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Linear and non-linear behaviour of structures

Short answer

Linear behaviour has a straight line force-displacement graph through the origin, constant stiffness, valid superposition, and double load gives double response. Non-linear behaviour has a curved graph, changing stiffness and no superposition; it arises from material yielding/cracking or large displacements and needs incremental or iterative solution. Linear analysis underlies classical methods; nonlinear analysis is used for collapse, slender and cable structures.

Differentiate between linear and non-linear behaviour of structures with suitable force-displacement diagrams and explain their uses in theory of structures.

A structure shows linear behaviour when displacements are directly proportional to the applied loads, and non-linear behaviour when this proportionality does not hold.

 Force                       Force
  |        /                  |       ___---
  |      /                    |   _--
  |    /  slope = k           | /   slope changes
  |  /                        |/
  +-------------- Disp.       +-------------- Disp.
   (a) Linear                  (b) Non-linear
PointLinear behaviourNon-linear behaviour
Force-displacementStraight line through originCurve, slope (stiffness) changes with load
Stiffness kkConstantVaries with load or displacement
Principle of superpositionValidNot valid
Governing equationsLinear algebraic equationsNon-linear equations, solved iteratively
Result with double loadResponse doublesResponse is not doubled
CausesHooke's law, small displacementMaterial yielding, large displacement, changing support conditions
Analysis effortSimple, one-stepLoad applied in increments, Newton-Raphson iteration
UnloadingReturns along the same lineResidual deformation may remain (if inelastic)

Types of non-linearity

  • Material non-linearity: stress-strain curve is not straight (steel after yield, concrete).
  • Geometric non-linearity: large displacements change the geometry, e.g. cables, slender columns.

Uses in theory of structures

  • Linear analysis is the basis of most classical methods (moment area, virtual work, influence lines, slope-deflection), used for service-load design.
  • Superposition allows complex loadings to be solved as combinations of simple cases.
  • Non-linear analysis is used to find collapse load and ductility, to design slender and cable structures, and to study post-yield behaviour in earthquake design.

✓ Answer checked by Claude Opus 5.5 (high effort)

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