﻿506 
  Mr. 
  R. 
  V. 
  Southwell 
  on 
  the 
  

  

  when 
  held 
  by 
  the 
  collapsing 
  pressure 
  in 
  a 
  position 
  of 
  slight 
  

   distortion, 
  defined 
  by 
  the 
  analysis 
  leading 
  to 
  equation 
  (3). 
  

   The 
  form 
  of 
  the 
  cross-section, 
  before 
  and 
  after 
  distortion, 
  

   is 
  shown 
  in 
  fig. 
  1 
  : 
  the 
  curvature 
  is 
  increased 
  and 
  decreased 
  

  

  Figr. 
  1. 
  

  

  The 
  thick 
  line 
  shovrs 
  form 
  of 
  cross-section 
  after 
  distortion. 
  

  

  at 
  different 
  parts 
  of 
  the 
  section, 
  and 
  at 
  four 
  equidistant 
  

   points 
  (A, 
  B, 
  C, 
  D 
  in 
  the 
  figure) 
  it 
  remains 
  unchanged 
  by 
  

   collapse. 
  Moreover, 
  in 
  the 
  initial 
  configuration 
  the 
  action 
  

   on 
  any 
  section 
  of 
  the 
  tube 
  wall 
  is 
  very 
  nearly 
  a 
  uniform 
  

   compressive 
  stress. 
  Hence 
  the 
  tube 
  may 
  be 
  regarded 
  as 
  

   composed 
  of 
  four 
  circular 
  struts, 
  pin-jointed 
  at 
  their 
  ex- 
  

   tremities, 
  and 
  of 
  circumferential 
  length 
  \-rrd. 
  If 
  we 
  consider 
  

   a 
  unit 
  length 
  of 
  the 
  tube, 
  the 
  cross-section 
  of 
  each 
  strut 
  is 
  a 
  

   rectangle, 
  of 
  dimensions 
  Ixf, 
  and 
  the 
  thrust, 
  which 
  at 
  every 
  

   point 
  acts 
  along 
  the 
  centre 
  line 
  of 
  the 
  strut, 
  is 
  j>1$ 
  c 
  d. 
  Euler's 
  

   theory 
  would 
  give 
  for 
  the 
  collapsing 
  load 
  

  

  -3M=tt 
  2 
  E.— 
  .^, 
  

  

  ^=§ 
  E 
  '-I> 
  (6) 
  

  

  and 
  the 
  amended 
  equation, 
  for 
  cases 
  in 
  which 
  elastic 
  break- 
  

   down 
  precedes 
  collapse, 
  would 
  be 
  

  

  32 
  E 
  t 
  3 
  

  

  *" 
  T 
  'K7IP 
  m 
  

  

  where 
  E 
  ; 
  corresponds 
  to 
  p, 
  the 
  circumferential 
  compressive 
  

  

  stress 
  t 
  or 
  ~ 
  ;P 
  C 
  - 
  ), 
  in 
  the 
  manner 
  described 
  above. 
  

  

  Comparing 
  the 
  expression 
  (G) 
  with 
  (3), 
  we 
  see 
  that 
  it 
  is 
  

   correct 
  in 
  form, 
  but 
  needs 
  to 
  be 
  multiplied 
  by 
  the 
  factors 
  

  

  9 
  2 
  

  

  - 
  and 
  9 
  ., 
  . 
  The 
  former 
  represents 
  the 
  relative 
  weakness 
  

   4 
  m 
  2 
  — 
  1 
  L 
  

  

  of 
  the 
  circular 
  as 
  compared 
  with 
  a 
  straight 
  strut, 
  and 
  does 
  

  

  