7: Statically Indeterminate Beams and Plane Frames

7.7: Example - Frame Example 2

Determine all of the reactions for the following frame:

Figure

Figure 7.7-1: Frame Example 2

This example is the same as Example 1 of the previous section. This time we will choose a moment redundant, rather than a force redundant.

1: Statical determinacy

The frame is 1 degree statically indeterminate.

2: Identify redundants

Figure

Figure 7.7-2: Free body of real structure

Choose the moment at a ($M_a$) as the redundant.

3: Analyze the primary structure

Figure

Figure 7.7-3: Determinate structure with real loads

4: Apply unit values of the redundants

Figure

Figure 7.7-4: Determinate structure with unit redundants

5: Compute Displacements in the primary structure

Figure

6: Compute flexibilty coefficients

Figure

7: Write compatibilty equations

The real vertical displacement at point c is zero, therefore:

\[\begin{split} 0 &= \theta_{10} + M_a f_{11}\\ &= -\frac{10 P L^2}{E I_0} + Ma \times \frac{9 L}{2 E I_0} \end{split}\]

8: Solve for the redundant

\[M_a = \frac{20}{9} P L\]

9: Other reactions by superposition

\[\begin{split} V_c &= V_{c0} + M_a v_{ac} \\ &= \frac{4P}{3} + \frac{20 P L}{9} \times -\frac{1}{3L} \\ &= \frac{16}{27} P\\ H_a &= H_{a0} + M_a h_{a1} \\ &= P + \frac{20 PL}{9} \times 0 \\ &= P\\ V_a &= V_{a0} + M_a v_{a1} \\ &= -frac{4P}{3} + \frac{20 P L}{9} \times \frac{1}{3L} \\ &= -\frac{16}{27} P\\ \end{split}\]

10: Summary

Figure

Figure 7.7-5: Summary of Frame Example 2