Q: Simple bending in beams: How does the normal bending stress vary with distance from the neutral axis in a prismatic beam under pure (simple) bending?

Introduction / Context: The flexure (bending) formula provides the stress distribution across a beam's section under pure bending. Understanding how stress varies with depth is essential for sizing sections and placing reinforcement. Given Data / Assumptions: Beam is prismatic and behaves elastically. Plane sections remain plane (Bernoulli-Euler assumption). No shear (pure bending) at the section considered. Concept / Approach: The bending stress is given by σ = M * y / I, where M is the bending moment about the neutral axis, y is the distance from the neutral axis, and I is the second moment of area. Hence, σ varies linearly with y: zero at the neutral axis and maximum at the extreme fibers. Step-by-Step Solution: Write flexure formula: σ = M * y / I.For fixed M and I, σ ∝ y.Thus, stress is directly proportional to the distance from the neutral axis. Verification / Alternative check: Stress diagram is triangular over the depth, changing sign across the neutral axis for sagging vs hogging moments. Why Other Options Are Wrong: Inversely proportional: contradicts σ = M y / I.Curvilinear/constant/zero at extreme fibers: all inconsistent with linear distribution and boundary conditions. Common Pitfalls: Confusing bending stress with shear stress, which has a parabolic distribution in rectangular sections. Final Answer: Directly proportional to the distance from the neutral axis.

Q: Strength of Materials — Cantilever beam with end load: effect of doubling the width on tip deflection A cantilever beam of rectangular cross-section carries a single concentrated load at the free end. If the breadth (width) b of the section is doubl…

Introduction / Context: For a cantilever beam with a point load at the free end, the tip deflection depends inversely on the flexural rigidity EI. For a rectangular section, the second moment of area I is proportional to the width b and to the cube of the depth h. This question checks understanding of how geometric changes alter stiffness and deflection. Given Data / Assumptions: Cantilever length L; end load P. Rectangular section with width b and depth h. Material modulus E is unchanged; only b is doubled. Small deflection Euler–Bernoulli beam theory applies. Concept / Approach: The tip deflection of a cantilever with an end load is given by δ = PL^3 / (3EI). For a rectangle, I = bh^3/12 about the strong axis. If b doubles, I doubles; hence δ becomes half. No other parameter changes, so the ratio of new to old deflection is 1/2. Step-by-Step Solution: I_old = bh^3/12I_new = (2b)h^3/12 = 2I_oldδ_old = PL^3 / (3EI_old)δ_new = PL^3 / (3EI_new) = (1/2)*δ_old Verification / Alternative check: Doubling any parameter that increases I linearly will halve the deflection, consistent with δ ∝ 1/I. A quick dimensional check confirms consistency. Why Other Options Are Wrong: 2 or 8 imply the deflection increases, which contradicts increased stiffness. 1/8 would require I to increase by a factor of 8, which does not occur by doubling b. 3 has no basis in the formula. Common Pitfalls: Confusing the roles of width and depth; deflection is far more sensitive to h since I ∝ h^3. Applying the cantilever formula for a different loading or support case. Final Answer: 1/2.

Q: Volumetric strain under equal normal stresses on all faces A solid cube is subjected to equal normal stresses on all six faces (hydrostatic pressure/tri-axial equal loading). The volumetric strain will be x times the linear strain measured along any…

Introduction / Context: When a body is subjected to equal normal stresses in three mutually perpendicular directions, the deformation is isotropic and the total volumetric change relates simply to the linear strains along each axis. This is a fundamental concept in strength of materials and elasticity relevant to pressure vessels and soil mechanics (volumetric strain). Given Data / Assumptions: Equal normal stress applied along x, y, z directions. Small strains; linear elasticity. Linear strain ε is the same in magnitude along each principal axis (for equal stress), ignoring Poisson coupling for the ratio asked (since we compare total to any one component). Concept / Approach: Volumetric strain ε_v is defined as the sum of linear strains in three orthogonal directions: ε_v = ε_x + ε_y + ε_z. If all three linear strains are equal (ε_x = ε_y = ε_z = ε), then ε_v = 3ε. Therefore, the volumetric strain is three times the linear strain measured along any direction. Step-by-Step Solution: Let ε_x = ε_y = ε_z = ε.Volumetric strain: ε_v = ε_x + ε_y + ε_z.Hence ε_v = ε + ε + ε = 3ε.Therefore, x = 3. Verification / Alternative check: This result is independent of elastic constants E and ν because the ratio is purely geometric for equal strains along orthogonal axes. Why Other Options Are Wrong: 1 or 2 underestimate the contribution from three directions. 4 and 5 overestimate because there are only three orthogonal components being summed. Common Pitfalls: Confusing the algebraic sum definition of volumetric strain with a product or average. Final Answer: 3.

