Consistent deformation method and its formula

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 $R_B=\dfrac{3wL}{8}$.