- Email: [email protected]
DYNAMIC ANALYSIS OF BEAMS ON AN ELASTIC FOUNDATION SUBJECTED TO MOVING LOADS
Journal of Sound and Vibration (1996) 198(2), 149–169
DYNAMIC ANALYSIS OF BEAMS ON AN ELASTIC FOUNDATION 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 treating the 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 the speed of the moving load, the foundation stiffness and the length of the beam on the response of the beam have been studied and dynamic amplifications of deflections and stresses have been evaluated. The technique is extended to the analysis of railway track structures, 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 and accurate results. Several numerical examples are presented. The technique and the findings will 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 or moving loads has attracted engineers and researchers for many decades. In his classic monograph, Hetenyi [1] has presented a closed form solution for an infinitely long beam on 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 applied to the analysis and design of railway tracks, with most of the research pertaining to static analysis [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 obtained from a static analysis of the structure subjected to the same loads. Although more complicated, the significance of dynamic analysis is thus evident [3]. Moreover, dynamic analysis is an important aspect in any complete structural investigation. A significant feature in the analysis of beams on elastic foundation is the rapid damping of the response away from the load [1, 2]. Timoshenko et al. [4] solved the governing differential equation for the dynamic analysis of a simply supported beam subjected to moving loads by mode superposition. They found that the maximum dynamic deflection was 1·5 times the static deflection when the travel time was half of the fundamental period of the structure. Warburton [5] analytically investigated the same problem and found that the maximum dynamic amplification in deflection was 1·743, and that this occurred when the travel time was 0·81 times the fundamental period of the structure. This finding was later confirmed by finite element analysis. Dynamic response of multiple-span beams has been studied by Ayre et al. [6] and Honda et al. [7], where the effects of span length and the number of spans on the dynamic response were examined. Many researchers have used the finite element method to 149 0022–460X/96/470149 + 21 $25.00/0
7 1996 Academic Press Limited
150
. .
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 governing equations of motion. Dynamic analysis of beams on an elastic foundation (BEF) under moving loads has received less attention, even though the results could be readily applied to the analysis and design of railway tracks. Timoshenko et al. [4] analytically solved for the free vibration of beams on elastic foundations. Ono and Yamada [10] presented a classic method for free and forced vibration analysis of BEF. Trochanis et al. [11] presented a method for analyzing beams on elastic foundations under moving loads and applied the results to the analysis of railway tracks. They used the fast Fourier transform (FFT) technique in their analysis. The beam on an elastic foundation under a moving load has also been treated by Kenny [12], Fryba [13] and Fryba et al. [14]. Thambiratnam and Zhuge [2, 15] have developed a simple finite element model to analyze simply supported beams on an elastic foundation (BEF) of any length. At first this model was employed in the static analysis of BEF and the results were applied to railway track structures, and then the model was used in free vibration analysis of BEF. In the present 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, free vibration and dynamic analyses of beams on an elastic foundation. Axial effects and variations in foundation properties along the length of the beam can be easily accommodated in the method, which will yet have a significant reduction in complexity in comparison to any of the presently available methods. In the finite element model, the elastic foundation is represented by springs with known stiffness. The moving concentrated load is assumed to travel along the beam with constant velocity. Newmark’s method is used in the numerical integration of the equations of motion. The length of the simply supported beam, the speed of the moving load and the magnitude of the foundation stiffness are the main parameters in the study. Time histories of deflections and stresses can be obtained, from which the resulting dynamic amplifications can be calculated. The influence of the suspension stiffness on the dynamic amplifications is also investigated. The effect of the length of the beam on the response was particularly studied in order to extend the application of the method to railway tracks. Finally, the effects of two moving loads travelling on a beam were studied. It is evident that by choosing an appropriate value for the length of the beam, and the number of elements to model the beam, a simple but efficient method can be obtained for the dynamic analysis of beams on 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 Figure 1, having a uniform width b and a linearly varying thickness h(x). It will be a simple matter to consider an element having a linearly varying width if the need arises. Neglecting axial deformations, this beam on an elastic foundation element has two-degrees-of-freedom per node; 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 of the element is obtained by adding the (4 × 4) stiffness matrices kB and kF pertaining to the usual beam bending and foundation stiffness respectively. Since there are four end
151
displacements (or degrees of freedom), a cubic variation in displacement is assumed, in the form v = Aa, (1) where A = (1 x x 2 x 3) and aT = (a1 a2 a3 a4 ). The four degrees of freedom corresponding to the displacements v1 , v3 and the rotations v2 , v4 at the longitudinal nodes are given by q = Ca, (2) where q T = (v1 v2 v3 v4 ) and C is the connectivity matrix for an element ij between x = 0 and x = L (Figure 1). From equations (1) and (2), (3) v = AC−1q. If E is the Young’s modulus, and I = bh(x)3/12 is the second moment of area of the beam cross-section about an axis normal to the plane of the paper, the bending moment M in the element is given by (4) M = D d2v/dx 2 = DBC−1q, where D = EI(x) and B = d2A/dx 2 = (0 0 2 6x). The potential energy UB due to bending is UB =
g
1 2
l
d 2v M dx dx 2
0
(5)
which, upon using equations (3) and (4), becomes
6g
l
UB = 12 q T(C−1)T
7
B TDB dx C−1q.
