Question 1

Basic definition check: the resistance per unit area that develops inside a material when external forces cause deformation is called what?

Accepted Answer

Introduction / Context: Understanding the distinction between stress and strain is foundational in mechanics of materials. When loads are applied, a body deforms and internal forces develop to resist that deformation. We quantify this internal resistance using stress. Given Data / Assumptions: A continuum body under external forces. Uniform cross-section considered for simplicity. Elastic range assumptions for clarity. Concept / Approach: Stress is the internal force per unit area that develops on an imagined cut through a body. Normal stress acts perpendicular to the area; shear stress acts tangentially. By contrast, strain is the geometric measure of deformation (change in length per original length, change in angle, etc.). Step-by-Step Solution: Identify the internal resisting force F over a section.Compute average normal stress: sigma = F / A (for axial loading).Recognize that “resistance per unit area” is the definition of stress. Verification / Alternative check: In shear, tau = V / A_s, still “resistance per unit area” but tangential. This reaffirms the general definition. Why Other Options Are Wrong: Strain measures deformation, not resistance.Pressure is external normal force per unit area applied on a surface, not the internal resistance within a solid.Modulus of elasticity is a material constant (E) relating stress to strain, not a measure of resistance by itself.Toughness is an energy absorption measure, not force per area. Common Pitfalls: Confusing stress (cause) with strain (effect); mixing pressure (external) with stress (internal state). Final Answer: stress

Question 2

Neutral axis behavior: at the layer passing through the centroid (neutral axis) of a loaded beam section, the fiber is in which state?

Accepted Answer

Introduction / Context: When a beam bends under transverse loading, the top fibers and bottom fibers usually experience opposite normal stresses (compression versus tension). Between them is a unique surface—the neutral surface—whose intersection with the cross-section is the neutral axis. Given Data / Assumptions: Prismatic beam, linearly elastic behavior. Plane sections remain plane (Bernoulli–Euler assumption). Symmetric bending without significant axial force. Concept / Approach: The bending stress varies linearly with distance y from the neutral axis: sigma = M * y / I. At y = 0 (neutral axis), sigma = 0. Therefore, at the neutral axis, the fiber carries neither tension nor compression in bending. Step-by-Step Solution: Locate the neutral axis through the centroid of the section (for symmetric, homogeneous sections).Use sigma = M * y / I; set y = 0 at the neutral axis.Conclude sigma = 0 at the neutral axis, so no bending normal stress there. Verification / Alternative check: Stress diagrams for standard cases (simply supported beam under point or uniform load) always show zero normal stress at the neutral axis. Why Other Options Are Wrong: Tension/Compression occurs away from the neutral axis at extreme fibers.“Shear only” is not correct as a general statement; while transverse shear is maximum at or near the neutral axis in many sections, the question asks about normal stress state due to bending. Common Pitfalls: Confusing normal bending stress with transverse shear stress distribution; the neutral axis refers to zero normal bending stress, not necessarily zero shear stress. Final Answer: neither in tension nor in compression

Question 3

Deflection comparison: two simply supported beams of equal length l and breadth b carry the same central point load W. Beam B has double the depth (2d) compared to beam A (depth d). State whether the deflection of beam B is “double” that of beam A.

Accepted Answer

Introduction / Context: The midspan deflection of a simply supported beam under a central point load depends on span, stiffness, and load. Changing the cross-section (depth) changes the second moment of area I and therefore the deflection. This question checks understanding of how deflection scales with depth. Given Data / Assumptions: Both beams: simply supported, same length l and breadth b. Same central point load W and same material (same E). Beam B has depth 2d; beam A has depth d. Concept / Approach: The midspan deflection for a simply supported beam with a central point load is delta = W * l^3 / (48 * E * I). For a rectangle, I = b * d^3 / 12. Deflection is inversely proportional to I and thus to d^3 for constant b. Step-by-Step Solution: I_A = b * d^3 / 12.I_B = b * (2d)^3 / 12 = 8 * (b * d^3 / 12) = 8 * I_A.delta_B / delta_A = I_A / I_B = 1 / 8.Therefore, beam B deflects one-eighth of beam A, not double. Verification / Alternative check: Use section modulus or quick proportionality: deflection ∝ 1 / d^3 for rectangular sections; doubling depth reduces deflection by 2^3 = 8. Why Other Options Are Wrong: “Correct”: opposite of the actual scaling.Conditions about E or cantilevers are irrelevant; the stated comparison is for like-for-like beams and loading. Common Pitfalls: Thinking deflection scales with 1/d or 1/d^2 instead of 1/d^3; forgetting the cubic effect of depth on I. Final Answer: Incorrect

Question 4

Under uniaxial direct tensile stress σ, the planes on which the shear stress is maximum are inclined at what angles to the normal of the section?

