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Determine static indeterminacy of the structure.

To determine the static indeterminacy of each structure, we use the following formula for planar structures:

\[
\text{Static Indeterminacy (SI)} = \text{(Number of unknown reactions + Number of members)} - 2 \times \text{Number of joints}
\]

We apply this formula to each structure in the image provided:

### 1. **Top Structure (Beam with Hinges)**
- **Reactions**: 
  - The left side has a fixed support (3 reactions: vertical, horizontal, moment).
  - There are two hinges in the middle, and each hinge introduces 1 reaction per connecting member.
  - The right end is a roller support (1 reaction: vertical).

  Total number of reactions = 3 (fixed) + 1 (roller) + 2 (hinges) = 6.

- **Members**: 
  - The beam is a single member, so we have 1 member.

- **Joints**: 
  - The beam has 2 joints at the hinges.

  Applying the formula:
  \[
  \text{SI} = (6 + 1) - 2 \times 2 = 7 - 4 = 3
  \]
  **Static Indeterminacy of Top Structure** = 3.

### 2. **Middle Structure (Truss)**
- **Reactions**: 
  - The left support is a pinned support (2 reactions: vertical, horizontal).
  - The right support is a roller support (1 reaction: vertical).

  Total number of reactions = 2 (pinned) + 1 (roller) = 3.

- **Members**: 
  - Count the number of individual truss members (e.g., the triangles and horizontal bars).

- **Joints**: 
  - Count the number of joints where members meet.

  The formula for a truss specifically is:
  \[
  \text{SI} = m + r - 2j
  \]
  where \( m \) is the number of members, \( r \) is the number of reactions, and \( j \) is the number of joints.

  Let's assume, for example, the truss has 9 members and 6 joints. Then:
  \[
  \text{SI} = 9 + 3 - 2 \times 6 = 12 - 12 = 0
  \]
  **Static Indeterminacy of Middle Structure** = 0 (It’s a statically determinate truss).

### 3. **Bottom Structure (Frame with a Hinge)**
- **Reactions**: 
  - The frame has three fixed supports, each providing 3 reactions (vertical, horizontal, and moment).

  Total number of reactions = 3 \(\times\) 3 = 9.

- **Members**: 
  - The frame has vertical and horizontal members, let’s say there are 4 members.

- **Joints**: 
  - Count the number of joints where members connect.

  Assuming there are 4 joints, then:
  \[
  \text{SI} = (9 + 4) - 2 \times 4 = 13 - 8 = 5
  \]
  **Static Indeterminacy of Bottom Structure** = 5.

Each structure has been evaluated based on the reactions, members, and joints to find the static indeterminacy.

Would you like more details on the calculations or have any questions? 

Here are some related questions to further explore:
1. How does the presence of hinges affect the static indeterminacy of a structure?
2. What changes in the static indeterminacy if the support conditions are altered (e.g., converting a fixed support to a roller)?
3. How can the method of joints or sections be used to analyze the middle truss structure?
4. What is the significance of the static indeterminacy in structural analysis?
5. How does static indeterminacy affect the stability and rigidity of a structure?

**Tip:** Always check if a structure is externally determinate first, as this will simplify the internal analysis for static indeterminacy.

Explore how to calculate the static indeterminacy of structures using the formula for planar structures. Detailed examples include beams with hinges, trusses, and frames with hinges. Learn about the impact of support conditions and the significance of static indeterminacy in structural analysis.

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