Q: Highway engineering — standard single-lane carriageway width What is the recommended width of carriageway for a single-lane roadway as per common Indian highway design practice?

Introduction / Context: Carriageway width standards ensure sufficient lateral clearance for vehicle envelopes, steering wander, and safety margins. For single-lane roads, the carriageway must comfortably accommodate one design vehicle with allowances for mirrors and minor weave while minimizing construction cost. Given Data / Assumptions: Design vehicle corresponds to standard Indian practice for rural single-lane roads. Shoulders are provided separately and are not included in the carriageway width. Normal traffic mix; not a special corridor for oversize vehicles. Concept / Approach: Indian Roads Congress (IRC) recommendations specify carriageway widths by the number of lanes. A single-lane road has one traffic lane; two-lane two-way roads typically provide 7.0 m (two lanes of 3.5 m each), while a single lane is roughly half that plus a small allowance, commonly standardized as 3.75 m. Shoulders provide additional recovery space and passing opportunities at designated bays. Step-by-Step Solution: Recall typical lane widths: multilane roads commonly adopt 3.5–3.75 m per lane.For a single-lane carriageway, use the standard single-lane width: 3.75 m.Select option 3.75 m. Verification / Alternative check: IRC geometric design guidelines list 3.75 m as the carriageway width for single-lane roads, with shoulders specified separately depending on terrain and classification. Why Other Options Are Wrong: 7.5 m and 7.0 m: Correspond to two-lane roads. 5.5 m: Often associated with intermediate-width two-lane roads, not single-lane. 4.75 m: Wider than standard for single-lane and not typical in practice. Common Pitfalls: Confusing carriageway width with roadway (carriageway + shoulders). Assuming urban curb-to-curb values apply to rural standards. Final Answer: 3.75 m

Q: Sewer design terminology — what is an inverted siphon? Which of the following best describes an inverted siphon in wastewater conveyance systems?

Introduction / Context: Sewers are ordinarily designed to flow by gravity following the hydraulic grade line (HGL). Where natural or man-made obstacles exist (rivers, utilities, road/rail embankments), the sewer may need to dip below the HGL and then rise again, forming what is commonly called an inverted siphon or depressed sewer. Given Data / Assumptions: Gravity conveyance is preferred; pumps are avoided if possible. Continuous flow must be maintained without clogging. Air release/venting and flushing velocities are considered in design. Concept / Approach: An inverted siphon is not a true siphon; it operates under pressure flow because the sewer crown is below the HGL. It usually consists of one or more parallel barrels, designed to maintain scouring velocities over a wide flow range. Entry and exit structures allow flow splitting and recombination, and air valves or vents prevent air locking. Step-by-Step Solution: Identify the need: crossing under an obstacle while maintaining conveyance.Describe configuration: depressed sewer segment below HGL with suitable inlet/outlet structures.Confirm that normal sewer ventilation and maintenance concerns are addressed by design features. Verification / Alternative check: Standard wastewater engineering texts depict inverted siphons as depressed pressure-flow sections used to pass beneath obstructions without pumps. Why Other Options Are Wrong: Septic distribution devices and overflow preventers relate to on-site systems and water tanks, not sewers. Crown corrosion mitigation involves ventilation and material choices, not a siphon. Vacuum pumps are different equipment altogether. Common Pitfalls: Assuming it functions by siphonic action (it does not; it is pressure flow). Undersizing barrels, leading to sediment deposition and maintenance issues. Final Answer: A section of sewer dropped below the hydraulic grade line to pass under an obstacle

Q: Superposition of 2D flows — resultant streamline family from stream functions Two planar, incompressible flow patterns have stream functions ψ1 = x^2 + y^2 and ψ2 = 2xy. If these patterns are superposed, what is the family of streamlines represented…

Introduction / Context: Stream functions add linearly for incompressible, planar flows, allowing construction of complex patterns by superposing simpler ones. Interpreting the resulting streamline equation helps visualize the flow field and classify it (e.g., uniform flow, source/sink, vortex, shear, stagnation flow). Given Data / Assumptions: ψ1 = x^2 + y^2; ψ2 = 2xy. Superposition applies: ψ = ψ1 + ψ2. Cartesian coordinates; streamlines are given by ψ = constant. Concept / Approach: Compute the combined stream function: ψ = x^2 + y^2 + 2xy = (x + y)^2. A streamline is defined by ψ = C, so (x + y)^2 = C. Hence x + y = ±√C, which represents a family of straight lines with slope −1 and different intercepts. Therefore, the combined pattern consists of parallel straight streamlines. Step-by-Step Solution: Form ψ = ψ1 + ψ2 = (x + y)^2.Set ψ = C → (x + y)^2 = C.Take square root: x + y = constant (±√C) → parallel straight lines. Verification / Alternative check: Velocity components from ψ are u = ∂ψ/∂y = 2(x + y) and v = −∂ψ/∂x = −2(x + y). Thus v = −u, confirming streamlines of the form x + y = constant. Why Other Options Are Wrong: Circles, parabolas, hyperbolas, ellipses do not satisfy (x + y)^2 = constant; they arise from different polynomial relations. Common Pitfalls: Confusing stream function with velocity potential; only the former gives streamlines ψ = constant. Dropping the cross term and misclassifying the pattern. Final Answer: A family of parallel straight lines

Q: Dimensional analysis of flexural rigidity (E * I) for a beam element Express the dimensions of flexural rigidity EI using mass (M), length (L), and time (T) base dimensions.

