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Civil Engineering 2023 Paper I 50 marks Compulsory Solve

Paper I — Q5

(a) The flow rate of water over a weir is 3 m³/s. A 1 : 10 scale model of the weir is tested in a water channel. Answer the…

(a)
(i)

The flow rate of water over a weir is 3 m³/s. A 1 : 10 scale model of the weir is tested in a water channel. Answer the following : What flow rate should be used for the model ?

(ii)

If a force of 15 N is experienced on the model, what force would be expected on the prototype ? 10 marks

(b)

A rectangular wing on a small airplane has a 1.3 m chord and a 10 m span. When flying in air at 250 km/hour, the wing experiences a total aerodynamic force of 20 kN. If the lift to drag ratio is 3, what would be the lift coefficient of the wing ? Take density of air as 1.20 kg/m³. 10 marks

(c)

An idealized radial turbine is rotating at 140 rev/min as shown in the figure. The absolute flow enters at 30° and leaves radially inward. The flow rate is 4.0 m³/s of water at 20°C. The blade thickness is constant at 10 cm. If density of water is 1000 kg/m³, what would be the theoretical power developed by the turbine ? [Figure: outer radius 70 cm, inner radius 40 cm, b = 10 cm] 10 marks

(d)

The shear stress induced at a depth of 7.0 m due to construction of a nearby foundation is 50 kN/m². The soil properties at the site are: unit weight γ = 18 kN/m³, effective cohesion C′ = 12 kN/m², effective friction angle φ′ = 30°. Compute the factor of safety against shear failure assuming water table located far below the point. Also compute the percentage reduction in factor of safety if water table rises to the ground level. Take unit weight of water = 9.81 kN/m³. 10 marks

(e)

Excavation is made in a soil whose porosity is 35% and specific gravity of soil grains is 2.65. A 3.0 m layer of this soil is subjected to an upward seepage head of 4.0 m. What factor of safety exists against boiling (piping) ? If a factor of safety of 2 is required against boiling, what depth of gravel is required to be placed above the soil stratum ? Assume unit weight of gravel and the soil to be the same and loss of head in the layer to be negligible. Assume γw = 9.81 kN/m³. 10 marks

हिंदी में प्रश्न पढ़ें
(i)

(क) एक वीयर के ऊपर जल की प्रवाह दर 3 m³/s है । वीयर के एक 1 : 10 अनुमाप निदर्श का परीक्षण एक जल वाहिका में किया जाता है । निम्नलिखित के उत्तर दीजिए : निदर्श के लिए किस प्रवाह दर का उपयोग किया जाना चाहिए ?

(ii)

आदिप्ररूप पर कितना बल प्रत्याशित होगा, यदि निदर्श पर 15 N के एक बल का अनुभव किया जाता है ? (10 अंक)

(ख) एक छोटे हवाई जहाज पर एक आयताकार विमान पंख की जीवा 1.3 m और विस्तृति 10 m है । 250 km/hour पर वायु में उड़ने पर विमान पंख 20 kN के सकल वायुगतिक बल का अनुभव करता है । विमान पंख का उत्थान गुणांक क्या होगा, यदि उत्थान-विकर्ष अनुपात 3 है ? वायु का घनत्व 1.20 kg/m³ लीजिए । (10 अंक)

(ग) चित्र में दर्शाए अनुसार, एक आदर्शीकृत त्रिज्यीय टरबाइन 140 परिक्रमण/मिनट पर घूर्णित है । निरपेक्ष प्रवाह 30° पर प्रवेश करता है और त्रिज्यीय अंतर्मुख निकलता है । 20°C पर जल की प्रवाह दर 4.0 m³/s है । ब्लेड की मोटाई 10 cm पर नियत है । टरबाइन द्वारा उत्पन्न की जाने वाली सैद्धांतिक शक्ति क्या होगी, यदि जल का घनत्व 1000 kg/m³ है । [चित्र: बाहरी त्रिज्या 70 cm, भीतरी त्रिज्या 40 cm, b = 10 cm] (10 अंक)

(घ) निकट में एक नींव के निर्माण के कारण 7.0 m की गहराई पर उत्पन्न अपरूपण प्रतिबल 50 kN/m² है । स्थल पर मृदा गुण ये हैं : एकक भार γ = 18 kN/m³, प्रभावी संसंजन C′ = 12 kN/m², प्रभावी घर्षण कोण φ′ = 30° । भौम जल स्तर को बिन्दु से बहुत अधिक नीचे मानते हुए अपरूपण विफलता के विरुद्ध सुरक्षा गुणक की गणना कीजिए । सुरक्षा गुणक में प्रतिशत कमी की गणना भी कीजिए, यदि भौम जल स्तर भूमि तल तक आ जाए । जल का एकक भार = 9.81 kN/m³ लीजिए । (10 अंक)