Q: Pure bending of a prismatic beam under constant bending moment If a constant-section (prismatic) beam is subjected to a uniform bending moment throughout its length (no shear force), the deflected shape of the centroidal axis is:

Introduction / Context: Pure bending refers to a region in a beam where the bending moment is constant and the shear force is zero. Understanding the resulting curvature and deflected shape under pure bending is fundamental to beam theory and the derivation of flexure formulas. Given Data / Assumptions: Prismatic beam of constant E (modulus of elasticity) and I (second moment of area). Uniform bending moment M along the entire length; shear force V = 0. Small deflection Euler–Bernoulli assumptions. Concept / Approach: The curvature κ of a beam is given by κ = M / (EI) under small deflection theory. If M, E, and I are constants, κ is constant. A curve with constant curvature is a circular arc. Therefore, the neutral axis of the beam bends into a part of a circle under pure bending. Step-by-Step Solution: κ = M / (EI) (constant) ⇒ radius R = 1/κ = EI / M (constant).A curve with constant radius R is a circle.Thus, the deflected shape is a circular arc. Verification / Alternative check: Differential equation of the elastic curve: d^2y/dx^2 = M/(EI) = constant integrates to y being a quadratic in x only when slope is small; however, the geometric curve corresponding to constant curvature is circular to first order. Why Other Options Are Wrong: Parabolic or catenary shapes arise under different loading (e.g., UDL for parabolic moment diagram; cable under uniform self-weight for catenary). “None” or “straight line” contradict constant nonzero curvature. Common Pitfalls: Confusing the parabolic bending-moment diagram under UDL with the geometric deflected shape. Final Answer: A circular arc.

Question 1

Simply supported timber beam with central point load: A 150 cm span wooden beam (rectangular section 16 cm × 24 cm, depth taken as 24 cm) carries a concentrated load at midspan. Permissible bending stress ft = 75 kg/cm² and permissible shear stress …

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Introduction / Context: Design of simply supported beams under a central point load requires checking both bending and shear against permissible stresses. The lower governing value determines the safe load. Given Data / Assumptions: Span L = 150 cm. Section b × d = 16 cm × 24 cm (d = vertical depth). Permissible bending stress ft = 75 kg/cm². Permissible shear stress fs = 10 kg/cm². Load W acts at midspan. Concept / Approach: For a central point load on a simple span: maximum bending moment M = W * L / 4. Section modulus for a rectangle Z = b * d² / 6. Maximum support shear V = W / 2; for a rectangular section, τ_max = 1.5 * V / (b * d). Step-by-Step Solution: Compute section modulus: Z = 16 * 24^2 / 6 = 1536 cm³.Bending check: ft ≥ M / Z = (W * L / 4) / Z → W ≤ 4 * Z * ft / L = 4 * 1536 * 75 / 150 = 3072 kg (≈ 3075 kg).Shear check: τ_max = 1.5 * (W / 2) / (b * d) = 1.5 * W / (2 * 16 * 24) → fs ≥ τ_max gives W ≤ (2 * b * d * fs) / 1.5 = (2 * 16 * 24 * 10) / 1.5 = 5120 kg.Governing value is bending: W_safe ≈ 3072 kg ≈ 3075 kg. Verification / Alternative check: Recalculate M = 3075 * 150 / 4 = 115,312.5 kg·cm; σ = M/Z ≈ 75.1 kg/cm², matching allowable within rounding. Why Other Options Are Wrong: Values near 3025–3100 kg are distractors; only 3075 kg aligns with the bending limit and rounding of 3072 kg.Higher loads would exceed ft; lower loads are conservative but not the computed safe limit. Common Pitfalls: Using average shear instead of τ_max for rectangular beams; treating 16 cm as depth rather than breadth. Final Answer: 3075 kg.

Question 2

Truss analysis basics: For a statically determinate simple truss, which methods can be used to determine member forces under given loading?

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Introduction / Context: Determining axial forces in truss members is fundamental in structural analysis and design. Multiple classical methods exist, each with its strengths for different member groups and load cases. Given Data / Assumptions: Truss is statically determinate and properly supported. Loads are applied at joints; members are two-force members. Self-weight and secondary effects are neglected for basic analysis. Concept / Approach: The method of joints solves for unknowns by enforcing equilibrium at each joint (ΣFx = 0, ΣFy = 0). The method of sections cuts through the truss and applies global equilibrium (ΣFx, ΣFy, ΣM = 0) to solve selected members efficiently. Graphical methods (Cremona/Maxwell diagrams) provide visual force polygons suitable for hand-drawn solutions and checks. Step-by-Step Solution: Decide the analysis scope: all members or a few target members.Pick method: joints for systematic full solution; sections for targeted members; graphical for visual cross-check.All three are valid for statically determinate simple trusses. Verification / Alternative check: Solutions obtained by different methods should agree numerically if assumptions are consistent. Why Other Options Are Wrong: Any single method alone is valid but the question asks which can be used; the inclusive answer is “All of the above.” Common Pitfalls: Forgetting to check joint solvability order; misreading tension/compression sign; cutting more than three unknowns with method of sections. Final Answer: All of the above.

Question 3

Horizontal members in a loaded truss: Depending on loading and geometry, what can be the nature of axial force in the horizontal members shown (tension/compression/zero)?

