Question 1

Riveted joint terminology: definition of pitch between rows  “The perpendicular distance between the centre lines of successive rows is called pitch.” Evaluate this statement for standard riveted joint terminology.

Accepted Answer

Introduction / Context: Clear terminology in riveted joints avoids fabrication and inspection errors. Two commonly confused terms are pitch and back pitch (or transverse pitch). Given Data / Assumptions: Pitched rows of rivets forming either chain or zig-zag patterns. Center-to-center distances define spacing parameters. Concept / Approach: Standard definitions: pitch (p) is the center-to-center distance between adjacent rivets in the same row, measured along the joint line. The perpendicular distance between adjacent rows is called back pitch or transverse pitch, not pitch. Step-by-Step Solution: Along-row spacing → pitch p.Perpendicular spacing between rows → back pitch (p_b) or transverse pitch.Therefore, the given statement mislabels back pitch as pitch and is incorrect. Verification / Alternative check: Consult typical joint layout diagrams: dimensions are marked as p (longitudinal pitch), p_b (back pitch), and sometimes diagonal pitch p_d for zig-zag arrangements. Why Other Options Are Wrong: “Agree” is incorrect by definition. Limiting correctness to zig-zag or chain joints still misuses the term. “Diamond” refers to group geometry, not spacing terminology. Common Pitfalls: Using “pitch” generically for any spacing; forgetting to distinguish back pitch from diagonal pitch in staggered rows. Final Answer: Disagree

Question 2

Effective (equivalent) length factor for a pin–pin (both ends hinged) column  For a column with both ends hinged (pinned), are the effective length L and actual length l the same?

Accepted Answer

Introduction / Context: Euler buckling load depends on effective length, which accounts for end conditions. Knowing the effective length factor K allows quick evaluation of buckling capacity and slenderness. Given Data / Assumptions: Column with ideal hinged (pinned) ends allowing rotation but no translation. Straight prismatic member; elastic buckling analysis. No intermediate bracing. Concept / Approach: Effective length L equals K * l, where l is the actual unsupported length and K is the effective length factor determined by end restraints. For a pin–pin column, K = 1.0, so L = l. Step-by-Step Solution: Euler load: P_cr = π^2 E I / L^2.End condition pin–pin → K = 1 → L = K l = l.Thus, the effective length equals the actual length. Verification / Alternative check: Buckling half-sine mode shape for pin–pin has a node at each end and a single antinode at midheight. The half-wave equals the actual length, confirming L = l. Why Other Options Are Wrong: “No” contradicts K = 1.0. Conditions on prismatic shape or frictionless pins are not needed for the idealized definition—though real connections may modify K slightly, the standard model uses K = 1. Common Pitfalls: Confusing fixed–fixed (K = 0.5) or fixed–free (K = 2.0) with pin–pin; using braced lengths instead of unsupported lengths; mixing effective length with effective area. Final Answer: Yes

Question 3

Uniaxial loading and oblique planes: A prismatic body is subjected to a direct tensile stress σ in one plane. What is the tangential (shear) stress acting on an oblique section whose normal makes an angle θ with the direction of σ?