Accepted Answer

Introduction / Context: Under a uniaxial tensile state, the material experiences not only normal stress but also shear stress on oblique planes. Identifying the orientation of planes of maximum shear is crucial for understanding yielding and failure (e.g., ductile metals forming shear bands). Given Data / Assumptions: Direct tensile stress σ acts in one direction only. No other normal or shear components initially. Linear elastic continuum assumptions. Concept / Approach: Using stress transformation, the shear stress on a plane inclined at angle θ to the normal is tau(θ). For a uniaxial state, tau(θ) reaches its maximum when θ = 45° to the normal (i.e., plane oriented at 45° to the loading direction). The complementary plane at 135° also sees the same magnitude due to symmetry. Step-by-Step Solution: Transformation for shear: tau(θ) = (σ / 2) * sin(2θ).Maximum of sin(2θ) is 1 when 2θ = 90° ⇒ θ = 45°.Thus, planes at 45° and 135° to the normal experience maximum shear. Verification / Alternative check: On Mohr’s circle for uniaxial stress, the maximum shear equals σ/2 and occurs at 2θ = 90°, confirming θ = 45°. Why Other Options Are Wrong: Other angles do not maximize sin(2θ); 90° and 0° give zero shear for uniaxial tension. Common Pitfalls: Confusing plane angle θ with physical fiber angle; mixing “plane angle” versus “direction angle.” Final Answer: 45° and 135°

Question 5

Strain energy in torsion: for a hollow circular shaft (outer diameter D, inner diameter d) under elastic torsion, the total strain energy expressed using the maximum shear stress τ_max at the outer surface equals which of the following?

Accepted Answer

Introduction / Context: Strain energy in torsion is vital for shock and fatigue considerations. For circular shafts, the shear stress varies linearly with radius, so the average energy density differs from the simple τ^2 / (2G) expression that assumes uniform shear. A compact and widely used result expresses total strain energy in terms of τ_max and the volume. Given Data / Assumptions: Hollow circular shaft, outer diameter D, inner diameter d. Linear elastic torsion, Saint-Venant assumptions. Maximum shear stress τ_max occurs at r = D/2. Concept / Approach: Total strain energy U in torsion is U = (T * θ) / 2. Also, U = ∫(τ^2 / 2G) dV, but τ varies with r. Carrying out the radial integration for a circular shaft shows that the average energy density equals τ_max^2 / (4G), so the total is U = (τ_max^2 / 4G) * Volume. Step-by-Step Solution: Shear distribution: τ(r) = (τ_max / R) * r with R = D/2.Energy density: u(r) = τ(r)^2 / (2G) = (τ_max^2 / (2G R^2)) * r^2.Integrate over annulus: U = ∫ u(r) dV = ∫_{r_i}^{r_o} (τ_max^2 / (2G R^2)) * r^2 * (2π r L) dr.Simplify integration ⇒ U = (τ_max^2 / 4G) * (π (r_o^2 − r_i^2) L) = (τ_max^2 / 4G) × Volume. Verification / Alternative check: From U = T^2 L / (2 G J) and τ_max = T R / J; eliminate T and J to recover U = (τ_max^2 / 4G) × Volume. Why Other Options Are Wrong: (τ^2 / 2G) × Volume assumes uniform τ; not valid for circular torsion.Other coefficients (1/6, 1/8, 1) do not match the correct radial integration. Common Pitfalls: Using solid-shaft intuition incorrectly; forgetting τ varies with r; confusing τ_max with average shear. Final Answer: (τ_max^2 / 4G) × Volume of shaft

Question 6

Application of springs in instruments: in a spring balance, what is the primary function performed by the spring?

