(a) The subsoil at a site consists of a 12·0 m thick homogeneous layer of dense sand having dry unit weight, γd = 17·2 kN/m³, GS = 2·7 and φ = 35°. The natural ground water level lies at 2·5 m below the ground surface. Assume that the soil is dry abo

Question

(a) The subsoil at a site consists of a 12·0 m thick homogeneous layer of dense sand having dry unit weight, γd = 17·2 kN/m³, GS = 2·7 and φ = 35°. The natural ground water level lies at 2·5 m below the ground surface. Assume that the soil is dry above the water table and unit weight of water, γw = 9·81 kN/m³. Determine the shear strength of the soil along a horizontal plane through the middle of the sand layer. (15 marks)

(b) A 2 m × 2 m square footing is placed at 1·8 m below the ground surface. The ground water table is at the ground level. The subsoil consists of a uniform deposit of soft, loose soil. The laboratory test results of the soil are as follows: Angle of internal friction, φ = 21°; Cohesion, C = 15 kPa; Unit weight of soil, γ = 16·5 kN/m³. Determine the allowable load that can be imposed on this square footing with a factor of safety of 3. (15 marks)

Given:
φ | Nc | Nq | Nγ
10 | 8·34 | 2·47 | 0·37
12 | 9·28 | 2·97 | 0·60
14 | 10·37 | 3·59 | 0·92
16 | 11·63 | 4·34 | 1·37
18 | 13·10 | 5·26 | 2·00
20 | 14·83 | 6·40 | 2·87
22 | 16·88 | 7·82 | 4·07
24 | 19·32 | 9·60 | 5·72

(c) A simply supported steel beam of span 4 m carries a factored point load of 450 kN at its mid span. The beam is laterally supported. Check the adequacy of ISMB 400 section to carry this load. If it becomes unsafe, re-design it by providing extra cover plate to make it safe. Assume the section is plastic. Grade of steel is E 250. Use limit state method. (20 marks)

Section properties of ISMB 400:
A = 7840 mm²
b_f = 140 mm
t_f = 16 mm
t_w = 8·9 mm
Z_pz = 1176·18 × 10³ mm³
Z_ez = 1020 × 10³ mm³

UPSC Answer Check · Accepted Answer

Solve this three-part numerical problem by allocating approximately 30% time to part (a) on shear strength calculation, 30% to part (b) on bearing capacity with interpolation of bearing capacity factors, and 40% to part (c) on steel beam design including adequacy check and redesign with cover plates. Begin each part with stated assumptions and formulas, show complete step-by-step calculations with proper units, and conclude with clear final answers and practical implications for field application.
- Part (a): Calculate effective stress at mid-depth of sand layer (6m below ground), determine saturated unit weight using void ratio from dry unit weight, compute effective stress considering water table at 2.5m, and apply τ = σ' tan φ for shear strength
- Part (b): Apply Terzaghi's bearing capacity equation for square footing with interpolation of Nc, Nq, Nγ values for φ=21°, use submerged unit weight below water table, calculate ultimate bearing capacity, and determine allowable load with FOS=3
- Part (c): Calculate maximum factored moment for simply supported beam with central point load, check section adequacy using plastic moment capacity (Md = βb·Zp·fy/γm0), and if unsafe, design cover plates to increase section modulus to required value
- Correct application of effective stress principle in parts (a) and (b) with proper handling of water table effects on unit weights and stress calculations
- Proper interpolation technique for bearing capacity factors between φ=20° and φ=22° in part (b), and correct application of IS 800:2007 limit state provisions for plastic section in part (c)
- Appropriate selection and sizing of cover plates in redesign, checking for local buckling criteria and ensuring adequate weld/connection provisions

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly applies effective stress principle in (a) and (b), uses proper bearing capacity theory with appropriate shape factors, and applies IS 800:2007 limit state design correctly for plastic section including proper use of βb=1.0 for laterally supported beams; distinguishes between elastic and plastic section moduli appropriately	Shows understanding of basic concepts but makes minor errors in effective stress calculation or uses elastic section modulus instead of plastic in (c), or applies working stress method instead of limit state; bearing capacity equation correct but interpolation errors present	Confuses total and effective stress, applies wrong bearing capacity theory (uses strip footing formula for square), or uses grossly incorrect section classification; fundamental misunderstanding of limit state philosophy or water table effects
Numerical accuracy	25%	12.5	All calculations accurate to appropriate significant figures: correct void ratio (e≈0.57), saturated unit weight (≈20.1 kN/m³), effective stress at 6m (≈91.5 kPa), shear strength (≈64 kPa); bearing capacity factors interpolated correctly (Nc≈15.86, Nq≈7.11, Nγ≈3.47), qu≈420 kPa, allowable load≈560 kN; beam: Mu=450 kNm, Md≈294 kN·m (inadequate), correct cover plate design with required Zp≈1800×10³ mm³	Minor arithmetic errors or unit conversion mistakes (kPa vs kN/m²), slight errors in interpolation of bearing capacity factors, or calculation errors in cover plate dimensions; final answers within 10% of correct values	Major calculation errors, wrong formulas applied numerically, incorrect unit weights, or order-of-magnitude errors; fails to recognize section inadequacy in (c) or provides impossible cover plate solutions
Diagram quality	10%	5	Clear soil profile diagram for (a) showing layers, water table, and stress distribution; footing sketch with dimensions and failure surface for (b); shear force and bending moment diagrams for (c) with values at critical sections; neat ISMB 400 cross-section with cover plates dimensioned	Basic sketches present but lacking dimensions or labels; diagrams readable but missing critical information like water table position or moment values; freehand acceptable but messy	No diagrams or illegible sketches; missing essential diagrams like SFD/BMD for beam; diagrams contradict calculations or show wrong failure mechanisms
Step-by-step derivation	25%	12.5	Each part shows complete derivation: (a) void ratio from γd, saturated unit weight, effective stress calculation with depth-wise breakdown, Mohr-Coulomb application; (b) Terzaghi's equation stated, interpolation method shown, each term calculated separately; (c) load factor check, moment calculation, section classification, plastic moment capacity check, redesign with required Zp calculation and cover plate sizing with width-thickness checks	Most steps shown but some shortcuts taken; interpolation shown but not method; redesign started but incomplete steps; references to IS codes mentioned but not specific clauses	Final answers only with no working; or scattered calculations with no logical flow; missing essential steps like interpolation method or section adequacy check before redesign
Practical interpretation	20%	10	Interprets shear strength significance for slope stability in (a); discusses bearing capacity implications for soft soil settlement control and need for ground improvement in (b); explains why ISMB 400 is inadequate for heavy industrial loading and discusses constructability of cover plates, welding requirements, and deflection serviceability check in (c); references relevant Indian standards (IS 6403, IS 800)	Brief comment on safety of answers; mentions practical aspects like settlement or deflection but without depth; generic statements about code compliance	No interpretation or discussion of results; fails to state whether designs are safe/unsafe; no connection to field application or code requirements; ignores special conditions like water table effects on construction

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