Accepted Answer

Introduction / Context: When a member is subjected to a direct uniaxial tensile stress, the stresses on any oblique section can be obtained by resolving the applied stress into normal and shear components on that plane. This concept underpins strength-of-materials topics such as failure criteria and Mohr’s circle. Given Data / Assumptions: State of stress: uniaxial tension of magnitude σ. Plane of interest is an oblique section whose normal is inclined at angle θ to the loading direction. Material is linearly elastic; small deformation; classic continuum assumptions. Concept / Approach: The stress transformation relations for a plane making an angle θ with the load direction give the normal and shear components on that plane. For uniaxial stress, the shear component simplifies to a compact trigonometric form involving sin 2θ. The maximum shear on an oblique plane occurs at θ = 45° to the load direction, consistent with Mohr’s circle predictions. Step-by-Step Solution: Consider a plane whose normal makes angle θ with the load axis.Normal stress on the plane: σ_n = σ cos^2 θ.Tangential (shear) stress on the plane: τ = σ/2 sin 2θ.This follows from stress transformation or from Mohr’s circle where coordinates of a point at angle 2θ give σ_n and τ. Verification / Alternative check: From Mohr’s circle, the shear stress ordinate at an angle 2θ from the point representing the original stress equals τ = (σ/2) sin 2θ. At θ = 45°, sin 90° = 1, so τ = σ/2, which is the well-known maximum in-plane shear under uniaxial loading. Why Other Options Are Wrong: σ sin 2θ and σ cos 2θ miss the factor 1/2. σ/2 cos 2θ corresponds to transformed normal stress deviation, not shear. σ tan θ is not a correct transformation expression for shear on the oblique plane. Common Pitfalls: Confusing the angle of the plane with the angle to the plane’s normal; omitting the 1/2 factor; mixing normal and shear formulas from Mohr’s circle. Final Answer: σ/2 sin 2θ

Question 4

Neutral axis orientation in pure bending: For a transverse section of a prismatic beam, the neutral axis passes through the centroid. What is its orientation relative to the plane in which the beam bends?

Accepted Answer

Introduction / Context: Neutral axis (NA) is a fundamental concept in beam bending. It is the line in the cross-section where bending stress changes sign and is zero. Correctly identifying its location and orientation is essential for stress analysis and section design. Given Data / Assumptions: Prismatic beam under pure or predominant bending. Linear elastic material; plane sections remain plane. NA passes through the centroid of the section for homogeneous, isotropic beams. Concept / Approach: In pure bending about, say, the z-axis (causing curvature in the x–z plane), the bending stress varies linearly with distance from the NA, and the NA is oriented perpendicular to the plane of bending. Thus, if bending occurs in a vertical plane, the NA is horizontal across the section and vice versa. Step-by-Step Solution: Plane of bending is defined by the beam axis and the direction of curvature.The NA is the axis within the cross-section about which compressive and tensile areas balance.For homogeneous sections, this axis passes through the centroid and is perpendicular to the plane of bending. Verification / Alternative check: From flexure formula, σ = M*y/I for bending about the horizontal centroidal axis. The line y = 0 (through the centroid) carries zero stress and is perpendicular to the plane of bending, confirming the rule. Why Other Options Are Wrong: “In the same plane” misinterprets NA orientation; NA lies in the cross-section, not along the curvature plane. Fixed vertical or horizontal answers depend on which plane the beam bends in, so only the perpendicular statement is generally correct. Common Pitfalls: Confusing the neutral axis with the neutral surface; assuming a fixed direction regardless of the bending plane; ignoring composite-section effects that shift NA location (but orientation rule still holds relative to bending plane). Final Answer: at right angle to the plane in which the beam bends

Question 5

Riveted joints terminology: In plate riveting, the distance from the centre of a rivet hole to the nearest free edge of the plate is called what?

Accepted Answer

Introduction / Context: Correct terminology in riveted joints is vital for design, drafting, and inspection. Dimensions such as pitch, back pitch, diagonal pitch, and margin (edge distance) control joint strength and prevent tearing at edges. Given Data / Assumptions: Single- or multi-row riveted lap or butt joints. Terminology per standard mechanical design and boiler codes. Concept / Approach: The edge distance—often called the margin—is measured from the centre of the rivet hole to the nearest free edge of the plate. It is typically selected as a multiple of the rivet diameter to avoid edge tearing and to ensure adequate bearing area around the hole. Step-by-Step Solution: Identify the rivet-hole centre.Measure the shortest distance to the free edge of the plate.This distance is termed the margin (also called edge distance). Verification / Alternative check: Design handbooks specify minimum margin, commonly about 1.5 d to 2 d (d = rivet diameter), confirming industry usage of the term and its purpose in preventing edge failures. Why Other Options Are Wrong: Pitch is centre-to-centre distance of rivets along a row. Back pitch is the perpendicular distance between adjacent rows. Diagonal pitch is the centre-to-centre distance of rivets in adjacent rows measured diagonally. Common Pitfalls: Confusing margin with pitch; taking margin from hole edge instead of hole centre; ignoring minimum margin requirements leading to plate edge tearing. Final Answer: margin

Question 6

Axial deformation formula (Hooke’s law in tension/compression): For an axially loaded prismatic bar, what is the change in length Δl in terms of P, l, A, and E?

