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From fig. 6.7 we see that
dx
d  . (6.45)

Substituting the equation (6.45) in (6.44) and taking into account
Hooke's law under bending (6.10), we obtain
1 M 2
dU  z dx . (6.46)
п
2 EJ
z
The full potential energy that accumulates a beam with the
length l , written in the form
1 M 2
U   z dx . (6.46)
п
2 EJ
l z
If M  const , then the equation (6.46) takes the form
z
2
1 M l
U  z . (6.47)
п
2 EJ z

Questions for self-assessment

1. Give the definition of direct cross bending?
2. What is the pure bending?
3. What are the rules of signs are made for internal force factors?
4. How do you calculate the cross force in any section of the
beam?
5. How do you calculate the bending moment in an arbitrary cross-
section of the beam?
6. Write the differential equations of balance beams.
7. What is the neutral axis of the beam?
8. What is the curvature of the beam in pure bending?
9. How do you determine normal stresses in the cross section of
the beam?
10. How do the normal stresses change along the beam height?
11. What is a cross-sectional hardness in bending?

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