Flexibility Method - Compatibility Equation and Solution

Compatibility Equation and Solution

Next, we need to set up compatibility equations in order to find . The compatibility equations restore the required continuity at the cut sections by setting the relative displacements at the redundants X to zero. That is, using the unit dummy force method:

$\mathbf{r}_{X} = \mathbf{B}_X^T \mathbf{q} = \mathbf{B}_X^T \Big[ \mathbf{f} \Big( \mathbf{B}_R \mathbf{R} + \mathbf{B}_X \mathbf{X} + \mathbf{Q}_v \Big) + \mathbf{q}^{o} \Big] = 0 \qquad \qquad \qquad \mathrm{(7a)}$

where

$\mathbf{r}^o_X = \mathbf{B}_X^T \Big[ \mathbf{f} \Big( \mathbf{B}_R \mathbf{R} + \mathbf{Q}_v \Big) + \mathbf{q}^{o} \Big]$

Equation (7b) can be solved for X, and the member forces are next found from (5) while the nodal displacements can be found by

where

is the system flexibility matrix.

$\mathbf{r}^o_R = \mathbf{B}_R^T \Big[ \mathbf{f} \Big( \mathbf{B}_X \mathbf{X} + \mathbf{Q}_v \Big) + \mathbf{q}^{o} \Big]$

Supports' movements taking place at the redundants can be included in the right-hand-side of equation (7), while supports' movements at other places must be included in and as well.

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