Sleepers

IntroductionSleepers are the transverse ties that are laid to support the rails. They have an important role in the track as they transmit the wheel load from the rails to the ballast. Several types of sleepers are in use on Indian Railways. The characteristics of these sleepers and their suitability with respect to load conditions are described in this chapter.

Functions and Requirements of SleepersThe main functions of sleepers are as follows.

1. Holding the rails in their correct gauge and alignment2. Giving a firm and even support to the rails3. Transferring the load evenly from the rails to a wider area of the ballast4. Acting as an elastic medium between the rails and the ballast to absorb the blows and

vibrations caused by moving loads5. Providing longitudinal and lateral stability to the permanent way6. Providing the means to rectify the track geometry during their service life.7. To support the rails at proper level in straight tracks and at proper super elevation on

curvesApart from performing these functions the ideal sleeper should normally fulfil the following requirements.

1. The initial as well as maintenance cost should be minimum.2. The weight of the sleeper should be moderate so that it is convenient to handle.3. The designs of the sleeper and the fastenings should be such that it is possible to fix

and remove the rails easily.4. The sleeper should have sufficient bearing area so that the ballast under it is not

crushed.5. The sleeper should be such that it is possible to maintain and adjust the gauge

properly.6. The material of the sleeper and its design should be such that it does not break or get

damaged during packing.7. The design of the sleeper should be such that it is possible to have track circuiting.8. The sleeper should be capable of resisting vibrations and shocks caused by the

passage of fast moving trains.9. The sleeper should have anti-sabotage and anti-theft features.10. Fittings of the sleepers should be such that they can be easily adjusted during

maintenance operations such as easy lifting, packing, removal and replacement.

Sleeper Density and Spacing of SleepersSleeper density is the number of sleepers per rail length. It is specified as M + x or N + x, where M or N is the length of the rail in metres and x is a number that varies according to factors such as (a) axle load and speed, (b) type and section of rails, (c) type and strength of the sleepers, (d) type of ballast and ballast cushion, and (e) nature of formation. If the sleeper density is M + 7 on a broad gauge route and the length of the rail is 13 m, it means that 13 + 7 = 20 sleepers will be used per rail on that route. The number of sleepers in a track can also be specified by indicating the number of sleepers per kilometre of the track. For example, 1540 sleepers/km. This specification becomes more relevant particularly in cases where rails are welded and the length of the rail does not have much bearing on the number of sleepers required. This system of specifying the number of sleepers per kilometre exists in many foreign countries and is now being adopted by Indian Railways as well. The spacing of

sleepers is fixed depending upon the sleeper density. Spacing is not kept uniform throughout the rail length. It is closer near the joints because of the weakness of the joints and impact of moving loads on them. There is, however, a limitation to the close spacing of the sleepers, as enough space is required for working the beaters that are used to pack the joint sleepers.

Types of sleepers

1. Wooden sleepers2. Metal sleepers

a. Cast iron sleepersb. Steel Sleepers

3. Concrete sleepersa. Reinforced concrete sleepersb. Prestressed concrete sleepers

Timber or wooden sleepers

Advantages• Easy to manufacture and handling.• They have long life of 10-12 years depending upon the climate, condition, rain,

intensity, nature of traffic, quality of wood etc• Good insulators and hence good for track circuited railway tracks• Suitable for salty regions and coastal areas• They are not badly damaged in case of derailment• They are not corroded• Suitable for track circuited area.• Can be used with or without ballast.• Suitable for bridges, Points/Crossings.• Suitable for gauntleted track.• Alignment can be easily corrected.

Disadvantages1. Lesser life compared to other sleepers.2. Liable to damage by beater packing.3. Difficult to maintain the gauge.4. Susceptible to fire hazards.5. They do not resist creep6. They are affected by dry and wet rot7. Become expensive day by day

Steel Sleepers

About 27% of the track on Indian Railways is laid on steel sleepers. The increasing shortage of timber in the country and other economical factors are mainly responsible for the use of steel sleepers in India. Steel sleepers have the following main advantages/disadvantages over wooden sleepers.