Accepted Answer

Introduction / Context: Springs are versatile machine elements used to apply load, absorb energy, control motion, or sense force. A spring balance is a classic device where a spring’s deflection is used as a measure of an applied load. Given Data / Assumptions: Linear elastic behavior of the spring within its working range. Deflection is proportional to applied load (Hooke’s law). Scale calibrated to convert deflection into force units. Concept / Approach: Hooke’s law states F = k * x, where k is the spring stiffness and x is the deflection. In a spring balance, the measured deflection is mapped through calibration to indicate the magnitude of the applied force or weight. Step-by-Step Solution: Apply load to the spring; the spring elongates by x.The internal restoring force equals k * x in equilibrium with the applied load.The scale converts x into a force reading, thereby measuring the force. Verification / Alternative check: Recalibration with known weights yields linear scale marks if the spring behaves linearly. Why Other Options Are Wrong: “To apply forces”: Springs can apply preload, but in a balance the key role is measurement.“To absorb shocks”: That is a suspension spring function, not the primary purpose here.“To store strain energy”: True physically, but the device’s purpose is to measure force via deflection.“Constant torque” applies to spiral springs in some instruments, not a generic spring balance. Common Pitfalls: Ignoring creep or nonlinearity; neglecting temperature effects on stiffness for precision balances. Final Answer: to measure forces

Question 7

Compare strengths: for brittle materials, the compressive strength is generally how related to the tensile strength?

Accepted Answer

Introduction / Context: Brittle materials such as cast iron, concrete, ceramics, and some hard polymers exhibit markedly different behavior in tension versus compression. Understanding this asymmetry guides safe design and choice of failure criteria. Given Data / Assumptions: Brittle, low-ductility materials with limited plastic deformation. Quasi-static loading conditions. Uniform specimens and standard testing procedures. Concept / Approach: Brittle materials fail by crack initiation and propagation. In tension, microscopic flaws open and grow, leading to low tensile strength. Under compression, crack faces tend to close, friction and confinement improve resistance, and other failure modes (shear, microcracking) govern at higher loads, resulting in higher compressive strength. Step-by-Step Solution: Consider typical data: concrete f_c >> f_t; cast iron compressive strength several times its tensile strength.Mechanism: tensile loading opens flaws; compressive loading inhibits opening and allows higher loads before instability.Therefore, compressive strength exceeds tensile strength for brittle materials. Verification / Alternative check: Review material datasheets: e.g., gray cast iron and concrete consistently show f_c much greater than f_t, confirming the general rule. Why Other Options Are Wrong: “Equal” or “less than” contradict empirical evidence.“Half of” is not a general rule and varies widely by material and microstructure.“Unrelated” ignores tested correlations and mechanisms. Common Pitfalls: Applying ductile metal intuition (where tensile and compressive yield strengths are similar) to brittle materials; ignoring flaw sensitivity in tension. Final Answer: greater than

Question 8

Combined uniaxial normal stress σx and shear stress τxy: write the expression for the minimum principal (normal) stress at the point.

Accepted Answer

Introduction / Context: When a point in a member is subjected to a normal stress in one direction and an in-plane shear stress, the state of stress must be resolved to principal stresses to assess failure. The minimum (more compressive) principal stress determines safety for brittle materials and affects yielding criteria. Given Data / Assumptions: Plane stress with components σx and τxy; σy = 0 initially. Material linear-elastic; small deformations. Sign convention: tension positive; shear using standard Mohr’s circle sign. Concept / Approach: Principal stresses for a general plane stress state are given by σ1,2 = (σx + σy)/2 ± √( ((σx − σy)/2)^2 + τxy^2 ). Here σy = 0, so σ1,2 = (σx / 2) ± √( (σx/2)^2 + τxy^2 ). The minimum principal stress is the minus-root expression. Step-by-Step Solution: Start with σy = 0: σ_avg = σx / 2.Compute radius R = √( (σx/2)^2 + τxy^2 ).σ_min = σ_avg − R = (σx / 2) − √( (σx / 2)^2 + τxy^2 ). Verification / Alternative check: Construct Mohr’s circle with center at (σx/2, 0) and radius R; the leftmost intersection on the σ-axis equals σ_min. Why Other Options Are Wrong: Linear combinations like σx − τxy are not correct transformations.Adding the square-root gives σ_max, not σ_min.The radical expression without averaging does not represent a principal stress. Common Pitfalls: Forgetting σy term; sign mistakes in Mohr’s circle; swapping σ_max and σ_min by choosing the wrong sign. Final Answer: σ_min = (σx / 2) − √( (σx / 2)^2 + τxy^2 )

Question 9

Comparing torsional strength: two solid circular shafts A and B (same material) have diameters 50 mm and 100 mm respectively. The torsional strength of shaft B is how many times that of shaft A?