(ङ) एक मृदा, जिसकी सरंध्रता 35% और मृदा कणों का विशिष्ट घनत्व 2.65 है, में खुदाई की गई है । इस मृदा की एक 3.0 m परत पर 4.0 m की उपरिमुखी रिसन दाबोच्चता लगी है । क्वथन (पाइपिंग) के विरुद्ध सुरक्षा गुणक कितना है ? यदि क्वथन के विरुद्ध आवश्यक सुरक्षा गुणक 2 है, तो मृदा परत के ऊपर कितनी गहराई तक बजरी रखा जाना आवश्यक है ? बजरी और मृदा का एकक भार समान मान लीजिए और परत में दाबोच्चता में हानि नगण्य मान लीजिए । γw = 9.81 kN/m³ मान लीजिए । (10 अंक)

Q5 of the 2023 UPSC Mains Civil Engineering Paper I, as printed
The question as printed in the 2023 Civil Engineering paper
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The figure this question refers to, in words

The question paper is a scan and the diagram did not survive as text. This is the figure as read from the original page — every component, value and label — so the question can be worked from the text below.

(c) A schematic of an idealized radial-flow turbine runner shown in front view as two concentric circles. The outer radius of the runner is marked as 70 cm, and the inner radius is marked as 40 cm. Between the inner and outer circles, a single curved runner blade is shown. A vertical dashed radial line passes through the center to the top of the outer circle. At the outer periphery (top), a horizontal dashed tangent line is drawn; the absolute inlet velocity vector V2 enters at an angle of 30 degrees to this tangent line, directed inwards and to the right. At the inner periphery, the absolute outlet velocity vector V1 is directed strictly radially inward along the vertical dashed line. The runner blade thickness/width is annotated to the right as 'b = 10 cm'.

(c) A schematic diagram of an idealized radial turbine runner showing two concentric circles: an inner circle with radius 40 cm and an outer circle with radius 70 cm. A curved arrow inside the inner circle indicates counter-clockwise rotation. At the top (12 o'clock position) of the outer circle, flow enters with an absolute velocity vector V2 directed downwards and to the right at an angle of 30 degrees below a horizontal dashed tangent line. A vertical dashed line connects the top of the outer circle radially inwards to the inner circle, where a downward-pointing arrow labelled V1 indicates that the flow leaves radially inward toward the center. The runner width is labelled as b = 10 cm near the outer circle.

Model answer

Drafted with AI by UPSC Answer Check against this question's marking rubric, to the expected length. UPSC does not publish answers for Mains — this is one way to score well, not an official key. How we build rubrics and answers.

(a) Weir model: flow rate and force

A weir is a free-surface flow, so gravity dominates and Froude similarity applies. Length ratio L_r = 1/10, so velocity ratio V_r = √L_r.

(i) Discharge ratio Q_r = L_r² × V_r = L_r^(5/2) = (1/10)^2.5 = 1/316.2. Model flow Q_m = 3/316.2 = 0.00949 m³/s, about 9.49 L/s.

(ii) With the same fluid, force ratio F_r = ρ_r L_r³ = 1/1000. Prototype force F_p = 15 × 1000 = 15 000 N, 15 kN.

(b) Lift coefficient of the wing

V = 250 km/h = 69.44 m/s. Wing area S = 1.3 × 10 = 13 m².

The total aerodynamic force is the resultant of lift L and drag D, with D = L/3. So R = √(L² + D²) = L√10/3 = 20 000 N, giving L = 60 000/√10 = 18 974 N.

Dynamic force ½ρV²S = 0.5 × 1.20 × 69.44² × 13 = 37 616 N.

C_L = 18 974/37 616 = 0.504.

(c) Theoretical power of the radial turbine

ω = 2π × 140/60 = 14.66 rad/s. Inlet radius r₂ = 0.70 m, outlet radius r₁ = 0.40 m, width b = 0.10 m.

Inlet blade speed u₂ = 14.66 × 0.70 = 10.26 m/s. Inlet area = 2πr₂b = 2π × 0.70 × 0.10 = 0.4398 m², so radial velocity V_r2 = 4.0/0.4398 = 9.09 m/s.

The absolute flow makes 30° with the tangent, so the whirl component V_w2 = V_r2 cot 30° = 9.09 × 1.732 = 15.75 m/s. The exit is radial, so V_w1 = 0.

By the Euler turbine equation, P = ρQ(u₂V_w2 − u₁V_w1) = 1000 × 4.0 × (10.26 × 15.75) = 646 100 W, about 646 kW. (Since the flow leaves radially inward, the torque is taken on the inlet angular momentum alone.)

Diagram: Inlet and outlet velocity triangles of the idealised radial inward-flow turbine

(d) Factor of safety against shear failure

Shear strength τ_f = c′ + σ′ tan φ′ at z = 7 m; induced shear τ = 50 kN/m².