Accepted Answer

Introduction / Context: Determining axial deformation of bars under tension or compression is foundational in structural and machine design. The linear-elastic relation connects load, geometry, and material stiffness to elongation or shortening. Given Data / Assumptions: Bar is prismatic with constant cross-section A and length l. Applied axial load P; Young’s modulus E of the material. Small deformation; linear elasticity; uniform stress distribution. Concept / Approach: Hooke’s law in uniaxial form relates stress and strain: σ = E * ε. With σ = P/A and ε = Δl / l, combining these gives the classic deformation formula Δl = (P * l)/(A * E). Step-by-Step Solution: σ = P / Aε = Δl / lHooke’s law: σ = E * ε ⇒ P / A = E * (Δl / l)Rearrange: Δl = (P * l) / (A * E) Verification / Alternative check: Dimensional check: N * m / (m^2 * N/m^2) reduces to m, consistent with length. Experimental tests on tensile specimens confirm linear proportionality in the elastic range. Why Other Options Are Wrong: Other expressions scramble variables and units, failing dimensionally or physically. Common Pitfalls: Using incorrect area (net vs. gross); neglecting temperature or secondary effects; applying beyond the elastic limit where the relation is no longer linear. Final Answer: Δl = (P * l) / (A * E)

Question 7

Poisson’s ratio under tension vs. compression: A steel bar (2 m × 20 mm × 10 mm) carries 2 kN in tension or in compression within the elastic range. How does Poisson’s ratio in tension compare with that in compression?

Accepted Answer

Introduction / Context: Poisson’s ratio ν is a material constant in linear elasticity that relates lateral strain to axial strain. For small strains within the elastic limit, ν is independent of whether the load is tensile or compressive. Given Data / Assumptions: Same prismatic steel bar loaded by ±2 kN (tension or compression). Material behavior in the linear-elastic range without plasticity or damage. Isotropy and homogeneity assumed. Concept / Approach: By definition, ν = − ε_lateral / ε_axial in the elastic regime. For isotropic materials, ν is a constant for a given material over small strains, independent of load sign. Only once plasticity, microcracking, or other nonlinear effects occur could an apparent asymmetry develop, which is outside the problem scope. Step-by-Step Solution: Under tension: ε_axial > 0, ε_lateral < 0 ⇒ ν = − ε_lat / ε_axial.Under compression: ε_axial < 0, ε_lateral > 0 ⇒ ν = − ε_lat / ε_axial.In both cases, the magnitude of the ratio remains the same in linear elasticity. Verification / Alternative check: Standard elastic-constant relations (E, G, K, ν) and reciprocity do not depend on load sign within the elastic domain; experimental handbooks report a single ν value for a material like steel (e.g., ~0.3). Why Other Options Are Wrong: “Less than/greater than” implies nonlinearity or damage not stated. Dependence on bar dimensions or load magnitude conflicts with ν being an intrinsic material property. Common Pitfalls: Confusing elastic constants with geometric stiffness; assuming plastic effects at low loads; mixing up ν with lateral contraction per se. Final Answer: equal to

Question 8

Thin cylindrical shell under internal pressure: For a thin cylinder of diameter d, thickness t, and internal pressure p, what is the circumferential (hoop) stress in the shell?