Introduction: Flexural rigidity EI appears in beam deflection and vibration formulas. Dimensional analysis is a quick way to verify equations and to understand how physical quantities scale with geometry and material properties. This exercise asks for the base dimensions of EI in terms of M, L, and T. Given Data / Assumptions: E (Young’s modulus) has the dimensions of stress. I (second moment of area) reflects cross-sectional geometry. Standard SI base dimensions are used: force = M L T^-2. Concept / Approach: Young’s modulus E is stress/strain. Strain is dimensionless, so dimensions of E equal those of stress. Stress = force/area = (M L T^-2) / L^2 = M L^-1 T^-2. The second moment of area I has dimensions of L^4. Multiplying gives EI = (M L^-1 T^-2) * (L^4) = M L^3 T^-2. Step-by-Step Solution: 1) E → M L^-1 T^-2.2) I → L^4.3) EI → (M L^-1 T^-2) * (L^4) = M L^3 T^-2. Verification / Alternative check: Check with common beam equation y'' = w / (E I): w (load per length) has dimensions of force/length = M T^-2; y'' has L^-3; therefore EI must be M L^3 T^-2 to balance dimensions. Why Other Options Are Wrong: Options A and C omit the L^3 factor; D is inverted and dimensionally inconsistent; E adds an incorrect exponent on L and omits M. Common Pitfalls: Confusing area moment of inertia (L^4) with mass moment (M L^2); mixing force and mass units; forgetting strain is dimensionless. Final Answer: ML^3T^-2

Question 1

National Highways in India — indicative total length range (textbook-era data) Choose the range that best represents the traditionally quoted total length (km) of the National Highways network in India, as used in standard exam references.

Accepted Answer

Introduction / Context: Civil engineering competitive exams often use conventional, textbook-era statistics for transport infrastructure. This item asks for the broadly cited range of the total length of National Highways (NH) in India used in such references, not necessarily the current value (which changes over time due to upgrades and reclassification). Given Data / Assumptions: Question targets historically quoted range from standard exam manuals. Exact year is not specified; we rely on the traditional range commonly taught. Units are kilometres (km). Concept / Approach: In many legacy texts and older question banks, the NH network length is typically placed in the band of a few tens of thousands of kilometres, with the most common range cited as 45,000–55,000 km. This reflects the status before subsequent expansions. The task is recognition/recall rather than a calculation. Step-by-Step Solution: 1) Consider the set of provided ranges.2) Identify the historically prevalent figure (about fifty thousand kilometres).3) Choose the matching option 45,000 to 55,000 km. Verification / Alternative check: Cross-check with traditional exam prep books which tend to quote approximately fifty thousand kilometres for NH in legacy data tables. Why Other Options Are Wrong: Lower ranges (15–25k, 25–35k, 35–45k) understate the legacy figures; the NH system has long exceeded these bands. Common Pitfalls: Confusing present-day network lengths with the historical numbers used in older exam questions; not noting that actual figures evolve over time. Final Answer: 45,000 to 55,000

Question 2

Equivalent (effective) permeability for 2D flow through anisotropic soil For a soil with permeabilities kx and ky in x- and y-directions, determine the effective permeability k_eq applicable to two-dimensional flow analyses.

Accepted Answer

Introduction / Context: Natural soils often exhibit directional permeability due to stratification or fabric, leading to anisotropic hydraulic conductivities. For two-dimensional seepage, an equivalent isotropic permeability is used after a coordinate transformation, preserving Laplace’s equation form. Given Data / Assumptions: Anisotropic soil with conductivities kx and ky along orthogonal axes. Steady, incompressible, 2D flow conditions. Goal: find an equivalent permeability that makes transformed flow nets valid. Concept / Approach: By stretching coordinates x and y with factors proportional to sqrt(kx) and sqrt(ky), the anisotropic Laplace equation can be recast into an isotropic form. The effective permeability corresponding to this transformation equals the geometric mean of principal conductivities. k_eq = (kx * ky)^(1/2) Step-by-Step Solution: 1) Recognize anisotropy with principal values kx, ky.2) Apply coordinate transform: x' = x * (kx)^(1/2), y' = y * (ky)^(1/2).3) Under transform, governing equation becomes isotropic with k_eq = (kx ky)^(1/2).4) Use k_eq in discharge/head-loss computations for 2D flow. Verification / Alternative check: When kx = ky, the geometric mean reduces to kx, confirming consistency. For layered media, this result aligns with classical derivations of transformed flow nets. Why Other Options Are Wrong: Arithmetic sum or RMS do not arise from the governing equation; kx/ky is dimensionally inconsistent for permeability. Common Pitfalls: Confusing geometric mean (2D flow) with arithmetic/harmonic means used for series/parallel 1D layered systems; mixing axes after rotation. Final Answer: (kx ky)^(1/2)

Question 3

Instantaneous Unit Hydrograph (IUH) — relationship with the S-curve For a catchment, which statement correctly describes the ordinate of the IUH at any time t?

