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A free body of the beam, showing all the forces acting on it, is shown in Figure 2.26d.

The steps involved in the method of consistent deformations are

1. The number of redundants in the structure is determined.

2. The redundants required to form a determinate structure are removed.

3. The displacements that the applied loads cause in the determinate structure at the points where the redundants have been removed are then calculated.

4. The displacements at these points in the determinate structure due to the redundants are obtained.

5. At each point where a redundant has been removed, the sum of the displacements calculated in steps (3) and (4) must be equal to the displacement that exists at that point in the actual indeterminate structure. The redundants are evaluated using these relationships.

6. Once the redundants are known the remaining reactions are determined using the equations of equilibrium.

2.4.4.1 Structures with Several Redundants

The method of consistent deformations can be applied to structures with two or more redundants. For example, the beam in Figure 2.27a is indeterminate to the second degree and has two redundant reactions. If the reactions at B and C are selected to be the redundants, then the determinate structure obtained by removing these supports is the cantilever beam shown in Figure 2.27b. To this determinate structure we apply separately the given load (Figure 2.27c) and the redundants RB and RC one at a time (Figure 2.27d and e).

Since the deflections at B and C in the original beam are zero, the algebraic sum of the deflections in Figure 2.27c, d, and e at the same points must also vanish. Thus,

It is useful in the case of complex structures to write the equations of consistent deformations in the form

where dBC, for example, denotes the deflection at B due to a unit load at C in the direction of RC. Solution of Equations 2.35 gives the redundant reactions RB and —c.

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