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MATRIX ANALYSIS OF STRUCTURES (ECE012)
SECTION – A
(10 × 2 = 20 Marks | Short Answers)
(a) Relation between Flexibility and Stiffness
Flexibility matrix [F] is the inverse of stiffness matrix [K].
[F]=[K]−1[F] = [K]^{-1}[F]=[K]−1
Stiffness relates force to displacement, while flexibility relates displacement to force.
(b) Flexibility Matrix
Flexibility matrix gives displacements per unit force applied at degrees of freedom. It is used mainly in the force (flexibility) method.
(c) Stiffness Matrix
Stiffness matrix represents forces required to produce unit displacements at the degrees of freedom. It is the basis of the displacement method.
(d) Degree of Freedom (DOF)
Degree of freedom is the number of independent displacements (translations or rotations) required to define the deformed shape of a structure.
(e) Yielding of Supports
Yielding of supports refers to support settlement or rotation, which introduces additional displacements affecting internal forces.
(f) Displacement Method
In displacement method, displacements are treated as unknowns, and equilibrium equations are written in terms of stiffness.
(g) Matrix Inversion
Matrix inversion is the process of finding a matrix [A]⁻¹ such that:
[A][A]−1=[I][A][A]^{-1} = [I][A][A]−1=[I]
It is required to convert stiffness matrix to flexibility matrix.
(h) Translational Stiffness
Translational stiffness is the force required to produce unit linear displacement at a joint.
(i) Kinematic Indeterminacy
Kinematic indeterminacy is the number of independent joint displacements in a structure.
(j) Structural Stability
A structure is stable if it maintains equilibrium and does not undergo rigid body motion under applied loads.
SECTION – B
(Attempt Any Five | 5 × 10 = 50 Marks)
(a) Indeterminacy of 3-Span Continuous Beam (Fixed at Both Ends)
For continuous beams:
Static indeterminacy (DS) depends on number of reactions and equilibrium equations.
Fixed supports increase redundancy.
For a 3-span beam fixed at both ends, the structure is statically indeterminate.
Suitable method:
Stiffness (displacement) method, as it efficiently handles fixed supports and continuity.
(b) Stiffness Matrix of One-Span Beam (4 m, Fixed at Both Ends)
For a beam element with length L and constant EI:
[K]=EIL3[126L−126L6L4L2−6L2L2−12−6L12−6L6L2L2−6L4L2][K] = \frac{EI}{L^3} \begin{bmatrix} 12 & 6L & -12 & 6L \\ 6L & 4L^2 & -6L & 2L^2 \\ -12 & -6L & 12 & -6L \\ 6L & 2L^2 & -6L & 4L^2 \end{bmatrix}[K]=L3EI126L−126L6L4L2−6L2L2−12−6L12−6L6L2L2−6L4L2
Substitute L = 4 m to obtain numerical values.
(c) Static and Kinematic Indeterminacy of Pin-Jointed Frames
Static indeterminacy:
Depends on number of members, joints, and reactions.
Kinematic indeterminacy:
Equals number of independent joint displacements (translations only, no rotations in pin joints).
Pin-jointed frames generally have low kinematic indeterminacy.
(d) Transfer Matrix Method
This method relates forces and displacements at one end of a member to the other end using transfer matrices.
Advantages:
Suitable for computer programming
Useful for linear structures
Limitation:
Less efficient for large complex frames.
(e) Computer-Oriented Stiffness Method
In this method:
Element stiffness matrices are formed
Assembled into global stiffness matrix
Boundary conditions applied
Equations solved using matrix operations
This method is ideal for computer-based structural analysis.
(f) Substructure Technique
Large structures are divided into smaller substructures.
Advantages:
Reduces computational effort
Suitable for very large buildings and bridges
Enables parallel processing
(g) Stiffness Matrix of One-Span Beam (Pinned at Both Ends)
For pinned ends:
No moment resistance at supports
Rotational DOFs eliminated
The stiffness matrix reduces accordingly by removing rotational stiffness terms.
(h) Force Method
In force method:
Redundant forces are treated as unknowns
Compatibility equations are written
Flexibility matrix is used
It is effective for small statically indeterminate structures.
SECTION – C
(Attempt Any Two | 2 × 15 = 30 Marks)
Indeterminacy of a 10-Storey Building Frame
Given:
10 storeys
5 bays in one direction
8 bays in other direction
(a) Bases Fixed
Static indeterminacy: High due to fixed supports and multiple bays
Kinematic indeterminacy: Depends on number of joints × DOF per joint
(b) Bases Hinged
Static indeterminacy reduces
Kinematic indeterminacy remains high due to joint displacements
Such problems are best solved using the stiffness method.
Two-Span Beam (Each 4 m, Fixed Ends, UDL = 30 kN/m)
Steps:
Determine fixed-end moments
Assemble stiffness matrix
Apply boundary conditions
Solve for joint rotations
Compute final moments and reactions
Given EI = constant, stiffness method gives accurate results.
Settlement of Intermediate Support in Two-Span Beam
Procedure:
Assume unknown reaction at settled support
Write compatibility condition for settlement
Use flexibility or stiffness method
Calculate additional moments due to settlement
Support settlement introduces secondary stresses, which must be included in final design.