Q: Thin cylindrical pressure vessels – Governing stress for design For thin cylindrical shells under internal pressure, design is primarily based on which stress component?

Introduction / Context: Thin-walled cylinders (t ≪ d) such as boilers and tanks are designed against tensile stresses caused by internal pressure. Two principal membrane stresses arise: hoop (circumferential) and longitudinal (axial). Given Data / Assumptions: Uniform internal pressure p; diameter d; thickness t; thin-wall criterion satisfied. Membrane theory with negligible radial stress compared to tangential stresses. Isotropic, linear-elastic material. Concept / Approach: For a thin cylinder, hoop stress σ_h = p d / (2 t) and longitudinal stress σ_l = p d / (4 t). Hoop stress is twice the longitudinal stress; hence it governs the design for maximum tensile stress. Step-by-Step Solution: Derive σ_h by cutting the cylinder along a diametral plane: pressure force balanced by two hoop stress resultants → σ_h = p d / (2 t).Derive σ_l by cutting with a transverse plane through the heads: σ_l = p d / (4 t).Compare magnitudes: σ_h = 2 σ_l → hoop stress governs. Verification / Alternative check: Design codes limit circumferential stress more strictly; thickness sizing formulas are based on hoop stress with allowances for corrosion and joint efficiency. Why Other Options Are Wrong: (b) longitudinal stress is smaller; (c) and (d) use averages not employed in design; (e) radial stress in thin shells is small compared to tangential components. Common Pitfalls: Applying spherical vessel formulas; misusing diameter versus radius; forgetting joint efficiency in welded/riveted shells. Final Answer: hoop stress

Q: Simple bending theory – Identify the incorrect statement Which of the following statements about pure bending of beams is incorrect?

Introduction / Context: Elementary beam theory rests on specific assumptions that lead to linear strain variation and corresponding stress distribution. Spotting an incorrect statement tests conceptual clarity about neutral axis behavior and section properties. Given Data / Assumptions: Material is homogeneous and isotropic; Hooke's law valid. Plane sections remain plane; small deflection theory. No shear deformation in the pure bending region. Concept / Approach: Strain varies linearly with distance y from the neutral axis, and stress σ = E * ε = (M * y) / I. At y = 0 (neutral axis), both strain and stress are zero; maximum stress occurs at the extreme fibers where |y| is maximum. Step-by-Step Solution: Bending stress formula: σ = M * y / I.Neutral axis: y = 0 → σ = 0.Extreme fibers: y = ±c → |σ_max| = M * c / I = M / Z, where Z = I / c is section modulus.Hence bending stress is inversely proportional to Z, consistent with option (d). Verification / Alternative check: Plotting the linear stress diagram confirms zero at the neutral axis and maxima at the outermost fibers. Beam design maximizes Z to reduce stresses for a given bending moment M. Why Other Options Are Wrong: Option (c) is wrong because stress at the neutral axis is zero. Options (a), (b), and (d) are standard results. Option (e) is consistent with homogeneous materials. Common Pitfalls: Confusing neutral axis with centroidal axis location for unsymmetrical sections; forgetting linear distribution under elastic bending. Final Answer: At the neutral axis of a beam, the bending stress is maximum.

Q: Euler's column theory – End conditions factor C According to Euler, the critical (crippling) load for a long column is p = π^2 E I / (C l^2). For a column with one end fixed and the other end free, what is the value of C?

Introduction / Context: Euler's buckling formula expresses the critical load in terms of column stiffness and effective length. Different end conditions change the effective length through the factor C. Given Data / Assumptions: Long, slender column (Euler region) with modulus E and second moment I. Geometric length = l; one end fixed, other free (cantilever-like). Elastic stability analysis; initial imperfections neglected. Concept / Approach: Effective length L_e captures boundary restraint. For fixed–free, L_e = 2 l. Substituting L_e in Euler's base form p = π^2 E I / L_e^2 gives p = π^2 E I / (4 l^2). Comparing with p = π^2 E I / (C l^2) yields C = 4. Step-by-Step Solution: Identify end condition: fixed–free → L_e = 2 l.Base Euler: p = π^2 E I / L_e^2.Substitute: p = π^2 E I / (2 l)^2 = π^2 E I / (4 l^2).Match forms → C = 4. Verification / Alternative check: Standard effective length factors: pinned–pinned C = 1; fixed–pinned C = 2; fixed–fixed C = 4; fixed–free C = 4 (because L_e = 2 l). Many texts tabulate these values consistently. Why Other Options Are Wrong: (a) and (b) correspond to more restrained cases; (c) is for fixed–pinned; larger C reduces critical load, and fixed–free has the least capacity among common end conditions. Common Pitfalls: Mixing up the two notations: some authors write p = π^2 E I / (K l)^2, where K is the effective length factor (K = 2 for fixed–free). Converting to the given form requires care. Final Answer: 4

Q: Beam deflection under central load: A simply supported beam A of length l, breadth b, and depth d carries a central load W. Another identical beam B carries a central load of 2W. How does the midspan deflection of beam B compare with that of beam A?

Introduction / Context: For linearly elastic beams, midspan deflection under a central point load is directly proportional to the magnitude of that load when all geometric and material properties remain the same. This question checks recall of proportionality in classic beam-bending relations for a simply supported beam. Given Data / Assumptions: Beam A and Beam B are identical: same length l, breadth b, depth d, modulus of elasticity E, and moment of inertia I. Both beams are simply supported with a central point load. Loads: W on A, and 2W on B. Small deflection theory, linear elasticity, and no shear deformation effects. Concept / Approach: For a simply supported beam with a central point load P, the maximum deflection at midspan is given by delta_max = (P * l^3) / (48 * E * I) Since E, I, and l are unchanged between the two beams, the deflection ratio equals the load ratio. Step-by-Step Solution: For Beam A: delta_A = (W * l^3) / (48 * E * I)For Beam B: delta_B = (2W * l^3) / (48 * E * I)Take the ratio: delta_B / delta_A = (2W) / W = 2Therefore, delta_B = 2 * delta_A, i.e., double. Verification / Alternative check: A quick dimensional check confirms deflection dependence on load P linearly, so doubling P doubles the deflection when geometry and material remain constant. Why Other Options Are Wrong: One-fourth / one-half / four times / one and a half times: These contradict the direct proportionality of deflection to load for this case. Common Pitfalls: Confusing this with uniformly distributed load or cantilever formulae; mixing up how deflection scales with depth (via I) versus with load P. Final Answer: double

Q: Strength of materials definition: The ratio of the largest (maximum) load carried in a tensile test to the original cross-sectional area of the test piece is called what?

