--- title: "Real work and virtual work, limitations" subject: "Theory of Structures I" chapter: 3 key: "semester_4/theory_of_structures_i:real-work-virtual-work-define" source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/real-work-virtual-work-define site: BCT Engineering --- # Real work and virtual work, limitations Theory of Structures I · Chapter 3: Analysis by the Virtual Work Method - Asked in: 2072 Magh, 4 marks; 2070 Magh, 6 marks (asked 2 times) - Video: [Method of Virtual Work - Structural Analysis](https://www.youtube.com/watch?v=TlznZHClMLY) — structurefree - 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/real-work-virtual-work-define - Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-3 ## Question Define real work method and virtual work method for deformable structures with neat sketches. What are the limitations of the real work method? ## Short answer Real work method equates external work ½PΔ to strain energy U = ∫ M²/(2EI)dx for a single load, giving deflection only at its point. Virtual work method applies a unit load, using 1·Δ = ∫ (M m/EI)dx for any loading. Real work limitations: only one load, only deflection at load point, cannot handle temperature/settlement, requires linear elasticity. ## Answer #### Real work method The real work method equates the **external work** done by the actual loads (applied gradually) to the **strain energy** stored in the structure: $$ W_e = U \;\Rightarrow\; \tfrac12 P\,\Delta = \int \frac{M^2}{2EI}\,dx $$ The work is "real" because the loads move through their own actual displacements. Example: a cantilever with end load $P$: $\tfrac12P\Delta = \dfrac{P^2L^3}{6EI}$, so $\Delta = \dfrac{PL^3}{3EI}$. ```text P | /| | / | Real work = area = 1/2 P*D |/____| +-----+--- Displacement D ``` #### Virtual work method In the virtual work method, an imaginary (virtual) force system, usually a unit load at the point where the displacement is wanted, is applied. The external virtual work of the virtual forces moving through the real displacements equals the internal virtual work of the virtual internal forces on the real deformations: $$ 1\cdot\Delta = \int \frac{M\,m}{EI}\,dx \qquad\text{(beams, frames)} $$ $$ 1\cdot\Delta = \sum \frac{F\,k\,L}{AE} \qquad\text{(trusses)} $$ Here the virtual load is constant while the real displacement occurs, so the virtual work is $1\times\Delta$ (no factor of $\tfrac12$). ```text Real loads Virtual load P1 P2 1 (at the point) v v v ______________ ______________ -> M, F -> m, k ``` #### Limitations of the real work method 1. Only **one load** can act on the structure; with several loads the energy equation has several unknown displacements. 2. It gives the displacement **only at the point of application of the load**, and only in the direction of the load. 3. It cannot give the displacement of a point where no load acts, or a rotation, unless a fictitious load is added. 4. It cannot account for temperature change, lack of fit or support settlement. 5. It is applicable only to linearly elastic structures. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The real work method equates: A. Virtual work of a unit load to the strain energy B. External work of the actual gradually applied loads to the strain energy stored C. Shear energy to bending energy D. Potential energy to kinetic energy Answer: B. W_e = U, with W_e = (1/2) P Delta because the load increases gradually from zero. (Key point: Real work method) 2. Why does the virtual work method use 1 x Delta with no factor 1/2? A. Because the unit load is constant while the real displacement occurs B. Because the unit load is applied gradually C. Because the structure is non-linear D. Because Delta is small Answer: A. The real work uses a gradually increasing load (hence 1/2), but the virtual load stays constant while moving through the real displacement. (Key point: Factor 1/2) 3. A key limitation of the real work method is that: A. It works only for trusses B. It gives displacement only at the point and direction of the single load C. It cannot be used for beams D. It requires plastic material Answer: B. Only one load can act, and the method gives the displacement only where that load acts and in its direction. (Key point: Limitation of real work) 4. Which effect can the real work method NOT handle? A. Bending of a cantilever B. Temperature change, lack of fit or support settlement C. Linear elastic material D. End load on a cantilever Answer: B. Temperature, lack of fit and settlement do no work with loads, so they cannot be captured; the virtual work method can include them. 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