﻿EQUILIBRIUM 
  OF 
  ELASTIC 
  SOLIDS. 
  115 
  

  

  the 
  amount 
  of 
  sliding 
  is 
  determined 
  by 
  the 
  linear 
  elasticity 
  of 
  the 
  substance. 
  The 
  

   deflection 
  of 
  the 
  beam 
  thus 
  arises 
  partly 
  from 
  the 
  bending 
  of 
  the 
  whole 
  beam, 
  and 
  

   partly 
  from 
  the 
  sliding 
  of 
  the 
  planks 
  ; 
  and 
  since 
  each 
  of 
  these 
  deflections 
  is 
  small 
  

   compared 
  with 
  the 
  length 
  of 
  the 
  beam, 
  the 
  total 
  deflection 
  will 
  be 
  the 
  sum 
  of 
  the 
  

   deflections 
  due 
  to 
  bending 
  and 
  sliding. 
  

  

  Let 
  A= 
  

  

  =Mc 
  = 
  EL 
  2 
  dy 
  . 
  . 
  (65.) 
  

  

  A 
  is 
  the 
  stiffness 
  of 
  the 
  beam 
  as 
  found 
  in 
  Case 
  V., 
  the 
  equation 
  of 
  the 
  trans- 
  

   verse 
  section 
  being 
  expressed 
  in 
  terms 
  of 
  x 
  and 
  y, 
  y 
  being 
  measured 
  from 
  the 
  

   neutral 
  surface. 
  

  

  Let 
  a 
  horizontal 
  beam, 
  whose 
  length 
  is 
  2 
  /, 
  and 
  whose 
  weight 
  is 
  2 
  w, 
  be 
  sup- 
  

   ported 
  at 
  the 
  extremities 
  and 
  loaded 
  at 
  the 
  middle 
  with 
  a 
  weight 
  W. 
  

  

  Let 
  the 
  deflection 
  at 
  any 
  point 
  be 
  expressed 
  by 
  8 
  l 
  y, 
  and 
  let 
  this 
  quantity 
  be 
  

   small 
  compared 
  with 
  the 
  length 
  of 
  the 
  beam. 
  

  

  At 
  the 
  middle 
  of 
  the 
  beam, 
  8 
  1 
  y 
  is 
  found 
  by 
  the 
  usual 
  methods 
  to 
  be 
  

  

  8 
  ^ 
  = 
  T(^i 
  pw+ 
  l^ 
  w 
  ) 
  • 
  • 
  • 
  • 
  ( 
  66 
  ) 
  

  

  Let 
  B 
  =-=-lxdy= 
  ^ 
  (sectional 
  area). 
  . 
  . 
  (67.) 
  

  

  B 
  is 
  the 
  resistance 
  of 
  the 
  beam 
  to 
  the 
  sliding 
  of 
  the 
  planks. 
  The 
  deflection 
  

   of 
  the 
  beam 
  arising 
  from 
  this 
  cause 
  is 
  

  

  *»y=o 
  (w+W) 
  (68,) 
  

  

  The 
  quantity 
  is 
  small 
  compared 
  with 
  8 
  l 
  y, 
  when 
  the 
  depth 
  of 
  the 
  beam 
  is 
  

   small 
  compared 
  with 
  its 
  length. 
  

  

  The 
  whole 
  deflection 
  &y=8 
  r 
  y 
  + 
  8 
  2 
  y 
  

  

  l 
  3 
  /5 
  „A 
  I 
  

  

  A 
  * 
  = 
  6a(4 
  W 
  + 
  W 
  ) 
  + 
  2^ 
  + 
  W 
  ) 
  

  

  A 
  ^ 
  = 
  W 
  (24A 
  + 
  2b)' 
  +W 
  (6A 
  + 
  2b) 
  " 
  ( 
  69 
  '> 
  

  

  Case 
  XIII. 
  

  

  When 
  the 
  values 
  of 
  the 
  compressions 
  at 
  any 
  point 
  have 
  been 
  found, 
  when 
  

   two 
  different 
  sets 
  of 
  forces 
  act 
  on 
  a 
  solid 
  separately, 
  the 
  compressions, 
  when 
  the 
  

   forces 
  act 
  at 
  the 
  same 
  time, 
  may 
  be 
  found 
  by 
  the 
  composition 
  of 
  compressions, 
  

   because 
  the 
  small 
  compressions 
  are 
  independent 
  of 
  one 
  another. 
  

  

  It 
  appears 
  from 
  Case 
  I., 
  that 
  if 
  a 
  cylinder 
  be 
  twisted 
  as 
  there 
  described, 
  the 
  

   compressions 
  will 
  be 
  inversely 
  proportional 
  to 
  the 
  square 
  of 
  the 
  distance 
  from 
  

   the 
  centre. 
  

  

  VOL. 
  XX. 
  PART 
  I. 
  2 
  H 
  

  

  