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)}](http://upload.wikimedia.org/math/f/0/e/f0ef9db561e923543373be1c2c71f568.png)
- or
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]](http://upload.wikimedia.org/math/5/4/5/5452f892bfa6867839802b088fefac51.png)
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]](http://upload.wikimedia.org/math/3/c/4/3c406127f658d183b6e277080e0af59b.png)
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.
Read more about this topic: Flexibility Method
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