THEORY EXAMINATION (SEM–VI) 2016-17 MATRIX ANALYSIS OF STRUCTRES

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]=L3EI​​126L−126L​6L4L2−6L2L2​−12−6L12−6L​6L2L2−6L4L2​​

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.