Accepted Answer

Introduction / Context: The Instantaneous Unit Hydrograph (IUH) idealizes the direct runoff response to an infinitesimally short burst (unit impulse) of effective rainfall. It is central to linear systems analysis in hydrology, linking rainfall excess to runoff via convolution. Given Data / Assumptions: Linear time-invariant watershed response. Unit hydrographs and S-curves are defined for unit depth of effective rainfall. Effective rainfall intensity considered: 1 cm/hr for developing the S-curve. Concept / Approach: The S-curve (or S-hydrograph) is formed by continuously applying a unit hydrograph (of given duration) such that 1 cm of excess rainfall is supplied each hour. The IUH is the time derivative of the S-curve corresponding to a unit-depth impulse. Therefore, the IUH ordinate at time t equals the slope (time derivative) of the S-curve generated by 1 cm/hr effective rainfall input. Step-by-Step Solution: 1) Construct the S-curve by superposing unit hydrographs at 1-hour intervals for a 1 cm/hr effective rainfall.2) Recognize that differentiating this cumulative response yields the impulse response (IUH).3) Conclude: IUH(t) = d/dt [S-curve(t)] for the 1 cm/hr input.4) Select the statement that matches this definition. Verification / Alternative check: In linear systems terms, the S-curve is the step response; IUH is the impulse response; derivative of step equals impulse, validating the relationship. Why Other Options Are Wrong: (a) and (b) refer to slopes of finite-duration unit hydrographs, not the cumulative S-curve; (c) has no standard meaning in UH theory. Common Pitfalls: Confusing IUH with a short-duration unit hydrograph; mixing cumulative (S) and incremental (UH/IUH) concepts. Final Answer: The slope of the S-curve with effective rainfall intensity of 1 cm/hr

Question 4

Why COD exceeds BOD — interpreting oxygen demand tests Chemical Oxygen Demand (COD) of a water/wastewater sample is always greater than Biochemical Oxygen Demand (BOD) because COD represents:

Accepted Answer

Introduction / Context: Oxygen demand tests quantify the oxidizable load in water/wastewater. Biochemical Oxygen Demand (BOD5) measures the oxygen consumed by microorganisms to oxidize biodegradable organics over 5 days, whereas Chemical Oxygen Demand (COD) uses a strong chemical oxidant to oxidize a broader suite of compounds. Given Data / Assumptions: BOD5 reflects biologically oxidizable fraction under specific test conditions (20°C, 5 days). COD reflects chemically oxidizable matter using dichromate (or similar) in acidic conditions. Some organics are refractory/slowly biodegradable. Concept / Approach: COD ≥ BOD because COD includes both readily biodegradable and non-biodegradable (or slowly biodegradable within 5 days) organic compounds that the microbial test does not fully oxidize. Inorganic reducing species (e.g., sulfides, ferrous iron) can also contribute to COD, further widening the gap. Step-by-Step Solution: 1) Define BOD5: oxygen used by microbes to oxidize biodegradable organics over 5 days.2) Define COD: oxygen equivalent of chemical oxidant required to oxidize the total oxidizable content.3) Recognize the coverage difference: COD captures both biodegradable and non-biodegradable organics.4) Conclude: COD values are typically greater than BOD5 for the same sample. Verification / Alternative check: Typical municipal wastewater shows COD/BOD ratios between ~1.5 and 2.5, reflecting the presence of slowly degradable and non-biodegradable organics. Why Other Options Are Wrong: (a) describes BOD, not COD; (c) and (d) provide only partial or incorrect coverage of what COD measures. Common Pitfalls: Assuming COD equals total organic carbon (TOC); ignoring the influence of inorganic reducers on COD; misusing BOD for rapid process control where COD is preferable. Final Answer: biodegradable and non-biodegradable organic matter

Question 5

Structural analysis — rotational stiffness coefficient K11 at a joint with two members For a rigid-jointed frame joint “1” connected to two prismatic members, each having flexural rigidity EI and length l, with their far ends effectively restrained …

Accepted Answer

Introduction / Context: In the stiffness (displacement) method for rigid frames, the joint rotational stiffness coefficient K11 represents the total resisting moment developed at a joint per unit rotation at that joint, with all other joint rotations and chord rotations restrained. Correctly assembling K11 is fundamental to forming the global stiffness matrix and predicting member-end moments and reactions accurately. Given Data / Assumptions: Two prismatic members meet at joint 1; each has flexural rigidity EI and length l. Their far ends are effectively restrained for the unit-rotation case at joint 1 (a common modeling assumption when forming fixed-end stiffness contributions at the near end). Small deformations; linear elastic behavior; plane frame bending about the relevant axis. Concept / Approach: For a prismatic member with the near end at the joint and the far end fixed, the end rotational stiffness at the near end is 4EI/l for the near-end rotation and a carry-over of 2EI/l to the far end. However, when forming the joint rotational coefficient at a single joint with the far-end “locked” against rotation induced by the unit rotation at the joint, the effective contribution commonly used in joint-sum assembly is 3EI/l per member for the joint in question (equivalently, 4EI/l at the near end minus the carry-over 1EI/l returned by restraining the far end). With two such members, K11 is the sum of their contributions. Step-by-Step Solution: For each member at the joint: k_member ≈ 3EI/l (near-end contribution with the far end effectively restrained for the unit-rotation case).Total joint coefficient: K11 = Σ k_member = 3EI/l + 3EI/l = 6EI/l.Match with options → 6 EI/l. Verification / Alternative check: A quick matrix-based derivation using member-end stiffness with the far end rotation suppressed yields the same “3EI/l per member at the joint” contribution; summing over two members gives 6EI/l. Why Other Options Are Wrong: 5 EI/l, 7 EI/l, 8 EI/l, 4 EI/l: Do not equal the required sum of two 3EI/l contributions for the stated modeling condition. Common Pitfalls: Confusing the near-end stiffness 4EI/l with the joint-sum contribution when the far end is restrained. Forgetting to sum contributions from all members framing into the joint. Final Answer: 6 EI/l

