  -absolute  value  of  plate  along M  M  x  -bending  the  and  N  x  N  N  y  xy  -midsurface  -non-linear  P  -corner  P  c  u  -load v  load, at  z'  uniform  twist  axis moment  per  the  x-y  axes  forces,  per ^  unit  positive  point  of  unit  length  length 3  in  z  direction  i n s t a b i l i t y  w -displacements  y'  to  during  ratio  the x'  x-y  twisting  relative N  curvature  x,  -coordinate  of  the  y,  z  axes  plate  midsurface  parallel  to  axes parallel  to  the  edge  of  the  plate x  y  z  -coordinate  axes  parallel  to  the  diagonals  of  the  plate V  Q  -midsurface  strain  energy  stored  in  the  plate,  energy  stored  in  the  plate,  per  unit  volume V  -midsurface  strain  x i i  total  ACKNOWLEDGEMENTS  I would l i k e to thank P r o f e s s o r Ramsey f o r h i s guidance during this project and h i s comments i n the final p r e p a r a t i o n of t h i s thesis. I would l i k e to thank the t e c h n i c i a n s of the Department of Mechanical Engineering for their h e l p . In p a r t i c u l a r I would l i k e to thank Tony B e s i c for machining the p l a t e s t h a t were tested. L a s t , but most of a l l , I t h e i r many l o n g d i s t a n c e much a p p r e c i a t e d .  would phone  l i k e t o t h a n k my p a r e n t s for c a l l s . Their support is very  S u p p o r t f o r t h i s work was p r o v i d e d by the N a t i o n a l and E n g i n e e r i n g R e s e a r c h C o u n c i l of Canada.  x i i i  Science  1  1.  1.1  BACKGROUND  1.1.1  DESCRIPTION  Shown  in  loaded loads a  Figure  on  OF THE  1.1  l a t e r a l l y  uniform  is  with  adjacent twist,  or  curvatures  parallel  to  the  edge  the  deflection  the  the  can  load  is  of  and  be  the  plate  with  remains  of  the  free  edges  self-equilibrating theory equal  diagonals.  through  constraints  with  linear  shape, plate  verified  plate  opposite  Classical  the  increased  significant  becomes  stiffer.  anticlastic  takes  equal  square  predicts and  Any  straight  experiment linear  line during  as  long  theory  as  are  to.  noticeable, plate  thin  across  This  adhered  a  anticlastic  twisting. small  INSTABILITY  corners.  opposite  As  INTRODUCTION  place  section.  Due  to to  shape a  to  point  non-linear At is  a no  cylindrical the  a  symmetry  where  deviations  particular longer  the  load  stable  surface of  deflections  with plate  occur  and  and  a  and  the  curvature transition  circular and  become  cross-  loading,  the  INTRODUCTION  Figure  1.1,  A n t i c l a s t i c curvatures  plate either  along  with  the  deflection  can a r b i t r a r i l y  concave  ( F i g . 1.2a),  a)  Figure  shape  1.2,  up  Concave  cylindrical  and  take or  one of  concave  the  plate  surface,  a)  two  down  b)  of  opposite  diagonals.  up,  Deflection  down.  equal  after Concave  forms:  (Fig.  Concave  bifurcation up.  b)  1.2b)  down,  into  Concave  a  INTRODUCTION The  bi-stable  phenomenon  can  easily  cardboard  and  applying  small  sheet  of  one's  fingertips.  cylindrical  The  form  lateral  load  mode  continues  it  forces  the  number  of  more  working  stresses  factors  are  analysis  tensile  to  be  by  the  corner  forced  the  Once  it  in  that  deflect  from  taken  mode  using  load  a  with  one  application has  twisting  OF C O N S T I T U T I V E  light are  be  of  a  a  particular  under  corner  of  the  relationship,  the  plate  are  shear  from  the  sections  the  no  can  modulus  can  modern sheet  safety stress  material from  sufficient.  determined of  The  the  deduced  application or  sheet.  longer  be  structures  while  of  those  are  material the  thin  use  that  determined;  by  strength  increasing  requires  load-deflection  the  high  through  This  heavier  square  measuring  strains  accurately  and  COEFFICIENTS  fabricated  reduced  modulus a  weight  being  and  being  testing  shear  simple  other  centre.  techniques.  properties  By  the  can  demonstrated  alone.  increases,  The  the  DETERMINATION  1.1.2  As  at  to  sheet  be  3  through  corner  loads.  load-curvature be  calculated  using  theory.  Presently,  to  obtain  deflection  must  be  reliable  small  in  test  results,  comparison  to  the  the  corner  thickness  of  INTRODUCTION the  plate  testing  to  thin  exceeded the  the  self-weight  with  phenomenon, bending.  in  the  twist, the  isotropic  as  wood  or  composites  as  part  of  their  layered  1.2  what  is  uniform  the  testing  be  is  range  identified.  behavior  is  the  and  is In  the  i n s t a b i l i t y  produces is the  no  unsymmetrical  longer  maximum  orthotropic  tested  procedure Testing  easier  linear  In  in  a  load  state  and  test.  materials, can  is  range.  account.  plate  limiting  coefficients.  to  by in a  materials  twisting  thin  determining single  fabricate  ply,  than  a  such sheet  their or  lamina,  multi-  laminate.  LITERATURE  Kelvin  which  the  during  as  and  in  the  point  non-linear  above,  thereby  plate  material  size,  yield  taken  occurs  well  saves  be  deflection  non-linearities  As  constitutive  linear  reasonable  range,  must  this  the  material  described  Once  uniform  strain  of  deflection  Associated  of  within  plates  before  large  plate  stay  4  and  REVIEW  Tait[l]  expected  from  a n t i c l a s t i c  occurring  during  noted  in  the  shape,  large  1883  linear and  the  discrepancy between  theory,  the  deflections.  predicting  synclastic  a  deflection  the  INTRODUCTION  1.2.1  Many  MATERIAL  methods  modulus, discuss  G, 10  twisting length  to  between  square  in  Miyagawa,  their  corner  the  This  They  due  strains  One  They (a/h)  determining  and  of  in  plates  order  shear  briefly  method  small  10  the  Miller[2]  convenient used  show  the  would  be  Shibuya  square ranging  results and  clearly  is  in  plates from the  displays  another  large  plot  for  various  a/h  ratios  the  plate  Ueda[3]  with 1.0  to  is  the  with  a  reduce  the  the  at  mm.  the  26.6  Of  of the  centre  of  of  plate  the  synclastic  estimated  used,  from  relationship  the  sizes  This  3.2  transition  into  done  ratios  mm t o  strain the  have  a/h  typical  shape  considerable.  therefore,  and  principal  anticlastic  bifurcation to  measured;  of  load  uniform  curvature  plate. ratio  for  Ramberg  methods.  thickness  plate.  shape.  But,  a  suggested  material.  testing  and  interest  from  a  recently,  240  the  in  been  effects.  extensive to  have  thickness  non-linear  More  TESTING  different  of  5  point  plate  tested.  midsurface  non-linearity  curvature  of  can  was  not  not  be  calculated.  The find  idea its  of  measuring  constitutive  the  deflection  coefficients  of  was  a  twisted  extended  by  plate  to  Tsai[4]  INTRODUCTION to  orthotropic  two  twist  tests  independent results  region used  analysed  is  equation. to  not  The  1.2.2  Lee  based  OF  square region  on  critical  uniform  with of  the  to  incorporating  determine  orthotropic  the  4  materials.  The  verified.  plates  inplane  corner  into  the  An A i r y  stresses.  large  stress  The  respect  to  minimum  potential  deflection.  inplane  shear  the  The  deflection function  error  in  is  the  compatibility energy  stress  is  used  function  requirement  at  the  used  plate  INSTABILITY  Hsu[6,7]  twisted linear  the  tests  Chandra[5].  the  method  exactly.  POINT  and  by  principal  satisfy  boundary  bending  for  minimized  calculate  does  two  proposed a  orthotropic  represent  function  He  experimentally  square  is  to  with  constants  were  Twisting  plates.  6  the  synclastic dimensional  plate using  with a  of  von  after  takes  moment  at  displacement  free  edges  place. the  the  which  a  field  its  plate  Up  assumes  results  non-  analysis to  the  an into  give  bifurcation  a  unstable,  transformation  of  of  difference  equations.  Their  point  into  finite  Karman  deflection,  shape, shape  the  numerical  non-linear  value  anticlastic  calculate  as  the M  the non= 21.  INTRODUCTION  Ramsey[8]  also  non-linear twist  to  shell  in  which  rectangular  free  edge  Associated  of  with  an order  beyond  show  that  a  free  edge  in  conditions  of  of  the  the  same  bifurcation, c r i t i c a l  M c  r  =  but  value  uses  of  1.42 was  magnitude  less  a  than  the  stresses  1.3  PURPOSE  A N D SCOPE  The  purpose  of  resolve  for  the  short  edge.  He  shows  instability.  shape,  which  Fung  i s  develops the  occurs  adjacent  develop shape.  to  after  boundary  and W i t t r i c k [ 1 0 ] .  They  to  the  plate  satisfy  the  boundary  They  do n o t ,  however,  OF THE STUDY  a  i s  two  rather results  fold:  large for  discrepancy  the point  of  in  a  the  itself.  study  theoretical and  by  cylindrical  the  of  equations  he a p p r o x i m a t e s  instability,  zone  i n s t a b i l i t y  to  of  which  along  cylindrical point  von Karman  in  type  described  which  for  plate  conditions  transition  the  1)  of  the  the non-linear  the  phenomenon  examine  He f i n d s  twisted  boundary  existence  layer  i s  solves  long  loading  point  [7].  Reissner[9]  the  theory.  the  be Ka= 3 . 2 9 h / a . A v a l u e  calculated, result  calculates  7  some  bifurcation,  INTRODUCTION 2)  to  generate  into  the  experimental  non-linear  understanding  The  following 1)  To  is  the  design a)  the  of  the  apparatus  further  twisted  the  to:  and  loads  which  deflect  opposite the  incremental plate  into  corner  the  non-  range; only  no  allow  plates  project.  equal  with  to  for  phenomenon.  apply  apply  c)  range  this  scope  linear b)  of  data  8  lateral  moment  for  or  plates  point  loads  inplane of  at  the  forces;  various  corners  and  sizes  to  be  tested. 2)  To  determine  using  a  linear  deflection  range  finite  and  of  test  plate  differences,  non-linear  a/h  ratios,  which  effects  limit  under  self-  we i g h t . 3)  To  measure  and  To  a)  principal  b)  midsurface  To  for  increments  in  load  strains, strains,  and  curvatures.  find  strains 5)  strains  calculate:  c) 4)  surface  non-dimensional and  develop  curvatures an  groups which  approximate  midsurface  strain  curvatures  and  in  a  twisted  the  collapse  analytical  plate by  for  with  corner  calculated the  results.  solution  unequal forces.  for  INTRODUCTION To  compare  the  experimental  analytical  and  numerical  conclusions.  results  models  and  with derive  10  PLATE  2.  In  this  section  develop  2.1  An  both  the  the  THEORY  elastic  square  throughout  surface  with  diagonals.  This  equal  in  1.1.  Fig.  and  the  to  equivalent  along  the  two  +P/2 These  plate  are  to  loads  applied  l o a d s ,of  at  P/2  have  when  plate  a uniform  bent  into  used  twisting  an  state  stress  be  adjacent  opposing  a uniform  corner  corner  twisting  At  each  corner,  of  P/2.  Statically  small  interval  couples  as  can  the  shown  in  as  can  plate  be  by  shown  shown  distributed P is  equivalent  along  moment  produced  loads  load  the  loads  moment  the  theory.  anticlastic  along  of  to  bending  curvatures  the  form  is  [12]  opposite  opposite  edge  edge[12].  corner  and  of  non-linear  will  plate  simple  Along be  the  THEORY  convention and  plate  equal  applying  by  linear  LINEAR  M , , x y  sign  BENDING  free Fig.  replaced loads edges. 2.1.  of  PLATE  Figure  Along w i l l  2.1,  the be  which  Corner  x'  loads  edge,  subject  results  each  to  in  a  a  forming  plate  constant  couples  element  twisting  BENDING  moment  of  of  THEORY  magnitude  length  of  11  M ,  ,  x  Ay,  (P/2)Ay , r  twisting  moment  f  P  per  unit  length  of  M  x'y'  Similarly, of  the  Fig.  the  plate  same  =  method  producing  -  Ax'  can  uniform  be  ...(2.1)  2  Ax'  applied  twisting  2.2,  Constant  twisting  moment  M  the  moments  2.2.  Figure  to  ,  other as  edges  shown  in  PLATE  From be  equilibrium,  calculated  moment  can  be  the  internal  using  Mohr's  shown  to  occur  in  the  sides  of  the  plate,  value  of  the  moments  in  M  =  xy  moment-curvature  K  are  y  respectively, Poisson's given  is  ratio.  The  direction  M  = D(K  M  = D(K  of the  of  the  can  bending  ±45°  to  the  diagonals.  The  diagonals  are  (2.2)  are  + vK )  x  y  + \)K )  y  . . . (2.3)  x  M  = D(1-\0K  the  curvatures  xy  the  twist  flexure  in  in the  rigidity  the  x  and  y  directions  plate  and  v  is  of  the  1  plate,  D,  is  by D  where  of  relationships[12]  xy  and  angle  direction  y  x  an  maximum  plate  0  x  K  The  the  12  1  M  where  at  within  THEORY  = - M = M , , = - | y x'y 2  x  The  moments  c i r c l e .  