Material and geometrical non-linearity

Non-linearity arises when displacement is not proportional to load. The two main sources are the material and the geometry.

Material non-linearity

The stress-strain relation of the material is not linear, so the stiffness changes with stress even when displacements are small.

• Examples: steel beyond the yield point, concrete in compression, cracked concrete, soil.
• Equilibrium is still written on the undeformed geometry.

 stress
   |         ______________  yield (plastic)
   |       /
   |     /
   |   /   elastic
   | /
   +------------------------ strain
     elastic-plastic curve

Geometric non-linearity

Displacements are large enough to change the geometry of the structure, so equilibrium must be written on the deformed shape, even if the material is linear.

• Examples: slender columns (P-delta effect), cables and suspension bridges, thin shells, arches near buckling.

        P                     Load
        |   |<- Delta ->|      |            ________ approaches Pcr
        v                      |        _.-'
        .   .                  |     .-'
        |    .                 |   /
        |    .  <- H           |  /   stiffness falls
        |   .                  | /    as load rises
        |  .                   |/
      __|__                    +-------------------- deflection
      /////
   slender column: base moment = H*h + P*Delta

Here the extra moment $P\Delta$ makes the load-deflection curve non-linear: the stiffness decreases as the axial load increases.