Railway, airport and bridge basics

In short

This topic covers the three non-highway transport structures in the Civil syllabus: the railway permanent way (gauge, rails, sleepers, ballast, geometry, points and crossings), airport engineering (runway orientation, length, geometry and pavement classification) and bridge engineering basics (components, classification, loading standards and hydraulic design). It matters because these are capital-intensive public assets designed to codes written by Indian Railways/RDSO, DGCA/AAI (following ICAO Annex 14) and the Indian Roads Congress. It is examined in GATE Civil (Transportation Engineering section), in university Transportation Engineering-II papers, and in interviews for RRB/RVNL/RITES/IRCON, AAI junior executive and NHAI/L&T/AFCONS/Tata Projects bridge roles. The arithmetic is formula-heavy and predictable, so marks here are easy to secure.

Explained

Permanent way: components and gauge

• Permanent way = rails + sleepers + ballast + fastenings on a prepared formation; it guides the wheels and transfers load to the subgrade, spreading a ~10 t wheel load to a few kN/m² on the formation.
• Gauge = clear distance between inner faces of the two rails. Broad Gauge 1,676 mm, Metre Gauge 1,000 mm, Narrow Gauge 762 mm and 610 mm. *Project Unigauge* has converted almost all MG/NG routes to BG; new lines are BG only.
• Rails: flat-footed (Vignoles) rails are standard. Sections 60 kg/m (90 UTS) on high-density and 52 kg/m on other routes; designated by mass per metre. Rail length 13 m (BG) in fish-plated track, welded into long rails.
• Sleepers: functions - hold gauge, hold rails, distribute load, provide elasticity, allow track alignment. Types: PSC (pre-stressed concrete, now standard), steel trough, cast iron pot, wooden. Sleeper density = M + x (number of sleepers per rail length = rail length in metres + x); BG main line is M + 7 (1,660 sleepers/km with 13 m rails).
• Ballast: 25-65 mm broken stone; functions - transfer and spread load, drainage, lateral/longitudinal stability, elasticity. Minimum cushion under sleeper ~250 mm, 300 mm for LWR on BG.
• Fastenings: fish plates and bolts (fish-plated track), bearing plates, elastic rail clips (ERC) with grooved rubber pads and GFN liners on PSC sleepers, anti-creep anchors, keys and chairs (older).
• Coning of wheels (1 in 20) and tilting of rails (1 in 20) reduce lateral oscillation and let the wheelset negotiate curves; the wheel flange is the last line of defence, not the normal guiding surface.
• Formation: BG single-line embankment top width about 6.85 m (double line ~13.1 m); side slopes 2:1 in embankment, blanket layer of granular material over weak subgrade.
• Creep = longitudinal movement of rails relative to sleepers, caused by wave action, braking/traction and temperature; measured with creep posts, resisted by anchors, elastic fastenings and adequate ballast.
• LWR/CWR: a long welded rail eliminates joints (the weakest part of track). The central portion does not move; only the breathing length at each end expands, so rail stresses build up instead of expansion - hence switch expansion joints (SEJ), limits on minimum curve radius, destressing at a specified rail temperature, and the IR *LWR Manual* governs.

