--- title: "Stress-strain curves: linear and non-linear cases" subject: "Concrete Technology and Masonry Structure" chapter: 4 key: "semester_5/concrete_technology_and_masonry_structure:stress-strain-nonlinear-cases" source: https://www.bctengineer.com/q/semester_5/concrete_technology_and_masonry_structure/stress-strain-nonlinear-cases site: BCT Engineering --- # Stress-strain curves: linear and non-linear cases Concrete Technology and Masonry Structure · Chapter 4: Properties of hardened concrete - Asked in: 2082 Baisakh, 2+4 marks - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/concrete_technology_and_masonry_structure/stress-strain-nonlinear-cases - Whole chapter: https://www.bctengineer.com/md/semester_5/concrete_technology_and_masonry_structure/chapter-4 ## Question Draw stress-strain curves for the following cases and explain in detail the reason for the non-linearity relationship of the stress-strain curve of concrete: (i) linear and elastic, (ii) non-linear and elastic, (iii) linear and non-elastic, (iv) non-linear and non-elastic. ## Short answer Non-linear inelastic behaviour is actual concrete response. Microcracks at the interface grow above 30% of strength, joined by mortar cracks above 50–75%, progressively reducing stiffness. Creep, porosity, and stress concentrations add permanent strain, giving the curved, inelastic shape. ## Answer #### Four types of behaviour ```text (i) Linear elastic (ii) Non-linear elastic s | / s | _.-' | / | .-' | / loading and | .' loading and | / unloading on | .' unloading on | / the same line | / the same curve | / | / | / | / +-/------------------ e +/-------------------- e (iii) Linear inelastic (iv) Non-linear inelastic s | /| s | _.-'| | / | | .-' | | / | unloading | .' / unloading path | / / parallel | .' .' differs from the | / / | / .' loading path | / / | / .' | / / | / .' +-/--/----------- e +/--.'--------------- e ^ residual strain ^ residual strain ``` - **(i) Linear elastic:** stress proportional to strain, and on unloading the strain returns to zero along the same line (e.g. steel below yield). - **(ii) Non-linear elastic:** the curve is curved but unloading follows the loading path with no residual strain (e.g. rubber). - **(iii) Linear inelastic:** the loading line is straight, but on unloading a permanent strain remains (idealised plastic behaviour). - **(iv) Non-linear inelastic:** curved loading path and permanent strain on unloading; this is the **actual behaviour of concrete**. #### Reasons for non-linearity of concrete 1. **Heterogeneous, three-phase structure.** Aggregate is stiff and elastic, paste is softer and the interface is weak. Their different stiffness and strain capacity give a curved composite response. 2. **Microcracks.** Bond cracks already exist at the interface before load (from bleeding, shrinkage, temperature). Above 30% of $f_c$ they grow, and above 50 to 75% mortar cracks join them. Each new crack reduces stiffness, so the slope falls continuously. 3. **Creep and time-dependent flow of gel.** Part of the strain is viscous and not recovered, so unloading leaves residual strain. 4. **Porosity and moisture.** Water in the gel pores and capillaries moves under stress, adding to non-linear deformation. 5. **Stress concentration** at pores and voids leads to local yielding at stresses lower than the average strength. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In non-linear elastic behaviour: A. the curve is curved but unloading retraces the loading path with no residual strain B. a permanent strain remains C. the line is straight D. the unloading line is steeper Answer: A. Example: rubber. (Key point: Non-linear elastic) 2. Linear inelastic behaviour has: A. a curved loading line B. a straight loading line but permanent strain after unloading C. complete recovery D. no loading line Answer: B. Idealised plastic behaviour. (Key point: Linear inelastic) 3. Actual concrete behaviour is: A. linear and elastic B. non-linear and elastic C. linear and inelastic D. non-linear and inelastic Answer: D. Curved loading path and permanent strain. (Key point: Concrete behaviour) 4. A main reason for non-linearity is: A. constant stiffness of paste B. absence of pores C. growth of microcracks that reduce stiffness continuously D. perfect bond everywhere Answer: C. Each new crack reduces slope. 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