0
(6)
Upon using Lagrange’s equations, the (4 × 4) element stiffness matrix kB is obtained from the potential energy of the element as kB = (C−1)Tk B C−1,
(7)
where k b =
g
l
B TDB dx.
0
Figure 1. A beam on an elastic foundation element.
(8)
152
. .
The above expression can be evaluated explicitly. If the element has linearly varying thickness, 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 =
g
1 2
l
v Tkf v dx,
(9)
0
where kf is the stiffness of the foundation. Use of equation (3) in equation (9) will yield UF = 12 q T(C−1)T
6g
7
l
A Tkf A dx C−1q.
0
(10)
Using Lagrange’s equations, the (4 × 4) stiffness matrix kF pertaining to the foundation stiffness is given by kF = (C−1)Tk F C−1,
(11)
where k F =
g
l
A Tkf A dx.
(12)
0
The above expression can also be evaluated explicitly and, finally, the complete stiffness matrix for the element is k = kB + kF . (13) 2.2. For dynamic analysis, it is also necessary to derive the element mass matrix. The element mass matrix is a matrix of equivalent nodal masses that dynamically represent the actual distributed mass of the element. In this investigation, the mass matrix is derived by considering the kinetic energy due to lateral velocity. This is consistent with the derivation of the stiffness matrix where axial effects were ignored. The kinetic energy of the element shown in Figure 1 is given by T=
1 2
g
l
(v˙ )Tr dVv˙ ,
(14)
0
where the lateral velocity v˙ is given by the time derivative of the displacement v and r is the mass density. Further simplification gives
6g
r T = (q˙ )T(C−1)T 2
l
7
A Th(x)A dx C−1q˙ .
0
(15)
Using Lagrange’s equations, on the kinetic energy term above, the mass matrix m is given by m = (C−1)Tm¯C−1, (16) where m¯ = r
g
l
0
A Th(x)A dx.
(17)
153
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 moving concentrated force travelling along the beam. The beam has been discretized into a number of finite elements. Following the usual procedure for stiffness analysis of structures, the governing 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 structure stiffness matrix, [N ]T is the transpose of the shape functions for the beam element [6] which are 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 vectors respectively. In equation (18), damping effects are neglected and the shape functions can be represented as [N] = [0 0 0 . . . N1i N2i N3i N4i 0 0 0 . . . ],
(19)
where
01 01
N1i = 1 − 3
x l
2
+2
01 01
N3i = 3
x l
2
−2
3
x , l
N2i = x
3
x , l
N4i = x
0 1
2
x −1 , l
$0 1 1% x l
2
−
x l
,
(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 of non-zero entries within the n × 1 vector will be four. This 4 × 1 ‘‘sub-vector’’ is time dependent as the load moves from one position to another within an element. As the load moves to the next element, this sub-vector will shift in position corresponding to the degrees 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 study to solve the governing equation (18) and the numerical procedure is implemented in a FORTRAN program. The program has been run on a mainframe computer by using the CONVEX-220 system. 3. NUMERICAL EXAMPLES AND DISCUSSION
In this section numerical examples are treated to illustrate the procedure, and the effects of some parameters are investigated.
. .