Accepted Answer

Introduction / Context: In torsion, the strength of a circular shaft—its capacity to resist a given maximum shear stress—is proportional to its polar section modulus. For solid round shafts, this scales with the cube of diameter. Doubling the diameter therefore has a dramatic effect on strength. Given Data / Assumptions: Shaft A: d_A = 50 mm; Shaft B: d_B = 100 mm. Same material and allowable shear stress. Solid circular sections; elastic behavior. Concept / Approach: Polar section modulus Z_p for a solid round is Z_p = J / R = (π d^4 / 32) / (d/2) = π d^3 / 16. For given allowable shear stress τ_allow, torque capacity T_allow ∝ Z_p ∝ d^3. Thus the strength ratio equals the cube of the diameter ratio. Step-by-Step Solution: Strength ratio = (d_B / d_A)^3 = (100 / 50)^3 = 2^3 = 8.Therefore, shaft B is eight times as strong in torsion as shaft A. Verification / Alternative check: Compute Z_p explicitly: Z_pA = π (50)^3 / 16; Z_pB = π (100)^3 / 16; take the ratio to confirm 8. Why Other Options Are Wrong: One-half, double, or four times underestimate the cubic scaling.Sixteen times would correspond to a diameter ratio of 2.52^(something), not to a simple doubling. Common Pitfalls: Assuming strength scales with area (d^2) instead of with section modulus (d^3); ignoring the role of maximum radius R in Z_p. Final Answer: eight times

Question 10

Retaining wall with uniform surcharge: When retained soil is subjected to a uniform superimposed (surcharged) load of intensity p over the backfill surface, what is the total horizontal active pressure on a vertical wall of height h? (Use Rankine's …

Accepted Answer

Introduction / Context: In soil mechanics and retaining wall design, surcharge loads such as traffic, storage, or construction equipment add to the lateral earth pressure on the wall. Under Rankine’s active condition for level backfill, a uniform surcharge of intensity p produces an additional, height-uniform lateral pressure of magnitude Ka * p. Understanding how to compute the total horizontal force from this surcharge is essential for safe design. Given Data / Assumptions: Vertical retaining wall of height h with level ground surface behind it. Uniform surcharge (superimposed load) of intensity p on the backfill. Active earth pressure condition per Rankine; coefficient Ka = (1 - sin φ) / (1 + sin φ). Backfill is homogeneous, dry, and cohesionless; wall friction neglected; no seismic effects. Concept / Approach: Under a uniform surcharge p, the lateral earth pressure intensity added by surcharge is constant with depth and equals Ka * p (same at all elevations). The total horizontal force due to this uniform pressure is the pressure intensity multiplied by the wall area per unit out-of-plane width, which is simply the height h. The resultant acts at mid-height (h/2 from the base). Step-by-Step Solution: Lateral pressure intensity from surcharge: q_lat = Ka * p.Because q_lat is uniform over height, total force per unit width is F = q_lat * h.Therefore, F = (Ka * p) * h.Line of action: at mid-height, i.e., h/2 above the base. Verification / Alternative check: The shear resultant of a uniform pressure strip of height h is simply intensity times height. A quick units check: p has units of force/area; multiplying by h (length) gives force/length, which is correct for force per unit out-of-plane width. Ka is dimensionless, so units remain consistent. Why Other Options Are Wrong: p * h ignores the reduction due to earth pressure coefficient Ka. Ka * p * h / 2 and p * h / 2 are triangular-resultant expressions; surcharge-induced pressure is uniform, not triangular. Ka * p * h^2 has incorrect dimensionality and arises from confusing pressure intensity with depth-varying self-weight pressure. Common Pitfalls: Confusing uniform surcharge pressure (constant with depth) with soil self-weight pressure (linear with depth); forgetting the coefficient Ka; placing the resultant at h/3 instead of the correct h/2 for a uniform load block. Final Answer: Ka * p * h

Question 11

Tensile testing — which materials are typically evaluated in standard tension tests?