Question 6

Slope stability (cohesive slope) — factor of safety using Taylor stability number A deep cut of 7 m is to be excavated in a purely cohesive clay with unit weight γ = 16 kN/m^3 and cohesion c = 25 kN/m^2. The side slope is 30°. For a depth factor of …

Accepted Answer

Introduction / Context: Taylor’s stability charts provide a quick way to estimate the safety of homogeneous slopes by relating a non-dimensional stability number to slope geometry and soil strength. For purely cohesive soils (φ ≈ 0), the factor of safety (F) can be computed directly from the stability number when the slope height and unit weight are known. Given Data / Assumptions: Slope height H = 7 m; slope angle = 30°. Soil properties: c = 25 kN/m^2, γ = 16 kN/m^3, φ = 0 (cohesive). Taylor’s chart gives Sn = 0.178 for the specified depth factor and slope angle. Undrained (total stress) analysis appropriate for short-term stability. Concept / Approach: The stability number is defined for cohesive slopes as Sn = c / (γ * H * F). Rearranging gives F = c / (Sn * γ * H). Using the chart value of Sn ensures the geometric effect of the slope angle and depth factor is included. Step-by-Step Solution: Compute denominator: Sn * γ * H = 0.178 * 16 * 7 = 0.178 * 112 = 19.936.Factor of safety: F = c / (Sn * γ * H) = 25 / 19.936 ≈ 1.254.Rounded to two decimal places, F ≈ 1.25. Verification / Alternative check: F is modestly above unity, which is reasonable for a 30° slope in a low-to-moderate strength clay of the stated height. Why Other Options Are Wrong: 0.80 and 1.0: Too low; would imply near or actual failure with the given parameters. 1.1: Underestimates the computed F. 1.40: Overestimates safety for the stated Sn. Common Pitfalls: Using γ_submerged instead of γ when not warranted by groundwater conditions. Mistyping Sn from the chart or mixing up units for c and γ. Final Answer: 1.25

Question 7

Wastewater BOD calculation — dilution method at 20°C for 5 days (BOD5) In a BOD test, 5 mL of wastewater is diluted with 295 mL of aerated dilution water (total volume 300 mL). The diluted sample has initial dissolved oxygen (DO) = 7.8 mg/L and, aft…

Accepted Answer

Introduction / Context: The biochemical oxygen demand over 5 days at 20°C (BOD5) is a standard measure of biodegradable organic pollution. When the sample is diluted, the observed oxygen depletion in the diluted bottle must be scaled by the dilution factor to recover the BOD of the original wastewater. Given Data / Assumptions: Sample volume V_s = 5 mL; dilution water volume = 295 mL; total volume V_t = 300 mL. Initial DO = 7.8 mg/L; final DO after 5 days = 4.4 mg/L. Seeding/blank corrections are negligible (not stated). Concept / Approach: Oxygen depletion in the diluted bottle is ΔDO_diluted = DO_initial − DO_final. The dilution factor is DF = V_t / V_s. Then the BOD of the undiluted wastewater is BOD5 = ΔDO_diluted * DF. Step-by-Step Solution: Compute ΔDO_diluted = 7.8 − 4.4 = 3.4 mg/L.Compute DF = 300 / 5 = 60.BOD5 = 3.4 * 60 = 204 mg/L. Verification / Alternative check: The result lies in a typical range for municipal wastewater; if a seed correction existed, it would be subtracted before applying the dilution factor, but none is provided here. Why Other Options Are Wrong: 196, 200, 208 mg/L: Close distractors but not equal to the exact product 3.4 * 60 = 204 mg/L. 180 mg/L: Larger error; does not match the data. Common Pitfalls: Using the ratio V_s / V_t instead of V_t / V_s when scaling. Forgetting to apply seed/blank corrections when they are provided. Final Answer: 204 mg/L

Question 8

Plastic analysis — which cross-section has the largest shape factor? Among the following cross-section shapes subjected to bending about their centroidal axis, which one typically exhibits the largest shape factor (plastic section modulus / elastic …

Accepted Answer

Introduction / Context: The shape factor, defined as Zp/Z (plastic section modulus divided by elastic section modulus), measures a section’s reserve strength beyond first yield in bending. A larger shape factor indicates greater plastic redistribution capacity before full plastic collapse, which is central to plastic design of beams and frames. Given Data / Assumptions: Bending is about a centroidal principal axis. Material is ideally elastic–perfectly plastic. Sections considered are common shapes used in practice. Concept / Approach: The shape factor depends on how the area is distributed with respect to the neutral axis. Shapes concentrating more area away from the neutral axis tend to have higher elastic modulus but may not maximize plastic capacity. Among standard solid shapes, typical shape factors are approximately: rectangular ≈ 1.5, solid circular ≈ 1.7, I-sections ≈ 1.1–1.2 (depending on proportions). A square turned through 45° (diamond) develops a higher plastic-to-elastic ratio, yielding one of the largest shape factors among simple solid shapes, commonly taken as about 2.0 for idealized plastic analysis. Step-by-Step Solution: Compare known shape factors: I (≈1.1–1.2) < rectangular (≈1.5) < circular (≈1.7) < diamond (≈2.0).Identify the maximum among listed shapes → diamond.Select “Diamond (square turned through 45°)”. Verification / Alternative check: Plastic design handbooks enumerate these typical values; the diamond’s shape factor exceeds that of rectangle and circle for bending about a symmetry axis. Why Other Options Are Wrong: Rectangular and solid circular: Lower shape factors (≈1.5, ≈1.7). I-section: Typically around 1.1–1.2 due to flange–web distribution. Thin-walled tube: Depending on wall thickness may be competitive, but for common comparisons the diamond’s idealized factor is larger. Common Pitfalls: Confusing section efficiency in elastic design (high Z) with plastic reserve (high Zp/Z). Comparing about a non-principal axis, which changes numerical values. Final Answer: Diamond (square turned through 45°)

Question 9

Highway engineering — standard single-lane carriageway width What is the recommended width of carriageway for a single-lane roadway as per common Indian highway design practice?

