Problem Sheet 2

Question 1

What 3 components of a track have stiffness to take the load?

Answer 1

Ballast, sleeper and pads

Question 2

What is the step by step process to answer this question:

A railway track is to be designed to take axle loads (including dynamic effects) up to 220
kN. The steel rails each have a second moment of area (I) of 2400 cm4. The pad stiffness
is 80 kN/mm. Sleeper spacing (centre to centre) is 0.6m. What sleeper support stiffness
(in kN/mm) is required in order to limit the hogging bending moment in each rail to 10
kNm?

Answer 2

1 - Find what value of lander x satisfies d3y/dx3 so that it is 0.

2 - Plug into the equation for d2y/dx2 * EI = Moment

3 - The only unknown in the equation is the stiffness modulus, rearrange to find k

4 - Multiply k by the sleeper separation

5 - Minus the reciprocal of kpad from reciprocal of this k, to get the reciprocal of k,ballast

Question 3

What is imperative to rememeber when performing calculations with the 2 moment of area of a rail?

Answer 3

Multiply by 2, as there is 2 rails.

Question 4

Calculate the approximate maximum upward tensile force in each rail fixing in front of a
17.5 tonne train axle travelling at 200 km/hr. Assume UIC60 rails with a 2nd moment of
area of 3044 cm4, sleepers of mass 260 kg at 650 mm centre-to-centre spacing, supported by
a foundation that equates to a 40 kN/mm stiffness under each sleeper.

How would you answer this question?

Answer 4

Force = mass x acceleration

Therefore calculate the acceleration using the same method as used before ….

Question 5

How many pads are there for each sleeper?

Answer 5

2

Question 6

For this question, how would you work out the k value of the sleeper?

Calculate the maximum hogging moment in each rail of a track under a 28 t axle load.
Falling Weight Deflectometer (FWD) tests on individual (unclipped) concrete sleepers give
a typical deflection of 1.2 mm under a 100 kN impulse force. Rail seat pads have a stiffness
of 150 kN/mm. The rails are UIC60 type with a 2nd moment of area of 3055 cm4
. Assume a
sleeper spacing (centre-to-centre) of 0.6 m.

Answer 6

Divide force by the sleeper distance.

100kN / 0.6m.

This gives the force / m

Then divide this force by the deflection in m..

Which results in a force / m / m

Question 7

For this question, how do we calculate the effect of the pad ?

Calculate the maximum hogging moment in each rail of a track under a 28 t axle load.
Falling Weight Deflectometer (FWD) tests on individual (unclipped) concrete sleepers give
a typical deflection of 1.2 mm under a 100 kN impulse force. Rail seat pads have a stiffness
of 150 kN/mm. The rails are UIC60 type with a 2nd moment of area of 3055 cm4
. Assume a
sleeper spacing (centre-to-centre) of 0.6 m.

Answer 7

150 X 2 = 300 (as there is 2 pads for each sleeper

Then divide by 0.6 to get a force per m / m

Question 8

How would you find the k value for this question, which would allow you to solve the rest of the question:

Calculate the maximum hogging moment in each rail of a track under a 28 t axle load.
Falling Weight Deflectometer (FWD) tests on individual (unclipped) concrete sleepers give
a typical deflection of 1.2 mm under a 100 kN impulse force. Rail seat pads have a stiffness
of 150 kN/mm. The rails are UIC60 type with a 2nd moment of area of 3055 cm4
. Assume a
sleeper spacing (centre-to-centre) of 0.6 m.

Answer 8

1 - For the load under impulse force:

(100 / 0.6 ) / 0.0012= 166.67 MN / m / m

2 - From the effect of the pad

(150 x 2 ) /0.6 = 500 kN / m / m

3 - Take recipricol of both, add them, then unrecipricol to find the value of k

Question 9

When finding lander for a sleeper, do you multiply I by 2 like you would for a rail?

Answer 9

No

Question 10

What equation relates stress, total moment and height?

Answer 10

Stress = 6 * M / H^2

Question 11

For this question, what do you set x to be in both cases, when calculating d2y/dx2 ?

It is proposed that the thickness of a concrete slab in a rigid trackbed should be 200mm.
Your task is to check whether this is reasonable from a structural point of view. Assume a
5.0MPa flexural strength concrete, a modulus of subgrade reaction (k) of 500MNm-2, that
the load is transmitted from the rail to the concrete slab through booted sleepers at 0.5m
spacing. Also assume that an axle load directly over a sleeper divides 25%, 50%, 25% across
three adjacent sleepers, while one between sleepers divides 12.5%, 37.5%, 37.5%, 12.5%
across four adjacent sleepers.

Answer 11

On top of sleeper : x = 0.5

In between sleepers, x = 0.75

Question 12

How can you derive k from this question ?

Calculate the maximum load (in kN/m) experienced by a rigid concrete slab under a rail of
second moment of area 1500 cm4 seated on a continuous 150mm wide 5mm thick rubber
strip of effective stiffness modulus 5MPa if the wheel load is 60kN.

Answer 12

1 - We know that strain = P / E

2 - Deflection is the strain times the height of the rubber

3 - Pressure is force per metre / width

4 - k is the force per metre / deflection (y)

5 - Sub in and you can find that k = Er*w / h