﻿EQUILIBRIUM 
  OF 
  ELASTIC 
  SOLIDS. 
  

  

  101 
  

  

  1. 
  When 
  the 
  external 
  and 
  internal 
  pressures 
  are 
  equal, 
  or 
  h 
  x 
  =h 
  r 
  

  

  2. 
  When 
  the 
  external 
  pressure 
  is 
  to 
  the 
  internal 
  pressure 
  as 
  the 
  square 
  of 
  

   the 
  interior 
  diameter 
  is 
  to 
  that 
  of 
  the 
  exterior 
  diameter, 
  or 
  when 
  a* 
  h 
  t 
  =a 
  2 
  2 
  h 
  r 
  

  

  3. 
  When 
  the 
  cylinder 
  is 
  solid, 
  or 
  when 
  a 
  2 
  = 
  0. 
  

  

  4. 
  When 
  the 
  solid 
  becomes 
  an 
  indefinitely 
  extended 
  plate 
  with 
  a 
  cylindric 
  

   hole 
  in 
  it, 
  or 
  when 
  a 
  2 
  becomes 
  infinite. 
  

  

  5. 
  When 
  pressure 
  is 
  applied 
  only 
  at 
  the 
  plane 
  surfaces 
  of 
  the 
  solid 
  cylinder, 
  

   and 
  the 
  cylindric 
  surface 
  is 
  prevented 
  from 
  expanding 
  by 
  being 
  inclosed 
  in 
  a 
  

  

  strong 
  case, 
  or 
  when 
  — 
  =0. 
  

  

  6. 
  When 
  pressure 
  is 
  applied 
  to 
  the 
  cylindric 
  surface, 
  and 
  the 
  ends 
  are 
  re- 
  

   tained 
  at 
  an 
  invariable 
  distance, 
  or 
  when 
  — 
  =0. 
  

  

  X 
  

  

  1. 
  When 
  h 
  1 
  =h 
  2 
  , 
  the 
  equations 
  of 
  compression 
  become 
  

  

  — 
  =~(o 
  + 
  2 
  h 
  x 
  ) 
  + 
  A 
  (o-hj 
  

   x 
  9 
  fx 
  v 
  l/ 
  3 
  m 
  K 
  x/ 
  

  

  0= 
  (97T 
  + 
  3^) 
  + 
  2/il 
  (97I-3 
  m) 
  

  

  \ 
  

  

  fl 
  

  

  8r 
  1 
  , 
  n 
  1 
  \ 
  1/7 
  \ 
  

  

  3 
  m 
  

  

  °~\9/ji 
  3 
  m) 
  + 
  hl 
  \9fJL 
  + 
  Sm) 
  

  

  (27.) 
  

  

  When 
  h 
  t 
  =h 
  2 
  = 
  0, 
  then 
  

  

  

  

  8 
  x 
  8r 
  h 
  x 
  

   x 
  r 
  3 
  fj. 
  

  

  The 
  compression 
  of 
  a 
  cylindrical 
  vessel 
  exposed 
  on 
  all 
  sides 
  to 
  the 
  same 
  hy- 
  

   drostatic 
  pressure 
  is 
  therefore 
  independent 
  of 
  m, 
  and 
  it 
  may 
  be 
  shewn 
  that 
  the 
  

   same 
  is 
  true 
  for 
  a 
  vessel 
  of 
  any 
  shape. 
  

  

  2. 
  Wheno 
  1 
  2 
  A 
  1 
  =a 
  2 
  2 
  h. 
  2 
  , 
  

  

  8: 
  

  

  8x 
  _ 
  / 
  j^ 
  2 
  \ 
  

  

  * 
  \9fi 
  3 
  m) 
  

  

  $ 
  r 
  1 
  / 
  \ 
  1 
  ,0, 
  \ 
  

  

  ( 
  1_ 
  _1_\ 
  h± 
  

  

  °\V/JL 
  Zm) 
  + 
  m 
  

  

  

  (28.) 
  

  

  In 
  this 
  case, 
  when 
  0=0, 
  the 
  compressions 
  are 
  independent 
  of 
  //. 
  

   3. 
  In 
  a 
  solid 
  cylinder, 
  a 
  2 
  = 
  0, 
  

  

  p 
  = 
  q 
  = 
  h 
  v 
  

  

  The 
  expressions 
  for 
  — 
  and 
  — 
  are 
  the 
  same 
  as 
  those 
  in 
  the 
  first 
  case, 
  when 
  

  

  \ 
  = 
  h 
  2 
  

  

  