the  BENDING  E is  the  Young's  =  E  modulus  h  (2.4)  3  and  h  is  the  plate  thickness.  PLATE Combining  Eq.  curvatures,  (2.2),  we  (2.3)  = -K X  K  therefore  have  direction  of  x-y  these  If  axes,  the  plate  are  also  the  plate  the  and  solving  for  13 the  = -6(1+M)  y  t  equal  and  opposite  diagonals. are  and  the  Because  principal  the  deflection  ...(2.5)  h  =0  xy  deflections  small  (2.4)  THEORY  find  K  We  and  BENDING  are  there  in  the  is  no  twist  curvatures  of  the  small,  curvatures  curvatures  the  and  associated  twist  are  in  the  plate.  slopes  related  to  by  K  x  ax  y  = - f-f 8x  2  ...(2.6) J  8 w 9x3y 2  K  We  consider  downward  Using  the  plate  in  in  the  curvature  positive  the  direction  of  curvature x-y  xy  along  coordinates  the is  =  the  plate  is  convex  z.  x-axis  the  deflection  of  the  then -K  v  if  (x -y ) J  2  ...(2.7)  PLATE The of  distribution the  plate  assumption. the  can  of  stress  be  calculated  Strains  midsurface  of  vary  the  and  strain using  linearly  plate  and  x e  8x  Using the  Eq.  (2.6)  plate  and  (z=-h/2)  (2.8),  Eq.  circumflex  we  (2.5)  curvatures,  _  distance  z  from  by  ...(2.8)  3 w 3x3y 2  9  the  surface  = - ^ K 2 x  y  = - ^ K 2 y = +h K  signifies  demonstrated  with  given  plate  2  x  r 'xy  In  thickness  Kirchhoff  the  14  strains  on  the  top  of  are  e  the  the  2  xy  e  where  the  THEORY  = -z |~7 8x  y  = r  through  with  are  BENDING  no  twist,  ...(2.9)  xy  that  that  occur  it  is  equal along  a  and the  bending  strain.  opposite x-y  axes.  Therefore, A  e  A  = -e  x  y .(2.10)  = o  A r  'xy  The  strains,  and  lie  along  e  x  and  the  e  y  are  the  diagonals.  principal  strains  PLATE  By  measuring  load,  the  the  shear  materials  the  constants  is  principal modulus  surface  can  relationship  be  BENDING  strain  found.  between  for  For  the  THEORY  a  given  Eq.  (2.5),  corner  isotropic  three  elastic  G  Combining  15  ...(2.11)  (2.9)  and  (2.11),  and  rearranging,  we  find  G  ...(2.12) x  Another  method  the  plate  the  x-axis  of  along the  determining  one  of  the  deflection  G is  diagonals W  c  ,  using  -K c  w  Combining  with  Eq.  small  deflections  calculate  the  strain  deflection.  or  shear  in  Eq.  a  deflection  distance (2.7),  c.  and  an  modulus  of  Along  is  2  ...(2.13)  (2.11),  l ^ S i y 2 w h c  For  at  the  = —2— 2  c  (2.5)  through  we  get  ...(2.14)  3  elastic from  Conversely,  the we  material,  we  can  load  and  surface  predict  the  strain  corner can  PLATE and  deflection  a given  for  material  BENDING  knowing  16  THEORY  the  load  corner  appli ed.  2.1.1  In  LIMITATIONS  the  into  linear  OF THE L I N E A R  theory  consideration  neglects 1)  the  certain  before  limiting  using  Edge  not  this  effects is  are  caused  resisted  by  T  vary  along  the  section.  Saint  Venant's  rectangular of  theory.  T  , , x'y'  and  edges  overall  of  if  Midsurface deflected stresses  by x  ,  must The  length  the  the  of  stresses  will  a be  the  be  taken  theory  shows  shear  biggest  x  ratio occur  the and  the  for  a  distribution  , , y'z '  over  but  the  occurs is  at  negligible  large.  because  non-developable negligible  cross-  torsion  deviation  is  twisting does  plate  of  cross-section,  a/h  into  alone,  theory  transverse The  assuming ,  cross-section  cross-section. the  factors  following.  moment  2)  THEORY  if  the  plate  surface.  the  a/h  is The  ratio  is  small. 3)  The  deformation  neglected. large.  This  due is  to  transverse  negligible  if  shear the  a/h  stress  is  ratio  is  PLATE In  [2]  the  these  total  ratio  2.2  In a  of  effect  NON-LINEAR  surface  the  a  The  with  Plates with  be  was  strain  large  increase  taken  to  is the  in  load The  The  to  for  the  based  on  is  of  these  is  shown  with  twisted  surface,  curvature,  plate  linear  an  that a/h  that  i s ,  stretching  of  theory, the  if  the  thickness  of  the  as  diverge the  when  of  the  deriving  is  and used.  no  had surface.  deflections  are  plate.  from  non-linear  There  are  of  we  middle  the  midsurface  stretching  problem  equations  the  negligible  coupled  complicated.  these  it  minimized  stretching  consideration equations.  equations  Gaussian  no  differential  solution  is  non-developable  deflections  effect.  into  a  discussion  there  considerable  errors  and  occurs.  respect  with  the  into  surface  that  discussed  17  10.  non-zero  midsurface  small  these  plate  previous  assumed  are  THEORY  THEORY ,  with  middle  the  of  approximately  bending  In  limitations  BENDING  linear  forces  have  midsurface the  theory a  must  governing  differential  closed  approximate  form  solution  solutions  PLATE  2.2.1  GOVERNING  These 1910 f i r s t  are  and  terms.  Below  for  developed  use  future  The  a  It  can of  displacements directions  [12].  the  the  where  derived  These  problem  most  of  of  von  equations  Karman are  the  of  the  in  the  non-linearity  significant  summary  by  in  non-linear  equations  involved  Strain  of be  the shown  the u,  plate from  midsurface v  and  originally differential is  w which The  l i e s  in  the  midsurface  the  geometry  dependent are  in  on  the  x,  y  strain  x-y that  the  midsurface  and  z  equations  are  au 3x  . . . (2.15)  y  r.x y strains.  to  respectively.  ex  the  in  brief  Geometric  e  18  reference.  stretching  where  only  is  midsurface  plane.  equations  approximation  plates  THEORY  EQUATIONS  non-linear and  BENDING  au a y  overbar  signifies  that  these  are  midsurface  PLATE We  expect  the  inplane  displacements,  comparison  with  the  lateral  therefore,  drop  the  second  produce  the  BENDING  u  and  displacement, order  terms  v, w.  in  THEORY  to We  u  and  be  19  small  in  can, v  to  strains  x  = 9u 8x  y  - | Y 3y  I  i  'xy Typically,  when  the  midsurface  become  i 2  3u 3^  =  i  +  (  2  +  ;  | w ay  )  ...(2.16)  a  , 3v ^ , 3 w . , 3 w . 3x 3x 3y° +  slopes,  large,  aw ^3x  (  } (  3w/3x  the  and  3w/9y,  midsurface  of  the  strains  also  Hooke's  law  become  large.  When  Constitutive  the  assumption  relations  for  an  Relations  a  =  z  0  isotropic  is  made,  material  a  o  e  x  =  x E  = -v  reduce  to  y  v =r  L  E  a I-y  the  a  =ZE + E  ...(2.17)  T 'xy where  the  midsurface  overbar  on  the  quantities.  G stresses  signifies  that  these  are  PLATE  The  and  the  are  in  subject  second  adding,  we  +  midsurface terms  of  to  3x^  u  compatibility of v  "  3x3y  (  c ,  )  forces,  N x  = ho  N  = ho y  (2.17)  1_ Eh  from  into  Eq.  Eq.  ,_3_i ^3x 2  3  2  Eq.  By  (2.16),  (  T  x  N  3x  y  /  N y  2 }  (  3y  ..-(2.18)  2 }  can  be  N  by xy  expressed  J  x ...(2.19) y  N xy  Substitution  condition.  strains,  x  Eq.  20  producing  and  x  midsurface  the  and  " ~3x3y  stresses,  the  a  derivatives  eliminate  3yT"  The  THEORY  Compatibility  strains  taking  BENDING  = hT  ((2.19)  (2.18),  xy  into  Eq.  (2.17),  yields  the  result  V  ,3 w x 3x3y  and  2  k  ;  v  3x* ^3y* ;  ;  then  from  (2.20)  PLATE  For in  the  A  exact  solution,  unit no  in  the  equilibrium  small  N  x  forces  element  of  N  each  x,  w i l l  rectangle  length  body  THEORY  21  Equilibrium  equilibrium  (x-y)  BENDING  y  be  and  z  size  dx  by  the  the  plate  should  x  be  The  inplane  f i r s t .  dy  applied  applied,  of  directions.  examined  and  are  element  has as and  midsurface  shown y  in  forces  Fig.  2.3.  equilibrium  of  per If the  requires  3N  3N  8 X  3  3N  ...(2.21)  y  3N  - 3x T  2  +  IT* 3y  =  0  respectively.  The  higher  order  due  to  element  not  being  equilibrium, ^  the  midsurface  the  terms  that  might  be  planar,  included are  in  Eq.  small  and  N  N  (2.21),  have  been  neglected.  For  z  projected we  find  onto  the  z  axis.  forces  Neglecting  the  x  N  y  higher  xy  are  order  terms  PLATE BENDING THEORY  Figure  2.3, M i d s u r f a c e small  forces  element.  3w  and N  N ,  N^,  + N  §-7) d x d y  x  x y  applied  22  to  a  ,  2  XF  (N  x 3x  2  3N  The  quantities  therefore z  in  they  direction  + 2N  x y 3x3y  3N  the  are  y 3y  .  3  .(2.22)  3N  square  zero.  2  This  brackets leaves  3N  match the  Eq.  (2.21);  equilibrium  in  the  as  (2.23)  The  sum o f  the  can  be added  to  midsurface the  load  forces on the  projected element,  onto  the  q(dx)(dy),  z  axis  which  is  PLATE supported edges  a*  by  the  a y ^  +  "  Substituting  8*w ^ ax^  Eq.  there  3 v  a ^  +  four  "  =  N  x  along  the  element's  +  N  x  a ^  +  2  Haf  N x  y  +  N  y  • • •  relationships,  Eq.  (  2  '  2  through  a  A  )  (2.3)  D  (  q  to  , .. 8 w N  ax^  +  2  N  8w  x y a ^  familiar  8 w.  2  .  J  y  N  ^  -..(2.25)  bi-harmonic  equation  when  forces.  Function  equations  From  x  the  no m i d s u r f a c e  problem  .  J  +  are  that the  these  must  be  three  satisfied  equilibrium  and the  to  the  the  equations,  compatibility  we c a n s o l v e  solve  four  Eq.  equation,  unknowns  N  non-  Eq.  , N , x ' y '  and w.  y  Theses  If  q  1 , =  (2.21) a n d E q . ( 2 . 2 5 ) (2.20).  (  moment-curvature  reduces  Stress  linear  i^T  ^ 8*w  A  are  2  the  a x ^  2  (2.25)  The  shears  23  gives  (2.6),  +  and the  THEORY  giving  -  and  moments  BENDING  equations the  c a n be  reduced  introduction  function,  F(x,y),  i s  of  down  an A i r y  found  such  to  two  stress  that  equations function[11].  PLATE  BENDING  THEORY  24  8 F 3y 2  x  N  2  =hg  y  ...(2.26)  3 F = -h 3 x 3 y 2  Nx y The  x-y  equilibrium  satisfied into  3*w 3x"  the  ,  by  the  stress  remaining  , 3 i w  S^v 3x 3y 2  3y"  2  Eq.(2.21),  equations,  function.  equations,  D ^h  3y  3x  2  identically  Substituting  this  function  yields  h , f l ^ F ^ w ^ F ^ w _ .  B  are  3x  2  3y  2  2  3 F  3w  3x3y  3x3y  2  .  2  .  ;  and  a? Equations  + 2  (2.27) a n d  differential solution d i f f i c u l t  to  '  +  a3?8F  for  in  practical  The  the  boundary  conditions  (  a^ V  and  the  ••• -  )]  (2  coupled  As  F.  }  non-linear  mentioned  boundary  28)  earlier,  conditions  the is  problems.  CONDITIONS  to  w and  equations  2.2.2. B O U N D A R Y  solution  W  (2.28) f o r m t w o  equations these  E [ (  plate  bending  imposed  along  problem the  must  edges  of  satisfy the  the  plate.  PLATE BENDING THEORY A plate with to  f r e e e d g e s h a s i s no m i d s u r f a c e  forces  25  applied  t h e e d g e s . The s t r e s s f u n c t i o n must s a t i s f y t h e  following:  x=0,a  - z e r o normal  force = o  i ^ a y  -zero  shear  2  force ^ 3x3y  y=0,a  - z e r o normal  = 0 ...(2.29)  force a F 2  ax  -zero  shear  2  =  0  =  0  force a F 2  3x3y  A l o n g t h e f r e e e d g e s t h e b e n d i n g moment a n d K i r c h h o f f are  zero.  x=0,a  From t h e s e  r e q u i r e m e n t s we g e t :  - z e r o b e n d i n g moment a w +^v ix^ -zero Kirchhoff  a w7=  2  axT y=0,a  shear  +  j  aT  n  0  shear  ( 2  " ^ a^yT  "  0  ...(2.30)  - z e r o b e n d i n g moment aw W  aw _ w ~  2  2  +  v  -zero Kirchhoff aw 3  W  n 0  shear ,0  ""  (2  . }  aw 3  n  =  0  PLATE  The  last  applied  boundary to  the  plate  found  for  a  along  the  edges.  2.3  A  corner  APPROXIMATE  number  of  using  In  the  and  Of  P,  the  a  to  variety  the  twisting From  moment  Eq.  moment  26  (2.1)  was  we  + 2M  ,  between  curvatures  along  the  midsurface  have  developed twisted  plate  methods.  opposite  an  midsurface  a n t i c l a s t i c the  strain plate  diagonals.  The  strain  varied  with  the  midsurface  strain  changes  and  with  equal  results square  of  curvature.  interest  the  twisting  non-linear  for  plate  load.  [2,5,7,9]  of  relationship  that  the  THEORY  SOLUTIONS  calculated  plate in  corner  is  showed the  relates  the  load,  solutions  problem  curvature  by  investigators  approximate  [2],  condition  BENDING  after plate  is  how  the are  the  point no  of  bifurcation,  longer  equal  in  where  the  magnitude.  in  the  curvatures  PLATE  ANALYTICAL  2.3.1  WITH  The  UNEQUAL  approach  used  in  SOLUTION  used  [2], but  is  similar  the the  This  degree  of  the  loss  forms UBC  a  of more  Reduce,  general a  the  curvatures freedom  increases  symmetry  to  displacement  allows  substantially  FOR M I D S U R F A C E  27  THEORY  STRAIN  IN  PLATES  CURVATURES  deflection extra  BENDING  in  in  function to  be  the  the  displacements  algebraic  approach  the in  magnitude.  