Journal of Sound and Vibration (1996) 198(2), 149–169

DYNAMIC ANALYSIS OF BEAMS ON AN ELASTICFOUNDATION SUBJECTED TO MOVING LOADS

D. T Y. Z

School of Civil Engineering, Queensland University of Technology, G.P.O. Box 2434,Brisbane, Queensland 4001, Australia

(Received 18 November 1995, and in final form 2 April 1996)

A simple procedure based on the finite element method has been developed for treatingthe dynamic analysis of beams on an elastic foundation subjected to moving point loads,where the foundation has been modelled by springs of variable stiffness. The effect of thespeed of the moving load, the foundation stiffness and the length of the beam on theresponse of the beam have been studied and dynamic amplifications of deflections andstresses have been evaluated. The technique is extended to the analysis of railway trackstructures, where the effect of the spring stiffness of the moving load is also incorporated.The entire analysis has been programmed to run on a microcomputer and gives fast andaccurate results. Several numerical examples are presented. The technique and the findingswill be useful in railway track design.

7 1996 Academic Press Limited

1. INTRODUCTION

Investigation of the response of beams on an elastic foundation subjected to static ormoving loads has attracted engineers and researchers for many decades. In his classicmonograph, Hetenyi [1] has presented a closed form solution for an infinitely long beamon an elastic foundation under static loads and series solutions for the cases of finite beams.The outcome of Hetenyi’s work and the subsequent work of others has been mainly appliedto the analysis and design of railway tracks, with most of the research pertaining to staticanalysis [2]. However, it is well known that when a structure is subjected to moving loads,there will be amplifications in the deflections and stresses in comparison to those obtainedfrom a static analysis of the structure subjected to the same loads. Although morecomplicated, the significance of dynamic analysis is thus evident [3]. Moreover, dynamicanalysis is an important aspect in any complete structural investigation. A significantfeature in the analysis of beams on elastic foundation is the rapid damping of the responseaway from the load [1, 2].

Timoshenko et al. [4] solved the governing differential equation for the dynamic analysisof a simply supported beam subjected to moving loads by mode superposition. They foundthat the maximum dynamic deflection was 1·5 times the static deflection when the traveltime was half of the fundamental period of the structure. Warburton [5] analyticallyinvestigated the same problem and found that the maximum dynamic amplification indeflection was 1·743, and that this occurred when the travel time was 0·81 times thefundamental period of the structure. This finding was later confirmed by finite elementanalysis. Dynamic response of multiple-span beams has been studied by Ayre et al. [6] andHonda et al. [7], where the effects of span length and the number of spans on the dynamicresponse were examined. Many researchers have used the finite element method to

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investigate the dynamic response of beams under moving loads [3, 8, 9]. The Wilson u

method or the Newmark method was used in the numerical integration of the governingequations of motion.

Dynamic analysis of beams on an elastic foundation (BEF) under moving loads hasreceived less attention, even though the results could be readily applied to the analysis anddesign of railway tracks. Timoshenko et al. [4] analytically solved for the free vibrationof beams on elastic foundations. Ono and Yamada [10] presented a classic method for freeand forced vibration analysis of BEF. Trochanis et al. [11] presented a method foranalyzing beams on elastic foundations under moving loads and applied the results to theanalysis of railway tracks. They used the fast Fourier transform (FFT) technique in theiranalysis. The beam on an elastic foundation under a moving load has also been treatedby Kenny [12], Fryba [13] and Fryba et al. [14].

Thambiratnam and Zhuge [2, 15] have developed a simple finite element model toanalyze simply supported beams on an elastic foundation (BEF) of any length. At first thismodel was employed in the static analysis of BEF and the results were applied to railwaytrack structures, and then the model was used in free vibration analysis of BEF. In thepresent paper, the technique is extended to the analysis of BEF subjected to moving loads.This extension has resulted in one simple procedure being now available for the static, freevibration and dynamic analyses of beams on an elastic foundation. Axial effects andvariations in foundation properties along the length of the beam can be easilyaccommodated in the method, which will yet have a significant reduction in complexityin comparison to any of the presently available methods.

