AN ANALYTICAL APPROACH FOR DYNAMIC RESPONSE

OF TIMBER RAILROAD BRIDGES

A. S. Uppal1, R. B. Pinknel and S. H. Rizkalli

Abstract

In the 1970's it was reported that there were approximately 2300 track miles of timber railroad bridges in the United States and Canada. For short spans, they offer an attractive alternative to other types of bridges as they are economical, faster to construct and easy to maintain. Current design prac­tices do not allow an independent consideration of the effects of the dynamic loads in sizing of the bridge components because very little information is available on the subject.

Dynamic tests were carried out in 1986 on timber bridge spans at two test sites using test trains consisting of a locomotive unit, two loaded hopper cars and a caboose.

This paper gives a brief description of the analytical approach employed for determining the dynamic response of timber bridge spans under railway vehicles travelling at a constant speed. The model comprises a multi-degree-of-freedom system with each vehicle having bounce, pitch and roll movements. Two parallel chords, each having its distributed mass lumped at discrete points, were used to idealize the bridge spans. A computer program developed on this basis was used to predict the loads at the wheel­rail interfaces and the vertical displacements at the discrete points on the spans.

The predicted loads were found to be in good agreement with measured values while maximum predicted displacements were up to 20% higher than measured ones.

The program was utilized to study the effect of speed and other factors on the dynamic response of open-deck and ballast deck bridges.

Introduction

The problem of the dynamic response of bridges has interested researchers since the middle of the nineteenth century. In 1851 Willis[1] gave an approximate solution for the case of a single constant load over a beam of a negligible mass. An exact solution of the equation he formulated was obtained by Stokes[2] in 1883. Later Timoshenko[3] in 1922 pointed out three major causes of vibrations in railroad bridges: the live load effect of a smoothly rolling load, the impact effect of the balance weights of the locomotive driving wheels, and the impact effect due to irregularities of the track and fiat spots in the wheels. He examined two possible extreme cases of the live load effect: that the mass of the moving load is either large or small in comparison to the mass of the beam, for a simply supported beam using the

1aegional Engineer - Bridges and Structures CN Rail Prairie Region 460-123 Main St. Winnipeg, Manitoba R3T 2P8

2 Associate Professor - Civil Engineering Dept., University of Manitoba, Wmnipeg, Manitoba, R3T 2N2

2

method of expansion of the eigenfunctions. Lowan[4) solved the later case with the aid of Green's func­tion. Timoshenko also solved the problem of the effect of the balance weights by using a harmonic force moving over a beam at a constant speed.

The problem involving both the load mass and the beam mass, being more complicated, was ex­amined much later by Jeffcott[5] whose iterative method became divergent in some cases. Different ap­proaches were attempted by Fryba[61, Wen[7] and Bolotin[8].

Inglis[9] in 1934 used harmonic analysis to solve several practically important cases of dynamics of railway bridges traversed by steam locomotives, i.e., motion of a concentrated force, sprung and un­sprung masses and harmonic forces acting on a beam, etc. His results were in excellent agreement with the experimental findings and were later compared by Chilver[10] with those arrived at by Mise and Kunii[ll] with the aid of elliptical functions.

Up to that time the vehicle had been idealized by a single mass point. However, around the early 1950's idealization of the vehicle as a sprung and unsprung mass was attempted. Hillerborg{12] was the first to obtain the solution of the motion of sprung masses on a beam by means of Fourier's method and the method of numerical differences. Further advances were made possible by the arrival of digital com­puters. The formulation involving both sprung and unsprung masses was solved by Biw et aI.[131 using Inglis' method and by Tung et aI.[14] using Hillerborg's method and was applied to vibration of highway bridges'

The use of high speed computers has allowed significant progress in research into the dynamic response of both highway and railway bridges. The vehicles as well as the bridge components have been idealized as multi-degree-of-freedom systems, with the equations of motions having been derived using d' Alembert's principle or Lagrangean energy equations and solved by finite element or other numerical methods. The work by Hathout[16f Chu[15) et aI, W'uiyachai et al(30), and others are significant.

