---
title: "Steps of virtual (unit) load method"
subject: "Theory of Structures I"
chapter: 3
key: "semester_4/theory_of_structures_i:unit-load-method-steps"
source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/unit-load-method-steps
site: BCT Engineering
---

# Steps of virtual (unit) load method

Theory of Structures I · Chapter 3: Analysis by the Virtual Work Method

- Asked in: 2073 Magh, 4 marks; 2073 Bhadra, 4 marks; 2070 Magh, 6 marks; 2066 Kartik, 6 marks (asked 4 times)
- Video: [Unit Load Method - Cantilever Beam -UDL](https://www.youtube.com/watch?v=WEc4W7iNie4) — praveen suvarna
- 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/unit-load-method-steps
- Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-3

## Question

Explain with a simple example the steps involved in determining the displacement of a point in a structural system applying the virtual work (unit load) method.

## Short answer

Steps: 1) Analyse structure under real loads for M, N, etc. 2) Apply unit virtual load at desired displacement point and direction; find virtual forces m, n. 3) Use Δ = ∫(M m/EI)dx (beams) or Σ(F k L/AE) (trusses). Example: cantilever tip deflection = PL³/(3EI).

## Answer

The virtual work (unit load) method finds the displacement of a chosen point by applying a unit virtual load at that point, in the direction of the required displacement, on the same structure.

#### Steps
1. Analyse the structure under the **real loads** and find the internal forces: $M$ (bending moment), $N$, $V$ (for trusses, the member forces $F$).
2. Remove the real loads and apply a **unit load** (or unit couple for rotation) at the point and in the direction of the required displacement.
3. Analyse the structure under this unit load and find the virtual internal forces $m$ (and $n$ or $k$ for a truss).
4. Write the virtual work equation: external virtual work = internal virtual work.
5. Integrate (or sum) over all members to get the displacement.

For beams and frames (bending only):

$$
1\cdot\Delta = \int \frac{M\,m}{EI}\,dx
$$

For trusses:

$$
1\cdot\Delta = \sum \frac{F\,k\,L}{AE}
$$

6. A positive result means the displacement is in the direction of the unit load; a negative result means the opposite.

#### Example
Find the deflection at the free end of a cantilever of length $L$ with a point load $P$ at that end.

- Real moment: $M = -Px$ (x from the free end).
- Apply a unit load at the free end: $m = -1\cdot x$.
- Then

$$
\Delta = \int_0^L \frac{(-Px)(-x)}{EI}dx = \frac{PL^3}{3EI}
$$

The sign is positive, so the deflection is downward, in the direction of the unit load.

Other effects (temperature, lack of fit, support settlement) are included by using the virtual forces with the corresponding deformations: $\Delta = \sum k\,\alpha\,\Delta T\,L$ for temperature and $\sum k\,e$ for lack of fit.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In the unit load method the virtual load is applied:
   A. Together with the real loads at all supports
   B. At the point and in the direction of the required displacement, with real loads removed
   C. At the point of maximum moment
   D. At the centroid of the structure
   Answer: B. The real loads are removed and a unit load (or unit couple for rotation) is applied at the point and in the direction of the wanted displacement. (Key point: Unit load placement)

2. For beams and frames (bending only) the displacement is:
   A. Delta = integral of M m dx / EI
   B. Delta = integral of M^2 dx / (2EI)
   C. Delta = sum of F k L / AE
   D. Delta = integral of m dx / EI
   Answer: A. 1.Delta = integral of M m/EI dx, where M is real and m is from the unit load; the truss form is the sum of FkL/AE. (Key point: Virtual work equation)

3. A positive result from the unit load method means:
   A. The displacement is opposite to the unit load
   B. The displacement is in the direction of the unit load
   C. The structure is unstable
   D. The answer must be doubled
   Answer: B. A positive value shows the displacement is in the assumed direction of the unit load; negative shows it is opposite. (Key point: Sign of result)

4. For a cantilever with end load P, M = -Px and m = -x. The deflection at the free end is:
   A. PL^3/(8EI)
   B. PL^3/(3EI)
   C. PL^3/(48EI)
   D. PL^2/(2EI)
   Answer: B. The integral of x^2/EI from 0 to L times P gives PL^3/(3EI), downward. (Key point: Cantilever example)

## Related in Theory of Structures I

- [Truss horizontal displacement at C by virtual work](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Atruss-virtual-work-horizontal-c) — Theory of Structures I, chapter 3
- [Truss deflection at E by virtual work](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Atruss-vw-e-2071o) — Theory of Structures I, chapter 3
- [Real work and virtual work, limitations](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Areal-work-virtual-work-define) — Theory of Structures I, chapter 3, asked 2×
- [Derive virtual work principle](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Avirtual-work-principle-derive) — Theory of Structures I, chapter 3
- [Truss joint L2 by virtual work and temperature](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Atruss-vw-l2-temperature-2069p) — Theory of Structures I, chapter 3
- [Overhang beam free end deflection by virtual work](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Abeam-vw-free-end-d-2065) — Theory of Structures I, chapter 3

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