Advantages

• Have a useful life of 20-25 years.• Free from decay and are not attacked by vermins• Connection between rail and sleeper is stronger• Connection between rail and sleeper is simple• More attention is not required after laying• Having better lateral rigidity• Good scrap value• Suitable for high speeds and load• Easy to handle• Good resistance against creep

Disadvantages• Liable to corrosion by moisture and should not because in salty regions• Bad insulators and hence cannot be used in track circuited regions• Should not be laid with all types of Ballast• Very costly• Can be badly damaged under derailments• The rail seat is weaker• Not a good shock absorber as there is no cushion between rail foot and ballast

Cast iron Sleepers

Advantages1. Less corrosion2. Less probability of cracking at rail seat3. Easy to manufacture4. Higher scrap value5. Long life upto 50-60 years- High scrape value as they can be remolded6. Can be manufactured locally - Provided sufficient bearing area7. Much stronger at the rail seat - Prevent and check creep of rail8. They are not attacked by vermin

Disadvantages1. Gauge maintenance is difficult as tie bars get bent2. Provides less lateral stability3. Unsuitable for track-circuited lines4. Not very suitable for mechanical maintenance and/or MSP because of rounded bottom5. Susceptible to breakage6. They are prone to corrosion and cannot be used in salty formations and coastal areas -

Not suitable for track circuited portions of railways7. Can badly damage under derailment8. Require a large number of fastening materials - Difficult to handle and may be easily

damaged - Lack of good shock absorber - They are expensive9. Not suitable for high speed route.10. Not fit for track circuited area.

Concrete SleepersThese types of sleepers were promoted because of shortage of good wooden sleepers and need for better design and economy of sleepers on sustainable basis.These sleepers are mainly of two types

1. Reinforced cement concrete sleepers2. Pre-stressed concrete sleepers

Experiments have shown that concrete sleepers are idela material for sleeper for the following reasons

1. They are made of a strong homogeneous material, impervious to effect of moisture and is unaffected by the chemical attack of atmospheric gases or sub –soil salts

2. It is easily moulded to size and shape required by scientific investigation, to withstand the stresses produced by fast and heavy traffic

Advantages and disadvantagesConcrete sleepers have the following advantages and disadvantages.Advantages

• Durable with life range from 40-50 years• They can be produced on large quantities locally by installing a plant• Heavier than all other types thus giving better lateral stability to the track• Good insulators and thus suitable for use in track circuited lines• Efficient in controlling creep - They are not attacked by corrosion• Free from attacks of vermin and decay, suitable for all types of soils• Most suitable for welded tracks- Prevent buckling more efficiently• Initial cost is high but proves to be economical in long run• Effectively and strongly hold the track to gauge• Inflammable and fire resistant

Disadvantages(a) Handling and laying concrete sleepers is difficult due to their large weights.Mechanical methods, which involve considerable initial expenditure, have to be adopted for handling them.(b) Concrete sleepers are heavily damaged at the time of derailment.(c) Concrete sleepers have no scrap value.(d) Concrete sleepers are not suitable for beater packing.(f) Concrete sleepers should preferably be maintained by heavy ‘on track’ tampers.

CST-9 sleepersThe CST-9 (Central Standard Trial No 9) sleeper is a standard sleeper and is being most extensively used on Indian Railways (IR). It is called CST-9 produced by the Central Standard Of a combination of pot, plate, and box sleepers. It consists of two triangular inverted pots on either side of the rail seat, a central plate with a projected keel, and a box on top of the plate. The two CI plates are connected by a tie bar with the help of four cotters. The rails are held to the sleeper by two-way keys provided at each rail seat on the side of the gauge face. The gauge is adjusted to a value of ± 5 mm by altering the relative positions of the four cotters. The rail seat of a CST-9 sleeper is 115 mm wide along the length, and this narrow bearing tends to reduce the rocking of the sleeper under the wave motion of the rail. The sleeper is designed to provide a firm support to the rail and provides fairly good lateral and longitudinal stability to the rails. The sleeper provides a bearing area approximately equal

to the effective bearing area of a standard BG wooden sleeper i.e., 5 sq. ft, for both the plates. CST-9 plates are also available with reverse jaws (T-443 type) to serve as an anti-sabotage measure; a few of these are provided in each rail length. Normally, three reverse jaw CST-9 sleepers are provided per rail to serve anti-sabotage purposes. The weight of a CST-9 sleeper assembly along with fastenings for BG is 102 kg and for MG is 58 kg. The CST-9 sleeper is one of the most popular sleepers on Indian Railways at present. The sleeper has, however, certain limitations when combined with the modern track as mentioned in the following.CST-9 Sleeper has got following features

