Size of Flexibility Matrix to Determine Slope and Deflection at Free End of Cantilever Beam

To determine the slope and deflection at the free end of a cantilever beam, a flexibility matrix is used. The flexibility matrix relates the applied loads to the resulting displacements and rotations at various points along the beam. The size of the flexibility matrix depends on the number of unknown displacements or rotations that need to be determined.

Determining the Number of Unknowns

In this case, the beam is subjected to a concentrated load at the free end. The unknowns that need to be determined are the slope and deflection at the free end. Since the beam is cantilevered, the other end is fixed and does not experience any displacements or rotations. Therefore, the only unknowns are the slope and deflection at the free end.

Size of Flexibility Matrix

The flexibility matrix is typically square and its size is determined by the number of unknowns. Since we have two unknowns (slope and deflection at the free end), the flexibility matrix will be a 2x2 matrix.

Explanation

The flexibility matrix is derived from the stiffness matrix, which relates the applied loads to the resulting displacements and rotations. The flexibility matrix is the inverse of the stiffness matrix. In this case, the stiffness matrix is not required since we are only interested in the slope and deflection at the free end.

By using the flexibility matrix, we can determine the relationship between the applied load and the resulting slope and deflection at the free end. This allows us to calculate the slope and deflection based on the known load.

The flexibility matrix can be used in conjunction with the equations of equilibrium and boundary conditions to solve for the unknowns. By solving the system of equations, we can determine the slope and deflection at the free end of the cantilever beam.

In conclusion, the size of the flexibility matrix to determine the slope and deflection at the free end of a cantilever beam is a 2x2 matrix. This matrix relates the applied load to the resulting displacements and rotations at the free end of the beam.