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The document contains a series of engineering problems focused on the stiffness method for analyzing structures. Below are the steps to address each problem:

### Problem 1: Degrees of Freedom (GDLs) for Figure P1
1. **Identify nodes and possible displacements:**
   - Traverse the nodes in alphabetical order.
   - For each node, assign the translational GDLs in X and Y directions, followed by the rotational GDL.
   - Apply symmetry or other constraints to minimize the number of GDLs.

2. **Node "d" GDLs:**
   - Assign GDLs for node "d" in the sequence mentioned. For example, if node "d" has its translational displacement in X as 1, in Y as 2, and rotational as 3, it should be represented as {1;2;3}.

### Problem 2: Computational GDLs for Figure P2
1. **Free displacements \({Ul}\):**
   - Start with free displacements in alphabetical order, first for X, then Y, and finally rotational.
   
2. **Restricted displacements \({Ur}\):**
   - Follow the same order after free displacements.

3. **Node "f" GDLs:**
   - Similarly, number the displacements for node "f" based on the sequence defined. For instance, if node "f" has displacements numbered as 15, 16, and 1, it would be written as {15;16;1}.

### Problem 3: Settlement in Node “h”
1. **Vector \({Ur}\):**
   - Identify the restricted displacements for the structure, considering the settlement.
   - Calculate and report the vector \({Ur}\).

### Problem 4: Simplified Structure for Figure P4
1. **Matrix \([K_{ll}]\):**
   - Calculate the stiffness matrix for the simplified structure.
   - Report the (1,1) position of the matrix in kN/m.

### Problem 5: Force Vector \({F_l}\) for Figure P5
1. **Load Calculations:**
   - Calculate the effect of the settlement and the applied loads.
   - Report the first row of the force vector in kN/rad.

### Problem 6: Moment \(M_{ab}\) in Element “ab”
1. **Moment Calculation:**
   - Use the provided displacement vector and load conditions.
   - Calculate and report the moment \(M_{ab}\) in kN/m.

### Problem 7: Horizontal Reaction at Support “d”
1. **Reaction Calculation:**
   - Use

Explore how to solve engineering problems using the stiffness method. This detailed guide covers topics such as calculating stiffness matrices, analyzing structural nodes, and determining displacements and reactions. Suitable for advanced engineering students and professionals.

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