﻿58 
  

  

  Mr. 
  W. 
  H. 
  Macaulay 
  on 
  the 
  Stresses 
  

  

  one 
  end 
  of 
  the 
  beam 
  and 
  y 
  from 
  the. 
  other 
  end 
  (the 
  unit 
  of 
  

   length 
  being 
  the 
  length 
  of 
  a 
  bay) 
  is 
  

  

  &W\t*xy(W 
  + 
  xy). 
  ..... 
  (41) 
  

  

  In 
  comparing 
  this 
  result 
  with 
  the 
  expression 
  (40), 
  we 
  must 
  

   suppose 
  that 
  x 
  and 
  y 
  can 
  only 
  have 
  integral 
  values 
  from 
  1 
  to 
  

   N 
  — 
  1 
  ; 
  thus 
  the 
  value 
  of 
  my 
  ranges 
  from 
  N— 
  1 
  to 
  £N 
  2 
  or 
  to 
  

   j(N 
  2 
  — 
  1) 
  according 
  as 
  N 
  is 
  even 
  or 
  odd. 
  Accordingly 
  (41) 
  

   comprises 
  what 
  may 
  be 
  expected 
  to 
  be 
  generally 
  the 
  most 
  

   important 
  part 
  of 
  (40). 
  

  

  However, 
  if, 
  as 
  will 
  naturally 
  be 
  the 
  case, 
  /jls 
  2 
  > 
  § 
  Ai 
  2 
  , 
  the 
  

   remaining 
  terms 
  of 
  (40), 
  which 
  may 
  be 
  by 
  no 
  means 
  insig- 
  

   nificant, 
  will 
  certainly 
  be 
  positive 
  ; 
  whereas 
  the 
  terms 
  omitted 
  

   in 
  (41) 
  by 
  neglecting 
  the 
  moment 
  of 
  inertia 
  of 
  the 
  web 
  will 
  

   be 
  negative. 
  

  

  Let 
  us 
  take 
  a 
  numerical 
  example. 
  Let 
  the 
  cross-sections 
  

   be 
  8f 
  for 
  horizontal 
  bars, 
  2£for 
  diagonals, 
  4f 
  for 
  end 
  verticals, 
  

   and 
  JJ 
  for 
  intermediate 
  verticals; 
  let 
  N 
  = 
  9 
  and 
  = 
  45°, 
  so 
  

   that 
  £ 
  = 
  1 
  and 
  s= 
  */2. 
  Then, 
  taking 
  the 
  depth 
  of 
  the 
  girder 
  

   or 
  the 
  length 
  of 
  a 
  horizontal 
  bar 
  to 
  be 
  unity 
  as 
  before, 
  and 
  

   writing 
  E 
  for 
  Young's 
  modulus, 
  

  

  \ 
  = 
  

  

  8?E 
  

  

  ^~2?E' 
  v 
  ~~fE' 
  

  

  v== 
  

  

  4?E' 
  

  

  /3=3-078, 
  w=--205, 
  ^=-4*873, 
  

   1 
  =4, 
  </>=3-38, 
  ^=-10-62, 
  ©=1-78, 
  

  

  and 
  the 
  values 
  of 
  the 
  several 
  terms 
  of 
  our 
  formula 
  are 
  given 
  

   by 
  the 
  following 
  table 
  : 
  — 
  

  

  

  x=l, 
  y=8. 
  

  

  x=2, 
  y=7. 
  

  

  x=3, 
  y=Q. 
  

  

  #=4, 
  y=5. 
  

  

  l\t 
  2 
  xy(W-\-xy)ZE, 
  

  

  297 
  

  

  10-6 
  

  

  1-7 
  

  

  554 
  

  

  18-6 
  

  

  1-4 
  

  

  74-2 
  

  

  23-9 
  

  

  1-4 
  

  

  84-2 
  

  

  26-6 
  

  

  1-4 
  

  

  (jus 
  2 
  -tXi 
  2 
  )a'?/£E 
  

  

  j/(l+w)(l-wz)(l-«y)w£E 
  ... 
  

  

  We 
  see 
  that 
  in 
  this 
  case 
  the 
  term 
  (fj,s 
  2 
  —%\t 
  2 
  )a?y 
  is 
  of 
  con- 
  

   siderable 
  importance, 
  and 
  that 
  to 
  make 
  it 
  small 
  compared 
  

  

  