1. The tie bar is fastened to the plate by means of four standard cotters. Small variation in gauge can be corrected by these four cotters

2. The shape of cast iron support is such as to give a stable base for the rail, and high lateral and longitudinal stability to the track

3. The sleeper may be used in sections of track in corrosive conditions such as saline soil, industrial waste etc

4. The bearing area is approximately equal to the effective bearing area of a wooden sleeper

5. This sleeper forms the rigid track subjected to vibrations under moving loads without any damage or absorption

6. The cantilever ends of the rails are long which lead to battering, and ultimately hogging of the rail end and deterioration of ballast under the joint which finally need the replacement of sleeper.

7. This type of sleeper is suitable for speeds upto 110 KMPH8. If used for tracks above speed of 110 kmph, due to the shallow depth of the bowl,

packing becomes loose under vibrations at high speedAs the sleeper does not have a flat bottom, it is not quite suitable for mechanical maintenance with tie tamers.

1. The suitability of a CST-9 sleeper on Long Welded Rails (LWRs), particularly on the breathing lengths, is doubtful because of rigid fastenings and the inability of the fastenings to hold the rail with a constant toe load.

2. The rail seat wears out quickly causing the keys to come loose.3. The sleeper has only limited longitudinal and lateral strength to hold LWRs

particularly in the breathing length.4. Due to the use of less metal under rail seat, the shocks and vibrations are directly

transmitted to the ballast, resulting in poor retention of packing (loose packing) and hence an increased frequency of attention.

Track Fittings and Fastenings

IntroductionThe purpose of providing fittings and fastenings in railway tracks is to hold the rails in their proper position in order to ensure the smooth running of trains. These fittings and fastenings are used for joining rails together as well as fixing them to the sleepers, and they serve their purpose so well that the level, alignment, and gauge of the railway track are maintained within permissible limits even during the passage of trains. The important fittings commonly used in a permanent way are the following

1. Fish plates2. Spikes3. Bolts4. Chairs5. Blocks6. Keys7. Plates

Fish Platesis a metal bar that is bolted to the ends of two rails to join them together in a track. The top and bottom edges are tapered inwards so the device wedges itself between the top and bottom of the rail when it is bolted into place. The name ‘fish plate’ derives from the fish-shaped section of this fitting. The function of a fish plate is to hold two rails together in both the horizontal and vertical planes.

Requirements of fish plates1. They must support the underside of the rail and top of the foot2. They should allow a free movement of rails for expansion and contraction for this

purpose, they should not touch web of the rail3. They must be of such a section as to bear the stresses due to lateral and vertical

bending moments without getting distorted and must absorb the shock caused by the jumping of wheel over the gap

4. They should hold the end of rail both laterally in line and vertically in level5. They should be provided against the wear of fish plate due to impact expansion and

contraction6. Strength should be atleast 58 to 67 kg/cm2

7. Minimum Elongation required is 20%

8. Sectional area of fish plate and rail section should be same9. Strength should approximately be of 55% of rail strength

Failure of fish plates1. Abrasion on top of fish plate, especially along central half length2. Reversal of stresses due to large length of fish plates results into cracking along

section3. Cracking develops along the section and this may even extend upto top or bottom of

fish plate.SpikesThe requirements of a good spike are

1. First of all, the spike should be strong enough to hold the rail in position and it should have enough resistance against motion to retain its original position so that it does not lead to creep

2. The spike should be as deep as possible, for better holding power3. The spike should be easy in fixing and removal from the sleepers4. The spike should be cheap in cost5. It should be capable of maintaining the gauge

Dog spikes

For holding the F.F. Rails to a wooden sleeper, dog spikes are commonly used. These are simply stout nails to hold rail flanges with timber sleepers. The shape of head of spike resembles with ear of the dog and hence its nomenclature as dog spike. The section of the spike is square-shape and bottom part is either pointed,blunt or chisel shaped. They are cheapest, easy in fixing and removing from sleepers and maintain a better gauge than scre spikes.