Introduction / Context: Tensile testing provides a stress–strain curve and key strength parameters. The identification of elastic limit, yield stress, ultimate stress, and breaking (fracture) stress is fundamental to material selection and design safety factors. Given Data / Assumptions: The 'largest load' refers to the maximum force the specimen sustains during the test before necking reduces the load capacity. Stress referenced to the original cross-sectional area (engineering or nominal stress). Standard tensile test conditions. Concept / Approach: Engineering stress is defined as load / original area. The highest value of this engineering stress during the test is the ultimate tensile stress (UTS). Yield stress is the stress at which plastic flow begins (e.g., 0.2% offset). Breaking stress refers to stress at fracture load on the original area, which is typically less than the UTS for ductile materials. Step-by-Step Solution: Let A0 be original area; let P_max be largest load.Engineering stress at maximum load = P_max / A0.By definition, this is the ultimate stress (UTS). Verification / Alternative check: Compare with the stress at fracture: P_fracture / A0, which can be less than UTS because the load typically drops after necking in ductile materials. Why Other Options Are Wrong: Elastic limit / yield stress occur at lower loads before the maximum. Breaking stress is at fracture load, not the maximum load necessarily. True stress at maximum load uses instantaneous area; the question specifies original area → engineering UTS. Common Pitfalls: Confusing yield strength with UTS; mixing true stress (uses current area) with engineering stress (uses original area). Final Answer: ultimate stress

Q: Elastic constants relationship check: If the modulus of elasticity E equals twice the modulus of rigidity G (i.e., E = 2G), then Poisson's ratio ν is equal to zero. Is this statement correct?

Introduction / Context: Elastic constants of isotropic materials are interrelated. Recognizing the standard relationship among E (Young's modulus), G (shear modulus), and ν (Poisson's ratio) is essential in stress–strain analysis and finite element modeling. Given Data / Assumptions: Isotropic, homogeneous, linear elastic material. Given that E = 2G. Concept / Approach: The standard relation for isotropic materials is E = 2 * G * (1 + ν) Rearranging gives ν = (E / (2G)) − 1. Substituting E = 2G directly yields ν = 0. Step-by-Step Solution: Start with E = 2 * G * (1 + ν)Given E = 2G → 2G = 2G * (1 + ν)Divide both sides by 2G (nonzero): 1 = 1 + ν → ν = 0 Verification / Alternative check: Typical metals have ν around 0.25 to 0.35, which would give E closer to about 2.5G to 2.7G. The special case ν = 0 indeed makes E exactly 2G, matching the provided statement. Why Other Options Are Wrong: “Incorrect” conflicts with the exact algebraic relation. “Depends on temperature only” and “Correct only for plastics” are irrelevant; ν is determined from the relation, not material class per se. “Cannot be determined” is false because E = 2G fully determines ν = 0 for isotropy. Common Pitfalls: Applying the relation to anisotropic materials (where it does not hold) or confusing true values with approximate empirical ranges for specific materials. Final Answer: Correct

Q: Columns and effective length: Which end condition gives the maximum equivalent (effective) length and therefore the least buckling load for the same physical length?

Introduction / Context: Euler buckling load depends inversely on the square of the effective length. Recognizing which boundary condition produces the largest effective length is crucial for stability design of slender columns. Given Data / Assumptions: Slender, straight column, initially perfect, elastic behavior. Same physical length L and cross-section for all cases. Classical Euler effective length factors K apply. Concept / Approach: Euler buckling load P_cr = (pi^2 * E * I) / (Le^2), where Le = K * L. Larger Le gives a smaller buckling load. Standard K values: both ends fixed → K = 0.5; both hinged → K = 1.0; one end fixed, other hinged → K ≈ 0.7; one end fixed, other free (cantilever) → K = 2.0. Thus, the fixed–free case has the largest effective length (Le = 2L). Step-by-Step Solution: List K values for each end condition.Compute Le = K * L. Largest Le occurs when K is largest.Fixed–free: K = 2.0 → Le = 2L (maximum). Verification / Alternative check: Compare Euler loads qualitatively: P_cr ∝ 1 / K^2, so fixed–free has the smallest P_cr and is the easiest to buckle, aligning with intuition for a cantilevered column. Why Other Options Are Wrong: Hinged–hinged (K = 1) and fixed–hinged (K ≈ 0.7) have smaller Le; fixed–fixed (K = 0.5) is the stiffest (smallest Le). The guided case is not a standard basic boundary condition in Euler's classic set. Common Pitfalls: Mixing up K for fixed–hinged vs hinged–hinged; assuming fixed–fixed is worst, when it is actually best for buckling resistance. Final Answer: One end fixed and the other end free

Q: Two bars, same size and tensile force: If unit elongation of material A : material B = 2 : 5, what is the ratio of their moduli of elasticity E_A : E_B?

