Matrix Method of Analysis GATE Civil Engineering Notes by Krishna Yadav Sir

The Matrix Method of Analysis is a fundamental topic in Structural Analysis for GATE Civil Engineering. It provides a systematic approach to analyse statically indeterminate structures such as trusses, beams, and frames. It relies on fundamental concepts of stiffness and flexibility to formulate system equations. 

This method is crucial for understanding how forces and displacements interrelate within a structural system, forming the basis for advanced computational analysis. Master Matrix Method of Analysis by Krishna Yadav Sir to boost your GATE Civil Engineering preparation.

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Matrix Method of Analysis: Stiffness vs. Flexibility

Stiffness and flexibility are important matrix methods of structural analysis, but for GATE, the Stiffness Method is far more important. Here’s a clear, exam-oriented comparison.

Check: GATE Civil Engineering Notes

Relationship between Stiffness and Flexibility

From their definitions, a direct relationship exists between stiffness and flexibility.

Given the stiffness equation F = k ⋅ δ, we can rearrange it for displacement: δ = (1/k) ⋅ F.

Comparing this with the flexibility equation δ = f ⋅ F, it is clear that f = 1/k.

This relationship highlights that stiffness and flexibility are inversely proportional to each other. Consequently, the flexibility matrix [F] and the stiffness matrix [K] for a given structure are inverses. If one matrix is known, the other can be derived by inversion:

• [K] = [F]⁻¹

• [F] = [K]⁻¹

Properties of Stiffness and Flexibility Matrices

Any stiffness or flexibility matrix developed for a structure consistently exhibits three important properties, useful for verification and problem-solving:

1. Square Matrix: Both stiffness and flexibility matrices are always square matrices. Their order is n x n, where 'n' represents the number of degrees of freedom or redundant forces.

2. Positive, Non-Zero Diagonal Elements: The diagonal elements (K11, K22, F11, F22, etc.) are always positive and non-zero. This is because applying a force or displacement in a specific direction always produces a corresponding displacement or force in that same direction.

3. Symmetric (Maxwell's Reciprocal Theorem): The matrices are symmetric about their main diagonal. This means diagonally opposite (off-diagonal) elements are equal in both magnitude and sign (e.g., F12 = F21, K23 = K32). This property is a direct consequence of Maxwell's Reciprocal Theorem.

How to Develop the Flexibility Matrix

The flexibility matrix method is a force method, where the primary unknowns are the redundant reactions in an indeterminate structure. The steps to develop the flexibility matrix are explained below:

Step 1: Determine the Order of the Matrix

• Calculate the Degree of Static Indeterminacy (DS) of the structure.

• The order of the flexibility matrix is DS x DS. For a fixed-fixed beam, DS = 3, resulting in a 3x3 flexibility matrix.

Step 2: Define the Primary Structure and Coordinate Directions

• A Primary Structure is formed by removing the redundant reactions, ensuring the resulting structure is determinate and stable. For a fixed-fixed beam, removing reactions at one support creates a cantilever beam, which is a valid primary structure.

• Coordinate Directions are assigned along the directions of the chosen redundant reactions. For a cantilever, these might be vertical force (1), horizontal force (2), and moment (3) at the free end.

Step 3: Calculate the Matrix Elements

• Each element Fij represents the displacement in direction i caused by a unit force applied in direction j.

Example Calculation (for a Cantilever Primary Structure with length L, flexural rigidity EI, axial rigidity AE):

• Column 1 (Unit vertical force in direction 1):

• F11: Vertical displacement in direction 1 = L³ / 3EI

• F21: Horizontal displacement in direction 2 = 0

• F31: Rotation in direction 3 = -L² / 2EI (negative due to opposite direction)

• Column 2 (Unit horizontal force in direction 2):

• F12: Vertical displacement in direction 1 = 0

• F22: Horizontal displacement in direction 2 = L / AE

• F32: Rotation in direction 3 = 0

• Column 3 (Unit moment in direction 3):

• F13: Vertical displacement in direction 1 = -L² / 2EI (negative due to opposite direction; F13 = F31, confirming symmetry)

• F23: Horizontal displacement in direction 2 = 0

• F33: Rotation in direction 3 = L / EI

Final Flexibility Matrix [F]:

How to Develop the Stiffness Matrix

The stiffness matrix method is a displacement method, where the primary unknowns are the possible joint displacements (deflections and rotations).

Step 1: Determine the Order of the Matrix

• Calculate the Degree of Kinematic Indeterminacy (DK) of the structure.

• The order of the stiffness matrix is DK x DK. For a cantilever beam, the free end has three possible displacements (horizontal, vertical translation, and rotation), so DK = 3, resulting in a 3x3 stiffness matrix.

Step 2: Define the Primary Structure and Coordinate Directions

• In the stiffness method, the original structure itself serves as the primary structure.

• Coordinate Directions are assigned along the possible unknown joint displacements. For the cantilever example, these are horizontal displacement (1), vertical displacement (2), and rotation (3) at the free end.

Step 3: Calculate the Matrix Elements

• Each element Kij represents the force developed in direction i when a unit displacement is applied in direction j, while all other coordinate directions are restrained.

Example Calculation (for a Cantilever Beam with length L, flexural rigidity EI, axial rigidity AE):

• Column 1 (Unit horizontal displacement in direction 1, restraining others):

• K11: Horizontal force in direction 1 = AE / L

• K21: Vertical force in direction 2 = 0

• K31: Moment in direction 3 = 0

• Column 2 (Unit vertical displacement in direction 2, restraining others):

• K12: Horizontal force in direction 1 = 0

• K22: Vertical force in direction 2 = 12EI / L³

• K32: Moment in direction 3 = 6EI / L²

• Column 3 (Unit rotation in direction 3, restraining others):

• K13: Horizontal force in direction 1 = 0

• K23: Vertical force in direction 2 = 6EI / L² (K23 = K32, confirming symmetry)

• K33: Moment in direction 3 = 4EI / L

Final Stiffness Matrix [K]:

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