(a) (i) Observations were made with a theodolite from the points A and B of baseline AB having horizontal length of 200 m to the two inaccessible points P and Q at the same side of the line AB.

The following observations are done :

∠PAB = 80°, ∠QBA

Question

(a) (i) Observations were made with a theodolite from the points A and B of baseline AB having horizontal length of 200 m to the two inaccessible points P and Q at the same side of the line AB.

The following observations are done :

∠PAB = 80°, ∠QBA = 90°, ∠QAB = 50°, ∠PBA = 45°

The angle of elevation of P from A = 20°

The angle of elevation of Q from A = 12°

Calculate the horizontal distance PQ and difference in elevation between P and Q.

10

(ii)

A pair of overlapping vertical photographs was taken with an aerial camera of focal length 150 mm from an altitude of 3000 m above datum. The mean principal base measured is equal to 80 mm. In the common overlap, a flagpole with its base 100 m above datum is observed. What will be the height of the flagpole if the parallax difference between the top and bottom point of the flagpole is 4·6 mm? Also, find the scale of photograph at datum and the distance covered by the aircraft between two successive exposures.

10

(b) (i) Explain the required properties of a good ballast material for railway track. Explain with reasons which ballast material is best suitable for high-speed railway track.

10

(ii) Calculate the minimum depth of ballast required for a broad gauge railway track having sleeper density of M+6, length of rail of 13·0 m and width of sleeper of 25 cm.

5

(c) Briefly discuss the effect of the following on the properties of mortar used for construction work :

(i) Alkali water and seawater

(ii) Low temperature

(iii) Sand and water

15

UPSC Answer Check · Accepted Answer

(a)(i) 
Take A = (0, 0), B = (200, 0) m, with P and Q on the same side of AB.

For P, in ΔABP: ∠PAB = 80°, ∠PBA = 45°, so ∠APB = 180° − 80° − 45° = 55°.
By sine rule: AP/sin45° = AB/sin55°.
AP = 200 sin45°/sin55° = 172.644 m.
Hence P = (AP cos80°, AP sin80°) = (29.979, 170.021) m.

For Q, in ΔABQ: ∠QAB = 50°, ∠QBA = 90°. Since AB is adjacent to 50° in the right triangle,
AQ = AB/cos50° = 200/cos50° = 311.145 m.
Q = (AQ cos50°, AQ sin50°) = (200.000, 238.351) m.

Therefore, Δx = 200.000 − 29.979 = 170.021 m and Δy = 238.351 − 170.021 = 68.330 m.
PQ = √(170.021² + 68.330²) = 183.238 m.

Elevation of P above A = AP tan20° = 172.644 × 0.363970 = 62.837 m.
Elevation of Q above A = AQ tan12° = 311.145 × 0.212557 = 66.136 m.
Difference = 62.837 − 66.136 = −3.299 m.
Thus Q is higher than P by 3.299 m ≈ 3.30 m.

(a)(ii) 
Using the vertical-photograph parallax equation, with f = 150 mm, H = 3000 m, mean principal base b = 80 mm, base elevation h_b = 100 m, and parallax difference Δp = 4.6 mm.

Scale at datum = f/H = 150 mm / 3000 m = 150 / 3,000,000 = 1:20,000.

Distance between successive exposures:
B = bH/f = 80 mm × 3000 m / 150 mm = 1600 m.

Parallax at flagpole base:
p_b = bH/(H − h_b) = 80 × 3000/(3000 − 100) = 82.7586 mm.

Height of flagpole:
h = Δp(H − h_b)/(p_b + Δp)
= 4.6 × 2900/(82.7586 + 4.6)
= 13340/87.3586
= 152.704 m ≈ 152.70 m.

This is valid for vertical photographs, nearly level ground, and small parallax difference.

(b)(i) 
A good ballast material for railway track should be:
strong, hard, tough and wear-resistant to resist crushing and abrasion under moving loads;
angular or cubical with rough surfaces to interlock and prevent lateral movement;
well-graded and free from dust, clay, loam, organic matter and soft particles;
free-draining so that water does not accumulate and weaken the track;
weather-resistant and chemically inert, not affected by rain, frost or salts;
elastic and resilient enough to absorb shock and vibration;
economical and easily available.

For high-speed railway track, crushed granite is generally best suitable. It has high compressive strength and toughness, low water absorption, good angularity, excellent drainage, and high resistance to attrition under repeated high-speed loads. Crushed basalt or quartzite may be used where granite is unavailable, but granite is preferred because it maintains gradation, interlock and drainage for a longer time.

(b)(ii) 
Sleeper density M + 6 for rail length 13.0 m:
Number of sleepers = 13 + 6 = 19 per rail length.
Sleeper spacing S = 1300/(19 − 1) = 1300/18 = 72.22 cm.

Width of sleeper b = 25 cm.
Gap between sleepers = S − b = 72.22 − 25 = 47.22 cm.

Minimum depth of ballast is given by load-spread criterion:
D = (S − b)/2 × cot θ.
Taking the angle of repose of ballast θ = 30°, cot30° = √3 = 1.732.
D = (47.22/2) × 1.732 = 23.61 × 1.732 = 40.90 cm.

Exact: D = 425√3/18 cm = 40.9 cm ≈ 0.409 m.

(c)(i) 
Alkali water and seawater contain sodium, potassium, chloride and sulfate salts. They may accelerate initial setting and give early strength, but later cause efflorescence, dampness, sulfate expansion, cracking, alkali–aggregate reaction, reduced bond, and lower long-term strength and durability. Chlorides promote corrosion of reinforcement. Therefore such water is prohibited for reinforced, prestressed or important mortar work; for plain mortar it may be used only after testing and with precautions.

(c)(ii) 
Low temperature slows cement hydration. Setting and hardening are retarded, early strength is low, and if mixing water freezes before setting, ice expansion causes cracks, loss of strength, bond and durability. Protective measures such as heated water/aggregates, accelerators, antifreeze admixtures, insulation and avoiding frozen subgrade are required.

(c)(iii) 
Sand affects gradation, water demand, strength, shrinkage and bond. Well-graded, clean, angular, coarse-to-medium sand gives dense, strong, durable mortar. Fine sand or excess silt/clay/organic matter increases water demand, shrinkage and permeability and reduces strength and bond.
Water quantity controls the water/cement ratio. More water improves workability but lowers strength, increases shrinkage, permeability and efflorescence. Water should be potable and free from harmful acids, alkalis, salts, organic matter and oil so that setting and durability are not harmed.

(a) (i) Observations were made with a theodolite from the points A and B of baseline AB having horizontal length of 200 m to the…

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