(SEM V) THEORY EXAMINATION 2023-24 STRUCTURAL ANALYSIS

Subject Code: KCE502

Subject Name: Structural Analysis

Course: B.Tech (Semester V)

Duration: 3 Hours

Maximum Marks: 100

Sections: A, B, and C

Paper Type: Analytical + Theoretical + Numerical

SECTION A – Short Answer Questions (2 × 10 = 20 Marks)

Attempt all questions briefly.

Focus for Section A:
Memorize key formulas and short definitions:

Strain Energy=12σεV\text{Strain Energy} = \frac{1}{2} \sigma \varepsilon VStrain Energy=21​σεV

Influence line = graphical representation of structural response (shear/moment) at a specific section.

Castigliano’s theorem: “Partial derivative of strain energy with respect to load gives corresponding deflection.”

SECTION B – Medium-Length Questions (10 × 3 = 30 Marks)

Attempt any three.

Find the SI & KI (Static and Kinematic Indeterminacy) of the given truss and frame (see figure on page 1).

Analyze a truss using the Tension Coefficient Method for given loads (80 kN and 60 kN, span = 8 m, angles 60°).

State and prove Maxwell’s Reciprocal Theorem.

Draw Influence Lines for shear and bending moment for a 30 m girder carrying a 150 kN moving load (section 10 m from left).

For a three-hinged semicircular arch, derive the horizontal thrust and maximum bending moment due to UDL www.

Key Concepts:

SI (Static Indeterminacy):
SI=r−(2j−m)\text{SI} = r - (2j - m)SI=r−(2j−m) for 2D truss.

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

Maxwell’s Theorem: δij=δji\delta_{ij} = \delta_{ji}δij​=δji​

Influence Line Formula: xL\frac{x}{L}Lx​ or 1−xL1 - \frac{x}{L}1−Lx​, depending on section location.

SECTION C – Analytical / Long Questions (10 × 5 = 50 Marks)

Q3. Cables

a. A 100 m cable with one end 4 m higher carries UDL w=10kN/mw = 10 kN/mw=10kN/m. Sag = 6 m.

Find horizontal tension and maximum tension.
b. Derive length of cable formula when ends are at same level.
L=l2+8h23L = \sqrt{l^2 + \frac{8h^2}{3}}L=l2+38h2​​ (approximation for small sag).

Q4. Truss Analysis

a. Explain Method of Substitution and Tension Coefficient Method.
b. Find forces in members for the truss shown (70 kN central load, span 6 m).

Q5. Deflection & Theorems

a. Determine vertical deflection at point C in a frame (given E=200kN/mm2,I=30×106mm4E = 200 kN/mm^2, I = 30 \times 10^6 mm^4E=200kN/mm2,I=30×106mm4).
b. Use Unit Load Method to find deflection and rotation in a cantilever (page 3 figure).

Formula Reminder:

δ=PL33EI\delta = \frac{PL^3}{3EI}δ=3EIPL3​ (for cantilever, tip deflection)

θ=PL22EI\theta = \frac{PL^2}{2EI}θ=2EIPL2​ (for rotation)

δ=∫M⋅mEIdx\delta = \int \frac{M \cdot m}{EI} dxδ=∫EIM⋅m​dx (unit load method)

Q6. Influence Lines and Moving Loads

a. State propositions for moving loads on simply supported beams; prove first proposition.
b. For UDL of 30 kN/m, length = 15 m, beam span = 60 m:

Find max B.M. at section 20 m.

Find absolute maximum shear and B.M.

Key Formula:

Max B.M. under UDL = wL28\frac{wL^2}{8}8wL2​

Max bending under partial load: Mmax=wl2(L−l2)M_{max} = \frac{w l}{2} (L - \frac{l}{2})Mmax​=2wl​(L−2l​)

Q7. Arches

a. Prove parabolic shape is funicular for a three-hinged arch under UDL.
b. For span 60 m, rise 12 m, and UDL = 30 kN/m over left half + point load 120 kN at quarter span:

Find bending moment, normal thrust, and radial shear at 15 m from left.

Important Formulas:

Horizontal thrust for 3-hinged parabolic arch:
H=wL28hH = \frac{wL^2}{8h}H=8hwL2​

Bending Moment at x:
M=Hy−wx(L−x)/2M = H y - w x (L - x)/2M=Hy−wx(L−x)/2

Key Topics to Study

Structural Basics

Types of structures and loads (dead, live, wind, seismic)

Determinacy and Stability                                        Types of supports and reactions

Energy Methods

Strain energy in bending and axial load                  Castigliano’s and Maxwell’s theorems

Unit load and virtual work methods

Deflection and Slope

Moment-area and conjugate beam methods          Unit load method applications

Deflection of beams, frames, and trusses

Influence Lines

ILD for simply supported beams                         Shear force and bending moment under moving loads

Absolute maximum conditions

Arches & Cables

Parabolic arches and funicular shapes                   Horizontal thrust and bending in 3-hinged arches

Cable tension and sag relationship

Study Strategy

Revise short theory for Section A — they appear directly from class notes.

Practice numerical derivations — cables, arches, influence lines.

Draw diagrams neatly — truss analysis, deflection shapes, parabolic arches.

Memorize key relationships:

H=wL28h\text{H} = \frac{wL^2}{8h}H=8hwL2​

δ=MmEI\delta = \frac{M m}{EI}δ=EIMm​

SI=r−(2j−m)SI = r - (2j - m)SI=r−(2j−m)