displacement  number  To  for  unequal  the  solution.  symbolic  Rayleigh-Ritz  of  field  coefficients, u  and  manipulate  v.  the  manipulation  due  to  This equations,  program,  was  used.  Using  the  midsurface  deflections be  of  estimated  strain  plates,  by  Eq.  applying  equations (2.16),  the  for  the  principle  the  large  midsurface of  strain  minimum  can  potential  energy.  It  is  Fig. K  x  of  assumed 1.1,  is  and  Ky  the  plate  that bent  along is  the into  the then  plate, an  w  the  anticlastic  diagonals given  with  of  the  coordinates shape  plate.  with The  shown  in  curvatures deflection  by  ...(2.31)  PLATE The  midsurface  strains,  displacements  in  deflection  The  by  the  w.  the  Eq.  u  (2.16),  and  v  in-plane  BENDING  depend  directions  on  as  displacement  THEORY the  well  can  28  as  be  the  approximated  polynomials 00  u  ^  =  2  . 1  ,  ...(2.32)  J  _  =  .  2  d , x  k,l  where  The  i,  j ,  k,  conditions  1  are  of  symmetry  y  IJ  00  v  1 1  C.. x  .  K  positive  and  k 1 y  i  integers.  antisymmetry  u(x,y)  = u(x,-y)  u(x,y)  =  are  (2.33) -u(-x,y)  and v(x,y)  =  v(x,-y) (2.34)  v(x,y)  These c^j  conditions  and  d ^  are  are zero  =  satisfied  v(-x,y)  if  some  of  the  coefficients  giving  i  =  1,3,5,...  j  =  0,2,4,...  ...(2.35)  and  PLATE  k  =  0,2,4,...  1  =  1,3,5,...  BENDING  THEORY  29  ...(2.36)  Expanding power  the  series  terms  gives  u  = c  1 0  x  + c  1 2  v  = d  0 1  y  + d  2 1  xy  of  + c  Eq.  (2.32)  xy*  + c  3 0  x  x*y  + d  0 3  y  1 A  3  up  to  and  + c  x y  2  3 2  + d  x y  3  2 3  3  including  +  C  +  d  5  q  X  f i f t h  5  ...(2.37)  Substituting Eq.  (2.16)  =  c  x  =  d  y  x  y  =  2  d  Using £  io  +  oi  +  2 !  the  c  d  y  t l  3  displacements,  Eq.  : y  2  2i^  ^  +  the  +  c  A i  d  i,y"  d  +  J  x  3  +  4 1  Y  x*  +  2  following  . = x/a,  rt  3c  3 0  + 3d  d  23 y x  x  y  0 3  3  +  + 3c  2  + 3d  2  2C  1 2  notation = y/a,  x y 2  3 J  xy  x y  c  0 5  ys  and  (2.37)  into  = K a ,  c  d  2  1  = D / a ,  d „  = D /a*  d  2 3  = D,/a\  d  = D /a*  c  3 0  = C /a ,  c  d  0 1  = D ,  0 3  = D /a ,  2  0  2  3  x  = C /a*.  c  2  i  a  4  2  3  +  y*  0 5  +  2C  |(Kx)  ( K^ y ) (K  + \  x y 3  3 J  2  2  + K K  ...(2.38)  xy  p  = K b y  y  = C /a*  ltt  5  xy  x"  convenience  3 2  =  3  1 A  c  1 0  5 0  + 5d  2  + 4C  for P  + 5c  = C^/a ,  c  d  (2.31)  2  2  2 3  x c  2  gives  2  x  x=y + d  2  0  0 5  = C  5  / a *  2  6  ...(2.39)  PLATE leads  BENDING  30  THEORY  to  ...(2.AO)  =  To  2Dtin + 4D &*n x  2  satisfy  plate,  these  edges  For  along  w i l l  related T  xy  of  3  one of  imposed  have  zero  + 2c^  along  stresses  along  then  t h e edge  to to  also  we w i l l  equilibrium  parallel  and shear  stresses  Ac^n  2 c^n +  conditions  t h e symmetry  A r b i t r a r i l y zero  3  the boundary  due to  other  +  2D,c:n  the normal  setting then  +  3  n + P P ^n x  y  the edges  must  be z e r o .  the edges  to  of the By  be  zero,  on t h e d i s p l a c e m e n t s , normal  s e t ~o ,  and shear  and T ,  .  the  stresses.  to  x'= a / 2 .  a triangular  the x and y  axes,  the midsurface  edge  element  the boundary  stresses  ~a , x  a  y  with  sides  stresses are and  by o  T  Substituting strains,  Hooke's  E q . (2.40)  = X  x'y'  o +o V x  xy  •2  2  law, Eq. (2.17), into  ...(2.41)  o -o x y  E q . (2.41),  and the and using  generalized the  notation  PLATE  gives  the midsurface  coefficients polynomials  2(l-v) + D  and position i n  E x'  {C  + D^  0  stresses  i n terms  BENDING  of  i n the plate.  THEORY  t h e 12  The f o r t h  + C ^  0  2  + C n * + 3C-,cy» + 3 C A ^ n  3  J  2  + D,£* + 3 D 3 n  2  jC  2  1  + 5C c;* + 5 8  2  + 3 D £ n » ' + 5D 5 n" + ^ P n  + |  2  -  + 2D En  +  3  4  2C lr\  + 4C2£n  x  3  +  ^,V^ .  (D  0  + D ;  + D E  2  2  IC  4  + 3D3n  2  + 3D,c;'n  2  • • • (2.43)  + 5D5n"  n )] 2  P  y  Along  the edge,  zero,  the non-dimensional  x'=a/2,  w h e r e CT , x  n  Substituting from  2  a  4  + PxPy^nl )  p^  order  £, a n d n a r e  + cxUD^r, + A D : n  + |  31  this  into  the expression  and f , x  coordinates  , are  are expressed by  = ±z - I V2  E q . (2.43)  gives  ...(2.44)  and factoring  the I  PLATE = gTJZ^y  + D  ["2aD,  - 4aD  - 2aC, - 4aC  2  + 5 C + 3 C , + C ] + 4V2 V  2  5  -  1 0 D - 3 D , - 3 C , - 2 C ] + 2V  -  12aC  5  x  + p + 2Dj + 2 C + p + 30D x y 2  2  2  5  - A a D , - AaC  3  2  0  2  - 6aD,  1  y  3  10D - 6D - 2 C ] + [4C + 2 C + 4D + p 5  + a C , + 6aC  2  + 2aDj + 2aCj + a D , + 2aC  x  t  32  6  + 3D, + 6D + 3C, + 6C  2  a  + 6 C ] + 272 £ [ a p p  -  [~2ap P  2  THEORY  + 5D + 3D,  2  [3aD, + 2aD  8  BENDING  ...(2.45)  3  - 2 C - p.2  2  X  + 5D + 6D + C,]}  2  0  5  3  and  y'  =  8(THO  tlOD  [  "  5  ° " 5  3  D  * " °  + 3D, - 3 C , - 2C ] + 2 E  g  2  +  5  C  S  +  3  "  C  +  C  '  ]  +  4  [p£ - 2 D + 2 C - p  2  2  a  2  V  L  V  -  y  30D  5  (2.46) - 3D, - 6D  + 3 C , + 6 C + 6 C ] + 2V2 ll-lC^  3  3  + 6D - 2C ] + [4C + 2C - 4D 3  For  2  0  the normal  boundary,  each  must  (2.46)  X  0  and shear term  i n  be z e r o .  P  y  5  stresses  form  Solving  these  becomes  unwieldily  at  point.  assumed  the  y  ,  b e 1/3  constants  other p  to  constants,  we g e t  this making  be z e r o  equations  i=l  along  of  equations  1/2. to  5,  with  in  them,  CQ a n d DQ a n d t h e c u r v a t u r e s ,  P  v  for  terms  of the x  and  t h e 12  a general  Solving i n  the  E q . (2.45)  To s i m p l i f y  a equal  and D ^ , where  5  2  brackets  10  constants.  6  to  the square  These  y  - 5D - 6D + C ] )  unknown  is  + P + 10D  2  and  form  PLATE  Yk  12 C  = ^  3  (  "  1 1 6  (68C  C  (-4C  k ^  Eq.  (2.47)  coefficients Placing Eq.  a as  -  °  (  2  0  0  certain might  0  y  + 3p (-4p -p x  x  )]  -  ..(2.47)  ^  116D  -  0  »  C  x  +  + 68D  (AC  )  3p p )  20D  2 0 D  0  1_ [-20C 72  In  44 +  y  p x  33  0  ^2  to  3  THEORY  + 4D )  +  (44C  ° ~ P  4 4 D  -  0  0  °  ( A C  +  -  0  [-52C  j  °  BENDING  0  -  +  0  52D  0  3p p ) x  3p p ) v  v  x' y  +  3  P y  above  results  (2.40),  gives  the  x  0  4  ° o  +  3 p  x y p  }  antisymmetry  the  (-p  4D )  +  be  y  expected back  strains  from  into as  exists  a  the  this  in type  strain  function  the  of  of  problem.  equations, position  as  PLATE  i  [2q* ( 3 p p  = J7  x  + n  x  2  ("3P P x  "  y  + E (-3pxPy -  = y  [2n*  X  2  V  I  =  W  ("3P P  y  + AAC  (3p  + 3r,£(p P x  To  solve  20C  xy P  -  y  +  energy  energy  per  -  20D )  0  0  0  52D )  -  116D )  terms  V  of  o  The  elastic  the  volume  + 20D )  0  0  2  (C  -  0  D ) 0  (3pxPy -  +  2 n E  in  volume,  0  '  (3p  xy " p  52C  °  +  7 6 D o )  the V  C  and  Q  plate  be  D ,  we  Q  a  require  minimum.  The  that  the  strain  is  Q  =2 - i (xoxe + yoye + xy'xy ? r )  only  the  + 68D )  0  0  coefficients  energy  AAC  12D )]  (2.A9)  strains  2(1+\J)  of  + AC  y  0  5 2 D o )  = T 7 T T - T [^r-  0  0  + 2AD ]  0  stored  unit  + 2i>  0  ° "  n  in  2  + 96n E  0  -  V0  or  x  + D )  0  0  + AD )  0  (3p p  21"  +  (-C  + 2AC ]  0  -  0  the  strain  + AAD )  75C  12C  for  0  2  + 96n S  0  34  ...(2.A8)  "  L  2  AAD )  THEORY  y  y  x  -  0  + 20C  (-3P P x  + V  0  (3p p  ^  + n  H6C  52C  2  i  + 68C  y  BENDING  1-M  (  stored, plate  £ 2  x  V  + 2\)e  ,  x  is  e  + e) 2  y  y  found  + \  2  by  r  2  'xy  )]  ...(2.50)  integration  over  PLATE  = A J"  m  J"  x=0  In  terms  of  THEORY  35  (a//2)-x  a/V2 V  BENDING  V  h dydx  0  . . . (2.5.1)  y=0  non-dimensional  coordinates  1//2 V  . = Aha  m  J  2  J"  1=0 Upon  substitution  equations  blossom  Eq.  in  size.  h  a  J  E  2  minimize  this  cj +  by  dndE  ...(2.52)  =o  (2.48)  and  After  (2.50)  2  101152  CD + 0  0  into  integration,  0 0  r  + 1386A0  0  n  [333 p p - 7776 p p C 60A.800 . x y "x y  m  We  of  V  -  we  7776 p p D "x y  1386A  r  (2.52),  the  get  0  D*]  ...(2.53)  setting  9V —52 8 C  =  ...(2.5A)  av —51  When  we  carry  equations  for  out  these  CQ a n d  DQ,  C  0  0  °  =  0  operations, we  and  0  the  two  find  = 3/1A6  pp ...(2.55)  7  D  solve  = 3/1A6  p p x  y  PLATE which we  are equal.  find  Placing  the following  e  x  midsurface  = [17011" -  109n + 1701" - 109c?  2  = [i70n3E  x y  relationship  no v  +  + eirtf]  n  conclusions  with  v=  n)/  are  c a n be made  of  point  XY  strains  strains,  at  a l l points curvature  surfaces  have  the centre  (3/146)p '  p  x x  The amount the  and  curvatures  e  in x  surface  the  and e  y  ,  equal.  Gaussian  At  the  on the p l a t e  the diagonals,  The m i d s u r f a c e  Y ,  about  strains  e x  4)  ...(2.56)  1/3.  the midsurface  direction  3)  P P  12] TIT^  +  ^  the midsurface  For any p a r t i c u l a r (£,  2)  E q . (2.48),  jgf-  + 12]  + 170&* "  2  between  materials 1)  into  36  JOH  few i n t e r e s t i n g  for  THEORY  strains  109n  y  A  back  = [17CV "  e  r  the constants  BENDING  vary of  , ~z a n d y  linearly  the plate.  no m i d s u r f a c e  the plate * a n d Y = 0. xy  of  thickness  of  midsurface of  the  the  Cylindrical  strain. e = "e = x y  strain  plate.  with  i s  independent  of  37  3.  This  section  is  DESCRIPTION OF EXPERIMENT  divided  into  two  subsections,  apparatus  and  procedure.  The It  f i r s t  section  covers:  the  design with  loads  the  plate  used;  and  the  design  numerical  analysis  The  section  It  other covers  the  operations  3.1  describes of  the  apparatus  base  plate,  corner  forces;  of  plates  used  the in  describes  i n i t i a l  completed  the  the  the  in  the  tests.  supports  and  instrumentation  under  test  the  before  and  plate  experimental  preparation during  which  the  determining  used  the  the size.  procedure tests  and  used. the  test.  APPARATUS  3.1.1  DESIGN  Shown  in  consists  Fig.  3.1  of  base,  a  is  a two  schematic supports  of and  the a  apparatus. loading  It  system.  The  Figure  3.1,  Schematic  of  the  plate  twisting  apparatus  DESCRIPTION apparatus of  equal  two  is  height,  base  is  diagonal support Along two  of  cables heads  straddle  with  the  from  a  to  test  the  the  loading  gap  system  of  between f i t t i n g  levelling  test its  is  variable  two  tables  between  of  the  plate.  For  a  set  corners,  attaching  the  and  of  holes  the  Along  of  The  stabilized  using  the  vertically hanger  rests  posts  the  different  the  support  of  allowing  holes  base  one  plates.  each  allowing  restrictions.  leveled  sizes  series  has  plate.  positioning  a  base  free  from  bolts  and  on  the  from the  two  screws.  plate  is  supported  with  two  6"  posts  at  either  end  diagonal.  Pinned  Support  hemispherical aligns  the  steel  different  are  through  plate  free  the  allowing  diagonal  plate,  be  30"x30"xl/4"  accommodate  pass  test  to  slots  to  other  underneath  The  two  posts  the  below  made  are  cables  size  of  to  39  EXPERIMENT  tables.  The  The  designed  OF  a  function providing  seat.  matching is  Post,  to  point  A  ball  seat  stop  shown  in  the  support.  in  Fig.  bearing the  plate  plate from  sits  3.2a,  has  in  the  above.  The  moving  a  seat  and  b a l l ' s  laterally  while  DESCRIPTION  OF E X P E R I M E N T  PLATE CORNER •BALL BEARING  PINNED SUPPORT POST  PLATE CORNER BALL BEARING  ROLLING SUPPORT POST  Figure  3.2,  Details Support  of  the  Post,  b)  two  support  Rolling  posts,  Support  a) Post.  Pinned  40  DESCRIPTION The of  Rolling circular  with to  Support  the  Post,  cross-section.  matching  groove  move  inward  during  induced  during  bending.  The  plate  Each  is  cable  corners  loaded  is  of  shown  the  test  Cable  is  in  ball the  bending,  via  passed  A  two  plate. thus  a  and  Fig.  3.2b,  bearing  1/16"  through plate  in  OF E X P E R I M E N T  aligns  This  no  has  allows  inplane  then  hole  in  clamped  groove  this  groove  the  plate  forces  multi-stranded  small  a  41  are  cables.  opposite  to  prevent  its  prevents  the  removal.  The  Fixed  test  plate  loaded.  It  through  a  corner runs hole  protruding longer the  loop  the  cable  The  Hanger  to  the  base  hanger,  on  hanger  to  plate  plate, plate.  plate  hole  the  hole the  achieve  long  and  the  other  corner  ending The  a  Anchor  preventing  is  vertically  in  diameter,  corner  loop  Rod,  is  the  down  just  which  placed  is  through  extraction  of  hole.  the  carries  weights  6"  while  base  connects  which  placing  test  base  through  Cable  by  weight.  the  l i f t i n g  the  the  across  back  loaded the  in  the  and  from  from  below  than  approximately  of the  other the 1,  corner  weights. 