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terms concentrated moment M that is located at a distance a
M
from the left end of the beam, will be multiplied by the multiplier
0
x a M  equal to one. Thus, the equation of bending moments
will have the form

M  x a  0 P  x a p  1  q x a q  2
M   x  M  P x  M   ,
z 0 0 a p a q a
0! 1! 2!
M
where a , a , a – abscissas of points of applying in respect
M P q
with the concentrated moment M , concentrated force P and
points of beginning of the action of the distributed load q .
Present the resulting expression in generalized form:

 M , if k  0;
k
  x a  
M   x  M  P x   , where    P , if k  1;
z 0 0 a 
! k 
 q , if k  2.
We write the differential equation of beam curved axis and
integrate it twice (perform the integration without opening
brackets):
  x a  k
EJ y  M  P x   ,
z 0 0 a 
! k
P x 2   x a  k 1
EJ    x  M x  0    C , (6.39)
z 0 a 
2 (k  1)!
M x 2 P x 3   x a  k 2
EJ y   x  0  0    Cx D . (6.40)
z a 
2 6 (k  2)!
We see that x  : C  EJ    0  EJ  – angle of rotation of
0
z z 0
a cross-section at the origin (in units of hardness);
D  EJ y   0  EJ y – beam deflection at the origin.
z z 0
So, universal equations for the angles of rotation of the beam
deflection are as follows:
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