--- title: "Linear and non-linear behaviour of structures" subject: "Theory of Structures I" chapter: 1 key: "semester_4/theory_of_structures_i:linear-nonlinear-behaviour" source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/linear-nonlinear-behaviour site: BCT Engineering --- # Linear and non-linear behaviour of structures Theory of Structures I · Chapter 1: Introduction - Asked in: 2073 Magh, 4 marks; 2069 Poush, 6 marks; 2075 Baisakh, 4 marks (asked 3 times) - Video: [Linear & Non Linear Behavior of Structures](https://www.youtube.com/watch?v=6AkDziDPNzg) — P GRID - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/linear-nonlinear-behaviour - Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-1 ## Question Differentiate between linear and non-linear behaviour of structures with suitable force-displacement diagrams and explain their uses in theory 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. ## Answer A structure shows **linear behaviour** when displacements are directly proportional to the applied loads, and **non-linear behaviour** when this proportionality does not hold. ```text Force Force | / | ___--- | / | _-- | / slope = k | / slope changes | / |/ +-------------- Disp. +-------------- Disp. (a) Linear (b) Non-linear ``` | Point | Linear behaviour | Non-linear behaviour | |---|---|---| | Force-displacement | Straight line through origin | Curve, slope (stiffness) changes with load | | Stiffness $k$ | Constant | Varies with load or displacement | | Principle of superposition | Valid | Not valid | | Governing equations | Linear algebraic equations | Non-linear equations, solved iteratively | | Result with double load | Response doubles | Response is not doubled | | Causes | Hooke's law, small displacement | Material yielding, large displacement, changing support conditions | | Analysis effort | Simple, one-step | Load applied in increments, Newton-Raphson iteration | | Unloading | Returns along the same line | Residual 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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In linear behaviour the force-displacement curve is: A. A straight line through the origin with constant stiffness B. A curve whose slope increases with load C. A horizontal line after the elastic limit D. A straight line that does not pass through the origin Answer: A. Linear means displacement is proportional to load, so k is constant. (Key point: Linear behaviour) 2. Which statement about superposition is correct? A. Valid for non-linear, not for linear B. Valid for linear behaviour, not valid for non-linear C. Valid for both equally D. Not valid for either Answer: B. Superposition requires proportionality. (Key point: Superposition) 3. How is a non-linear structure typically analysed? A. One-step solution of linear algebraic equations B. Only by the moment area method C. Load applied in increments with iteration such as Newton-Raphson D. By doubling the single-load result Answer: C. Non-linear equations are solved incrementally and iteratively. (Key point: Analysis effort) 4. Non-linear analysis is mainly used to: A. Do routine service-load design of simple beams B. Draw influence lines C. Replace the use of equilibrium D. Find collapse load and ductility and design cable or slender structures Answer: D. Linear analysis covers service loads; non-linear covers collapse and post-yield behaviour. 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