--- title: "Derive virtual work principle" subject: "Theory of Structures I" chapter: 3 key: "semester_4/theory_of_structures_i:virtual-work-principle-derive" source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/virtual-work-principle-derive site: BCT Engineering --- # Derive virtual work principle Theory of Structures I · Chapter 3: Analysis by the Virtual Work Method - Asked in: 2079 Chaitra, 4 marks - Video: [Principle of Virtual Work](https://www.youtube.com/watch?v=qR8NlrP7_VM) — Structure Pro - 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/virtual-work-principle-derive - Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-3 ## Question Derive the virtual work principle. ## Short answer Apply a virtual unit load at the desired displacement; let it produce virtual internal forces m, n, v. External virtual work 1·Δ equals internal virtual work ∫ (m·dθ + n·dδ + v·dγ). Substituting elastic deformations dθ=Mdx/EI, etc., yields Δ = ∫ (mM/EI + nN/AE + α vV/GA)dx. ## Answer **Principle of virtual work (for deformable bodies):** If a deformable structure in equilibrium under a real load system is given a virtual displacement field (compatible, small and imaginary), the external virtual work done by the real forces equals the internal virtual work done by the real internal stresses on the virtual strains. In the dual form (virtual forces), virtual loads are applied and move through the real displacements; this is the form used below. #### Derivation (unit load form) Consider a structure under real loads $P_1, P_2,\dots$ producing real internal forces and real deformations. To find the displacement $\Delta$ at a point, apply a **virtual unit load** at that point in the required direction. Let this virtual load produce internal forces: moment $m$, axial force $n$ and shear $v$. 1. **External virtual work.** The unit load, kept constant, moves through the real displacement $\Delta$: $$ W_e = 1\times\Delta $$ 2. **Internal virtual work.** In an element of length $dx$, the virtual internal forces move through the real deformations of the element. For a linearly elastic member, the real deformations are $$ d\theta = \frac{M\,dx}{EI},\qquad d\delta_{axial} = \frac{N\,dx}{AE},\qquad d\gamma = \frac{\alpha V\,dx}{GA} $$ so the internal virtual work of the element is $$ dW_i = m\,d\theta + n\,d\delta_{axial} + v\,d\gamma = \frac{mM}{EI}dx + \frac{nN}{AE}dx + \frac{\alpha vV}{GA}dx $$ 3. **Equate** $W_e = W_i$ (the virtual force system is in equilibrium, and the real deformations are compatible): $$ 1\cdot\Delta = \int \frac{mM}{EI}dx + \int\frac{nN}{AE}dx + \int\frac{\alpha vV}{GA}dx $$ For beams and frames, usually only the first term is kept. For trusses, $\Delta = \sum \dfrac{kFL}{AE}$. The general principle ($W_e = W_i$, i.e. $1\cdot\Delta = \int m\,d\theta + \int n\,d\delta + \int v\,d\gamma$) holds for any material behaviour, provided the real deformations are small and compatible and the virtual forces satisfy equilibrium. The final form with $EI$, $AE$ and $GA$, however, uses $d\theta = M\,dx/EI$ and similar relations, so it applies only to linearly elastic members. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The principle of virtual work (force form) states that: A. Virtual external work equals virtual internal work B. Real work equals the kinetic energy C. Internal work is always zero D. External work equals twice the internal work Answer: A. External virtual work done by virtual loads on real displacements equals internal virtual work of the virtual forces on real deformations. (Key point: Principle statement) 2. The external virtual work of a unit load at the point is: A. (1/2) x 1 x Delta B. 1 x Delta C. Delta^2 D. EI x Delta Answer: B. The unit load stays constant while moving through the real displacement Delta, so W_e = 1 x Delta. (Key point: External virtual work) 3. The internal virtual work in an element from bending is: A. m x M dx / EI B. M^2 dx / (2EI) C. m dx / M D. EI m dx / M Answer: A. The virtual moment m acts through the real rotation d(theta) = M dx/EI, giving mM dx/EI. (Key point: Internal virtual work) 4. What conditions does the derivation require? A. Linear elastic material only B. Virtual forces in equilibrium and real deformations compatible and small C. Only static loads and no deformations D. Constant EI throughout Answer: B. The relation holds for any material as long as the virtual forces satisfy equilibrium and the real deformations are small and compatible. 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