﻿and 
  Deflection 
  of 
  Braced 
  Girders. 
  61 
  

  

  Then 
  

  

  r 
  =(- 
  l)*+i, 
  ra(r=?v= 
  (_i)^- 
  % 
  *•«, 
  = 
  *•*,= 
  -(-1)*-**, 
  

  

  and 
  for 
  a 
  single 
  load 
  at 
  the 
  lower 
  joint 
  ra, 
  n, 
  the 
  p's 
  are 
  the 
  

   same 
  as 
  before. 
  For 
  the 
  single 
  load 
  W, 
  R 
  is 
  given 
  by 
  the 
  

   equation 
  

  

  R(e 
  + 
  & 
  2 
  ) 
  + 
  W%=0, 
  .... 
  (42) 
  

  

  and 
  R 
  having 
  been 
  determined, 
  all 
  the 
  other 
  tensions 
  can 
  be 
  

   written 
  down 
  at 
  once 
  by 
  resolving 
  the 
  forces 
  at 
  each 
  joint, 
  

   and 
  by 
  summing 
  we 
  can 
  deduce 
  results 
  for 
  any 
  system 
  of 
  

   loading. 
  

  

  Although 
  no 
  case 
  presents 
  any 
  difficulty, 
  it 
  is 
  well 
  to 
  assume 
  

   some 
  symmetry 
  of 
  structure 
  in 
  order 
  to 
  obtain 
  results 
  which 
  

   are 
  concise 
  enough 
  to 
  be 
  of 
  interest. 
  It 
  will 
  be 
  noticed 
  that 
  

   if 
  N 
  is 
  even 
  and 
  the 
  structure 
  and 
  loading 
  are 
  both 
  symme- 
  

   trical 
  about 
  the 
  vertical 
  line 
  through 
  the 
  centre 
  of 
  the 
  girder 
  

   (the 
  loading 
  being 
  wholly 
  on 
  the 
  lower 
  boom), 
  T 
  c 
  p- 
  and 
  

   T^pr+1 
  must 
  both 
  be 
  zero; 
  accordingly, 
  the 
  bars 
  c, 
  ^NT 
  and 
  

   rf, 
  ^N 
  -f- 
  1 
  may 
  be 
  removed 
  without 
  any 
  effect 
  upon 
  the 
  tensions 
  

   of 
  the 
  other 
  bars. 
  But 
  this 
  removal 
  reduces 
  the 
  frame 
  to 
  the 
  

   state 
  of 
  being 
  just 
  stiff, 
  so 
  all 
  the 
  tensions 
  are 
  independent 
  of 
  

   the 
  distribution 
  of 
  extensibilities, 
  and 
  can 
  be 
  found 
  by 
  re- 
  

   solving 
  the 
  forces 
  at 
  the 
  joints. 
  

  

  Let 
  us 
  suppose 
  that 
  the 
  girder 
  is 
  symmetrical 
  about 
  the 
  

   horizontal 
  line 
  through 
  its 
  centre 
  and 
  in 
  its 
  plane. 
  With 
  

   this 
  assumption 
  very 
  simple 
  expressions 
  can 
  be 
  found 
  for 
  the 
  

   tensions. 
  We 
  will 
  also 
  suppose 
  that 
  it 
  is 
  symmetrical 
  about 
  

   the 
  vertical 
  line 
  through 
  its 
  centre, 
  since 
  all 
  actual 
  girders 
  

   have 
  this 
  property. 
  We 
  have 
  then 
  e 
  a 
  <r—eb<r, 
  let 
  us 
  call 
  each 
  

   of 
  these 
  X^ 
  ; 
  we 
  have 
  also 
  ^ 
  co 
  .= 
  ^o-, 
  let 
  us 
  call 
  each 
  of 
  these 
  

   /V. 
  And 
  by 
  virtue 
  of 
  the 
  symmetry 
  about 
  the 
  vertical 
  line 
  

   Xo-=^x-o-+ij 
  and 
  /ub 
  cr 
  =fi^_ 
  <T+ 
  i 
  ; 
  also 
  e 
  = 
  e, 
  let 
  us 
  call 
  each 
  of 
  

   these 
  v. 
  

  

  Then 
  for 
  a 
  single 
  load 
  at 
  the 
  lower 
  joint 
  m, 
  n, 
  we 
  have 
  

  

  -| 
  + 
  [2JJ2 
  \ 
  + 
  ^2 
  A. 
  

  

  -*{j-wK*+*;< 
  +/ 
  

  

  m 
  

  

  = 
  vr 
  p 
  + 
  |j 
  (t*t\ 
  + 
  s 
  2 
  Sfi) 
  

  

  m 
  m 
  

  

  _(-i)»(f*2 
  x 
  + 
  s*t 
  fi), 
  

  

  