Accepted Answer

Introduction / Context: Carriageway width standards ensure sufficient lateral clearance for vehicle envelopes, steering wander, and safety margins. For single-lane roads, the carriageway must comfortably accommodate one design vehicle with allowances for mirrors and minor weave while minimizing construction cost. Given Data / Assumptions: Design vehicle corresponds to standard Indian practice for rural single-lane roads. Shoulders are provided separately and are not included in the carriageway width. Normal traffic mix; not a special corridor for oversize vehicles. Concept / Approach: Indian Roads Congress (IRC) recommendations specify carriageway widths by the number of lanes. A single-lane road has one traffic lane; two-lane two-way roads typically provide 7.0 m (two lanes of 3.5 m each), while a single lane is roughly half that plus a small allowance, commonly standardized as 3.75 m. Shoulders provide additional recovery space and passing opportunities at designated bays. Step-by-Step Solution: Recall typical lane widths: multilane roads commonly adopt 3.5–3.75 m per lane.For a single-lane carriageway, use the standard single-lane width: 3.75 m.Select option 3.75 m. Verification / Alternative check: IRC geometric design guidelines list 3.75 m as the carriageway width for single-lane roads, with shoulders specified separately depending on terrain and classification. Why Other Options Are Wrong: 7.5 m and 7.0 m: Correspond to two-lane roads. 5.5 m: Often associated with intermediate-width two-lane roads, not single-lane. 4.75 m: Wider than standard for single-lane and not typical in practice. Common Pitfalls: Confusing carriageway width with roadway (carriageway + shoulders). Assuming urban curb-to-curb values apply to rural standards. Final Answer: 3.75 m

Question 10

Sewer design terminology — what is an inverted siphon? Which of the following best describes an inverted siphon in wastewater conveyance systems?

Accepted Answer

Introduction / Context: Sewers are ordinarily designed to flow by gravity following the hydraulic grade line (HGL). Where natural or man-made obstacles exist (rivers, utilities, road/rail embankments), the sewer may need to dip below the HGL and then rise again, forming what is commonly called an inverted siphon or depressed sewer. Given Data / Assumptions: Gravity conveyance is preferred; pumps are avoided if possible. Continuous flow must be maintained without clogging. Air release/venting and flushing velocities are considered in design. Concept / Approach: An inverted siphon is not a true siphon; it operates under pressure flow because the sewer crown is below the HGL. It usually consists of one or more parallel barrels, designed to maintain scouring velocities over a wide flow range. Entry and exit structures allow flow splitting and recombination, and air valves or vents prevent air locking. Step-by-Step Solution: Identify the need: crossing under an obstacle while maintaining conveyance.Describe configuration: depressed sewer segment below HGL with suitable inlet/outlet structures.Confirm that normal sewer ventilation and maintenance concerns are addressed by design features. Verification / Alternative check: Standard wastewater engineering texts depict inverted siphons as depressed pressure-flow sections used to pass beneath obstructions without pumps. Why Other Options Are Wrong: Septic distribution devices and overflow preventers relate to on-site systems and water tanks, not sewers. Crown corrosion mitigation involves ventilation and material choices, not a siphon. Vacuum pumps are different equipment altogether. Common Pitfalls: Assuming it functions by siphonic action (it does not; it is pressure flow). Undersizing barrels, leading to sediment deposition and maintenance issues. Final Answer: A section of sewer dropped below the hydraulic grade line to pass under an obstacle

Question 11

Superposition of 2D flows — resultant streamline family from stream functions Two planar, incompressible flow patterns have stream functions ψ1 = x^2 + y^2 and ψ2 = 2xy. If these patterns are superposed, what is the family of streamlines represented…

Accepted Answer

Introduction / Context: Stream functions add linearly for incompressible, planar flows, allowing construction of complex patterns by superposing simpler ones. Interpreting the resulting streamline equation helps visualize the flow field and classify it (e.g., uniform flow, source/sink, vortex, shear, stagnation flow). Given Data / Assumptions: ψ1 = x^2 + y^2; ψ2 = 2xy. Superposition applies: ψ = ψ1 + ψ2. Cartesian coordinates; streamlines are given by ψ = constant. Concept / Approach: Compute the combined stream function: ψ = x^2 + y^2 + 2xy = (x + y)^2. A streamline is defined by ψ = C, so (x + y)^2 = C. Hence x + y = ±√C, which represents a family of straight lines with slope −1 and different intercepts. Therefore, the combined pattern consists of parallel straight streamlines. Step-by-Step Solution: Form ψ = ψ1 + ψ2 = (x + y)^2.Set ψ = C → (x + y)^2 = C.Take square root: x + y = constant (±√C) → parallel straight lines. Verification / Alternative check: Velocity components from ψ are u = ∂ψ/∂y = 2(x + y) and v = −∂ψ/∂x = −2(x + y). Thus v = −u, confirming streamlines of the form x + y = constant. Why Other Options Are Wrong: Circles, parabolas, hyperbolas, ellipses do not satisfy (x + y)^2 = constant; they arise from different polynomial relations. Common Pitfalls: Confusing stream function with velocity potential; only the former gives streamlines ψ = constant. Dropping the cross term and misclassifying the pattern. Final Answer: A family of parallel straight lines

