﻿Collapse 
  of 
  Tubes 
  by 
  External 
  Pressure. 
  505 
  

  

  I 
  have 
  shown 
  * 
  that 
  Euler 
  s 
  theory 
  may 
  be 
  extended 
  so 
  as 
  

   to 
  apply 
  to 
  cases 
  in 
  which 
  the 
  collapsing 
  thrust 
  is 
  more 
  than 
  

   sufficient 
  to 
  produce 
  elastic 
  break-down 
  of 
  the 
  strut, 
  and 
  it 
  is 
  

   found 
  that 
  by 
  this 
  extension 
  close 
  agreement 
  with 
  experi- 
  

   mental 
  results 
  may 
  be 
  obtained 
  for 
  struts 
  of 
  any 
  length 
  and 
  

   material. 
  Euler's 
  formula, 
  for 
  a 
  free-ended 
  strut, 
  is 
  

  

  M=- 
  2 
  ^ 
  ...!... 
  (4) 
  

  

  where 
  

  

  p 
  is 
  the 
  intensity 
  of 
  compressive 
  stress 
  at 
  collapse 
  ; 
  

  

  A 
  is 
  the 
  area, 
  

  

  and 
  I 
  is 
  the 
  moment 
  of 
  inertia 
  of 
  the 
  cross-section 
  of 
  the 
  

   strut 
  ; 
  

  

  I 
  is 
  the 
  length, 
  

  

  and 
  E 
  is 
  Young's 
  modulus 
  for 
  the 
  material 
  of 
  the 
  strut. 
  

  

  This 
  is 
  amended 
  by 
  reducing 
  /, 
  the 
  length 
  which, 
  according 
  

   to 
  (4), 
  corresponds 
  to 
  a 
  given 
  compressive 
  stress 
  p, 
  in 
  the 
  

   ratio 
  

  

  2 
  

  

  where 
  E 
  r 
  is 
  the 
  ratio 
  

   Lt. 
  

  

  [Increase 
  of 
  stress 
  ~| 
  

   Increase 
  of 
  strainJ 
  

  

  (5) 
  

  

  for 
  the 
  material 
  under 
  consideration, 
  corresponding 
  to 
  the 
  

   stress 
  p 
  ; 
  it 
  is 
  given 
  by 
  the 
  slope 
  of 
  the 
  ordinary 
  stress- 
  

   strain 
  diagram 
  for 
  a 
  compression 
  test 
  at 
  the 
  point 
  which 
  

   corresponds 
  to 
  p, 
  and 
  has 
  the 
  value 
  E, 
  Young's 
  modulus 
  for 
  

   the 
  material, 
  when 
  p 
  is 
  below 
  the 
  elastic 
  limit 
  of 
  the 
  material. 
  

   The 
  analysis 
  leading 
  to 
  this 
  amendment 
  is 
  fully 
  explained 
  in 
  

   the 
  papers 
  cited 
  above. 
  

  

  Since 
  an 
  infinitely 
  long 
  tube 
  collapses 
  under 
  external 
  

   pressure 
  in 
  a 
  manner 
  which 
  has 
  many 
  points 
  of 
  resemblance 
  

   to 
  strut 
  failure, 
  it 
  is 
  reasonable 
  to 
  hope 
  for 
  equally 
  good 
  

   results 
  from 
  a 
  similar 
  extension 
  of 
  Bryan's 
  formula. 
  The 
  

   two 
  problems, 
  in 
  fact, 
  are 
  almost 
  identical 
  f. 
  This 
  is 
  made 
  

   clear 
  by 
  a 
  consideration 
  of 
  the 
  form 
  of 
  the 
  cross-section 
  

  

  * 
  < 
  Engineering/ 
  Aug. 
  23, 
  1912 
  ; 
  Phil. 
  Trans. 
  Roy. 
  Soc. 
  A. 
  vol. 
  ccxiii. 
  

   pp. 
  187-244(1913). 
  

  

  t 
  Cf. 
  W. 
  C. 
  Unwin, 
  Proc. 
  Inst. 
  C.E. 
  vol. 
  xlvi. 
  p. 
  225 
  (1875); 
  

   W. 
  E. 
  Lilly, 
  Trans. 
  Inst. 
  Civ. 
  Eng. 
  Ireland 
  (1910). 
  

  

  