Matrix Method | Stiffness Method for Structural Analysis

MATRIX METHOD OFSTRUCTURE ANALYSIS

* The number of equations required over and above the equations of static equilibrium for the

analysis of a structure is known as the degree of static indeterminacy or degree of redundancy.

* The number of equilibrium conditions required to find the displacement components of all joints of

the structure are known as the degree of kinematic indeterminacy or degree of freedom of the

structure.

* The systematic development of consistent deformation method has led to flexibility method which is

also known as the force method or compatibility method.

* The systematic development of slope deflection method in the matrix form has led to stiffness matrix

method which is also known as displacement or equilibrium method.

* The element dij of a flexibility matrix is the displacement in coordinate i due to a unit force applied

at coordinate j.

* The element of stiffness matrix kij

is the force at coordinate i due to unit displacement at coordinate

j.

The flexibility and stiffness matrices are inverse of each other.

Steps involved in flexibility method

1. Determine the degree of static indeterminacy.

2. Choose the redundant.

3. Assign the coordinates to the redundant force directions.

4. Remove the restraints to redundant forces to get basic determinate structure.

5. Determine the deflections in the coordinate directions due to given loading in basic determinate

structure.

6. Determine flexibility matrix.

7. Apply the compatibility conditions:

P = [d]

–1

[D – Di

]

8. Knowing the redundant forces, compute the member forces.

Stiffness matrix method

1. Determine the degree of kinematic indeterminacy.

2. Assign the coordinate numbers to the unknown displacements.

3. Impose restraints in all coordinate directions to get a fully restrained structure.

4. Determine the forces developed in each of the coordinate directions of a fully restrained structure

(PL

).

5. Determine stiffness matrix k.

6. Note the final forces.

7. Form and solve the stiffness equation [k] [D] = P – P2 and find D.

8. Compute member forces using these displacements.

Element approach of stiffness matrix

1. Give coordinate directions to displacements in all Cartesian directions at the end of each element.

2. For each element, find transformation matrix and element stiffness matrix in local coordinate

directions.

3. Transform local stiffness matrix to global system by using the relation [k] = [R]

T

[Ke

] [R] and place

it in the global matrix.

4. Assemble load vector of the system.

5. Write stiffness equations.

6. Impose boundary conditions.

7. Solve the equations.

8. Calculate the required member forces.

9. Calculate support reactions.