In the finite element model, the elastic foundation is represented by springs with knownstiffness. The moving concentrated load is assumed to travel along the beam with constantvelocity. Newmark’s method is used in the numerical integration of the equations ofmotion. The length of the simply supported beam, the speed of the moving load and themagnitude of the foundation stiffness are the main parameters in the study. Time historiesof deflections and stresses can be obtained, from which the resulting dynamicamplifications can be calculated. The influence of the suspension stiffness on the dynamicamplifications is also investigated.

The effect of the length of the beam on the response was particularly studied in orderto extend the application of the method to railway tracks. Finally, the effects of twomoving loads travelling on a beam were studied. It is evident that by choosing anappropriate value for the length of the beam, and the number of elements to model thebeam, a simple but efficient method can be obtained for the dynamic analysis of beamson an elastic foundation or of railway tracks.

2. FORMULATION OF THE METHOD

2.1.

Consider an element ij of length L of a beam on an elastic foundation as shown in Figure1, having a uniform width b and a linearly varying thickness h(x). It will be a simple matterto consider an element having a linearly varying width if the need arises. Neglecting axialdeformations, this beam on an elastic foundation element has two-degrees-of-freedom pernode; a lateral translation and a rotation about an axis normal to the plane of the paper,and thus possesses a total of four degrees of freedom. The (4×4) stiffness matrix k ofthe element is obtained by adding the (4×4) stiffness matrices kB and kF pertaining to theusual beam bending and foundation stiffness respectively. Since there are four end

151

displacements (or degrees of freedom), a cubic variation in displacement is assumed, inthe form

v=Aa, (1)

where A=(1 x x2 x3) and aT = (a1 a2 a3 a4). The four degrees of freedomcorresponding to the displacements v1, v3 and the rotations v2, v4 at the longitudinal nodesare given by

q=Ca, (2)

where qT = (v1 v2 v3 v4) and C is the connectivity matrix for an element ij between x=0and x=L (Figure 1).

From equations (1) and (2),v=AC−1q. (3)

If E is the Young’s modulus, and I= bh(x)3/12 is the second moment of area of the beamcross-section about an axis normal to the plane of the paper, the bending moment M inthe element is given by

M=D d2v/dx2 =DBC−1q, (4)

where D=EI(x) and B=d2A/dx2 = (0 0 2 6x).The potential energy UB due to bending is

UB =12 g

l

0

d2vdx2 M dx (5)

which, upon using equations (3) and (4), becomes

UB = 12q

T(C−1)T6gl

0

BTDB dx7C−1q. (6)

Upon using Lagrange’s equations, the (4×4) element stiffness matrix kB is obtained fromthe potential energy of the element as

kB =(C−1)Tk�BC−1, (7)

where

k�b =gl

0

BTDB dx. (8)

Figure 1. A beam on an elastic foundation element.

. . 152

The above expression can be evaluated explicitly. If the element has linearly varyingthickness, this can be taken into consideration by using the appropriate expression for I.

The potential energy UF due to the foundation stiffness is given by

UF =12 g

l

0

vTkfv dx, (9)

where kf is the stiffness of the foundation. Use of equation (3) in equation (9) will yield

UF = 12q

T(C−1)T6gl

0

ATkfA dx7C−1q. (10)

Using Lagrange’s equations, the (4×4) stiffness matrix kF pertaining to the foundationstiffness is given by

kF =(C−1)Tk�FC−1, (11)

where

k�F =gl

0

ATkfA dx. (12)

The above expression can also be evaluated explicitly and, finally, the complete stiffnessmatrix for the element is

k= kB + kF . (13)

2.2.

