Page 180 - Aircraft Stuctures for Engineering Student
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164  Structural instability














                                  Y+
                  Fig. 6.10  Beam-column supporting a point load.


                  Eq. (6.34) becomes

                                              d2v
                                                           Wa
                                             -+A     v= --
                                              dz2         EII
                  the general solution of which is
                                                               Wa
                                         w = A cos Xz + Bsin XZ - -z                 (6.36)
                                                               PI
                  Similarly, the general solution of Eq. (6.35) is
                                                           W
                                                             (I
                                    v = C cos Xz + D sin Az - - - a)(, - z)          (6.37)
                                                          PI
                  where A, By C and D are constants which are found from the boundary conditions as
                  follows.
                    When z = 0, v = 0, therefore from Eq. (6.36) A  = 0. At z = I, w = 0 giving, from
                  Eq.  (6.37),  C = -DtanXI.  At  the point  of  application  of  the load  the deflection
                  and slope of  the beam  given by  Eqs (6.36) and  (6.37) must  be  the  same. Hence,
                  equating deflections
                                   Wa                                          Wa
                     B sin X(1 - a) - - (I - a) = D[sin X(1-  a) - tan XI cos X(1 - a)] - - (I - a)
                                   PI                                          PI
                  and equating slopes
                                      Wa                                     W
                       BXcosX(1-a)  --=     DX[cosX(I-a)+tanXIsinX(I-a)]  +-(I-a)
                                       PI                                    PI
                  Solving the above equations for B and D and substituting for A, By C and D in Eqs
                  (6.36) and (6.37) we have
                              W sin Xa       Wa
                          V=          sink--z      for  z < I-a                      (6.38)
                              PA  sin XI      PI
                              W sin X(1 - a)           W
                          V=               sinX(I-2)--(I-a)(/-z)      for  z2l-a     (6.39)
                                 PA  sin XI           PI
                  These equations for the beam-column deflection enable the bending moment  and
                  resulting bending stresses to be found at all sections.
                    A  particular  case  arises when  the  load  is  applied  at  the  centre  of  the  span.
                  The  deflection curve  is  then  symmetrical with  a  maximum  deflection under  the
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