Problem Sheet 3

Question 1

What is the area of influence under a sleeper. Assuming a load spread angle of 35 degrees ?

Answer 1

Area = ( L + 2H tan 35 ) ( W + 2H tan 35)

Question 2

To achieve a stress of 30kPa at the subgrade, with a force 120kN on the sleeper, what must the area of load be?

Answer 2

4 m2

Question 3

This is the question: what would the thickness of tracked (ballast & subballast) need to be if the maximum sleeper load was 120kN, and the ,and the max allowable stress on the subgrade must not exceed 30kPa?

What would the step by step procedure?

Answer 3

1 - take load angle to be 35 degrees, and estimate the dimensions of the sleeper to be 2.5 by 0.25m

2 - find an equation for A with H

3 - calculate A using load and stress, which can be used to calculate H

Question 4

What other methods can you use to reduce stress on subgrade, other than increasing thickness of trackbed?

Answer 4

Reduce pad stiffness, increase rail bending stiffness ( larger value of I), possible reduce sleeper spacing

Question 5

What does frequency (Hz) equal?

Answer 5

x * (2ý/rs)^0.5. /. 2 pi

Where x is ut

Question 6

If the a,plitude of motion from the klingel motion is 5mm, how do you find max amount of lateral acceleration?

What represents the amplitude of motion?

Answer 6

Y represents amplitude of motion, to find acceleration of motion differentiate twice

Question 7

Given that the lateral acceleration from klingel motion has already been calculated, for this question what steps do you need to take?

‘If the unsparing mass over a given axle is 4 tonnes, whereas the spring mass is 12 tonnes, approximately what is the frictional force that has to be generated at the wheel rail contact to avoid further outward slip of the rail’?

Answer 7

1 - Assume that the unsparing mass follows the Klingel Motion

2 - Max force required per axle = mass * acceleration

Question 8

What is the 3 step method to answer this question?

‘If the max friction angle at the contact angle is 10 degrees, what is the max speed possible before slip occurs ?’

Answer 8

1 - find the vertical force
2 - limiting horizontal force = vertical force * tan theta
3 - horizontal force = unsprung mass * acceleration ( solve for v)

Question 9

What is hunting motion calculated as ?

Answer 9

Acceleration

Question 10

You are told that the mass of a bogie is 6t and the dead weight from the wagon to the bogie is 12t.

How would you find the max sideways force, and vertical force?

In the question also ‘ assume all friction has to be supplied through one rail ‘

Answer 10

Max sideways force = mass (of bogie) * hunting acceleration

Vertical force = 1/2 * ( bogie mass + dead weight from wagon ) * 9.81

Question 11

You are told to resolve the sideways and vertical force in the contact plane, whose tangent is 0.2.

How would you find the slip force? How would you find the normal force?

Answer 11

Slip force = Fsideways * 0.981 - Fvertical * tan-1(0.2)

Normal force = Fsideways * tan-1(0.2) + Fvertical * 0.981

Question 12

You have calculated the slip force and normal force. From this how do you calculate the coefficient of friction? And therefore the friction angle ?

Answer 12

Coefficient of friction must be better than slip force / normal force

Taking tan -1 of this value gives us the friction angle.

Question 13

What is the specific weight, 𝛾, of ballast?

Answer 13

18 kN / m3

Question 14

This is the question: calculate the approximate sleeper end resistance offered by ballast that is heaped about 100mm above the level of the sleeper top.
Assume sleeper end dimensions of 0.24 m deep by 0.25m wide and a ballast internal friction of 50 degrees.

What is the process to calculate this ?

Answer 14

1) Calculate dead load stress at mid point of sleeper

2)Calculate stress ratio at failure

3) Horizontal stresss = stress ratio * dead load stress at mid point of sleeper

4)This gives a sleeper end force = horizontal stress * sleeper height * sleeper width

Question 15

What is the stress ratio at failure?

Answer 15

1 + sin angle / 1 - sin angle

Question 16

What does ‘r’ represent, and what is the equation to find r?

Answer 16

r represents the transverse reaction force from the ballast

The equation of r is 2ETA*alpha / R

Question 17

What value to we take for alpha when calculating the transverse reaction force ?

Answer 17

10^-5

Question 18

The question wants us to find the increase in temperature over the ‘stress-free temperature’ If we have calculated the sleeper end force in a previous question, how can we find this temperature?

Answer 18

Make it equal to the transverse reaction force from the ballast, and solve for T

Question 19

How do we derive r, the transverse reaction force from the ballast?

Answer 19

1) Londitiudanl stress = E* alpha * T

2)Longitudanal Force = 2 * E * alpha * T * A

3) Balance radically:

2 * longitudinal force * angle / 2 = r * R * angle

Solve for r

Question 20

How do we answer this question (fairly simple)

What additional force per sleeper will be generated by a train cornering at 100 km / hr, given a train mass per metre?

Answer 20

Outward = Mv2 / R

Question 21

A runaway train is travelling at 100km/hr. It is approaching a switch with a min radius of 800m and no transition. Track gauge = 1.5m. Centre of gravity is 1.8m above rail level.

What does gauge mean?
How do you calculate the angle of resultant to vertical?
How,do you calculate then max angle of resultant to vertical so that the train stays between the rails?

Answer 21

Gauge is the distance from the centre of rail to the centre of other rail.

Calculate the lateral acceleration and divide it by g. Take the inverse tan of this.

For these values, to find the max angle of resultant to vertical it is inverse tan ( 0.75/ 1.8 )

Question 22

A track is on a curve of radius R = 1500m.
You have just calculated the lateral resistance of ballast acting against sleeper end.
What increase in temperature above the stress free temperature is possible before the resistance of sleeper ends is overcome?
Estimate the extra resistance available from the friction against the underside of the sleepers.

What step by step method do you take to get the answer.

Answer 22

1 - find r in terms of T, assume sleeper spacing to find r in terms of T per sleeper

2 - total resistance = lateral resistance of ballast against sleeper end + specific weight of concrete * volume of concrete * tan 30

3 - make r equal to total resistance and solve for T