Problem Sheet 1

Question 1

What is lander in railway analysis?

Answer 1

The characteristic length

Question 2

What is lander measure in ?

Answer 2

m^-1

Question 3

For this question: 1 - Load distribution on individual sleepers from isolated train axles on elastic foundation.

What is the 4 step process to find the approximate load on the sleepers?

Also, when caclulating lander, why do you multiply the second moment of intertia by 2?

Answer 3

1 - Use the elastic foundation equation, and input values to find the value of lander

2 - Looking at the equation of y, it is apparent that ymax occurs when x=0, therefore find the value of ymax

3 - The stress at a max value of y can be found using the equation R = k x y

4 - As we now know the force / m, we just need to divide this by the distance between sleepers to find the load on an individual sleeper.

You multiply the second moment of intertia by 2 as there is 2 rails.

Question 4

From the original question: For this question: 1 - Load distribution on individual sleepers from isolated train axles on elastic foundation.

If axle pairs are spaced at 1.8m, how (approximately) does this affect the maximum load to
an individual sleeper?

What is the iterative way to find this?

Answer 4

Find y at x = 0.0, and 1.8
The x = 0.2 and 1.6
…. so on

The max value of y1 + y2 is the new largest deflection, which can be used to find the highest load on an individual sleeper

Question 5

What is the difference between sagging and hogging bending moments ?

Answer 5

Sagging is when a beam bends downward, creating tension at the bottom and compression at the top (positive moment, U-shape), while hogging is when it bends upward, causing tension at the top and compression at the bottom (negative moment, inverted U-shape)

Question 6

From the elastic foundation equation, how do you find curvature?

Answer 6

Curvature is approximately d2y / dx2

Question 7

What is the beam bending moment formula?

Answer 7

M = EI x curvature (d2y/dx2)

Question 8

What is the difference of finding the curvature for sagging and hogging?

Answer 8

Sagging is easy, you just calculate d2y/dx2 at x=0

Hogging is harder ,need to find location of max curvature, which occurs at d3y/dx3 = 0 … input that x value into d2y/dx2 equation.

Question 9

What is the maximum shear force on a rail?

Answer 9

It is just half of the wheel load

Question 10

For a beam in bending, what is the strain?

Answer 10

Distance from neutral axis x radius of curvature (d2y/dx2)

Question 11

What is the equation relating stress and Young’s Modulus?

Answer 11

Stress = strain * E

Question 12

What is the approximate maximum tensile stress in the lower flange of the rail directly under a wheel load. Assume a distance of 70mm from the neutral axis and a rail cross section of 75cm2.

How would you snawer this question?

Answer 12

Strain = distance from neutral axis * d2y/dx2 (when x=0)

Easy, you can then find there stress to be the value of strain calculated X E

Question 13

What does dy/dt equal?

Answer 13

dy/dx * dx/dt

Question 14

What is dx/dt?

Answer 14

The horizontal velocty

Question 15

From this question, what is the step by step process to answer it? If the train is travelling at 150 km/hr, with what downward velocity will the sleeper pass
through its initial level following uplift in front of the axle? (This could potentially generate
an impact between the sleeper and the ballast)

Answer 15

Find dx/dt, which is just the horizontal velocity in m/s

Find dy/dx through differentiating, and need to find slope when y = 0, therefore sin lander x + cos lander x = 0 ….

Plug in to find dy/dt

Question 16

What is the inertial force ?

Answer 16

The mass * the acceleration

Question 17

What is d2y / dt2, and the equation to find it?

Answer 17

It is the acceleration.

= d2y/dx2 * (dx/dt)^2

Question 18

How would you answer this question: What is the maximum inertial resistance to downward deflection offered by the sleeper if it has
dimensions 2.5m × 0.25m × 0.22m and it is made of concrete?

Answer 18

1 - Use equation of intertial force

2 - Find the acceleration, d2y/dt2 = d2y/dx2 * (dx/dt)^2

3 - Inertial force = mass * acceleration

Question 19

When does a maximum lateral tilt of a sleeper occur during a single axial pass?

Answer 19

When dy/dx is at a maximum, meaning that d2y/dx2 = 0

Question 20

How would you answer this question: What is the maximum lateral tilt of the sleeper during a single axle pass assuming that it is rigidly attached to the rail?

Answer 20

Occurs when dy/dx is at a maximum, so find the value of x when d2y/dx2 = 0, and input in to equation for dy/dx.