Question 12

Allowable bearing capacity of shallow foundations In geotechnical engineering practice, the permissible (allowable) bearing capacity of a foundation is governed by two independent criteria. Identify the correct pair of criteria that must both be sat…

Accepted Answer

Introduction: Designing the allowable bearing capacity of a shallow foundation requires attention to both ultimate limit states and serviceability limit states. The soil–foundation system must not reach a shear (bearing) failure under factored or working loads, and at the same time, the settlement under service loads must remain within tolerable limits to protect the superstructure from damage and functional impairment. Given Data / Assumptions: Shallow foundation on natural soil or engineered fill. Working (service) load cases considered for settlement; ultimate resistance considered for shear/bearing. Allowable bearing pressure is derived by applying factors of safety and/or serviceability checks. Concept / Approach: The classic bearing capacity problem addresses two distinct checks. First, shear failure (general or local) must not occur in the soil beneath the foundation. Second, settlements (both total and differential) must be within permissible limits for the supported structure. The allowable bearing capacity is often the lesser pressure arising from these two checks: one from dividing ultimate capacity by a factor of safety, the other from settlement analysis and permissible limits (e.g., 25 mm to 75 mm depending on structure type). Step-by-Step Solution: 1) Compute ultimate bearing capacity using an appropriate theory (e.g., Terzaghi, Meyerhof) and soil parameters.2) Apply a factor of safety to obtain a shear-controlled allowable pressure.3) Independently estimate immediate and consolidation settlements for the service load and compare to permissible thresholds.4) Select the smaller of the shear-controlled and settlement-controlled allowable pressures as the design value. Verification / Alternative check: Back-analyses of case histories show structures that meet shear safety can still suffer distress from excessive settlement. Conversely, a very stiff soil may pass settlement criteria while shear governs at lower pressure. Hence both checks are necessary and complementary. Why Other Options Are Wrong: Options A and B include ‘‘tensile failure’’ which is not a conventional criterion for soil bearing capacity. Option C mentions ‘‘bond failure’’ which is not relevant to soil–foundation interface at footing levels. Option E pertains to hydraulic or uplift scenarios, not the general bearing capacity criteria. Common Pitfalls: Using only a factor of safety on ultimate capacity and ignoring settlement; neglecting differential settlement between supports; relying on undrained parameters for long-term settlement predictions where drained behavior controls. Final Answer: shear failure and settlement

Question 13

Anchorage (development) length of a straight reinforcing bar in compression For a straight bar in compression with nominal bar diameter φ, compressive stress in steel fs, and design bond stress of concrete fbd, what is the required anchorage length …

Accepted Answer

Introduction: Anchorage or development length ensures that reinforcing steel can safely develop the required stress by bonding with surrounding concrete. In compression, bond transfer is generally more favorable than in tension, but the fundamental relationship between steel stress, bar size, and design bond stress remains at the core of the calculation. Given Data / Assumptions: Straight deformed bar embedded in concrete. Variables: φ (bar diameter), fs (design compressive stress in steel), fbd (design bond stress). Limit state design context; fbd already accounts for material and safety factors. Concept / Approach: The basic equilibrium is that the total bond force mobilized along the embedded perimeter over length La must equal the steel force to be developed. Idealizing uniform bond along the bar, this leads to a proportionality with bar circumference and development length. In common code formulations, this resolves to La = (φ * fs) / (4 * fbd) for straight bars (compression case uses the same form when fbd is the design value for compression). Step-by-Step Solution: 1) Steel force to develop: T_s = As * fs; for bar-based form, use diameter φ.2) Bond capacity over La: τ * perimeter * La = fbd * (π * φ) * La.3) Equate and simplify (per standard derivation): La = (φ * fs) / (4 * fbd). Verification / Alternative check: Codes may allow reduced La in compression by adopting higher fbd in compression than in tension; when fbd already represents the compressive bond design value, the expression above holds directly. Why Other Options Are Wrong: Options B, C, and E alter the denominator and would underestimate or overestimate anchorage. Option D ignores the 4 factor from derivation, giving an unconservative length. Common Pitfalls: Confusing characteristic bond stress with design bond stress; forgetting to adjust for hooks, confinement, or closely spaced bars; misapplying tension formula without considering compressive enhancement rules for fbd. Final Answer: La = (φ * fs) / (4 * fbd)

Question 14

Dimensional analysis of flexural rigidity (E * I) for a beam element Express the dimensions of flexural rigidity EI using mass (M), length (L), and time (T) base dimensions.

