Question

What is true for flexibility and stiffness matrix?

a. They are square matrix

b. The diagonal elements are non-zero and having positive values

c. Element ij = Element ji

d. They are inverse of each other

1. a and b
2. c and d
3. a, c and d
4. All of the above

Answer 1

Correct Answer - Option 4 : All of the above

Explanation:

Flexibility and stiffness matrix:

Flexibility matrix

Stiffness matrix

Forces are taken as unknowns (i.e., shear force and bending moment) and equations are expressed in terms of these forces. Additional equations are called compatibility conditions are developed to find all the unknown forces.  The flexibility matrix is a square symmetric matrix because of Maxwell’s reciprocal theorem. Hence Element ij = Element ji In this matrix, diagonal elements are always positive (from Castigliano’s first theorem). If the structure is unstable, then a large deformation will occur and the flexibility matrix will not exist. The flexibility matrix is the inverse of the stiffness matrix.

Displacements are taken as unknowns (i.e., slope and deflections) and equations are expressed in terms of these unknown displacements. The additional joint equilibrium equation is developed to find the unknown displacements. The stiffness matrix is a square symmetric matrix because of Maxwell’s reciprocal theorem. Hence Element ij = Element ji In this matrix, diagonal elements are always positive (from Castigliano’s first theorem). The stiffness matrix can be developed only when the structure is stable. The stiffness matrix is the inverse of the flexibility matrix.

Hence, all the statements are correct.