Introduction / Context: Axial deformation under the same load relates inversely to the modulus of elasticity when geometry is the same. This question reinforces the proportionality between elongation and 1/E for bars of identical dimensions under identical forces. Given Data / Assumptions: Two bars have the same length L and cross-sectional area A. Same tensile force P acts on both bars. Unit elongation (strain) ratio A:B = 2:5. Linear elastic behavior: delta = P * L / (A * E). Concept / Approach: For equal P, L, and A, the elongation (and strain) is inversely proportional to E. Therefore, if strain_A : strain_B = 2 : 5, then E_A : E_B = 5 : 2. Step-by-Step Solution: delta ∝ 1 / E for same P, L, A.Given delta_A : delta_B = 2 : 5 → (1/E_A) : (1/E_B) = 2 : 5.Invert the ratio: E_A : E_B = 5 : 2. Verification / Alternative check: Pick numbers: if E_A = 50 and E_B = 20 (arbitrary units), then delta_A ∝ 1/50 and delta_B ∝ 1/20, giving a ratio 0.02 : 0.05 = 2 : 5, confirming. Why Other Options Are Wrong: They either invert or distort the correct inverse relationship; 1:1 would imply equal elongations, which contradicts the data. Common Pitfalls: Forgetting to invert the ratio; confusing modulus with strength; assuming larger E gives larger elongation (it is the opposite). Final Answer: 5 : 2

Q: Energy methods terminology: The total strain energy stored in a body due to external loading (within the elastic range) is termed as what?

Introduction / Context: Several energy-based terms are used in strength of materials: strain energy, resilience, proof resilience, and modulus of resilience. Correct identification supports design against shock and fatigue, and informs material selection for energy absorption. Given Data / Assumptions: Loading remains within the elastic limit of the material. We are concerned with the total energy stored under the applied loading. Concept / Approach: Strain energy is the energy stored in a body due to deformation. When this is considered up to the elastic limit, it is called resilience. Proof resilience is specifically the maximum strain energy stored per unit volume up to the elastic limit. Modulus of resilience is that proof resilience expressed per unit volume. Toughness (for comparison) relates to total energy absorption up to fracture (area under the entire stress–strain curve). Step-by-Step Solution: Identify the domain: elastic range only.Term for total energy stored in this range: resilience.Differentiate from proof resilience and modulus of resilience: the latter is an energy per unit volume at the elastic limit. Verification / Alternative check: Consulting standard definitions: resilience (overall stored energy within elastic limits), proof resilience (maximum at elastic limit), modulus of resilience (per unit volume measure at elastic limit). Why Other Options Are Wrong: Proof resilience: specifically the maximum energy at yield, not just any total stored energy. Modulus of resilience: energy per unit volume measure, not the total for the body. Impact energy / toughness: concern energy to fracture, typically beyond elastic range. Common Pitfalls: Using the terms interchangeably; forgetting the 'per unit volume' qualifier for modulus of resilience; confusing toughness with resilience. Final Answer: resilience

Question 1

Tensile testing – Breaking stress versus ultimate tensile stress  In a standard tensile test on a ductile material, the engineering breaking (fracture) stress is how related to the ultimate tensile stress?

Accepted Answer

Introduction / Context: The engineering stress–strain curve for ductile metals shows a peak (ultimate tensile stress) followed by necking until fracture. Understanding the difference between ultimate and breaking stress is vital for safe design and correct interpretation of material data. Given Data / Assumptions: Engineering stress is calculated using original area. After UTS, localized necking reduces actual area. Material is ductile so necking occurs prior to fracture. Concept / Approach: Ultimate tensile stress (UTS) is the maximum engineering stress attained. After necking starts, load typically drops while the original area is still used to compute engineering stress, so the engineering stress value at fracture (breaking stress) is lower than the UTS. Step-by-Step Solution: Before necking: load rises → stress increases.UTS reached at maximum load divided by original area.Necking: load decreases, original area constant for engineering stress → stress value drops.Breaking stress (at fracture) < UTS for ductile materials. Verification / Alternative check: True stress (based on instantaneous area) may increase during necking, but engineering stress (based on original area) decreases; testing standards and typical curves confirm breaking stress lower than UTS. Why Other Options Are Wrong: (a) equal to is false for ductile materials; (c) greater than contradicts the curve; (d) it can be compared using definitions; (e) is an unnecessary conditional—necking generally occurs in ductile materials. Common Pitfalls: Mixing true stress with engineering stress; misreading the stress–strain curve endpoints. Final Answer: less than

Question 2

Thin cylindrical pressure vessels – Governing stress for design  For thin cylindrical shells under internal pressure, design is primarily based on which stress component?

Accepted Answer

Introduction / Context: Thin-walled cylinders (t ≪ d) such as boilers and tanks are designed against tensile stresses caused by internal pressure. Two principal membrane stresses arise: hoop (circumferential) and longitudinal (axial). Given Data / Assumptions: Uniform internal pressure p; diameter d; thickness t; thin-wall criterion satisfied. Membrane theory with negligible radial stress compared to tangential stresses. Isotropic, linear-elastic material. Concept / Approach: For a thin cylinder, hoop stress σ_h = p d / (2 t) and longitudinal stress σ_l = p d / (4 t). Hoop stress is twice the longitudinal stress; hence it governs the design for maximum tensile stress. Step-by-Step Solution: Derive σ_h by cutting the cylinder along a diametral plane: pressure force balanced by two hoop stress resultants → σ_h = p d / (2 t).Derive σ_l by cutting with a transverse plane through the heads: σ_l = p d / (4 t).Compare magnitudes: σ_h = 2 σ_l → hoop stress governs. Verification / Alternative check: Design codes limit circumferential stress more strictly; thickness sizing formulas are based on hoop stress with allowances for corrosion and joint efficiency. Why Other Options Are Wrong: (b) longitudinal stress is smaller; (c) and (d) use averages not employed in design; (e) radial stress in thin shells is small compared to tangential components. Common Pitfalls: Applying spherical vessel formulas; misusing diameter versus radius; forgetting joint efficiency in welded/riveted shells. Final Answer: hoop stress

Question 3

Simple bending theory – Identify the incorrect statement  Which of the following statements about pure bending of beams is incorrect?