Accepted Answer

Introduction / Context: The standard uniaxial tensile test is the most common mechanical test for determining material properties like yield strength, ultimate tensile strength, elastic modulus, and ductility. It is widely used in quality control, design, and materials research. Given Data / Assumptions: The question asks which class of materials is typically evaluated using a routine tensile test. Specimens follow standardized geometries (e.g., proportional gauge length) and are pulled until failure. Concept / Approach: While both ductile and brittle materials can be tested in tension, tensile testing is most associated with ductile materials because they undergo appreciable plastic deformation, enabling clear identification of yield point, uniform elongation, and reduction of area. Brittle materials often fail abruptly with minimal plastic strain, making compression or bend tests preferable for them in many practical contexts. Step-by-Step Solution: Identify materials that exhibit significant plasticity before fracture (metals like mild steel, aluminum alloys, some polymers).Recognize tensile test outputs: yield strength, UTS, % elongation, % reduction in area.Conclude: standard tensile tests are primarily conducted on ductile materials. Verification / Alternative check: In codes and handbooks, tensile properties are the baseline for metals selection and are reported prominently for ductile metals. Brittle materials (cast iron, ceramics) are frequently characterized in compression or flexure because tensile gripping and alignment are more challenging and failures are sudden. Why Other Options Are Wrong: Brittle: can be tested, but not the typical or preferred primary method. Malleable and plastic: these are property descriptors, not standard testing categories; also malleability is linked to large plastic deformation under compression rather than tension. Elastomeric: special extensometry and low-rate tests are used, but the canonical tensile test discussion centers on ductile metals. Common Pitfalls: Assuming brittle materials are never tested in tension (they can be, but with caution); conflating ductility with malleability or plasticity terminology. Final Answer: ductile

Question 12

Thin riveted cylindrical shell under internal pressure: For a shell of diameter d and thickness t subjected to internal pressure p, what is the longitudinal (axial) membrane stress in the shell plate (neglecting joint efficiency)?

Accepted Answer

Introduction / Context: Thin-walled cylindrical pressure vessels develop two principal membrane stresses: circumferential (hoop) and longitudinal (axial). Correct formulas are crucial for safe design of boilers, tanks, and pipelines, including riveted or welded shells when joint efficiency is not specifically introduced. Given Data / Assumptions: Thin shell: t ≪ d (thin-wall assumption). Uniform internal pressure p. Homogeneous, isotropic plate; joint efficiency effects neglected unless stated. Concept / Approach: Equilibrium of half the cylinder by a longitudinal cut gives the hoop stress; equilibrium of a segment cut by a transverse plane across the cylinder gives the longitudinal stress. Standard results: σ_hoop = p d / (2 t), and σ_long = p d / (4 t). The longitudinal stress is half the hoop stress in a thin cylinder. Step-by-Step Solution: Consider a transverse cut exposing a circular area of diameter d.Pressure force tending to separate the cylinder: F_p = p * (π d^2 / 4).Resisting area of the wall (two sides): A_r = 2 * (t * π d / 2) = π d t.Equilibrium: F_p = σ_long * A_r ⇒ p (π d^2 / 4) = σ_long (π d t).Solve: σ_long = p d / (4 t). Verification / Alternative check: Compare with hoop stress derivation; ratio σ_hoop : σ_long = 2 : 1 for thin cylinders, consistent with classic results. Why Other Options Are Wrong: p d / (2 t) is the hoop (circumferential) stress, not longitudinal. p t / d and 2 p t / d invert variables and give wrong trends. p d^2 / (8 t) has incorrect dimensions for stress. Common Pitfalls: Mixing hoop and longitudinal formulas; introducing joint efficiency without being asked; applying thin-wall formulas outside their validity range (t not small compared to d). Final Answer: σ_long = p d / (4 t)

Question 13

Riveted joint strength calculations: When calculating joint strength in tearing, shearing, and crushing, which diameter should be used?

