Q: Working-stress design in reinforced concrete: If the permissible compressive stress in bending for concrete is C (in kg/cm²), what is the appropriate modular ratio m to be used for transformed-section analysis?

Introduction / Context: The modular ratio m is a cornerstone of the working-stress method in reinforced concrete design. It converts steel area into an equivalent concrete area for transformed-section analysis under service loads. Classical codes and handbooks define m in terms of a constant divided by the permissible compressive stress in bending for concrete, reflecting creep and long-term effects. Given Data / Assumptions: Permissible compressive stress in bending for concrete = C (kg/cm²). Working-stress (service) design framework. Standard long-term modular ratio expression used in many exam problems. Concept / Approach: The long-term (effective) modular ratio is taken as m = 2800/(3C) when C is in kg/cm². The 2800 factor and the divisor 3 represent empirical adjustments for creep and sustained loading, leading to a reduced effective stiffness of concrete under service conditions compared to its instantaneous modulus. Step-by-Step Solution: Identify the required formula: m = 2800/(3 * C).Check units: C is in kg/cm², so 2800/(3C) is dimensionless, appropriate for a ratio of moduli.Select the matching option: 2800/3C. Verification / Alternative check: Some references list an instantaneous m ≈ Ec/Es, but for working-stress design, the long-term expression with 2800/(3C) is traditionally used for service calculations and matches the exam-style answer set. Why Other Options Are Wrong: 2800/C: Omits the divisor 3; too large for long-term effective m. 2300/2C and 700/C: Do not reflect the standard constant or divisor. 2800/C²: Incorrect dimensional form; m must be dimensionless, not vary with 1/C². Common Pitfalls: Confusing instantaneous modulus ratio with the long-term modular ratio used in WSM. Mixing units (MPa vs kg/cm²). Keep C in kg/cm² to use the 2800 constant correctly. Final Answer: 2800/3C

Q: A uniform prismatic pile (length L) is to be lifted at two points so that the maximum hogging moment at supports equals the maximum sagging moment at midspan. The optimal distance of each suspension point from either end is closest to:

Introduction / Context: Long members such as piles or pipelines are often lifted or transported using two-point suspension. To minimize maximum bending stress, the suspension points are positioned so that hogging (over supports) and sagging (at midspan) moments are equal in magnitude, giving the most economical arrangement. Given Data / Assumptions: Uniform self-weight w per unit length. Two identical suspension points located symmetrically at distance x from each end. Static, simply supported behavior between the lifting points. Concept / Approach: For a uniformly loaded beam with overhangs of length x and central span of L − 2x, bending moments include hogging at supports (negative) and sagging at the center span (positive). Setting these maxima equal yields the “economic” lifting distance. Step-by-Step Solution: 1) Let reactions be equal due to symmetry: R = wL/2.2) Max sagging in central span (simply supported) occurs at midspan: M_sag = R*(L/2 − x) − w*(L − 2x)^2/8.3) Max hogging at each support (due to overhang) occurs just inside the support: M_hog = Rx − wx^2/2.4) Set |M_hog| = |M_sag| and solve for x/L, which yields approximately 0.207. Verification / Alternative check: Published lifting tables for uniform beams and piles list the two-point suspension distance as about 0.207 L from each end for equal extreme moments, confirming the calculation. Why Other Options Are Wrong: 0.107 L / 0.307 L / 0.407 L / 0.500 L: These do not produce equality of peak hogging and sagging; they either over-stress midspan or the supports. Common Pitfalls: Ignoring overhang effects or assuming the best position is at quarter points (0.25 L), which is not optimal for equalizing extremes with uniform load and two-point lift. Final Answer: 0.207 L

Q: For a reinforced concrete square column with 16 mm diameter longitudinal bars, what is the minimum diameter required for the lateral ties as per common code limits (not less than one-quarter of the largest longitudinal bar and an absolute minimum)?

Introduction / Context: Lateral ties (transverse reinforcement) in reinforced concrete columns confine the core concrete and prevent buckling of longitudinal bars. Codes specify minimum diameters based on the size of longitudinal reinforcement and absolute minima for durability and constructability. Given Data / Assumptions: Square column; longitudinal bars are 16 mm diameter. Standard code-based rule: tie diameter ≥ 1/4 of the largest longitudinal bar diameter and also ≥ an absolute minimum value. Typical practice based on IS-style provisions. Concept / Approach: For 16 mm bars, one-quarter is 16/4 = 4 mm. However, most codes specify an absolute minimum (for example, 6 mm) to ensure adequate stiffness and robustness of ties in handling and placement. Therefore, the controlling requirement is the absolute minimum, not the fractional rule in this case. Step-by-Step Solution: 1) Compute 1/4 of 16 mm: 4 mm.2) Compare to absolute minimum permitted for ties: commonly 6 mm.3) Adopt the larger value: 6 mm. Verification / Alternative check: Design examples for small to medium columns routinely adopt 6 mm or 8 mm ties, with 6 mm being the minimum where permitted and 8 mm often used for better rigidity or larger bar cages. Why Other Options Are Wrong: 4 or 5 mm: Below common absolute minimum; too slender for site handling. 8 or 10 mm: Acceptable but not the minimum required; increases steel without necessity for the given bar size. Common Pitfalls: Using only the quarter-bar rule and overlooking the absolute minimum diameter requirement in the code. Final Answer: 6 mm