Accepted Answer

Introduction / Context: Elementary beam theory rests on specific assumptions that lead to linear strain variation and corresponding stress distribution. Spotting an incorrect statement tests conceptual clarity about neutral axis behavior and section properties. Given Data / Assumptions: Material is homogeneous and isotropic; Hooke's law valid. Plane sections remain plane; small deflection theory. No shear deformation in the pure bending region. Concept / Approach: Strain varies linearly with distance y from the neutral axis, and stress σ = E * ε = (M * y) / I. At y = 0 (neutral axis), both strain and stress are zero; maximum stress occurs at the extreme fibers where |y| is maximum. Step-by-Step Solution: Bending stress formula: σ = M * y / I.Neutral axis: y = 0 → σ = 0.Extreme fibers: y = ±c → |σ_max| = M * c / I = M / Z, where Z = I / c is section modulus.Hence bending stress is inversely proportional to Z, consistent with option (d). Verification / Alternative check: Plotting the linear stress diagram confirms zero at the neutral axis and maxima at the outermost fibers. Beam design maximizes Z to reduce stresses for a given bending moment M. Why Other Options Are Wrong: Option (c) is wrong because stress at the neutral axis is zero. Options (a), (b), and (d) are standard results. Option (e) is consistent with homogeneous materials. Common Pitfalls: Confusing neutral axis with centroidal axis location for unsymmetrical sections; forgetting linear distribution under elastic bending. Final Answer: At the neutral axis of a beam, the bending stress is maximum.

Question 4

Euler's column theory – End conditions factor C  According to Euler, the critical (crippling) load for a long column is p = π^2 E I / (C l^2). For a column with one end fixed and the other end free, what is the value of C?

Accepted Answer

Introduction / Context: Euler's buckling formula expresses the critical load in terms of column stiffness and effective length. Different end conditions change the effective length through the factor C. Given Data / Assumptions: Long, slender column (Euler region) with modulus E and second moment I. Geometric length = l; one end fixed, other free (cantilever-like). Elastic stability analysis; initial imperfections neglected. Concept / Approach: Effective length L_e captures boundary restraint. For fixed–free, L_e = 2 l. Substituting L_e in Euler's base form p = π^2 E I / L_e^2 gives p = π^2 E I / (4 l^2). Comparing with p = π^2 E I / (C l^2) yields C = 4. Step-by-Step Solution: Identify end condition: fixed–free → L_e = 2 l.Base Euler: p = π^2 E I / L_e^2.Substitute: p = π^2 E I / (2 l)^2 = π^2 E I / (4 l^2).Match forms → C = 4. Verification / Alternative check: Standard effective length factors: pinned–pinned C = 1; fixed–pinned C = 2; fixed–fixed C = 4; fixed–free C = 4 (because L_e = 2 l). Many texts tabulate these values consistently. Why Other Options Are Wrong: (a) and (b) correspond to more restrained cases; (c) is for fixed–pinned; larger C reduces critical load, and fixed–free has the least capacity among common end conditions. Common Pitfalls: Mixing up the two notations: some authors write p = π^2 E I / (K l)^2, where K is the effective length factor (K = 2 for fixed–free). Converting to the given form requires care. Final Answer: 4

Question 5

Beam deflection under central load: A simply supported beam A of length l, breadth b, and depth d carries a central load W. Another identical beam B carries a central load of 2W. How does the midspan deflection of beam B compare with that of beam A?

Accepted Answer

Introduction / Context: For linearly elastic beams, midspan deflection under a central point load is directly proportional to the magnitude of that load when all geometric and material properties remain the same. This question checks recall of proportionality in classic beam-bending relations for a simply supported beam. Given Data / Assumptions: Beam A and Beam B are identical: same length l, breadth b, depth d, modulus of elasticity E, and moment of inertia I. Both beams are simply supported with a central point load. Loads: W on A, and 2W on B. Small deflection theory, linear elasticity, and no shear deformation effects. Concept / Approach: For a simply supported beam with a central point load P, the maximum deflection at midspan is given by delta_max = (P * l^3) / (48 * E * I) Since E, I, and l are unchanged between the two beams, the deflection ratio equals the load ratio. Step-by-Step Solution: For Beam A: delta_A = (W * l^3) / (48 * E * I)For Beam B: delta_B = (2W * l^3) / (48 * E * I)Take the ratio: delta_B / delta_A = (2W) / W = 2Therefore, delta_B = 2 * delta_A, i.e., double. Verification / Alternative check: A quick dimensional check confirms deflection dependence on load P linearly, so doubling P doubles the deflection when geometry and material remain constant. Why Other Options Are Wrong: One-fourth / one-half / four times / one and a half times: These contradict the direct proportionality of deflection to load for this case. Common Pitfalls: Confusing this with uniformly distributed load or cantilever formulae; mixing up how deflection scales with depth (via I) versus with load P. Final Answer: double

Question 6

Strength of materials definition: The ratio of the largest (maximum) load carried in a tensile test to the original cross-sectional area of the test piece is called what?

Accepted Answer

Introduction / Context: Tensile testing provides a stress–strain curve and key strength parameters. The identification of elastic limit, yield stress, ultimate stress, and breaking (fracture) stress is fundamental to material selection and design safety factors. Given Data / Assumptions: The 'largest load' refers to the maximum force the specimen sustains during the test before necking reduces the load capacity. Stress referenced to the original cross-sectional area (engineering or nominal stress). Standard tensile test conditions. Concept / Approach: Engineering stress is defined as load / original area. The highest value of this engineering stress during the test is the ultimate tensile stress (UTS). Yield stress is the stress at which plastic flow begins (e.g., 0.2% offset). Breaking stress refers to stress at fracture load on the original area, which is typically less than the UTS for ductile materials. Step-by-Step Solution: Let A0 be original area; let P_max be largest load.Engineering stress at maximum load = P_max / A0.By definition, this is the ultimate stress (UTS). Verification / Alternative check: Compare with the stress at fracture: P_fracture / A0, which can be less than UTS because the load typically drops after necking in ductile materials. Why Other Options Are Wrong: Elastic limit / yield stress occur at lower loads before the maximum. Breaking stress is at fracture load, not the maximum load necessarily. True stress at maximum load uses instantaneous area; the question specifies original area → engineering UTS. Common Pitfalls: Confusing yield strength with UTS; mixing true stress (uses current area) with engineering stress (uses original area). Final Answer: ultimate stress

Question 7

Elastic constants relationship check: If the modulus of elasticity E equals twice the modulus of rigidity G (i.e., E = 2G), then Poisson's ratio ν is equal to zero. Is this statement correct?

