Lecture 10

MSc Transportation Engineering

Railroad Engineering

College of Engineering

University of Babylon

General Principles of Structural Design of Travelways

• Step 3 Determine Minimum Size and/or Minimum Thickness for Each Structural Component

Lecturer : Prof. Dr. Hussein A. Ewadh 2017-2018

Step 3 Determine Minimum Size and/or Minimum Thickness for Each Structural Component

• This step involves the determination of the minimum size and/or minimum thickness of each structural component such that the stress and/or strain within any component does not exceed the maximum allowable.

• This depends on the load transmitted from the wheels of the vehicles using the travelway and the strength of the subgrade.

• The design procedure presented is the AREMA method for railway tracks.

Design Parameters

• The track’s structural ability is its capability to withstand the stresses caused by the dynamic loading applied by the train wheels.

- The design parameters are;

• the dynamic load applied onto the track by the wheels,

• the rail support modulus,

• the maximum allowable tie-ballast bearing pressure,

• the maximum allowable stress on the subgrade,

• the contact stress between the tie plate and the tie,

• and the stresses due to flexure and fatigue on the rail.

• The rail support modulus (kr) is defined by AREMA as the load (in pounds) that causes a 25.4 mm vertical rail deflection/linear inch of track.

• Factors that influence the value of kr are the quality of the ties, tie spacing, tie dimensions, the quality of the ballast in terms of its cleanliness, moisture content, temperature, compaction and depth, and the load-carrying capacity of the subgrade.

Design Parameters

AREMA recommends the following maximum allowable stresses:

• Maximum allowable tie-ballast bearing pressure 0.4485 N/mm2

• Maximum allowable stress on the subgrade 0.1725 N/mm2

(it is recommended that lower values be used even for good subgrades but should be definitely reduced for poor-quality subgrades)

Design Parameters

• Maximum allowable contact stress between tie plate and tie (for wood) 1.38 N/mm2 (since tests have shown that this varies from 2.76 N/mm2 for hard wood to 1.725 N/mm2 for softwood).

• Maximum allowable bending stress in the rail 172 .5 N/mm2.

• The allowable bending and fatigue stresses in continuous rail steel are obtained from the yield stress and the endurance strength of the rail steel after adjustments have been made for the influencing factors .

Design Parameters

Design Parameters

AREMA reduction factors for each of the influencing factors and the temperature stress. AREMA recommended reduction factors to obtain permissible rail stress for continuously welded rails

Depth of Rail Track Ballast

• The minimum total depth of the ballasts (ballast and subballast) required below the ties can be determined by using either the Talbot equation, the Boussinesq equation, or Love’s formula.

• The Talbot equation is given as:

where

Pc = maximum intensity of pressure on the subgrade N/mm2 (maximum value 0.1725 N/mm2)

h = depth of ballast below ties, mm

Pm = intensity of pressure on ballast (2q)/Ab N/mm2 (maximum value 0.4485 N/mm2)

• Generally, half of overall ballast depth should be highest quality available, 6” minimum depth for subballast (AREMA)

Width of Ballast Shoulder at Ends of Ties

• In order to provide lateral support for the track, the width of the ballasts should be extended beyond the ends of the ties.

• AREMA has noted that if a tie is buried 102 mm into the ballast with a 152 mm ballast shoulder and carrying no vertical load, a force of approximately 446.4 kg/m will be required to move the tie 25.4 mm.

• However, the width at curves depends on the lateral force that is produced by the continuously welded rail on the curved track due to temperature changes.

• This force is given as

PL = 1.1812D(^T)

PL = lateral force, lb/linear m , D = degree of curve , ^T = temperature change, C

• The total lateral force acting between ties is therefore given as the lateral force/linear ft, PL multiplied by the tie spacing (ft).

• The additional width is obtained by dividing this force by the force that will cause the tie to move 25.4 mm.

• AREMA therefore suggests that wider ballast shoulders may be required to facilitate adequate lateral stability.

• Experience and local conditions should be used to determine when this is necessary.

Width of Ballast Shoulder at Ends of Ties

• The analysis of track behavior under vertical load is traditionally based on the presumption that the stresses and deformations in track elements can be determined by the application of the Winkler’s Hypothesis.

• The rail is considered a beam on a continuous elastic foundation. The basis of the hypothesis is a presumption about the proportionality between the load and deflection.

• However, it is empirically known that the track and track foundation elements in a real environment during the railway exploitation behave neither linearly, nor completely elastically.