For dynamic analysis, it is also necessary to derive the element mass matrix. The elementmass matrix is a matrix of equivalent nodal masses that dynamically represent the actualdistributed mass of the element. In this investigation, the mass matrix is derived byconsidering the kinetic energy due to lateral velocity. This is consistent with the derivationof the stiffness matrix where axial effects were ignored.

The kinetic energy of the element shown in Figure 1 is given by

T=12 g

l

0

(v)Tr dVv, (14)

where the lateral velocity v is given by the time derivative of the displacement v and r isthe mass density. Further simplification gives

T=r

2(q)T(C−1)T6g

l

0

ATh(x)A dx7C−1q. (15)

Using Lagrange’s equations, on the kinetic energy term above, the mass matrix m is givenby

m=(C−1)TmC−1, (16)

where

m= r gl

0

ATh(x)A dx. (17)

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Figure 2. A moving load on a simply supported beam on an elastic foundation.

2.3.

Consider a beam on an elastic foundation as shown in Figure 2, with a movingconcentrated force travelling along the beam. The beam has been discretized into a numberof finite elements. Following the usual procedure for stiffness analysis of structures, thegoverning equation of motion for the beam can be represented as

[M ]{q}+[C ]{q}+[K ]{q}= { f }=[N ]Tf0, (18)

where [M ] is the structure mass matrix, [C ] is the damping matrix, [K ] is the structurestiffness matrix, [N ]T is the transpose of the shape functions for the beam element [6] whichare evaluated at the position of the force, f0 is the magnitude of the concentrated force,and {q}, {q} and {q} denote the displacement, velocity and acceleration vectorsrespectively.

In equation (18), damping effects are neglected and the shape functions can berepresented as

[N]= [0 0 0 . . . N1i N2i N3i N4i 0 0 0 . . . ], (19)

where

N1i =1−30xl12

+20xl13

, N2i = x0xl −112

,

N3i =30xl12

−20xl13

, N4i = x$0xl12

−xl1%, (20)

in which i is the number of the element on which the load is acting.[N]T is a vector with zero entries except for those corresponding to the nodes of the

element on which the load is acting [3]. For a beam element in this study, the number ofnon-zero entries within the n×1 vector will be four. This 4×1 ‘‘sub-vector’’ is timedependent as the load moves from one position to another within an element. As the loadmoves to the next element, this sub-vector will shift in position corresponding to thedegrees of freedom of the element on which the load is then positioned.

The Newmark method of direct step-by-step integration is employed in the present studyto solve the governing equation (18) and the numerical procedure is implemented in aFORTRAN program. The program has been run on a mainframe computer by using theCONVEX-220 system.

3. NUMERICAL EXAMPLES AND DISCUSSION

In this section numerical examples are treated to illustrate the procedure, and the effectsof some parameters are investigated.

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3.1.

3.1.1. A simply supported beam without foundation stiffnessTo validate the method, a simply supported beam without an elastic foundation

subjected to a concentrated force moving with constant velocity, is analyzed and the resultsare compared with those from the existing analytical solution of Warburton [5] and finiteelement analysis of Lin and Trethewey [3], where the set of second order differentialequations have been solved by the Runge–Kutta numerical integration scheme.

Results for the dynamic amplification factors fD , defined as the ratio of the maximumdynamic and static deflections at the center of the beam, are computed and compared inTable 1, for different values of t/t, where t denotes the travelling time of the force movingfrom the left end of the beam to the right end, while t denotes the time after the movingload enters the beam from the left end.

It can be seen that the present results, obtained with only four elements modelling thebeam, compare quite well with the results of the others and thereby confirm the validityof the proposed numerical procedure. The maximum value of the dynamic amplificationis about 1·7 and occurs at t/t=1·234. The maximum discrepancy between the presentresults and those of others is less than 5%, which is acceptable, especially as the resultsin reference [3], obtained from a numerical model, and those in reference [5], obtained byconsidering only the first mode, may not be exact. With a finer mesh, results obtained fromthe present procedure, match more closely the results from the numerical approach inreference [3]. This problem is not directly relevant to the scope of the present study, butit was treated as a test case to validate the numerical procedure. The technique is extendedto beams on an elastic foundation in the next section.