Railway geometry, gradients and points

• Superelevation (cant) raises the outer rail so the resultant of weight and centrifugal force is perpendicular to the plane of the rails. Equilibrium cant is provided for the weighted average (equilibrium) speed, not the maximum speed.
• Cant deficiency = cant needed for the fastest train − cant actually provided; limit 75 mm on BG (100 mm for nominated stock with CRS approval). Cant excess (slow goods trains) limited to 75 mm. Maximum cant on BG 165 mm (up to 185 mm on Group A/B routes with approval).
• Negative superelevation arises on a branch line turning out in the opposite direction from a main-line curve; the main line governs the cant, so the branch speed is restricted.
• Degree of curve D = 1750/R (BG, 30.5 m chord). Maximum ~10° on BG, 16° on MG. Transition curve (cubic parabola on IR) introduces curvature and cant gradually; cant gradient normally not steeper than 1 in 720, rate of change of cant/cant deficiency 35 mm/s.
• Gradients: *ruling gradient* is the steepest gradient a locomotive must haul its full load on (1 in 150-250 in plains, 1 in 100-150 in hills); *momentum gradient* can be steeper because the train arrives with kinetic energy; *pusher/helper gradient* needs a banking locomotive; *station yards* are nearly level (flatter than 1 in 400).
• Grade compensation on curves: reduce the ruling gradient by 0.04% per degree of curve on BG (0.03% MG, 0.02% NG) to offset extra curve resistance.
• Points and crossings let a train move from one track to another. Parts: stock rail, tongue (switch) rail, heel and toe of switch, lead rails, crossing (V-shaped - point rail + splice rail forming the nose), wing rails, check rails.
• Crossing number N = cot α (α = crossing angle). Common turnouts 1 in 8.5, 1 in 12, 1 in 16 and 1 in 20 - the flatter the crossing, the longer the turnout and the higher the permissible speed through it. Turnout geometry is computed by the *centre-line* or *Cole's* method.
• Track junctions and layouts: turnout, crossover, diamond crossing, scissors crossover, triangle, turntable, gauntlet track; station types - wayside, junction, terminal; yards - goods, marshalling, locomotive.
• Signalling and safety: absolute block and automatic block systems; semaphore (legacy) and multiple-aspect colour-light signals; panel, route-relay and electronic interlocking; Kavach, the indigenous Automatic Train Protection system, is being progressively installed on IR routes (coverage is still expanding, so quote it as "under deployment" rather than complete).
• Legal/institutional frame: the Railways Act, 1989 (amended 2024, which subsumed the Railway Board Act, 1905), the Commissioner of Railway Safety under the Ministry of Civil Aviation sanctions new lines and speed raising, and RDSO issues the technical specifications, schedules of dimensions and maintenance manuals. Track maintenance is mechanised (tampers, ballast cleaning machines, rail grinders).

Airport engineering: site, orientation and runway length

• Aircraft characteristics that drive design: all-up weight (pavement), wheel/undercarriage configuration (pavement thickness), wing span and outer main-gear wheel span (taxiway and clearance widths), ARFL - aeroplane reference field length (runway length), turning radius and approach speed.
• Aerodrome reference code (ICAO Annex 14): a number 1-4 by ARFL (1: <800 m, 2: 800-1200, 3: 1200-1800, 4: ≥1800 m) and a letter A-F by wing span/outer main-gear span (A smallest; E = B777/B747; F = A380). Geometric standards follow from the code - e.g. code 4E runway width 45 m.
• Runway orientation is fixed by the wind rose: the runway is aligned along the direction of the prevailing wind so that the crosswind component stays within limits for at least 95% of the time (usability factor). ICAO crosswind limits are 37 km/h (20 kt) for ARFL ≥1500 m, 24 km/h (13 kt) for 1200-1500 m and 19 km/h (10 kt) for <1500 m shorter fields; some texts/FAA quote lower figures, so state the source used.
• Wind rose types: Type I (percentage of time wind blows from each direction and speed range - gives the best direction); Type II (also allows direct reading of crosswind coverage with a transparent template).
• Basic runway length is from aircraft performance: it must satisfy normal landing, normal take-off and the engine-failure cases (accelerate-stop distance and take-off distance with one engine out).
• Corrections to basic length (ICAO):
• Elevation: +7% per 300 m rise above MSL (thinner air, longer take-off run).
• Temperature: +1% per °C by which the airport reference temperature exceeds the standard atmosphere temperature at that elevation (ISA: 15 − 0.0065 h °C, h in m). Applied on the elevation-corrected length.
• Gradient: +20% per 1% effective gradient, applied on the elevation- and temperature-corrected length.
• If the elevation + temperature corrections together exceed 35%, the length must be checked by a specific operational study, not by the rule of thumb.
• Declared distances: TORA (take-off run available), TODA = TORA + clearway, ASDA = TORA + stopway, LDA (landing distance available). A clearway is an obstacle-free area beyond the runway (air only, not load-bearing); a stopway is paved and load-bearing for an abandoned take-off.

