﻿60 
  Mr. 
  W. 
  H. 
  Macaulay 
  on 
  the 
  Stresses 
  

  

  In 
  the 
  general 
  case 
  let 
  us 
  suppose 
  that 
  w 
  N 
  is 
  negligible 
  

   compared 
  with 
  unity, 
  then 
  

  

  i 
  (N 
  _ 
  1)+ 
  i(i±^y_ 
  *• 
  

  

  2 
  J 
  a 
  \l 
  — 
  u) 
  (1 
  — 
  u) 
  

  

  and 
  

  

  1 
  

  

  w 
  = 
  J 
  

  

  -(1 
  + 
  m) 
  — 
  u 
  

  

  a 
  

  

  To=-iW 
  1 
  -i±^{ 
  i 
  (N-l)- 
  I 
  ^}. 
  

  

  Thus 
  the 
  condition 
  that 
  — 
  -T 
  is 
  equal 
  to 
  a 
  quarter 
  of 
  the 
  

   total 
  load, 
  namely 
  i(N 
  — 
  1)W, 
  is 
  

  

  i(N-l) 
  = 
  -^, 
  

  

  or 
  

  

  

  Thus, 
  when 
  N 
  = 
  9 
  and 
  w=— 
  *205, 
  — 
  T 
  will 
  be 
  equal 
  to 
  a 
  

   quarter 
  of 
  the 
  total 
  load 
  if 
  the 
  end 
  verticals 
  are 
  of 
  about 
  

   6 
  times 
  the 
  strength 
  of 
  the 
  intermediate 
  ones; 
  when 
  N=9 
  

   and 
  it=—'39 
  the 
  same 
  result 
  will 
  be 
  secured 
  if 
  the 
  end 
  

   verticals 
  are 
  about 
  9 
  times 
  the 
  strength 
  of 
  the 
  intermediate 
  

   ones. 
  By 
  means 
  of 
  such 
  results 
  as 
  these, 
  we 
  may 
  form 
  an 
  

   estimate 
  of 
  the 
  conditions 
  under 
  which 
  the 
  method, 
  sometimes 
  

   employed, 
  of 
  calculating 
  the 
  tensions 
  of 
  the 
  bars 
  of 
  a 
  re- 
  

   dundant 
  frame 
  by 
  considering 
  it 
  as 
  the 
  result 
  of 
  the 
  super- 
  

   position 
  of 
  two 
  just 
  stiff 
  frames, 
  can 
  be 
  regarded 
  as 
  in 
  any 
  

   degree 
  trustworthy. 
  

  

  Certain 
  results 
  for 
  the 
  girder 
  

  

  can, 
  as 
  

  

  we 
  remarked 
  above, 
  be 
  deduced 
  from 
  the 
  case 
  already 
  con- 
  

   sidered 
  by 
  making 
  the 
  extensibilities 
  of 
  the 
  intermediate 
  

   vertical 
  bars 
  all 
  infinite. 
  But, 
  as 
  we 
  have 
  in 
  this 
  frame 
  

   only 
  one 
  redundant 
  member, 
  we 
  can, 
  without 
  inconvenience, 
  

   take 
  a 
  more 
  general 
  case 
  than 
  we 
  have 
  hitherto 
  dealt 
  with 
  

   with 
  regard 
  to 
  the 
  sections 
  or 
  extensibilities 
  of 
  the 
  bars, 
  or 
  

   even 
  suppose 
  these 
  to 
  be 
  all 
  different. 
  Let 
  us 
  use 
  the 
  same 
  

   notation 
  as 
  before 
  for 
  distinguishing 
  the 
  bars 
  of 
  the 
  frame, 
  

   and 
  take 
  as 
  our 
  redundant 
  bar 
  the 
  vertical 
  bar 
  which 
  bounds 
  

   the 
  Nth 
  panel, 
  its 
  extensibility 
  being 
  e 
  and 
  its 
  tension 
  R. 
  

  

  