Question 1

Shear Stress Distribution – Rectangular Section For a prismatic member with a rectangular cross-section under transverse shear, what is the ratio of maximum shear stress to average shear stress over the section?

Accepted Answer

Introduction / Context: The shear stress distribution in beams depends on cross-section shape. For rectangles, the distribution is parabolic with a maximum at the neutral axis and zero at the top and bottom fibers. Engineers frequently use the ratio of maximum to average shear to relate nominal design values to peak stresses. Given Data / Assumptions: Cross-section: rectangle, width b, depth h. Transverse shear force = V. Linear elastic behavior, small deformations, Euler–Bernoulli kinematics. Average shear stress defined as V divided by area b * h. Concept / Approach: Exact shear stress distribution comes from the elastic formula tau = V * Q / (I * b), where Q is the first moment of area about the neutral axis of the portion above or below the point, I is the second moment of area about the neutral axis, and b is the local width. For a rectangle, this distribution is parabolic with a maximum at mid-depth. Step-by-Step Solution: Average shear stress: tau_avg = V / (b * h)For a rectangle: tau_max = 3/2 * tau_avgTherefore, ratio R = tau_max / tau_avg = 3/2 = 1.5 Verification / Alternative check: Integrating the parabolic distribution over the depth reproduces the total shear force V, confirming that the average equals V / (b * h) and the peak equals 1.5 times this value. Why Other Options Are Wrong: 1 implies a uniform distribution, which is not valid for rectangles under elastic theory. 1.25 and 2.0 are typical of other contexts but not for rectangular sections. 2.5 is excessively high and contradicts the derived parabolic profile. Common Pitfalls: Confusing rectangular results with those for circular or I-sections, using shear formula with incorrect Q, or evaluating b at the wrong location. Another mistake is to average the parabolic expression incorrectly, leading to wrong peak factors. Final Answer: 1.5

Question 2

Energy in Close-Coiled Helical Spring under Axial Load A close-coiled helical spring with mean coil diameter D, wire diameter d, and n active coils carries an axial load W. For material shear modulus G, what is the strain energy stored in the spring?

Accepted Answer

Introduction / Context: Close-coiled helical springs deform primarily by torsion of the wire. Under an axial load W, both deflection and stored energy can be expressed using torsion theory and the spring rate. Energy expressions are essential for impact, vibration, and fatigue assessments. Given Data / Assumptions: Mean coil diameter = D, wire diameter = d, active coils = n. Axial load = W, shear modulus = G. Close-coiled assumption, negligible direct shear component on energy, linear elastic behavior. Concept / Approach: The axial deflection of a close-coiled spring is delta = 8 W D^3 n / (G d^4). Strain energy U equals one half times load times deflection, or equivalently W^2 divided by twice the spring stiffness k. Step-by-Step Solution: Spring deflection: delta = 8 W D^3 n / (G d^4)Energy: U = 1/2 * W * deltaSubstitute delta: U = 1/2 * W * (8 W D^3 n / (G d^4))Simplify: U = 4 W^2 D^3 n / (G d^4) Verification / Alternative check: Using stiffness k = G d^4 / (8 D^3 n), compute U = W^2 / (2 k) which gives the same value: 4 W^2 D^3 n / (G d^4). Why Other Options Are Wrong: W^2 D n / (2 G d^3) has incorrect powers of D and d. W^2 D^3 n / (2 E d^4) uses the wrong modulus and coefficient. W^2 / (2 k) is correct in form but not a final expression in basic variables for direct comparison. 8 W^2 D^3 n / (G d^4) is off by a factor of 2. Common Pitfalls: Confusing E with G, forgetting the 1/2 factor in energy, or missing the close-coiled assumption that yields the standard deflection formula. Final Answer: 4 W^2 D^3 n / (G d^4)

Question 3

Columns – Effective Length with One End Fixed and the Other Free A straight column of actual length L has one end fixed and the other end free. What is the Euler equivalent length Le used in buckling calculations?

Accepted Answer

Introduction / Context: Euler buckling analysis uses an equivalent length to represent how end restraints affect the critical load. Different end conditions change effective length factors and thus the stability of columns. Given Data / Assumptions: Column of actual length L. One end fixed, the other end free. Perfectly straight, centrally loaded, elastic behavior, pinned-plane assumption for flexure. Concept / Approach: Equivalent length Le is defined such that the Euler critical load Pcr = pi^2 E I / (Le^2) reproduces the correct buckling load for the given boundary conditions. For fixed–free, the effective length factor K equals 2. Step-by-Step Solution: End condition: fixed–free implies K = 2Equivalent length: Le = K * L = 2 LEuler load: Pcr = pi^2 E I / (Le^2) = pi^2 E I / (4 L^2) Verification / Alternative check: Mode shape for fixed–free is a quarter sine, matching the K = 2 value seen in standard tables and energy methods. Why Other Options Are Wrong: L corresponds to pinned–pinned. L / 2 and L / sqrt(2) are too short and overpredict stability. pi * L / 2 is not an effective length factor used for Euler K. Common Pitfalls: Using the wrong K for fixed–pinned or fixed–fixed, or confusing actual length with effective length when computing slenderness ratio. Final Answer: 2 L

Question 4

Area Moment of Inertia – Circle About Any Diameter For a circular cross-section of diameter D, what is the second moment of area about any centroidal diameter in its plane?