Accepted Answer

Introduction / Context: Accurate strength assessment of riveted joints requires correct geometric parameters. In practice, the plate is weakened by the hole, not by the rivet shank size alone. Hence, the diameter used in tearing, shearing, and bearing (crushing) checks must reflect the hole actually present in the plate. Given Data / Assumptions: Hole is drilled or punched to a specified diameter slightly larger than the rivet shank to allow driving/expansion. Joint efficiency calculations consider plate net section, rivet shear area, and bearing area. Concept / Approach: For tearing of plate, the net section is based on hole diameter because material is removed at the hole. For shearing of rivet and crushing (bearing) between rivet and plate, areas are also dependent on the hole size since contact and shear planes are governed by the hole geometry after fabrication. Step-by-Step Solution: Tearing strength of plate: based on (pitch − hole diameter) * plate thickness.Shear strength of rivet: area = π/4 * (d_hole)^2 for single shear per rivet.Crushing (bearing) strength: projected area = d_hole * plate thickness per rivet. Verification / Alternative check: Design codes specify hole diameter in joint efficiency calculations to incorporate clearance and fabrication tolerances. Using rivet shank diameter unconservatively overestimates capacity and ignores weakening due to hole enlargement or fitting clearances. Why Other Options Are Wrong: Rivet diameter underestimates the removed area and overestimates strength. Mean or smaller of the two diameters lacks physical basis in the stress paths. Nominal (pre-driving) diameter does not reflect finished hole size. Common Pitfalls: Confusing nominal rivet size with finished hole size; neglecting allowance (typically hole ≈ rivet + 1 mm to 1.5 mm depending on size and practice). Final Answer: actual diameter of hole drilled for rivet

Question 14

Shear of a rivet in double shear per pitch length: If the permissible shear stress is τ and the rivet shank diameter is d, what is the pull required to shear the rivet in double shear per pitch length?

Accepted Answer

Introduction / Context: Riveted joints can experience single or double shear depending on the connection geometry. Determining the shear capacity per pitch length is a standard step in joint design to ensure safety against rivet shearing. Given Data / Assumptions: Rivet diameter = d, allowable shear stress = τ. Double shear condition (two shear planes). Static loading and uniform shear distribution across the rivet cross-section. Concept / Approach: Shear capacity is shear stress times resisting area on all active shear planes. For a circular rivet shank, area per plane is A = π/4 * d^2. With two planes in double shear, the total resisting area is 2A. Multiply by τ to obtain the force to cause shearing. Step-by-Step Solution: Area per shear plane: A = π/4 * d^2.Double shear ⇒ total resisting area = 2 * A = 2 * (π/4 * d^2) = π/2 * d^2.Required pull to reach shear limit: P = τ * (π/2 * d^2) = π/2 * d^2 * τ. Verification / Alternative check: Compare to single shear: P_single = τ * (π/4 * d^2). Doubling the number of planes doubles the capacity, yielding P_double = 2 * P_single, which is the same expression obtained above. Why Other Options Are Wrong: Expressions with σt relate to tension, not shear. π/4 * d^2 * τ is for single shear only. π * d^2 * τ overestimates by a factor of 2. Common Pitfalls: Using rivet nominal diameter instead of finished hole diameter where applicable; confusing single vs. double shear; mixing allowable stresses (shear vs. tensile or bearing). Final Answer: π/2 * d^2 * τ

Question 15

Effect of percentage elongation on ductility: In a standard tensile test, how does material ductility change as the percentage elongation decreases?

Accepted Answer

Introduction / Context: Ductility is a measure of a material’s ability to undergo significant plastic deformation before fracture. It is commonly quantified by percentage elongation and percentage reduction in area obtained from standard tensile tests. Given Data / Assumptions: Ductility indicators: % elongation and % reduction in area at fracture. Comparison at similar gauge lengths and testing standards. Concept / Approach: By definition, higher percentage elongation means greater capacity for plastic deformation. Therefore, a decrease in percentage elongation indicates reduced ability to plastically deform before failure, i.e., reduced ductility. Step-by-Step Solution: Observe that % elongation reflects plastic strain accumulated before fracture.If % elongation decreases, the strain capacity to failure is lower.Thus, ductility decreases as % elongation decreases. Verification / Alternative check: Handbooks and materials data sheets consistently associate higher % elongation with ductile behavior (e.g., mild steel) and low % elongation with brittle behavior (e.g., cast iron). Why Other Options Are Wrong: “Increases/remains same” contradict the fundamental definition of ductility. “First increases then decreases” implies a non-monotonic relationship not supported for a single material comparison. “Undefined” is incorrect; ductility remains definable at all % elongations. Common Pitfalls: Comparing values taken with different gauge lengths or test standards; conflating ductility with toughness (energy absorption), which also depends on strength and strain behavior. Final Answer: decreases

Question 16

Section modulus of a beam: The section modulus Z used in flexural design is expressed in terms of the second moment of area I and the distance to the extreme fiber y. What is Z?

