--- title: "Degree of freedom and static determinacy" subject: "Strength of Materials" chapter: 1 key: "semester_3/strength_of_materials:degree-of-freedom-supports" source: https://www.bctengineer.com/q/semester_3/strength_of_materials/degree-of-freedom-supports site: BCT Engineering --- # Degree of freedom and static determinacy Strength of Materials · Chapter 1: Axial Forces, Shearing Forces and Bending Moments - Asked in: 2081 Bhadra, 2+4 marks - Video: [Determinate vs Indeterminate Structures - Intro to Structural Analysis](https://www.youtube.com/watch?v=ml26FjTl-C8) — Structures with Prof. H - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/degree-of-freedom-supports - Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-1 ## Question Define degree of freedom of a system and write its values for roller, hinge and fixed support for a plane structure. Is it possible for a statically determinate or indeterminate structure to be geometrically unstable? Explain with suitable example. ## Short answer Degree of freedom (DOF) is the number of independent displacements. For a plane structure: roller leaves 2 DOF, hinge 1, fixed 0. A structure can be geometrically unstable even if statically determinate or indeterminate, e.g., three vertical rollers provide no horizontal restraint, making it unable to resist lateral loads. ## Answer #### Degree of freedom The degree of freedom (DOF) of a system is the number of independent displacement components (translations and rotations) by which the position of the system can change. For a plane structure a free body has 3 DOF: two translations ($x$ and $y$) and one rotation about the axis perpendicular to the plane. A support removes the DOF that it restrains; the number of reactions of a support equals the number of DOF it removes. | Support (plane structure) | DOF left at the support | Reactions provided | |---|---|---| | Roller | 2 (translation along the support surface and rotation) | 1 (normal to the surface) | | Hinge (pin) | 1 (rotation) | 2 (horizontal and vertical) | | Fixed | 0 | 3 (two forces and a moment) | #### Can a statically determinate or indeterminate structure be geometrically unstable? Yes. Static determinacy only compares the number of reactions $r$ with the equations of equilibrium (3 for a plane structure). Geometric stability depends on how the supports are arranged. Even if the count is right, the structure is unstable if the reactions cannot resist every possible load. **Example 1: determinate but unstable.** A beam on three rollers (all vertical reactions), $r = 3 = $ equations, so it looks determinate. But there is no horizontal restraint; $\Sigma F_x = 0$ cannot be satisfied for a horizontal load, so the beam can slide. Another case: a beam with three reactions that are parallel or meet at one point (concurrent) cannot resist a moment about that point. ```text ○──────────────○──────────────○ all reactions vertical: ▲ ▲ ▲ horizontal load → beam moves ``` **Example 2: indeterminate but unstable.** A beam supported on four vertical rollers has $r = 4 > 3$, so it is indeterminate by one degree (vertical reactions only), yet it is horizontally unstable. Hence $r \ge 3$ is a necessary condition for stability, not a sufficient one. The supports must also be arranged so that the reactions are neither all parallel nor all concurrent, and no part of the structure forms a mechanism. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How many degrees of freedom does a free body have in a plane? A. 2 B. 3 C. 4 D. 6 Answer: B. A plane body can translate in x and y and rotate about the perpendicular axis, giving 3 DOF. (Key point: Degrees of freedom of a plane body) 2. Which row correctly gives the DOF left and reactions provided? A. Roller: 1 DOF left, 2 reactions B. Hinge: 2 DOF left, 1 reaction C. Fixed: 1 DOF left, 2 reactions D. Hinge: 1 DOF (rotation) left, 2 reactions Answer: D. A hinge restrains both translations (2 reactions) and leaves rotation free (1 DOF). A roller removes only one DOF, and a fixed support removes all 3. (Key point: Support DOF and reactions) 3. A beam on three vertical rollers has r = 3. Why is it still unstable? A. Because three reactions are too many B. Because rollers cannot carry vertical load C. Because there is no horizontal restraint, so a horizontal load makes it slide D. Because the beam is statically indeterminate Answer: C. All reactions are vertical, so sum Fx = 0 cannot be satisfied for a horizontal load even though r equals the number of equations. (Key point: Determinate but unstable) 4. What does r >= 3 mean for stability of a plane structure? A. It is necessary but not sufficient, because the reactions must also be neither all parallel nor all concurrent B. It is sufficient for stability C. It guarantees the structure is determinate D. It is unrelated to stability Answer: A. Enough reactions are needed, but their arrangement matters. Parallel or concurrent reactions cannot resist every load, so the structure can still be unstable. 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