154
3.1. 3.1.1. A simply supported beam without foundation stiffness To validate the method, a simply supported beam without an elastic foundation subjected to a concentrated force moving with constant velocity, is analyzed and the results are compared with those from the existing analytical solution of Warburton [5] and finite element analysis of Lin and Trethewey [3], where the set of second order differential equations have been solved by the Runge–Kutta numerical integration scheme. Results for the dynamic amplification factors fD , defined as the ratio of the maximum dynamic and static deflections at the center of the beam, are computed and compared in Table 1, for different values of t/t, where t denotes the travelling time of the force moving from the left end of the beam to the right end, while t denotes the time after the moving load enters the beam from the left end. It can be seen that the present results, obtained with only four elements modelling the beam, compare quite well with the results of the others and thereby confirm the validity of the proposed numerical procedure. The maximum value of the dynamic amplification is about 1·7 and occurs at t/t = 1·234. The maximum discrepancy between the present results and those of others is less than 5%, which is acceptable, especially as the results in reference [3], obtained from a numerical model, and those in reference [5], obtained by considering only the first mode, may not be exact. With a finer mesh, results obtained from the present procedure, match more closely the results from the numerical approach in reference [3]. This problem is not directly relevant to the scope of the present study, but it was treated as a test case to validate the numerical procedure. The technique is extended to beams on an elastic foundation in the next section. 3.1.2. The effect of foundation stiffness A 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, the Poisson ratio n = 0·3 and the second moment of area I = 1·844e–4 m4. The moving load has a constant velocity V = 16·7 m/s (60 km/h), and the foundation stiffness kf is varied from 0 to 1·14e8 N/m2, and includes values typical for railway tracks. Results for the dynamic amplification of mid-span deflections are shown in Figure 3. It can be seen that the dynamic amplifications fD initially increase as kf increases. However, when kf is increased beyond the value of 1·14e6 N/m2, the dynamic amplifications fD decrease. The dynamic amplification is largest for kf = 1·14 e6 N/m2, and this value is about three times the value for kf = 0. T 1 Dynamic amplifications, fD
t/t 0·1 0·5 1·0 1·234 1·5 2·0
fD ZXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXV This study Warburton [5] Lin and Trethewey [3] 1·040 1·330 1·710 1·723 1·630 1·500
1·040 1·250 1·710 1·740 1·710 1·550
1·053 1·252 1·705 1·730 1·704 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, as shown in Figure 4, where the dynamic amplifications at the beam center are plotted with respect to the foundation stiffness. These results indicate that when a concentrated force travels along a beam, the dynamic effects are greatly influenced by the foundation stiffness.
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, 60 and 100 km/h; kf = 1·14e7.
3.1.3. The effect of travelling speed The 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 varied from 4·1 m/s (15 km/h) to 27·8 m/s (100 km/h). The results for the dynamic amplifications in mid-span deflections are shown in Figure 5. It is interesting to note that the dynamic amplifications are reasonably constant for various travelling speeds, within the range considered. However, at smaller values of the foundation stiffness, the dynamic amplification increases slightly with travelling speed, as shown in Figure 6, which also shows a slight decrease in the dynamic amplification for the largest value of the foundation stiffness treated. Hence, when the foundation stiffness is relatively large, as in the case of foundations of railway tracks, the influence of the travelling speed, in the range 15–100 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 beam In 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. The moving load has a velocity V = 8·1 m/s (29 km/h). The length of the beam is varied from 5 m to 40 m and, for each case, the dynamic amplifications on the mid-span deflections are shown in Figure 7. It can be seen that when the span length L e 10 m, these dynamic amplifications on the mid-span deflection remain constant. However, at lower values of kf this is not the case, as shown in Figure 8. When the foundation stiffness kf is reduced to 1·14e2 N/m2, the peak dynamic amplification increases with the span, as shown in the figure, and no convergence is observed for the range of L considered here. Practical values of kf are greater than 1·14e5 N/m2, and for this range the dynamic amplifications converge (to a value of approximately 3·6) for L e 20 m, and the 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 important considerations are the allowable bending stress and allowable vertical deflection. In the previous section, dynamic amplifications in deflections of beams on an elastic foundation subjected 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 bending stresses. For this purpose, the dynamic amplifications in stress fS can be defined as the ratio of the maximum dynamic stress to the maximum static stress at the center of the beam. The maximum bending stress sS in a beam may be calculated using the simple applied 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 beam A 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. The length of the beam is varied from 5 m to 40 m and the resulting dynamic amplifications in stresses fS are shown in Figure 9. It can be seen that the values of the dynamic amplifications 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 dynamic amplifications increase with the span length of the beam and no convergence is noticed for the range of span lengths considered. When kf = 1·14e5 N/m2, the dynamic amplifications sensibly converge when L e 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.
