Validity of bending equations under different conditions

[mtrl]

Hi all,

I want to ask you a question about the bending moment equation:

σ= M * y / I

Does this equation have the same form if we assume,

-Plasticity instead of elasticity.
-Viscoelasticity instead of elasticity.
-Nonlinearity instead of linearity.
-Anisotropy instead of isotropy.

Can you comment on the validity of the equation with these assumptions please.

Thanks.

 

[SteamKing]

All of the qualities you mentioned (plasticity, viscoelasticity, nonlinearity, ans anisotropy) cause the Euler-Bernoulli bending equation to deviate from an accurate solution.

This equation pre-supposes linear elastic behavior of the beam material with the added requirement that the slope of the beam under bending is also << 1.

The E-B equation, by the fact of its linearity, is quite useful for calculation because it permits the use of superposition when calculating the effect of multiple simultaneous beam loadings.

For materials which are not linearly elastic, different methods must be employed to calculate beam responses.