﻿and 
  Deflection 
  of 
  Braced 
  Girders. 
  63 
  

  

  and 
  

  

  2vr 
  0i 
  > 
  = 
  -^2(-l)"+ 
  ^tm+{-iy^$n 
  

  

  = 
  ivN{(-l) 
  N 
  + 
  l}, 
  

   thus 
  

  

  Thus 
  we 
  get 
  

  

  n_ 
  i 
  NW 
  , 
  i{i-(-i) 
  y 
  ^v+^(-i) 
  N 
  -%+^(-if-V, 
  w 
  

  

  K_ 
  4iNVV+ 
  2(v-K 
  a 
  2* 
  + 
  *%*)" 
  " 
  

  

  = 
  -i(N- 
  7 
  )W 
  (43) 
  

  

  When 
  N 
  is 
  even, 
  S(-l) 
  N 
  " 
  <r 
  X 
  <r 
  andS(-l) 
  N 
  ">cr 
  are 
  hoth 
  zero 
  

   on 
  account 
  of 
  the 
  symmetry 
  about 
  the 
  vertical 
  line 
  through 
  

   the 
  centre 
  of 
  the 
  girder, 
  and 
  thus 
  y 
  is 
  zero. 
  

   When 
  N 
  is 
  odd, 
  

  

  Ny-^(-l>;w-^(-l)>, 
  

   ^- 
  i 
  v 
  + 
  ftX 
  + 
  s'lp, 
  •• 
  • 
  (44 
  -> 
  

  

  The 
  values 
  of 
  the 
  remaining 
  tensions 
  can 
  readily 
  be 
  de- 
  

   duced 
  ; 
  they 
  are 
  as 
  follows 
  : 
  — 
  

  

  , 
  1^-iWti(«r-l)(N-»)+*N+i(-l)M. 
  

  

  T 
  =-iW(N- 
  7 
  ). 
  

  

  These 
  expressions 
  for 
  the 
  tensions 
  do 
  not 
  depend 
  upon 
  the 
  

   assumption 
  of 
  any 
  symmetry 
  about 
  the 
  vertical 
  through 
  the 
  

   centre 
  of 
  the 
  girder, 
  except 
  that 
  the 
  two 
  end 
  vertical 
  bars 
  are 
  

   equal, 
  and 
  to 
  suppose 
  them 
  to 
  be 
  unequal 
  would 
  not 
  introduce 
  

   any 
  complication. 
  

  

  If 
  v 
  is 
  not 
  too 
  great 
  compared 
  with 
  the 
  X's 
  and 
  /as, 
  that 
  is 
  

   to 
  say 
  if 
  the 
  vertical 
  bars 
  are 
  strong 
  enough, 
  we 
  may 
  expect 
  

   7 
  to 
  be 
  of 
  a 
  magnitude 
  less 
  important 
  than 
  N 
  ; 
  and 
  so 
  we 
  can 
  

   choose 
  the 
  sections 
  of 
  bars 
  roughly 
  proportional 
  to 
  the 
  ten- 
  

   sions 
  throughout, 
  and 
  then 
  calculate 
  7 
  and 
  determine 
  the 
  

   exact 
  values 
  of 
  the 
  tensions, 
  and 
  so 
  decide 
  whether 
  it 
  is 
  

  

  