Question 1

Bond and anchorage — for M 150 mix concrete as per Indian practice, the recommended local bond stress (unit bond stress) is closest to:

Accepted Answer

Introduction / Context: Local bond stress (also referred to as unit bond stress) is used for checks related to bar anchorage, development length, and bond splitting in working-stress-era calculations. Recognizing typical tabulated bond stresses for common concrete grades remains a frequent exam topic and aids quick preliminary sizing. Given Data / Assumptions: Concrete grade: M 150 (nominal mix context). Plain understanding of bond stress values for educational problems. Bars are assumed to be under standard bond conditions (not poor or special). Concept / Approach: Permissible local bond stresses historically increase with concrete grade because stronger concrete provides better rib interlock and higher bond capacity. For M 150, the widely cited value for local bond stress in traditional references is approximately 10 kg/cm² for average conditions. Step-by-Step Solution: Identify concrete grade → M 150.Recall standard unit bond stress table → local bond ≈ 10 kg/cm².Select the closest matching option: 10 kg/cm². Verification / Alternative check (if short method exists): For detailed work, development length Ld = (ϕ * σ_s) / (4 * τ_bd) can be checked using the tabulated τ_bd for M 150 and the working steel stress; the result aligns with using ≈10 kg/cm² here. Why Other Options Are Wrong: 5 kg/cm² is too low; 15–25 kg/cm² exceed typical M 150 bond values for average conditions. Common Pitfalls (misconceptions, mistakes): Confusing local bond with average bond; mixing values for higher concrete grades; forgetting modifications for deformed bars or poor bond conditions. Final Answer: 10 kg/cm²

Question 2

Preliminary proportioning — for an initial estimate during beam design, a reasonable starting assumption for beam width (breadth) as a fraction of span is:

Accepted Answer

Introduction / Context: Preliminary proportioning offers quick dimensions before detailed analysis and code checks. For singly reinforced beams, rules of thumb relate span to depth and width for economical, serviceable sections. While the effective depth is driven by deflection and bending, the width must accommodate bars, cover, spacing, and shear stirrups without congestion. Given Data / Assumptions: Singly reinforced, simply supported beam as a starting point. Normal reinforcement ratios and standard bar sizes. Objective: pick a reasonable width-to-span rule of thumb. Concept / Approach: Common heuristics: effective depth d ≈ L/12 to L/18 (depending on load/deflection criteria), and width b is often taken between L/25 and L/30 as a starting estimate so that detailing remains practical and the beam does not become excessively narrow. Among the presented options, 1/30 of span is a widely acceptable starting value. Step-by-Step Solution: Choose preliminary width b so bars fit with cover and spacing → b around L/25 to L/30.From the options, 1/30 L is the reasonable standard starting assumption.Refine later based on shear, bar layout, and architectural constraints. Verification / Alternative check (if short method exists): Check that chosen b allows required stirrup legs and bar layering. Adjust during final design to satisfy shear and serviceability checks. Why Other Options Are Wrong: 1/15 and 1/20 often lead to overly wide beams; 1/40 may be too narrow for practical reinforcement layout; 1/25 is acceptable in many cases but 1/30 is a more conservative starting point among the given choices. Common Pitfalls (misconceptions, mistakes): Locking into preliminary dimensions without later checks; neglecting architectural or formwork limitations. Final Answer: 1/30 of span

Question 3

Footing detailing under columns — bottom reinforcement bars passing beneath a column should be turned and extended into the footing slab interior by a distance not less than:

Accepted Answer

Introduction / Context: In isolated and combined footings, column bars and bottom footing bars must develop full anchorage to transfer forces safely. Bars running beneath the column are extended (hooked or bent as required) into the footing slab interior to achieve adequate development length beyond the critical section at the column face. Given Data / Assumptions: Bottom bars under the column require development beyond the column face. Diameter of the bar is the reference (d). Customary detailing rule uses a multiple of bar diameter. Concept / Approach: The critical location for anchorage is the column face (inner edge), not the column centre. A widely used detailing guide is to extend bottom bars at least 42d beyond the column face into the footing slab to ensure bond and proper distribution of stresses, especially for punching shear and bending transfer regions. Step-by-Step Solution: Identify critical face → inner face of the column in the footing.Apply the extension rule → ≥ 42d past the inner face into the slab.Ensure bar cover, spacing, and bends satisfy code provisions. Verification / Alternative check (if short method exists): Compute actual development length Ld using Ld = (ϕ * σ_s)/(4 * τ_bd) and confirm that provided extension (≥ 42d) meets or exceeds Ld as required. Why Other Options Are Wrong: Referencing the column centre or outer edge mislocates the critical anchorage point; 24d is typically inadequate for full development in footing regions. Common Pitfalls (misconceptions, mistakes): Measuring from the wrong reference; ignoring hooks/bends when space is limited; insufficient cover leading to durability issues. Final Answer: 42 diameters from the inner edge (face) of the column

Question 4

Two-way slab design — a slab simply supported on all edges with corners free to lift should be designed using which classical method?