Screw spikes

These are tapered screws with V threads used to fasten the rails with timber sleepers. The head is circular with square projection

Screw spike has double the holding power as that of dog spike an can also resist lateral thrust in a better way as compared to dog spikes. However, the screw spikes are costly and with that their use, the gauge maintenance become difficult

Round spikes

Round spikes with a head either cylindrical or hemispherical are used for fixing chairs of B.H. Rails to wooden sleepers for fixing slide chairs of points and crossings. These have a blunt end and limited use

Standard spikes

These are used for cast iron chairs only to fix them with timber sleepers

Elastic spikes

Actually the main disadvantage with the dog head spikes is that due to wave motion the spike comes out hence the fastening will get loose. To over come this disadvantage elastic spikes are introduced which will absorb the wave motion without getting loose

Dog Spikes• It is used to hold flat footed rails to a wooden sleeper.• The shape of the top head resembles that of a dog, hence the name.• The section of the spike is square shaped and the bottom part is either pointed, blunted

or chisel shaped.• They are cheap and easy to install, but they move out of place due to wave motion,

resulting in creep.

SCREW SPIKE• These are tapered screws with V threads used to fasten rails with timber sleepers.• The head is circular with a square projection.• Screw spikes have more than double the holding power of dog spikes but they are

costly and maintenance is more difficult.

Bolts1. Dog bolts, where sleepers rest directly on a girder, they are fastened to the top flange

of the girder by bolts called dog bolts2. Fish bolt: the fish bolts have to undergo shear due to heavy transverse stresses, fish

bolts are made of medium or high carbon steel. Usually a bolt of 2.5 cm dia and 12.7 cm length is used. Generally the length of the bolt depends on the type of plate used.

CI Chairs• Chairs are used for holding Double Headed and Bull Headed rails.• The rails are placed between the two jaws of a chair and pressed against the inner jaw

by inserting tapered keys. • B.H. rails are supported on Cast Iron Chairs fixed to the sleepers by round spikes.• In case of cast iron sleepers, the chairs are casted together with the sleepers.• In case of steel sleepers, the chairs are welded to the steel sleepers.

BlocksWhen two rails run very close as in case of check rails, etc. small blocks are inserted in between the two rails and bolted to maintain the required distance or spacing. Bearing PlatesBearing Plates are rectangular plates of Mild Steel or Cast Iron that are used below Flat Footed rails to distribute the load on a larger area of the sleepers.

Advantages of Bearing Plates• They distribute the load coming from the rails to the sleepers over a larger area.• They prevent the destruction of sleepers due to rubbing action of the rail.• No adzing(tilting) of sleepers is required since the bearing surfaces are canted at 1 in

20.• They help in firm holding of spikes to the sleepers, which in turn prevents the shifting

of rails.• Better maintenance of gauge, if bearing plates are used, is possible

Disadvantages of Bearing Plates• The plates rattle when loose.

• When a spike is damaged, a new hole is to be created for it, which required new holes for all the other spikes. This reduces the holding strength of the spikes.

• When the bearing plates are loose, they admit moisture, which results in increase of mechanical wear of the sleepers.