Accepted Answer

Introduction / Context: Section modulus Z is a geometric property that links bending moment to maximum normal stress at the outer fiber of a beam. It is widely used to size beams in structural and machine design. Given Data / Assumptions: Linear elastic behavior and Bernoulli–Euler beam theory. Maximum stress occurs at the extreme fiber located a distance y from the neutral axis. Concept / Approach: The flexure formula is σ_max = M * y / I. Rearranging gives M / Z = σ_max where Z = I / y. Thus for a given allowable stress, the required section modulus is M / σ_allow, translating directly into geometry selection. Step-by-Step Solution: Flexure formula: σ = M * y / I.At the extreme fiber: σ_max corresponds to y = y_max.Define Z = I / y_max so that σ_max = M / Z.Therefore, Z = I / y. Verification / Alternative check: For standard shapes (rectangular, circular, I-sections), published Z values equal I / y_max, confirming the definition. Example: for a rectangle b × h, I = b h^3 / 12 and y = h/2, giving Z = b h^2 / 6, a well-known result. Why Other Options Are Wrong: I * y and y / I have incorrect dimensions. M / I is unrelated to Z; it appears in curvature κ = M / (E I). I / y^2 does not match flexure relations for stress at the extreme fiber. Common Pitfalls: Using the wrong y (not the extreme fiber); confusing elastic section modulus with plastic section modulus (Zp), which is different and used in plastic design. Final Answer: I/y

Question 17

Automotive suspension springs: Is it correct to say that the springs in cars are used to store strain energy for ride comfort and load management?

Accepted Answer

Introduction / Context: Vehicle suspension systems use springs (coil, leaf, torsion bar, or air springs) to improve ride comfort, maintain tire contact, and protect the chassis. The fundamental role of a spring is to store and release strain energy when subjected to loads and road irregularities. Given Data / Assumptions: Conventional passive suspension with springs and dampers (shock absorbers). Normal driving conditions with variable road inputs. Concept / Approach: Springs store strain energy U = ∫ F dx, smoothing load transmission from the road. Dampers dissipate energy to control oscillations. Together, they filter road disturbances; the spring’s energy storage is crucial for transient load buffering and for maintaining tire-ground contact. Step-by-Step Solution: Road bump applies an upward force; the spring compresses, storing energy.As the bump passes, the spring releases energy, returning the wheel toward equilibrium.The damper converts part of this kinetic/strain energy to heat, preventing sustained oscillations. Verification / Alternative check: From elementary vibration theory, a sprung mass–spring–damper system exhibits reduced transmissibility in a target frequency range, with the spring’s compliant energy storage central to performance. Why Other Options Are Wrong: “Incorrect” ignores the fundamental mechanical role of a spring. Speed- or vehicle-type-limited statements are false; the principle applies across use cases. “Only static loads” is wrong; dynamic inputs are the main reason for springs. Common Pitfalls: Confusing the roles of spring (stores energy) and damper (dissipates energy); assuming stiffer springs always improve handling without considering comfort and tire grip trade-offs. Final Answer: Correct

Question 18

Cantilever with triangular (linearly varying) load: A cantilever beam of length l carries a load varying from zero at the free end to w per unit length at the fixed end. What is the shear force at the fixed end?