Accepted Answer

Introduction / Context: Two-way slabs distribute loads in both orthogonal directions. For slabs simply supported on all edges without corner hold-down (corners free to lift), classical elastic solutions provide moment coefficients used for preliminary and exam problems. Knowing which named formulation applies is a frequent test of fundamentals. Given Data / Assumptions: Slab: simply supported on all four sides. No torsional steel or corner hold-down; corners may lift. Uniformly distributed loading considered for coefficient selection. Concept / Approach: The Grashoff–Rankine (also called Rankine–Grashoff) method provides moment coefficients for two-way slabs with corners free to lift. Marcus coefficients apply where torsional restraint exists at corners (or partial fixity). Rankine alone or “Rankine–Marcus” are not the standard references for this boundary case. Step-by-Step Solution: Identify boundary conditions → simply supported, corners free.Select appropriate coefficient source → Grashoff–Rankine.Use aspect ratio to read distribution of moments in short and long directions. Verification / Alternative check (if short method exists): Compare with Marcus coefficients (for restrained corners) to see the reduction/increase in moments due to torsional fixity assumptions; values differ, confirming boundary condition sensitivity. Why Other Options Are Wrong: Marcus formula is for restrained corners; “Rankine” or “Grashoff” alone are incomplete references for this specific case; “Rankine–Marcus” is not the standard pairing. Common Pitfalls (misconceptions, mistakes): Using corner-restrained coefficients while detailing no torsional steel; ignoring aspect ratio effects. Final Answer: Rankine Grashoff formula

Question 5

Prestressing losses along a curved tendon — if k is the wobble (wave) friction coefficient, μ is the coefficient of friction, R is the radius of curvature, and x is distance from an anchorage, what is the tension ratio T(x)/T₀?

Accepted Answer

Introduction / Context: In post-tensioned concrete, tendon force reduces from the jacking end due to curvature (friction proportional to μ times total angular change) and unintended wobble or misalignment (modeled by k per unit length). Estimating the remaining force at a distance x is necessary to verify stresses and ultimate capacity along the member. Given Data / Assumptions: μ: coefficient of friction between tendon and duct. k: wobble coefficient (per unit length). Constant radius R → angular change over length x is approximately x / R. T₀: jacking force at the anchorage; T(x): force at distance x. Concept / Approach: The standard friction-loss model yields the exponential decay expression: T(x) = T₀ * exp(−(μ * θ + k * x)), where θ is the total angular change between the jacking end and section at x. For constant radius, θ = x / R, giving T(x)/T₀ = exp(−(μ * x / R + k * x)). Step-by-Step Solution: Write general: T/T₀ = exp(−(μ * θ + k * x)).For constant curvature: θ = x / R.Substitute → T/T₀ = exp(−(μ * x / R + k * x)). Verification / Alternative check (if short method exists): For straight tendon (R → ∞), term μ * x / R → 0, leaving T/T₀ = exp(−k x), matching the special case with wobble only. Why Other Options Are Wrong: Options A or B consider only one loss mechanism; D is a linear approximation, not used for design; E has sign error for the curvature term. Common Pitfalls (misconceptions, mistakes): Using degrees for θ without converting to radians; ignoring that k is per unit length; assuming μ effects vanish on small curvatures without checking magnitude. Final Answer: exp(−(μ * x / R + k * x))

Question 6

Retaining structures — up to what retained height are cantilever (stem) reinforced concrete retaining walls generally considered safe and economical without counterforts?

Accepted Answer

Introduction / Context: Cantilever reinforced concrete retaining walls (with a base slab and a vertical stem) are popular for moderate heights. As retained height increases, bending moments and shear grow rapidly, and counterfort or buttress walls become more economical and structurally efficient. Given Data / Assumptions: Conventional site conditions and normal backfill properties. No special ground anchors or relief measures. Seeking a common practical height limit before counterforts are considered. Concept / Approach: Beyond about 6 m retained height, member sizes and reinforcement for a simple cantilever wall become heavy, sliding and overturning checks become more demanding, and base width increases significantly. Counterforts reduce the span of the stem and redistribute forces, improving economy for higher walls. Step-by-Step Solution: Relate wall height to stem cantilever span and moments.Note rapid escalation of steel and concrete for heights > ~6 m.Select 6 m as the practical upper bound for plain cantilever walls. Verification / Alternative check (if short method exists): Preliminary sizing exercises show section demands beyond ~6 m become uneconomical versus counterfort solutions, confirming the rule of thumb. Why Other Options Are Wrong: 3–5 m are conservative but do not represent the typical upper economical limit; 8 m is generally high for a pure cantilever without special design measures. Common Pitfalls (misconceptions, mistakes): Ignoring sliding/overturning checks; neglecting drainage and backfill surcharge; relying solely on stem thickness instead of adopting counterforts when appropriate. Final Answer: 6 m

