8-5-1 Straight Beams

The displacement of a beam carrying a concentrated force P = 1 is known as the Green's function or influence function for the displacement. When the Green's function for the displacement is given, one can find, by means of integration, the displacement of the same beam due to an arbitrary load. Let us consider, for example, a single-span beam with still unspecified boundary conditions, carrying the load P = 1 at x = a (Fig. 8-54). We shall derive the expression for the Green's function by integrating the differential equation for deflection [21, p. 273],

Figure 8-54

We begin our analysis by representing the concentrated load in the form (see Section 8-1-3)

Substituting Eq. (8-200) into Eq. (8-199), we get

By integrating this and using Eqs. (8-15) and (8-21) we obtain

Consecutive integrations of Eq. (8-202) give

Next the constants C ₁, C ₂, C ₃, and C ₄ are determined for various cases of support.

1. Simply supported beam The boundary conditions are

   

   Therefore,

   

   so that

   

   Consequently the Green's function for a simply supported beam is

   

2. Beam clamped on both ends Now with the boundary conditions

   

   we obtain

   

   Solving these equations we get

   

   so that the Green's function becomes

   

This and other results obtained in a similar way are listed in the Table 8-3. It can be shown, and this is left for the reader, that G(