Page 337 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)
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324                      Introduction to the Finite Element Method   Chap. 10










                                                                     Figure  10.5-5.

                                  can  also  add  row  3  to  row  1 and  rewrite  the  stiffness  matrix  as  a  3  X  3  matrix:

                                                                 24  6/  61
                                                     M,          6/  8/'  21"  <
                                                                 6/  21"  SI"  6, I
                                                                             V -)

                                       Comparing  the  external  loads  of Fig.  10.5-5  with  those  of the  global  system,  we
                                  have
                                                                      /  \   /
                                                    + F'lv             "i      U


                                                    M,   >=  <           > -   < -6
                                                             - ^ 2 /          -e


                                  With  ^ 2 ^ =  0 and  F, ^ =  5^,  the  stiffness  matrix in  terms of the  given coordinates  and
                                  given  loads  is

                                                            ■ 24  -6 1  -6 1
                                                          E/
                                                             -6 1  SI"  21"
                                                   ■'^2 )    -6 1  21"  SI"

                       10.6  SPRING CONSTRAINTS ON  STRUCTURE

                              In  Chapter 9,  spring constraints were  treated  by virtual work  as generalized forces.
                              The  same  concept  applies  in  the  finite  element  approach.  The  point  of application
                              of the  spring must  be  chosen  here  as  a joint  station.  Thus,  the  load  on  the  original
                              structure  in  global  coordinates  is  supplemented  by  the  spring  force.
                                  Because  the  spring force  is  always opposite  to  the  displacement,  the  force  or
                              moment  at  the  joint  is  decreased  by  -/cr,   -KO^.  Thus,  these  terms,  when
                              shifted  to  the  other  side  of  the  equation,  become  additions  to  the  corresponding
                              stiffness  term.
                              Example  10.6-1
                                  Determine  the  stiffness  matrix  for  the  uniform  beam  with  a  linear  and  rotational
                                  spring,  as  shown  in  Fig.  10.6-l(a).
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