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Introduction / Context: In truss systems, members are idealized as carrying only axial forces (tension or compression). The sign and magnitude depend on load placement, support conditions, and geometry. Given Data / Assumptions: Planar pin-jointed truss with loads at joints. Members are two-force members; shear and bending within members are neglected. “Horizontal members” refers to chords/ties that may change nature with load reversal. Concept / Approach: Equilibrium at joints and of sections decides the internal axial force. For example, the top chord of a simply supported, uniformly loaded truss is typically in compression, while the bottom chord is in tension. Under different load patterns or support changes, a segment can become zero-force or switch sign. Step-by-Step Solution: Analyze joint equilibrium or cut the truss with a section.Compute axial forces; classify as tension (+) or compression (−).Recognize special zero-force conditions (e.g., two non-collinear members meeting at an unloaded joint next to a collinear member). Verification / Alternative check: Use method of sections to cross-check a member found tensile by joints; results must match. Why Other Options Are Wrong: “Always compressive/tensile/zero” is false; the nature depends on the specific loading and geometry.“Always shear only” contradicts the truss two-force member assumption. Common Pitfalls: Confusing sign convention; neglecting that load reversals (e.g., wind uplift) can flip member action. Final Answer: Any of the above depending on the case.

Question 4

Stress transformation under axial tension: A member of normal cross-section A is subjected to a tensile force P. What is the normal stress on an oblique plane inclined at angle θ to the transverse (normal) plane?

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Introduction / Context: Determining stresses on inclined planes helps predict failure planes and understand combined normal and shear actions even when the external load is uniaxial. Given Data / Assumptions: Tension P on a uniform member with area A. Angle θ is measured from the transverse cross-section. Linear elasticity and small deformations. Concept / Approach: Under axial tension, the traction on an inclined plane has magnitude (P/A) * cos θ and decomposes into normal and shear components. Standard results: σ_n = (P/A) * cos^2 θ and τ = (P/A) * sin θ * cos θ. Step-by-Step Solution: Find inclined area A_inclined = A / cos θ.Traction magnitude p = P / A_inclined = (P/A) * cos θ.Normal component σ_n = p * cos θ = (P/A) * cos^2 θ. Verification / Alternative check: Check limits: θ = 0° → σ_n = P/A; θ = 90° → σ_n = 0, consistent with mechanics. Why Other Options Are Wrong: sin^2 θ form swaps boundary values; shear expression sin θ cos θ is not a normal stress; sec^2 θ and zero are not generally correct. Common Pitfalls: Confusing angle definition or mixing up σ_n and τ. Final Answer: (P / A) * cos^2 θ.

Question 5

Vertical bar under self-weight and end load: A long bar is held vertically from its upper end and carries a load at its lower end. Considering the additional stress due to the bar's own weight, where will the maximum normal stress occur?

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Introduction / Context: Members oriented vertically experience a stress gradient due to self-weight. When an additional load is applied at the free (lower) end and the bar is supported at the upper end, the support section carries the greatest force. Given Data / Assumptions: Bar is long, prismatic, and vertical. Supported (held) at the top; load applied at the bottom end. Self-weight acts uniformly along the length. Concept / Approach: Internal axial force at a cross-section equals the sum of the applied end load plus the weight of the portion of bar below that section. The portion of bar contributing to the force is greatest at the top section, hence the maximum stress occurs at the upper support. Step-by-Step Solution: Let w be weight per unit length and P the end load.Axial force at a section located y below the top: N(y) = P + w * y.This is maximum at y = L (top section reaction), giving N_max = P + w * L and σ_max = N_max / A at the top. Verification / Alternative check: Plot N(y) versus y: a straight line increasing toward the top; the maximum occurs at the support. Why Other Options Are Wrong: Lower/mid sections carry less cumulative self-weight, hence lower stress.Uniform stress is not possible with self-weight. Common Pitfalls: Confusing a hanging bar (top support) with a bar supported from below (then the gradient reverses). Final Answer: At the built-in upper cross-section.

Question 6

Thin cylinder criterion: A cylindrical shell may be treated as “thin” if the ratio of wall thickness t to diameter D is less than which limit?

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Introduction / Context: Formulas for thin-walled pressure vessels assume a uniform membrane stress through the thickness and neglect radial stress variation. A practical geometric limit separates “thin” from “thick.” Given Data / Assumptions: Cylindrical shell with diameter D and thickness t. Membrane theory valid when t/D is small. Elastic, isotropic material; moderate internal pressure. Concept / Approach: For thin cylinders, hoop and longitudinal stresses are evaluated using σ_hoop = p * D / (2 * t) and σ_long = p * D / (4 * t). These rely on negligible radial stress gradient. A commonly adopted limit is t/D < 1/20 (or equivalently D/t > 20). Step-by-Step Solution: Check criterion: if t/D < 1/20, thin-wall equations apply.For t/D ≥ 1/20, switch to thick-cylinder (Lame) theory. Verification / Alternative check: Codes and textbooks often use 1/20 to 1/10 as transition ranges; 1/20 is the conventional conservative threshold for thin shells. Why Other Options Are Wrong: 1/10 or 1/15: too thick for thin-wall assumptions.1/25: more restrictive than needed (still acceptable but not the standard conventional cutoff).1/5: far too thick for thin-wall theory. Common Pitfalls: Applying thin-wall formulas outside their validity range, leading to unsafe stress estimates. Final Answer: 1/20.

Question 7

Simple bending in beams: How does the normal bending stress vary with distance from the neutral axis in a prismatic beam under pure (simple) bending?