Accepted Answer

Introduction: Flexural rigidity EI appears in beam deflection and vibration formulas. Dimensional analysis is a quick way to verify equations and to understand how physical quantities scale with geometry and material properties. This exercise asks for the base dimensions of EI in terms of M, L, and T. Given Data / Assumptions: E (Young’s modulus) has the dimensions of stress. I (second moment of area) reflects cross-sectional geometry. Standard SI base dimensions are used: force = M L T^-2. Concept / Approach: Young’s modulus E is stress/strain. Strain is dimensionless, so dimensions of E equal those of stress. Stress = force/area = (M L T^-2) / L^2 = M L^-1 T^-2. The second moment of area I has dimensions of L^4. Multiplying gives EI = (M L^-1 T^-2) * (L^4) = M L^3 T^-2. Step-by-Step Solution: 1) E → M L^-1 T^-2.2) I → L^4.3) EI → (M L^-1 T^-2) * (L^4) = M L^3 T^-2. Verification / Alternative check: Check with common beam equation y'' = w / (E I): w (load per length) has dimensions of force/length = M T^-2; y'' has L^-3; therefore EI must be M L^3 T^-2 to balance dimensions. Why Other Options Are Wrong: Options A and C omit the L^3 factor; D is inverted and dimensionally inconsistent; E adds an incorrect exponent on L and omits M. Common Pitfalls: Confusing area moment of inertia (L^4) with mass moment (M L^2); mixing force and mass units; forgetting strain is dimensionless. Final Answer: ML^3T^-2

Question 15

Minimum pitch (spacing) of rivets in one row as per IS 800:1984 According to IS 800:1984 recommendations for riveted joints, the minimum pitch of rivets in a row should not be less than how many times the rivet diameter?

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Introduction: Proper rivet pitch ensures adequate load distribution, prevents plate tearing, and allows room for driving or forming heads. Codes prescribe minimum spacing to safeguard against joint weakening and excessive stress concentrations. Given Data / Assumptions: Riveted steel plate or built-up member following IS 800:1984. Pitch refers to center-to-center distance of adjacent rivets along a row. Diameter denotes nominal rivet shank diameter. Concept / Approach: Minimum pitch limits are set to avoid plate splitting and to maintain adequate net section. IS 800:1984 recommends a minimum pitch of 2.5 times the rivet diameter in a row, with higher values often required for practical considerations such as edge distances, stagger, and hole tolerances. Step-by-Step Solution: 1) Identify code clause governing minimum pitch.2) Apply the recommended minimum: p_min = 2.5 * d.3) Check additional detailing rules (edge distance, stagger, maximum pitch) during final design. Verification / Alternative check: Design handbooks reflecting IS 800:1984 and traditional riveted practice corroborate the 2.5 d minimum spacing for durability and constructability. Why Other Options Are Wrong: 2.0 d may be too close, risking tearing; 3.0 d or 4.0 d are acceptable but not the minimum; 1.5 d is clearly inadequate. Common Pitfalls: Ignoring edge distance limits; not accounting for hole oversize; using uniform pitch without checking net-section efficiency. Final Answer: 2.5

Question 16

Irrigation outlet discharge to meet evapotranspiration An outlet serves 20 hectares of cropped area. The evapotranspiration requirement is 20 mm, occurring uniformly over 20 days. Neglecting other field losses, determine the continuous discharge req…

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Introduction: Irrigation flow sizing often reduces to converting a depth of water over an area and time into a steady discharge. This problem illustrates the straightforward volumetric approach for evapotranspiration (ET) demands when losses are neglected and application is uniform over the design period. Given Data / Assumptions: Area A = 20 ha = 200,000 m^2. Total ET depth over period = 20 mm = 0.020 m. Duration T = 20 days; assume continuous supply. Losses like conveyance and field application are neglected. Concept / Approach: Compute required daily (or total) volume and divide by time to obtain discharge. Since ET occurs uniformly over 20 days, the daily depth is 1 mm (0.001 m). Multiply by area to get daily volume; convert to L/s using 1 m^3 = 1000 L and 1 day = 86400 s. Step-by-Step Solution: 1) Daily depth = 20 mm / 20 days = 1 mm = 0.001 m.2) Daily volume V = A * depth = 200,000 * 0.001 = 200 m^3/day.3) Convert to L/day: 200 m^3/day = 200,000 L/day.4) Discharge Q = 200,000 / 86,400 ≈ 2.3148 L/s → 2.31 L/s. Verification / Alternative check: Alternatively, compute total volume (A * 0.020 m = 4000 m^3) and divide by 20 days to obtain the same daily rate; conversion yields 2.31 L/s again. Why Other Options Are Wrong: 2.52 and 2.01 result from rounding or area mis-conversions; 1.52 assumes a smaller daily depth; 3.10 is too high for the given ET. Common Pitfalls: Forgetting that 1 ha = 10,000 m^2; confusing mm/day with cm/day; neglecting to convert days to seconds correctly when expressing L/s. Final Answer: 2.31

Question 17

Delta–Duty–Base period relation (consistent units) For irrigation planning with Delta Δ in cm, Duty D in hectare per cumec, and Base period B in days, select the correct relationship connecting these parameters.

Accepted Answer

Introduction: The Delta–Duty–Base period relation converts continuous discharge availability into seasonal depth over an area. Getting the unit conversions right is essential in irrigation scheduling and canal capacity design. Given Data / Assumptions: Δ in centimeters. D in hectare per cumec (ha per m^3/s). B in days. Concept / Approach: One cumec supplies 86,400 m^3 in a day. Spread over 1 hectare (10,000 m^2), this gives a depth of 8.64 m per day. Converting meters to centimeters multiplies by 100, giving 864 cm per day. Therefore, for B days and duty D (ha per cumec), the seasonal depth is Δ = (864 * B) / D. Step-by-Step Solution: 1) Depth from 1 cumec over 1 ha in 1 day: 86,400 / 10,000 = 8.64 m = 864 cm.2) For B days: depth = 864 * B cm.3) For D hectares per cumec: divide by D → Δ = 864 * B / D. Verification / Alternative check: If Δ is required in meters, the relation becomes Δ(m) = 8.64 * B / D, consistent with standard texts. Why Other Options Are Wrong: Option D uses the meter-based constant 8.64 with Δ in cm—unit mismatch. Other options rearrange incorrectly or multiply instead of divide by D. Common Pitfalls: Mixing hectare with square meter, forgetting to multiply by 100 when converting m to cm, or inverting duty inadvertently. Final Answer: Δ = 864 * B / D

Question 18

Common applications of well (caisson) foundations Well foundations are most commonly adopted as foundations for which of the following types of structures?