Accepted Answer

Introduction / Context: Thin-walled pressure vessels experience two principal membrane stresses: hoop (circumferential) and longitudinal. Correctly computing these stresses is critical for safe vessel and pipe design. Given Data / Assumptions: Thin cylinder: t ≪ d (thin-wall assumption). Internal pressure p uniformly distributed. Membrane theory (neglect bending through thickness). Concept / Approach: Balancing forces on a diametral cut of the cylinder gives the hoop stress expression. For longitudinal stress, a different free-body leads to σ_long = pd/(4t). Here we specifically need the circumferential (hoop) stress σ_hoop = pd/(2t). Step-by-Step Solution: Consider equilibrium of half-cylinder: pressure force p * d * L (projected area) balanced by 2 * σ_hoop * t * L.Thus, p * d * L = 2 * σ_hoop * t * L.Cancel L: σ_hoop = p d / (2 t). Verification / Alternative check: Textbook relation: σ_hoop = p r / t with r = d/2, giving σ_hoop = p d / (2 t). Longitudinal stress check: σ_long = p d / (4 t) is half of hoop stress, consistent with thin-cylinder theory. Why Other Options Are Wrong: pd/t and pd/6t overestimate; pd/4t corresponds to longitudinal stress, not hoop. 2pt/d has wrong dependence and units for stress. Common Pitfalls: Confusing hoop and longitudinal formulas; using diameter vs. radius incorrectly; applying thin-wall formulas when t is not small compared to d. Final Answer: pd/2t

Question 9

Planes of maximum shear under biaxial stress: For a biaxial state with principal planes defined, what is the orientation of the planes of maximum shear stress?

Accepted Answer

Introduction / Context: Understanding the orientation of maximum shear stresses relative to principal planes is fundamental for failure analysis and for interpreting Mohr’s circle in biaxial stress states. Given Data / Assumptions: Biaxial stress condition with defined principal planes (no in-plane shear on those planes). Linear elasticity; standard sign conventions. Concept / Approach: Principal planes carry only normal stress; shear is zero there. Maximum shear planes occur midway between principal planes. In physical space, this corresponds to a 45° inclination to principal planes, and two orthogonal maximum-shear planes exist, 90° apart from each other. Step-by-Step Solution: On Mohr’s circle, principal planes lie at the horizontal intercepts where τ = 0.Maximum shear occurs at the top and bottom of the circle (τ = R), 90° on Mohr’s circle, which translates to 45° in the physical plane.Hence, the two planes of maximum shear are orthogonal and each is 45° to the principal planes. Verification / Alternative check: Transforming stresses via the stress transformation equations and differentiating τ(θ) with respect to θ yields the same 45° condition for extremum shear. Why Other Options Are Wrong: Any angle other than 45° contradicts the transformation result. Coincidence with principal planes is impossible because shear there is zero by definition. Common Pitfalls: Mixing Mohr’s-circle angle (2θ) with the physical angle θ; forgetting the two orthogonal maximum-shear planes; sign convention errors. Final Answer: True

Question 10

Uniaxial loading and oblique planes (normal stress): A prismatic body is subjected to a direct tensile stress σ. What is the normal stress on an oblique section whose normal makes an angle θ with the load direction?

Accepted Answer

Introduction / Context: Stress transformation allows us to find the normal stress acting on any inclined plane under a uniaxial load. This is essential for analyzing potential failure planes and deriving Mohr’s circle relations. Given Data / Assumptions: Uniaxial tensile stress σ. Plane with normal at angle θ to the load axis. Linear elasticity; small deformation; continuum assumptions. Concept / Approach: Using transformation formulas, the normal stress σ_n on a plane whose normal is at angle θ to the stress direction is σ_n = σ cos^2 θ. Equivalently, using the double-angle identity, σ_n = σ/2 (1 + cos 2θ). Step-by-Step Solution: Resolve the uniaxial stress vector on the plane: component along the plane’s normal is σ cos θ.Stress is proportional to projected area; combining projections gives σ_n = σ cos^2 θ.Identity check: cos^2 θ = (1 + cos 2θ)/2, so σ_n = σ/2 (1 + cos 2θ), consistent. Verification / Alternative check: Mohr’s circle yields the same result: the abscissa at angle 2θ from the loading point gives σ_n = σ/2 (1 + cos 2θ). Why Other Options Are Wrong: σ cos θ and σ sin θ are not correct for normal stress on the oblique plane. σ sin^2 θ is the complementary form for a plane normal at 90°−θ. While σ/2 (1 + cos 2θ) is mathematically equivalent, the explicitly squared form is the standard direct expression; both are correct, but the single best choice requested is σ cos^2 θ. Common Pitfalls: Confusing angle to the plane with angle to the plane’s normal; mixing normal and shear stress expressions; sign errors in transformation. Final Answer: σ cos^2 θ

Question 11

Plastic region behavior in tensile testing: Beyond the elastic limit on a stress–strain curve, how does tensile strain vary compared to applied stress?