Accepted Answer

Introduction / Context: Elastic constants of isotropic materials are interrelated. Recognizing the standard relationship among E (Young's modulus), G (shear modulus), and ν (Poisson's ratio) is essential in stress–strain analysis and finite element modeling. Given Data / Assumptions: Isotropic, homogeneous, linear elastic material. Given that E = 2G. Concept / Approach: The standard relation for isotropic materials is E = 2 * G * (1 + ν) Rearranging gives ν = (E / (2G)) − 1. Substituting E = 2G directly yields ν = 0. Step-by-Step Solution: Start with E = 2 * G * (1 + ν)Given E = 2G → 2G = 2G * (1 + ν)Divide both sides by 2G (nonzero): 1 = 1 + ν → ν = 0 Verification / Alternative check: Typical metals have ν around 0.25 to 0.35, which would give E closer to about 2.5G to 2.7G. The special case ν = 0 indeed makes E exactly 2G, matching the provided statement. Why Other Options Are Wrong: “Incorrect” conflicts with the exact algebraic relation. “Depends on temperature only” and “Correct only for plastics” are irrelevant; ν is determined from the relation, not material class per se. “Cannot be determined” is false because E = 2G fully determines ν = 0 for isotropy. Common Pitfalls: Applying the relation to anisotropic materials (where it does not hold) or confusing true values with approximate empirical ranges for specific materials. Final Answer: Correct

Question 8

Columns and effective length: Which end condition gives the maximum equivalent (effective) length and therefore the least buckling load for the same physical length?

Accepted Answer

Introduction / Context: Euler buckling load depends inversely on the square of the effective length. Recognizing which boundary condition produces the largest effective length is crucial for stability design of slender columns. Given Data / Assumptions: Slender, straight column, initially perfect, elastic behavior. Same physical length L and cross-section for all cases. Classical Euler effective length factors K apply. Concept / Approach: Euler buckling load P_cr = (pi^2 * E * I) / (Le^2), where Le = K * L. Larger Le gives a smaller buckling load. Standard K values: both ends fixed → K = 0.5; both hinged → K = 1.0; one end fixed, other hinged → K ≈ 0.7; one end fixed, other free (cantilever) → K = 2.0. Thus, the fixed–free case has the largest effective length (Le = 2L). Step-by-Step Solution: List K values for each end condition.Compute Le = K * L. Largest Le occurs when K is largest.Fixed–free: K = 2.0 → Le = 2L (maximum). Verification / Alternative check: Compare Euler loads qualitatively: P_cr ∝ 1 / K^2, so fixed–free has the smallest P_cr and is the easiest to buckle, aligning with intuition for a cantilevered column. Why Other Options Are Wrong: Hinged–hinged (K = 1) and fixed–hinged (K ≈ 0.7) have smaller Le; fixed–fixed (K = 0.5) is the stiffest (smallest Le). The guided case is not a standard basic boundary condition in Euler's classic set. Common Pitfalls: Mixing up K for fixed–hinged vs hinged–hinged; assuming fixed–fixed is worst, when it is actually best for buckling resistance. Final Answer: One end fixed and the other end free

Question 9

Two bars, same size and tensile force: If unit elongation of material A : material B = 2 : 5, what is the ratio of their moduli of elasticity E_A : E_B?

Accepted Answer

Introduction / Context: Axial deformation under the same load relates inversely to the modulus of elasticity when geometry is the same. This question reinforces the proportionality between elongation and 1/E for bars of identical dimensions under identical forces. Given Data / Assumptions: Two bars have the same length L and cross-sectional area A. Same tensile force P acts on both bars. Unit elongation (strain) ratio A:B = 2:5. Linear elastic behavior: delta = P * L / (A * E). Concept / Approach: For equal P, L, and A, the elongation (and strain) is inversely proportional to E. Therefore, if strain_A : strain_B = 2 : 5, then E_A : E_B = 5 : 2. Step-by-Step Solution: delta ∝ 1 / E for same P, L, A.Given delta_A : delta_B = 2 : 5 → (1/E_A) : (1/E_B) = 2 : 5.Invert the ratio: E_A : E_B = 5 : 2. Verification / Alternative check: Pick numbers: if E_A = 50 and E_B = 20 (arbitrary units), then delta_A ∝ 1/50 and delta_B ∝ 1/20, giving a ratio 0.02 : 0.05 = 2 : 5, confirming. Why Other Options Are Wrong: They either invert or distort the correct inverse relationship; 1:1 would imply equal elongations, which contradicts the data. Common Pitfalls: Forgetting to invert the ratio; confusing modulus with strength; assuming larger E gives larger elongation (it is the opposite). Final Answer: 5 : 2

Question 10

Energy methods terminology: The total strain energy stored in a body due to external loading (within the elastic range) is termed as what?