Airport geometric design, zoning and pavements

• Runway gradients: maximum longitudinal 1.25-1.5% for code 3/4 (2% for code 1/2); effective gradient (max elevation difference ÷ runway length) kept within about 1%; rate of change of gradient limited (≈0.1% per 30 m for code 3/4); transverse (camber) 1-1.5%.
• Runway strip, shoulders and RESA: the graded strip extends beyond and either side of the runway (e.g. 150 m each side of the centre line for instrument code 3/4); RESA (runway end safety area) beyond the strip end reduces overrun damage.
• Taxiways are designed for lower speed: smaller radii, longitudinal gradient ≤1.5%, fillets at junctions, and exit (rapid) taxiways at about 30° to free the runway quickly and raise capacity. Holding bays/aprons near the runway end let aircraft wait and allow overtaking. Apron holds gates (nose-in parking with aerobridges), with taxilanes and clearances per the code letter.
• Imaginary surfaces / obstruction clearance: approach surface, take-off climb surface, transitional surface, inner horizontal surface, conical surface and the balked-landing surface; objects penetrating them must be removed, lowered or lit and marked. Zoning laws around Indian airports restrict building heights (NOC from AAI), and the obstacle limitation surfaces define the protected airspace.
• Visual aids: runway designation markings (magnetic azimuth ÷ 10, e.g. runway 09/27), centre line, threshold and touchdown-zone markings; approach lighting system, runway edge/centre-line/threshold/end lights, PAPI for glide-path guidance, taxiway blue edge and green centre-line lights, wind direction indicator; ILS CAT I/II/III for low-visibility operations (several Indian airports including Delhi have CAT III-B capability).
• Pavements: designed for very high single-wheel loads and channelised traffic - flexible pavements by the CBR/FAA method (now FAARFIELD software), rigid pavements by Westergaard-based edge-stress analysis; grooved surfaces for skid resistance and drainage.
• Pavement strength reporting: the long-standing ACN-PCN system (aircraft/pavement classification number, with subgrade category A-D, tyre pressure code and evaluation method) has been replaced in ICAO practice by the ACR-PCR method from late 2024; both appear in documents during the transition, so check the AIP entry.
• Regulators: DGCA (licensing of aerodromes, CAR series), AAI (operator/ANS provider), BCAS (security), with ICAO Annex 14 as the technical parent. Capacity is governed by runway configuration, aircraft mix and ATC separation.

Bridges: components and classification

• Components: *superstructure* - deck slab, girders/arch/truss, wearing coat, railing/crash barrier, expansion joints; *substructure* - piers, abutments, wing/return walls, pier caps; *foundations* - open, pile, well (caisson); plus bearings, approach slabs, river training works (guide bunds, spurs, aprons).
• By span (IRC): culvert up to 6 m, minor bridge 6-60 m, major bridge above 60 m, long-span above 120 m.
• By structural form: slab, slab-and-girder (T-beam), box girder (often prestressed, cast-in-situ or segmental), arch, truss, cable-stayed, suspension, extradosed; balanced cantilever for medium-long spans.
• By position of deck: deck type (traffic on top), through type, half-through (semi-through, pony truss).
• By water level: high-level bridge (above HFL), submersible bridge/causeway (overtopped in floods), vented causeway, movable bridges (lift, bascule, swing - e.g. the new Pamban vertical-lift sea bridge).
• By material/degree of fixity: masonry, RCC, prestressed concrete, steel, composite (steel girder + RC deck, IRC:22); simply supported, continuous, cantilever, integral.
• Bearings: elastomeric pads (IRC:83 Part II), POT-PTFE and spherical bearings (Part III) for large movements, rocker-roller (steel). Expansion joints: filler, slab seal, strip seal, modular.

