﻿and 
  Deflection 
  of 
  Braced 
  Girders. 
  57 
  

  

  1 
  f 
  N 
  — 
  1 
  $ 
  + 
  ^u 
  2 
  \ 
  __ 
  

  

  and 
  let 
  # 
  and 
  ?/ 
  be 
  the 
  number 
  of 
  bays 
  from 
  A 
  and 
  B 
  re- 
  

   spectively 
  to 
  centre 
  of 
  bar 
  in 
  the 
  case 
  of 
  a 
  tension 
  formula, 
  

   or 
  to 
  point 
  of 
  deflection 
  in 
  the 
  case 
  of 
  the 
  deflection 
  formula, 
  

   then 
  

  

  T 
  a 
  =^itW[^-i 
  + 
  (u^ 
  + 
  uy^)co^^^yy. 
  (35) 
  

   T^^wl^-i-C^+i 
  + 
  ^^^ 
  + 
  ^^j, 
  . 
  (36) 
  

  

  T 
  c 
  =i 
  s 
  w{K^-y)+(^ 
  +i 
  +^ 
  +i 
  )«-(i5^} 
  J 
  • 
  ( 
  37 
  ) 
  

  

  T<z=i*W 
  {i(y-^) 
  + 
  (^+* 
  + 
  uy+*)»-p^}, 
  . 
  (38) 
  

  

  RsJWII 
  + 
  uJI-^ 
  + 
  ^w 
  + 
  t^s}, 
  . 
  . 
  .(39) 
  

  

  Deflection 
  = 
  £W 
  { 
  £\fti!y 
  (N 
  2 
  + 
  a# 
  - 
  2) 
  + 
  f*s 
  2 
  #y 
  

  

  + 
  v(l 
  + 
  w)(l-w 
  x 
  )(l-^)o)|. 
  . 
  . 
  . 
  (40) 
  

  

  Here, 
  as 
  before, 
  the 
  expression 
  (39), 
  when 
  #=0 
  and 
  y 
  = 
  N 
  } 
  

   is 
  equal 
  to 
  aT 
  , 
  which 
  is 
  equal 
  to 
  aRj$. 
  

  

  These 
  results 
  appear 
  to 
  be 
  sufficiently 
  simple 
  and 
  concise. 
  

   The 
  numerical 
  values 
  of 
  u 
  and 
  co 
  can 
  readily 
  be 
  calculated 
  

   for 
  a 
  given 
  girder. 
  Referring 
  to 
  the 
  quadratic 
  equation 
  

   which 
  u 
  satisfies, 
  we 
  see 
  that 
  

  

  2u 
  2 
  /l 
  + 
  u\* 
  @ 
  

  

  9 
  \1-J 
  " 
  

  

  {l-uf 
  4 
  + 
  /3> 
  \l-uJ 
  4 
  + 
  /3 
  J 
  

  

  and 
  u 
  = 
  - 
  l-i£ 
  + 
  i/3*(4 
  + 
  £)*. 
  

  

  The 
  data 
  of 
  the 
  question, 
  admitting 
  only 
  four 
  different 
  

   sections 
  of 
  members 
  of 
  the 
  frame, 
  do 
  not 
  permit 
  the 
  sections 
  

   to 
  be 
  all 
  proportioned 
  to 
  the 
  tensions. 
  The 
  results 
  obtained 
  

   can, 
  however, 
  be 
  compared 
  with 
  those 
  given 
  by 
  other 
  less 
  

   accurate 
  methods. 
  

  

  Let 
  us 
  regard 
  the 
  girder 
  as 
  a 
  uniform 
  beam, 
  loaded 
  with 
  

   N 
  — 
  1 
  weights, 
  each 
  equal 
  to 
  W, 
  at 
  equal 
  inters'als 
  along 
  it, 
  

   and 
  apply 
  the 
  ordinary 
  Bernoulli-Euler 
  method 
  for 
  deter- 
  

   mining 
  the 
  deflection 
  at 
  any 
  point. 
  If, 
  in 
  calculating 
  the 
  

   moment 
  of 
  inertia 
  of 
  a 
  section 
  of 
  the 
  beam, 
  we 
  take 
  account 
  

   only 
  of 
  the- 
  two 
  horizontal 
  booms, 
  neglecting 
  the 
  web, 
  we 
  

   readily 
  find 
  that 
  the 
  deflection 
  at 
  a 
  point 
  at 
  a 
  distance 
  x 
  from 
  

  

  