Lateral oscillation of rail vehicles is a major concern on narrow or metre gauge track because of short-wheelbase bogies. This problem could be addressed by modifying the Prud'homme limit for the vehicles and by improving the track panel resistance

Sundararaja Gopalakrishnan
Manager (Track Works & Systems), Rawang - Ipoh Electrified Double Track Project,
UEM Construction Malaysia Sdn Bhd

ROUGH RIDING of passenger and freight vehicles is a common problem on metre gauge railways. Lateral movements are a particular problem causing discomfort to passengers, and in severe cases operating speeds may have to be reduced. Over time, damage may be caused to track and trains.

A possible cure for the problem would be to reduce the Prud'homme limit which defines the allowable repeated lateral loading exerted on the track. This formula was evolved for standard gauge track and vehicles, but many railways and rolling stock manufacturers presume that it is also applicable to metre gauge as well. Note that the term 'metre gauge' is used here to cover 1 000 mm gauge track as in Malaysia and Thailand as well as the 'Cape gauge' of 1 067 mm which is used in southern Africa, parts of Australia and elsewhere.

Based on existing information, without any field tests or computer simulations, I believe that it is possible to modify the Prud'homme limit for metre gauge railways, but further investigations will be needed to confirm the effects of lateral loading on metre gauge track. Inevitably, several simplistic assumptions have been made, but it is unlikely that these would alter the conclusions significantly.

Lateral forces and track-train dynamics

Increased dynamic lateral forces between vehicle and track may be generated by rail and wheel profiles, which are designed to control lateral wheel movements and to optimise wheel-rail contact. Lateral suspension and the ride characteristics of a vehicle also govern the lateral forces, and this form of lateral oscillation can be controlled through good design and maintenance. The ride characteristics depend on the springs and dampers, allowable clearances between wheelset, bogie and vehicle body, and the effect of any self-steering axles.

Track geometry is a critical element. A well-maintained alignment is essential to reduce lateral oscillation, and this is facilitated by a good standard of construction, a high mass per unit length and an appropriate track modulus.

Other track and vehicle defects may also result in excessive rolling, pitching and bouncing, which ultimately amplify the lateral forces. Under such excessive forces, the track may yield elastically, causing temporary misalignment, or it may deform and not return to its original position, depending on the severity of the lateral force.

Any remaining misalignment is called 'residual deflection'. If such misalignments accumulate because of the frequent passage of poorly-maintained rolling stock, the track is likely to deteriorate progressively and induce lateral oscillation in other vehicles. This leads to a vicious circle of further wear and tear.

Limiting the lateral forces

Research carried out by André Prud'homme at SNCF during the 1950s and 1960s1 concluded that if the lateral force exerted repeatedly by an axle on the track does not exceed a certain value H, the residual deflections do not add up indefinitely and the final deflection stabilises within acceptable limits.

The limiting value is represented by H where H = 10 + 0·33P, and H kN represents the lateral force and P kN is the static axleload. This relationship was initially evolved for track with timber sleepers and newly-tamped ballast free of any thermal forces. H is also regarded as the limiting track panel strength to satisfy the criterion of acceptable residual deflection under repeated lateral loading concurrent with the vertical loading P. Prud'homme later extended his research to take into account thermal forces in welded track and subsequently applied a multiplying factor of 0·85, and the final form of the Prud'homme limit is H = 0·85 (10 + 0·33P).

The results of tests conducted in the 1990s by SNCF with concrete sleeper track, welded rails and ballast compacted by a dynamic track stabiliser on the Paris - Lyon high speed line2 did not require any significant changes to be made to the Prud'homme formula, as reflected in current SNCF practice3. H is defined as the maximum repeated lateral force which is permitted to be exerted on the track by the vehicle.

According to SNCF, the Prud'homme limit was not exceeded during several subsequent measurements made to evaluate the maximum lateral forces exerted on the track during tests at 408 and 482 km/h. This provided further validation for the use of the Prud'homme limit by designers and manufacturers of 1 435 mm gauge rolling stock, and it provides the basis for the allowable maintenance tolerances for vehicles.

