MCQ Structural Engineering

MCQ Structural Engineering

Topics: Strength of materials (SF/BM, stress, bending, torsion, columns) · Structural analysis (determinate & indeterminate, force & displacement methods, influence lines, matrix methods)

Q1. At a point of contraflexure in a beam, the bending moment is:

Maximum Minimum but non-zero Zero Equal to shear force

Q2. For a simply supported beam with a central point load W and span L, maximum BM is:

WL/4 WL/8 WL/2 WL

Q3. Shear force is equal to the:

Rate of change of bending moment along the beam Integral of BM Always zero at supports Same as axial force

Q4. Hooke's law is valid within the:

Plastic range only Elastic limit / proportional limit Ultimate stress region Fracture region only

Q5. Principal stresses are:

Maximum and minimum normal stresses on planes of zero shear Always equal to shear stresses Only compressive Independent of orientation

Q6. On Mohr's circle, the centre represents:

Average normal stress (σx+σy)/2 Maximum shear only Principal shear Zero stress

Q7. Maximum shear stress in plane stress equals:

(σ₁ − σ₂)/2 σ₁ + σ₂ σ₁ σ₂ Zero always

Q8. Simple bending theory assumes:

Plane sections remain plane Material is plastic only Shear deformation dominates Large deflections only

Q9. Flexural formula is:

σ = My/I σ = P/A τ = VQ/(Ib) σ = Eε only

Q10. Shear stress in a beam due to transverse shear is given by:

τ = VQ/(Ib) τ = My/I τ = P/A τ = Eγ

Q11. Shear centre is the point in the cross-section through which shear force produces:

Only bending, no torsion Only torsion No stress Only axial stress

Q12. In a thin cylindrical pressure vessel, hoop stress is:

pd/(2t) pd/(4t) pd/t 2pd/t

Q13. Torsion formula for a circular shaft is:

T/J = τ/r = Gθ/L σ = My/I τ = VQ/(Ib) P/A = σ

Q14. Euler's critical load for a column hinged at both ends is:

π²EI/L² π²EI/(2L)² 2π²EI/L² πEI/L²

Q15. For a column fixed at both ends, effective length is:

L L/2 2L 0.7L approximately in design codes sometimes

Q16. Degree of static indeterminacy of a plane truss with m members, j joints and r reactions is:

m + r − 2j m + r − 3j 2j − m − r 3j − m − r

Q17. Influence line for a reaction at a support of a simply supported beam is:

A straight line varying from 1 at that support to 0 at the other A parabola Constant 1 Zero everywhere

Q18. Three-hinged arch is:

Statically determinate Indeterminate to 1 degree Indeterminate to 2 degrees Unstable always

Q19. Method of consistent deformation (flexibility method) is a:

Force method Displacement method Finite element only None

Q20. Slope-deflection method treats as unknowns the:

Member end moments related to slopes and displacements Only reactions Only axial forces Redundant forces only

Q21. Moment distribution method was developed by:

Hardy Cross Castigliano Maxwell Euler

Q22. Castigliano's second theorem is used to find:

Displacements from strain energy (∂U/∂Pi = δi) Only forces Buckling load only None

Q23. In moment distribution, the distribution factor for a member is:

Its stiffness / sum of stiffnesses at the joint Always 0.5 Inverse of length only Zero

Q24. Carry-over factor for a prismatic beam member fixed at far end is:

+1/2 −1/2 +1 0

Q25. Stiffness method (displacement method in matrix form) uses:

K u = F F u = K Flexibility matrix only None

Q26. Flexibility method uses as primary unknowns:

Redundant forces Joint displacements Only temperatures None

Q27. For combined axial load P and moment M on a short column, maximum stress is:

P/A ± My/I P/A only My/I only V/A

Q28. Unsymmetrical bending occurs when:

Load plane does not coincide with a principal axis of the section Section is circular Load is pure axial None

Q29. Maxwell's reciprocal theorem states that:

δij = δji (displacement at i due to load at j equals displacement at j due to load at i) Forces are always equal Only for indeterminate structures None

Q30. Kinematic indeterminacy (degrees of freedom) of a fixed beam is:

0 1 2 3

Structural Engineering – Theory & Formulae

📖 Shear Force & Bending Moment

$$ \frac{dV}{dx} = -w,\quad \frac{dM}{dx} = V $$

Sign conventions must be consistent. Point of contraflexure: \(M=0\) and changes sign. For SS beam, central point load \(W\): \(M_{\max}=WL/4\); UDL \(w\): \(M_{\max}=wL^2/8\).

