--- title: "Discrete vs continuum structure; stability" subject: "Applied Mechanics" chapter: 6 key: "semester_1/applied_mechanics:discrete-continuum-structure-stability-determinacy" source: https://www.bctengineer.com/q/semester_1/applied_mechanics/discrete-continuum-structure-stability-determinacy site: BCT Engineering --- # Discrete vs continuum structure; stability Applied Mechanics · Chapter 6: Analysis of Beams and Frames - Asked in: 2067 Asar, 6 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_1/applied_mechanics/discrete-continuum-structure-stability-determinacy - Whole chapter: https://www.bctengineer.com/md/semester_1/applied_mechanics/chapter-6 ## Question Define discrete and continuum structure. Also discuss about stability, indeterminacy, and determinacy of structures with suitable examples. ## Short answer Discrete structures consist of separate members joined at nodes (e.g., trusses, frames). Continuum structures have continuous material (e.g., dams, slabs). Stability requires sufficient non-parallel, non-concurrent reactions. Determinacy: D = r - (3 + c) for frames; for trusses, m + r = 2j. Determinate structures are solvable by statics alone; indeterminate ones need compatibility. ## Answer #### Discrete and continuum structures - **Discrete structure:** made of separate, clearly identifiable members joined at joints (nodes), so it can be analysed member by member with a finite number of unknowns. Examples: trusses, beams and frames. - **Continuum structure:** the material is continuous and cannot be divided into members; stresses vary continuously through the body. Examples: dams, slabs, shells, retaining walls, a deep beam. They are analysed by elasticity theory or finite elements. #### Stability A structure is **stable** if it can carry any system of loads without rigid-body motion or collapse, and the supports prevent movement of the whole body. It is **unstable** if the reactions are fewer than needed, or if they are arranged so that they cannot resist some loading: - Fewer than 3 independent reactions in 2D (e.g. a beam on two rollers). - All reactions parallel (two rollers on horizontal ground cannot resist horizontal load). - All reactions concurrent at one point (the moment about that point cannot be resisted). - Internal arrangement forming a mechanism (e.g. a four-bar panel of a truss without a diagonal). #### Determinacy and indeterminacy - **Beams and frames (2D):** degree of indeterminacy D = r − (3 + c), where r is the number of reaction components and c the number of extra equations from internal hinges (each hinge gives one). D = 0 determinate, D > 0 indeterminate, D < 0 unstable. - **Trusses:** externally, D_e = r − 3. Internally, D_i = m − (2j − 3). Total: m + r − 2j (m members, j joints, r reactions). | Structure | r | Equations | Result | |---|---|---|---| | Simply supported beam (pin + roller) | 3 | 3 | Determinate | | Propped cantilever (fixed + roller) | 4 | 3 | Indeterminate to 1st degree | | Beam with pin and roller plus an internal hinge | 3 | 3 + 1 | Unstable (one equation too many) | | Triangular truss, 3 members, 3 joints, pin + roller | m + r = 6 = 2j | | Determinate and stable | | Truss with an extra diagonal in a square panel (m + r = 2j + 1) | | | Indeterminate to 1st degree | Statically determinate structures can be solved by equilibrium alone and do not develop stresses from support settlement or temperature change; indeterminate structures need compatibility conditions but are stiffer and have lower moments. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which is an example of a continuum structure? A. Truss B. Portal frame C. Dam or slab D. Continuous beam made of members Answer: C. Continuum structures like dams, slabs and shells cannot be divided into members and need elasticity theory or finite elements. (Key point: Discrete vs continuum) 2. For a 2D beam or frame, the degree of indeterminacy is: A. D = r - 3 - c, where c is the number of internal hinges B. D = r + 3 C. D = m + r - 2j D. D = 3 - r Answer: A. D = r - (3 + c). D = 0 is determinate, D > 0 indeterminate, D < 0 unstable. The formula m + r - 2j is for trusses. (Key point: Degree of indeterminacy) 3. Which of these is unstable? A. A pin plus a roller support B. A fixed support on a cantilever C. A propped cantilever D. A beam supported on two rollers on horizontal ground Answer: D. All reactions would be parallel, so no horizontal load can be resisted and the beam can move. (Key point: Unstable arrangement) 4. A beam on a pin and roller with an internal hinge is: A. Determinate B. Indeterminate to first degree C. Unstable, as there are 3 reactions but 4 equations D. Stable and over-braced Answer: C. The hinge adds one more equation (3 + 1) than the 3 reactions, so D = -1 and the beam is a mechanism. (Key point: Beam with internal hinge) ## Related in Applied Mechanics - [Determinacy and stability of frame](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Aframe-determinacy-stability-check) — Applied Mechanics, chapter 6, asked 2× - [Truss forces in CE, BE, BD, DE; stability](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Atruss-forces-ce-be-bd-de-stability-determinacy) — Applied Mechanics, chapter 7 ## Related in other subjects - [Static determinacy and stability of a truss](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Astatic-determinacy-stability-truss) — Strength of Materials, chapter 1 - [Indeterminacy and stability of truss and frame](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aindeterminacy-stability-truss-and-frame-bhadra) — Theory of Structures II, chapter 1 - [Indeterminacy of truss and beam-cable structure](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aindeterminacy-extensible-truss-and-beam-cable) — Theory of Structures II, chapter 1 - [Indeterminacy and stability of frame and truss](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aindeterminacy-stability-frame-and-truss-baishakh) — Theory of Structures II, chapter 1