Based on more recent investigations, SNCF set limits for the lateral force/strength for the vehicle/track system on TGV lines. These specify that the repeated dynamic lateral axleload exerted by the vehicle must not exceed H = 0·85 (10 + 0·33P). Concrete-sleepered track must be designed such that the static lateral panel resistance is not less than H = (24 + 0·41P) for tamped track and H = (38 + 0·63P) for track treated by a dynamic stabiliser. The lateral panel resistance represents the maximum stationary lateral load that can be sustained by the track under the vehicle vertical loads without causing any permanent deflection of the track panel.

As an example, where axleload P is 150 kN, the limiting values for standard gauge track are:

  • maximum lateral force exerted by an axle = 0·85 (10 + 0·33P) = 51 kN
  • minimum track panel resistance in tamped condition = 24 + 0·41P = 85·5 kN
  • minimum track panel resistance after use of stabiliser = 38 + 0·63P = 132·5 kN.

A reasonable safety margin between panel resistance and lateral thrust improves ride comfort and enhances safety, while ensuring reliability and optimal maintainability of track and vehicles. Consequently, the Prud'homme concept has become a meeting point for track and vehicle design engineers.

The US Federal Railroad Administration specifies in Part 213.333 of Track Safety Standards (January 2002) with respect to high speed railways4 that a vehicle must be approved for use on its intended route and that wheel-rail forces must be measured in a representative vehicle on an annual basis. One of the safety limits of track-vehicle interaction is that the vehicle may not produce a lateral axleload to static vertical load ratio (H/P) exceeding 0·5. For a nominal axleload of 150 kN, permissible horizontal thrust is therefore 75 kN, which is 50% more than the Prud'homme limit. But the FRA requirement is for a safety limit. Hence the values quoted for lateral load and track panel resistance as per SNCF norms may be considered acceptable for design and operation.

Similar research has been carried out by other organisations, including Netherlands Railways5 and the US Department of Transportation's Volpe Center6.

Metre gauge track limit

I suggest that a simple mathematical analysis will provide a quick solution for modifying the Prud'homme limit for metre gauge track, starting from the research carried out by SNCF. There are two possible independent approaches - one based on the lateral stiffness of the track and the other on proportional wheel offloading.

Taking the first approach, the lateral stiffness of track is made up of ballast resistance and frame resistance.

Ballast resistance comprises the shoulder ballast resistance acting on the sleeper ends and frictional resistance acting on the sides and undersides of the sleepers. The shoulder ballast resistance can be assumed to be the same for both standard and metre gauge track as the cross-sectional areas of the sleeper end-faces are similar.

The sleeper friction resistance for metre gauge is assumed to be 0·8 of that for standard gauge, based on the ratio of the sleeper lengths of 2·0 m and 2·5 m. Therefore, overall ballast resistance in metre gauge track can be assumed to be 0·9 of that in standard gauge track using the simple calculation (1·0 + 0·8)/2 = 0·9.

The frame resistance can be assumed to vary approximately in the same ratio as the distance between the rail centres, say 1 065 mm to 1 500 mm, giving a figure of 0·71:1. Therefore the frame resistance of metre gauge track can be assumed to be 0·7 of standard gauge.

Space does not permit a rigorous analysis to decide the relative contribution of the ballast and frame resistances in preventing small deflections of the track in any case many studies have been conducted on the theory of track buckling. As a simplified approach, equal weightings have been assigned to the two components. The overall lateral stiffness of metre gauge track is therefore assumed to be (0·9 + 0·7)/2 or 0·8 of standard gauge.

Tests conducted by Indian Railways' Research, Design & Standards Organisation on the relative lateral stiffnesses of metre and broad gauge track show that the suggested approach for comparison is likely to be valid. Therefore the Prud'homme limit to be applied for metre gauge would be H = 0·80 x 0·85 (10 + 0·33P) = 0·68 (10 + 0·33P).

Fig 1 illustrates the second approach, based on proportional wheel offloading. Axleload P results in reactions of P/2 = Q (wheel load) at each rail. Horizontal force H acting at the centre of gravity at height c from rail top is resisted by horizontal forces H1 and (H - H1) acting at rail level. It also creates a variation in the wheel load R = Hc/g, where g is the distance between the rail centres (1 500 mm for standard gauge and 1 065 mm for metre gauge).