Accepted Answer

Introduction / Context: The second moment of area, also called the area moment of inertia, measures a section resistance to bending. For symmetric shapes like circles, closed-form expressions are standard and widely tabulated. Given Data / Assumptions: Section is a solid circle of diameter D. Axis is any centroidal diameter within the section plane. Homogeneous, prismatic, small deflection elastic bending context. Concept / Approach: For a circle, the polar second moment is J = pi D^4 / 32 about the centroid. By the perpendicular axis theorem, J equals the sum of the two equal in-plane centroidal moments about any pair of orthogonal diameters, so each in-plane moment is J/2. Step-by-Step Solution: Polar moment about centroid: J = pi D^4 / 32By symmetry, I_x = I_yPerpendicular axis: J = I_x + I_y = 2 I_xTherefore I_x = J / 2 = (pi D^4 / 32) / 2 = pi D^4 / 64 Verification / Alternative check: Equivalent expression in radius R = D / 2 is I = pi R^4 / 4. Substituting R reproduces pi D^4 / 64. Why Other Options Are Wrong: pi D^4 / 32 is the polar moment, not the planar diameter moment. pi D^3 / 64 and pi D^5 / 64 have wrong dimensions. pi D^4 / 128 is off by a factor of 2. Common Pitfalls: Confusing polar and planar moments or mixing radius and diameter forms without proper conversion. Final Answer: pi D^4 / 64

Question 5

Deflection Comparison – Point Load at Midspan vs Equivalent Uniform Load Two identical simply supported beams have span L and flexural rigidity E I. Beam A carries a point load W at midspan. Beam B carries a uniformly distributed load of intensity w…

Accepted Answer

Introduction / Context: Serviceability checks often compare deflections under different loading patterns for the same total load. Understanding how load distribution affects deflection is crucial for realistic performance predictions. Given Data / Assumptions: Simply supported beams, identical L and E I. Beam A: central point load W. Beam B: uniform load with total W, so w = W / L. Linear elastic, small deflection, prismatic sections. Concept / Approach: Use standard closed-form deflection formulas and take the ratio. The maximum deflection occurs at midspan for both cases under the given loads. Step-by-Step Solution: Beam A: delta_A = W L^3 / (48 E I)Beam B: delta_B = 5 w L^4 / (384 E I) with w = W / LSubstitute: delta_B = 5 (W / L) L^4 / (384 E I) = 5 W L^3 / (384 E I)Form ratio: (delta_A) / (delta_B) = (W L^3 / 48) / (5 W L^3 / 384) = 384 / 240 = 8 / 5 Verification / Alternative check: Energy methods or numerical integration give the same ratio because both deflections scale with W L^3 / (E I), and only coefficients differ. Why Other Options Are Wrong: 5 / 8 is the reciprocal. 4 / 5 and 3 / 2 are not obtained from standard coefficients. 2 ignores the correct constants in the solution. Common Pitfalls: Confusing total load with intensity, applying W as w directly, or forgetting that the UDL deflection coefficient is 5/384. Final Answer: 8 / 5

Question 6

Springs and Elasticity: The load per unit deflection of a spring is called what?

Accepted Answer

Introduction: The question tests basic spring terminology in mechanics of materials. When a spring is loaded, it deflects. The proportionality between the applied load and the resulting deflection is a fundamental property used in design and testing of springs. Given Data / Assumptions: Linear elastic behavior holds (Hooke's law regime). Deflection is small and recoverable (no plastic deformation). We are naming the parameter defined as load per unit deflection. Concept / Approach: For a linear spring, load P and deflection δ follow P = k * δ. Here, k is the spring rate, also known as spring stiffness. Therefore, load per unit deflection equals k. Other terms in the options are standard but mean different things in material testing. Step-by-Step Solution: Identify definition: load per unit deflection ⇒ P/δ.Relate to Hooke’s law for springs: P = k * δ ⇒ P/δ = k.Therefore, the required term is spring stiffness (spring rate). Verification / Alternative check: Units check: If P is in N and δ in m, k is N/m, which is the standard unit of stiffness. This dimensional confirmation verifies the identification. Why Other Options Are Wrong: Proof resilience: Energy per unit volume up to elastic limit, not load/deflection.Proof stress: Stress used to define yield-like behavior (e.g., 0.2% offset), unrelated to spring rate.Proof load: A specified test load in quality checks, not a ratio.Spring index: Ratio of mean coil diameter to wire diameter, a geometry parameter. Common Pitfalls: Confusing stiffness (k) with strength-related terms (proof stress) or energy terms (proof resilience). Another common error is mixing geometric quality factors (spring index) with mechanical response parameters. Final Answer: stiffness.