Experimental work also followed the theoretical work. The first dynamic tests were reported by Robinson[l7] in 1883. Later the American Railway Engineering Association conducted extensive tests on railway bridges which were reported by Turneaure[18) in 1911 and Hunley[19] in 1936. Until that time all the tests were on steel bridges. Around the late forties, the Association of American Railroads[201 at the request of AREA Subcommittee No.7, conducted tests on timber railway bridges for the first time as a part of their total bridge dynamic test program. Tests on timber bridges have been reported by Ruble(27], Drew [28] and Uppal and Rizkalla[21]'

The purpose of this paper is to describe briefly the analytical approach used for determining the dynamic response of timber railway bridge spans. The predicted loads at the wheel-rail interfaces, verti­cal displacements and accelerations at mid-points of a ballast -deck span and an open-deck span are com­pared with those obtained experimentally.

The analysis is used further to study the effects of other parameters, such as the train speed and the train consist, etc., on the dynamic behaviour of the spans.

Analytical System

A multi-degree-of-freedom vehicles--span model was considered for the analytical system. It consisted of a maximum of four railway vehicles coupled one to another with universal joints to simulate the test train (made up of a locomotive, two open-top hopper cars loaded with ballast and a caboose) used for the experimental work. Each vehicle in the train was assumed to possess three degrees of freedom, i.e., bounce, roll and pitch. The bridge spans consisted of two parallel chords. Each chord was divided into a number of equal segments and it was assumed that the distributed masses of the chords were lumped at discrete segment connection points or nodes.

---------- -- -

3

Vehicle Model

Each vehicle of the system comprises a car body supported by dual axle trucks at each end. The body rests on the bolster centre plate with or without stops mounted on the side frames. The analysis con­sidered the car body as a rigid body.

The major truck components[22,29] are the two side frames, the bolster and the two wheel sets. The wheel set has two wheels rigidly connected by an axle which is assumed to be isolated from the truck frame by a primary suspension system, which consists of the bearing box and the side frame, and by the flexibility of the side frame itself. The only flexibility in this connection is due to the bending of the side frame, while damping is provided through friction of the bearing boxes sliding vertically in their guides.

The secondary suspension consists of the coil springs between bolsters and side frames, friction snubbers that also act between side frames and bolsters and friction at the center plate that resists rota­tion of the truck relative to the car body. The side frames also prevent the car body from rolling exces­sively. See Figure 1.

Assumptions:

Each vehicle has been idealized as a rigid body and four axle-sets with three degrees of freedom corresponding to bounce, )'b, pitch, %, and roll, Ilt., as shown in Figure 2. The two dual-axle trucks are assumed to be part of the vehicle body. The mass of the body is considered to be concentrated at the centre of gravity of the vehicle. The vertical springs in the primary suspension (i.e. between the wheel­axle set and the truck frame, with spring constant, kyp) and the secondary suspension system (i.e. between the vehicle body and the truck frame, with spring constant,kys) are treated as linear springs acting in series with an equivalent spring constant of kyo

The damping in the suspension systems of the vehicle is small and assumed not to change while the vehicle traverses a short bridge span and is therefore neglected. The effects of lateral or longitudinal movements in the vehicle components caused by hunting, sway or braking action are neglected. The cou­plings between the vehicles are provided by universal joints and so no motion is transferred from one vehicle to another. All vehicles in a train cross the bridge span at a constant speed.

Equations of Motion:

Assuming no damping in the suspension systems and using Newton's second law of motion, the equations of motion for a vehicle with three degrees of freedom can be expressed as follows:

Vertical Displacement

Pitch Displacement (1)

Roll Displacement

where: i. i

Yr = ()'b1 ± II %1 ± de 6 bl - Ub) and

I Ub = the vertical displacement of the wheel rail contact point for the ilb wheel at any time t.

Bolster

Side Frame

Wheel Sets

4

V .... I. Bod,

+ I'IiInarJ s.por.ioo ~----ko, -, 5 ....

Suspension System

Figure 1. Detail of typical bolster, side frame and wheel sets.

Bounce 'Yb

-x

7 Pitch

Figure 2. Idealized vehicle model.

5

By substituting the value of Yri and rearranging the terms, Equation 1 can be represented in the following matrix form:

o o

=

or simply

(2)

where 1 refers to vehicle number 1.