Bridge loading, hydraulics and codes

• IRC:6 (loads and load combinations) prescribes Class 70R (tracked 70 t / wheeled / bogie), Class AA (tracked 70 t, wheeled 40 t) for urban and industrial routes, Class A (the standard train of wheeled vehicles) and the lighter Class B; the SV (special vehicle) loading of 385 t applies to designated corridors. Number of lanes loaded with reduction factors for 3+ lanes; carriageway 3.5 m per lane.
• Other loads: dead load, SIDL, impact, longitudinal forces (braking, bearing friction), centrifugal force, wind, seismic (IRC:SP:114), water current and hydrodynamic force, buoyancy, earth pressure, temperature, shrinkage/creep, collision on piers, erection loads.
• Impact factor (IRC:6): RC bridges 4.5/(6+L); steel bridges 9/(13.5+L), L = span in m; tracked vehicles 25% up to 5 m span reducing to 10% at 9 m and above. For railway bridges the coefficient of dynamic augment CDA = 0.15 + 8/(6+L) under the *IRS Bridge Rules* (BG loading standards: Modified BG Loading-1987 and the 25 t loading-2008).
• Main IRC codes: IRC:5 (general features of design), IRC:6 (loads), IRC:112 (concrete road bridges, limit state - supersedes the old working-stress IRC:21), IRC:22 (composite), IRC:24 (steel), IRC:78 (foundations and substructure), IRC:83 (bearings), IRC:SP:13 (small bridges and culverts), IRC:SP:114 (seismic), IRC:SP:35/40 (inspection and rehabilitation). Railway bridges follow the IRS codes and RDSO drawings.
• Hydraulic design: estimate design discharge (rational method, Dickens/Ryves/Inglis empirical formulas, flood-frequency or area-velocity for large catchments, usually 50-100 year return period); fix the linear waterway (Lacey regime width); compute normal and maximum scour depth; set foundation depth below maximum scour; provide afflux and vertical clearance/freeboard (roughly 0.6-1.5 m depending on discharge, per IRC:5/SP:13).
• Site selection: straight reach, stable and well-defined channel, firm non-scourable bed, narrowest section, square crossing, no confluence/meander immediately upstream, good approaches.
• Inspection and management: routine and detailed inspections, bridge inventory and rating under the Indian Bridge Management System (IBMS) of MoRTH; load testing, strengthening, bearing and joint replacement.

Key points

• Gauge - inner face to inner face of rails; BG = 1,676 mm, and Unigauge has made BG the Indian standard.
• Sleeper density - M + x sleepers per rail length; BG main line M + 7.
• Coning of wheels (1 in 20) - gives self-centring and easier curve negotiation; the flange is only a safeguard.
• Equilibrium cant - provided for the equilibrium (weighted average) speed, not the maximum speed.
• Cant deficiency - limited to 75 mm on BG; maximum cant 165 mm (185 mm with approval).
• Degree of curve - D = 1750/R on BG; BG maximum about 10°.
• Grade compensation - 0.04% per degree of curve on BG, deducted from the ruling gradient.
• Creep - longitudinal rail movement from wave action, braking and temperature; controlled by anchors and elastic fastenings.
• LWR - stress replaces expansion; only the breathing lengths move, SEJs provided at the ends.
• Crossing number - N = cot α; a 1 in 12 turnout is flatter and faster than 1 in 8.5.
• Wind rose - fixes runway orientation for 95% usability within the permissible crosswind component.
• Runway length corrections - elevation +7%/300 m, temperature +1%/°C above ISA, gradient +20%/1% effective gradient; elevation + temperature capped at 35%.
• Aerodrome reference code - number (ARFL) + letter (wing span/gear span) drives every geometric standard.
• Declared distances - TORA, TODA (+clearway), ASDA (+stopway), LDA.
• ACR-PCR - ICAO's current pavement strength reporting method, replacing ACN-PCN from late 2024.
• Impact factor - 4.5/(6+L) for RC and 9/(13.5+L) for steel road bridges (IRC:6); CDA = 0.15 + 8/(6+L) for railway bridges.
• Lacey regime width - W = 4.8√Q gives the economical linear waterway for an alluvial river.
• Maximum scour at a pier - about twice the Lacey normal scour depth; foundations are taken well below it (IRC:78).

Formulas and examples

Example 1 - Cant, cant deficiency and transition length (BG). A BG curve of R = 700 m carries a maximum sanctioned speed of 100 km/h with an equilibrium speed of 80 km/h. Take G = 1.676 m.

• Cant for the equilibrium speed: e = 1.676 × 80² / (127 × 700) = 10,726.4 / 88,900 = 0.1207 m → C_a = 121 mm (within the 165 mm limit).
• Cant theoretically needed at 100 km/h: 1.676 × 100² / 88,900 = 0.1885 m = 188.5 mm.
• Cant deficiency C_d = 188.5 − 121 = 67.5 mm < 75 mm → acceptable.
• Degree of curve D = 1750/700 = 2.5°.
• Transition length: 0.008 × 121 × 100 = 96.8 m; 0.008 × 67.5 × 100 = 54.0 m; 0.72 × 121 = 87.1 m → governing L ≈ 97 m, provide 100 m.
• Check by Martin's formula: V = 4.35 √(700 − 67) = 4.35 × 25.16 = 109 km/h > 100 km/h, so the curve is safe for the sanctioned speed.

