﻿and 
  Deflection 
  of 
  Braced 
  Girders. 
  55 
  

  

  JWW» 
  |£(N»-m»-P-i) 
  +iW^ 
  

  

  4- 
  1 
  Wy 
  1 
  + 
  " 
  / 
  (1 
  -«") 
  (<-«*) 
  (ft 
  + 
  ^»"+ 
  1 
  ) 
  (0W-* 
  + 
  fo 
  tM+I) 
  

  

  + 
  ^ 
  (V* 
  + 
  ««) 
  (1 
  -« 
  N 
  ) 
  + 
  || 
  (u"-«") 
  - 
  J 
  («*-«•) 
  ] 
  , 
  

   1 
  4-u 
  

  

  + 
  iWv 
  T 
  I 
  —(u 
  k 
  - 
  m 
  —u 
  m 
  - 
  k 
  ) 
  (32) 
  

  

  A 
  it 
  

  

  We 
  can 
  obtain 
  another 
  expression 
  for 
  the 
  deflection 
  at 
  a 
  point 
  

   at 
  which 
  k 
  >mhj 
  interchanging 
  m, 
  n 
  and 
  I, 
  k 
  throughout 
  the 
  

   expression 
  (31). 
  

  

  We 
  are 
  now 
  in 
  a 
  position 
  to 
  find 
  by 
  a 
  process 
  of 
  summation 
  

   the 
  tensions 
  of 
  the 
  bars 
  and 
  the 
  deflection 
  at 
  any 
  point 
  for 
  the 
  

   case 
  of 
  uniform 
  loading, 
  a 
  weight 
  W 
  being 
  attached 
  to 
  each 
  

   of 
  the 
  lower 
  joints, 
  the 
  total 
  load 
  being 
  (N 
  — 
  1)W. 
  It 
  will 
  be 
  

   seen 
  that 
  the 
  results 
  in 
  this 
  case 
  are, 
  in 
  consequence 
  of 
  the 
  

   symmetry 
  of 
  the 
  loading, 
  somewhat 
  simpler 
  than 
  those 
  

   obtained 
  for 
  the 
  case 
  of 
  a 
  single 
  load. 
  To 
  obtain 
  B 
  p 
  for 
  this 
  

   case 
  we 
  have 
  to 
  sum 
  the 
  values 
  of 
  the 
  expression 
  (25) 
  for 
  a 
  

   single 
  load 
  for 
  all 
  values 
  of 
  m 
  from 
  1 
  to 
  N 
  — 
  1, 
  n 
  varying 
  from 
  

   N-ltol. 
  The 
  result 
  is 
  

  

  iw 
  jltg't-ff:'.) 
  { 
  *.-'-+»--') 
  S" 
  - 
  (N-l).-.(l-.) 
  } 
  

  

  which 
  reduces 
  to 
  

  

  (1 
  + 
  iQ(hp 
  + 
  !**-*) 
  fN-1 
  4> 
  + 
  W 
  \ 
  1W 
  /1+*V. 
  m] 
  

  

  this 
  is 
  «T 
  , 
  which 
  is 
  equal 
  to 
  aR^j 
  when 
  p 
  is 
  or 
  N. 
  

  

  We 
  have 
  some 
  simple 
  relations 
  as 
  before 
  between 
  the 
  other 
  

   tensions, 
  but 
  these 
  it 
  is 
  hardly 
  necessary 
  to 
  use, 
  for 
  the 
  

   principal 
  summation 
  is 
  the 
  same 
  for 
  all 
  of 
  them, 
  that 
  is 
  to 
  say 
  

   we 
  have 
  to 
  evaluate 
  

  

  I 
  m=TS— 
  1 
  1 
  m=p— 
  1 
  

  

  —^— 
  S 
  (Ai^-p-^ 
  + 
  Bup-")— 
  fW-= 
  t 
  (u 
  m 
  -P 
  +1 
  —u^ 
  m 
  ). 
  

  

  Call 
  this 
  expression 
  S, 
  then 
  

  

  n=N— 
  p 
  ^ 
  m=p— 
  1 
  

  

  T 
  op 
  =- 
  S 
  (f)-i)^«W- 
  2 
  (N-p 
  + 
  iJ^tW 
  + 
  ffi 
  

  

  n=l 
  x> 
  »»=1 
  -^ 
  

  

  = 
  -H(N-p+i)(p-i)-i}«W 
  + 
  t2. 
  

  

  