2  and  desired  5  of The  lb.  load  up  the  test  plate  sizes to  30  plate  is  in lb.  series in  DESCRIPTION With  this  method  condition  3.1.2  with  of  plate  uniform  curvature  of  the  principal  surface  the  linear  sides  be  measured  were  plate.  the  diagonals  described  the in  free  edge  Section  2.1.  on  in  be  either of  side  of  surface  range,  order  calculated  to  the  knowing plate.  strains  surface  In  is  strains  eliminate  the  on  effect  strains,  mounted  at  gauges of  the  the  were plate.  rotational  two  rectangular  centre,  aligned A  one  with  Mohr's  on  strain  either  side  the  90°  arms  circle  can  then  placement  accuracy  and  numbering  is  in  to  of  along be  find  used the  strains.  placement  Strain  following  suitable  easily  non-linear  surface  The  the  gauge The  achieve  strain.  the  the  The  as  measurement  must  measure  principal  one the  To  can  strain  in  midsurface  check  plate  but  of  rosettes  the  range,  sufficient,  to  twisting  we  42  INSTRUMENTATION  The  both  loading,  OF E X P E R I M E N T  Gauge items  strain  and  shown  Fig.  3.4.  Selection were  gauge:  considered  in  the  selection  of  a  DESCRIPTION 1)  strain-sensitive  2)  backing  3)  gauge  length,  4)  gauge  pattern,  5)  self  6)  grid  7)  The  material,  compensation,  resistance,  selected  was It  a  is  temperature  completely  general  corrected in  terminals  following  Micro-Measurements a  encapsulated  copper-coated the  alloy,  temperature  CEA-13-125UR-350. is  43  encapsulation.  gauge  which  OF E X P E R I M E N T  purpose  for  easy  number  constantan  aluminium.  polyimide,  for  gauge  with  It  is  large  soldering.  The  rosette  a  gauge,  integral, gauge  has  characteristics:  1)  gauge  length  of  2)  gauge  pattern  .125",  is  a  45°  in  co-planar  orientation, 3)  resistance  4)  gauge  The  data  lots  of  The  sheets strain  strain  of  factor are  of  in  2.0.  Appendix  A . l  following  the  for  the  two  used.  were  mounted  instructions[13].  The  wires  prescribed  manner  for  1/4  loops  formed  in  were  and  approximately  included  gauges  gauges  3508,  the  were  bridge wires  soldered  on  operation.  adjacent  to  manufactures in  the  Strain the  gauge  relief and  the  DESCRIPTION wires  were  taped  firmly  to  the  plate  to  OF E X P E R I M E N T prevent  44  induced  strain.  A  Strain  Vishay  was  Indicator  Instruments  used  with  a  model  Vishay  specifications  appear  The  gauges,  in  six a  quarter  Unit. and be  strain  This  bridge  allowed  switched read  The  to  from  three 3.3,  was  with  high  quality  compared  used.  of  because to  the  from  the  gauges  strain  digital  Fig.  neglected  Appendix  It  the  gauge  and  two  to  Strain  Balance  Indicator  Unit.  Their  A.2.  rosettes,  to be  the  were  Switch  where  the  connected  and  independently  indicator  Balance  balanced  strain  could  display.  bridge is  both  temperature the  Configuration  Digital  Switch  configuration  quarter  desensitization  in  Bridge  P-350A  SB-1  the  the  the  wire  and  gauge, wire  configuration, simple  and  shown  accurate  in when  compensated  gauges.  due  resistance,  to  resistance  resistance.  wire was  used  The  insignificant  was when  DESCRIPTION  STRAIN  OF E X P E R I M E N T  GAUGE  —  DUMMY  Figure  3.3,  3.1.3  PLATE  Shown  in  bridge  3.4  is  the  dimensional  at  "A"  at  description  configuration.  DESIGN  associated  applied  V  RESISTOR  Quarter  Fig.  corners  45  by  the  corners of  design  of  parameters. support  "B"  by  the  loading  is  in  the  plate  Upward  posts  and  cables. Section  tested  loads  were  downward  The  with  applied  loads  detailed  3.1.1.  a TOP  the  B  VIEW c-  2 (5)  a  NUMBERS IN BRACKETS D E S I G N A T E G A U G E N U M B E R S ON B O T T O M S I D E .  Figure  3.4,  Design  of  plate  and  location  of  strain  gauges.  DESCRIPTION  It  was  that  found  the  from  i n i t i a l  considerable; plate  for by  The  amount  strain large  Eq.  we  if  at  z=  the  was loads  self  weight  was  approximately  clearly  its  found  plate  +h/2.  plate  small,  neglect  (2.10),  own  plate  were not  applied,  1/2 the  linear.  weight  proper  1  prompted  lengths  and  effect.  Non-Linearity  was  midsurface  the  (2.16),  its  unwanted  non-linearity  the  terms  versa,  to  ascertain  Plate  loaded  was  24 " x 2 4 " x . 1 2 5 "  46  at  with  the  free  time edges  of  the  and  research  supported  corners.  of  then  of  a  corner  by  this  solution  a uniformly opposite  the  to  reduce  Estimate  comparing  If  to  of  deflection  caused  investigation  analytical  due  relationship  non-linearity  thicknesses  No  centre  t h i c k n e s s e s . When  further  tests  deflection  the  load-curvature  This  i n i t i a l  OF E X P E R I M E N T  it  the  and  w i l l  f i r s t  of  the  be  maximum  to  the  If  ratio  of  these  the  term  the  can  terms  be  by  plate  strain  deflection  divide  ratio  in  the  w i l l  be  estimated  bending  terms  non-linear,  is and  vice  linear.  of  the  midsurface  surface  maximum  bending  strains  w i l l  strain, strain, be  by  Eq.  DESCRIPTION  (w = .' hw  N*  where the  A  the  comma  amount  linear  of  the  midsurface  The  in  shown  With  deflection equal  difference  Appendix  the  ... (3.1)  ,xx  program  using to  Eq.  must  to  the  be  zero.  The  procedure  From  boundary  to the  conditions  midsurface  explained  and  force.  the  results  (w,  ) 'x  into  and  (w,  max  Eq.  (3.1)  to  )' 'xx'max  can  determine  that  measure  the  used. in  so  method  The far  used  value as  a  of  is N  larger  very w i l l  rough give  number  B.3,  the  maximum  slope  and  curvature  due  a  represents  non-linearity.  Appendix  are  ratio.  approximations  greater  developed  zero  is  determined,  substituted  c l a r i f i e d  quantitative a  and  non-linear  It  represents  B.  deflections  calculated  the  N  (2.25), w i t h  ' be  and  was  2.2.2, a g a i n w i t h  Section  in  2  X  differentiation  difference  forces  f i n i t e  47  non-linearity.  f i n i t e  calculate  shown  denotes  )  OF E X P E R I M E N T  are  DESCRIPTION  (w  (w  where is  q  the  is  the  flexural  ) , x max  =  ) , x x max  =  plate  self-load,  rigidity.  a  ^ ...(3.2)  Sii D  is  the  Substituting  N*  edge  Eq.  length,  and  into  Eq.  (3.2)  plate  self-load  = .119 ^  D  where  p  is  the  Substituting  material  this  ...(3.3)  is  q  and  =  ...(3.4)  ph  density.  flexural  rigidity,  Eq.  (2.4),  into  Eq.  gives  (3.3)  N*  Eq.  .357  D  48  gives  (3.1)  The  .206  OF E X P E R I M E N T  is  (3.5)  values  of  increases  (l-\>»)  and  graphed  rearranged  a/h,  self-weight.  = 1.43  As  there the  is  l i t t l e  ratio  exponentially.  ...(3.5)  in  Fig.  non-linearity  increases,  the  3.5. due  For to  small its  non-linearity  DESCRIPTION  OF  EXPERIMENT  49  CN  0  0  0.0  Figure  The  Graph  3.5,  i n i t i a l  where  24"  the  non-linear  plate  pronounced,  reduce  of  40 .0  tested,  had a/h=  non-linearity  80.0  PLATE SIZE, a/h ratio  in  v s . plate  which  the  By r e d u c i n g  192.  ratio  120.0  by over  size.  non-linearities a / h by a half  an order  of  magni tude. S i z e  A l l  plates  thickness used  in  material  and  were of  a l l  Material  made  .125". the  from  T h e same  specimens  properties  6061-T6  material  due t o  a r e shown  aluminum  in  i t s  with  a  nominal  and t h i c k n e s s a v a i l a b i l i t y .  Table  3.1.  were The  we  DESCRIPTION  Density  -  Ultimate  Strength,  Strength,  Tensile  Tensile Shear  Elastic  Moduli,  Poisson's  Ratio  displayed  i s in  Table  p s i  G  -  3.75 x 1 0  6  p s i  0.33  -  properties  for  •  a n d 6061 T6 varying  properties,  sizes  of  Length,  plates  a  0.125" Non - l i n e a r  *  Ratio,  24"  2.16  18  0.683  12  0.1350  10  0.0651  8  0.0262  6  0.0084  3.2, Plate  the  A l u m i n i u m 6061-T6  Thickness:  Edge  x  3.2.  Material:  Table  20 K s i  -  6  .125"  calculated  35 K s i  -  * N  24 K s i  -  10  the of  38 K s i  -  10.0  Using value  3  -  3.1, 6061-T6 M a t e r i a l  of  size  50  E  Table  thickness  l b / i n  .100  Shear Yield  OF E X P E R I M E N T  and n o n - l i n e a r i t y  ratio.  N  and  DESCRIPTION  Based  on  were  3.4  this  analysis  the  and are  chosen  The  variation  were  f  6"  .125"  0.6"  0.3"  8  .125  0.8  0.4  10  .125  1.0  0.5  12  .125  0.9  0.5  from  examined.  to  design  on  the  For  the  and  0.1%  c r i t i c a l  dimensions  respectively.  2%  stock  largest  manufacturing  in  Fig.  parameters.  nominal  selected  shown  51  3.3.  e  approximately  machined  Table  h  Plate  3.3,  parameters  listed in  a  Table  design  OF E X P E R I M E N T  where  plate  the  was a p p r o x i m a t e l y  the  The  flatness  i n i t i a l  a  and  plates was  +6%  of  were  f i r s t  deflection  +0.007" o r  h,  due  the  thickness.  3.2  3.2.1  The  EXPERIMENTAL  INITIAL  i n i t i a l  PROCEDURE  PREPARATION  preparation  calibration  check  the  and  weights  of  the  hanger.  prior  to  strain  testing, indicator  consisted and  of  weighing  a of  DESCRIPTION  The  strain  indicator  internal  calibration  external  precision  described  The  Switch  Indicator  The  on  0.1  and  an  TEST  strain  Unit.  Each  quarter Switch  check  ensure were  resistor.  Balance  Unit  the  and  electronic  the  The  instruction  system  was  done  indicator  using  and  calibration  the  an  method  is  manual[14].  was  connected  schematics  weights balance.  on  were  the  to  the  Unit.  calibrated  Each  Strain  by  was  measured  the  Switch  measuring to  within  PROCEDURE  gauges gauge  corresponded  A  in  check  52  gram.  3.2.2  The  the  circuit  following  hanger  them  in  calibration  OF E X P E R I M E N T  to  bridge and  the  recorded.  is  to  identified  channel  hookup  Balance  same  connected  number,  l i s t / d a t a the  were  to  by  which  described  by  Fig.  it  was  the  and  Balance  3.4, connected.  schematic  The  on  the  Unit.  recording  procedure  sheet  was  was  used  followed.  The  for  each  test  following  to  items  DESCRIPTION  1)  Date.  2)  Plate  size.  3)  Gauge  factor  for  the  OF E X P E R I M E N T  particular  lot  of  53  gauges  used. 4)  Strain  5)  The  six  controls:  a)  gauge  b)  calibration  c)  balance  switch  off,  d)  quarter  bridge  switch  Room  strain  Indicator factor  set  to  2.00,  switch  off, and  temperature.  gauges  must  be  balanced  unstrained  position.  To  bending- of  the  plate  under  its  vertical  plane  during  the  on  test  held  The ball The  in  the  plate  was  bearings ball  positioned centre  The  then  of  at the  strain  the  the  which end  plate sits  of  the  strain  own  weight,  the  the  plate  the  plate  induced the  shown  Rolling groove  an  the  plate  was  process.  apparatus as  in  by  balancing  corners on  the  with  with  in  Fig.  Support  nearest  the 3.2.  Post  to  was  the  plate.  with  zero  cables  were  Strain  measurements  lb.  corner  of  negate  mounted  under  bearing,  on.  then  corner  attached  load.  were  load  and made  was  the at  measured.  strain 2  lb.  was  The  again  increments  two recorded. up  to  30  DESCRIPTION  During were  the  taken  unloading  of  periodically  the to  plate check  the for  OF  strain  EXPERIMENT  measurements  measurement  d r i f t .  55  4.  In  this  It  is  broken  strains the  section  on  into  both  amount  Each  of  section  been  for  i n i t i a l the  In  of  plate  sections, of  plate,  along  the  under  the  raw  results.  In  and  strains  and  curvatures  the  or  have  the  principal  other  in  an  describing  no  uncorrected  correction  curvatures  its  described.  diagonals.  form  the  are  containing  displayed  strains  plate  results  plate  uncorrected  i n i t i a l the  One  the  results  the  RESULTS  own  corrected been  has  caused  weight.  and  by  These  form  subtracted  are  the from  a l l  results.  The to  the  the  deflection  essentially  two  sides  has  form.  the  individual  curvature  corrected made  the  EXPERIMENTAL  Gl the  and  plate  corners  and  comparison curvature when  directions  G3  diagonals between  are  negative.  graphed  on  the  following  graphs  running  between  the  hanger  corners  respectively.  the  purposes,  shown  the  absolute  with  the  value  sign  of  shown  support  the in  post  strain  the  refer  For and  legend  box  EXPERIMENTAL  4.1  PLATE  The  data  PRINCIPAL  from  computer.  the  Using  lot  2)  calculation  3)  The  gauge  4.1a-d.  The  be  by  seen  increases  G3  The  in  size  in  entered  following  into  was  a  done:  correction,  strains  in  the  axes  of  the  strain  at  due  zero  Gl  results to.  the  corner  offsets  larger.  