---
title: "Consistent deformation method and its formula"
subject: "Theory of Structures II"
chapter: 2
key: "semester_5/theory_of_structures_ii:what-is-consistent-deformation-method"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/what-is-consistent-deformation-method
site: BCT Engineering
---

# Consistent deformation method and its formula

Theory of Structures II · Chapter 2: Force method

- Asked in: 2067 Asar, 5 marks
- Video: [introduction to Consistent Deformation Method](https://www.youtube.com/watch?v=CbPyYtO2wBM) — Civil Engineering By Amit Kusanale
- 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/what-is-consistent-deformation-method
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-2

## Question

What is the consistent deformation method? Derive the formula.

## Short answer

Choose redundants equal to the degree of indeterminacy, release them to a determinate primary structure, compute load and unit-redundant displacements, then impose compatibility: Δ_i0 + Σ f_ij X_j = δ_i. Solve for X; final actions are M = M_0 + Σ X_j m_j.

## Answer

#### Meaning
The **consistent deformation method** (also called the method of consistent displacements or the force/flexibility method) analyses a statically indeterminate structure by:
1. choosing a suitable number of reactions or internal forces equal to the degree of indeterminacy as **redundants**,
2. removing them to get a stable, statically determinate **primary (released) structure**,
3. finding the displacement of the released structure at each redundant due to the loads, and
4. writing that the redundant forces must restore the displacements to the **value that actually exists in the real structure** (zero at a rigid support, equal to the settlement if the support settles). These are the compatibility, or consistent-deformation, conditions.

#### Derivation (one redundant)
Let a beam be indeterminate to the first degree, with the support reaction $R$ at B as redundant.

1. **Released structure under the loads.** Remove the support at B. The loads cause a deflection $\Delta_{B0}$ at B (call it $\Delta_{10}$, positive in the direction of $R$).
2. **Released structure under a unit load at B** in the direction of $R$. The deflection at B is the flexibility coefficient $f_{11}$ (deflection per unit force). If the force is $R$, the deflection is $f_{11}R$ (linear elasticity, superposition).
3. **Superpose.** The actual deflection at B is
$$
\Delta_B=\Delta_{10}+f_{11}R
$$
4. **Compatibility.** In the actual structure the support prevents deflection (or allows a known settlement $\delta_B$):
$$
\Delta_{10}+f_{11}R=\delta_B\quad(\delta_B=0\text{ for an unyielding support})
$$
$$
R=\frac{\delta_B-\Delta_{10}}{f_{11}}
$$

where $f_{11}=\int\dfrac{m_1^2}{EI}dx$ and $\Delta_{10}=\int\dfrac{m_1M_0}{EI}dx$, with $m_1$ the moment due to unit load at B, and $M_0$ the moment due to the loads in the released structure.

#### Several redundants
For $n$ redundants $X_1,\ldots,X_n$:
$$
\Delta_{i0}+\sum_{j=1}^{n}f_{ij}X_j=\delta_i\quad(i=1,\ldots,n)\qquad\Longrightarrow\qquad [F]\{X\}=\{\delta\}-\{\Delta_0\}
$$
$f_{ij}=f_{ji}$ (Maxwell). After finding $X$, the final moment is $M=M_0+\sum X_jm_j$, and the remaining reactions follow from equilibrium.

#### Example
Propped cantilever of span $L$ with UDL $w$, redundant $R_B$ at the roller: $\Delta_{10}=-\dfrac{wL^4}{8EI}$, $f_{11}=\dfrac{L^3}{3EI}$, so $R_B=\dfrac{3wL}{8}$.

## Check yourself

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

1. What is the central idea of the consistent deformation method?
   A. Choose redundants, release them, then require them to restore the actual displacements of the real structure
   B. Assume all joints rotate equally
   C. Solve using only equilibrium equations
   D. Distribute fixed-end moments until balanced
   Answer: A. Redundants are removed to get a determinate structure and then chosen so deformations are consistent with the real supports. (Key point: Meaning of the method)

2. For one redundant R at B with settlement delta_B, which equation holds?
   A. Delta_10 + f11 R = delta_B
   B. Delta_10 - f11 R = R
   C. Delta_10 f11 = delta_B
   D. Delta_10 + R = 0 always
   Answer: A. Superposing load deflection Delta_10 and f11 R gives the actual deflection delta_B (zero for an unyielding support). (Key point: Compatibility equation)

3. For a propped cantilever with UDL, Delta_10 = -wL^4/8EI and f11 = L^3/3EI. What is R_B?
   A. 3wL/8
   B. 5wL/8
   C. wL/2
   D. wL/8
   Answer: A. R = -Delta_10/f11 = (wL^4/8EI)/(L^3/3EI) = 3wL/8. (Key point: Example propped cantilever)

## Related in Theory of Structures II

- [Consistent deformation: frame BMD with 20 kN](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aconsistent-deformation-frame-bm-20kn) — Theory of Structures II, chapter 2
- [Consistent deformation: frame, BMD, SFD, normal thrust](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aconsistent-deformation-frame-15kn-10knm-sfd) — Theory of Structures II, chapter 2
- [Consistent deformation: frame with 12 kN, SFD and BMD](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aconsistent-deformation-frame-12kn-sfd-bmd) — Theory of Structures II, chapter 2
- [Consistent deformation: frame, BMD, SFD, normal thrust](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aconsistent-deformation-frame-1knm-3kn) — Theory of Structures II, chapter 2
- [Consistent deformation: chair-frame with couple](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aconsistent-deformation-chair-frame-couple) — Theory of Structures II, chapter 2
- [Consistent deformation: bent frame with support movement](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aconsistent-deformation-bent-frame-settlement-20knm) — Theory of Structures II, chapter 2

## Related in other subjects

- [Derive Newton-Raphson formula](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Anewton-raphson-derive) — Numerical Method, chapter 2, asked 2×
- [Horizontal reaction of cable, derive H formula](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Acable-horizontal-reaction-formula) — Theory of Structures I, chapter 7
- [Sabine's formula and good acoustics](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Asabine-formula-acoustic-hall) — Engineering Physics, chapter 3, asked 5×
- [Euler's formula: both ends hinged](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aeuler-both-ends-hinged) — Strength of Materials, chapter 8, asked 4×