3.1.2. The effect of foundation stiffnessA simply supported beam of length 10 m resting on a uniform elastic foundation, as

shown in Figure 2, is considered. The elastic modulus of the beam, E=2·05e11 N/m2, thePoisson ratio n=0·3 and the second moment of area I=1·844e–4 m4. The moving loadhas a constant velocity V=16·7 m/s (60 km/h), and the foundation stiffness kf is variedfrom 0 to 1·14e8 N/m2, and includes values typical for railway tracks. Results for thedynamic amplification of mid-span deflections are shown in Figure 3. It can be seen thatthe dynamic amplifications fD initially increase as kf increases. However, when kf isincreased beyond the value of 1·14e6 N/m2, the dynamic amplifications fD decrease. Thedynamic amplification is largest for kf =1·14 e6 N/m2, and this value is about three timesthe value for kf =0.

T 1

Dynamic amplifications, fD

fD

ZXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXVt/t This study Warburton [5] Lin and Trethewey [3]

0·1 1·040 1·040 1·0530·5 1·330 1·250 1·2521·0 1·710 1·710 1·7051·234 1·723 1·740 1·7301·5 1·630 1·710 1·7042·0 1·500 1·550 1·550

155

Figure 3. The effect of foundation stiffness on the dynamic amplification in mid-span deflection: V=60 km/h.

When the velocity is changed to V=8·1 m/s (29 km/h), similar results are obtained, asshown in Figure 4, where the dynamic amplifications at the beam center are plottedwith respect to the foundation stiffness. These results indicate that when a concentratedforce travels along a beam, the dynamic effects are greatly influenced by the foundationstiffness.

Figure 4. The variation of the peak value of the dynamic amplification in deflection with foundation stiffness:V=8·1 m/s; length=10 m.

. . 156

Figure 5. The effect of travelling speed on the dynamic amplification in mid-span deflections V=15, 30, 60and 100 km/h; kf =1·14e7.

3.1.3. The effect of travelling speedThe same beam as treated in section 3.1.2 is considered, but with the beam resting on

a foundation having a stiffness kf =1·14e7 N/m2. The velocity of the moving load is variedfrom 4·1 m/s (15 km/h) to 27·8 m/s (100 km/h). The results for the dynamic amplificationsin mid-span deflections are shown in Figure 5. It is interesting to note that the dynamicamplifications are reasonably constant for various travelling speeds, within the rangeconsidered. However, at smaller values of the foundation stiffness, the dynamicamplification increases slightly with travelling speed, as shown in Figure 6, which alsoshows a slight decrease in the dynamic amplification for the largest value of the foundationstiffness treated. Hence, when the foundation stiffness is relatively large, as in the case offoundations of railway tracks, the influence of the travelling speed, in the range15–100 km/h, is quite insignificant as amply demonstrated by the results in Figures 5 and 6.

3.1.4. The effect of the span length of the beamIn order to study the influence of the length of beam on its dynamic response, a simply

supported beam with a constant foundation stiffness kf =1·14e7 N/m2 is considered. Themoving load has a velocity V=8·1 m/s (29 km/h). The length of the beam is varied from5 m to 40 m and, for each case, the dynamic amplifications on the mid-span deflectionsare shown in Figure 7. It can be seen that when the span length Le 10 m, these dynamicamplifications on the mid-span deflection remain constant.

However, at lower values of kf this is not the case, as shown in Figure 8. When thefoundation stiffness kf is reduced to 1·14e2 N/m2, the peak dynamic amplification increaseswith the span, as shown in the figure, and no convergence is observed for the range ofL considered here. Practical values of kf are greater than 1·14e5 N/m2, and for this rangethe dynamic amplifications converge (to a value of approximately 3·6) for Le 20 m, andthe effect of speed is not significant for such cases.

157

3.2.