strains.  strain  strains  the  the  in  are  plate  load.  both  shown  the  Fig.  self-weight  As Gl  in  the and  plate G3  Under  the  self-weight  of  direction  are  larger  those  than  can  the  plate in  the  direction.  corrected  results,  strains  figures  a  the  due  to  progression  plate  strain, one  principal  become  strains  the  principal  offset  i n i t i a l  to  of  were  and  i n i t i a l the  directions the  factor  calculation  uncorrected  sheet  spreadsheet  1)  gauge,  56  STRAIN  recording  a  RESULTS  size  depending  can on  half  the  value  symbols.  The  greater  change  in  slope  of  shown  in  the  be  seen.  the  principal The  have  can the  of  be  in  the  black  the  these  relation  midsurface  found  strain,  representing  had  Comparing  strains  amount  between  midsurface  line  4.2a-d,  removed.  direction,  strain  the  Fig.  self-weight  the  of  in  by  or  the  taking  white  greater  the  non-linearity.  EXPERIMENTAL  RESULTS  57  PLATE PRINCIPAL STRAINS UNCORRECTED TOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: 6'X6'X.125'  0.0  100.0  200.0  300.0  -300.0  500.0  600.0  700.0  800.0  PRINCIPAL STRAIN (MICRO-STRAIN) Figure  4 . 1 a ,P l a t e weight.  principal 6"  plate.  strains,  uncorrected  for  plate  EXPERIMENTAL  RESULTS  58  PLATE PRINCIPAL STRAINS UNCORRECTED FOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: 8"X8'X.125"  LD  O • rO  /  LD • CM  CO G O  o  CE r\j O  f  LJ 2  o  or •  o £ o  y  LJ CE  5?°  • o • n  LEGEND (NEG.) - Gl DIR.,TOP = G2 D I R . , T O P - G l DIR.,BOTTOM •= R ' D I R . , B O T T O M ( N E G . ) 2  O  0.0  100.0  200.0  300.0  PRINCIPAL  Figure  4 . 1 b ,P l a t e weight.  STRAIN  principal 8"  plate  400.0  500.0  600.0  700.0  (MICRO-STRAIN)  strains,  uncorrected  f o r  plate  EXPERIMENTAL  RESULTS  59  PLATE PRINCIPRL STRAINS UNCORRECTED FOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: 1•'XI••X.125'  o  in m  •  I  I  j  I  I  200.0  300.0  400.0  500.0  I  CD  0.0  100.0  600.0  700.0  PRINCIPAL STRAIN (MICRO-STRAIN) Figure  4.1c,  Plate principal  weight.  10"  plate.  strains,  uncorrected  for  plate  EXPERIMENTAL  RESULTS  60  PLATE PRINCIPAL 5TRRIN5 UNCORRECTED TOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: 12"X12'X.125"  o  m i  i  ro  0.0  Figure  I  100.0  200.0  i  i  300.0  i  400.0  500.0  PRINCIPAL STRAIN (MICRO-STRAIN)  4.1d, Plate weight.  principal 12"  plate.  strains,  uncorrected  i  600.0  for  700.0  plate  EXPERIMENTAL  The  6"  that by  plate,  diverge  the  Fig. with  increase  two  sets  the  curvatures  equal,  The  values  of  both  diverge.  The  the  the  same.  The  and  10"  are  plates  are  in  which  which  the  same,  the  fact  for  a  where  the  6"  i n i t i a l l y  caused  that  these  that  strains  given  characteristics to  is  indicates  midsurface  similar  lines  divergence The  61  both are  load.  at  small  plate.  After  coincident  also  strains  along  the  diagonals  are  the  curvatures  along  the  diagonals  are  no  plates,  between  has  coincident  formed  diagonals  are  midsurface  longer  which  the  The  strain. are  and  4.2d,  lines  therefore,  8"  of  almost  load.  lines  equal,  Fig.  the  two  midsurface  are  strain  point  same,  in  plate,  has  increasing  coincident  along  12"  this  of  4.2a,  RESULTS  Fig.  the  two  showing  4.2b,c,  have  extremes. there  is  characteristics  The a  thickness  dependence  of  on  the  edge  length.  4.2  PLATE  CURVATURE  The  curvature  due  to  is  bending.  following  was  related To  done:  directly  calculate  the  to  the  strain  curvature  in  in  the  the plate,  plate the  62  EXPERIMENTAL RESULTS  PLATE PRINCIPAL STRAINS CORRECTED FOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: 6'X6'X.125'  D.O  100.0  200.0  300.0  400.0  500.0  600.0  700.0  BDO.O  PRINCIPAL STRAIN (MICRO-STRAIN) Figure  4.2a, Plate weight.  principal 6"  plate.  strains,  corrected  for  plate  EXPERIMENTAL  RESULTS  63  PLRTE PRINCIPAL STRRINS CORRECTED FOR PLRTE WEIGHT MATERIAL: ALUMINIUM SIZE: 8'X8'X.125"  0.0  100.0  200.0  300.0  400.0  500.0  600.0  700.0  800.0  PRINCIPAL STRAIN (MICRO-STRAIN) Figure  4 . 2 b ,Plate weight.  principal 8"  plate.  strains,  corrected  for  plate  EXPERIMENTAL  RESULTS  64  PLATE PRINCIPAL 5TRRIN5 CORRECTED FOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: 10•X]•'X.125"  0.0  100.0  200.0  300.0  400.0  500.0  600.0  700.0  800.0  PRINCIPAL STRAIN (MICRO-STRAIN) Figure  4.2c,  Plate principal  weight.  10"  plate.  strains,  corrected  f o r  plate  EXPERIMENTAL  RESULTS  65  PLATE PRINCIPAL STRRINS CORRECTED FOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: 12'XI2'X.125'  o Ln ro  i  i  0.0  i  i  100.0  200.0  |  300.0  100.0  |  500.0  I  600.0  700.0  PRINCIPAL STRAIN (MICRO-STRAIN) Figure  4.2d,  Plate principal  weight.  12"  plate.  strains,  corrected  for  plate  EXPERIMENTAL  1)  Find the  the  bending  magnitude  diagonal 2)  Use  Eq.  3)  Subtract  and  strain  of on  (2.9)  the  taking  principal  opposite  to  off  by  the  i n i t i a l  the  of  the  66  average  strains  sides  calculate  RESULTS  the  of  along  the  plate.  curvature.  curvatures  due  to  plate  wei ght.  Shown  in  Fig.  the  different  the  i n i t i a l  negative  due  6"  and  opposite  along  order  With  the  of  other  along  the  after  which  the  The  curvature  slight  is  the  that  are  the  Only  only  expected maximum  diverge  for  are  corrected  for  the  G3 d i r e c t i o n  is  curvatures at  the  are  equal  largest  showing  the  onset  only  of  slightly  curved  producing  from  linear  theory  the  curvature.  4.3b,c  equal  relationship  used.  divergence  is  Fig.  in  system  shows  line  lines  results  diagonals.  plates,  diagonals the  a  what  10% a t  The  4.3a,  The  from  load-curvature  coordinate  there  i n s t a b i l i t y .  the  plates.  the  Fig.  is  deviation  the  to  plate,  curvatures  a  sized  is  curvature.  The  the  4.3a-d  and  d,  for  small  showing  the  curvatures curvatures,  bifurcation.  in  EXPERIMENTAL  RESULTS  67  PLATE CURVATURE CORRECTED FOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: 6'X 6•X.125"  in ro  /  o  ro  LD  «  1  r  CO CD o  £ CE C°M-  O  LJ  J* G3 HIDDEN  2 o ct: • o o L J  cr  /  / LEGEND • - GR UGE 1 DIR. o - GA UGE 3 DIR. (NEG.)  D LD "  /  D D.  0.000  0. D02  0. D04  0.006  0. 008  0.010  0.012  0.  CURVATURE (I /IN. ) Figure  4.3a, P l a t e plate.  curvature,  corrected  for plate  weight.  6"  EXPERIMENTAL  RESULTS  68  PLRTE CURVATURE CORRECTED FOR PLRTE WEIGHT MATERIAL: ALUMINIUM SIZE: 8'X8."X.125"  Ul  o•  in • (M  CO CD  o  /  O  LJ •  CD O  /  LJ f—  cr o  LEGEND • - GF1UGE 1 DIR. o ^ - o r1UGE 3 DIR. (NEG.)  o  in  o D  0.000  0.002  0.004  0.006  0.0DB  0.010  0.012  0.014  CURVATURE (1/IN.) Figure  4.3b,  Plate plate.  curvature,  corrected  for  plate  weight.  8"  EXPERIMENTAL  69  RESULTS  PLRTE CURVRTURE CORRECTED FOR PLRTE WEIGHT MRTER]RL: RLUM1NIUM SIZE: 1•'XIO•X.125 *  0.000  0.004  0.002  0.006  0.012  0.008  0.014  CURVATURE ( 1 / 1 N . ) Figure  4.3c,  Plate  10"  curvature,  plate.  corrected  for plate  weight.  EXPERIMENTAL  RESULTS  70  PLATE CURVATURE CORRECTED FOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: ]2•XI2'X.125*  0.000  0.004  0.002  CURVATURE Figure  4.3d,  Plate  12"  curvature,  plate.  0.014  0.008  0.006  ll/IN,  corrected  for plate  weight.  EXPERIMENTAL The  effect  plates.  of  For  curvature  non-linearity  small  values  relationship  become  stiffer  Due  to  self-weight,  the  Gl  direction  expected Gl  8"  after  direction  would  in  the  after  did  is  the  linear,  along  curvature  the  of  of  the  manufacturing.  The  slight  large  enough  a  to  the  corner  which  larger load-  the  larger  plates  curvature  G3 d i r e c t i o n .  plates  found  in  71  region.  It  while  the  exhibited  that  this  i n i t i a l  trigger  G3  the  other  the  curvature  was  curvature  be  that  this,  plate  in  would  bifurcation  increase  was  was  the  curvature  would  seen  after  have  It  direction  not.  curvature  plates  point  Three  of  clearly  non-linear  than  decrease.  plate  the  is  RESULTS  but not  in  mode  the flat  the  G3  after  bi furcation.  When  the  lateral could  plates load  be  its  was  created  cardboard. to  where  In  i n i t i a l  a l l  loaded  applied as  can  cases  shape.  to be  past see  done  this  the  if  the  with  failed  point  a and  other  of  stable  twisted the  bifurcation mode  sheet  plate  of  returned  a  72  5. COMPARISON  OF RESULTS WITH THEORETICAL AND  NUMERICAL  MODELS  The  results,  shown  non-dimensional be  made  about  limited graphs  to  the  then  Section  form.  plates  are  in  This  4,  allows  deformation of  of  particular  compared  are  to  compiled  more  and  general  the  plate  size.  The  theoretical  shown  in  statements  which  are  to  not  non-dimensional  and  numerical  results.  5.1  COLLAPSING  The  principal  corrected to  a  which  strain  for  form  from  curvature  form  coincident this  in  F i g .  in  which  curves.  collapse  graphs, 4.2  The  come  FORM  the  and  which 4.3,  results  were are  from  non-dimensional  partly  from  theory  reduced the  four  groupings and  observation.  non-dimensional the  and  NON-DIMENSIONAL  self-weight  achieve  partly  from  INTO  non-dimensional  plates  The  DATA  analysis  in  groups  for  Section  the 2.3.1.  midsurface  strain  come  COMPARISON  For to  convenience, their  5.1.1  From  the terms  non-dimensional  PRINCIPAL  E q . (2.12)  curvature  quantities  i n this  we g e t t h e p r o p o r t i o n a l  G with  and strain  relationship  - h T G  E q . (2.11)  To  keep  •••  for isotropic  the non-dimensional  by  any non-dimensional to multiply  2 by (a/h)  a l l the materials  dimensional  Shown vs.  i n  load  principal  sides  of  the  tested  was taken  F i g .5.1a-d strain plate.  form  materials,  (  5  -  1  )  we g e t  ...(5.2)  we c a n m u l t i p l y  quantity.  hT"  As  refer  section.  e « (1+v) £ 7 ^ '  need  73  STRAINS  e  Replacing  load,  OF RESULTS  To c o l l a p s e  both  sides  the results  we  giving  tt  have  h^E  a \>=.33,  ...(5.3)  the  n o n -  2 4 as Pa / E h .  are the non-dimensional  graphs  along  and on  the two diagonals  for both  load  COMPARISON  OF  RESULTS  P R I N C I P A L STRAIN TOP GAUGE 1 DIRECTION CORRECTED TOR PLATE WEIGHT MATERIAL: ALUMINIUM o o  p  /  x x LJ  \ OO  LEGEND • - 12 X12 X.125' o - ]0"X]0'X.]25" A8'X 8"X. 125" + - 6"X 6"X.]25'  LD  ,  X X  D D_  ,  / /  O — CE O  /  LJ  -z.  or o o  /  LINE/01 THEORY  9  y  CE t\  y  O  CO ZZL  j y  LJ o tn I  CD  o o 0.0  ].0  2.0  3.0  4.0  5.0  NON-DIMENSIONAL PRINCIPAL STRAIN, £ a x x 2 / h * K 2 Figure  5 . 1 a ,Non-dimensional di  rection.  principal  strain.  TopG l  6.0  COMPARISON  OF R E S U L T S  P R I N C I P A L STRAIN TOP GAUGE 3 DIRECTION CORRECTED FOR PLATE WEIGHT MATERIAL: ALUMINIUM  D LD  r\j  I  I  j  LD  x x  LEGEND 12 X12"X.125' 10 XIO'X.125' 8 X B'X.125" 6 X 6'X.125' 3 9VX3.94X  _c  LJ \ ro  x X  D  0.0  Figure  0.5  1.0  1.5  2.0  NON-DIMENSIONAL PRINCIPAL STRAIN,  5.1b, Non-dimensional di  rection.  principal  strain.  2.5 £a**2/b**2  T o p G3  75  COMPARISON  OF RESULTS  76  P R I N C I P A L STRRIN BOTTOM GAUGE 1 DIRECTION CORRECTED TOR PLATE WEIGHT MATERIAL: ALUMINIUM o in  —  0.0  0.5  1.0  1.5  2.0  2.5  3.0  NON-DIMENSIONAL PRINCIPAL STRAIN, £ax*2/hxK2 Figure  5.1c,  Non-dimensional  principal  strain.  Bottom G l  3.5  COMPARISON  OF  77  RESULTS  P R I N C I P A L STRRIN BOTTOM GAUGE 3 DIRECTION CORRECTED FOR PLATE WEIGHT MATERIAL: ALUMINIUM o o  0.0  3.0  2.0  3.0  4.0  NON-DIMENSIONAL PRINCIPAL STRAIN, Figure  5.1d,  Non-dimensional  principal  strain.  5.0  6.0  £o**2/h**2  Bottom  G3  COMPARISON  There  is  excellent the  In  results  collapse  onto  one  coincident  In  5.1b  5.1c  the  results  and  linear  between  plates.  Fig.  both  agreement  and  non-dimensionalized  and  the  edge  thickness  the  present  This  length  comparison  results  groups.  from  to  this  the  agreement thinner  Up  a  to  This  of  of  the  sections  four  the  curve.  from  the  tests  Fig.  4  plate  of  has  smallest  check  strain  plate  done  identified  of  are  [3]  are  about  plate  here.  a  radical  plates  the  change  occurs  at  a  the  the  good  After  half  used  in  1.5,  the  in  takes  point, [3],  rate  as  is  the For  shape  the  with  which  place.  anticlastic  greater  non-  agreement  this  point  departure  from  on  approximately  in  bifurcation  and  shape  further a  smaller  ends  synclastic  non-linear  plotted.  provides  experimental  close  results  78  experiment.  