﻿and 
  Jjefleclion 
  of 
  Braced 
  Girders. 
  45 
  

  

  respectively. 
  Distinguish 
  the 
  four 
  bars 
  of 
  a 
  panel 
  57 
  of 
  the 
  

  

  just 
  stiff 
  frame 
  by 
  the 
  letters 
  a, 
  b, 
  c 
  y 
  d; 
  a 
  belonging 
  to 
  the 
  top 
  

   bar, 
  b 
  to 
  the 
  bottom 
  bar, 
  c 
  to 
  the 
  one 
  which 
  slopes 
  upwards 
  

   from 
  left 
  to 
  right, 
  and 
  d 
  to 
  the 
  one 
  which 
  slopes 
  downwards 
  

   from 
  left 
  to 
  right. 
  The 
  tension 
  of 
  a 
  bar 
  of 
  the 
  just 
  stiff 
  frame 
  

   being 
  denoted 
  by 
  T 
  with 
  a 
  suitable 
  suffix, 
  T 
  al 
  is 
  written 
  for 
  

   the 
  tension 
  of 
  the 
  bar 
  a 
  in 
  the 
  panel 
  1. 
  The 
  jo's 
  of 
  Maxwell's 
  

   equation 
  can 
  be 
  distinguished 
  by 
  the 
  same 
  double 
  suffixes, 
  

   and 
  the 
  r's 
  by 
  triple 
  suffixes, 
  of 
  which 
  the 
  first 
  indicates 
  the 
  

   redundant 
  bar 
  referred 
  to, 
  the 
  second 
  the 
  position 
  in 
  the 
  

   panel, 
  while 
  the 
  third 
  is 
  the 
  suffix 
  of 
  the 
  panel. 
  The 
  bar 
  is 
  

   of 
  course 
  an 
  exceptional 
  case, 
  requiring 
  only 
  the 
  single 
  

   suffix 
  0. 
  

  

  A 
  number 
  of 
  relations 
  among 
  the 
  tensions 
  can 
  be 
  written 
  

   down 
  without 
  reference 
  to 
  the 
  strengths 
  of 
  the 
  bars 
  ; 
  in 
  fact 
  

   the 
  whole 
  number 
  of 
  bars 
  being 
  5N 
  + 
  1, 
  we 
  can 
  find 
  4N 
  + 
  1 
  

   such 
  relations 
  independent 
  of 
  one 
  another. 
  

  

  Using 
  the 
  method 
  of 
  sections 
  we 
  get 
  

  

  p 
  = 
  1 
  to 
  N 
  T 
  ap 
  + 
  T 
  6p 
  + 
  (T 
  cp 
  + 
  T 
  dp 
  ) 
  sin 
  6 
  = 
  (N 
  equations), 
  . 
  (4) 
  

  

  p 
  = 
  ltom 
  T 
  cp 
  -T, 
  p 
  =- 
  5 
  ^W 
  j 
  

  

  Y 
  (N 
  equations), 
  . 
  » 
  . 
  . 
  (5) 
  

   p 
  = 
  wi 
  + 
  ltoN 
  T 
  cp 
  -T, 
  p 
  = 
  5 
  ^W 
  I 
  

  

  p=ltom 
  T 
  6p 
  -T 
  flp 
  =(2/>-l)4w 
  ] 
  

  

  equations), 
  (6) 
  

  

  

  p 
  = 
  m 
  + 
  ltoN 
  T 
  Jp 
  -T 
  ap 
  = 
  j2(N-p) 
  + 
  l^W 
  I 
  

  

  To 
  + 
  T^cos^O 
  N 
  

  

  n 
  

  

  p=ltom 
  R 
  p 
  + 
  (T 
  dp 
  + 
  T* 
  p+ 
  i) 
  cos 
  = 
  ~ 
  W 
  

  

  p=m 
  + 
  l 
  toN-1 
  R 
  p+ 
  (T, 
  p 
  + 
  T,, 
  p+1 
  )cos^=-^W 
  ) 
  ( 
  N 
  + 
  le 
  ^ 
  ati 
  ™s)- 
  (?) 
  

  

  iN 
  / 
  

  

  We 
  want 
  N 
  more 
  equations 
  to 
  enable 
  us 
  to 
  determine 
  the 
  

   tensions 
  of 
  all 
  the 
  bars 
  of 
  the 
  frame, 
  and 
  these 
  are 
  furnished 
  

  

  by 
  Maxwell's 
  system 
  of 
  N 
  linear 
  equations 
  for 
  Rj 
  R 
  2 
  R^. 
  

  

  Let 
  us 
  now 
  solve 
  these 
  equations, 
  replacing 
  F 
  by 
  W. 
  

  

  The 
  first 
  step 
  is 
  to 
  calculate 
  the 
  coefficients 
  for 
  our 
  particular 
  

   case. 
  The 
  r's 
  and 
  p's 
  are 
  readily 
  found, 
  the 
  only 
  difficulty 
  

   being 
  to 
  write 
  them 
  down 
  systematically, 
  with 
  due 
  regard 
  to 
  

  

  