Accepted Answer

Introduction / Context: Several energy-based terms are used in strength of materials: strain energy, resilience, proof resilience, and modulus of resilience. Correct identification supports design against shock and fatigue, and informs material selection for energy absorption. Given Data / Assumptions: Loading remains within the elastic limit of the material. We are concerned with the total energy stored under the applied loading. Concept / Approach: Strain energy is the energy stored in a body due to deformation. When this is considered up to the elastic limit, it is called resilience. Proof resilience is specifically the maximum strain energy stored per unit volume up to the elastic limit. Modulus of resilience is that proof resilience expressed per unit volume. Toughness (for comparison) relates to total energy absorption up to fracture (area under the entire stress–strain curve). Step-by-Step Solution: Identify the domain: elastic range only.Term for total energy stored in this range: resilience.Differentiate from proof resilience and modulus of resilience: the latter is an energy per unit volume at the elastic limit. Verification / Alternative check: Consulting standard definitions: resilience (overall stored energy within elastic limits), proof resilience (maximum at elastic limit), modulus of resilience (per unit volume measure at elastic limit). Why Other Options Are Wrong: Proof resilience: specifically the maximum energy at yield, not just any total stored energy. Modulus of resilience: energy per unit volume measure, not the total for the body. Impact energy / toughness: concern energy to fracture, typically beyond elastic range. Common Pitfalls: Using the terms interchangeably; forgetting the 'per unit volume' qualifier for modulus of resilience; confusing toughness with resilience. Final Answer: resilience

Question 11

Combined biaxial normal stresses with in-plane shear: For stresses σx and σy on perpendicular planes with shear τxy, what is the maximum shear stress τ_max in the material?

Accepted Answer

Introduction / Context: State of stress at a point in a plane generally consists of two perpendicular normal stresses and an in-plane shear. Determining the maximum shear stress is key for yielding criteria (e.g., Tresca) and failure analysis. This is a staple result derived conveniently via Mohr’s circle or stress transformation equations. Given Data / Assumptions: Plane stress state with normal stresses σx and σy, and in-plane shear τxy. Material is homogeneous and isotropic; linear elastic relations used for transformation. Sign convention consistent with standard mechanics texts. Concept / Approach: Using Mohr’s circle for plane stress, the circle center is at (σx + σy)/2 and radius R equals 0.5 * sqrt( (σx − σy)^2 + 4 * τxy^2 ). The maximum in-plane shear stress equals the radius of Mohr’s circle. Out-of-plane considerations are neglected here (plane stress). Step-by-Step Solution: Center C = (σx + σy) / 2 on the normal stress axis.Radius R = 0.5 * sqrt( (σx − σy)^2 + 4 * τxy^2 ).Maximum shear stress τ_max equals the radius R.Therefore, τ_max = 0.5 * sqrt( (σx − σy)^2 + 4 * τxy^2 ). Verification / Alternative check: Special cases: if τxy = 0 → τ_max = |σx − σy| / 2, matching the uniaxial/biaxial no-shear case. If σx = σy → τ_max = |τxy|, as expected for pure shear superposed on hydrostatic stress. Why Other Options Are Wrong: (σx + σy)/2 is the circle center, not τ_max. |σx − σy|/2 applies only when τxy = 0. sqrt( σx^2 + σy^2 + τxy^2 ) is not a recognized invariant for τ_max. 2 * τxy is incorrect except under nonstandard conditions. Common Pitfalls: Confusing radius with diameter; forgetting the factor 0.5; neglecting the 4 * τxy^2 term, which is critical when shear is significant. Final Answer: 0.5 * sqrt( (σx − σy)^2 + 4 * τxy^2 )

Question 12

Stability of masonry dam (sliding check): The force of friction at the base should be at least what multiple of the total horizontal water pressure per metre length to avoid sliding?

Accepted Answer

Introduction / Context: For gravity and masonry dams, sliding stability is ensured by providing adequate frictional resistance at the dam–foundation interface against the horizontal water thrust. Design is often done with a prescribed factor of safety against sliding to account for uncertainties in loads, material properties, and uplift forces. Given Data / Assumptions: Total horizontal water pressure per metre length is P_w. Resultant resisting friction is μ * W, where W is effective vertical force (including self-weight and considering uplift, if any). A standard factor of safety against sliding is adopted in preliminary design. Concept / Approach: The no-sliding condition requires μ * W ≥ P_w. For design safety, codes and classic practice adopt a factor of safety F_s, so μ * W ≥ F_s * P_w. A commonly used minimum F_s is about 1.5 for normal loading conditions, implying frictional resistance at least 1.5 times the driving water thrust. Step-by-Step Solution: Define F_s = (available friction) / (driving thrust) = (μ * W) / P_w.Adopt F_s ≥ 1.5 for normal load combinations.Therefore, μ * W ≥ 1.5 * P_w → friction at least 1.5 times water pressure. Verification / Alternative check: Some practices may adopt larger factors for unusual loads (e.g., earthquake, flood). For the baseline question with typical options, 1.5 is the accepted minimum. Why Other Options Are Wrong: Equal to P_w gives F_s = 1.0 (no safety margin). Double or 2.5 times are conservative beyond the common minimum; not typically specified as the minimum. 0.5 times is unsafe (F_s < 1). Common Pitfalls: Ignoring uplift or sediment pressure; confusing overturning checks with sliding; using peak instead of long-term friction coefficients without safety factors. Final Answer: 1.5 times the total water pressure

Question 13

Relationship between shear force and bending moment: If there is no increase or decrease in shear force between two points (i.e., zero load intensity over that region), what does it indicate about the bending moment between those points?