Similar expressions can be derived for other vehicles in the train and the entire train can be rep­resl:nted in matrix form.

Bridge Span Model

A timber railroad bridge[23] consists of relatively short spans supported by bents. The spans are made up of structural members called stringers which run parallel to the track. The stringers may be simp­ly supported or, alternatively, may be continuous over the bents and may be spaced apart or closely pack­ed together in a chord under each rail. The spans are often classified according to the type of deck they carry, i.e., a ballast deck or an open deck as shown in F"tgure 3. In a ballast deck, the track ties are par­tially embedded in ballast which is laid between the rails and wooden flooring planks secured to the stringers whereas in an open deck the ties are laid transversely between the rails and the stringers.

Assumptions:

A bridge span comprises two parallel chords (i.e., beams) which are simply supported over bents as shown in Figure 4. Each chord was divided into a number of equal segments approximating the tie spacing in the case of an open deck. The distributed mass of chord, deck and track was considered to be concentrated (or lumped) at the segment connections or nodes. Only a vertical degree of freedom was assigned to each node and only the fundamental mode of vibration was considered. All displacements were assumed to be small. The effect of rotary inertia was neglected. The span material was assumed to possess linear behaviour. The experimental work confirmed this to be valid within the limits of the operat­ing loads. The span was considered to have viscous damping, which depends on the velocity of vibration.

The bridge span was assumed to be at rest before the train of vehicles entered the span.

Equations of Motion:

For a dynamic system possessing stiffness and damping such as a stringer chord with lumped mas­ses, the following equations of motion were obtained by means of d' Alembert's principle.

[Me] {~t)} + [Dc){u(t)} + [Kc]{u(t)} = {Fc(x,t)} (3)

in which

[Me] = mass matrix of chord with m masses lumped at the nodal points. This is a diagonal matrix with m=(w+Agp)ls/g

[Kc] = stiffness matrix of chord. The stiffness matrix of a chord was obtained by inversion of the flexibility matrix, the elements of which were obtained by summation of the flexibility influence coefficients [24]. This is a symmetric matrix.

8

[Dc] = damping matrix of chord. The viscous damping was taken as a linear combination of (Me] and [Ke], i.e.,

(Dc] = a (Me] + [3[Kc)

whlch for normal modes gives

(Dc] = 2 E (J)n (Me] (4)

where:

E = damping coefficient of chord as a fraction of critical damping

(J)n = circular frequency for nth mode. For n = 1, it is the fundamental circular frequency.

u = vertical displacement at nodal points.

Similarly the equations of motion for the second chord were derived and the two sets were com­bined to form the equations of motion for the bridge span.

Vehicle--Bridge Interaction

The vehlcle--bridge interaction takes place at the wheel--rail contact surfaces. The load that a wheel exerts on a rail is a function of the masses and the suspension systems of the vehicle and the charac­teristics of the span. These loads at the wheel-rail interfaces fluctuate continuously as the vehicles move over the bridge spans.

Assumptions:

The wheels of the vehicle were assumed to remain in contact with the rails at all times. The sur­faces of the wheel treads were assumed smooth and round. The track surface irregularities were assumed to be small and negligible. The rails and bridge ties for the open deck and the flooring planks for the bal­last deck were assumed to be pin-connected at nodal points to the stringers.

. Let us consider the ith wheel. There are two masses, a sprung mass (i.e., part of the vehicle body), Ms', supported by a spring system of stiffness ky (damping in vehicle assumed to be zero) and the unsprung mass, M~ (i.e., the wheel and half of the axle), which is always in contact with the rail.

The load at the wheel--rail interface pi for the ilh wheel is given by the following expression[25):

(5)

or, by rearranging the terms, . 1'1 i i

F' = Mu(g-u,) + ~ yr+ Msg

where

y~ = 6'br ± li%r ± dc~r - u:,) and r refers to the vehicle number. i .. ..

The dispacement lib can also be expressed in terms of nodal displacements ul, U'j + 1, ul', u'l + 1 using linear interpolation by the following relationship.