3.2.2. The effects of foundation stiffness A simply supported beam of length L = 10 m is considered, on which the load travels with a velocity of V = 16·7 m/s (60 km/h). The stiffness of the foundation is varied from 0 to 1·14e8 N/m2. This range covers the typical foundations of railway tracks. For each case the dynamic amplifications fS were calculated and are presented in Figure 11. It can be seen that initially, the dynamic amplification increase with kf . When kf is increased
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.
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 . These results are similar to those obtained for deflections. However, it is evident that the dynamic amplifications in the stresses are less than those in the deflections. The maximum values of 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 study of dynamic amplifications in bridges due to moving loads [21], and could be explained as follows. Stresses considered in these studies are those due to bending effects and are obtained from the curvatures or second derivatives of the vertical deflections. It is not necessary for a function (deflection) and its second derivative (stress) to have the same dynamic amplification and in the present case amplifications in the second derivatives of the deflection are smaller than those in the deflection. Beams on an elastic foundation with other support (boundary) conditions can be treated with equal ease by the numerical procedure established in this paper. In order to illustrate the versatility of the present model, the problem treated above is re-analyzed with the beam having fixed supports. Dynamic amplifications in the bending stresses fS are shown in Figure 12 for various values of foundation stiffness. It can be seen that the trends are similar to those for a simply supported beam, but with the maximum value occurring for kf = 1·14e6 kN/m2. 3.2.3. The effect of travelling speed A 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 from 60 km/h to 120 km/h. The results for the dynamic amplifications in stresses fS are shown in Figure 13. The values of the dynamic amplifications in the mid-span stress remains more or less constant at the different speeds. It is evident that these results are similar to those for the deflections shown in Figure 5. The maximum dynamic amplification can be observed at a speed of 100 km/h. Again, the dynamic effects on the stresses are smaller than those on the deflections, the average peak dynamic amplification being about 1·6 for stresses and 3·7 for deflections.
161
In Figure 14 is shown the effect of the travelling speed on the dynamic amplifications in the mid-span stress for different values of foundation stiffness. It can be seen that when kf is reduced to 1·14e2 N/m2, dynamic amplifications increase with travelling speed, up to about 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 usually depends to a large extent upon the engineering experience of the designer [17, 18]. The available design expressions are at best empirical. This is because the methods of analyzing railway tracks are complicated, as the tracks are infinitely long and supported on beds the properties of which can vary along the track length. Moreover, the dynamic response characteristics of the tracks are not well enough understood to form a rational design method. Although research in seeking simpler and/or improved methods of track analysis has been going on, rail selection procedures and design have remained relatively static for over 100 years [19]. This is exemplified by the continuous common adoption of the quasi-static design 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 in severity of operating conditions. Tew et al. [19] have given a comprehensive coverage of some of the more important quasi-static methods. Kerr [20] has presented an interesting compilation 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 satisfying several criteria for strength of the individual components [18, 19]. The important criteria for 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, over 100 years ago. Using the BEF model, Hetenyi [1] analyzed railway track under static loads by solving the governing differential equation. Hetenyi’s classic solution is still used, often in the form of a computer program, to analyze and design railway tracks. The approach is quasi-static where the design load is obtained by multiplying the static wheel load by one of many available impact factors [17, 19] to account for dynamic effects. There are several such formulae, each with its merits and/or limitations. For want of a more comprehensive dynamic analysis procedure, this is the method used at present. There are, however, some shortcomings in the present method of analyzing and designing railway tracks: the analysis is tedious and not well understood; and the effects of moving loads cannot be fully accounted for by using impact factors in a quasi-static approach and variations in track and/or foundation properties along the length cannot be accounted for. In an earlier paper by Thambiratnam and Zhuge [2], a special technique was proposed to handle infinitely long beams (or tracks) by using an equivalent finite beam for static analysis. The length of this beam was chosen so that when it is subjected to a concentrated load the response curves for the deflection, the bending moment and the shear force are rapidly damped away from the load [1]. This equivalent beam of finite length has been tested to give converging results. The stiffness of the spring represents the stiffness of the rail 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 to identify the required span length of the beam on an elastic foundation. The rapid damping of the beam response away from the load enables the track to be modelled as a finite beam on an elastic foundation. The point load moving at a constant speed is representative of the 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 sections of the paper can be used to obtain this finite span length, with the appropriate value for the foundation stiffness kf . For rail foundations, the range of values for this stiffness is from 5·2e6 N/m2 to 3·54e7 N/m2. From the results in the earlier sections, it can be seen that, for this range of kf , dynamic amplifications in both deflections and stresses converge for span lengths L e 10 m. The corresponding value of the dynamic amplification in stress is about 1·6, in the absence of damping.