Question 7

Two parallel bars (bronze and copper), equal length: Identify the correct combined relations among total load, shared loads, stress, and modulus

Accepted Answer

Introduction: This problem concerns load sharing in a composite member made of two prismatic bars (bronze and copper) connected in parallel (equal length, loaded together). The key concepts are compatibility of deformation and equilibrium of forces, which together determine the stresses and load carried by each bar. Given Data / Assumptions: Two bars: bronze (area Ab, modulus Eb) and copper (area Ac, modulus Ec). Equal initial length and common axial deformation (parallel arrangement). Total external load is P, shared as Pb (bronze) and Pc (copper). Linear elastic behavior and perfect bonding (same axial strain). Concept / Approach: Compatibility in parallel requires equal axial strain: εb = εc. Using Hooke’s law, ε = σ/E, so σb/Eb = σc/Ec. Force equilibrium requires total load P to equal the sum of the component loads: P = Pb + Pc. The load in each bar equals its area times its stress: Pb = Abσb and Pc = Acσc. Step-by-Step Solution: Write compatibility: σb/Eb = σc/Ec (equal strain).Use equilibrium: P = Pb + Pc.Relate stress to load in each bar: Pb = Abσb and Pc = Acσc.Combine these statements to identify the option that lists all correct relations. Verification / Alternative check: If Eb ≠ Ec, equal strain implies different stresses. When Ab and Ac are known, the system of equations can be solved for σb and σc for any given P, confirming internal consistency. Why Other Options Are Wrong: Option b: Incorrect—equal lengths in parallel enforce equal strain, not unequal.Option c: Uses σb twice; misses σc term, so it violates Pb + Pc definition.Option d: Equal areas do not force equal loads unless Eb also matches; stiffness matters.Option e: Equal stress would only occur if Eb = Ec under equal strain; not generally true. Common Pitfalls: Assuming equal stress instead of equal strain for members in parallel; overlooking the role of modulus in load sharing; mixing series vs parallel rules. Final Answer: σb/Eb = σc/Ec, P = Pb + Pc, Pb = Abσb, Pc = Acσc.

Question 8

Solid circular shaft under torsion: The shear stress τ is not directly proportional to which quantity?

Accepted Answer

Introduction: The question probes your understanding of torsion in circular shafts and which variables directly govern shear stress distribution. Knowing the exact proportionalities helps avoid design errors and separates material response from geometric and loading effects. Given Data / Assumptions: Homogeneous, isotropic, circular shaft in pure torsion. Linear elastic behavior and Saint-Venant torsion theory apply. Polar moment of inertia J is for a solid circle: J = (π/32) * d^4. Concept / Approach: For a circular shaft under applied torque T, shear stress varies linearly with radius: τ(r) = T * r / J. This expression shows which parameters directly control τ for a given T: radius r (linearly), polar section property J (inversely), and torque T (linearly). Length L and modulus G do not appear in τ = Tr/J; they govern angle of twist, not stress, via θ = TL/(GJ). Step-by-Step Solution: Start from τ(r) = T * r / J (no L or G present).Therefore, τ is directly proportional to r and T and inversely to J.Angle of twist depends on L and G: θ = TL/(GJ), not τ.Hence τ is not directly proportional to shaft length L. Verification / Alternative check: Dimensional analysis of τ = Tr/J confirms independence from L and G for stress under given torque. Why Other Options Are Wrong: Radius of the shaft: τ increases linearly with radius.Angle of twist: Although related to T, τ is not directly proportional to θ unless L and G are fixed; the primary stress formula does not use θ.Modulus of rigidity: G influences θ, not τ = Tr/J.Applied torque: τ is directly proportional to T. Common Pitfalls: Mixing the stress formula (τ = Tr/J) with the twist formula (θ = TL/(GJ)); assuming that longer shafts have higher stress under the same torque—stress is geometry and load dependent, not length dependent. Final Answer: length of the shaft.

Question 9

Inertia about axes inclined to the principal axes: The locus of the area moment of inertia is what?