Accepted Answer

Introduction / Context: The stress–strain curve of ductile materials shows an elastic region (linear, Hookean) followed by a plastic region. Understanding how strain evolves relative to stress beyond the elastic limit is crucial for safe design and for predicting permanent deformations. Given Data / Assumptions: Standard uniaxial tensile test on a ductile metal specimen. Observation focuses on the region beyond the elastic limit (onset of plasticity). Monotonic loading; room temperature; engineering stress–strain definition. Concept / Approach: In the elastic region, stress is proportional to strain (σ = E ε). Once the elastic limit is exceeded, plastic deformation begins. Additional strain accrues at a faster rate for comparatively small increases in stress, especially near yield where the curve flattens (yield plateau in some steels). Step-by-Step Solution: Before yield: ε = σ / E (linear).At/after yield: curve deviates; large ε increment for small σ increment.Thus, strain increases more quickly than stress beyond elastic limit. Verification / Alternative check: Typical stress–strain plots for steels show a yield plateau or gradual hardening: significant strain develops with little to moderate stress increase, confirming the qualitative relationship. Why Other Options Are Wrong: Decreasing strain with increasing stress contradicts physical behavior. “Remains proportional” is only valid in the elastic region, not beyond. Common Pitfalls: Confusing engineering and true stress–strain beyond necking; misreading the yield plateau as constant strain; assuming linearity throughout loading. Final Answer: increases more quickly

Question 12

Pressure vessels – definition of a thick shell A pressure vessel is said to be a thick shell when which geometric condition holds true?

Accepted Answer

Introduction / Context: In strength of materials and machine design, pressure vessels are classified as thin or thick shells to choose appropriate stress formulas. Thin-shell (membrane) theory assumes nearly uniform stress across thickness, while thick-shell theory (Lame’s equations) accounts for radial stress variation. Correct classification hinges on the ratio of wall thickness to diameter. Given Data / Assumptions: D = internal (or mean) diameter of the cylindrical or spherical vessel. t = wall thickness of the vessel. Elastic, isotropic material; small deformations. Classification threshold used in standard design practice. Concept / Approach: When the wall is relatively thin compared to diameter, stress variation through the wall is small and thin-cylinder formulas apply. As thickness increases, the radial stress and hoop stress vary significantly across the wall, requiring thick-cylinder theory. Step-by-Step Solution: Classification criterion widely used: thin shell if t / D ≤ 1/20 to 1/10 (engineering practice uses conservative bounds).If the wall is relatively thick, i.e., t / D > 1/10, thin-shell assumptions break down.Therefore, a pressure vessel is treated as a thick shell when t / D is greater than 1/10. Verification / Alternative check: Another common rule uses t / r (r = radius). If t / r > 0.05, thick formulas (Lame’s) are needed. Since D = 2 r, t / D > 0.1 corresponds to t / r > 0.2, a conservative threshold; both indicate the necessity of thick-wall analysis. Why Other Options Are Wrong: “Manufactured from thick sheets” and “internal pressure very high” do not, by themselves, dictate stress distribution; geometry governs theory choice. “t / D less than 1/10” indicates thin shell, not thick. A specific value like 1/20 is a thin-wall guideline, not thick. Common Pitfalls: Confusing service pressure with geometry; using thin-wall hoop stress (σ_hoop = p D / (2 t)) for thick shells, leading to unsafe designs. Final Answer: t / D is greater than 1/10

Question 13

Simply supported beam – central point load For a simply supported beam of span l carrying a point load W at the centre, what is the maximum bending moment?

