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3
d y Q y   x
 , (6.34)
dx 3 EJ z
4
d y q   x
 . (6.35)
dx 4 EJ z
The deflection y (x) can be found by solving any of the
equations (6.33) - (6.35), depending on whether the value M , Q
z y
or q set and what is more convenient from a mathematical point
of view.
For example, following the integration of (6.33) once we obtain
the equation for the angles of rotation
dy 1
   x    M z   x dx C . (6.36)
dx EJ
z
Integrating the second time, we get the equation for the
deflection
1
dx M
y
  x    z   x dx Cx D  , (6.37)
EJ
z
where C and D – constant of integration, found from the
boundary conditions of a beam and boundary conditions within its
areas.
Keep in mind that the equation (6.33) can be used only in the
case of small displacements. Exact differential equations used in
determining displacements in very flexible beams has the form
y   x
EJ  M   x ,
z 3 z (6.38)
 1   2  2
y
 
where the known expression from differential geometry for
calculating the curvature of the line is used
1 y
 .
  y 2  3 2
 1   
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