In the design of beams on an elastic foundation or of railway tracks, the most importantconsiderations are the allowable bending stress and allowable vertical deflection. In theprevious section, dynamic amplifications in deflections of beams on an elastic foundationsubjected to moving loads were treated and the influence of certain parameters was studied.In this section a similar treatment is presented for the dynamic amplifications in bendingstresses. For this purpose, the dynamic amplifications in stress fS can be defined as theratio of the maximum dynamic stress to the maximum static stress at the center of thebeam. The maximum bending stress sS in a beam may be calculated using the simpleapplied mechanics formula in equation (21):

sS =Mm/Z0, (21)

where Mm is maximum bending moment (kN m) and Z0 is the section modulus of the beam(m3).

3.2.1. The effects of the span length of beamA simply supported beam resting on a foundation with a stiffness kf =1·14e7 N/m2 is

considered. A concentrated load moves on this beam with a velocity V=80 km/h. Thelength of the beam is varied from 5 m to 40 m and the resulting dynamic amplificationsin stresses fS are shown in Figure 9. It can be seen that the values of the dynamicamplifications of the mid-span stress are reasonably constant for different span lengths.However, as shown in Figure 10, when kf was reduced to 1·14e2 N/m2, these dynamicamplifications increase with the span length of the beam and no convergence is noticedfor the range of span lengths considered. When kf =1·14e5 N/m2, the dynamicamplifications sensibly converge when Le 10 m, as shown in this figure.

Figure 6. Variation of peak values of dynamic amplification in deflection with travelling speed. +, kf =1·14e2;r, kf =1·14e4; ×, kf =1·14e5; e, kf =1·14e7.

. . 158

Figure 7. The effect of the span length of beam on the dynamic amplification in mid-span deflections.

Figure 8. The variation of peak values of dynamic amplification in deflection with span length. r, kf =1·14e2;×, kf =1·14e5; e, kf =1·14e7.

3.2.2. The effects of foundation stiffness

A simply supported beam of length L=10 m is considered, on which the load travelswith a velocity of V=16·7 m/s (60 km/h). The stiffness of the foundation is varied from0 to 1·14e8 N/m2. This range covers the typical foundations of railway tracks. For eachcase the dynamic amplifications fS were calculated and are presented in Figure 11. It canbe seen that initially, the dynamic amplification increase with kf . When kf is increased

159

Figure 9. The effect of the span length on the dynamic amplification in mid-span stress.

beyond the value of 1·14e5 N/m2, the dynamic amplifications decrease with kf . Theseresults are similar to those obtained for deflections. However, it is evident that the dynamicamplifications in the stresses are less than those in the deflections. The maximum valuesof the dynamic amplifications are about 3·6 for deflection, and less than 2 for stress. This

Figure 10. The variation of peak values of dynamic amplification in stress with span length. r, kf =1·14e2;×, kf =1·14e5; e, kf =1·14e7.

. . 160

Figure 11. The effect of foundation stiffness on the dynamic amplification in mid-span stress. V=16·7 m/s.

trend (viz., a smaller dynamic amplification for stress) has also been observed in the studyof dynamic amplifications in bridges due to moving loads [21], and could be explained asfollows. Stresses considered in these studies are those due to bending effects and areobtained from the curvatures or second derivatives of the vertical deflections. It is notnecessary for a function (deflection) and its second derivative (stress) to have the samedynamic amplification and in the present case amplifications in the second derivatives ofthe deflection are smaller than those in the deflection.

Beams on an elastic foundation with other support (boundary) conditions can be treatedwith equal ease by the numerical procedure established in this paper. In order to illustratethe versatility of the present model, the problem treated above is re-analyzed with the beamhaving fixed supports. Dynamic amplifications in the bending stresses fS are shown inFigure 12 for various values of foundation stiffness. It can be seen that the trends aresimilar to those for a simply supported beam, but with the maximum value occurring forkf =1·14e6 kN/m2.