Accepted Answer

Introduction / Context: Bending moment evaluation is fundamental in beam design. A classic case is a simply supported beam with a central point load, which appears in bridges, crane girders, and machine frames. Knowing the closed-form maximum bending moment allows quick sizing and verification. Given Data / Assumptions: Straight, prismatic beam of span l. Point load W applied at midspan. Supports are simple (pin and roller), so end moments are zero. Linearly elastic behavior; small deflections. Concept / Approach: For symmetric loading, reactions at supports are equal. The internal bending moment diagram features a peak under the load. The maximum bending moment occurs where shear force crosses zero, which, for symmetry, is at the midspan. Step-by-Step Solution: Support reactions: R_A = R_B = W / 2.Section at a distance x from the left: M(x) = R_A * x for x ≤ l/2; beyond midspan, M(x) decreases symmetrically.At x = l/2: M_max = (W / 2) * (l / 2) = W l / 4. Verification / Alternative check: Shear force V(x) is W / 2 up to midspan, then −W / 2 after midspan; V changes sign at the centre, confirming the location of M_max. Standard tables also list M_max = W l / 4. Why Other Options Are Wrong: W l / 2 and W l are too large; W l^2 / 4 has wrong dimensions; W l / 8 is the maximum reaction times half span error, not the correct bending moment. Common Pitfalls: Mixing up total load and reaction; writing W l^2 / 8 (a deflection-like form) by mistake; forgetting symmetry. Final Answer: W l / 4

Question 14

Generalized Hooke’s law in shear Within the elastic limit, shear stress is __________ shear strain.

Accepted Answer

Introduction / Context: Hooke’s law extends to shear: in the elastic range, material response is linear, and shear stress is proportional to shear strain. This underpins torsion formulas, beam shear relations, and finite element constitutive models. Given Data / Assumptions: Material is homogeneous and isotropic. Deformations are small; elastic limit not exceeded. Shear modulus G (also written as C) is constant in this range. Concept / Approach: Generalized Hooke’s law in shear states τ = G * γ, where τ is shear stress and γ is engineering shear strain. The direct proportionality is characterized by the slope G of the τ–γ line in the elastic region. Step-by-Step Reasoning: Identify the elastic region (no permanent deformation).Relate stress and strain: τ = G * γ implies linear proportionality.Therefore, within elastic limit, τ varies directly with γ. Verification / Alternative check: For torsion of circular shafts: τ = T r / J and γ = r θ / L. Eliminating r shows τ ∝ γ with slope G when Torsion equation G J θ / L = T is used, reaffirming linearity. Why Other Options Are Wrong: “Equal to” ignores units (stress vs strain). “Less than” or “inversely proportional” contradicts basic elasticity. “Independent of” is inconsistent with elastic behavior. Common Pitfalls: Confusing proportionality (τ ∝ γ) with equality; mixing G with E (Young’s modulus) or K (bulk modulus). Final Answer: directly proportional to

Question 15

Principal planes – definition check Planes in a stressed body that carry no shear stress are known as principal planes. State whether this is true or false.

Accepted Answer

Introduction / Context: Principal stresses and principal planes are central to failure theories and stress analysis. Identifying planes with zero shear simplifies complex stress states and enables use of criteria like maximum normal stress or von Mises. Given Data / Assumptions: Continuum, small deformations. At a given point, a triad of mutually orthogonal principal planes exists. Material behavior not restricted to ductile or brittle for the definition. Concept / Approach: For any 2D or 3D stress state, there are orientations where the stress tensor becomes diagonal (no shear components). Those orientations define principal planes, and the normal stresses on them are the principal stresses. Step-by-Step Reasoning: Resolve stresses on an arbitrary plane using transformation equations.Set shear stress on that plane to zero and solve for the plane angle(s).Solutions correspond to principal directions; the associated planes are principal planes with zero shear. Verification / Alternative check: Mohr’s circle: principal planes occur at points where the circle crosses the σ-axis; at these points, shear τ = 0 by construction. Why Other Options Are Wrong: The statement is general and not limited to specific materials, torsion, or Poisson’s ratio conditions. Common Pitfalls: Confusing maximum shear planes with principal planes; assuming principal planes always align with geometric axes (not true for unsymmetrical loading). Final Answer: True