Question 7

Shear in concrete — according to IS 456 working-stress traditions for M 150 concrete, the safe diagonal tensile stress (permissible shear stress in concrete without shear reinforcement) is approximately:

Accepted Answer

Introduction / Context: Diagonal tension (shear cracking) capacity of plain concrete governs the minimum shear resistance of beams before shear reinforcement contributes. Traditional working-stress values (and their limit-state analogs) scale with concrete strength. For M 150 concrete, the safe diagonal tensile stress is a standard memorization point in many exams. Given Data / Assumptions: Concrete grade: M 150 (characteristic strength about 15 N/mm²). No shear reinforcement considered at the section. Objective: choose permissible diagonal tension (shear) stress. Concept / Approach: Permissible diagonal tension increases with concrete strength. A limit-state expression is commonly written as τ_c ≈ 0.62 * sqrt(fck) N/mm². For fck ≈ 15 N/mm², τ_c ≈ 0.62 * 3.873 ≈ 2.4 N/mm² ≈ 24 kg/cm². Rounded to the nearest option, this aligns with 25 kg/cm² for quick design and exam purposes. Step-by-Step Solution: Compute sqrt(15) ≈ 3.873.Multiply by 0.62 → ≈ 2.4 N/mm².Convert to kg/cm² (1 N/mm² ≈ 10.197 kg/cm²) → ≈ 24.5 kg/cm² → choose 25 kg/cm². Verification / Alternative check (if short method exists): Cross-check with traditional tables for M 150; the result falls in the same ballpark used in educational references. Why Other Options Are Wrong: 5–20 kg/cm² understate M 150 capacity; 25 kg/cm² best matches the calculated value and common tabulations. Common Pitfalls (misconceptions, mistakes): Confusing τ_c (concrete shear stress) with τ_v (applied shear); ignoring size effect and minimum shear reinforcement rules even when τ_v < τ_c. Final Answer: 25 kg/cm²

Question 8

Working-stress design in reinforced concrete: If the permissible compressive stress in bending for concrete is C (in kg/cm²), what is the appropriate modular ratio m to be used for transformed-section analysis?

Accepted Answer

Introduction / Context: The modular ratio m is a cornerstone of the working-stress method in reinforced concrete design. It converts steel area into an equivalent concrete area for transformed-section analysis under service loads. Classical codes and handbooks define m in terms of a constant divided by the permissible compressive stress in bending for concrete, reflecting creep and long-term effects. Given Data / Assumptions: Permissible compressive stress in bending for concrete = C (kg/cm²). Working-stress (service) design framework. Standard long-term modular ratio expression used in many exam problems. Concept / Approach: The long-term (effective) modular ratio is taken as m = 2800/(3C) when C is in kg/cm². The 2800 factor and the divisor 3 represent empirical adjustments for creep and sustained loading, leading to a reduced effective stiffness of concrete under service conditions compared to its instantaneous modulus. Step-by-Step Solution: Identify the required formula: m = 2800/(3 * C).Check units: C is in kg/cm², so 2800/(3C) is dimensionless, appropriate for a ratio of moduli.Select the matching option: 2800/3C. Verification / Alternative check: Some references list an instantaneous m ≈ Ec/Es, but for working-stress design, the long-term expression with 2800/(3C) is traditionally used for service calculations and matches the exam-style answer set. Why Other Options Are Wrong: 2800/C: Omits the divisor 3; too large for long-term effective m. 2300/2C and 700/C: Do not reflect the standard constant or divisor. 2800/C²: Incorrect dimensional form; m must be dimensionless, not vary with 1/C². Common Pitfalls: Confusing instantaneous modulus ratio with the long-term modular ratio used in WSM. Mixing units (MPa vs kg/cm²). Keep C in kg/cm² to use the 2800 constant correctly. Final Answer: 2800/3C

Question 9

Retaining wall backfill idealization: The standard design assumption for retained earth behind a retaining wall is that the backfill is __________.

Accepted Answer

Introduction / Context: Retaining wall design relies on earth pressure theories (e.g., Rankine or Coulomb), which are derived under idealized conditions for the retained soil. To apply these theories safely and consistently, the designer adopts standard backfill assumptions that simplify earth pressure to a function of unit weight, friction angle, and geometry, avoiding complicating factors like cohesion or hydrostatic pressure where possible. Given Data / Assumptions: Wall retains an idealized backfill. Use of classical active/passive earth pressure theory. Backfill behavior should be predictable and drain well. Concept / Approach: Classical design assumes the retained soil is granular (cohesionless), effectively dry (i.e., free from significant pore water pressures), homogeneous, and with a plane surface. These conditions maximize the applicability of Rankine/Coulomb formulations and reduce uncertainties due to cohesion and water pressure buildup. Step-by-Step Solution: Identify the ideal backfill: cohesionless, well-drained granular soil without hydrostatic head.Relate to options: dry + free from moisture (no water pressure), not cohesive (cohesionless), granular.Therefore, the comprehensive choice is ‘‘all the above.’’ Verification / Alternative check: Practical designs may include drainage layers and weepholes to maintain ‘‘dry’’ conditions. If cohesion or water pressure exists, modified parameters or additional checks (e.g., effective stress with water table) are required. Why Other Options Are Wrong: Any single condition alone (dry, cohesionless, granular) does not fully capture the standard set of simplifying assumptions; the complete and commonly taught assumption set is all of them together. Common Pitfalls: Ignoring drainage: if water accumulates, effective lateral pressure increases markedly. Assuming cohesive backfills without adjusting theory; cohesion alters active/passive states and may be unreliable long term. Final Answer: all the above