Accepted Answer

Introduction / Context: Beam diagram interpretation hinges on the differential relations: dV/dx = w and dM/dx = V, where w is load intensity, V is shear force, and M is bending moment. Understanding how constancy or variation of one quantity affects the others is essential for quick diagram sketching and checking. Given Data / Assumptions: Between two points along the span, there is no distributed load: w = 0. As a result, the shear force does not increase or decrease in that region (V is constant; may be zero or nonzero). Classical beam theory, small deflections. Concept / Approach: If w = 0, then dV/dx = 0 → V is constant. Next, dM/dx = V: if V is constant but nonzero, M varies linearly; if V = 0 specifically, M is constant. The phrase 'no increase or decrease in shear force' is typically used to denote w = 0, and especially the common case where V = 0, leading to locally constant bending moment. In many textbook statements: 'Where shear is zero, bending moment is maximum or minimum (locally constant slope zero).' For a region with V identically zero, M is constant across that region. Step-by-Step Solution: No load between the points → w = 0.Hence dV/dx = 0 → V is constant (often zero in unloaded segments of idealized problems).If V = 0 in that region → dM/dx = 0 → M is constant. Verification / Alternative check: In a simply supported beam with a pure couple applied inside the span, the adjacent unloaded region has zero V and thus constant M, matching this conclusion. Why Other Options Are Wrong: Zero M is not implied by constant shear; only zero shear implies constant M (not necessarily zero in value). Parabolic variation arises from linearly varying shear, not constant or zero shear. Infinite bending moments do not occur in standard statics problems. Common Pitfalls: Confusing 'constant shear' (nonzero) with 'zero shear'; forgetting that M'(x) = w, so w = 0 makes M linear unless V = 0 makes it flat (constant). Final Answer: Bending moment is constant between those points

Question 14

Elongation of a vertical bar due to self-weight A prismatic bar of length l and diameter d is rigidly fixed at the top and hangs freely under its own weight. If w is the weight per unit volume and E is Young's modulus, what is the total axial elonga…

Accepted Answer

Introduction / Context: This problem tests the fundamentals of strength of materials related to axial deformation under a nonuniform load. A hanging bar experiences a linearly varying axial stress due to its own weight, which produces a definite elongation that must be accounted for in tall members, tie rods, and hangers. Given Data / Assumptions: Prismatic bar: constant cross-sectional area A = (π/4) * d^2. Length = l; weight per unit volume = w (units N/m^3). Material follows Hooke’s law with Young’s modulus E; small deformations. Bar fixed at the top and hanging vertically in air; self-weight only. Concept / Approach: The axial force at a section is the weight of the portion below that section. Hence stress varies linearly along the length. Total elongation equals the integral of strain over the length, where strain = stress / E. Step-by-Step Solution: Let y be the distance measured downward from the top (y = 0 at top, y = l at bottom).Force at section y is the weight of the segment below: P(y) = w * A * (l − y).Stress at y: σ(y) = P(y) / A = w * (l − y).Strain at y: ε(y) = σ(y) / E = w * (l − y) / E.Elongation: δ = ∫[0→l] ε(y) dy = (w/E) * ∫[0→l] (l − y) dy = (w/E) * (l^2/2) = (w * l^2) / (2 * E). Verification / Alternative check: Dimensional check: w has N/m^3; multiplied by l^2 gives N/m; divide by E (N/m^2) gives meters, consistent with elongation. Why Other Options Are Wrong: Expressions without the 1/2 overestimate the elongation; terms involving d appear when using weight per unit length, not weight per unit volume; stating zero contradicts mechanics. Common Pitfalls: Confusing w as weight per unit length; forgetting that stress is not uniform along the bar; omitting the integral. Final Answer: (w * l^2) / (2 * E)

Question 15

Shear in a T-section beam A beam with a T-shaped cross-section is subjected to a vertical shear force F. At which location within the section does the maximum shear stress occur?

Accepted Answer

Introduction / Context: Understanding the distribution of shear stress in non-rectangular sections such as T-sections is critical in structural and machine design. Shear flow is highest where the first moment of area about the neutral axis is large and the local thickness is small. Given Data / Assumptions: Prismatic T-section, homogeneous material, linear elastic behavior. Vertical shear force F acting on the beam. Bernoulli–Euler beam theory (plane sections remain plane). Concept / Approach: The shear stress at a level y is τ = V * Q / (I * b), where V is shear force, Q is the first moment of area above (or below) the level, I is the second moment of area about the neutral axis, and b is the local thickness at that level. For most practical T- and I-sections, τ peaks near the neutral axis and within the web where b is small. Step-by-Step Solution: Locate the neutral axis (NA) of the T-section by composite area method.Note the web at the NA has minimal thickness b compared to the flange.Compute τ(y) qualitatively using τ ∝ Q / b; Q is significant near the NA, and b is the web thickness, yielding high τ there.Thus the maximum shear stress typically occurs at or very close to the NA within the web. Verification / Alternative check: Detailed τ distribution plots for T/I-beams consistently show a maximum at the NA in the web, decreasing toward zero at free surfaces. Why Other Options Are Wrong: Top/bottom extremes are free surfaces where τ → 0. The junction of web and flange can have a local change in slope of τ, but not the absolute maximum if NA lies within the web. Common Pitfalls: Assuming peak τ at flange–web junction by intuition; forgetting the influence of local thickness b in τ = VQ/(Ib). Final Answer: At the neutral axis of the section (within the web)

Question 16

Euler buckling – Equivalent length comparison A column of clear length l is fixed at both ends. It is equivalent, for buckling strength, to which effective length if the end conditions were hinged at both ends?

Accepted Answer

Introduction / Context: Critical load for elastic buckling depends on the effective length, which is a function of end restraints. Comparing different end conditions using effective length is central to column design. Given Data / Assumptions: Same cross-section and material for both cases. Case 1: fixed–fixed ends; Case 2: hinged–hinged ends. Euler buckling regime (long, slender column), linear elastic behavior. Concept / Approach: Euler load: P_cr = π^2 * E * I / (L_e^2), where L_e = K * l. For fixed–fixed, K = 0.5; for hinged–hinged, K = 1.0. Equivalence of buckling strength requires equal L_e values. Step-by-Step Solution: Fixed–fixed: L_e = 0.5 * l.Hinged–hinged: L_e = 1.0 * l.Thus a fixed–fixed column of length l has the same buckling strength as a hinged–hinged column of effective length l/2. Verification / Alternative check: For the same P_cr, set π^2 E I /(0.5 l)^2 = π^2 E I /(L_eq)^2, giving L_eq = l/2. Why Other Options Are Wrong: Values like l/8 or l/4 overstate restraint; l or 2l ignore the increased stiffness due to fixity. Common Pitfalls: Confusing clear length with effective length factor K; mixing end conditions (fixed–free, fixed–hinged, etc.). Final Answer: l/2

Question 17

Flexure theory – Variation of strain with distance from the neutral axis When a straight prismatic beam is subjected to a bending moment within the elastic range, how does the normal strain in any layer vary with its distance from the neutral axis?

