﻿Collapse 
  of 
  Tubes 
  by 
  External 
  Pressure. 
  503 
  

  

  relative 
  strengths 
  of 
  long 
  and 
  short 
  tubes 
  are 
  concerned. 
  

   The 
  theoretical 
  formula 
  is 
  not, 
  however, 
  satisfactory 
  as 
  a 
  

   basis 
  for 
  practical 
  design, 
  since 
  the 
  results 
  of 
  careful 
  experi- 
  

   ments 
  * 
  have 
  shown 
  that 
  it 
  over-estimates, 
  often 
  by 
  a 
  very 
  

   considerable 
  amount, 
  the 
  resistance 
  of 
  actual 
  tubes 
  to 
  collapse. 
  

   I 
  propose 
  in 
  the 
  present 
  paper 
  to 
  investigate 
  whether 
  these 
  

   discrepancies 
  can 
  be 
  accounted 
  for. 
  If 
  so, 
  it 
  may 
  be 
  possible 
  

   to 
  substitute 
  rational 
  estimates 
  of 
  strength 
  for 
  the 
  purely 
  

   empirical 
  formulas 
  which 
  are 
  at 
  present 
  employed. 
  

  

  1 
  shall 
  here 
  consider 
  in 
  detail 
  only 
  tubes 
  which 
  exceed 
  

   what 
  Professor 
  Carman 
  has 
  called 
  u 
  the 
  critical 
  length." 
  

   This 
  term 
  may 
  be 
  explained 
  by 
  a 
  reference 
  to 
  the 
  general 
  

   expression 
  for 
  the 
  collapsing 
  pressure 
  f, 
  which 
  may 
  be 
  

   written 
  as 
  follows 
  : 
  — 
  

  

  ^JiWM^^ 
  2 
  - 
  1 
  ^] 
  • 
  (1) 
  

  

  where 
  

  

  3P 
  is 
  the 
  collapsing 
  pressure, 
  in 
  pounds 
  per 
  square 
  inch 
  ; 
  

   E 
  is 
  Young's 
  modulus, 
  in 
  pounds 
  per 
  square 
  inch, 
  

  

  and 
  — 
  is 
  Poisson's 
  ratio, 
  for 
  the 
  material 
  of 
  the 
  tube 
  ; 
  

   m 
  

  

  t 
  is 
  the 
  thickness, 
  

  

  d 
  is 
  the 
  diameter, 
  

  

  and 
  I 
  is 
  the 
  length 
  of 
  the 
  tube 
  between 
  end-constraints 
  ; 
  

  

  k 
  is 
  the 
  number 
  of 
  lobes 
  characterizing 
  the 
  cross-section 
  

   after 
  collapse, 
  

  

  and 
  Z 
  is 
  a 
  constant, 
  depending 
  upon 
  the 
  type 
  of 
  the 
  end- 
  

   constraints. 
  

  

  It 
  is 
  easily 
  seen 
  | 
  that 
  as 
  the 
  length 
  I 
  is 
  increased, 
  the 
  

   pressure 
  required 
  to 
  produce 
  collapse, 
  for 
  any 
  definite 
  value 
  

   of 
  k, 
  falls 
  rapidly 
  to 
  a 
  sensibly 
  constant 
  value, 
  given 
  by 
  

  

  ' 
  -. 
  »=|^ 
  I 
  B(P-1)|, 
  .... 
  (2) 
  

  

  and 
  since 
  this 
  expression 
  has 
  its 
  least 
  possible 
  value 
  when 
  

   k=2, 
  we 
  are 
  led 
  by 
  theory 
  to 
  predict 
  that 
  all 
  tubes 
  of 
  

  

  * 
  A. 
  P. 
  Carman, 
  University 
  of 
  Illinois 
  Bulletin, 
  vol. 
  iii. 
  No. 
  17 
  ; 
  

   R 
  T. 
  Stewart, 
  Am. 
  Soc. 
  Mech. 
  Eng. 
  Transactions, 
  pp. 
  730-820 
  (1906). 
  

   t 
  Cf. 
  my 
  papers 
  cited 
  above. 
  

  

  