Ballast Deck Bridlle Spa. N' 53

7

Open Deck Bridlle Span N" 52

F1gU1"e 3. Typical bridge spans.

Two porollel chords with moss •• I~",ped 01 node,

Figure 4. Idealized bridge model span.

8

III • uL UJ+l'

FJgUI'e 5. Interaction between wheel and rail.

Differentiating the above equation twice

ii, = :y (uit1j + 1 + ~~) + "y (uiti(1-l + ~i~~') (6)

The contributions of the effect of the ilh and i + 1st wheels on the chord segments j, j + 1 and j; fi- 1 were obtained assuming linear interpolation function and generalized coordinates[26] for rigid body masses, stiffness, damping and interaction forces.

Overall Dynamic System

Each chord was divided into no = n + 1 equal segments or n effective nodal points. Every node was assigned one degree of freedom, namely, the vertical displacement. Therefore a bridge span pos­sessed 2n degrees of freedom. Further, there were three degrees of freedom assigned to each vehicle, so a train consisting of lear number of cars had 31 car degrees of freedom. The overall dynamic system there­fore comprised 2n + 31 car degrees of freedom.

From Equations 1 to 6, the equations of motion for the overall train--bridge span system may be expressed as follows:

" . [Mo] {D} + [Co) {O} + [Ko) {O} = {Fo} (7)

in which

9

[Mo] , [Co] and [Ko] are, respectively, the overall matrices of mass, dam~ an~ stiffness and {Fo} is the vector offorce including the effect of vehicle-bridge span interaction. {O} {O} and {O} are the vectors of accelerations, velocities and displacements, respectively at the nodal points and other locations. The sizes of the matrices and the vectors for the overall system depend on the number of seg­ments a bridge span is divided into and on the number of vehicles considered in a train.

Computer Program

A computer program was developed to solve the equations of motion for the overall dynamic sys­tem using different methods of numerical integration. The outline of the procedure of analysis employed in the program is given in the following steps:

i) Compute the constant parameters of the system and construct the mass and stiffness matrices for each vehicle and each chord individually from Equations 2 and 3, respectively.

ii) Obtain the fundamental circular frequency of the chords by eigenvalue analysis, assuming undamped harmonic motion. Choose the damping coefficient for the chord and construct the damping matrix for the span using Equation 4.

iii) Establish the distance vectors from configuration of wheels in each vehicle and distances between the vehicles as shown in Figure 6.

iv) Choose a time step and calculate the position of the wheels by algebraically adding Y1 = v t to all the terms of the distance vector and determine the number of wheels on a chord segment.

v) FO.r every whee~ determine its position with respect to the chord segment it occupies, i.e., the distance r , from node j and j + 1 (or j and j + 1 for the other chord) as shown in Ftgure 5 and, using the general coordinates for mass, stiffness and interacting force, determine the contributions of the wheel posi­tions to be added to the overall mass, damping and stiffness matrices and force vectors.

vi) Formulate the equations of motion of the overall dynamic system by constructing the overall mass, damping and stiffness matrices and the force vector.

vii) Solve the equations of motion for the overall system by numerical integration using a) Newmark's 13-method, b) Wilson's O-method or c) Houbolt's method to find the dynamic displacements, velocities and accelerations, etc.

viii) Choose the next time step and repeat the above procedure until the last axle of the train has gone past the span.

The computer program, written in FORTRAN, is quite flexible in that it can be used for any length of span having uniform or variable geometric properties. Though at present no provision exists for track irregularities, the program could be adapted to incorporate the track line and surface irregularities. Up to four vehicles are currently included in a train but the program can be expanded to include more than four vehicles. Initial values of displacements, velocities and accelerations for different degrees of freedom can be specified both for vehicles and span for predicting the dynamic response of the system.

Numerical Example

The numerical example is based on span no. 3 of the ballast deck and span no. 2 of the open deck test bridges, respectively, as shown in Ftgure 7, for which the measured data[21] are available under the test train no. 2 shown in Figure 8.

The data on the spans and the test train used as input for the program are given in Tables 1 and 2.

•

• '.1. .. T

;r

10

.. ~~~ .., 1 2! :H '!! !' 12 10

T.-$. ~ 1 i 2, 127 2!1 ....