dimensional  the  and  the  OF R E S U L T S  to  load  the is  increased.  The  experimental  principal while  up  agreement to the  the  strain to is  2  on due  bending  second.  results  agree  of  the  the to  1  on Gl  the  strains  Top  G3 and  with  the  linear  Top  and  Gl  G3 B o t t o m  midsurface in  the  f i r s t  Bottom graphs.  strains case  line  are  and  up  to  a  graphs, This  being  added  subtracted  in  COMPARISON 5.1.2  The  PRINCIPAL  79  CURVATURE  non-dimensional  relationship  OF R E S U L T S  come  quantities  from  Eq.  for  2.5.  the  By  load-curvature  multiplying  both  sides  by  2 a  /h  we  get  the  proportional  Ka IT  which  collapse  As  the  in  have  a  v=.33,  Using  these  directions  quantities are  shown  result  shown  approximately and  on  case,  the  •••  a l l  the  materials  non-dimensional  curvature  of  directions the  3.0  c  ( 5  -  4 )  load  tested  was  bifurcate  linear  Up  linear the  showing  a n t i c l a s t i c  the  to  a  theory  Gl  shape  in  curvature  load-curvature  4.0  in  and  G3  5.2.  the  graph.  the  and  curvatures  Fig.  using  the  to  the  in  the  3.0  corresponds  between  "  strain  therefore  expected also  2  2 4 Pa / E h .  as  is  , ,x  , , Pa iVE  J  results.  principal  taken  The  the  relationship  theory  closely. in  the  the  of  the  start the  2.1  of  relationship  curvature  and  Section  is  Between Gl  and  linear a G3  transition  synclastic  shape.  COMPARISON  OF R E S U L T S  80  CURVATURE  CORRECTED TOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: .125' THICK o  12 o = 12 A = 10 + = 10 X = 8 o = 8 V = 6 B = 6  • =  LD  D  a tr o  0.0  Figure  5.2,  ] .0  LEGEND X12" GAUGE XI2' GAUGE XI •" GAUGE X10' GAUGE X 8' GAUGE X 8' GAUGE X 6' GAUGE X 6' GAUGE  2.0  3.0  1 3 1 3 1 3 1 3  DIR. DIR. DIR. DIR. DIR. DIR. DIR. DIR.  4.0  •  •  5.0  6.0  NON-DIMENSIONAL CURVATURE  Non-dimensional  curvature.  7.0  8.0  COMPARISON It  is  within  which the  the  the  in  the  thickness  From  from  react  in  lines  to  of  the  non-linear  the  theory  region, same  the  After  linear  the  plates  manner  as  region  bifurcation  increases  with the  non-dimensional  the  81 in  point  quickly.  same  curvatures  form  form.  STRAIN  midsurface  non-dimensional found  the  in  MIDSURFACE  the  occurs.  non-linear  coincident  5.1.3  section  bifurcation  deviation  Even  f i r s t  OF R E S U L T S  strain  analysis  relationship  at  the  in  Section  centre  of  2.3.1 the  the  plate  was  be  e  =  x  e  y  «  p K  x  p  K  ...(5.5)  y  whe r e p x  = K a x  p y  = K a y  K  K  The  midsurface  Gaussian  curvature  relationship amount  of  strain  for  in  a l l  scatter.  is  plotted  Fig. the  5.3.  plates  (5.6)  against The  non-dimensional  results  tested  with  show  a  linear  just  a  small  COMPARISON  OF R E S U L T S  82  MIDPLRNE STRRINS UNCORRECTED FOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: .125' THICK  o o  D. I  O CD.  o  o• cn i CE  LEGEND 12'X12 GAUGE o = 12'X12 GAUGE A = 10'XIO GAUGE + = 10X10 GAUGE X = 8'X 8 GAUGE O = 8'X 8 GAUGE V -= 6 X 6 GAUGE 6 X 6 GAUGE • =  DIR. DIR. DIR. DIR. DIR.. DIR. DIR. DIR.  o•  ZS  7°  CO I  co i o oc  o  i  +/  XL-  CD-  f CE f— CO  • X  o•  "0>5  3  X"  I  LJ ZZ.  <^v  °  CE o-  x  vx  CL Q _ 1  O • ro  CD CDCM I  CD CD  I O CD  0.0000.000 -0.002 -0.003 -0.004 -0.005 -0.006 -0.007 -0.008 -0.009 -0.0)0  NON-DIMENSIONAL GAUSSIAN CURVATURE Figure  5.3,  Non-dimensional  midsurface  strains.  COMPARISON Linear Using  regression this  line  was  we  get  e  The  linear  one  explained  analytical  The of  to  and  not is  the  but 2  field  in  through  shown  and  the  of  in  u  v  less The  used  than  line.  the  discrepancy in  freedom  [2]  this  which  the  is  number  of  displacement field  more  increased,  decrease, in  exact  can  developing  the  an  decrease,  the  overall  unequal  degrees  of  would As  of  in  was  only freedom  the  number  plate  plate  sense  the  calculated true  displacement  of  curvatures  strain  the  the of  strain  true  the  for  represent  the  opposite  it  (i.e.  degrees  freedom  the  calculated  on  degrees  problem  equal  on  converging  normal  the  12  solution.  f i e l d  converging  of  had  requirements  closely.  the  in are  number  the  the  i n i t i a l l y  symmetries  not  with  fit  . . . (5.7)  40%  analysis  therefore  allowing  shape  of  equilibrium  Therefore  best  2  (2.55).  the  displacement  also  Eq.  is  satisfy  problem,  would  .0124  in  type  degrees  increasing  freedom  of  the  equation  = .0124 K K a x y  y  3/146  calculate  83  result.  DOF). As  does  By  by  freedom,  one  By  of  displacement  reduced  = e  x  to  the  coefficient  analytical be  used  OF R E S U L T S  true  degrees  of  energy  energy. strains  would  strains.  the as  plate in  to  Section  assume 2.3.1,  a  COMPARISON the  point  of  the  total  potential  the  degree  have  to  As  f i r s t  a  good. off,  5.1.4  be  To  predicted  achieve  an  of  the  inplane  approximation  the  theoretical  by  84  minimizing  accurate  result  displacements  would  increased.  showed  the  magnitude  of  the  calculation  linear  there  was  a  curvature,  which  was  experimentally  this  plate  relationship  w i l l  COMPARISON  be  used  is  linear  relationship  independent  was  coefficient  that  that  the  could  energy.  freedom  Although  Gaussian  of  of  be  it  fact  bifurcation  OF R E S U L T S  with  v e r i f i e d .  of  the  was the  The  thickness  later.  OF L O A D - C U R V A T U R E  RESULTS  WITH  APPROXIMATE  best  fit  SOLUTIONS  The  dotted  line  load-curvature experimental  in  Fig.  results  results  5.4 in  are  Chandra[5],  Reissner[9],  Hsu[7).  The  result  dashed  line.  All  the  results  curvature  from  are  approaches  represents  Fig. the  5.2.  and  linear  asymptotic zero.  Compared  approximate  Ramberg the  the  with  to  theory  is  and  of Lee  plotted  linear  the  these  solutions  Miller[2]  the  of  result  and  with  a  as  the  COMPARISON  OF R E S U L T S  85  COMPRRISON OF RESULTS CORRECTED FOR PLRTE WEIGHT MATERIAL: ALUMINIUM  o o  LEGEND EXPERIMENTAL Gl DIR. EXPERIMENTAL G3 DIR. LINEAR THEORY CHANDRA REISSNER RAMBERG AND MILLER v - LEE AND HSU  o o A + x o  in  O  Ln  O CE O _} _  or or o o CE  -  9  II I  4  //  i  /  /  OJ  '  /*  o o  Ii  >  / / / s  £  *  D I  Z.  /. I-  /  CO  O  /p  1  J  Ln  o UJ  0  I  i  * /* • ' '•  o  in  A  y y  >  in  o o  0.0  1.0  2.0  3.0  4.0  5.0  6.0  7.0  8.0  NON-DIMENSIONAL CURVATURE Figure  5.4,  Comparison results.  of  approximate  and  experimental  9.0  COMPARISON  Taking  RESULTS  86  Chandra[5]  the  load-deflection  converting we  OF  it  to  the  form  relationship to  be  in  graphed,  [5],  with  and v=.33,  get  Pa Eh"  1 Ka  2  The  f i r s t  part  of  term  the  the  equation  2  square and  ...(5.8)  [1 + .0270 (^-) ]  h  8  in  2  2  brackets  the  second  represents  term  the  the  linear  non-linear  part.  As  can  be  seen  experimental  in  Fig.  5.4,  data  very  well.  corresponding  In  Chandra's  represent The  vary  This  stresses  but  over  the  along  the  only  extra  strain  imposed  midsurface  strain  energy  plate  results.  Airy  the  stress of  fit  of  6,  function  the  are  the the  plate  are  not.  and  satisfied  in  not  curvature  boundary  edge  on  does  the the  high.  stresses  along is  a  conditions  shear  condition  (5.8)  At  40%  the  boundary  periodically  boundary  is  analysis,  the  normal  exactly,  load  Eq.  in  zero an  boundary plate  and  very  not well.  satisfied  The  the  does  shear shear  overall distorts an  overly  stresses stress sense. the s t i f f  COMPARISON  Reissner  solves  into  the  non-linear  this  solution  the  E q . (31)  form  with  the  of  [9]  \>=.33, w e  2  equation  is  strongly  from  Reissner  solves  he  along he  of It  a  long  is  rectangular  interesting  experimental  two  of  results  to  plate compare  from  twisting  the  the  of  boundary  conditions  are  constraint  is  not  although  a  non-dimensional  short  the  the  plate.  the  compared a  deviates  square  are better  equations.  Along  in  the  to  the  plate than  To do  conditions the  other  an o v e r a l l  results  imposed with  met w i t h  it  boundary  conditions  when  they  where  von Karman  principal  sides  ...(5.9)  results.  satisfy  sides  if  not  F i g . 5.4  non-linear  Venant's the  in  exactly  opposite  into  [1 + .0222 ( M i ) ' ] h  I Mi 8 h  experimental  meaningful  match,  rearranged  get  =  the  Saint  is  graphed  can only  satisfies  view  region.  with  Pa Eh*  this  twisting  plates.  When  This  87  RESULTS  Reissner[9]  square  OF  w i l l  latter  sides  sense. only  be  boundary  former.  This  and the  results  [5].  In  do  COMPARISON  By  Ramberg  taking  Eq.  dimensional  and  (67)  form,  RESULTS  [2]  we  get  =  | ^  and p u t t i n g  it  into  non-  2  This to  equation  due as  to  the  the  the  When  the  Figure theory.  Taking  dimensional Eq.  missing.  (14)  of  midsurface  strain  of  freedom  the  section,  close  results  up  6.  to  the  Lee and Hsu were an e r r o r  shows the  a  line  slope  quantities  of  for  respectively,  Either  experimental  of  the  is  incorrect  displacements,  energy  actual  as  stored a  on  good  produced.  results [7]  2  Hsu[7]  results  8 of  is  is  of  ...(5.10)  (~-) ]  of  previous  strain  f i t  the  curvature  degree  the  Lee and  experimental  and  in  midsurface  experimental  with  distribution  limited  discussed  [1 + .00917  corresponds  a non-dimensional  Although  88  Miller[2]  of  Pa Eh*  OF  both  the  compared  was found  in  representing this  line  deflection we  find  graphs  the  their  work.  linear  plate  and e q u a t i n g and moment,  there  axes,  to  is  a  the nonEq.  factor  F i g . 8 and 9,  (5) of  have  two  COMPARISON been  labeled  wrong  or  has  been  incorrectly  the  4  the  This  in  change  one  of  typed.  denominator  the  diraensionless  Most  of  Eq.  OF R E S U L T S  l i k e l y (14)  it  provides  the  closest  fit  to  correction  the  results  are  shown  quantities  was  should  the  have  the  89  latter;  been  a  2.  experimental  results.  With  this  coincides  with  dimensional showing Fig.  8  seen  in  As  seen  for  a  could  5.2  A  grid  Fig.  and  scheme  in  9  of  for  the  CURVATURE  was  those  [7], of  12.  B.  then  point  at  a  to  a  starts  of  much  up  to  It  nondeviate  bifurcation higher  5.4.  load  shown than  in  what  is  results.  the  moment  This  ITS  between  predicted  Appendix  results  Fig.  has  along  not  with  yet  converged  machine  errors  discrepancy.  UNDER  made  and  The  occurs  division  comparison  weight  also  5,  experimental  account  PLATE  of  stiffness.  [7]  the  in  experimental  curvature  less of  the  in  OWN W E I G H T  the  from  curvatures  the  linear  due  finite  to  selfdifference  COMPARISON  OF  RESULTS  90  CURVATURE DUE TO PLATE WEIGHT COMPARISON OP EXPERIMENTAL AND PD RESULTS MATERIAL: ALUMINIUM SIZE: .325' THICK  « 'o  o  in to.  • o A +  = = -  LEGEND EXPERIMENTAL DATA, EXPERIMENTAL DATA, FINITE DIFFERENCE, FINITE DIFFERENCE,  Gl G3 Gl G3  •  DIR. DIR. DIR. DIR.  / /  >  or  ZD CD  _J CE  CDCM  zz  o  c n  LJ ^ Q I  O 2  o o d:  o  O in'  o d.  1  0.0  2.0  4.0  6.0  8.0  10.0  NON-DIMENSIONAL PLATE LOAD, Figure  5.5,  Curvature  due  to  plate  12.0  hqoxK2/D  weight.  14.0  *10  -5  COMPARISON The  deflection  at  the  plate,  used  diagonals values  at  the  Gl  and  is  experimental  10%  the  less  unreasonable  1)  the for  The  5.3  In  as  and  section  Hsu[7]  and  experimentally  are  strain  COMPARISON  this  the  the  were  as  the  is  in  external gauge  be  determined  shown  to  finite 5.5.  the  The  results  error  the  Fig.  posts,  is  the  The  in  expected.  result  This  converging  Fig. are  error  closely.  approximately is  not  as  the  grid  is  B.2.1. very  small  making  comparatively  factors  such  as  the  large.  the  weight  of  wires.  theoretical  Ramsey[8]  along  of  reasons:  strains  reading  are  . 91  portion  compared  numerical  s t i l l  shown  curvature  supporting  numerical  following  centre  measurements.  would the  the  then  results  OF THE C U R V A T U R E  the  in  strain  follow  measured  There the  the  between  solution  instrument 3)  These  experimental. the  refined 2)  calculate  largest  results  The  points,  experimental  the  than  to  from  direction  G3  grid  centre.  direction,  curvature  In  the  calculated  difference  In  was  the  OF R E S U L T S  w i l l  be  point.  