Example 2 - Grade compensation. A BG hill section has a ruling gradient of 1 in 100 (1.0%) and a 4° curve on that grade.

• Compensation = 0.04 × 4 = 0.16%.
• Permissible gradient on the curve = 1.00 − 0.16 = 0.84%, i.e. about 1 in 119 - the grade must be eased through the curve.

Example 3 - Runway length and crosswind. An aircraft needs an ARFL of 2,000 m. The airport is at 300 m elevation, the airport reference temperature is 32 °C, and the effective gradient is 0.5%.

• Elevation correction = 2,000 × 0.07 × (300/300) = 140 m → 2,140 m.
• ISA temperature at 300 m = 15 − 0.0065 × 300 = 13.05 °C; rise ΔT = 32 − 13.05 = 18.95 °C.
• Temperature correction = 2,140 × 0.1895 = 405.5 m → 2,545.5 m.
• Check: (2,545.5 − 2,000)/2,000 = 27.3% < 35% → the rule of thumb may be used.
• Gradient correction = 2,545.5 × (0.20 × 0.5) = 2,545.5 × 0.10 = 254.6 m → corrected runway length ≈ 2,800 m.
• Crosswind check: a 45 km/h wind at 40° to the centre line gives crosswind = 45 sin 40° = 28.9 km/h, within the 37 km/h limit for a code 4 runway; headwind = 45 cos 40° = 34.5 km/h.

Example 4 - Waterway, scour and foundation level. A river has a design discharge Q = 1,500 m³/s and silt factor f = 1.0; HFL is at RL 100.0 m.

• Lacey regime width W = 4.8 √1,500 = 4.8 × 38.73 = 185.9 m (provide ~186 m of clear waterway).
• If the waterway is constricted to 150 m, q = 1,500/150 = 10 m³/s per m → d_sm = 1.34 (10²/1.0)^1/3 = 1.34 × 4.642 = 6.22 m below HFL.
• Maximum scour at a pier ≈ 2 × d_sm = 12.4 m → scour level RL 100.0 − 12.4 = RL 87.6 m.
• A well foundation would be sunk below this with a grip of about one-third of the maximum scour depth (IRC:78); an open foundation needs a minimum embedment below the scour level, so founding level ≈ RL 83-84 m - the exact grip must be taken from IRC:78 and bearing-capacity checks.

Example 5 - Impact factor and economic span. (a) A simply supported RC slab bridge of 10 m span: impact factor = 4.5/(6 + 10) = 0.281 (28.1%), so a 554 kN Class A train axle group is designed for 554 × 1.281 ≈ 710 kN. A 10 m steel girder bridge: 9/(13.5 + 10) = 0.383 (38.3%) - steel bridges attract higher impact. A 10 m railway bridge: CDA = 0.15 + 8/(6 + 10) = 0.65. (b) One pier with its foundation costs ₹1.20 crore; the superstructure costs ₹30,000 per metre per metre of span (so a span L costs 30,000 L² rupees). L = √(1.2 × 10⁷ / 3 × 10⁴) = √400 = 20 m economic span; at that span superstructure cost = 30,000 × 400 = ₹1.20 crore, equal to the pier cost, as the theory requires.

Where it is used

• Railway construction and maintenance: Indian Railways P.Way engineers, RVNL, IRCON, RITES, Konkan Railway and contractors on doubling, electrification, Dedicated Freight Corridors and high-speed rail; track machines, LWR destressing, speed-raising certification.
• Metro and urban rail: DMRC, MMRDA, BMRCL and L&T/Afcons packages use the same cant, turnout and crossing theory at tighter radii, with ballastless track.
• Airports: AAI and private operators (GMR Delhi/Hyderabad, Adani Mumbai/Ahmedabad) for runway extensions, parallel taxiways, apron expansion and the UDAN regional airports; obstacle-limitation and height-clearance NOC work.
• Bridge design and construction: NHAI/MoRTH, state PWDs, and contractors on projects such as the Chenab rail arch bridge in J&K, the Bogibeel rail-road bridge in Assam, Atal Setu (MTHL) in Mumbai and the new Pamban vertical-lift sea bridge.
• Job roles: P.Way/bridge section engineer, design engineer (IRC:112/IRS), airport pavement and airside planning engineer, quantity surveyor for substructure and well sinking, bridge inspection and IBMS rating.
• Exams and recruitment: GATE Civil, ESE, RRB JE/SSE and AAI Junior Executive (Civil) papers draw heavily on cant, runway length corrections and IRC loading.