📖 Stress–Strain & 2D Stress

Hooke: \(\sigma=E\varepsilon\) within proportional limit. Plane stress transformation:

$$ \sigma_n = \frac{\sigma_x+\sigma_y}{2}+\frac{\sigma_x-\sigma_y}{2}\cos 2\theta + \tau_{xy}\sin 2\theta $$

$$ \tau = -\frac{\sigma_x-\sigma_y}{2}\sin 2\theta + \tau_{xy}\cos 2\theta $$

Principal stresses: \(\sigma_{1,2}=\frac{\sigma_x+\sigma_y}{2}\pm\sqrt{\big(\frac{\sigma_x-\sigma_y}{2}\big)^2+\tau_{xy}^2}\); \(\tau_{\max}=|\sigma_1-\sigma_2|/2\).

Mohr's circle: centre \((\frac{\sigma_x+\sigma_y}{2},0)\); radius = \(\tau_{\max}\).

📖 Simple Bending, Shear & Shear Centre

$$ \frac{\sigma}{y} = \frac{M}{I} = \frac{E}{R},\quad \sigma=\frac{My}{I} $$

$$ \tau = \frac{VQ}{Ib} $$

Assumptions: homogeneous isotropic material, plane sections remain plane, linear elasticity, pure bending (or approx. for slender beams).

Unsymmetrical bending: resolve \(M\) on principal axes or use \(\sigma = \frac{M_y I_z - M_z I_{yz}}{I_y I_z - I_{yz}^2}z + \cdots\).

Shear centre: point in the cross-section through which transverse shear causes bending without torsion (important for open thin-walled sections).

📖 Thin Pressure Vessels, Torsion, Columns

Thin cylinder: \(\sigma_h = pd/(2t)\), \(\sigma_l = pd/(4t)\). Thin sphere: \(\sigma = pd/(4t)\).

Circular shaft torsion:

$$ \frac{T}{J} = \frac{\tau}{r} = \frac{G\theta}{L} $$

Euler buckling:

$$ P_{cr} = \frac{\pi^2 EI}{(KL)^2} $$

\(K\): both hinged 1.0; both fixed 0.5; fixed–free 2.0; fixed–hinged ≈ 0.7.

Combined axial + bending: \(\sigma = P/A \pm My/I\) (short members; check buckling for slender compression members).

📖 Determinate Structures

Plane truss: \(D_s = m + r - 2j\). Space truss: \(m + r - 3j\).

Beams/frames: count unknowns vs equilibrium equations (+ internal releases).

Three-hinged arch/cable: determinate. Influence lines for determinate beams: use unit load and equilibrium (or Müller-Breslau qualitative shape).

📖 Indeterminate Analysis

Force (flexibility) methods: choose redundants; compatibility of displacements; Castigliano / virtual work for flexibility coefficients. Maxwell: \(f_{ij}=f_{ji}\).

Displacement methods:

• Slope-deflection: \(M_{ab}=M_{ab}^F + \frac{2EI}{L}(2\theta_a+\theta_b-3\psi)\)
• Moment distribution (Hardy Cross): distribution factor \(DF_i=K_i/\sum K\); carry-over factor \(+1/2\) for prismatic member fixed at far end; balance joints iteratively.

Matrix methods:

• Stiffness: \([K]\{u\}=\{F\}\) — displacement method; standard in software.
• Flexibility: \([F]\{X\}=\{\Delta\}\) — force method.

Kinematic indeterminacy \(D_k\) = number of independent joint displacement DOFs; static indeterminacy \(D_s\) = number of redundant forces.

📖 Energy Methods (brief)

Strain energy \(U=\int\frac{M^2 dx}{2EI}+\int\frac{P^2 dx}{2AE}+\cdots\). Castigliano: \(\delta_i=\partial U/\partial P_i\). Unit load (virtual work) method for displacements in determinate structures.

Yogendra Gopal Borse

Civil Engineer | Assistant Engineer Grade-I, Maharashtra PWD

B.Tech (Civil) from VJTI Mumbai. Experienced in bridge design, road works, estimation, project monitoring and digital engineering tools. Creator of YogiPWD – practical technical resources for civil engineers.

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