Accepted Answer

Introduction / Context: The flexure (bending) formula provides the stress distribution across a beam's section under pure bending. Understanding how stress varies with depth is essential for sizing sections and placing reinforcement. Given Data / Assumptions: Beam is prismatic and behaves elastically. Plane sections remain plane (Bernoulli-Euler assumption). No shear (pure bending) at the section considered. Concept / Approach: The bending stress is given by σ = M * y / I, where M is the bending moment about the neutral axis, y is the distance from the neutral axis, and I is the second moment of area. Hence, σ varies linearly with y: zero at the neutral axis and maximum at the extreme fibers. Step-by-Step Solution: Write flexure formula: σ = M * y / I.For fixed M and I, σ ∝ y.Thus, stress is directly proportional to the distance from the neutral axis. Verification / Alternative check: Stress diagram is triangular over the depth, changing sign across the neutral axis for sagging vs hogging moments. Why Other Options Are Wrong: Inversely proportional: contradicts σ = M y / I.Curvilinear/constant/zero at extreme fibers: all inconsistent with linear distribution and boundary conditions. Common Pitfalls: Confusing bending stress with shear stress, which has a parabolic distribution in rectangular sections. Final Answer: Directly proportional to the distance from the neutral axis.

Question 8

Strength of Materials — Cantilever beam with end load: effect of doubling the width on tip deflection A cantilever beam of rectangular cross-section carries a single concentrated load at the free end. If the breadth (width) b of the section is doubl…

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Introduction / Context: For a cantilever beam with a point load at the free end, the tip deflection depends inversely on the flexural rigidity EI. For a rectangular section, the second moment of area I is proportional to the width b and to the cube of the depth h. This question checks understanding of how geometric changes alter stiffness and deflection. Given Data / Assumptions: Cantilever length L; end load P. Rectangular section with width b and depth h. Material modulus E is unchanged; only b is doubled. Small deflection Euler–Bernoulli beam theory applies. Concept / Approach: The tip deflection of a cantilever with an end load is given by δ = PL^3 / (3EI). For a rectangle, I = bh^3/12 about the strong axis. If b doubles, I doubles; hence δ becomes half. No other parameter changes, so the ratio of new to old deflection is 1/2. Step-by-Step Solution: I_old = bh^3/12I_new = (2b)h^3/12 = 2I_oldδ_old = PL^3 / (3EI_old)δ_new = PL^3 / (3EI_new) = (1/2)*δ_old Verification / Alternative check: Doubling any parameter that increases I linearly will halve the deflection, consistent with δ ∝ 1/I. A quick dimensional check confirms consistency. Why Other Options Are Wrong: 2 or 8 imply the deflection increases, which contradicts increased stiffness. 1/8 would require I to increase by a factor of 8, which does not occur by doubling b. 3 has no basis in the formula. Common Pitfalls: Confusing the roles of width and depth; deflection is far more sensitive to h since I ∝ h^3. Applying the cantilever formula for a different loading or support case. Final Answer: 1/2.

Question 9

Beams with equal overhangs under uniformly distributed load A simply supported beam has a clear span l and equal overhangs a on both sides (total length l + 2a). It carries a uniformly distributed load over the whole length including the overhangs. …

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Introduction / Context: Beams with overhangs exhibit both sagging and hogging bending moments under a uniformly distributed load (UDL). Whether the moment within the supported span changes sign depends on the relative magnitude of the overhangs compared to the span. This determines if negative (hogging) effects from the overhangs penetrate into the main span sufficiently to create a contraflexure point. Given Data / Assumptions: Equal overhangs of length a at both ends; clear span between supports is l. Uniformly distributed load of intensity w acting along the entire length (l + 2a). Linear elastic, small deflection theory. Concept / Approach: Because of symmetry, the reactions at the supports are equal. The overhangs produce negative moments at the supports which compete with the positive (sagging) moments in the main span. A sign change within the span occurs when the negative influence from the overhangs is strong enough, which happens when the overhangs are relatively long compared to the span. Step-by-Step Solution (qualitative): Compute reactions by symmetry: R_A = R_B = w*(l + 2a)/2.The support sections carry hogging moments due to the UDL on the adjacent overhangs.As a increases, hogging at supports grows; the positive span moment is reduced and eventually a contraflexure point appears within the span.The classical criterion for a sign change in the span under full-length UDL with equal overhangs is l < 2a. Verification / Alternative check: Detailed shear and moment equations show zero-moment locations moving into the span once the overhang load effect dominates, consistent with the rule l < 2a. Why Other Options Are Wrong: l > 2a: overhangs are too short; the span moment remains of one sign (sagging). l = 2a is the limiting case; the question asks when a sign change occurs, which is strictly when l is less than 2a. l = 3a or 4a again correspond to shorter overhangs relative to the span. Common Pitfalls: Confusing the condition for zero support moment with the condition for a span contraflexure point. Final Answer: l < 2a.