Accepted Answer

Introduction: Well (caisson) foundations are large, hollow substructures sunk to substantial depths, often under water. They are designed to transfer loads to competent strata while resisting scour and lateral forces. Knowing where they are typically used aids in selecting suitable foundation systems for major civil works. Given Data / Assumptions: Rivers or estuaries with deep alluvium. High vertical loads combined with horizontal forces and moments (e.g., from water flow or traffic). Need for construction below water level using pneumatic sinking or dredging methods. Concept / Approach: Bridges commonly require deep foundations for piers and abutments situated in waterways. Well foundations provide adequate bearing, settlement control, and resistance to scour and lateral loads. They can be constructed in segments and sunk to the design depth, then filled with concrete to form a robust substructure. Step-by-Step Solution: 1) Identify structural demands of bridge piers in rivers: heavy axial loads and lateral/environmental forces.2) Match foundation type with environment: wells perform well in deep, loose alluvium under water.3) Select ‘‘Bridges’’ as the most common application. Verification / Alternative check: Case histories of major river bridges (e.g., caisson-founded piers) confirm widespread use of well foundations for such structures. Why Other Options Are Wrong: Water tanks and buildings often use raft or pile foundations; reciprocating machines prefer block foundations with vibration isolation; rock sites for wind turbines typically use direct anchorage or piles, not wells. Common Pitfalls: Confusing well foundations with open caissons for dry land structures; underestimating scour depth in design. Final Answer: Bridges

Question 19

Bearing capacity factor Nc for piles in clay (Meyerhof) According to Meyerhof's recommendations for deep foundations in cohesive soils, what value is taken for the cohesion bearing capacity factor Nc for piles?

Accepted Answer

Introduction: End-bearing capacity of piles in clay is commonly expressed using a cohesion factor Nc multiplying the undrained shear strength. Different theories propose different Nc values for shallow versus deep foundations. Recognizing the appropriate factor for piles prevents over- or underestimation of capacity. Given Data / Assumptions: Deep foundation behavior (pile base in undrained clay). Short-term undrained condition controls end bearing. Meyerhof's empirical recommendations for piles. Concept / Approach: For shallow foundations under general shear, Terzaghi's Nc is 5.14. For deep foundations (piles) in undrained clay, higher confinement and failure mechanisms justify a larger Nc. Meyerhof recommended Nc ≈ 9 for pile toe capacity in clay, reflecting deep foundation behavior. Step-by-Step Solution: 1) Identify the deep foundation scenario (pile end bearing in clay).2) Use Meyerhof's pile bearing expression: qb = Nc * su (with su = undrained strength).3) Take Nc = 9.0 for design per Meyerhof's guidance. Verification / Alternative check: Comparisons with field load tests show that Nc = 9 gives reasonable predictions for pile base resistance, whereas Nc = 5.14 would be conservative for deep foundations. Why Other Options Are Wrong: 5.14 is for shallow foundations; 6.2 and 7.5 are not standard for piles; 12.0 is overly high for typical practice. Common Pitfalls: Applying shallow-foundation factors to piles; mixing drained and undrained parameters; ignoring setup or remolding effects near the pile base. Final Answer: 9.0

Question 20

Plate load test – settlement criterion for modulus of subgrade reaction When deriving the modulus of subgrade reaction (k) from a plate bearing test for pavement design, the standard settlement used to compute k is:

Accepted Answer

Introduction: The modulus of subgrade reaction k is the proportionality between bearing pressure and surface deflection used in slab-on-grade and rigid pavement analysis. Standardizing the deflection at which k is determined allows consistent comparison across sites and tests. Given Data / Assumptions: Static plate load test performed on compacted subgrade. Pressure–settlement curve recorded for a standard plate. Rigid pavement design context (e.g., Westergaard). Concept / Approach: In pavement engineering practice, k is usually taken as pressure divided by settlement at a small reference deflection. A commonly adopted standard is 1.25 mm (0.05 in). This value provides a representative stiffness near the origin of the load–settlement curve and avoids nonlinearities at larger settlements. Step-by-Step Solution: 1) Conduct plate load test and plot pressure–settlement response.2) Read pressure p at settlement s = 1.25 mm.3) Compute k = p / s with s in meters to keep units consistent (e.g., kN/m^3). Verification / Alternative check: Design guides and standards for rigid pavements reference k at 1.25 mm as the benchmark; correlations to CBR or resilient modulus also rely on this convention. Why Other Options Are Wrong: Using larger settlements (2.50 or 3.75 mm) shifts k to a lower tangent/slope region and is not the standard reference; 1.75 or 5.00 mm are non-standard for the k-value definition. Common Pitfalls: Not correcting for plate size; using total (plastic + elastic) settlement without considering rebound; mixing units when converting mm to m in k calculations. Final Answer: 1.25 mm

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