Traction and tractive resistanceThere are a number of forces which resist the movement and speed of a train. The tractive force employed by a locomotive should be adequate enough to overcome the resistance offered by the locomotive, trailer load and other agencies against its movement.Resistances can be classified as

1. Train resistance2. Resistance due to track profile3. Resistance due to starting and acceleration4. Wind resistance

Train resistanceTrain resistance can be classified into following categories

1. Resistance independent of speed2. Resistance dependent of speed3. Atmospheric resistances

Resistances independent of speed:This resistance is due to the following reasons

Resistance due to friction is the resistance offered by the friction between the internal parts of locomotives and wagons as well as between the metal surface of the rail and the wheel to a train moving at a constant speed. This resistance is independent of speed and can be further broken down into the following parts.Journal friction This is dependent on the type of bearing, the lubricant used, the temperature and condition of the bearing, etcInternal resistance This resistance is consequential to the movement of the various parts of the locomotive and wagons.Rolling resistance This occurs due to rail-wheel interaction on account of the movement of steel wheels on a steel rail. The total frictional resistance is given by the empirical formula

Rt1= 0.0016WResistance Dependent of speed

1. Track irregularities: if the track is not properly maintained due to irregularities, additional resistance has to be overcome

2. Due to vertical movements: Due to improper joints and poor maintenance of track, vertical movements of wheels on rails occurs creating resistance

Atmospheric resistanceThis is resistance which is developed on the ends and sides of the train when wind velocity is considered to be zero. It is low at lower speeds, but increases as the square of velocity as the speed increases.

Rt3=0.0000006wv2

Resistance due to track profile1. Resistance due to gradient

It may be noted here that when a train ascends a slope, extra effort is required to overcome the resistance offered by the gradient.

Rg=w*slope2. Resistance due to curves

When a train negotiates a horizontal curve, extra effort is required to overcome the resistance offered by the curvature of the track. Curve resistance is caused basically because of the following reasons (a) The vehicle cannot adapt itself to a curved track because of its rigid wheel base. This is why the frame takes up a tangential position as vehicle tries to move in a longitudinal direction along the curve. On account of this, the flange of the outer wheel of the leading axle rubs against the inner face of the outer rail, giving rise to resistance to the movement of the train(b) Curve resistance can sometimes be the result of longitudinal slip, which causes the forward motion of the wheels on a curved track. The outer wheel flange of the trailing axle remains clear and tends to derail. The position worsens further if the wheel base is long and the curve is sharp.(c) Curve resistance is caused when a transverse slip occurs, which increases the friction between the wheel flanges and the rails.(d) Poor track maintenance, particularly bad alignment, worn out rails, and improper levels, also increase resistance. Inadequate superelevation increases the pressure on the outer rail and, similarly, excess superelevation puts greater pressure on the inner rails, and this also contributes to an increase in resistance.

For B.G. Rc=0.0004wDM.G. Rc=0.0003wDN.G. Rc=0.0002wD

Resistance due to starting and AcceleratingTrains face these resistances at stations when they start, accelerate, and decelerate.The values of these resistances are as follows:Resistance on starting, Rs = 0.15W1+ 0.005W2

Resistance due to acceleration, Rac= 0.028aW where W1 is the weight of the locomotive in tonnes, W2 is the weight of the vehicles in tonnes, W is the total weight of the locomotive and vehicle in tonnes

Wind resistanceWhen a vehicle moves with speed, a certain resistance develops, as the vehicle has to move forward against the wind. Wind resistance consists of side resistance, head resistance, and tail resistance, but its exact magnitude depends upon the size and shape of the vehicle, its speed, and wind direction as well as velocity. Wind resistance depends upon the exposed area of the vehicle and the velocity and direction of the wind.

Rw= 0.000017AV

BallastThe ballast is a layer of broken stones, gravel, moorum, or any other granular material placed and packed below and around sleepers for distributing load from the sleepers to the formation. It provides drainage as well as longitudinal and lateral stability to the track. Different types of ballast materials and their specifications are discussed in this chapter.Functions of BallastThe ballast serves the following functions in a railway track.

3. Provides a level and hard bed for the sleepers to rest on.

4. Holds the sleepers in position during the passage of trains.5. Transfers and distributes load from the sleepers to a large area of theformation.6. Provides elasticity and resilience to the track for proper riding comfort.7. Provides the necessary resistance to the track for longitudinal and lateralstability.8. Provides effective drainage to the track.9. Provides an effective means of maintaining the level and alignment of the track.

Requirements of a Good BallastBallast material should possess the following properties.