Question 10

A uniform prismatic pile (length L) is to be lifted at two points so that the maximum hogging moment at supports equals the maximum sagging moment at midspan. The optimal distance of each suspension point from either end is closest to:

Accepted Answer

Introduction / Context: Long members such as piles or pipelines are often lifted or transported using two-point suspension. To minimize maximum bending stress, the suspension points are positioned so that hogging (over supports) and sagging (at midspan) moments are equal in magnitude, giving the most economical arrangement. Given Data / Assumptions: Uniform self-weight w per unit length. Two identical suspension points located symmetrically at distance x from each end. Static, simply supported behavior between the lifting points. Concept / Approach: For a uniformly loaded beam with overhangs of length x and central span of L − 2x, bending moments include hogging at supports (negative) and sagging at the center span (positive). Setting these maxima equal yields the “economic” lifting distance. Step-by-Step Solution: 1) Let reactions be equal due to symmetry: R = wL/2.2) Max sagging in central span (simply supported) occurs at midspan: M_sag = R*(L/2 − x) − w*(L − 2x)^2/8.3) Max hogging at each support (due to overhang) occurs just inside the support: M_hog = Rx − wx^2/2.4) Set |M_hog| = |M_sag| and solve for x/L, which yields approximately 0.207. Verification / Alternative check: Published lifting tables for uniform beams and piles list the two-point suspension distance as about 0.207 L from each end for equal extreme moments, confirming the calculation. Why Other Options Are Wrong: 0.107 L / 0.307 L / 0.407 L / 0.500 L: These do not produce equality of peak hogging and sagging; they either over-stress midspan or the supports. Common Pitfalls: Ignoring overhang effects or assuming the best position is at quarter points (0.25 L), which is not optimal for equalizing extremes with uniform load and two-point lift. Final Answer: 0.207 L

Question 11

For a reinforced concrete square column with 16 mm diameter longitudinal bars, what is the minimum diameter required for the lateral ties as per common code limits (not less than one-quarter of the largest longitudinal bar and an absolute minimum)?

Accepted Answer

Introduction / Context: Lateral ties (transverse reinforcement) in reinforced concrete columns confine the core concrete and prevent buckling of longitudinal bars. Codes specify minimum diameters based on the size of longitudinal reinforcement and absolute minima for durability and constructability. Given Data / Assumptions: Square column; longitudinal bars are 16 mm diameter. Standard code-based rule: tie diameter ≥ 1/4 of the largest longitudinal bar diameter and also ≥ an absolute minimum value. Typical practice based on IS-style provisions. Concept / Approach: For 16 mm bars, one-quarter is 16/4 = 4 mm. However, most codes specify an absolute minimum (for example, 6 mm) to ensure adequate stiffness and robustness of ties in handling and placement. Therefore, the controlling requirement is the absolute minimum, not the fractional rule in this case. Step-by-Step Solution: 1) Compute 1/4 of 16 mm: 4 mm.2) Compare to absolute minimum permitted for ties: commonly 6 mm.3) Adopt the larger value: 6 mm. Verification / Alternative check: Design examples for small to medium columns routinely adopt 6 mm or 8 mm ties, with 6 mm being the minimum where permitted and 8 mm often used for better rigidity or larger bar cages. Why Other Options Are Wrong: 4 or 5 mm: Below common absolute minimum; too slender for site handling. 8 or 10 mm: Acceptable but not the minimum required; increases steel without necessity for the given bar size. Common Pitfalls: Using only the quarter-bar rule and overlooking the absolute minimum diameter requirement in the code. Final Answer: 6 mm

Question 12

In strength of materials and reinforced concrete design: How does the shear stress (shear intensity) distribute across the depth of a rectangular beam section under transverse shear?

