﻿64 
  Mr. 
  W. 
  H. 
  Macaulay 
  on 
  the 
  Stresses 
  

  

  desirable 
  to 
  make 
  any 
  corrections 
  in 
  the 
  assumed 
  sections, 
  the 
  

   process 
  being 
  repeated 
  with 
  corrected 
  values 
  of 
  sections 
  if 
  

   necessary. 
  In 
  the 
  limiting 
  case 
  in 
  which 
  the 
  vertical 
  

   members 
  are 
  very 
  strong 
  y 
  will 
  certainly 
  be 
  small, 
  and 
  T 
  

   will 
  be 
  nearly 
  equal 
  to 
  |NW. 
  

  

  The 
  case 
  of 
  the 
  common 
  type 
  of 
  girder 
  

  

  / 
  /I 
  / 
  ./ 
  

  

  / 
  / 
  1/ 
  / 
  /s 
  \ 
  \ 
  \1 
  S 
  

  

  \ 
  S. 
  \ 
  IS 
  

  

  is 
  very 
  simple, 
  since 
  Maxwell's 
  problem 
  has 
  only 
  to 
  be 
  applied 
  

   to 
  the 
  centre 
  panel, 
  and 
  the 
  results 
  are 
  hardly 
  worth 
  writing 
  

   down. 
  

  

  In 
  practice 
  the 
  joints 
  of 
  a 
  girder 
  are 
  generally 
  rivetted 
  

   and 
  stiffened 
  with 
  gussets, 
  and 
  the 
  horizontal 
  booms 
  are 
  

   continuous. 
  The 
  effect 
  of 
  stiffness 
  of 
  the 
  angles 
  is 
  doubtless 
  

   a 
  matter 
  of 
  secondary 
  importance, 
  but 
  the 
  effect 
  of 
  continuity 
  

   of 
  the 
  horizontal 
  booms 
  and 
  their 
  resistance 
  to 
  bending, 
  when 
  

   they 
  are 
  no 
  longer 
  regarded 
  as 
  lines, 
  seems 
  to 
  deserve 
  more 
  

   attention. 
  Accordingly, 
  let 
  us 
  see 
  how 
  the 
  comparison 
  of 
  

   results 
  (40) 
  and 
  (41) 
  is 
  modified 
  by 
  this 
  consideration. 
  

  

  Let 
  the 
  lines 
  which 
  we 
  have 
  hitherto 
  taken 
  for 
  the 
  horizontal 
  

   members 
  in 
  our 
  original 
  girder 
  be 
  replaced 
  by 
  the 
  neutral 
  

   lines 
  of 
  continuous 
  booms. 
  Let 
  J 
  be 
  the 
  area 
  of 
  cross-section 
  

   of 
  each, 
  and 
  k 
  the 
  radius 
  of 
  gyration 
  of 
  the 
  area 
  about 
  the 
  

   horizontal 
  line 
  in 
  its 
  plane 
  through 
  its 
  centre 
  of 
  gravity. 
  

   Then 
  to 
  obtain 
  the 
  Bernoulli-Euler 
  deflection, 
  neglecting 
  the 
  

   web 
  as 
  before, 
  we 
  must 
  multiply 
  the 
  expression 
  (41) 
  by 
  

  

  i^(i 
  + 
  ^ 
  2 
  ) 
  ;thusweget 
  

  

  ^W\t\vy(N 
  2 
  + 
  zy)(l 
  + 
  4:l 
  2 
  k 
  2 
  )-\ 
  . 
  . 
  (45) 
  

  

  here 
  2tk 
  is 
  certainly 
  less 
  than 
  the 
  ratio 
  of 
  the 
  depth 
  of 
  a 
  boom 
  

   to 
  the 
  depth 
  of 
  the 
  girder 
  between 
  the 
  neutral 
  lines. 
  

  

  Now 
  consider 
  the 
  girder 
  as 
  a 
  frame, 
  but 
  with 
  the 
  bottom 
  

   boom 
  continuous, 
  and 
  let 
  us 
  aim 
  at 
  adding 
  such 
  loading 
  to 
  

   the 
  bottom 
  boom, 
  at 
  the 
  joints, 
  as 
  shall 
  give 
  it 
  at 
  those 
  points 
  

   the 
  deflection 
  (40). 
  Write 
  8 
  for 
  the 
  deflection 
  (41), 
  S 
  + 
  S' 
  

   for 
  the 
  deflection 
  (40), 
  and 
  S 
  1? 
  8/ 
  for 
  the 
  greatest 
  values 
  of 
  8 
  

   and 
  8'. 
  First 
  add 
  a 
  uniform 
  loading 
  2f 
  2 
  FW, 
  this 
  by 
  the 
  

   Bernoulli-Euler 
  theory 
  will 
  give 
  the 
  boom 
  by 
  itself 
  the 
  de- 
  

   flection 
  8. 
  Now 
  we 
  want 
  to 
  add 
  a 
  loading 
  which 
  will 
  give 
  

   the 
  boom 
  bv 
  itself 
  a 
  deflection 
  8'; 
  this 
  will 
  not 
  be 
  a 
  uniform 
  

  

  8 
  ' 
  

   loading, 
  but 
  a 
  uniform 
  loading 
  2t 
  2 
  k 
  2 
  W 
  -±- 
  which 
  gives 
  the 
  

  

  central 
  deflection 
  correctly 
  will 
  give 
  nearly 
  the 
  deflection 
  8' 
  

  

  