Water table far below: σ′ = γz = 18 × 7 = 126 kN/m². τ_f = 12 + 126 × tan 30° = 12 + 72.75 = 84.75 kN/m². F₁ = 84.75/50 = 1.70.

Water table at ground level: σ′ = (18 − 9.81) × 7 = 57.33 kN/m². τ_f = 12 + 57.33 × 0.5774 = 45.10 kN/m². F₂ = 45.10/50 = 0.90.

Reduction = (1.695 − 0.902)/1.695 × 100 = 46.8%. Since F₂ is below 1, the soil would fail at this point once the water table rises to ground level.

(e) Factor of safety against piping; gravel thickness

Void ratio e = n/(1 − n) = 0.35/0.65 = 0.538. Critical hydraulic gradient i_c = (G − 1)/(1 + e) = 1.65/1.538 = 1.0725. Actual gradient i = 4.0/3.0 = 1.333.

F = i_c/i = 1.0725/1.333 = 0.80, which is below 1, so the soil boils.

With gravel of thickness d (same submerged unit weight γ′ = i_c γ_w = 10.52 kN/m³ and no head loss in it), the excess uplift at the soil base is γ_w H = 9.81 × 4 = 39.24 kN/m². F = γ′(L + d)/(γ_w H) = 2 gives L + d = 2 × 39.24/10.52 = 7.46 m.

d = 7.46 − 3.0 = 4.46 m of gravel.

How this answer will be evaluated

What earns full marks: All parts show complete working with correct formulas, units, and final values with proper interpretation.

Evaluation rubric

Each sub-part is marked on its own, against the marks and word limit printed on the paper.

  1. (a) Find the model flow rate and the prototype force for the 1:10 weir model. 10 marks

    calculate— given → formula → substitution → result with units → interpretation

    Must cover

    • Free-surface flow, so Froude similarity
    • Q_r = L_r^(5/2); Q_m = 9.49 L/s
    • F_r = L_r³ = 1/1000 for the same fluid
    • F_p = 15 kN

    Loses marks

    • Using velocity or length ratio instead of the discharge ratio.

    Earns more

    • Marks for scale ratios and both results with units.

    Extra mark

    • Stating why Froude rather than Reynolds similarity applies.
  2. (b) Find the lift coefficient of the wing. 10 marks

    calculate— given → formula → substitution → result with units → interpretation

    Must cover

    • V = 69.44 m/s and S = 13 m²
    • Resultant R = L√10/3 from D = L/3, so L = 18 974 N
    • Dynamic force ½ρV²S = 37 616 N
    • C_L = 0.504

    Loses marks

    • Taking 20 kN as the lift.

    Earns more

    • Marks for correct resolution of the total force and final C_L.

    Extra mark

    • Noting that the 20 kN is the resultant, not the lift.
  3. (c) Find the theoretical power developed by the radial turbine. 10 marks

    calculate— given → formula → substitution → result with units → interpretation

    Must cover

    • u₂ = 10.26 m/s and V_r2 = 9.09 m/s
    • V_w2 = V_r2 cot 30° = 15.75 m/s; V_w1 = 0
    • Euler equation P = ρQ u₂V_w2
    • P ≈ 646 kW

    Loses marks

    • Using the 30° from the radial direction or ignoring the blade width in the inlet area.

    Earns more

    • Marks for the velocity triangle, Euler equation and result.

    Extra mark

    • Explaining that the radial exit gives no outlet whirl.
  4. (d) Find the factor of safety against shear failure and its percentage reduction when the water table rises. 10 marks

    calculate— given → formula → substitution → result with units → interpretation

    Must cover

    • τ_f = c′ + σ′ tan φ′
    • Dry case: σ′ = 126, τ_f = 84.75, F = 1.70
    • Submerged case: σ′ = 57.33, τ_f = 45.10, F = 0.90
    • Reduction = 46.8%

    Loses marks

    • Using total stress in the submerged case.

    Earns more

    • Marks for both cases and the percentage.

    Extra mark

    • Commenting that F < 1 means failure.
  5. (e) Find the factor of safety against piping and the gravel depth needed for F = 2. 10 marks

    calculate— given → formula → substitution → result with units → interpretation

    Must cover

    • e = 0.538 and i_c = 1.0725
    • i = 4/3, F = 0.80 (unsafe)
    • F = γ′(L + d)/(γ_w H) = 2
    • d ≈ 4.46 m

    Loses marks

    • Using total unit weight instead of submerged unit weight.

    Earns more

    • Marks for i_c, F and gravel depth.

    Extra mark

    • Noting that the gravel acts as a surcharge filter.
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Choose the method, then carry it through to a final answer. Identifying what kind of problem this is and why that method applies is the first thing marked; a correct figure arrived at invisibly earns almost nothing.

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