Accepted Answer

Introduction: This conceptual question checks whether you recognize the geometric representation of second moments of area about axes rotated from the principal directions. The variation of moment of inertia with axis orientation has a characteristic locus used in advanced section analysis. Given Data / Assumptions: A plane area with well-defined principal moments I1 and I2 and product of inertia zero in principal axes. We consider axes through the centroid rotated by an angle θ relative to principal axes. Concept / Approach: The transformation of second moments for a rotated axis is I(θ) = (I1 + I2)/2 + (I1 − I2)/2 * cos(2θ). When you plot I(θ) in orthogonal components corresponding to the two axis directions, the tip of the vector describing the second moment traces an ellipse, known as the ellipse of inertia. Step-by-Step Solution: Start from inertia transformation equations for rotated axes.Note the cos(2θ) and sin(2θ) terms that generate elliptical parametric relations.Conclude that the locus of I for all θ is an ellipse whose axes align with the principal directions. Verification / Alternative check: In the special case I1 = I2 (axisymmetric area), the ellipse degenerates to a circle, which is consistent with the general elliptical form. Why Other Options Are Wrong: Straight line / parabola / hyperbola: Do not match the trigonometric (cos 2θ, sin 2θ) parametric form of inertia transformation.Circle: Only for I1 = I2; not general. Common Pitfalls: Confusing Mohr’s circle for stresses with inertia transformations; although both use angle-doubling relations, inertia locus is the ellipse of inertia, not a circle except in the symmetric case. Final Answer: ellipse.

Question 10

Column effective length: One end fixed and the other end hinged—what is the Euler equivalent length?

Accepted Answer

Introduction: Euler buckling analysis uses an effective length Le = K*L that depends on end restraints. Knowing K for common end conditions is essential in column design to estimate critical load and slenderness. Given Data / Assumptions: Prismatic column of length L. One end perfectly fixed (no rotation/translation), the other end hinged (pinned, free to rotate, no translation). Euler elastic buckling regime, small deflection theory. Concept / Approach: For standard end conditions, the effective length factor K is tabulated. Values include: pinned–pinned K = 1.0, fixed–free K = 2.0, fixed–fixed K = 0.5, and fixed–pinned K ≈ 0.699 (often rounded to 0.7). The Euler critical load is Pcr = π^2 * E * I / (Le^2) = π^2 * E * I / (K^2 * L^2). Step-by-Step Solution: Identify end condition: fixed–pinned.From standard tables, K ≈ 0.699.Compute Le = K * L ≈ 0.699L, commonly rounded to 0.7L.Select the option stating 0.7L (approximately 0.699L). Verification / Alternative check: Energy or exact solution methods yield the same K. Many design codes provide this value directly; rounding to 0.7 is accepted practice for hand calculations. Why Other Options Are Wrong: 2L: Fixed–free cantilever, not fixed–pinned.L: Pinned–pinned case.0.5L: Fixed–fixed case.1.5L: Not a standard Euler end condition factor. Common Pitfalls: Memorizing only the extreme cases (pinned–pinned or fixed–free) and guessing for mixed end conditions; mixing up effective length with unsupported physical length. Final Answer: 0.7L (approximately 0.699L).

Question 11

Mohr’s construction: The locus of the end point of the resultant of normal and tangential stresses on an inclined plane is a ____.

Accepted Answer

Introduction: This question targets your understanding of Mohr’s circle, a powerful graphical tool for plane stress transformation. It asks about the geometric path traced by the resultant stress vector as the physical plane rotates through different inclinations. Given Data / Assumptions: Biaxial (plane) stress state defined by σx, σy, and τxy at a point. We examine stresses on all possible plane orientations passing through that point. Linear elasticity and standard sign conventions apply. Concept / Approach: When the plane is rotated by an angle θ in the body, the normal and shear stresses on that plane (σn, τn) satisfy parametric equations that, in σ–τ coordinates, describe a circle. This construction is Mohr’s circle, with center at (σavg, 0), where σavg = (σx + σy)/2, and radius R = √[( (σx − σy)/2 )^2 + τxy^2 ]. Step-by-Step Solution: Write transformation equations: σn = σavg + ((σx − σy)/2) * cos(2θ) + τxy * sin(2θ).Also τn = −((σx − σy)/2) * sin(2θ) + τxy * cos(2θ).Eliminate θ to obtain (σn − σavg)^2 + τn^2 = R^2, which is the equation of a circle.Therefore, the locus of the stress resultant endpoint is a circle. Verification / Alternative check: Special cases: pure uniaxial stress gives a circle passing through (σx, 0) and (0, 0); pure shear gives a circle centered at the origin. Why Other Options Are Wrong: Parabola / ellipse / straight line / hyperbola: None satisfy the derived relation (σn − σavg)^2 + τn^2 = R^2. Common Pitfalls: Confusing stress transformation with strain transformation or inertia ellipse; forgetting the angle-doubling (2θ) relation in Mohr’s construction. Final Answer: circle.

Theory of Structures Questions