The augmented wheel load is represented by Q + R, and the offloaded wheel load by Q - R. The proportional offloading is therefore calculated as R/Q = Hc/Qg.

When H reaches the Prud'homme limit on standard gauge, this will result in a certain value of offloading, which has been shown to cause neither ride discomfort nor derailment. It would therefore be appropriate to adopt the same degree of offloading as the upper limit for metre gauge track.

The expression representing this comparison is: HMcM / QMgM = HScS / QSgS, where suffix M represents metre gauge parameters and suffix S represents standard gauge parameters. As wheel load QM = QS for comparison with the same value of axleload P, it follows that HM/HS = [gM/gS] ÷ [cM/cS].

The first factor is the ratio of the rail centre gauges (g), which is 0·71. The second is the ratio of the heights of the centre of gravity (c) above rail level. The height of the centre of gravity does not vary significantly between standard and metre gauge vehicles, although it would normally be lower for metre gauge stock. In any case, even for the same wagon, c will vary according to the load.

It is therefore considered acceptable to assume that cM/cS = 0·9. Substituting the values HM/HS = 0·71 ÷ 0·9 = 0·8, the horizontal force for metre gauge can be taken as 0·8 of standard gauge, to limit the degree of lateral loading at the same level in both cases.

Both these analyses give similar results for adapting the Prud'homme limit to metre gauge. It should be noted that this limit cannot be increased merely by improving the lateral strength of the track, as offloading effects may also be critical. In this case, any strengthening of metre gauge track should be complemented by lowering of the centre of gravity of the vehicles.

On metre gauge railways passenger and freight trains normally share the same track. Freight service tends to cause track to deteriorate, leading to rough riding for passenger trains and possible economic consequences. Therefore the modified Prud'homme limit must be implemented uniformly for passenger coaches, freight wagons and locomotives.

Using this information, the static lateral panel resistance of metre gauge track with concrete sleepers may be computed by multiplying the corresponding value for standard gauge by 0·8. The computed values for metre gauge track are (19 + 0·33P) for newly-tamped track and (30 + 0·5P) for track that has been stabilised.


Rolling stock manufacturers are perhaps unaware of how to apply the Prud'homme limit to track gauges other than 1 435 mm. It is important that railways amend their specifications for metre gauge track and consider the implementation of the formula H < 0·68 (10 + 0·33P). The required improvement to suspension systems may cost more initially, but advancing technology and market forces will ensure that the changes are financially viable over the longer term. It is worth noting that various innovative designs of standard gauge track-friendly' bogie have appeared in recent years.

Some reports suggest that operation of tilting trains on metre gauge routes does not offer the anticipated benefits in terms of ride quality. This may be due to the lateral forces exerted on the track reaching the levels intended for standard gauge. Tilting mechanisms are intended solely to improve passenger comfort by compensating for the unbalanced centrifugal force and they do not reduce lateral forces acting on the track. Therefore adopting a modified lateral force limit for metre gauge track may also improve ride comfort for passengers in tilting trains - and improve safety.

It is possible that the speed limits applicable to metre gauge track could be raised by creating conditions conducive to an increase of the multiplying factor to more than 0·8. This would be possible by increasing the lateral strength of the track. For example, transverse ribs could be cast monlithic with the concrete sleepers to generate more lateral ballast resistance.

Another option would be to lower the centre of gravity of the vehicles. For passenger cars, a well underframe may be feasible this could also facilitate the use of double-deck coaches. For freight, well wagons could be more widely adopted.

It would be helpful if research organisations could conduct field tests and computer simulations to assess the lateral stiffness of metre gauge track more accurately, and to verify the somewhat simplistic assumptions made in this article.

  • CAPTION: KTM operates several batches of 120 km/h EMU on its electrified network around Kuala Lumpur. A new design of 160 km/h inter-city EMU is envisaged for services to Ipoh
  • CAPTION: Queensland Rail operates both diesel and electric tilt trains on its 1 067 mm gauge main line north from Brisbane. QR is to return its tilt train services to their 160 km/h maximum speed this year following approval from safety regulator Queensland Transport. The trains had been restricted to 100 km/h following the Bundaberg derailment in November 2004
  • CAPTION: Fig 1. Proportional wheel offloading is one option that would permit the Prud'homme limit to be adopted for metre gauge vehicles

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