Interview questions

• Q: Why are wheels coned at 1 in 20? A: Coning keeps the wheelset centred on straight track and lets the outer wheel roll on a larger effective diameter on curves, reducing flange wear and lateral oscillation.
• Q: Superelevation is provided for which speed, and why not the maximum? A: For the equilibrium (weighted average) speed, because cant set for the fastest train would give large cant excess and severe inner-rail wear for slow goods trains.
• Q: What is cant deficiency and what is its limit? A: The shortfall between the cant required by the fastest train and the cant provided; on BG it is limited to 75 mm (100 mm for nominated stock with approval) to control passenger discomfort and lateral forces.
• Q: Why do long welded rails not need expansion joints along their length? A: The central length is restrained by ballast and fastenings so temperature change produces stress, not movement; only the breathing lengths at the ends move, and SEJs take that movement.
• Q: What does a 1 in 12 turnout mean, and why is it preferred over 1 in 8.5? A: The crossing spreads 1 unit laterally for 12 along the track (N = cot α), so it is a flatter, longer turnout permitting higher speed through the diverging route.
• Q: How is the orientation of a runway decided? A: From a wind rose, aligning the runway so that the crosswind component stays within the permissible limit for at least 95% of the time; a crosswind runway is added if one direction cannot achieve it.
• Q: Why does a hot, high-altitude airport need a longer runway? A: Lower air density reduces engine thrust and lift, so the take-off run increases - hence +7% per 300 m of elevation and +1% per °C above the standard temperature.
• Q: What is the difference between a clearway and a stopway? A: A clearway is an obstacle-free area beyond the runway used only for the initial climb (it adds to TODA); a stopway is paved and load-bearing so an aborted take-off can stop on it (it adds to ASDA).
• Q: What is ACN-PCN and what has replaced it? A: It was ICAO's system of matching aircraft load (ACN) with pavement strength (PCN); ICAO has moved to the ACR-PCR method from late 2024, though PCN values still appear during the transition.
• Q: Name the live load classes in IRC:6 and when Class 70R is used. A: Class 70R, Class AA, Class A and Class B, plus SV 385 t; Class 70R (or AA) is used on national and state highways and in industrial/urban areas where heavy vehicles ply, while Class A is the normal standard.
• Q: Why is the impact factor higher for a steel bridge than an RC bridge of the same span? A: Steel bridges are lighter and more flexible, so the same moving load produces larger dynamic amplification - hence 9/(13.5+L) against 4.5/(6+L).
• Q: How deep do you take a bridge foundation in a sandy river bed? A: Compute the Lacey scour depth, take the maximum scour at a pier as about twice the normal value, and found below that with the grip or embedment required by IRC:78, also checking bearing capacity and settlement.
• Q: What is a submersible bridge and when is it justified? A: A bridge designed to be overtopped in high floods, used on low-traffic roads across seasonal rivers where a high-level bridge is not economical; it needs scour protection and traffic closure during floods.

Common mistakes

• Using the maximum speed instead of the equilibrium speed when computing the cant to be provided, then wrongly concluding the cant exceeds the limit.
• Mixing up the gauge values in e = GV²/(127R) - and forgetting that e comes out in metres, not millimetres.
• Applying the runway temperature correction to the basic length instead of to the elevation-corrected length, or applying the gradient correction before the other two.
• Taking the standard temperature as 15 °C regardless of elevation, instead of 15 − 0.0065 h.
• Using the plain crossing angle as the crossing number - the number is cot α, so a *larger* number means a *flatter*, faster turnout.
• Quoting the withdrawn working-stress code for concrete road bridges; design is now by limit state to IRC:112.
• Confusing normal scour depth with maximum scour at a pier (about twice), and ignoring the extra foundation grip required below it.
• Treating a stopway as usable for take-off distance, or adding a clearway to ASDA.

Related topics

Highway engineering and pavement design · Pavement materials and testing · Traffic engineering · Urban transport planning and metro systems · Tunnel engineering · Foundation engineering (well and pile foundations) · Structural analysis and design of concrete and steel structures