Question 16

Helical springs – stiffness versus number of turns For a closely-coiled helical spring, the stiffness (k) varies with the number of turns (n) as:

Accepted Answer

Introduction / Context: Helical compression and tension springs are designed to achieve a target stiffness (spring rate). Understanding how geometry affects stiffness enables quick design adjustments. The number of active turns has a direct, simple influence on k. Given Data / Assumptions: Closely-coiled helical spring (helix angle small). Wire diameter d, mean coil diameter D, shear modulus G. n = number of active coils. Concept / Approach: The standard spring rate formula for a closely-coiled helical spring loaded axially is k = (G d^4) / (8 D^3 n). This shows linear inverse dependence on n, holding other parameters constant. Step-by-Step Solution: Start from k = (G d^4) / (8 D^3 n).Treat G, d, D as constants for a given design iteration.Therefore, k ∝ 1 / n. Increasing the number of active turns reduces stiffness. Verification / Alternative check: Doubling n halves k; halving n doubles k. This proportionality matches both analytical derivations and empirical spring catalogs. Why Other Options Are Wrong: Options claiming k ∝ n, n^2, or independent of n contradict the formula; k ∝ 1 / n^2 is also incorrect because the dependence is first power. Common Pitfalls: Forgetting to count only active coils (end coils may be inactive); changing D or d simultaneously and misattributing the effect to n. Final Answer: k is inversely proportional to n

Question 17

Transverse shear – comparing cross-sections of equal area The ratio of maximum shear stress in a rectangular beam to that in a circular beam of the same cross-sectional area is:

Accepted Answer

Introduction / Context: Transverse shear stress distributions depend on cross-sectional shape. For beams of equal area under the same shear force V, maximum shear differs by shape. Knowing these ratios helps in preliminary sizing and in selecting efficient sections. Given Data / Assumptions: One beam has a rectangular cross-section; the other is circular. Both have the same area A and carry the same shear force V. Elastic theory of shear in beams applies; small deformations. Concept / Approach: Maximum shear stress for a rectangle is τ_max(rect) = 1.5 * V / A. For a circle, τ_max(circ) = 4/3 * V / A. Taking the ratio eliminates V and A, leaving a pure number. Step-by-Step Calculation: τ_max(rect) = (3/2) * V / A.τ_max(circ) = (4/3) * V / A.Ratio = τ_max(rect) / τ_max(circ) = (3/2) / (4/3) = (3/2) * (3/4) = 9/8. Verification / Alternative check: Numerical example: choose V = 1 and A = 1. Then τ_max(rect) = 1.5 and τ_max(circ) = 1.333..., giving 1.125 = 9/8, confirming the ratio. Why Other Options Are Wrong: 2/3 and 3/4 are less than 1 and contradict known results; 1 would imply identical maxima; 4/3 in the choices is the circular multiplier itself, not the ratio asked. Common Pitfalls: Comparing shapes with unequal areas; confusing average shear V / A with maximum shear. Final Answer: 9 / 8

Question 18

Beam of triangular section (base horizontal) Where does the maximum transverse shear stress occur?

