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action. For example, for the deflections and moments
we obtain

For a distributed load the summation should be replaced
by the integral.

n P n P
z 
y (x y )  3 i  1  x i  ; M  i  2  x i  .
i 1 8 EJ z i 1 4

8.4 Short beams on elastic foundation

In the calculation of short beams it is convenient to use
the method of initial parameters. While any number of sites, this
method allows the constant integrations to express in terms of the
initial parameters y , , M and Q - the deflection, the rotation
0 0 0 0
angle, the moment and the transverse force at the origin.
We write the solution of equation (8.3) in the form

y   x  AK  x  BK  x  CK  x  
1 2 3
(8.25)
 DK  x  y x *   ,
4
where A , B ,C , D - constants, which are determined from the
boundary conditions of the problem; - A.N.Krylov’s functions
that have the form
K   chx  x  cos ; x 
1
1
K  x  ch x sin x  sh x cos x ;
2
2
1 (8.26)
K  x  sh x sin ; x 
3
2
1
K   x  ch x sin x  sh x cos . x 
4
2
Krylov’s rule of function differentiation is schematically
shown in Fig. 8.5. The arrow in the diagram indicates where the
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