Question 10

Propped cantilever under full uniformly distributed load A cantilever beam of length L carrying a uniformly distributed load over its full length is propped at the free end so that the free end remains level with the fixed end. The bending moment is…

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Introduction / Context: A propped cantilever combines features of both a fixed-free cantilever and a simple support at the free end. Under a uniformly distributed load (UDL), the internal bending moment diagram has zeros at locations where the curvature changes sign, known as points of contraflexure. This question probes the qualitative position of that second zero moment within the span. Given Data / Assumptions: Beam length L; uniformly distributed load w over 0 to L. Fixed at x = 0; propped (vertical reaction) at x = L so that deflection at x = L is zero (same level as fixed end). Linear elastic behavior; small deflections. Concept / Approach: Because the tip is propped to zero deflection, bending moment vanishes at both ends: M(0) = 0 (due to compatibility for this statically indeterminate system) and M(L) = 0 at the prop. The UDL creates a hogging domain near the fixed support and a sagging domain closer to the free end with a transition (M = 0) inside the span. Classical analysis or standard charts place this interior zero approximately near the quarter point from the free end. Step-by-Step Solution (outline): Establish equilibrium with unknown prop reaction R and fixed-end actions.Apply compatibility: deflection at x = L is zero.Solve for R, then form M(x) along the span.Find the interior root of M(x) = 0; it lies close to x ≈ 0.75L from the fixed end, i.e., ≈ L/4 from the free end. Verification / Alternative check: Standard propped-cantilever tables for a full-length UDL show a contraflexure near the quarter point from the free end, corroborating the qualitative result required by the MCQ. Why Other Options Are Wrong: Fixed end and mid-span are not zero-moment points for this case. Three-quarters from free end corresponds to near the fixed region, which is not where the second zero occurs. L/3 is not supported by canonical solutions. Common Pitfalls: Assuming the free end moment must be nonzero as in a basic cantilever; the prop alters boundary conditions. Final Answer: At about one-quarter of the length from the free end (≈ L/4).

Question 11

Volumetric strain under equal normal stresses on all faces A solid cube is subjected to equal normal stresses on all six faces (hydrostatic pressure/tri-axial equal loading). The volumetric strain will be x times the linear strain measured along any…

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Introduction / Context: When a body is subjected to equal normal stresses in three mutually perpendicular directions, the deformation is isotropic and the total volumetric change relates simply to the linear strains along each axis. This is a fundamental concept in strength of materials and elasticity relevant to pressure vessels and soil mechanics (volumetric strain). Given Data / Assumptions: Equal normal stress applied along x, y, z directions. Small strains; linear elasticity. Linear strain ε is the same in magnitude along each principal axis (for equal stress), ignoring Poisson coupling for the ratio asked (since we compare total to any one component). Concept / Approach: Volumetric strain ε_v is defined as the sum of linear strains in three orthogonal directions: ε_v = ε_x + ε_y + ε_z. If all three linear strains are equal (ε_x = ε_y = ε_z = ε), then ε_v = 3ε. Therefore, the volumetric strain is three times the linear strain measured along any direction. Step-by-Step Solution: Let ε_x = ε_y = ε_z = ε.Volumetric strain: ε_v = ε_x + ε_y + ε_z.Hence ε_v = ε + ε + ε = 3ε.Therefore, x = 3. Verification / Alternative check: This result is independent of elastic constants E and ν because the ratio is purely geometric for equal strains along orthogonal axes. Why Other Options Are Wrong: 1 or 2 underestimate the contribution from three directions. 4 and 5 overestimate because there are only three orthogonal components being summed. Common Pitfalls: Confusing the algebraic sum definition of volumetric strain with a product or average. Final Answer: 3.

Question 12

Pure bending of a prismatic beam under constant bending moment If a constant-section (prismatic) beam is subjected to a uniform bending moment throughout its length (no shear force), the deflected shape of the centroidal axis is:

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Introduction / Context: Pure bending refers to a region in a beam where the bending moment is constant and the shear force is zero. Understanding the resulting curvature and deflected shape under pure bending is fundamental to beam theory and the derivation of flexure formulas. Given Data / Assumptions: Prismatic beam of constant E (modulus of elasticity) and I (second moment of area). Uniform bending moment M along the entire length; shear force V = 0. Small deflection Euler–Bernoulli assumptions. Concept / Approach: The curvature κ of a beam is given by κ = M / (EI) under small deflection theory. If M, E, and I are constants, κ is constant. A curve with constant curvature is a circular arc. Therefore, the neutral axis of the beam bends into a part of a circle under pure bending. Step-by-Step Solution: κ = M / (EI) (constant) ⇒ radius R = 1/κ = EI / M (constant).A curve with constant radius R is a circle.Thus, the deflected shape is a circular arc. Verification / Alternative check: Differential equation of the elastic curve: d^2y/dx^2 = M/(EI) = constant integrates to y being a quadratic in x only when slope is small; however, the geometric curve corresponding to constant curvature is circular to first order. Why Other Options Are Wrong: Parabolic or catenary shapes arise under different loading (e.g., UDL for parabolic moment diagram; cable under uniform self-weight for catenary). “None” or “straight line” contradict constant nonzero curvature. Common Pitfalls: Confusing the parabolic bending-moment diagram under UDL with the geometric deflected shape. Final Answer: A circular arc.