.I.. -T 15 13 I " •

I • • I

VEHICLE 4 VEHICI.E ;, VEHICI.E 2

PLAN VIEW

Figure 6. Train model.

Table 1. Bridge spans data.

Particulars

1. Span length, l(in) 2. No. of stringers per chord 3. Effective no. of stringers per chord 4. Nominal size of stringers (in x in) 5. Density of Douglas Fir, p (Ib/in") 6. Weight of track and deck per chord,

III (Ibfm) 7. Damping coefficient as percentage of

critical damping, ~ 8. No. of segments Ichord, tis 9. Center to center spacing of chords, d (in) 10. Dist. ofl'! rail to near side chord, dn (in) 11. Dist. of 2nd rail to near side chord, dr (in) 12. Modulus of elasticity of Douglas Fir, E

(Ib/in2)

Ballast-deck bridge, span 3

144.00 5 4

8x16 0.34722 x 10-1

%.00

9.8

10 60.28 0.97

59.97 1.65 x 106

Notes: i) Acceleration due to gravity, g = 386.4 in/sec2

ii) One inch = 25.4 mm

(~~ !H-I I

•. 1. 6 4

-r.o. T' .~ ~. • • 11. ..., 1'\ 5 S

, VEHICI.£ I

Open-deck bridge, span 2

138.00 4 3

8x16 0.34722 x 10-1

23.00

6.2

10 60.97 1.03

60.03 1.65 x 106

i! .

I

END VIEW

h-...

11

I Track

I

t if a t=

FJgIlfe 7(a). Ballast deck bridge.

r Track

Figure 7(b). Open deck bridge.

Table 2. Vehicle train data.

Particulars Locomotive OTHCar OTHear Caboose CN#5608 CN#090159 CN#090151 CN #79715

1. Body mass, Mb (lb-sec2(m) 654.14 484.42 529.50 157.40 2. Sprung mass, Ms associated 70.72 56.02 61.66 15.10

with each wheel (lb-sec2(m) 3. Unsprung mass, Mu associated 11.05 4.53 4.53 4.53

with each wheel (lb-sec2(m) 1.98xl07 1.66 x107 1.66 xl07 0.27x107 4. Body pitch moment of

inertia, Ib (lb-in-sec2) 1.17 xl06 1.28xl<f 1.28xl<f 0.24xl06 5. Body roll moment of

inertia, Jb (lb-in-seJ) 6. VertiallspringstiftDe~hee~ 3,324. 11,020. 11,020. 1,600.

Kv (lb(m) 7. Half OOt. between 204.00 190.25 109.25 164.88

truck centers, It (in) 8. Half OOt. between two wheel-axle 54.00 34.00 34.00 34.00

sets of a truck, lw (in) 9. Half 001. between two wheel-rail 29.50 29.50 29.50 29.50

points of a wheel-axle set, de (in) 10. Dist. between last axle of one vehicle 0.00 138.25 82.50 118.50

and fIrst axle of following vehicle, Iv (in)

Notes: i) The vertiall damping constant of vehicle(s), Cv taken as 0 lb-sec(m. ii) Time step used for speeds less than 30 mph = 0.005 seconds or more and for speeds 30 mph

and over = 0.001 seconds.

---------------------------------------- ---------

12

Computer Output

Figures 9 and 10 show typical computed displacement versus time plots for the mid-point of spans 3 and 2, respectively, for a train speed of 30 mph_ The maximum values of displacement occur at axle 9 and axle 4, respectively.

Figures 11 and 12 show typical computed acceleration versus time plots for the above cases. The maximum ranges of acceleration values for the ballast-deck and open-deck spans were + 0.86, -0.82 g and + 1.70, -1.49 g, respectively.

The effect of the train consist, i.e., the number of vehicles on the mid-point displacement of span 52 of the open-deck bridge, was also computed and the maximum values at a speed of 50 mph were found to be as follows:

i) ii) iii) iv)

Table 3A. Computed vertical displacements (mm) at mid-point of open-deck bridge, span 52.