1. It should be tough and wear resistant.2. It should be hard so that it does not get crushed under the moving loads.3. It should be generally cubical with sharp edges.4. It should be non-porous and should not absorb waterIt should resist both attrition and

abrasion.5. It should be durable and should not get pulverized or disintegrated under adverse

weather conditions.6. It should allow for good drainage of water.7. It should be cheap and economical.

Types of BallastThe different types of ballast used on Indian Railways are described in the following.

Sand ballastSand ballast is used primarily for cast iron (CI) pots. It is also used with wooden and steel trough sleepers in areas where traffic density is very low. Coarse sand is preferred in comparison to fine sand. It has good drainage properties, but has the drawback of blowing off because of being light. It also causes excessive wear of the rail top and the moving parts of the rolling stock.

Moorum ballastThe decomposition of laterite results in the formation of moorum. It is red, and sometimes yellow, in colour. The moorum ballast is normally used as the initial ballast in new constructions and also as sub-ballast. As it prevents water from percolating into the formation, it is also used as a blanketing material for black cotton soil.

Coal ash or cinderThis type of ballast is normally used in yards and sidings or as the initial ballast in new constructions since it is very cheap and easily available. It is harmful for steel sleepers and fittings because of its corrosive action.

Broken stone ballastThis type of ballast is used the most on Indian Railways. A good stone ballast is generally procured from hard stones such as granite, quartzite, and hard trap. The quality of stone should be such that neither is it porous nor does it flake off due to the vagaries of weather. Good quality hard stone is normally used for high-speed tracks. This type of ballast works out to be economical in the long run.

Other types of ballastThere are other types of ballast also such as the brickbat ballast, gravel ballast, kankar stone ballast, and even earth ballast. These types of ballast are used only in special circumstances.

Minimum depth of Ballast sectionThe load on the sleeper is transferred through the medium of the ballast to the formation. The pressure distribution in the ballast section depends upon the size and shape of the ballast and the degree of consolidation. Though the lines of equal pressure are in the shape of a bulb, yet for simplicity, the dispersion of load can be assumed to be roughly 45° to the vertical. In order to ensure that the load is transferred evenly on the formation, the depth of the ballast should be such that the dispersion lines do not overlap each other. For the even distribution of load on the formation, the depth of the ballast is determined by the following formula Sleeper spacing = width of the sleeper + 2 × depth of ballast If a BG track is laid with wooden sleepers with a sleeper density of N + 6, then the sleeper spacing would be 68.4 cm. If the width of the sleeper is 25.4 cm, then the depth of the ballast cushion would be

A minimum cushion of 15–20 cm of ballast below the sleeper bed is normally prescribed on Indian RailwaysAdvantages and disadvantages of different types of ballast Type of ballast Advantages Disadvantages SuitabilitySand ballast Good drainage

properties Cheap „

No noise produced „ on the track

Good packing „

material for CI sleepers

Causes excessive wear

Blows off easily Poor retentively

of packing Track cannot be

maintained to highstandards

Suitable for CI pot sleeper tracks

Not suitable for high-speed tracks

Moorumballast

Cheap, if locally available

Prevents water from percolating

Provides good aesthetics

Very soft and turns into dust

Maintenance of track the difficult

Quality of trackaverage

Used as a sub-ballast

Initial ballast for new construction

Coal ash or

cinder

Easy availability on railways „ Very cheap „ Good drainage

Harmful for steel sleepers Corrodes rail

bottom and steel sleepers

Soft and easily pulverized

Normally used in

yards and sidings

Suitable for repairs

of formations

Maintenance is difficult

floods andemergencies

Not fit for high-speed tracks

Broken stoneballast

Hard and durable when procured from hard rocks

Good drainage properties

Is stable, and resilient to the track

Economical inthe long run

Initial cost is high Difficulties in

procurement Angular shape

mayinjure woodensleepers

Suitable for high speedtracks

ProblemsCalculate the maximum permissible load that a BG locomotive with three pairs of driving wheels bearing an axle load of 22 t each can pull on a straight level track at a speed of 80 km/h. Also calculate the reduction in speed if the train has to run on a rising gradient of 1 in 200. What would be the further reduction in speed if the train has to negotiate a 4° curve on the rising gradient? Assume the coefficient of friction to be 0.2