Accepted Answer

Introduction / Context: Shear stress in beams is not uniform and depends on the first moment of area Q at the level considered. For non-rectangular sections such as triangles, understanding where the maximum shear occurs is important for checking web stresses and potential shear failures. Given Data / Assumptions: Triangular cross-section with base horizontal; width varies linearly with height. Beam subjected to a transverse shear force V. Elastic theory applies; small deformations. Concept / Approach: Shear stress at a level y from a reference is τ = V Q / (I b), where Q is first moment of the area above (or below) the level about the neutral axis, I is second moment of area, and b is width at that level. For a triangle, b varies linearly with y, and Q is a quadratic function of y, so τ becomes a cubic-like function with a single maximum within the depth. Step-by-Step Reasoning (qualitative): 1) At the extreme fibers (base or apex), Q = 0 or b → 0, giving τ = 0 at both extremes.2) Between extremes, Q increases from zero, while b also changes; the ratio Q / b peaks roughly at the mid-depth.3) Detailed derivations show the maximum shear for a triangular section occurs at mid-height (i.e., halfway between base and apex). Verification / Alternative check: Carrying out the full τ(y) = V Q / (I b) derivation with b(y) linear in y and integrating to find Q(y) leads to a stationary point at mid-depth, which is the maximum τ. Why Other Options Are Wrong: Apex and base are extreme fibers where τ = 0; centroid is not the location of maximum τ for a triangle; one-third from base is not the maximum for this shape. Common Pitfalls: Assuming maximum shear always occurs at the neutral axis; for triangles, the neutral axis is at h/3 from the base, but τ_max is at mid-depth, not at the centroidal axis. Final Answer: At mid-height of the section

Question 19

Torsion of circular shafts – shear stress at the centre What is the shear stress at the centre of a solid circular shaft subjected to torsion?

Accepted Answer

Introduction / Context: In torsion of circular shafts, shear stress varies linearly with radius. Knowing where the maximum and minimum shear occur is vital for design against yielding and fatigue. Given Data / Assumptions: Solid circular shaft under torque T. Linear elastic behavior; Saint-Venant torsion theory. r = radial distance from the centre; R = outer radius. Concept / Approach: The torsion formula states τ(r) = T r / J, where J is the polar second moment of area (J = π R^4 / 2 for a solid circle). Shear stress is directly proportional to r, so it increases from the centre to the surface. Step-by-Step Reasoning: At r = 0 (the centre): τ(0) = T * 0 / J = 0.At r = R (outer surface): τ_max = T R / J, which is the maximum value.Therefore, the centre experiences zero shear stress under pure torsion. Verification / Alternative check: Shear strain distribution is γ(r) = r θ / L, also zero at the centre and maximum at the surface. Since τ = G * γ, τ follows the same radial linearity. Why Other Options Are Wrong: “Minimum but non-zero” conflicts with τ(0) = 0; “maximum” and “infinity” contradict the linear relation; “equal to average shear V / A” is a beam shear concept, unrelated to torsion at the centre. Common Pitfalls: Confusing torsional shear distribution with transverse shear in beams; forgetting that τ varies with radius in torsion. Final Answer: Zero

Question 20

Beam theory – shear stress at the neutral axis At the neutral axis of a beam subjected to transverse shear, the shear stress is:

Accepted Answer

Introduction / Context: Understanding the shear stress distribution across a beam depth is essential for web design, particularly in rectangular and I-sections. The neutral axis is the line of zero bending normal stress, but its role in shear distribution is different. Given Data / Assumptions: Beam carrying a transverse shear force V. Cross-sections like rectangles and I-beams with non-zero web thickness at the neutral axis. Elastic, small-deflection conditions. Concept / Approach: The shear stress at a level y is given by τ = V Q / (I b), where Q is the first moment of area about the neutral axis, I is the second moment of area, and b is the width at that level. For common sections, Q is maximum at the neutral axis; consequently, τ is also typically maximum there if b does not spike excessively. Step-by-Step Reasoning: At extreme fibers, Q = 0, hence τ = 0.Moving toward the neutral axis, Q increases, reaching its maximum at the neutral axis.Since b is finite at the neutral axis for rectangular and I-sections (web thickness), τ attains a maximum there. Verification / Alternative check: For a rectangle: τ_max = 1.5 * V / A and occurs at the neutral axis. For an I-beam, the maximum is in the web at the neutral axis region. Why Other Options Are Wrong: Zero or minimum applies at extreme fibers, not at the neutral axis; infinity is non-physical; “equal to bending stress” mixes different stress components—bending normal stress is zero at the neutral axis, not equal to shear stress. Common Pitfalls: Assuming “neutral axis” means all stresses are zero; only bending normal stress is zero there, not shear. Final Answer: Maximum

Strength of Materials Questions