Train Consist Left rail Right rail

Locomotive 4.04 4.05 Locomotive and one car 4.94 4.91 Locomotive and two cars 5.23 5.21 Locomotive, two cars and caboose 5.20 5.22

From the above values, it can be seen that, at a constant speed of 50 mph. as the consist increases in length the vertical displacements increase.

Table 3B. Computed accelerations (g) at mid-point of open-deck bridge, span 52.

Train Consist Left rail Right rail

i) Locomotive Max +5.22 +6.76 Min -4.89 -6.57

ii) Locomotive and one car Max +6.34 +8.04 Min -5.54 -10.14

iii) Locomotive and two cars Max +7.43 +732 Min -7.80 -7.39

iv) Locomotive, two cars and caboose Max +3.70 +3.69 Min -3.03 -2.89

The acceleration values do not seem to indicate a definite relationship with the train consist.

The displacement versus time plots for a locomotive, a locomotive and a car, a locomotive and two cars and for the full train are given in Figures 13A, B, C and D, respectively.

Comparison with Measured Data

Loads at Wheel--Rail Interfaces

The maximum value of the loads at the wheel--rail interface, as predicted by the analytical model and as measured from the tests in the field under test train 2 at the mid-point of the spans, are given in Table 4.

------------ ---

Speed (mph)

1 10 30 50

Speed (mph)

1 10 30 50

13

Table 4. Maximum loads at wheel-rail interfaces (kips).

Left rail

31.23 31.77 34.70 36.47

Left rail

31.21 3U19 33.68 37.54

A) Ballast-deck bridge, span S3.

Predicted

Right rail

31.78 32.02 35.00 36.68

B) Open-deck, bridge span S2.

Predicted

Right rail

31.18 31.62 34.31 37.42

Measured

Left rail Right rail

34.54 34.24 3555 36.04 35.00 35.70 36.00

Measured

Left rail Right rail

34.62 33.06 35.60 34.18 40.77 36.04 31.20 34.57

The predicted values of maximum loads at the wheel-rail interfaces are based on absolutely smooth wheel and rail surfaces which in fact have irregularities, however small they may be. These ir­regularities affect the loads. Further, the number of measured values is not sufficient to lend a fair com­parison. Despite this, in most cases the difference between the two is quite smaIL

Vertical Displacements

The maximum values of the predicted and measured net displacements at the mid-points of the spans for the above cases are as follows:

Table 5. Vertical displacements (mm).

A) Ballast-deck bridge, span S3.

Speed (mph) Predicted Measured net

Left rail Right rail Avg Left rail Right rail Avg

1 3.11 3.18 3.15 3.35 2.16 2.76 10 3.42 3.49 3.42 3.38 2.24 2.81 30 3.48 3.56 352 359 2.13 2.78 50 3.99 4.07 4.03 3.52 2.30 2.81

•

Speed (mph)

Left rail

1 4.41 10 439 30 4.56 50 5.20

Note: i)

14

B) Open-deck bridge, span S2.

Predicted Measured Net

Right rail Avg Left rail Right rail Avg

4.42 4.41 3.10 3.17 3.14 4.36 4.38 3.30 3.17 3.23 4.57 4.53 435 3.24 3.71 5.22 5.21 4.92 5.13 5.02

Measured net displacement is equal to the actual measured displacement less displacement due to compliance. i.e. tightening of the components of a span. The compliance for the open-deck span = 3.19 mm and for the ballast-deck span = 1.87 mm.

From the above it may be noted that the maximum predicted displacements increase in value with increase in speed and their average values are up to 20% higher than the measured displacements. This was expected because the analytical model assumes the spans to be simply supported whereas, in actual fact, they were'semi-continuous over their supports.

Locomotive Hopper Cor N°·1 Hopper Cor NO'2 Caboose

._~4JiJ~YmI-IJ&J~ADonn , . , . • . - - t--t' ~

• ••

. L . , '-

, 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16

1'.,- ."-0" IS'·O· ~..,. ,t .. t. s· ... Z6'-O~ rt·~ "«>'l rtf/" 1'''' "-10%- 1'''' 11'·.""- rt ..

Figure 8. Test train.

Accelerations:

The maximum values of the computed and the measured accelerations at the mid-points of the spans for the above cases are as follows:

Table 6. Maximum & minimum accelerations (g).

