7: Statically Indeterminate Beams and Plane Frames

7.6: Example - Frame Example 1

Determine all of the reactions for the following frame:

Figure 7.6-1: Frame Example 1

1: Statical determinacy

The frame is 1 degree statically indeterminate.

2: Identify redundants

Figure 7.6-2: Free body of real structure

Choose the vertical reaction at c ($V_c$) as the redundant.

3: Analyze the primary structure

Figure 7.6-3: Determinate structure with real loads

4: Apply unit values of the redundants

Figure 7.6-4: 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:

\[\begin{split} 0 &= \Delta_{10} + V_c f_{11}\\ &= -\frac{24 P L^3}{E I_0} + V_c \times \frac{81 L^3}{2 E I_0} \end{split}\]

8: Solve for the redundant

\[V_c = \frac{16}{27} P\]

9: Other reactions by superposition

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

10: Summary

Figure 7.6-5: Summary of Frame Example 1