Determination of Track Rail Cross-Section

Determination of Track Rail Cross-Section

• The cross-section of the rail is selected to ensure that the bending stresses within the rail do not exceed the maximum allowable.

• Winkler’s model for Beams on elastic foundations is the simplest both in terms of representation of the soil reaction at the footing soil interface as well as analysis of the resulting soil–structure interaction

• The bending moment and deflection are determined from the basic differential equation given as:

• in which k = spring or the soil constant, to be evaluated from suitable laboratory and field tests

Determination of Track Rail Cross-Section

• Since we deal only with point loads, the distributed load q(x) will not be considered here (q ≡ 0). The discrete wheel load P will be introduced later as boundary condition.

• After introduction of the short notation for spatial derivatives to x it follows:

The boundary conditions for the part x > 0 are: •𝑤(∞)=0; •𝑤′(0)=0; •𝑤"′(0)=𝑄/2𝐸I

Determination of Track Rail Cross-Section

Solving equation, the magnitude of the deflection w(x)at point X and the bending moment m(x) for A single wheel load can be given as:

The quantity L in these equations is the so-called characteristic length determined by:

Application of Winkler Equation

• Separate elastic supports may be replaced by an imaginary continuous foundation if we have at least four of the supports in the characteristics wave length (L).

•

• Since the maximum deflection and therefore the maximum bending stress occurs at the top of the rail that is immediately underneath the wheel, the maximum deflection and maximum bending moment occur at x = 0, which gives:

Determination of Track Rail Cross-Section

• If there are several wheel loads, as in the case of bogies, then the resulting deflection and bending moment are found by means of superposition.

Determination of Track Rail Cross-Section

Table for functions Ø (λ) for deflection, Ψ (µ )for moment

• Example: Consider the situation where an infinitely

long track is loaded by a two-axle with a half wheel-

base ℓ = 1.25 m. The characteristic length L is fixed

at 100 cm. UIC 54 rail with a bending stiffness of EI =

4.93 kNcm² is used. For a single wheel load Q = 125

kN, find the maximum deflection and the maximum

bending moment the deflection and bending

moment for the rail at points A and B.

• The maximum dynamic bending stress in the rail is given as:

• and for design purposes the section modulus of a suitable rail cross-section is obtained from:

Determination of Track Rail Cross-Section

Determination of Tie Plate Size

• It is necessary to determine the required size of the tie plate to ensure that the contact stress between the tie plate and tie is not higher than the specified maximum value.

• The area of the tie plate is obtained from Equation:

• The rail seat load is a function of the intensity of the continuously distributed load, p, against the underside of the rail and the deflection at the point that is the maximum deflection, and the elastic modulus of rail support for one rail.

• For a single wheel load, it is given as:

Determination of Tie Plate Size

Determination of Effective Bearing Area of Tie

• It is also necessary to determine the minimum effective bearing area of the tie to ensure that the contact pressure between tie and ballast is not greater than the maximum allowable.

• The maximum contact pressure occurs at the rail seat and the minimum is at the tie center.

• In order to simplify the calculations, the pressure distribution along the tie length shown in Figure 7.52 is assumed.

• The effective length (Leff) of the tie is therefore considered to be one-third of its length (L) and the effective bearing area (Ab) of the tie is given as

Pressure distribution along length of tie.

Determination of Effective Bearing Area of Tie

Practical Design

•Load distribution: – Bearing area is ⅔ of tie length (tamping

zone), so A'b= ⅔Lb

and unit load on ballast will be

pa= 2Qo/A’b

Thus pa= 3Qo/Lb

Where pa= unit tie pressure on ballast (<65 psi wood, <85 psi

concrete)

A'b= total tie bearing area, in2

L = tie length, inches

b = tie width, inches

Qo= Rail seat load, lbs (static or dynamic), based on trial tie

spacing

Determination of Effective Bearing Area of Tie

Subgrade Allowable Stress (Pressure)

•"Traditional" design value of 20 psi, AREMA recommends limiting stress to 25 psi

•Using soil strength with a factor of safety

-AREMA recommends a factor of safety of at least 2 and as much as 5 or more depending on the traffic (wheel loads and load repetitions) and soil conditions.

-Company design standards will dictate (e.g. Army allows a design unconfined compressive strength (qu) of 1.0 qu for "normal" traffic levels -less than 5 MGT/yr -and design of 0.8 quwhen traffic levels exceed 5 MGT.)

-Hay recommends factor of safety of 1.5 as applied to an ultimate bearing capacity of <2.5 qu, thus allowable stress <1.67 qu.