Hauling power of the locomotive = number of pairs of driving wheels × wt exerted on each pair × coefficient of friction = 3 × 22 × 0.2 = 13.2 t

1. Maximum Permissible load The total resistance negotiated by the train on a straight level track at a speed of 80 km/h:R = Resistance independent of speed + resistance dependent on speed + resistance due to wind= 0.0016W + 0.00008WV + 0.0000006WV2

Substituting the value of V = 80 km/h= 0.0016W + 0.00008 X 80W + 0.0000006 X 802 WR = 0.01184WAssuming total resistance = hauling power,W × 0.01184 = 13.2 t

W= 1115 t2. Reduction in speed if the train has to run on an up gradient of 1in 200

On a gradient of 1 in 200, there will be an additional resistance due to gradientequal to W × % of slope. R = Resistance independent of speed + resistance dependent on speed + resistance due to wind+ W × % of slope.= 0.0016W + 0.00008WV + 0.0000006WV+W x slopeSubstituting the value of w = 1115 t and slope = 1/200= 0.0016 X 1115 + 0.00008 X 1115 X V + 0.0000006 X 1115 X V2+W x 1/200Since hauling power = total resistance,13.2=0.0016 X 1115 + 0.00008 X 1115 X V + 0.0000006 X 1115 X V2+W x 1/200V=48.13Reduction in speed = 80 – 48.13 = 31.87 km/h = 32 km/h

3. Along with the gradient of 1in 200 if there is a curve of 4° the reduction in speed is calculated as

On a curve of 4° on a rising gradient of 1 in 200, curve resistance will be equal toR == 0.0016W + 0.00008WV + 0.0000006WV+W x slope +0.0004 × degree of curve × W (B.G resistance due to curve is = 0.0004× degree of curve × W)= 0.0016 X 1115 + 0.00008 X 1115 X V + 0.0000006 X 1115 X V2+W x 1/200+ 0.0004 × 4 × W = 0.0016WHauling power of locomotive = total resistance. Therefore,13.2 = 0.0016W + 0.00008WV + 0.0000006WV2 + 0.005W + 0.0016WBy substituting the value of W = 1115t in the equation and solving further,

V = 43.68 km/hFurther reduction in speed = 48.13 – 43.68 = 4.45 km/h. Therefore,

Maximum permissible train load = 1115 tReduction in speed due to rising gradient = 31.87 tFurther reduction in speed due to curvature = 4.45 km/hCompute the steepest gradient that a train of 20 wagons and a locomotive can negotiate given the following data: weight of each wagon = 20 t, weight of locomotive = 150 t, tractive effort of locomotive = 15 t, rolling resistance of locomotive = 3 kg/t, rolling resistance of wagon = 2.5 kg/t, speed of the train =60 km/h

Rolling resistance due to wagons = rolling resistance of wagon × weight ofwagon × number of wagons= 2.5 × 20 × 20 = 1000 kg = 1 tRolling resistance due to locomotive= rolling resistance of locomotive × wt of locomotive= 3 × 150 = 450 kg = 0.45 ttotal rolling resistance = rolling resistance due to wagons + rolling resistancedue to locomotive = 1.00 + 0.45 t = 1.45 ttotal weight of train = weight of all wagons + wt of locomotive= 20 × 20 + 150 = 550 total train resistance = rolling resistance + resistance dependent on speed + resistance due to wind + resistance due to gradient= 1.45 + 0.00008WV + 0.0000006WV2 + W/g= 1.45 + 0.00008 × 550 × 60 + 0.0000006 × 550 × 60+ (550/g)= 1.45 + 2.64 + 1.19 + (550/g) = 5.28 + (550/g)Where g is the gradient.Tractive effort of locomotive = Total train resistance15 = 5.28 + (550/g)= 56.5= 1/56 = 1 in 56Therefore, the steepest gradient that the train will be able to negotiate is 1 in 56