Test: Matrix Method of Analysis - Civil Engineering (CE) MCQ

Concept:

Stiffness matrix (K) is inverse of a flexible matrix (F) or vice versa.

K = [F]^-1 or F = [K]^-1

Calculation:

Concept:

Development of stiffness matrix:

To develop jth column of a stiffness matrix – unit displacements will be given in jth coordinate direction without giving displacement in any other coordinate directions and forces developed in all the coordinate direction are found out.

Ist Column:

K₁₃ = 0

K₂₃ = 0

Important Point:

• Stiffness Matrix is a displacement method of analysis in which Degree of Kinematic Indeterminacy is found and number of coordinate direction are chosen accordingly. The structure is analysed by developing stiffness matrix.
• Flexibility Matrix is a force method of analysis in which Degree of Static Indeterminacy is found and number of coordinate direction are chosen accordingly. The structure is analysed by developing flexibility matrix.

Given: 

Flexibility matrix is inverse of the stiffness matrix
So, A-1 is the stiffness matrix of A

∴The flexibility matrix of given stiffness matrix is 

Stiffness Matrix Method

1. This method is also known as displacement method or equilibrium method.

2. It is suitable if static indeterminacy (D_s) > kinematic indeterminacy (D_k­) for the structure.

Stiffness is defined as Force required to produce unit displacement, k = P/Δ

K₁₂ → Force at 2 due to unit displacement at 1

Flexibility Matrix Method

1. This method is also known as force method or compatibility method.

2. It is suitable if kinematic indeterminacy (D_k) > static indeterminacy (D_s) for the structure.

Flexibility is defined as Displacement caused due to unit Force.

δ₁₂ → Displacement at 2 due to Force at 1

∴ To generate a j^th column of flexibility matrix a unit force is applied at coordinate j and the displacement are calculated at all coordinates.

Concept:

Stiffness Matrix:

The elements of the stiffness matrix are the forces that are required to hold the restrained structure with a unit displacement at one of the coordinates and with zero displacements at all other coordinates are determined.

Hence,

For the fixed beam shown below -

Let, the end A is given translational displacement in an upward direction keeping all other coordinates fixed,

The following figure shows the forces required to hold the structure in the given position -

Hence, from the above figure, the end forces generated at B are -

• Vertical force = 12EI/L³ (Downward) = - 12EI/L³

Negative sign because opposite to the direction of displacement.

• Moment = 6EI/L2 (Anticlockwise)

If we denote stiffness matrix as M and flexibility matrix as Δ

It is stiffness matrix, and then flexibility matrix is: Δ = K^-1

Calculation:

The difference between flexibility matrix and stiffness matrix is as follows:

Flexibility and stiffness matrix:

Stiffness Matrix Method:

• The Stiffness method provides a very systematic way of analyzing determinate and indeterminate structures.
• These local (member) force-displacement relationships can be easily established for all the members in the truss, simply by using the given material and geometric properties of the different members.
• In the stiffness matrix method, nodal displacements are treated as the basic unknowns for the solution of indeterminate structures.

The properties of the stiffness matrix are

•  It is a symmetric matrix
•  The sum of elements in any column must be equal to zero. 
•  It is an unstable element therefore the determinant is equal to zero.
• The joint displacements are treated as basic unknowns
• The number of displacements involved is equal to the no of degrees of freedom of the structure
• The method is the generalization of the slope deflection method.
•  The same procedure is used for both determinate and indeterminate structures.
• Deflection and Rotation are the quantity taken as redundant.

Concept:

Flexibility matrix:

• The redundant forces are identified and assign one co-ordinate to each redundant.
• The redundant are released to obtained to obtain the primary structure.
• Due to external load, the displacements in the direction of various co-ordinate and at the location of co-ordinate are found out.
• Positive sign is taken when the direction of slope/deflection is the same as the original moment/force.
• Negative sign taken when the direction of slope/deflection is opposite as of original moment/force

Slope at the point A due to the moment at A is given by

θ_α = ML/3EI

Slope at point B due to moment at point A is given by

θ_b = ML/6EI

Calculation:

Given: EI = Constant

First, apply unit moment at A and draw its deflected shape

δ₁₁ = L/3EI (Same as of original moment) and δ₂₁ = L/6EI (same as of original moment)

Now apply unit moment at B and draw its deflected shape

δ₂₁ = L/6EI(Same as of original moment) and δ₂₂ = L/3EI(Same as of original moment)

So, The flexibility matrix will be

∴ The flexibility matrix is given by