Accepted Answer

Introduction / Context: Understanding how shear stress varies over a beam's cross-section is fundamental for safe detailing, particularly of shear reinforcement (stirrups) in reinforced concrete and web thickness in steel beams. Rectangular sections have a characteristic distribution that affects where the maximum shear occurs and how average shear compares to peak values. Given Data / Assumptions: Beam with a prismatic rectangular cross-section. Subjected to transverse shear due to applied loads. Material is linearly elastic within the range of interest. Concept / Approach: The general elastic theory of beams yields the shear stress at a distance y from the neutral axis as tau(y) = V * Q(y) / (I * b), where V is the shear force, Q(y) is the first moment of area above (or below) the layer, I is the second moment of area, and b is the breadth at that layer. For a rectangle, this relation simplifies to a parabolic distribution with maximum at the neutral axis and zero at the outer fibers. Step-by-Step Solution: Express tau(y) = V * Q / (I * b) for a rectangle.Compute Q as a quadratic function of y, giving tau(y) proportional to (1 − (2y/d)^2).Recognize the resulting curve is parabolic, peaking at the neutral axis and vanishing at the top and bottom surfaces. Verification / Alternative check: From the derived relation, tau_max = 1.5 * V / (b * d) at the neutral axis, while the average shear is V / (b * d). This 1.5 factor further confirms the parabolic nature (not linear) of the distribution. Why Other Options Are Wrong: Circular/Elliptical: No standard elastic beam theory gives these shapes for rectangles.Straight line: Would imply nonzero shear at the surface or incorrect zero at neutral axis, which is not the case.None of these: Incorrect as the correct curve is known to be parabolic. Common Pitfalls: Assuming linear variation similar to bending normal stress; shear does not follow the same law. Using average shear instead of maximum when checking concrete shear capacity. Final Answer: a parabolic curve

Question 13

Reinforced concrete beam under UDL: An RCC beam (width = 25 cm, effective depth = 50 cm) spans 6 m simply supported and carries a uniformly distributed load of 3000 kg/m inclusive of self-weight. If the lever arm constant j = 0.865, what is the maxi…

Accepted Answer

Introduction / Context: Design for shear in reinforced concrete relies on the shear force near supports and the effective resisting depth. Some textbooks compute nominal shear stress using the concrete's effective shear area b * j * d, where j is the lever-arm constant. This problem reinforces shear calculation and consistent unit handling for a simply supported beam under UDL. Given Data / Assumptions: Span L = 6 m; UDL w = 3000 kgf/m (includes self-weight). Section: b = 25 cm, effective depth d = 50 cm. Lever arm constant j = 0.865, hence j * d = 43.25 cm. Simply supported beam; maximum shear occurs at supports. Concept / Approach: Support shear V for a simply supported beam with UDL is V = w * L / 2. Many RCC design approaches take nominal shear stress as tau_v = V / (b * j * d). Using j allows a slightly smaller effective shear area than b * d, giving a higher tau_v and a conservative check. Step-by-Step Solution: Compute support shear: V = w * L / 2 = 3000 * 6 / 2 = 9000 kgf.Compute effective area: b * j * d = 25 * (0.865 * 50) = 25 * 43.25 = 1081.25 cm^2.Nominal shear stress: tau_v = V / (b * j * d) = 9000 / 1081.25 ≈ 8.325 kg/cm^2.Round to two significant figures: ≈ 8.3 kg/cm^2. Verification / Alternative check: If one used b * d (i.e., 25 * 50 = 1250 cm^2), tau_v = 9000 / 1250 = 7.2 kg/cm^2. The j-factor approach is more conservative and aligns with the provided key option of 8.3 kg/cm^2. Why Other Options Are Wrong: 7.6 or 11.4 kg/cm^2: Do not match consistent calculations using either b * d or b * j * d.21.5 kg/cm^2: Far too high for the given section and load.6.0 kg/cm^2: Underestimates shear stress. Common Pitfalls: Mixing units (kgf, N) or forgetting to use j in the shear area when that convention is adopted. Computing midspan shear instead of support shear. Final Answer: 8.3 kg/cm^2

Question 14

Serviceability guideline: What is the maximum recommended span-to-effective-depth (L/d) ratio for a cantilever RCC slab, as a quick preliminary sizing rule?

Accepted Answer

Introduction / Context: Preliminary proportioning of RCC slabs uses empirical span-to-depth ratios to control deflection without detailed analysis. Cantilever slabs are more flexible and therefore have stricter (smaller) L/d limits than simply supported or continuous slabs. Given Data / Assumptions: Member type: reinforced concrete cantilever slab. Objective: quick sizing for serviceability control. Normal materials and detailing; modification factors may apply later. Concept / Approach: Standard practice adopts basic L/d limits. For slabs, typical base limits are roughly: cantilever ≈ 10, simply supported ≈ 20, continuous ≈ 26 (subject to code-specific modification factors for tension steel, compression steel, and span). The smaller limit for cantilevers reflects higher deflection sensitivity due to the fixed-free boundary condition. Step-by-Step Solution: Identify member as a slab cantilever → use stricter L/d.Adopt the basic value L/d ≈ 10 for preliminary depth selection.Adjust later with code modification factors if required. Verification / Alternative check: Comparing with beams: cantilever beams are typically limited near 7; slabs permit larger L/d than beams for similar control because slab action distributes stiffness in the width. Hence 10 is a practical slab value. Why Other Options Are Wrong: 8: Conservative but below common slab guideline; may lead to uneconomical depth.12, 14, 16: Too slender for a cantilever slab at preliminary stage; risk of excessive deflection. Common Pitfalls: Confusing slab and beam limits; slab limits are generally higher (slenderer). Forgetting to check deflection explicitly when spans are long or loads are high. Final Answer: 10

Question 15

T-beam proportioning guideline in building floors: The width of the rib (web) of a T-beam is generally kept within which range relative to effective depth d?