3.2.3. The effect of travelling speedA simply supported beam of span L=10 m and resting on a foundation having a

stiffness kf =1·14e7 N/m2 is considered. The velocity V of the moving load is varied from60 km/h to 120 km/h. The results for the dynamic amplifications in stresses fS are shownin Figure 13. The values of the dynamic amplifications in the mid-span stress remains moreor less constant at the different speeds. It is evident that these results are similar to thosefor the deflections shown in Figure 5. The maximum dynamic amplification can beobserved at a speed of 100 km/h. Again, the dynamic effects on the stresses are smallerthan those on the deflections, the average peak dynamic amplification being about 1·6 forstresses and 3·7 for deflections.

161

In Figure 14 is shown the effect of the travelling speed on the dynamic amplificationsin the mid-span stress for different values of foundation stiffness. It can be seen that whenkf is reduced to 1·14e2 N/m2, dynamic amplifications increase with travelling speed, up toabout V=60 km/h, a trend observed earlier with dynamic amplification in deflections.This figure also shows that convergence characteristics depend on the foundation stiffness.

4. MODELLING INFINITELY LONG BEAMS ON AN ELASTIC FOUNDATION:APPLICATION TO RAILWAY TRACKS

4.1.

Despite over 100 years of operating experience, the design of railway tracks usuallydepends to a large extent upon the engineering experience of the designer [17, 18]. Theavailable design expressions are at best empirical. This is because the methods of analyzingrailway tracks are complicated, as the tracks are infinitely long and supported on beds theproperties of which can vary along the track length. Moreover, the dynamic responsecharacteristics of the tracks are not well enough understood to form a rational designmethod.

Although research in seeking simpler and/or improved methods of track analysis hasbeen going on, rail selection procedures and design have remained relatively static for over100 years [19]. This is exemplified by the continuous common adoption of the quasi-staticdesign approach together with the beam on elastic foundation analysis of tracks. However,various safety factors have been introduced to keep pace with the gradual increase inseverity of operating conditions. Tew et al. [19] have given a comprehensive coverage ofsome of the more important quasi-static methods. Kerr [20] has presented an interestingcompilation of papers on various aspects pertaining to railway tracks.

Figure 12. The effect of foundation stiffness on the dynamic amplification in mid-span stress: fixed beam.V=60 km/h.

. . 162

Figure 13. The effect of travelling speed on the dynamic amplification in mid-span stresses.

The current practice for designing the railway track structure is based upon satisfyingseveral criteria for strength of the individual components [18, 19]. The important criteriafor the railway track are the allowable bending stress and the allowable vertical deflection.For the purpose of analysis, a railway track is treated as a beam on an elastic foundation

Figure 14. The variation of the peak values of dynamic amplification in stress with travelling speed.r, kf =1·14e2; ×, kf =1·14e4; e, kf =1·14e5; E, kf =1·14e7.

163

Figure 15. The effective length of BEF for modelling railway tracks. Q, k/kf =0; q, kf/kf =0·1; R, k/kj =0·2.

(BEF)—a model first proposed by Winkler in 1867 and used by Zimmermann in 1888, over100 years ago. Using the BEF model, Hetenyi [1] analyzed railway track under static loadsby solving the governing differential equation. Hetenyi’s classic solution is still used, oftenin the form of a computer program, to analyze and design railway tracks. The approachis quasi-static where the design load is obtained by multiplying the static wheel load byone of many available impact factors [17, 19] to account for dynamic effects. There areseveral such formulae, each with its merits and/or limitations. For want of a morecomprehensive dynamic analysis procedure, this is the method used at present.

There are, however, some shortcomings in the present method of analyzing anddesigning railway tracks: the analysis is tedious and not well understood; and the effectsof moving loads cannot be fully accounted for by using impact factors in a quasi-staticapproach and variations in track and/or foundation properties along the length cannot beaccounted for.