. .
164
4.3. It is necessary to address the applicability of the simple model developed herein to railway track analysis. Railways track which have been modelled and analyzed as beams on an elastic foundation have stood the test of time. However, the dynamic effects on the track response have not been fully understood, as mentioned earlier, and in this paper an attempt is made to investigate the dynamic amplifications, in order to provide a more realistic design. As with other track analysis models, the model proposed in this paper also relies on the same beam on elastic foundation concept. Real loads on railway tracks are sprung masses, although they have been simplified up to now as moving point loads. In this 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 damping and the mass of the wheel, the governing equations can be derived as
$
[M] 0
%6 7 $
[N]Tm1 m1
q¨ [K] 0 + y¨ −k[N] k
%6 7 6
7
q [N]Tm1 g = , y 0
(22)
where m1 is the sprung mass, k is the spring stiffness and y is its deflection, measured from the static equilibrium position before the moving load enters the beam. Double dots above quantities denote second time derivatives of those quantities. The other notations are as before. A convergence study was carried out to determine the effective length of the simply supported beam on an elastic foundation to be used in modelling railway tracks. In Figure 15 it is shown that when L e 10 m, dynamic amplification in the mid-span bending stress converges for stiffness ratios in the range 0 Q k/kf Q 0·20, where k is the stiffness of the sprung mass. Analogous results were obtained for fD , the dynamic amplification in deflection. 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 resting on an elastic foundation with kf = 1·14e7 N/m2 is considered, on which the mass moves with a velocity of 16·6 m/s (60 km/h). Dynamic amplifications in the mid-span deflections fD and the mid-span stresses fS are shown in Figures 16 and 17 for different values of the stiffness ratio k/kf . It can be seen that the effect of the spring stiffness is a reduction in 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 to superpose bending moments and deflections caused by adjacent wheel loads, as shown in Figure 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 is subjected to two wheel loads, P0 and P1 , at a constant distance apart of X1 = 3·2 m and moving 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 can be 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 f=1+
India
V 58·14(kf )0·5 V2 3 × 10 4
Germany (for speeds up to 100 km/h)
f=1+
South Africa
f = 1 + 4·92
Clarke
f=1+
AREA
f = 1 + 5·21
V D
10·65V D(kf )0·5 V D
f = (1 + 3·86 × 10−5V 2) 2/3
WMATA
either one or two wheel loads. However, the duration of the peak response, especially the deflection, is increased with two wheel loads acting on the beam. Therefore, in a real railway track structure, the duration of the peak response will depend on the speed of the train and its length, which in turn will determine the number of axles contributing to the maximum response. Similar results were obtained when the travelling speed was reduced to V = 60 km/h. 4.5. Dynamic loading on railway tracks has been subject of extensive investigation by railway authorities throughout the world. The main method adopted to cater for dynamic effects in 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 a dimensionless impact factor (q1). The expressions used for the calculation of the impact factor 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 train speed, the wheel diameter, the vehicle unsprung mass, the track condition (including track stiffness, geometry and joint condition), the track irregularities, the track construction, the static 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 speeds and foundation stiffnesses used in the examples treated in this paper. In all of the above formulae, 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 present study on dynamic analysis, it has been found that the foundation stiffness kf has a significant effect on the response of the beam. Impact factors calculated from the various formulae give an average value of about 1·5. This value is close to the maximum dynamic amplification in stress fS obtained in this paper. However, for vertical deflection, the dynamic analysis indicated a higher amplification, at about 3. When damping effects are considered, 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 spring stiffness of the wheel load is included in the analysis.
168
. . 5. CONCLUSIONS
A simple finite element method for the dynamic analysis of beams on an elastic foundation, 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 moving loads in general and, as shown in the paper, can be easily extended to treat railway track structures. The effects of some important parameters, such as the foundation stiffness, the travelling 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 and 3 respectively, when the moving load was modelled without any spring stiffness. These amplifications diminish with the spring stiffness. Impact factors calculated from several formulae 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 in the amplitudes of the responses. The entire analysis has been conveniently programmed and, with few elements, gives converging results which, for a limiting case, compare well with those from existing solutions.