Question 13

Short hollow circular column — maximum eccentricity to avoid tension For a short column with hollow circular cross-section of external diameter D and internal diameter d, what is the greatest eccentricity e (measured from the centroid) that a compre…

Accepted Answer

Introduction / Context: For concentrically compressed members with possible eccentricity, avoiding tensile stress anywhere on the section requires the load line to pass within the core (kern). The kern radius depends on the section geometry. For circular and hollow circular sections, closed-form expressions exist for the limiting eccentricity that ensures compressive stress across the whole section. Given Data / Assumptions: Short column (no instability effects; pure elastic stress distribution). Hollow circular section with outer diameter D and inner diameter d. Linear stress distribution due to combined axial load and bending. Concept / Approach: The limiting eccentricity equals the kern radius r_k measured from the centroid. For a hollow circular section, r_k = (I)/(AR_max), where I is second moment of area about any axis through the centroid, A is area, and R_max is the outer radius (D/2). Evaluation gives r_k = (D^2 + d^2)/(8D). A load with e ≤ r_k keeps the resultant compressive over the entire section; e > r_k would induce tension at the far edge. Step-by-Step Solution: Let R = D/2 and r = d/2.A = π(R^2 − r^2)I = (π/4)(R^4 − r^4)Kern radius r_k = I / (AR) = [(π/4)(R^4 − r^4)] / [π(R^2 − r^2) * R] = (R^2 + r^2) / 4Substitute R = D/2 and r = d/2 ⇒ r_k = (D^2 + d^2) / (8D) Verification / Alternative check: For a solid circular section (d = 0), the formula reduces to r_k = D/8, matching the well-known solid circle core radius. Why Other Options Are Wrong: (b) Uses D^2 − d^2, which corresponds to A terms, not the correct (R^2 + r^2) dependence. (c) Dividing by d instead of D gives a nonphysical increase as the hole shrinks. (d) The factor 16 is incorrect per derivation. (e) D/12 is not the kern radius for circular sections. Common Pitfalls: Using Z = I/(c) or section modulus directly; kern uses I/(A*R), not I/c. Final Answer: e = (D^2 + d^2) / (8D).

Question 14

Definition of shear force at a section of a beam For a simply supported (or any) beam, the shear force at a given cross-section is equal to the:

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Introduction / Context: Shear force and bending moment diagrams are constructed from equilibrium of a cut section. A precise definition of shear force at a section is essential before solving for reactions, internal forces, and drawing diagrams. Given Data / Assumptions: Beam subjected to transverse loading (point loads, UDLs, etc.). Statics: equilibrium of a free-body diagram (FBD) of part of the beam. Sign convention for shear: typically upward on left face positive (or another consistent convention). Concept / Approach: At a chosen section, we isolate one part of the beam. The internal shear force V at that section balances the algebraic sum of all external transverse forces on the isolated part. Therefore, V equals the algebraic sum (with sign) of transverse forces acting on either the left or the right segment of the cut, not the entire beam. Step-by-Step Solution: Cut the beam at the section of interest.Write ΣF_y = 0 for one side of the cut (left or right).Solve for internal shear V; this is the algebraic sum of transverse forces on that side. Verification / Alternative check: Consistency check: Shear force diagram slope equals negative of load intensity (dV/dx = −w), derived from this definition and equilibrium, reinforcing the sign-aware summation concept. Why Other Options Are Wrong: (a) and (e) are deformation measures, not force definitions. (b) takes forces from both sides and along all directions, which is not how shear is defined at a section. (c) omits algebraic sign, losing directionality necessary for equilibrium. Common Pitfalls: Summing forces on both sides of the cut, which trivially gives zero and is meaningless for finding internal forces. Final Answer: Algebraic sum of the transverse forces on one side of the section.

Question 15

“Force method” association in structural analysis In classical structural analysis, the approach often termed the “force method” (flexibility method) is closely associated with which of the following relations/tools for continuous beams?

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Introduction / Context: Two fundamental families of methods exist in structural analysis: displacement (stiffness) methods and force (flexibility) methods. Continuous beams traditionally were analyzed using the force method via classic relationships that link end moments and spans, the most famous being the three-moment equation. Given Data / Assumptions: Prismatic or piecewise-prismatic continuous beams on multiple supports. Small deflection linear behavior. Compatibility of rotations and deflections imposed at interior supports. Concept / Approach: The force method treats redundant reactions or end moments as unknown “forces” and enforces compatibility to solve them. Clapeyron’s three-moment equation directly relates support moments in two adjacent spans to loads and span properties, serving as a principal instrument of the force method for continuous beams. Step-by-Step Solution (conceptual): Select redundants (e.g., interior support moments).Write flexibility relations and compatibility conditions of rotations/deflections.Employ the three-moment equation to connect adjacent spans and solve for unknown moments.Construct shear and moment diagrams thereafter. Verification / Alternative check: Solutions obtained via the force method (three-moment equation) agree with displacement-method (slope-deflection/ stiffness) solutions for the same geometry and loading, validating the association. Why Other Options Are Wrong: Moment-area theorems are deformation evaluation tools, not the core of the force method for continuous beams. Maxwell’s reciprocal theorem describes symmetry of deflections, not a direct solving tool for continuous beams. Conjugate-beam analogy is a deformation technique, closer to displacement methods. “None” is incorrect since a specific relation is indeed tied to the force method. Common Pitfalls: Confusing three-moment (force method) with slope-deflection (displacement method); both solve continuous beams but via different primary unknowns. Final Answer: Three-moment equation (Clapeyron’s theorem for three moments).