In an earlier paper by Thambiratnam and Zhuge [2], a special technique was proposedto handle infinitely long beams (or tracks) by using an equivalent finite beam for staticanalysis. The length of this beam was chosen so that when it is subjected to a concentratedload the response curves for the deflection, the bending moment and the shear force arerapidly damped away from the load [1]. This equivalent beam of finite length has beentested to give converging results. The stiffness of the spring represents the stiffness of therail foundation, which comprises the sleepers, the ballast and the subgrade.

4.2.

To extend the present procedure to the analysis of railway tracks, it is necessary toidentify the required span length of the beam on an elastic foundation. The rapid dampingof the beam response away from the load enables the track to be modelled as a finite beamon an elastic foundation. The point load moving at a constant speed is representative ofthe usual wheel load, currently used in the analysis and design of railway tracks.Convergence studies for beam deflections and stresses carried out in the earlier sectionsof the paper can be used to obtain this finite span length, with the appropriate value forthe foundation stiffness kf . For rail foundations, the range of values for this stiffness isfrom 5·2e6 N/m2 to 3·54e7 N/m2. From the results in the earlier sections, it can be seenthat, for this range of kf , dynamic amplifications in both deflections and stresses convergefor span lengths Le 10 m. The corresponding value of the dynamic amplification in stressis about 1·6, in the absence of damping.

. . 164

4.3.

It is necessary to address the applicability of the simple model developed herein torailway track analysis. Railways track which have been modelled and analyzed as beamson an elastic foundation have stood the test of time. However, the dynamic effects on thetrack response have not been fully understood, as mentioned earlier, and in this paper anattempt is made to investigate the dynamic amplifications, in order to provide a morerealistic design. As with other track analysis models, the model proposed in this paper alsorelies on the same beam on elastic foundation concept. Real loads on railway tracks aresprung masses, although they have been simplified up to now as moving point loads. Inthis section the effect of modelling the moving load as a sprung mass will be investigated.Following a procedure similar to that used by Lin and Trethewey [3] and ignoring dampingand the mass of the wheel, the governing equations can be derived as

$[M]0

[N]Tm1

m1 %6qy7+$ [K]−k[N]

0k%6qy7=6[N]Tm1g

0 7, (22)

where m1 is the sprung mass, k is the spring stiffness and y is its deflection, measured fromthe static equilibrium position before the moving load enters the beam. Double dots abovequantities denote second time derivatives of those quantities. The other notations are asbefore. A convergence study was carried out to determine the effective length of the simplysupported beam on an elastic foundation to be used in modelling railway tracks. In Figure15 it is shown that when Le 10 m, dynamic amplification in the mid-span bending stressconverges for stiffness ratios in the range 0Q k/kf Q 0·20, where k is the stiffness of thesprung mass. Analogous results were obtained for fD , the dynamic amplification indeflection. Therefore, to illustrate the effect of the sprung mass on the dynamic

Figure 16. Dynamic amplifications in mid-span deflections: the sprung mass case.

165

Figure 17. Dynamic amplifications in mid-span stresses: the sprung mass case.

amplifications in the railway track response, a 10 m long simply supported beam restingon an elastic foundation with kf =1·14e7 N/m2 is considered, on which the mass moveswith a velocity of 16·6 m/s (60 km/h). Dynamic amplifications in the mid-span deflectionsfD and the mid-span stresses fS are shown in Figures 16 and 17 for different values ofthe stiffness ratio k/kf . It can be seen that the effect of the spring stiffness is a reductionin the dynamic amplifications. When k/kf =0·2, there are roughly 21% and 26%reductions in the deflection and stress amplifications respectively.

4.4.

To determine the worst effects at a point on the railway track, it may be necessary tosuperpose bending moments and deflections caused by adjacent wheel loads, as shown inFigure 18. To investigate the effect of adjacent wheel loads, a beam with length L=10 m,

Figure 18. Typical wheel loads on railway tracks.

