﻿EQUILIBRIUM 
  OF 
  ELASTIC 
  SOLIDS. 
  109 
  

  

  To 
  find 
  an 
  expression 
  for 
  the 
  curvature 
  produced 
  in 
  a 
  flat, 
  circular, 
  elastic 
  

   plate, 
  by 
  the 
  difference 
  of 
  the 
  hydrostatic 
  pressures 
  which 
  act 
  on 
  each 
  side 
  of 
  it, 
  — 
  

  

  Let 
  t 
  be 
  the 
  thickness 
  of 
  the 
  plate, 
  which 
  must 
  be 
  small 
  compared 
  with 
  its 
  

   diameter. 
  

  

  Let 
  the 
  form 
  of 
  the 
  middle 
  surface 
  of 
  the 
  plate, 
  after 
  the 
  curvature 
  is 
  pro- 
  

   duced, 
  be 
  expressed 
  by 
  an 
  equation 
  between 
  r, 
  the 
  distance 
  of 
  any 
  point 
  from 
  the 
  

   axis, 
  or 
  normal 
  to 
  the 
  centre 
  of 
  the 
  plate, 
  and 
  x 
  the 
  distance 
  of 
  the 
  point 
  from 
  the 
  

   plane 
  in 
  which 
  the 
  middle 
  of 
  the 
  plate 
  originally 
  was, 
  and 
  let 
  ds 
  =\/(dx) 
  2 
  + 
  (dr 
  2 
  ~). 
  

  

  Let 
  h 
  x 
  be 
  the 
  pressure 
  on 
  one 
  side 
  of 
  the 
  plate, 
  and 
  h 
  2 
  that 
  on 
  the 
  other. 
  

  

  Let 
  p 
  and 
  q 
  be 
  the 
  pressures 
  in 
  the 
  plane 
  of 
  the 
  plate 
  at 
  any 
  point, 
  p 
  acting 
  

   in 
  the 
  direction 
  of 
  a 
  tangent 
  to 
  the 
  section 
  of 
  the 
  plate 
  by 
  a 
  plane 
  passing 
  through 
  

   the 
  axis, 
  and 
  q 
  acting 
  in 
  the 
  direction 
  perpendicular 
  to 
  that 
  plane. 
  

  

  By 
  equating 
  the 
  forces 
  which 
  act 
  on 
  any 
  particle 
  in 
  a 
  direction 
  parallel 
  to 
  

   the 
  axis, 
  we 
  find 
  

  

  dr 
  dx 
  dp 
  dx 
  , 
  d 
  2 
  x 
  ., 
  , 
  .dr 
  . 
  

  

  tp 
  dsdl 
  + 
  tr 
  Ts 
  Ts 
  + 
  trp 
  d^ 
  + 
  r 
  ^- 
  h2) 
  T^ 
  

   By 
  making 
  p=0 
  when 
  r=0 
  in 
  this 
  equation, 
  

  

  *=-f,S>-« 
  • 
  • 
  • 
  < 
  61 
  -> 
  

  

  The 
  forces 
  perpendicular 
  to 
  the 
  axis 
  are 
  

  

  (dr\ 
  2 
  M 
  . 
  dp 
  dr 
  , 
  t 
  d 
  2 
  r 
  . 
  dx 
  

  

  Substituting 
  for 
  p 
  its 
  value, 
  the 
  equation 
  gives 
  

  

  (dr 
  dr 
  dx\ 
  (7^ 
  — 
  /j 
  2 
  ) 
  2 
  /dr 
  ds 
  d 
  2 
  x 
  ds 
  d 
  2 
  r\ 
  . 
  

  

  ds 
  dx 
  ds) 
  2t 
  \dx 
  dx 
  d 
  s' 
  J 
  dx 
  ds 
  2 
  )' 
  ^ 
  '' 
  

  

  t 
  

  

  The 
  equations 
  of 
  elasticity 
  become 
  

  

  ds 
  ~\9fx 
  3m) 
  \ 
  p 
  + 
  9+ 
  2 
  ) 
  + 
  m 
  

  

  r 
  \9fJL 
  3m) 
  \ 
  9 
  2 
  ) 
  + 
  m 
  

  

  d8 
  r 
  d 
  /S 
  

  

  Differentiating 
  -^ 
  = 
  -r- 
  I— 
  A, 
  and 
  in 
  this 
  case 
  

  

  ° 
  dr 
  dr 
  \ 
  r 
  J 
  

  

  d 
  d 
  r 
  -. 
  dr 
  dr 
  d 
  8 
  s 
  

   dr 
  ds 
  ds 
  ds 
  

  

  By 
  a 
  comparison 
  of 
  these 
  values 
  of 
  —7— 
  

  

  \ 
  ds) 
  \9fi 
  3m) 
  \ 
  P+9 
  2 
  )^m 
  dsni 
  \9fx 
  3m)\dr 
  + 
  dr) 
  

  

  r 
  da 
  dr 
  ^ 
  rt 
  

  

  + 
  — 
  -j 
  L 
  + 
  - 
  f 
  1 
  = 
  0. 
  

  

  m 
  dr 
  d 
  s 
  

  

  