﻿460 
  Dr. 
  J. 
  Morrow 
  on 
  the 
  Lateral 
  Deflexion 
  and 
  

  

  Section 
  V. 
  — 
  Loaded 
  Bars 
  ; 
  Mass 
  of 
  Bar 
  Neglected, 
  

  

  § 
  15. 
  When 
  the 
  mass 
  o£ 
  the 
  bar 
  is 
  neglected 
  and 
  vibrations 
  

   occur 
  under 
  the 
  action 
  of 
  a 
  concentrated 
  load 
  only, 
  the 
  exact 
  

   solutions 
  can 
  be 
  determined 
  directly. 
  

  

  If 
  there 
  be 
  no 
  longitudinal 
  force 
  and 
  the 
  mass 
  m 
  be 
  at 
  the 
  

   directed 
  end, 
  the 
  equation 
  of 
  equilibrium 
  is 
  

  

  m.'^^ 
  = 
  -~my,{l-x)-M. 
  

   12 
  EI 
  

  

  and 
  

  

  /c^ 
  

  

  (21) 
  

  

  § 
  16. 
  Longitudinal 
  Thrust. 
  — 
  If 
  in 
  addition 
  to 
  the 
  load 
  m 
  

   there 
  be 
  a 
  longitudinal 
  thrust 
  T 
  acting 
  on 
  the 
  massless 
  bar 
  

   as 
  represented 
  in 
  figure 
  5, 
  

  

  Fio:. 
  5. 
  

  

  By 
  virtue 
  of 
  the 
  conditions 
  that 
  at 
  x 
  = 
  0, 
  ?/ 
  = 
  -^ 
  = 
  0, 
  

  

  and, 
  at 
  x 
  = 
  I, 
  j 
  — 
  0? 
  ^^ 
  solution 
  may 
  be 
  written 
  

  

  mz/i 
  rcosnZ— 
  1 
  .^ 
  . 
  , 
  . 
  "1 
  

  

  ?/ 
  = 
  — 
  ^- 
  W 
  — 
  -. 
  7— 
  (1 
  —cos 
  nx) 
  +n.t' 
  — 
  sm 
  nx 
  > 
  

  

  ^ 
  nT 
  \ 
  sm 
  nl 
  ^ 
  ^ 
  ^ 
  J 
  

  

  in 
  which 
  n^ 
  = 
  T/EI 
  ; 
  and 
  putting 
  x 
  = 
  I, 
  

  

  T 
  nl 
  sin 
  nl 
  

  

  P 
  = 
  

  

  ml 
  2—2 
  cos 
  nl 
  — 
  nl 
  sin 
  nl 
  ' 
  

  

  ■ 
  (22) 
  

  

  This 
  result 
  might 
  have 
  been 
  deduced 
  from 
  that 
  for 
  a 
  bar 
  

   clamped 
  at 
  each 
  end 
  with 
  a 
  load 
  at 
  the 
  centre 
  ^. 
  

  

  * 
  Phil. 
  Mag. 
  September 
  1906, 
  p. 
  243. 
  

  

  