. . 166

Figure 19. Time histories of maximum deflections due to one- and two-wheel loads, v=120 km/h. L=10 m;x=3·2; V=120 km/h.

resting on a foundation having a stiffness kf =1·14e7 N/m2, is considered. The beam issubjected to two wheel loads, P0 and P1, at a constant distance apart of X1 =3·2 m andmoving with a velocity V=120 km/h.

The results for the maximum vertical deflections w and maximum bending stresses sS

in the beam, which occur at mid-span, are shown in Figures 19 and 20 respectively,together with those obtained for a single wheel load, travelling with the same speed. It canbe seen that the peak values of both deflections and stresses are more or less the same with

Figure 20. Time histories of maximum stresses due to one- and two-wheel loads, v=120 km/h. L=10 m;x=3·2; V=120 km/h.

167

T 2

Impact factors, f

India f=1+ V58·14(kf)0·5

Germany (for speeds up to 100 km/h) f=1+ V 2

3×104

South Africa f=1+4·92 VD

Clarke f=1+ 10·65VD(kf)0·5

AREA f=1+5·21 VD

WMATA f=(1+3·86×10−5V 2)2/3

either one or two wheel loads. However, the duration of the peak response, especially thedeflection, is increased with two wheel loads acting on the beam. Therefore, in a realrailway track structure, the duration of the peak response will depend on the speed of thetrain and its length, which in turn will determine the number of axles contributing to themaximum response. Similar results were obtained when the travelling speed was reducedto V=60 km/h.

4.5.

Dynamic loading on railway tracks has been subject of extensive investigation by railwayauthorities throughout the world. The main method adopted to cater for dynamic effectsin the design is to apply an impact factor to the static wheel load; i.e.,

P=f · PS , (23)

where P is the design wheel load (kN), PS is the static wheel load (kN) and f is adimensionless impact factor (q1). The expressions used for the calculation of the impactfactor have been determined empirically and are always expressed in terms of train speed.To develop expressions for the impact factor, the major factors have been [19] the trainspeed, the wheel diameter, the vehicle unsprung mass, the track condition (including trackstiffness, geometry and joint condition), the track irregularities, the track construction, thestatic wheel load and the vehicle condition.

Seven impact factor formulae are shown in Table 2, where D is the diameter of the wheel.Values of impact factors have been calculated from these formulae, by using the speedsand foundation stiffnesses used in the examples treated in this paper. In all of the aboveformulae, the travelling speed V has been considered to be the most important factor,whereas the effect of foundation stiffness is considered in only two formulae. In the presentstudy on dynamic analysis, it has been found that the foundation stiffness kf has asignificant effect on the response of the beam. Impact factors calculated from the variousformulae give an average value of about 1·5. This value is close to the maximum dynamicamplification in stress fS obtained in this paper. However, for vertical deflection, thedynamic analysis indicated a higher amplification, at about 3. When damping effects areconsidered, the amplifications can be expected to diminish. Moreover, as shown earlier,there will be a further reduction in the dynamic amplifications when the effect of springstiffness of the wheel load is included in the analysis.

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5. CONCLUSIONS

A simple finite element method for the dynamic analysis of beams on an elasticfoundation, subjected to a concentrated moving load, has been presented in this paper.This technique will be attractive for treating beams on an elastic foundation under movingloads in general and, as shown in the paper, can be easily extended to treat railway trackstructures. The effects of some important parameters, such as the foundation stiffness, thetravelling speed, the length of beam and the stiffness of the sprung mass, have been studied.Dynamic amplifications in stresses and deflections have been found to be about 1·6 and3 respectively, when the moving load was modelled without any spring stiffness. Theseamplifications diminish with the spring stiffness. Impact factors calculated from severalformulae gave an average value of about 1·5 for the parameters used in the present study.Damping is not included in the present formulation but its effect will be a reduction inthe amplitudes of the responses. The entire analysis has been conveniently programmedand, with few elements, gives converging results which, for a limiting case, compare wellwith those from existing solutions.

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