--- title: "Betti's law and Maxwell's reciprocal theorem" subject: "Theory of Structures II" chapter: 2 key: "semester_5/theory_of_structures_ii:betti-maxwell-theorems" source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/betti-maxwell-theorems site: BCT Engineering --- # Betti's law and Maxwell's reciprocal theorem Theory of Structures II · Chapter 2: Force method - Asked in: 2076 Asoj, 6 marks - Video: [Maxwell Betti Reciprocal Theorem](https://www.youtube.com/watch?v=5QcDIOhggsU) — RECKON CIVIL ACADEMY - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/betti-maxwell-theorems - Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-2 ## Question Enunciate Betti's law and Maxwell's reciprocal theorem and explain their uses. ## Short answer Betti’s law: in a linear elastic structure, the work of one force system through the displacements of another equals the reverse. Maxwell’s theorem: displacement at A from a unit load at B equals displacement at B from a unit load at A. Uses: ensuring flexibility/stiffness matrix symmetry, checking coefficients, constructing influence lines for deflections, and computing deflections via reciprocal loadings. ## Answer #### Betti's law **Statement.** In a linearly elastic structure acted on by two separate systems of loads, the work done by the forces of the first system through the displacements produced by the second system equals the work done by the forces of the second system through the displacements produced by the first system. If system 1 has loads $P_i$ giving displacements $\Delta_{i1}$ and system 2 has loads $Q_j$ giving displacements $\Delta_{j2}$ at the same points, then $$ \sum P_i\,\Delta_{i2} = \sum Q_j\,\Delta_{j1} $$ **Proof (outline).** Apply $P$ first, then $Q$: the total work is $U_P + U_Q + \sum P\,\Delta_{P(Q)}$. Apply $Q$ first, then $P$: the total work is $U_Q + U_P + \sum Q\,\Delta_{Q(P)}$. Strain energy depends only on the final loads, not on the order, so the two cross terms are equal. #### Maxwell's reciprocal theorem **Statement.** The displacement at point A in the direction of a unit load applied at B equals the displacement at B in the direction of a unit load applied at A: $$ \delta_{AB} = \delta_{BA}, \qquad \text{or in general } f_{ij} = f_{ji} $$ It follows from Betti's law by taking each system as a single unit load. It holds for linear displacements, rotations and mixed pairs (force and couple). **Example.** For a simply supported beam of span $L$, the deflection at the centre due to a unit load at $L/4$ equals the deflection at $L/4$ due to a unit load at the centre, $\dfrac{11L^3}{768EI}$. #### Uses 1. It makes the flexibility matrix $[F]$ and the stiffness matrix $[K]$ symmetric, which halves the work of finding the coefficients in the force method. 2. It checks the computed flexibility coefficients ($f_{12}$ against $f_{21}$). 3. It gives influence lines for deflection: the deflected shape of the structure under a unit load placed at A is the influence line for the deflection at A. This idea is the basis of the Muller-Breslau principle. 4. It lets the deflection at a point be found from the deflection curve of another loading, which is easier to compute. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does Betti's law state? A. The work done equals the strain energy for each system separately B. The total work is the sum of the two systems' works C. Displacements from two systems always cancel D. The work of the first load system through displacements due to the second equals the work of the second system through displacements due to the first Answer: D. sum P_i Delta_i2 = sum Q_j Delta_j1 for a linear elastic structure. (Key point: Betti's law) 2. Why are the two cross terms in Betti's proof equal? A. Strain energy depends only on the final loads, not the order of application B. Because both systems have equal loads C. Because displacements are zero at supports D. Because the structure is symmetric Answer: A. Loading in either order produces the same final energy, so the cross work terms must match. (Key point: Why it holds) 3. Which is a use of Maxwell's reciprocal theorem? A. It makes a structure statically determinate B. It makes the flexibility matrix symmetric, halving the coefficients to compute, and checks f12 = f21 C. It removes the need for compatibility equations D. It gives the yield stress of the material Answer: B. f_ij = f_ji, so only half the off-diagonal terms need calculation and the symmetry gives a check. (Key point: Use of Maxwell's theorem) ## Related in Theory of Structures II - [State and prove Maxwell's reciprocal theorem](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Amaxwell-reciprocal-theorem) — Theory of Structures II, chapter 2 - [Compatibility conditions; Maxwell's theorem](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acompatibility-conditions-maxwell) — Theory of Structures II, chapter 2 ## Related in other subjects - [Ampere's law, Maxwell's modification](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aampere-law-maxwell-modification-displacement) — Engineering Physics, chapter 9 - [Faraday's law and modified Maxwell equations](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Afaraday-law-correct-curl-e-maxwell) — Electromagnetism, chapter 3 - [Maxwell's equations and continuity equation](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Amaxwell-continuity-equation) — Engineering Physics, chapter 9, asked 2×