﻿and 
  Deflection 
  of 
  Braced 
  Girders. 
  43 
  

  

  The 
  problem 
  of 
  determining 
  the 
  tensions 
  of 
  the 
  members 
  

   of 
  a 
  stiff 
  frame 
  with 
  redundant 
  bars, 
  and 
  its 
  deformation 
  

   (assumed 
  to 
  be 
  small) 
  in 
  any 
  direction 
  under 
  the 
  action 
  of 
  

   given 
  forces 
  applied 
  at 
  the 
  joints_, 
  was 
  completely 
  solved 
  by 
  

   Clerk 
  Maxwell. 
  His 
  solution 
  was 
  published 
  in 
  the 
  Philo- 
  

   sophical 
  Magazine 
  iu 
  1864, 
  series 
  4, 
  vol. 
  xxvii., 
  and 
  is 
  

   reprinted 
  at 
  the 
  end 
  of 
  vol. 
  i. 
  of 
  his 
  collected 
  papers. 
  The 
  

   only 
  step 
  in 
  this 
  direction 
  previously 
  published 
  was 
  due 
  to 
  

   Clapeyron, 
  and 
  dealt 
  only 
  with 
  the 
  deformation 
  ot 
  a 
  frame 
  

   without 
  redundant 
  bars. 
  The 
  method 
  employed 
  is 
  sometimes 
  

   referred 
  to 
  as 
  Mohr's, 
  although 
  Mohr 
  made 
  bis 
  earliest 
  con- 
  

   tributions 
  to 
  5 
  the 
  subject 
  some 
  years 
  after 
  the 
  publication 
  of 
  

   Maxwell's 
  solution, 
  and 
  then 
  only 
  attempted 
  some 
  particular 
  

   examples 
  of 
  very 
  simple 
  character. 
  

  

  The 
  application 
  of 
  Maxwell's 
  solution 
  to 
  any 
  practical 
  case 
  

   involves 
  no 
  difficulty, 
  though 
  the 
  calculations 
  may 
  be 
  rather 
  

   long. 
  Several 
  examples 
  of 
  such 
  applications 
  have 
  been 
  given 
  

   from 
  time 
  to 
  time 
  by 
  Mr. 
  Max 
  am 
  Ende 
  in 
  the 
  ' 
  Engineer.' 
  

   The 
  primary 
  practical 
  use 
  of 
  the 
  method 
  is 
  for 
  such 
  applica- 
  

   tions 
  to 
  examples 
  with 
  particular 
  numerical 
  data, 
  and 
  in 
  this 
  

   connexion 
  it 
  deserves 
  rather 
  more 
  attention 
  than 
  it 
  has 
  received. 
  

   General 
  results, 
  sufficiently 
  concise 
  to 
  be 
  of 
  interest, 
  can 
  only 
  

   be 
  obtained 
  for 
  frames 
  of 
  rather 
  simple 
  and 
  symmetrical 
  types 
  

   with 
  simple 
  arrangements 
  of 
  loading. 
  The 
  object 
  of 
  the 
  

   present 
  investigation 
  is 
  to 
  obtain 
  from 
  Maxwell's 
  equations 
  

   some 
  general 
  results 
  for 
  one 
  or 
  two 
  simple 
  types 
  of 
  girders. 
  

  

  It 
  may 
  be 
  convenient 
  to 
  begin 
  with 
  a 
  statement 
  of 
  Maxwell's 
  

   solution 
  of 
  the 
  problem. 
  The 
  question 
  which 
  he 
  proposes 
  is 
  

   this 
  : 
  — 
  Let 
  K, 
  L, 
  M, 
  N 
  be 
  four 
  joints 
  of 
  a 
  stiff 
  frame, 
  and 
  let 
  the 
  

   forces 
  applied 
  to 
  it 
  consist 
  of 
  a 
  tension 
  F 
  between 
  K 
  and 
  L, 
  

   in 
  consequence 
  of 
  which 
  the 
  points 
  M 
  and 
  N 
  approach 
  each 
  

   other 
  through 
  a 
  small 
  distance 
  x 
  ; 
  to 
  find 
  x 
  and 
  the 
  tensions 
  

   of 
  all 
  the 
  bars 
  of 
  the 
  frame, 
  assuming 
  that 
  the 
  strains 
  are 
  all 
  

   very 
  small, 
  and 
  that 
  Hooke's 
  law 
  is 
  applicable. 
  And 
  the 
  

   answer 
  is 
  as 
  follows 
  : 
  — 
  Select 
  out 
  of 
  the 
  bars 
  of 
  the 
  frame 
  a 
  set 
  

   which 
  form 
  a 
  just 
  stiff 
  frame, 
  connecting 
  all 
  the 
  joints 
  of 
  the 
  

   given 
  frame; 
  let 
  the 
  extensibilities 
  of 
  these 
  be 
  e 
  ly 
  e 
  2 
  , 
  &c, 
  

   extensibility 
  being 
  the 
  ratio 
  of 
  extension 
  to 
  tension, 
  or, 
  for 
  a 
  

   uniform 
  bar, 
  length 
  -f- 
  (area 
  of 
  section 
  x 
  Young's 
  modulus) 
  ; 
  

   let 
  their 
  tensions 
  be 
  T 
  l5 
  To, 
  &c. 
  ; 
  let 
  e 
  1? 
  e 
  2 
  , 
  &c. 
  be 
  the 
  extensi- 
  

   bilities 
  and 
  Hi, 
  R 
  2 
  ? 
  & 
  c 
  '« 
  the 
  tensions 
  of 
  the 
  remaining 
  or 
  re- 
  

   dundant 
  bars 
  ; 
  let 
  p 
  lt 
  p 
  2 
  , 
  &c. 
  be 
  the 
  tensions 
  of 
  the 
  selected 
  

   bars, 
  taken 
  alone 
  as 
  a 
  just 
  stiff 
  frame, 
  due 
  to 
  a 
  unit 
  tension 
  

   between 
  K 
  and 
  L, 
  g 
  ly 
  q 
  2 
  , 
  &c. 
  their 
  tensions 
  due 
  to 
  a 
  unit 
  

   tension 
  between 
  M 
  and 
  N, 
  r/, 
  r{' 
  y 
  & 
  c 
  « 
  their 
  tensions 
  due 
  to 
  

   a 
  unit 
  tension 
  in 
  the 
  line 
  of 
  the 
  first 
  redundant 
  bar, 
  r 
  2 
  , 
  r 
  2 
  \&<\ 
  

  

  