A) Ballast deck-bridge, span S3.

Speed (mph) Predicted Measured Left rail Right rail Left rail Right rail

1 Max -0.08 +0.02 +0.75 +0.08 Min +0.07 -0.02 -1.06 -0.13

10 Max +0.21 +0.21 +4.52 +1.11 Min -0.24 -0.25 -5.18 -0.81

30 Max +0.86 +0.72 +4.10 +5.86 Min -0.82 -0.73 -4.86 -2.09

50 Max +137 +1.49 * +3.16 Min -1.38 -1.43 -7.00 -4.65

.. ·s Axle Spacing

15

Table 6. (continued). B) Open-deck bridge, span S2.

Speed (mph) Predicted Left rail Right rail

Measured Left rail Right rail

1 Max +0.19 +0.06 +0.23 +0.33 Min -0.19 -0.05 -0.21 -0.38

10 Max +0.45 +0.46 +5.78 +3.07 Min -0.43 -0.42 -3.63 -2.51

30 Max +1.71 +1.49 • • Min -1.60 -1.44 • •

50 Max +3.70 +3.69 • • Min -3.03 -2.89 • •

• ± 10.08 g was the limit set for measurement; these values exceeded the limit. The predicted values of the accelerations were very low compared to the measured ones. This is because the computed values were for the chords whereas, the measured ones are for stringers only that are located directly under the rails.

4

3.5 -

.3 -

E 2.5

:: ~ 2 c -• to ~ v 1.5 0 -0. ~

0 -

0.5 -

0

-0.5

o

h. I! •. il •. f" rv' rv' .

, , , 2 3

Time (So<or>ds)

Ftgure 9. Computed displacement versus time. Ballast-deck bridge, span S3 - train at 30 mph.

~ !\ \. J \ ,.

··1-- I

4

0.9

0.8

0.7

0.6

O.~

0.4

0.3

,:~ 0.2

.~ 0.1

" 0 i -0.1 .J ., :i -0.2

-0.3

-0.4

-0.5

-0.6

-0.7

-O.B

-0.9

0

16

I I

! If' /\ ! I \. I j . ~ I \ --III,w...I---4--'oI--'----1'O/i"-.J.-- . -.--.~--.-.

2

lime (Ser.vn..:i'S)

Figure 10. Computed displacement versus time. Open-deck bridge, span S2 - train at 30 mph.

2

nrne (S.-cor\ds)

FIgUI"e 11. Computed acceleration versus time. Ballast -deck bridge, span S3 - train at 30 mph.

4

4

17

\'B

1.6

\.'

1.2

0.8

0.6 ,~ 0.'

j 0.2 '5 .! 0 • 'J ., -0.2 ~

-0.4

-0.6

-0.8

-1

-1.2

-1.4

-1.6

0 2 •

Figure 12. Computed acceleration versus time. Open-deck bridge, span S2 - train at 30 mph.

4.5

4

3.5

-

r A A 1\

3 -

£ 2.5 -:;. -

c ~ 2 ~ • c' 0 ~ 1.5 -• ;5

0.5 -0 "LA -'" 1>-"" \

v ~

-0.5 . a 0.2 D.' 0.6 a.B

Time (Secondt)

rIglll'e 13A. Computed displacement versus time. Open-deck bridge, span S2 -locomotive at 50 mph.

18

5

4 -1\

E 3 - -! c • ~ E 2 • v 0 0. • is

-

\ -

~AA 11\". V'¥

VV y o

, , . -1

o 0.2 0.4 0.6 O.B 1.:2 1.4

llme (S&conds)

Figure 13B. Computed displacement versus time. Open-deck bridge, span S2 - locomotive and one car at 50 mph.

6

~

4 -.~

E

c 3

• E • v 2 0

. .

~ .

V V J J -a • Ci

-

o !u U~~AI 1 UI UYU

. , . , . . -1

o 0.2 OA 0.6 O.B 1.2 r.4 1.6 1.6 2

Time (SecOndi)

FIgure 13C. Computed displacement versus time. Open-deck bridge, span S2 - locomotive and two cars at 50 mph.

