(SEM V) THEORY EXAMINATION 2024-25 STRUCTURAL ANALYSIS

Subject Code: BCE502
Maximum Marks: 70
Time: 3 Hours
Paper ID: 310458

Question Paper Overview

SECTION A (2 × 7 = 14 Marks)

(Short theoretical questions — definitions, principles, and basic concepts)

a. Classify structures based on stability.
b. Where should loads be applied on a pin-jointed truss?
c. Give the classification of trusses.
d. List the assumptions made while finding forces in a truss.
e. What is the use of Maxwell’s Reciprocal Theorem?
f. State the advantages of Influence Line Diagrams (ILDs).
g. Explain the principle of Eddy’s Theorem.

SECTION B (Attempt any three × 7 = 21 Marks)

(Medium-length derivation or application-based problems)

a. Determine the degree of indeterminacy of the truss shown in figure.
b. Explain simple trusses (with sketch) that are stable in form independent of support conditions.
c. A continuous beam ABC of equal spans l undergoes settlement:

Support B sinks by δ₁, and support C sinks by δ₂.

Find reactions using the strain energy method.
d. A UDL of 15 kN/m (3 m long) crosses a 10 m span girder — find the maximum shear force (S.F.) and bending moment (B.M.) at a section 4 m from the left support.
e. Formulate the equations for a three-hinged parabolic arch and a three-hinged circular arch.

SECTION C (Attempt one part from each question × 7 = 35 Marks)

Q3

(a) Define static indeterminacy and give at least two examples in trusses.
OR
(b) A cable of 100 m span with ends 8 m above its lowest point carries a UDL of 10 kN/m.

Find horizontal and vertical reactions at supports and the length of cable.

Q4

(a) For a Warren-type cantilever truss, determine member forces using the method of tension coefficients.
OR
(b) State the conditions for zero-force members in a truss.

Q5

(a) A simply supported beam (6 m) has a 45 kN load at 2 m from left support.

Find deflection under the load using unit load method.

Take E = 200×10⁶ kN/m² and I = 14×10⁻⁶ m⁴.
OR
(b) A beam ABCD (30 m) is simply supported at A and D, with I-values:

I₁ = I (AB = 10 m),

I₂ = 3I (BC = 10 m),

I₃ = 2I (CD = 10 m).

Loads: 150 kN at B, 300 kN at C.

Using Conjugate Beam Method, determine slopes and deflections at A, B, C, D.

Given: E = 200 kN/mm², I = 2×10¹⁰ mm⁴.

Q6

(a) Explain Müller–Breslau’s Principle and verify with an example.
OR
(b) Draw influence line diagrams (ILDs) for S.F. and B.M. at a section 3 m from left end of a 12 m simply supported beam, and determine max S.F. and max B.M. due to a 5 m UDL of 2 kN/m.

Q7

(a) Define normal thrust and radial shear force for three-hinged arches. Derive expressions for both.
OR
(b) A three-hinged parabolic arch (span = 20 m, rise = 4 m) carries UDL = 20 kN/m on half span.

Find maximum bending moment for the arch.

Key Topics for Revision

1. Classification of Structures

Based on stability:

Stable, Unstable, Determinate, Indeterminate.

Static determinacy condition:

• Ds=r+m−2jD_s = r + m - 2jDs​=r+m−2j

where rrr = reactions, mmm = members, jjj = joints.

2. Trusses

Loads in truss: Applied only at joints (assumed pin-connected).

Types: Simple, Compound, Complex.

Assumptions:

Members are pin-jointed.

Loads act only at joints.

Self-weight neglected.

Members carry only axial force (tension or compression).

Zero-force members:

At a joint with two non-collinear members & no load, both are zero-force.

At a joint with three members and two are collinear, the third is zero-force.

3. Maxwell’s Reciprocal Theorem

States that:

• δAB=δBA\delta_{AB} = \delta_{BA}δAB​=δBA​

Deflection at point A due to unit load at B = deflection at B due to unit load at A.

Used in flexibility methods.

4. Influence Line Diagrams (ILDs)

Show variation of internal forces (S.F., B.M., reaction) at a specific point as a unit load moves across the structure.

Applications: Analysis of moving loads, bridges, cranes, etc.

Advantages:

Determine max values under any moving load.

Essential for bridge and railway design.

5. Theorems of Structural Analysis

Eddy’s Theorem: Relation between strain energy and deflection.

Müller–Breslau’s Principle:
Shape of ILD for a function = deflected shape of structure when that function is given a unit displacement.

6. Bending and Deflection Methods

Unit Load Method:

• δ=∫MmEIdx\delta = \int \frac{M m}{EI} dxδ=∫EIMm​dx

where MMM = moment due to actual load, mmm = moment due to unit load.

Conjugate Beam Method: Converts real beam → imaginary beam to find slopes and deflections.

7. Arches

Normal Thrust (N): Component of arch reaction along its axis.

Radial Shear (Q): Component perpendicular to axis.

For Parabolic Arch:

• y=4hL2x(L−x)y = \frac{4h}{L^2}x(L-x)y=L24h​x(L−x)

For Circular Arch:

• R=L2+4h28hR = \frac{L^2 + 4h^2}{8h}R=8hL2+4h2​

8. Cables

Parabolic cable carrying UDL (w per unit length):

Horizontal thrust:

• H=wL28hH = \frac{wL^2}{8h}H=8hwL2​

Maximum tension:

• Tmax=H2+(wL/2)2T_{max} = \sqrt{H^2 + (wL/2)^2}Tmax​=H2+(wL/2)2​

Length of cable:

• l=L+8h23Ll = L + \frac{8h^2}{3L}l=L+3L8h2​

9. Indeterminacy

Static Indeterminacy (Dₛ):

• Ds=r+m−2jDₛ = r + m - 2jDs​=r+m−2j

Kinematic Indeterminacy (Dₖ): Number of independent joint displacements.

Examples: Fixed beam, continuous beam, redundant truss.

10. Important Formulas for Quick Revision