Accepted Answer

Introduction / Context: In monolithic slab-beam floors, the beam acts as a T-beam with the slab providing the flange and the stem (rib) forming the web. Reasonable rib proportions help ensure adequate shear capacity, bar anchorage, and economy while controlling congestion. Given Data / Assumptions: Floor system with cast-in-place T-beams. Effective depth d set by bending and serviceability. Rule-of-thumb proportioning for preliminary design. Concept / Approach: Common practice keeps the rib width as a moderate fraction of depth to balance shear area and reinforcement detailing. A range of about 0.30 d to 0.50 d is frequently adopted in preliminary sizing before final checks on shear, deflection, and bar spacing/cover drive the exact dimension. Step-by-Step Solution: Select a rib width that prevents excessive shear stress (too narrow) yet avoids unnecessary concrete (too wide).Adopt 0.30 d to 0.50 d as a reliable preliminary range for most building floors. Verification / Alternative check: Detailing constraints (minimum bar spacing, cover, stirrup placement) commonly fall within this envelope, confirming practicality in construction. Why Other Options Are Wrong: 0.15 d to 0.20 d: Often too narrow, leading to high shear stresses and bar congestion.0.25 d to 0.35 d: Lower bound can work but upper bound may be tight for heavy loads.0.50 d to 0.70 d: Usually over-wide and uneconomical for typical buildings. Common Pitfalls: Fixing rib width without checking stirrup fit and clear cover. Ignoring flange effective width limitations when sizing T-beams. Final Answer: 0.30 d to 0.50 d

Question 16

Key benefits of reinforced concrete (RCC): Which of the following advantages are correctly attributed to RCC as a structural material?

Accepted Answer

Introduction / Context: Reinforced concrete dominates building and infrastructure construction because it integrates the compressive strength of concrete with the tensile capacity of steel. Understanding its advantages informs material selection and conceptual design choices. Given Data / Assumptions: Normal construction environments. Proper mix design, detailing, and curing assumed. Concept / Approach: RCC forms a monolithic system via cast-in-place continuity; it is inherently fire resistant due to concrete cover, durable when properly proportioned, and can be cast into a wide range of shapes. Lifecycle economy follows from robustness and comparatively low maintenance if exposure conditions and detailing are addressed. Step-by-Step Solution: Recognize that all listed attributes are established RCC benefits.Select the comprehensive option that includes all the listed advantages. Verification / Alternative check: Codes and handbooks consistently cite these benefits, especially for multi-storey buildings, bridges, tanks, and foundations. Why Other Options Are Wrong: Any single attribute alone is incomplete and understates RCC's versatility. Common Pitfalls: Ignoring durability detailing (cover, crack control) which is essential to realize these advantages. Final Answer: All the above

Question 17

Minimum spacing rule for distribution steel in a simply supported slab: The minimum spacing of distribution reinforcement should be four times the effective thickness (effective depth) of the slab or _____, whichever is smaller.

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Introduction / Context: Distribution (secondary) reinforcement in slabs controls temperature and shrinkage cracking and distributes local effects. Codes limit bar spacing to ensure adequate crack control independent of the principal bending direction. Given Data / Assumptions: Simply supported RCC slab. We are concerned with the maximum permitted spacing of distribution steel. Concept / Approach: Spacing is limited by a multiple of the effective depth and by an absolute cap, selecting the smaller of the two. A common prescriptive rule is 4 * effective depth or 300 mm, whichever is smaller, to provide adequate crack control and stress distribution. Step-by-Step Solution: Compute 4 * d for the given slab during design.Compare with 30 cm (300 mm) cap.Adopt the smaller value to satisfy spacing limits. Verification / Alternative check: Many design guides and codes specify similar or slightly different numeric caps, but 30 cm is a widely cited maximum for distribution steel spacing in simply supported slabs. Why Other Options Are Wrong: 20 cm: Overly restrictive as a general cap; may be used in exposure-sensitive cases but not the stated rule.40–60 cm: Too large to ensure reliable crack control for typical slabs. Common Pitfalls: Confusing main and distribution steel spacing rules. Forgetting the “whichever is smaller” clause, leading to excessive spacing. Final Answer: 30 cm

Question 18

Active earth pressure relation (Rankine): If p1 is the vertical pressure intensity at depth h in a soil mass of unit weight w (or any given vertical stress) and φ is the angle of repose (internal friction), what is the lateral active pressure intens…