E ~ C • E • v • 1i • 0

19

3

2

a 0.4 0.8 1.2 1.6 2

Tun. (Second.)

FJglUe 13D. Computed displacement versus time. Open-deck bridge, span S2 - full train at 50 mph.

Summary & Conclusions

2.4

i) An analytical approach has been presented for predicting the dynamic response of timber rail-road bridge spans. The computer program developed using this approach can be used for simply supported spans of steel or concrete bridges as well. The program could be expanded to include any number of vehicles in a train. With further modifications, the effects of wheel­surface and track-surface irregularities could be incorporated into the program.

ii) The predicted values of the maximum loads at the rail-wheel interfaces, the maximum vertical displacements and the accelerations were compared to those measured in the field and the results were as follows:

a)The predicted maximum loads were in good agreement with the measured values.

b)The maximum predicted vertical displacements were up to 20 % higher than the measured maximum values. This was expected because the analytical model assumes the spans to be simply supported whereas, in actual fact, they were semi-continuous over their supports.

c)The predicted values of the accelerations were very low compared to the measured ones. This is probably because the computed values were for the chords whereas the measured ones were for the stringers only located directly under the rails.

iii) For a constant speed, the maximum displacement values increase with an increase in train length.

20

Notation

1. General:

v "" Speed of train in inches/second g "" Acceleration due to gravity 386.4 in/sec2

t "" Time for the 1st axle of train taken to travel a distance of x inches from the left end of the bridge span in seconds

2. Bridge Span:

p "" Mass density of the material of chord in Ib-sec2rm' (d == Dead weight of track & deck material in lbsfm. Ag Gross cross-sectional area of chord in sq.in. I == Moment of inertia of chord material in inch 4.

E == Modulus of elasticity of chord material in Ibs/sq.in. ~ == Dampening coefficient of chord (as percentage of critical dampening). I == Length of span (center to center of support points) in inches. ls = Length of chord segment in inches. d == Distance center to center of chords in inches. dn = Distance between rail! (R.R.) and Chord 1 (R.Chord). dj = Distance between rail 2 (L.R.) and Chord 1 (R.Chord). r Distance of the ith wheel from node j.

(i == hls 'Y dn/d 6 == dUd

• pi == 1 _ IIi :y == 1-dn/d == 1-'Y

• '6 == 1-dr/d == 1-6

n == No. of active nodes Os == No. of equal segments in chord == n + 1

3. Vehicle (s):

Subscript r where r == 1.2 •... 4 is used to indicate the vehicle number. and superscript i where i == 1,2, ... nw is used to indicate the wheel number

WT= Ml> ==

It MS==

M~ :: )bt == Ib == ~t == Jbr = qbt = kVr =

Scale weight of a vehicle (locomotive or car) in Ibs. Body mass of vehicle r inel. truck frames in Ib-se~rm. Sprung mass associated with wheel i of vehicle MbI8 in lb-sec2 lin. Unsprung mass per wheel i of vehicle (ie. half the mass of axle-set) in Ib-sec2rm. Vertical displacement ofvehicle r in inches. Pitch moment of inertia of vehicle r in lb-in-sec2•

Pitch displacement ofvehicle r in radians. Roll moment of inertia of vehicle r in lb-in-sec2

Roll displacement of vehicle r in radians. Equivalent vertical spring stiffness per wheel (The vertical springs in the primary and secondary suspension are considered as linear springs in series) of vehicle r in Ibs(m.

Itt == One-half distance between the truck centers of vehicle r in inches. lwt == one-half distance between the wheel base (ie. between two wheel-axle sets of a truck)

of vehicle r in inches.

de =

Ii = r Iv.

uj =

21

One -half distance between the wheel-rail contact points of a wheel-axle set. 112( dc-dn) in inches. Distance of the centroid of vehicle r to the ith wheel in inches. Distance between the last axle of the 1st vehicle r and the 1st axle ofthe rear vehicle (i.e. r + 1) in inches. Vertical displacement of node j due to wheel i on segment betweeen nodes j and j + 1.

References

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22

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Acknowledgement

This study was carried out in the Department of Civil Engineering at the University of Manitoba with the financial assistance from the Transportation Institute of the University of Manitoba.