BIFURCATION  points  of  compared  POINT  bifurcation with  the  of  Lee  COMPARISON  5.3.1  The  EXPERIMENTALLY  bifurcation  experiment curvature  points  were  found  graphs,  dimensional  DETERMINED  form  of by  Fig. of  the  [8]  plates  examining  in  92  POINT  four  4.3a-d.  OF R E S U L T S  the  They  Fig.  in  corrected  are  5.6.  tested  plotted  Square  this  plate in  the  symbols  non-  mark  the  points.  Using  the  curvature  experimentally  determined  and  strain,  information work  It  of  was  the  deduced  was  strains  are of  in  the  Up  the  point in  along  had  opposite  2.3.1  edge  of  sign.  the and  is  the  was  further  extracted  length,  bifurcation, Eq.  5.1.3  that  of by  only not  the Eq.  on the  the  (5.7)  of  of  given  but  strain  results  direction are  their  diagonals  and  dependent  Using  (5.7),  point  placed  Section in  Eq.  between  from  the  [3].  with  strain  and  opposite  they  plate  plate  to  al  results  equal  the  midsurface  et  and  their  shown  bifurcation  that  sides  comparing  centre  on  Miyagawa  opposite  It  midsurface  relationship  the  this  plate  the  by  midsurface  diagonals (5.7).  on  experiment.  the  at  the  The  curvature  of  the  thickness.  curvatures  gives  gauges  are  equal  and  COMPARISON  OF RESULTS  93  CURVATURE BIFURCATION POINT MATERIAL: ALUMINIUM LD CD  • oA + x-  oo  O  o  —  o -  LJ  v •=  O  LEGEND .125" THICK .118' .098' .079' .059' .039' BEST FIT LINE  o  +  y  >  or ZD  o  o  CO  o  2: o  A  o  CO  LJ  n  ° .  r: o CD  X  X  1  / < >  >  s  o  * N/  y ^ i  .  A  A  >  o  A  "t  X  o  •  X x  A  -<r  y  o /  I  O  +  A  A A  A  •  •  •  CO  o  0.000  0.005  0.010  0.015  0.020  0.025  THICKNESS-LENGTH RATIO, h/o-  Figure  5 . 6 , Bifurcation  point.  0.030  COMPARISON  i  where  K  is  the  Substituting Eq.  (2.9)  the  curvature  surface  strain  where  the  total  surface  the  diagonal.  strain  relationship  of  '  . . . (5.12)  the  strain  Eq.  = £ + i  circumflex  strain,  with  (^rr-)  = -.0124  is  asterisk,  midsurface  Combining  along  gives  e*  the  94  ...(5.11)  2  curvature-bending  i  The  = -.0124 K a  positive  in  OF R E S U L T S  '  and  bending  ...(5.13)  overbar strain  represent  at  the  the  surface  and  respectively.  (5.12),  we  get  the  non-linear  relationship  e*  Shown  in  Fig.  bifurcation strain the  at  5  for the  bending  curvatures,  of  = e -  [3]  various  point  strain, using  of  is  the  sizes  ( ^ ^ )  Eq.  (2.10).  ...(5.14)  surface of  strain  plates.  bifurcation,  using  Eq.  .0124  we  (5.14),  at  the  Taking  the  can and  then then  point  surface  solve the  of  for  COMPARISON  This The  wascompleted solid  line  andthe results  represents  OF RESULTS  a r e shown  the least  square  i n Fig.  f i t  9  5  5.6.  giving the  equation  Ka  The  scatter  i s moderately  with  estimating  Fig.  4.3,  slowly.  that  the  the lines error  bifurcation with  from  the errors  representing  would  point.  . . . (5.15)  of bifurcation.  i n strain  calculated,  associated  high  the point  A small  curvature  = 3.64 h/a  A certain  c a nbe seen  the curvatures  measurement,  produce  t h e measurement  It  associated  diverge  a n dhence t h e  significant  amount  i n  changes  of error  of the points  i s  from  i n  also Fig.  5 i n  [3].  By  taking  Eq.  (5.15)  andrearranging  i t  into  the non-  2 dimensional curvature noted 5.2,  occurs Eq.  5.1.2,  K a / h , we s e e t h e n o n - d i m e n s i o n a l  at the point  of bifurcation. As  the bifurcation  at a curvature  o f between  point,  3.0  shown  a n d 4.0,  i n Fig.  which  (5.15).  ANALYTICAL  Ramsey[8]  form  i s a constant  i n Section  v e r i f i e s  5.3.2  curvature  found  BIFURCATION  that  POINTS  the point  of bifurcation  occurred  at  COMPARISON  Lee and Hsu[7],  factor  of  2 noted  after  i n Section  is  an extrapolated  a factor  Comparing Eq.  5.4  In  more  c r  8 times  we f i n d  =21.6,  i n F i g .9 of  between  to  the difference  these  [7].  results.  the experimental i s  10% l e s s  There  for  result,  [8] and  for [7].  LOAD AND SURFACE  OF  MODULUS  SHEAR  the determination  relationship  through  test,  t h e method  Using  the results strain  calculated. test.  ...(5.17)  found  results  LIMITING  surface  of a, has  difference  the analytical  (5.15),  700%  of  M  the correction  =24.9 h/a  Ka  from  making  96  ...(5.16)  =3.29 h/a  Ka  while  OF R E S U L T S  i s  of  limited  obtained  values  DURING  G and the shear  the application  at which  These  STRAIN  of  by the onset in this  stress-strain the twisted of  plate  bifurcation.  experiment,  the i n s t a b i l i t y should  T H E MEASUREMENT  occurs  the load and c a nbe  n o t be exceeded d u r i n g  the  COMPARISON The  point  strain  of  at  bifurcation  this  i s given  curvature  OF R E S U L T S  by E q . (5.15).  c a n be found  using  97  The bending  E q . (2.9).  This  gives  -2^*4 2  c  Substituting  this  into  2 the (h/a) ,  factoring  To  i s  the surface  calculate  2  the surface  strain,  E q . (5.14),  and  we g e t  e* c  This  ...(5.18) a  = 1.66 (h/a)  strain  the load  at  ...(5.19)  2  at which  the point  relationship  between  corner  Substituting  the curvatures  load  at  bifurcation  of  occurs.  bifurcation  and curvature,  the point  of  we u s e t h e  E q .  (2.5).  bifurcation, E q .  (5.15),  P  Replacing  = 2D(l-\0 (3.64 h / a )  the flexural  rigidity,  c  We  now have  surface  the limiting  strain,  ...(5.20)  2  c  which  =  D,  gives  ^ 6 0 7 Eh* (1+v) a  ...(5.21)  2  values  should  of  the corner  load and  n o t be e x c e e d e d d u r i n g  the  COMPARISON test.  These  limiting  values  taken  to  apply  to  any  OF R E S U L T S  size  of  98  isotropic  plate.  Care the  be  instability.  with [2]  must  .  edge  effects  not  With and  make  ratios  the  larger  transverse  h/a  ratio  than  shear  large  about  will  be  1/10  to  avoid problems  encountered  99  6.  In  the  analysis  dimensional  more  deformation particular  The  experimental were  results. general of  the  For  found  to  which  be  a  number  collapse  sizes  are  conclusions  of  made not  made  plate  1)  The  principal  strains  l i e  along  diagonals  The  the  be  the  linear  in  can  of  2)  which  different  statements plate,  results  of  non-  the  plate  about  this  the  limited  to  plates  of  size.  following  strain  the  quantities  experimental allows  of  CONCLUSIONS  the  and  linear  non-linear  non-dimensional  about  and  at  principal  non-linear  the of  the  centre  the  regions:  of  plate  the  in  both  regions.  quantities,  which  collapse 2  load-strain  plate  relationship,  are  (l+v)Pa  the  4 /Eh  2 and  3)  e(a/h)  These  .  non-dimensional  results  for  the  same  quantities plate  collapse  thickness  in  the  both  the  CONCLUSIONS linear  and  non-linear  regions  with  very  100  l i t t l e  scatter.  4)  The  non-dimensional  thickness of  the  point  of  bifurcation, thickness  the  the  plate the  the  The  linear a  and  of  the  of  the  plate  the  linear  and  But,  after  rate  load  with  strain  be  the  for  at  to  made  linear  the the  The  of  thinner  it  changes  synclastic  shape  increased.  relationship  the of  different  plates  which  the  is  load-strain  can  in  of  collapse.  shape  coincides  conclusions  plates  results  not  the  non-dimensional  non-dimensional  following  curvatures  higher  of  region.  the do  anticlastic  start  in  non-linear  different  as  The  collapse  start  from  5)  also  results  linear  theory  approximately  about and  is  the  up  to  1.5.  principal  non-linear  regions:  1)  The of  principal the  plate  curvatures in  both  the  l i e  along  linear  the  diagonals  and  non-linear  which  collapse  regions.  2)  The  non-dimensional  load-curvature thickness,  in  quantities  relationship both  the  for  linear  plates  and  of  the  non-linear  the same  CONCLUSIONS regions, of  the  bi  are  same  linear  The  in  thickness  of  the  4  2 and Ka / h .  behave  Plates  similarly  load-curvature  and c o i n c i d e s  non-dimensional  following  strain  / E h  after  furcation.  3) T h e s t a r t  a  2  (l+v)Pa  101  conclusions  the  plate  in  with  the  curvature  c a n b e made  both  the  relationship  linear  of  theory  up  approximately  about  p r e - and  the  is to  3.0.  midsurface  post-bifurcation  regions.  1)  At  any p a r t i c u l a r  midsurface  point  strains,  e  on the and e  x equal  2)  in  the  direction  The m i d s u r f a c e  strains  of  , y'  the  vary  plate,  the  normal  are diagonals.  linearly  with  the non-  2 dimensional  3)  curvature  no  midsurface  strains  At  the  of  t  =  x  e  l i t t l e  4)  Gaussian  The  centre  the  .0124 K K a x y  y  scatter  in  experimentally  shown  above  determined  the  in  the  There  experimental  surfaces.  with results.  determined midsurface than the  i n c r e a s i n g the  are  relationship  was found  w a s 40% l e s s one. By  developable  plate, 2  K^K^a .  strain  analytically degree  of  CONCLUSIONS freedom  of  the  analytically converge  5)  The  non-dimensional and  solutions The in a  did  results f a i r l y  the  load-curvature  not  good  satisfy  Ramberg  in-plane  displacements  field  bifurcate.  I n i t i a l the  of  due  In  of  and  and  was  would  of  Rayleigh-Ritz  6  by  the  error or  the  methods, method,  of  in  could  as  of  are up  to  The  of  the  displacement [7]  might  use  deflections of  be  of  plate  finite  have  the  plate  non-linearity design  in  process.  elements  achieved  in  linear  calculating  amount  such  Hsu[7]  results  the  large  The  the  closely.  freedom  of  their  errors.  with  satisfactory  of  because  and  results  machine  during  fit  Lee  inability  the  results  respectively.  lack  the  experimental  conditions  5  self-weight.  sizes  poor  and  and  in  the  experimental  of  the  estimating  other  the  encountered  their  different  retrospect,  simple  to  a  to The  boundary  with  convergence  were  compared  Miller[2]  limited  The  differences  deflections, plates  was  problems  plates  f i n i t e  [2]  lack  independent  results.  the  curvature  in  to  are  produced  and  agreement  solution  due  were  Reissner[9]  of  strain  strains.  strains  solutions  non-dimensional  to  true  midsurface  the  thickness.  approximate  Chandra[5]  displacements,  determined  midsurface  plate  Four  on  in-plane  102  or  a  comparable  CONCLUSIONS or  better  less  results  time  I n i t i a l  and  curvatures  due  deflection  after  applying  a  curvature  The  analytic  less  and  The  to  at  of  was  the  [7]  point  of  corner  load,  which  * e = c  Three  1.66  areas  involves the less  point  a  are  scatter  with  the  point of  of  f a i r l y  The  Once  of  a  particular  not  be  was  Ka  Lee  and  Hsu[7]  Although  the  in  and  need  of  the  used  method  the  P = o  bifurcation  formed  in  the  The  results.  h/a.  are  10%  load-curvature  is  point  of  d i f f i c u l t  determination  be  The  only  surface  exceeded during 4 2 E h / ( l + \>)a  the  to  second  is  be  the used  strain  up and  test,  respectively,  plate  accurately  of  the  consideration.  measuring more  can  c r i t i c a l  further  can  3.64  results.  .607  of  =  predicted  relationship  not  of  and  good,  point  should  the  or  mode  could  bifurcation  Ramsey[8]  method  2  the  other  flatness  load.  bifurcation in  influence  to  process  bifurcation.  assumed,  is  in  better of  of  bifurcation.  (h/a)  due  manufacturing  stress-strain  the  are  the  scatter  plate  shear  non-linearity  whether  respectively.  not.  causing  elastic  plate  plate,  point  lateral  is  twisted  in  the  700% more  bifurcation  the  self-weight,  results  relationship  measure  to  mode  The  in  formed  deflection  displacement  estimating  effort.  imperfections  by  in  103  The  curvature  determined  the  f i r s t so with  development  CONCLUSIONS of  a  in  the  the of  testing  vertical  plate  under  orthotropic  The al  new  work [3].  both  apparatus plane, its  In  this  sides  of  weight.  And  l a s t l y ,  own  here  into  goes  plate  were  enabled  demonstration  for  midsurface  results  curvature,  occurs;  The  the  a l l  of  results  using  the  and and  which  to  of  in  future  the  twisted  of the  the  the  principal  of  twisting  Miyagawa  allowing  strains  on  This  dimensionless  quantities  determination  of  at  et  the  curvatures.  strain  of  region.  work  and  plate  which  expressions  which  bifurcation  novel.  this  thesis  analytical  measurement plate  is  the  measured,  strains  test  deflection  non-linear  beyond  surface  are  presented  verification importance  load  the  experiment,  of  collapse  the  unwanted  calculation the  hold  the  present  the  would  negating  materials  presented  which  104  test.  of  form  models  a  basis  and  are  constitutive  for of  the direct  relationships  105  7.  [1]  Kelvin  andTait P . G . Treatise on Natural Philosophy, Vol. 1, P a r t 2, C a m b r i d g e U n i v e r s i t y Press, 1883,  pp.  [2]  BIBLIOGRAPHY  203-204.  Ramberg W. a n d M i l l e r J . A . " T w i s t e d S q u a r e P l a t e Method and Other Methods f o r D e t e r m i n i n g t h e Shear Stress-Strain Relation of Flat Sheet", J . R e s . N a t ' l . B u r . S t a n d a r d s , V o l . 50, N o . 2, 1953, p p .  111-123. [3]  Miyagawa M . , S h i b u y a Y . , a n dUeda M. "Measurement o f Shear Modulus o f E l a s t i c i t y b y Means o f T w i s t e d Square P l a t e " , TheTwelfth Japan Congress on M a t e r i a l s R e s e a r c h - T e s t i n g Method and A p p a r a t u s ,  pp.261-264. [4]  Tsai  S.W. "Experimental Determination of the Elastic Behavior of Orthotropic Plates", Journal of E n g i n e e r i n g f o r I n d u s t r y , A u g . 1965, p p . 315-318.  [5]  Chandra R. "Ont w i s t i n g o f O r t h o t r o p i c Plates Large Deflection Regime", AIAA J o u r n a l ,  No. [6]  i n the V o l . 14,  8, 1976, p p . 1130-1131.  H s uC S . a n d L e e S . S . " S t a b i l i t y of Doubly Periodic Deformed C o n f i g u r a t i o n s of Plates and Shallow Shells", J . A p p l . M e c h . , V o l . 37, 1970, p p . 641-  650. [7]  LeeS . S . andHsu C S . " S t a b i l i t y of Saddle-like Deformed Configurations of Plates andShallow Shells", Int. J . N o n - L i n e a r M e c h a n i c s , V o l . 6, 1971, pp.221-236.  [8]  Ramsey  H. "A Rayleigh Quotient f o r the Instability of a Rectangular P l a t e w i t h Free Edges Twisted by Corner F o r c e s " , Journal de Mecanique Theorique e t  Appliquee, [9]  N o . 2,  1985,  pp.243-256.  Reissner E. "Finite Twisting andBending of t h i n Rectangular Elastic Plates", J . Appl. Mech., V o l .  24, [10]  V o l . 4,  Fung  1957, p p . 3 9 1 - 3 9 6 .  Y . C a n dW i t t r i c k W . H . " A Boundary Layer Phenomenon i n t h e Large D e f l e c t i o n of Thin Plates", Quart. J . M e c h . A p p l . M a t h . , V o l . 8,  1955, p p . 353-355.  BIBLIOGRAPHY  Goodier J . N . McGraw-Hill,  [11]  T i m o s h e n k o S . and 3rd e d i t i o n ,  [12]  Timoshenko S. and W o i n o w s k y - K r i e g e r Theory and S h e l l s , 2nd e d i t i o n , McGraw-Hill, 1959.  [13]  Measurements Group 200 A d h e s i v e , V i s h a y , USA.  [14]  Measurements Group I n s t r u c t i o n Strain Indicator, Vishay,  [15]  S z i l a r d R. T h e o r y a n d A n a l y s i s o f P l a t e s , C l a s s i c a l and Numerical Methods, P r e n t i c e - H a l l Inc., New J e r s e y , 1974, pp. 158-185.  [16]  James  S t r a i n Gage Instruction  T h e o r y of New Y o r k ,  106  E l a s t i c i t y , 1970. of Plates New Y o r k ,  Installation with Bulletin B-127-9,  Manual P-350A USA, 1980.  M-Bond  Digital  M.L., S m i t h G . M . , a n d W o l f o r d J . C . A p p l i e d Numerical Methods for D i g i t a l Computation, Harper a n d Row I n c . , New Y o r k , 1 9 7 7 , p p . 343-348.  107  8.  A.  INSTRUMENTATION  In  this  are  appendix  described.  gauges  used,  The  and  and  balance  A . l  S T R A I N GAUGE  Two  lots  were  the  the  of  sub-section  second  DATA  strain  The  in  the  gauges by A  the  were  with  the  indicator  general  A . l a  and  for  these  strain  and  switch  was  and  indicator  using  the is  experiment.  Group is  and  were  given  in  strain  gauges  They of  type  Section are  strain  Indicator  the  description  The  measured  in  Measurements  specifications  Tables  used  INSTRUMENT S P E C I F I C A T I O N S  The  instrumentation  SHEET  VISHAY  Unit.  the  deals  strain  A.2  Strain  for  unit.  CEA-13-125UR-350.  listed  specifications  f i r s t  manufactured  APPENDICES  the  Vishay  designed  Vishay  SB-1 for  P-350A  Switch  and  resistance  Digital Balance type  strain  APPENDICES gauges  and  strain, is  allows  from  designed  output  of  channel simplify  A.3  to  be  read  digital  display.  The  to  provide  a  of  channels be  data  specifications and  strain  a  10  can  the  on  method a  single  independently reduction for  these  and two  directly,  Switch  and  balanced  to  interpretation. units  are  shown  Unit  reading  indicator. zero  micro-  Balance  sequentially strain  in  108  the  Each  output  to  The in  Tables  A.2  APPENDICES  SMSM5  ENGINEERING DATA SHEET  u  THE INFORMATION APPEARING ON THIS SHEET HAS BEEN COMPILED SPECtFICALLV FOR THE GAGES CONTAINED IN THIS PACKAGE. THIS FORM IS PRODUCEO WITH ADVANCED EQUIPMENT ft PROCEDURES WHICH PERMIT COMPREHENSIVE QUALITY ASSURANCE VERIFICATION OF ALL DATA SUPPLIED HEREIN, SHOULD ANV QUESTIONS Aftis{ RELATIVE TO THESE GAGES. PLEASE MENTION GAGE TYPf. ITEM NUMBER. AND LOT NUMBER.  2.065  cr  © to  50 sen • S-*  Z ~~  So.  cn  cn  •0.9  I*  ft ft  3  2 1 " " o- o -o  •0.7  *©  a  SECl  PRECISION STRAIN GAGES  >  a  SEC1  coot  HALE IOH. NORTH CAROLINA  951516  MEASUREMENTS GROUP  s  2.135  M  N  •  . . jTZ Mlcro-M»a»ur»m»nt» Division  IP »o fi  S o ; *»m Z  if  | .  5-n o O ro 2 . cn &** 1  30 Cd  cn  IP  GENERAL INFORMATION: SERIES CEA FEATURE GAGES GENERAL DESCRIPTION: CCA M M «r* • a>n»wa CXXPOBI family of L W P I fpgs am H»lid with • fully ancapajhatae and and a  •  TEMPERATURE RANGE; - 10CTF f-»*C7 B *«0*F 1*208*0fart>  FATIGUE LIFE: Faiigua lit* « t ma«*ad functnn af aMoar tottn Hrmmxr. WW 3D' ANG a lif*«<llb* 10* t > o - « 1150ai<n/in f j*n/tr»/ M. Una MIA Sotaar. 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S | OAOC THICKNESS »'•«•••"• C R E E F A N D DRIFT F i a M w n ( I t w W i M N A t M l M*in»«i l  Table  A . l b , S t r a i n  Gauge  Data  Sheet,  Lot  2.  110  2.0 RANGE & DISPLAY  SENSITIVITY  READABILITY ACCURACY  CALIBRATION  INPUT CURCUITS  SPECIFICATIONS ±50,000 microstrain (ue): 10,000uc on the primary balance knob. P o l a r i t y and range extension on the range extender knob. Total reading i n single d i g i t a l display.  A l l bridge completion r e s i s t ors are standard self-temperature-compensated s t r a i n gages bonded to 2024 aluminum. BALANCE  10-turn lock knob provides approximately ±2000 micros t r a i n (GF=2, quarter and half bridge or 1200 f u l l bridge) for zeroing the d i g i t a l readout prior to load a p p l i c a t i o n . On-off switch provided.  BRIDGE EXCITATION  1.5 v o l t s RMS, 1000 Hz square wave.  Variable: Null meter deflects from zero to f u l l scale with 40 to 4000ue (nominal at GF=2). lue. 10.11 of reading or Sue, whichever i s greater, for R-120B and GF-2. •0.31 of reading or 5ut, whichever i s greater, for R=120n, GF-1.5-4.5. ±0.5» of reading or 5ue, whichever i s greater, for R=50-2000(J, GF=1.5-4.5. Self-contained shunt-calibration across internal 120fi and 350(1 dummy gages. Simulates +5000ut (±0.1%) on quarter bridge operation regardless of lead-wire resistance (GF=2). 50-20000 half or f u l l bridge internal dummy gage provided for 1200 and 3S0fi 3-wire quarter bridges.  LEAD WIRE CAPACITANCE T y p i c a l l y less than .05% loss of accuracy with .005 EFFECT microfarads. Negligible e f f e c t with 500 feet of lead wire. AMPLIFIER  AC t r a n s i s t o r i z e d .  GAGE FACTOR  Continuously variable from 0.10 to 10.00 (calibrated between 1.50 and 4.50 only).  OSCILLOSCOPE OUTPUT  Linear range ±250 mv DC. S e n s i t i v i t y variable from approximately 0.2 to 20 ue/mv Bandpass approx. DC to 60 Hz (±51) .  GALVANOMETER OUTPUT  F u l l scale ±1/2 ma unf i l t e r e d DC. Sensitivity variable from approximately 80 to 8000iiE/ma.  > PJ  o  n m  APPENDICES  2.0  SPECIFICATIONS 10  CAPACITY  channels  plus  OPEN  position. EXTERNAL  Will  CIRCUITS  or  accept  full  any  inputs  3-wire RESISTANCE  50  to  or  Quarter  may  be  2-  or  circuits. 2000  ohms.  Compatible  OUTPUT  half,  circuits  combination.  bridge  INPUT BRIDGE  quarter,  bridge  with  any  strain  indicator. BALANCE  RANGE  Quarter  (GF=2)  and  Half  Bridge:  uc  with  120-ohm  +2000 half Full  bridge  RESOLUTION  SWITCHING  (GF=2)  REPEATABILITY  ±2000  U E for  Range  proportional  1  pc.  Better  than  1  ye.  SIZE  9"w. [230  WEIGHT  5-1/2  A . 3 , Vishay  SB-1 Switch  specifications.  -  with x x  bridge.  to  resistance.  than  tight  Table  120-ohm  Better  Aluminum  CASE  indicator.  Bridge:  bridge BALANCE  in  dust  and  spray-  detachable  6"h. x 6"d. 1 5 0 x 150mm] pounds  12.4  and Balance  kg]  Unit  cover.  112  APPENDICES B.  FINITE  Linear  DIFFERENCE  f i n i t e  deflection corners. were  of  were  a  loaded  uniformly these  calculated  non-linearity  The  f i n i t e  linear  mesh  and  The  are  difference  to  calculate  plate  supported  deflections,  used  difference  points  used  in  the  Section  the by  curvatures  to  opposite  and  slopes  calculate  the  ratio.  form.  equations  ANALYSIS  differences  Having  113  method  is  quite  derivatives  in  the  replaced  along method  and is  by  the  discussed  its  basic  differential  quantities  plate  in  in  governing  difference  inside  simple  at  boundary.  selected  The  finite  [12,15,16].  2 A  central  was  used  difference to  approximate  equations.  The  midsurface  forces,  plate and  corners. were  A  solved  program  was  its  was  imposed  the  accuracy.  a  at set  to  equations.  error  equation, at  a l l  their of  Eq.  mesh  points  Eq.  simultaneous  generate  and  to  (2.27),  were  ,  with  zero  on  the (2.30)  boundaries equations  and which  plate.  solve  made  verify  Ax  (2.29),  respective  the  Tests  order  differential  conditions,  of  exist,  of  governing  deflection  written  solutions  with  applied  boundary  formed  for  simultaneous analytical  The  were  This  the  bi-harmonic  surface.  (2.1),  scheme,  on  the  these problems  program  where  and  test  APPENDICES B.l  DEFLECTION  The  non-dimensional  supported  The  by  RESULTS  deflection  opposite  supported  • 114  corners  corners  are  of is  shown  a  uniformly  shown  with  in  loaded  Fig.  zero  plate  B . l .  deflection.  At  the  4 centre  and  the  free  corners  the  deflections  are  .0703  qa  /D  4 and a  .0893  qa  the  edge  is  assumes of  almost  and  D the  cylindrical  curvature  where  q  is  the  flexural  shape  present  loading,  r i g i d i t y .  with  under  plate  only  this  a  The  small  plate amount  loading.  CONVERGENCE  determine  reasonable parallel series to  respectively,  length,  a n t i c l a s t i c  B.2  To  an  /D  10  by  the  tests 10,  in  The  deflection  B.2  and  size,  B.3  the  mesh.  the  plate were  mesh  and  .2%  edge  moment  there  and  was  examined  varying of  the  the  moment  a  and  for mesh  are  last  change  .1% r e s p e c t i v e l y . only  made  to  obtain  bending  moment  convergence. size  from  2  A by  2.  convergence For  be  deflection  was  made,  and  should  centre  increments  deflection  show  the  respectively.  approximately figures  fine  results,  to  of  how  small  plotted  increment was  in  Fig. plate  small,  The  error  in  for  convergence the  10  by  10  2  APPENDICES  DEFLECTIONS PLATE SUPPORTED BY TWO CORNER POSTS  Figure  B . l ,Deflection supported  of plate  by opposite  under  uniform  corners.  load and  115  APPENDICES  EFFECT OF GRID REFINEMENT -o CENTRE DEFLECTION  LO  > 1  O  ><^.  '—'r-  Q X  •  d C 3  — <M  CT  ZZ.  LU  1  I  L_ • LxJr^ CD  o  2.0  Figure  B.2,  4.0  GRID  Convergence mesh  sizes  of  ~~1 6.0  8.0  (N BY N)  centre  between  r  2  deflection  and  10.  10.0  for  varying  116  APPENDICES  117  EFFECT OF GRID REFINEMENT CM  ©  o CNTR. MOMENT  CM I  o „ X CD  2.0  6.0  8.0  centre  bending  GRID (N BY N)  Figure  B.3,  Convergence mesh  sizes  of  between  2  and  10.  10.0  moment  for  varying  APPENDICES  B.3  In  MAXIMUM V A L U E S  Section  required design  At  each  using of  for  of  the  the  mesh  axes  the  test  maximum  the  SLOPE  curvature  of  curvatures  with  and  twist  plate  edges.  calculated  using  a  ,xx max= )  3  5  7  <3  both  the  plate  diagonals.  slope  was  found  using  maximum  the  points  the  The  calculated  maximum  values  c i r c l e .  and  the  . . . ( B . l ) was  equal  geometric  in  the  relationship  giving  ( w  in  in  /°  of  This  corners  a 4  direction  [12]  supported  -  the  in  used  were  curvature  at  maximum  slope  where  Mohr's  found  The  and  non-linearity  the  ( w  was  AND  plate.  point  were  maximum  estimation  parallel  curvature  The  OF C U R V A T U R E  118  was  direction of  maximum  ,x max= )  also of  found  the  slope  -  2  at  0  6  be  3  the  diagonal can  <J* /  ...(B.2)  D  supported  between  clearly  the  seen  corners posts. in  F i g .  and The B . l .  was  


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