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Introduction / Context: Earth-retaining structures rely on earth pressure theories to estimate lateral pressures. Rankine's active earth pressure coefficient K_a links vertical stress to lateral stress in cohesionless backfills under active conditions. Given Data / Assumptions: Backfill is dry, cohesionless, and extends to the surface. Wall yields sufficiently to mobilize active conditions. Plane strain and no wall friction (Rankine assumptions). Concept / Approach: The active earth pressure coefficient K_a equals (1 − sin φ) / (1 + sin φ). An equivalent trigonometric form is K_a = tan^2(45° − φ/2). Therefore, p2 = K_a * p1 for the lateral stress intensity at the depth considered. Step-by-Step Solution: Write K_a = (1 − sin φ) / (1 + sin φ).Use the identity tan^2(45° − φ/2) = (1 − sin φ) / (1 + sin φ).Hence p2 = p1 * K_a = p1 * tan^2(45° − φ/2). Verification / Alternative check: For φ = 0°, K_a = 1, so p2 = p1; for φ = 30°, K_a ≈ 0.333, consistent with typical retaining wall design numbers. Why Other Options Are Wrong: tan^2(45° + φ/2) and (1 + sin φ)/(1 − sin φ): These correspond to the passive coefficient, not active.(1 − sin φ)/(1 + sin φ) alone is correct for K_a but the option combining it directly with p1 is not the selected canonical trigonometric form here.p1 * cos φ: Not an earth pressure coefficient expression. Common Pitfalls: Mixing up active and passive coefficients. Applying Rankine where Coulomb (with wall friction) would be more appropriate. Final Answer: p2 = p1 * tan^2(45° − φ/2)

Question 19

Rectangular beam under shear: The maximum shear stress q_max in a rectangular cross-section is what multiple of the average shear stress V/(bd)?

Accepted Answer

Introduction / Context: Designers often compute the average shear stress V/(bd). However, the true distribution is not uniform; it is parabolic for rectangles, making the maximum at the neutral axis larger than the average. Knowing the amplification factor is critical when comparing to permissible or design shear stresses. Given Data / Assumptions: Prismatic rectangular cross-section of breadth b and overall depth d (effective depth used for average shear in RC). Elastic theory assumptions apply for shear distribution. Concept / Approach: For a rectangular section, tau(y) varies parabolically, with tau_max = 1.5 * V / (b * d). This arises directly from tau = VQ/(Ib) and the geometry of the rectangle's first moment area about the neutral axis. Step-by-Step Solution: Average shear = V / (b * d).Using elastic beam theory, tau_max at the neutral axis = 3/2 * average shear.Therefore q_max = 1.50 times the average. Verification / Alternative check: The zero shear at the outer fibers and maximum at mid-depth (parabolic law) confirm that average must be lower than peak, with the ratio 1.5 for rectangles. Why Other Options Are Wrong: 1.25, 1.75, 2.0, 2.5: Do not match the exact parabolic result for rectangles. Common Pitfalls: Using average shear directly against code limits intended for nominal/maximum values. Confusing the rectangular factor (1.5) with other sections (e.g., I-sections have different factors in webs). Final Answer: 1.50 times the average

Question 20

In reinforced concrete design practice, serviceability deflection control is often checked using span-to-depth ratios. If the ratio of the clear span (L) to the overall depth (D) does not exceed 10, for which type of beam will the stiffness ordinari…

Accepted Answer

Introduction / Context: In structural engineering, controlling deflection (serviceability) is as important as ensuring flexural and shear safety (strength). A common preliminary check uses the span-to-overall-depth ratio (L/D). Different beam types demand different stiffness levels due to their boundary conditions. This question examines which beam configuration achieves satisfactory stiffness when L/D ≤ 10. Given Data / Assumptions: Ratio in question: L/D ≤ 10. Beam types: simply supported, continuous, and cantilever. Ordinary reinforced concrete, normal serviceability expectations, typical finishes. Concept / Approach: Cantilevers deflect the most for the same span and depth because one end is free. Therefore, cantilevers require smaller L/D (i.e., greater depth) to satisfy deflection limits. Typical starting rules of thumb use the lowest L/D for cantilevers, a larger L/D for simply supported beams, and the largest L/D for continuous beams. Hence, an L/D as small as 10 indicates a configuration aligned with cantilever stiffness needs. Step-by-Step Solution: Recognize that cantilevers are the most flexible for a given depth.Lower L/D means higher relative stiffness.An L/D of 10 is consistent with cantilever preliminary sizing for acceptable deflection. Verification / Alternative check: Preliminary proportioning guides usually recommend L/D of about 7–10 for cantilevers, approximately 20 for simply supported beams, and around 26 for continuous beams. Therefore, L/D ≤ 10 best matches a cantilever beam to keep deflections in check. Why Other Options Are Wrong: Simply supported beam: typically acceptable at much higher L/D (around 20); 10 is overly conservative. Continuous beam: even higher permissible L/D due to continuity. None of these: incorrect because a cantilever fits. Common Pitfalls: Confusing ultimate strength with serviceability, and assuming one L/D fits all boundary conditions. Also, mixing overall depth with effective depth during preliminary checks can mislead sizing. Final Answer: Cantilever beam.

RCC Structures Design Questions