Accepted Answer

Introduction / Context: Linearly varying distributed loads are common in structural members. For cantilevers with a triangular load intensity that is zero at the tip and maximum at the fixed end, the reaction shear and moment at the fixed support are obtained from resultants of the equivalent load diagram. Given Data / Assumptions: Cantilever length = l. Load intensity varies linearly from 0 (free end) to w (fixed end). Static equilibrium; prismatic member. Concept / Approach: The total load equals the area under the load-intensity diagram. For a triangle of base l and height w, the resultant load is (1/2) * w * l, acting at the centroid of the triangle located at a distance l/3 from the high-intensity end (i.e., from the fixed end) or 2l/3 from the free end. The reaction shear at the fixed end equals this resultant (taking the cantilever as a free body). Step-by-Step Solution: Equivalent resultant load: W = (1/2) * w * l.Location: from the fixed end, a distance l/3.Equilibrium of the beam: vertical reaction (shear) at fixed end V_fixed = W = w l / 2. Verification / Alternative check: Integrate the load intensity q(x) = (w / l) * x measured from the free end to the fixed end: ∫_0^l q(x) dx = ∫_0^l (w/l) x dx = (w/l) * (l^2 / 2) = w l / 2, confirming the resultant magnitude equals the support shear. Why Other Options Are Wrong: zero: contradicts presence of a distributed load. w l / 4: underestimates by half. w l and 2 w l: overestimates the triangular resultant. Common Pitfalls: Placing the resultant at l/3 from the free end (incorrect); confusing triangular versus uniform loads (for uniform, resultant would be w l with different symbols). Final Answer: w l / 2

Question 19

Riveting practice: Relative to the rivet shank diameter, how is the plate hole size selected for drilling before riveting?

Accepted Answer

Introduction / Context: In riveted joints, holes are made slightly larger than the rivet shank to facilitate insertion, accommodate manufacturing tolerances, and allow proper rivet expansion during setting. Using the correct hole size is essential for accurate strength calculations and reliable fabrication. Given Data / Assumptions: Conventional hot- or cold-driven rivets in metal plates. Standard allowances of approximately 0.5 mm to 1.5 mm above shank diameter depending on size and practice. Concept / Approach: The hole diameter must exceed the nominal shank diameter to permit insertion and to ensure the rivet upsets properly to fill the hole during driving. Design calculations for net section, rivet shear, and bearing use the hole diameter because it governs the effective areas after drilling/punching and reaming where applicable. Step-by-Step Solution: Select a hole drill size slightly larger than the rivet shank diameter.Account for fit, tolerance, and intended expansion during riveting.Use the hole diameter in strength checks to remain conservative. Verification / Alternative check: Fabrication standards and handbooks specify hole oversize; inspection gauges and reamers are chosen accordingly. Empirical practice aligns with using larger holes to avoid damage during driving and to ensure full seating of the rivet head and shank. Why Other Options Are Wrong: Equal or smaller holes hinder insertion and risk plate damage or misalignment. “Depends only on thickness or material” ignores the primary fit and tolerance requirement; while these affect allowance magnitude, the direction is always “greater than”. Common Pitfalls: Using rivet diameter instead of hole diameter in calculations; neglecting clearance leading to practical assembly issues and residual stresses. Final Answer: greater than

Question 20

Elastic constants relationship: if a material has modulus of elasticity E equal to twice its modulus of rigidity C (i.e., E = 2C), what is the corresponding bulk modulus K for that material?

Accepted Answer

Introduction / Context: Designers often convert between the three fundamental elastic constants for isotropic materials: Young’s modulus E, modulus of rigidity C (also written G), and bulk modulus K. This question checks your fluency in using the standard relations to move from one constant to another. Given Data / Assumptions: E = 2C (i.e., E = 2G). Material is homogeneous, linear-elastic, and isotropic. Standard relationships among E, C, K, and Poisson’s ratio ν apply. Concept / Approach: For isotropic materials, the following formulas hold: E = 2C * (1 + ν) E = 3K * (1 − 2ν) Alternatively, K = E / (3 * (1 − 2ν)) and C = E / (2 * (1 + ν)) Step-by-Step Solution: From E = 2C * (1 + ν) and E = 2C, we get 2C = 2C * (1 + ν).Therefore, 1 + ν = 1 ⇒ ν = 0.Use K = E / (3 * (1 − 2ν)). With ν = 0, K = E / 3.Since E = 2C, substitute to get K = (2C) / 3 = 2C/3. Verification / Alternative check: Using the well-known identity E = 9KC / (3K + C). Set E = 2C and solve for K to confirm K = 2C/3. Why Other Options Are Wrong: 2C, 3C: These ignore the required division by 3 from the E–K–ν relationship.3C/2, C/2: Do not satisfy the isotropic consistency conditions when E = 2C. Common Pitfalls: Forgetting that ν is implicated through E = 2C(1 + ν); if E = 2C, it forces ν = 0, not any arbitrary value. Always solve for ν first when constants are related in this way. Final Answer: 2C/3

Strength of Materials Questions