Question 16

Simply supported beam of span L under a uniformly distributed load (total load notation) A simply supported beam of clear span L carries a uniformly distributed load whose total magnitude over the span is W. What is the maximum bending moment M (sag…

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Introduction / Context: The maximum bending moment for a simply supported beam under a uniformly distributed load (UDL) depends on whether the symbol denotes load per unit length or total load. Many texts use w for intensity (force per length) and W for total load. This question explicitly states W is the total load acting over the span L. Given Data / Assumptions: Simply supported beam, span L. Total uniformly distributed load W across the span. Static equilibrium and small deflection theory. Concept / Approach: For a UDL of intensity w (force per unit length), the maximum bending moment is M_max = wL^2/8 at mid-span. If W is the total load, then W = wL, so M_max = (W/L)L^2/8 = WL/8. Step-by-Step Solution: Given W_total = W = wL ⇒ w = W / L.Standard result: M_max = wL^2 / 8.Substitute: M_max = (W/L)L^2 / 8 = WL / 8. Verification / Alternative check: Shear at mid-span is zero; bending moment diagram is parabolic, peaking at mid-span with the stated value, consistent with classical solutions. Why Other Options Are Wrong: W L / 4 doubles the correct value. W L^2 / 8 corresponds to using W as w (confusing total with intensity). W^2 L / 8 and W/(8L) have incorrect dimensions. Common Pitfalls: Mixing symbols W and w; remember W = w*L for a uniform distribution. Final Answer: M = W L / 8.

Question 17

Materials engineering — identifying malleability In materials science and civil engineering, what is the property of a material that allows it to be beaten or rolled into very thin sheets or plates during manufacturing processes such as rolling, ham…

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Introduction / Context: Malleability is a key mechanical property used to select materials for sheet production, cladding, and forming operations in civil and structural engineering. Understanding how malleability differs from related terms such as ductility, plasticity, and elasticity helps engineers choose suitable materials for processes like rolling, stamping, and forming thin plates for roofing, flashing, and façade panels. Given Data / Assumptions: The question asks for the name of the property that enables metals to be beaten or rolled into thin sheets. We compare it to other common properties: ductility, plasticity, and elasticity. Concept / Approach: Malleability is the ability of a material to undergo extensive plastic deformation under compressive loading (hammering, rolling, pressing) without cracking, which directly enables forming into thin plates. Ductility is the capacity to deform plastically under tensile loading, producing wires or elongated shapes. Elasticity is the tendency to return to original shape after removal of load. Plasticity is the broader ability to undergo permanent deformation without fracture, of which malleability and ductility are specific manifestations under compression and tension respectively. Step-by-Step Solution: Identify the loading mode implicit in the question: beating or rolling implies predominantly compressive deformation.Match definitions: property enabling large compressive plastic deformation without fracture is malleability.Conclude that the correct term corresponding to forming thin sheets is malleability. Verification / Alternative check: Common examples of malleable metals include gold, silver, aluminum, and copper. These can be rolled into foils or sheets used in building envelopes, flashing, and ducting. In contrast, very ductile metals (like copper) can also be drawn into wire, illustrating the distinction between the two properties by loading mode. Why Other Options Are Wrong: Elasticity: pertains to reversible deformation only; does not describe forming thin plates. Plasticity: a general term for permanent deformation; not specific to sheet-forming under compression. Ductility: deformation under tension to produce wires, not sheets. None of these: incorrect because malleability exactly matches the description. Common Pitfalls: Confusing ductility with malleability; both involve plastic deformation but under different dominant stress states (tension vs. compression). Another mistake is assuming elasticity helps in forming; elasticity actually resists permanent shape change. Final Answer: Malleability

Question 18

Three-hinged parabolic arch — horizontal thrust for partial uniformly distributed load A three-hinged parabolic arch (hinged at both springings and the crown) has a horizontal span L = 4.8 m and a central rise f = 1 m. It carries a uniformly distrib…

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Introduction / Context: Three-hinged arches are statically determinate. For a parabolic arch, the bending moment at the crown hinge must be zero. This condition allows direct calculation of the horizontal thrust H by comparing the bending moment at midspan for the equivalent simply supported beam with the arch rise at the crown. This is especially handy for unsymmetrical loading (such as a uniform load over half the span). Given Data / Assumptions: Span L = 4.8 m; rise at crown f = 1 m. UDL w = 0.75 t/m applied over left half: 0 ≤ x ≤ L/2 = 2.4 m. Arch is three-hinged (hinges at supports and crown), parabolic in profile. Concept / Approach: At the crown hinge of a three-hinged arch, internal bending moment is zero. Therefore: H * y_crown = M_beam at crown (with the same loading on a simply supported beam). Thus, H = M_beam(crown) / f. Compute the simply supported beam's reaction forces for the given partial UDL, then find M at midspan, and divide by f to obtain H. Step-by-Step Solution: Total load W = w * (L/2) = 0.75 * 2.4 = 1.8 t acting at 1.2 m from the left support.Simply supported reactions: R_A = W * (L − a)/L = 1.8 * (4.8 − 1.2)/4.8 = 1.35 t, R_B = W − R_A = 0.45 t.Beam bending moment at midspan x = 2.4 m (within loaded region): M(mid) = R_A * x − w * x^2 / 2 = 1.35 * 2.4 − 0.75 * (2.4)^2 / 2 = 3.24 − 2.16 = 1.08 t·m.Rise at crown f = 1 m, so horizontal thrust H = M(mid)/f = 1.08 / 1 = 1.08 t. Verification / Alternative check: Because only half the span is loaded, there is a nonzero vertical shear at midspan in the beam analogy, but the crown-hinge moment condition remains valid. Units check: t·m divided by m gives t, appropriate for thrust. Why Other Options Are Wrong: 1.8 t and 0.8 t: incorrect magnitudes relative to the computed beam moment and rise. 10.8 t: off by a factor of 10; likely from unit or decimal misplacement. None of these: unnecessary since 1.08 t is correct. Common Pitfalls: Forgetting to use the beam moment at the crown location, misplacing the resultant load of the partial UDL, or confusing rise f with total depth. Also, mixing tonne and kiloNewton units can cause errors if conversions are not tracked. Final Answer: 1.08 tonnes

