﻿and 
  Deflection 
  of 
  Braced 
  Girders. 
  65 
  

  

  elsewhere. 
  Thus 
  for 
  the 
  girder 
  considered 
  as 
  a 
  frame, 
  with 
  

   the 
  bottom 
  boom 
  only 
  continuous, 
  and 
  a 
  uniform 
  loading 
  

  

  W 
  ■< 
  1 
  -f 
  2t 
  2 
  k 
  2 
  ~^-s 
  — 
  — 
  > 
  , 
  we 
  have 
  a 
  deflection 
  agreeing 
  with 
  

  

  S 
  + 
  S 
  / 
  at 
  the 
  centre 
  and 
  very 
  nearly 
  equal 
  to 
  it 
  elsewhere. 
  

   Now 
  4t 
  2 
  k 
  2 
  is 
  a 
  small 
  fraction, 
  squares 
  of 
  which 
  cannot 
  be 
  

   worth 
  keeping, 
  so 
  the 
  reduction 
  in 
  deflection 
  under 
  loading 
  W, 
  

   due 
  to 
  the 
  bottom 
  boom 
  being 
  continuous, 
  is 
  very 
  nearly 
  

  

  2t 
  2 
  k 
  2 
  - 
  1 
  ~ 
  1 
  -(8 
  + 
  8 
  / 
  ). 
  And 
  if 
  we 
  suppose 
  the 
  effect 
  of 
  con- 
  

  

  tinuity 
  of 
  the 
  top 
  boom 
  to 
  be 
  about 
  the 
  same 
  as 
  for 
  the 
  bottom 
  

   one 
  (it 
  is 
  probably 
  less 
  and 
  hardly 
  can 
  be 
  greater) 
  , 
  the 
  whole 
  

   reduction 
  of 
  deflection 
  due 
  to 
  continuity 
  of 
  both 
  booms 
  is 
  

  

  about 
  4^ 
  2§ 
  ii^-(^ 
  + 
  §0. 
  

  

  1 
  

   Now 
  the 
  reduction 
  in 
  the 
  result 
  (41) 
  already 
  found 
  (45) 
  is 
  

  

  4t 
  2 
  k 
  2 
  S 
  } 
  and 
  the 
  difference 
  between 
  these 
  reductions 
  is 
  

  

  4* 
  2 
  f[o 
  / 
  +|- 
  / 
  (o- 
  + 
  o 
  / 
  )]-. 
  

  

  So 
  w 
  r 
  e 
  see 
  that 
  the 
  effect 
  of 
  taking 
  • 
  account 
  of 
  the 
  area 
  of 
  

   section 
  and 
  continuity 
  of 
  booms 
  in 
  both 
  calculations 
  is 
  to 
  

   reduce 
  the 
  divergence 
  between 
  the 
  results 
  (40) 
  and 
  (41) 
  by 
  

   the 
  fraction 
  

  

  ^{i 
  + 
  |'(i 
  + 
  |)} 
  

  

  of 
  itself. 
  We 
  shall 
  not 
  be 
  far 
  wrong 
  in 
  substituting 
  tor 
  this 
  

  

  the 
  fraction 
  4t 
  2 
  k 
  2 
  ( 
  2 
  + 
  -4- 
  \ 
  Thus, 
  if 
  we 
  calculate 
  the 
  deflection 
  

  

  of 
  the 
  girder 
  as 
  a 
  frame 
  with 
  continuous 
  booms, 
  and 
  compare 
  

   the 
  result 
  with 
  that 
  given 
  by 
  the 
  Bernoulli-Euler 
  theory 
  

   (leaving 
  out 
  the 
  web 
  in 
  the 
  moment 
  of 
  inertia 
  of 
  the 
  section), 
  

   we 
  still 
  have 
  substantially 
  the 
  divergence 
  already 
  noted, 
  

   though 
  it 
  is 
  reduced 
  by 
  a 
  small 
  percentage. 
  Moreover, 
  the 
  

   remark 
  made 
  above 
  as 
  to 
  the 
  omission 
  of 
  the 
  web 
  still 
  applies. 
  

   The 
  reduction 
  in 
  the 
  divergence 
  between 
  the 
  two 
  results, 
  

   since 
  it 
  can 
  be 
  roughly 
  estimated, 
  may 
  as 
  well 
  be 
  taken 
  account 
  

   of. 
  Thus, 
  if 
  the 
  depth 
  of 
  boom 
  is 
  -^ 
  of 
  the 
  depth 
  of 
  the 
  

   girder 
  between 
  the 
  neutral 
  lines, 
  the 
  reduction 
  will 
  be 
  about 
  

   2 
  per 
  cent, 
  or 
  a 
  little 
  more. 
  

  

  FUl. 
  Mag. 
  S. 
  5. 
  Yol. 
  45. 
  No. 
  272. 
  Jan. 
  1898. 
  F 
  

  

  