Accepted Answer

Introduction / Context: The linear strain distribution across a beam depth under pure bending is a central result of Bernoulli–Euler beam theory, leading to the classic flexure formula for stress. Given Data / Assumptions: Prismatic, homogeneous beam; small deflections. Plane sections remain plane after bending. Elastic, isotropic material with Young’s modulus E. Concept / Approach: If curvature is κ, the strain at a distance y from the neutral axis is ε = κ * y. Hence strain varies linearly from compression at one extreme to tension at the other, and is zero at the neutral axis. Step-by-Step Explanation: Assume a cross-section rotates to maintain plane sections (Bernoulli hypothesis).Fiber length change is proportional to its radius difference from the neutral surface.Therefore ε(y) = κ * y, indicating direct proportionality to distance from the neutral axis. Verification / Alternative check: Combining ε = κ y with σ = E ε yields σ = E κ y, the familiar linear stress distribution and the flexure formula σ = M y / I. Why Other Options Are Wrong: Constant or inverse relationships contradict ε = κ y; independence from distance is incorrect; maximum at the NA is false since ε = 0 at the NA. Common Pitfalls: Confusing strain with shear stress distribution; assuming non-linear distribution in the elastic range without justification. Final Answer: Directly proportional to the distance from the neutral axis

Question 18

Fillet weld design – Nature of loading for parallel fillet welds Evaluate the statement: “Parallel fillet welds are designed for bending strength.” Select the correct response.

Accepted Answer

Introduction / Context: Fillet welds are commonly classified by orientation relative to the applied load: parallel or transverse. Correctly identifying the governing stress (shear vs. normal) is essential for safe welding design. Given Data / Assumptions: Parallel fillet weld: weld axis is parallel to the applied force. Design per elementary weld mechanics (throat area concept). Concept / Approach: When the load is parallel to the weld axis, the weld primarily resists by shear along its throat area. Conversely, a transverse fillet weld (load perpendicular to weld axis) is governed by normal (tensile/compressive) stress—sometimes loosely termed “bending” in plates. Step-by-Step Reasoning: Identify orientation: parallel fillet weld → shear on throat.Design equation: τ_allow ≥ P / (Σ throat areas).Bending checks are associated with eccentricities or transverse fillet welds, not the basic parallel case. Verification / Alternative check: Standard weld design references classify stresses by orientation; parallel fillet welds are sized on allowable shear stress of the weld metal and throat area. Why Other Options Are Wrong: “Agree” statements misidentify the governing stress; adding conditions (intermittent/dynamic) does not change the fundamental shear nature. Common Pitfalls: Confusing fillet weld orientation; neglecting eccentric loads that can introduce additional moments, which is a separate check. Final Answer: Disagree

Question 19

Definition check – Strain components in uniaxial loading The deformation of a bar per unit original length measured along the direction of the applied force is called what?

Accepted Answer

Introduction / Context: Clear definitions of strain components are foundational in strength of materials and elasticity. Linear (axial) strain quantifies elongation or shortening along the load direction. Given Data / Assumptions: Small deformations; engineering strain definition. Uniaxial loading scenario for clarity. Concept / Approach: Linear strain ε_longitudinal = ΔL / L. Lateral strain is the contraction/expansion perpendicular to load. Volumetric strain is the sum of principal strains (approx. ΔV/V). Shear strain is the change in right angle (distortion) measured as an angle in radians. Step-by-Step Clarification: Identify the direction: along applied force → longitudinal.Apply definition: ε = change in length / original length.Hence it is linear (longitudinal) strain. Verification / Alternative check: Consistency with Hooke’s law: σ = E * ε_longitudinal in uniaxial tension/compression. Why Other Options Are Wrong: Lateral strain is transverse; volumetric includes all directions; shear strain is angular distortion; “Poisson strain” is not a standard term for the primary definition. Common Pitfalls: Mixing up lateral and linear strain, especially when discussing Poisson’s ratio. Final Answer: Linear strain

Question 20

Springs in instruments – Do watch springs absorb shocks and vibrations? Consider the function of springs in watches and delicate instruments. Are they used to absorb shocks and vibrations to protect and stabilize the mechanism?

Accepted Answer

Introduction / Context: Springs in precision instruments (e.g., balance springs, shock-protection systems) play a vital role in preserving timekeeping accuracy and protecting delicate components from impacts and vibrations. Given Data / Assumptions: General knowledge of mechanical watches and instruments. Use of compliant elements to mitigate shocks. Concept / Approach: Springs store and release energy, providing compliance that reduces peak forces transmitted to sensitive parts. In watches, hairsprings regulate oscillation, while dedicated spring mounts (e.g., shock settings) isolate jewels and pivots from sudden impacts. Step-by-Step Reasoning: Impact or vibration occurs.Elastic elements deflect, spreading force over time.Reduced peak acceleration protects pivots and maintains oscillation stability. Verification / Alternative check: Shock-protection systems use miniature springs to suspend jewel bearings; the balance spring also contributes to controlled oscillation and some disturbance recovery. Why Other Options Are Wrong: Limiting use to digital watches, temperature compensation only, or torque-only ignores the protective role of compliant spring elements in mechanical designs. Common Pitfalls: Confusing the regulation role of the hairspring with the separate shock-absorbing spring mounts; both are spring-based protections. Final Answer: Yes

Strength of Materials Questions