Question 19

Eccentric loading — kern (core) of a rectangular column section For a rectangular column with cross-section b × h (about the centroidal axes), the core (kern) region within which a compressive load must act to avoid tensile stress anywhere on the se…

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Introduction / Context: The kern (or core) of a section is the locus of load application points that keep the stress over the entire section compressive (no tension) when the member is axially compressed with eccentricity. For masonry and unreinforced concrete columns, staying within the kern avoids cracking from tension, making core geometry crucial in design checks and detailing guidelines. Given Data / Assumptions: Cross-section is rectangular with breadth b and depth h. Material behaves linearly elastic; stress varies linearly across the section. Load is compressive with possible biaxial eccentricities. Concept / Approach: The kern is defined by the condition that the resultant stress does not change sign anywhere on the section. For a rectangular section, the no-tension condition leads to eccentricity limits of ±b/6 about the x-axis and ±h/6 about the y-axis. Combining these for biaxial loading yields a rhombus in the cross-sectional plane whose diagonals are b/3 and h/3, centered at the centroid. Any load resultant within this rhombus produces compressive stress over the entire section. Step-by-Step Solution: For 1D bending about the axis parallel to b: allowable eccentricity range e_x ∈ [−h/6, +h/6].For 1D bending about the axis parallel to h: allowable eccentricity range e_y ∈ [−b/6, +b/6].Combine biaxial limits: the kern in the (e_x, e_y) plane is a rhombus centered at the origin with diagonals lengths h/3 and b/3.Map this rhombus back onto the section offset space to visualize the allowable load application area. Verification / Alternative check: Using the linear stress distribution formula, sigma = P/A ± M_x/Z_x ± M_y/Z_y, set the edge stress to zero to recover the e ≤ b/6 and h/6 bounds, confirming the rhombus kern. Why Other Options Are Wrong: Rectangle b/2 × h/2 or square b/2: overly conservative and not the theoretical kern. Rhombus with side length h/2: dimensionally and conceptually incorrect. None: incorrect because the rhombus with diagonals b/3 and h/3 is standard. Common Pitfalls: Confusing the kern with the centroidal core used in reinforced sections; mixing up diagonals versus side lengths; forgetting that the kern shrinks as section shape changes (e.g., circular sections have a circular kern of radius r/4). Final Answer: Rhombus whose diagonals are b/3 and h/3

Question 20

Basic mechanics — name of stress produced when equal and opposite forces elongate a member When equal and opposite axial forces are applied to a prismatic bar causing it to elongate along its length, what is the stress called?

Accepted Answer

Introduction / Context: Correctly identifying stress types is foundational in strength of materials and structural design. Members may be subjected to tension, compression, shear, bending, or torsion. Each stress state produces characteristic deformations and governs different design checks such as yielding, buckling, or rupture. Here the bar increases in length, a hallmark of axial tension producing tensile stress. Given Data / Assumptions: Forces are equal, opposite, and collinear with the member’s axis. Cross-section and material are uniform; deformations are small. End constraints allow axial extension. Concept / Approach: Stress is internal force per unit area. Under axial pulls, normal stress acts over the cross-section with its line of action along the member’s axis. A positive axial strain (elongation) corresponds to tensile stress. The magnitude is sigma_tension = P / A for a uniform bar within elastic limits, where P is the axial force and A is the cross-sectional area. Step-by-Step Solution: Recognize the described deformation: elongation along the axis.Associate elongation under axial load with tensile stress.State the stress type explicitly: tensile stress. Verification / Alternative check: Hooke’s law in the elastic range gives epsilon = sigma / E; a positive strain (lengthening) means sigma is positive in tension. Visual checks in labs (extensometer readings) confirm elongation under tension. Why Other Options Are Wrong: Compressive stress causes shortening, not elongation. Shear stress acts parallel to the area, leading to shear deformations, not pure axial elongation. Transverse stress is nonstandard terminology here; bending introduces normal stress that varies across the depth. Bearing stress is localized contact pressure, not uniform axial tension. Common Pitfalls: Confusing sign conventions; assuming any normal stress under axial load is compressive. Remember: pulling creates tensile stress; pushing creates compressive stress. Final Answer: Tensile stress

Strength of Materials Questions