Guiding Rules in the Conjugate Beam Method

Without adequate guiding rules and a good number
of typical examples, the conjugate beam method may likely appear as in accessible or esoteric
to many beginners. On the other hand, most beginners are pleasantly surprised to learn that there
are only two major steps in this method. The first step  is to set up an  additional beam, called
conjugate beam, besides the actual beam. The  second step  is to determine the “ shearing forces ”
and “bending moments ” in the conjugate beam using mainly concepts and skills in statics. In
the process, these two steps are most  effectively guided by the set of ten   rules synthesized by
Jong.

 These rules are natural and logical  extensions of the method using  moment-area theorems.

   Rule 1: The conjugate beam and the actual beam are of the  same length.

   Rule 2:  The  loading on the conjugate beam  is simply the distributed  elastic weight ,
which is given by the bending moment  M  in the actual beam  divided by the flexural
rigidity EI  of the actual beam. (The  elastic weight ,  M/EI, points upward if the  bending moment  is
positive — to cause the top fiber in compression — in beam convention.)


For each existing  support condition  of the actual beam, there is a  corresponding suppor
condition for the conjugate beam. The correspondence is given by  rules 3  through  7 listed i
Table 1, where a simple support is either a roller support or a hinge support, since a beam i
usually not subjected to axial loads.

Ta b l e  1  Corresponding support conditi on for the conjugate beam

  Existing support condition of the  actual beam      Corresponding  support condition for the  conjugate beam

 
Rule 3:  Fixed end  .................................................................Free end
Rule 4:  Free end .................................................................... Fixed end
Rule 5:  Simple support at the end  .............................Simple support at the end
Rule 6:  Simple support not at the end........................ Unsupported hinge
Rule 7:  Unsupported hinge ....................................... Simple support
 
The  slope and  deflection  of the actual beam are obtaine d by employing the following rules:

   Rule 8: The conjugate beam (hence its free body) is in static  equilibrium.

   Rule 9: The slope of (the centerline of ) the actual beam at any cross section is given by
the “shearing force” at that cross section of the conjugate beam. (This slope is positive, or
counterclockwise, if the “shearing force” is positive — tending to rotate the beam element clockwise — in
beam convention.)

   Rule 10:  The deflection of (the centerline of ) the actual beam at any point is given by the
“ bending moment” at that point of  the conjugate beam. (This deflection is upward if the
“bending moment” is positive — tending to cause the top fiber in compression — in beam convention. )