Flexibility coefficients are used in the force method.

The force method is a structural analysis technique used to determine the internal forces, displacements, and reactions in a structure by considering equilibrium of forces. It involves solving a set of simultaneous equations derived from the equilibrium conditions.

What are flexibility coefficients?

Flexibility coefficients, also known as flexibility matrices, are mathematical coefficients that relate the applied loads to the resulting displacements in a structure. They are used in the force method to determine the unknown displacements.

How are flexibility coefficients used in the force method?

1. Formation of equilibrium equations: The first step in the force method is to establish the equilibrium equations based on the structure's geometry and boundary conditions. These equations relate the applied loads to the internal forces and displacements.

2. Definition of flexibility coefficients: Flexibility coefficients are defined as the ratio of the displacement at a particular point in the structure to the load applied at that point, while all other loads are held constant. They represent the flexibility or deformability of the structure.

3. Assembly of flexibility matrix: The flexibility coefficients are organized into a matrix known as the flexibility matrix. Each entry in the matrix corresponds to the flexibility coefficient between two points in the structure. The flexibility matrix is typically a square matrix with dimensions equal to the number of degrees of freedom in the structure.

4. Application of boundary conditions: The boundary conditions, such as fixed supports or prescribed displacements, are applied to the flexibility matrix. This involves modifying the matrix entries to account for the constraints imposed by the supports.

5. Formation of compatibility equations: The compatibility equations are derived by equating the actual displacements to the displacements calculated using the flexibility matrix. These equations ensure that the structure remains compatible and does not undergo any rigid body motion.

6. Solution of simultaneous equations: The equilibrium equations and compatibility equations are solved simultaneously to determine the unknown displacements. This involves manipulating the flexibility matrix and applying matrix algebra techniques.

7. Calculation of internal forces: Once the displacements are known, the internal forces in the structure can be calculated using the equilibrium equations.

Conclusion:

Flexibility coefficients play a crucial role in the force method of structural analysis. They are used to establish the equilibrium equations, define the flexibility matrix, apply boundary conditions, and solve for the unknown displacements. By considering the flexibility of the structure, the force method provides a comprehensive analysis of the internal forces and displacements.