REFERENCES 1. M. H 1961 Beams on Elastic Foundation. Ann Arbor, MI: The University of Michigan Press. 2. D. P. T and Y. Z 1993 Journal of Microcomputers in Civil Engineering 8, 467–476. Finite element analysis of track structures. 3. Y. H. L and M. W. T 1990 Journal of Sound and Vibration 136, 323–342. Finite element analysis of elastic beams subjected to moving dynamic loads. 4. S. T, D. H. Y and W. W 1974 Vibration Problems in Engineering. New York: John Wiley; fourth edition. 5. G. B. W 1976 The Dynamic Behaviour of Structures. Oxford: Pergamon Press. 6. R. S. A, F. F and L. S. J 1950 American Society of Mechanical Engineers, Transactions of the Journal of Applied Mechanics 17, 1–12. Transverse vibration of a two-span beam under the action of a moving constant force. 7. H. H, T. K and H. Y 1986 International Association of Bridge and Structure Engineering 98, 57–75. Dynamic factors of highway steel girder bridges. 8. F. V. H 1978 Shock and Vibration Digest 10, 27–35. Finite element analysis of structures under moving loads. 9. J. H, T. Y and K. K 1986 Journal of Sound and Vibration 96, 45–53. A finite element method prediction of the vibration of a bridge subjected to a moving vehicle load. 10. K. O and M. Y 1989 Journal of Sound and Vibration 130, 269–297. Analysis of railway track vibration. 11. A. M. T, R. C and J. B 1987 Journal of Geotechnical Engineering 113, 879–895. Unified approach for beams on elastic foundations under moving load. 12. J. T. K 1954 Journal of Applied Mechanics 21, 359–364; 1955, 22, 436. Steady-state vibrations of beams on elastic foundation for moving load. 13. L. F 1972 Vibration of Solids and Structures under Moving Loads. Groningen: Noorhoff. 14. L. F, S. N and N. Y 1993 Journal of Sound and Vibration 163, 31–35. Stochastic finite elements for a random foundation with uncertain damping under a moving force. 15. D. P. T and Y. Z 1993 in Proceedings of Second Asia-Pacific Conference on Computational Mechanics, Sydney, Australia, 231–235. A simple finite element method for free vibration of beams on elastic foundation. 16. K. C. R, H. R. E, D. W. G and D. A. N 1983 The Finite Element Method. Oxford: BSP Professional Books. 17. C. E 1989 Modern Railway Track. Duisburg, Germany: MRT Productions. 18. N. F. D 1980 Railway Track Design: a Review of Current Practice. Canberra, ACT: Australian Government Publishing Serivce.
169
19. G. P. T, S. M and P. J. M 1991 A Review of Track Design Procedure. Melbourne, Victoria: BHP Research Laboratories for Railways; Volume 1, Rails. 20. A. D. K 1978 in Proceedings of a Symposium held at Princeton University, April 1975. Oxford: Pergamon Press. Railroad track mechanics and technology. 21. J. S, D. P. T and G. H. B 1996 Research Report, Physical Infrastructure Centre, Queensland University of Technology, Brisbane, Australia. Dynamic load testing of a curved bridge.
DYNAMIC ANALYSIS OF BEAMS ON AN ELASTIC FOUNDATION 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 treating the 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 the speed of the moving load, the foundation stiffness and the length of the beam on the response of the beam have been studied and dynamic amplifications of deflections and stresses have been evaluated. The technique is extended to the analysis of railway track structures, 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 and accurate results. Several numerical examples are presented. The technique and the findings will 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 or moving loads has attracted engineers and researchers for many decades. In his classic monograph, Hetenyi [1] has presented a closed form solution for an infinitely long beam on 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 applied to the analysis and design of railway tracks, with most of the research pertaining to static analysis [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 obtained from a static analysis of the structure subjected to the same loads. Although more complicated, the significance of dynamic analysis is thus evident [3]. Moreover, dynamic analysis is an important aspect in any complete structural investigation. A significant feature in the analysis of beams on elastic foundation is the rapid damping of the response away from the load [1, 2]. Timoshenko et al. [4] solved the governing differential equation for the dynamic analysis of a simply supported beam subjected to moving loads by mode superposition. They found that the maximum dynamic deflection was 1·5 times the static deflection when the travel time was half of the fundamental period of the structure. Warburton [5] analytically investigated the same problem and found that the maximum dynamic amplification in deflection was 1·743, and that this occurred when the travel time was 0·81 times the fundamental period of the structure. This finding was later confirmed by finite element analysis. Dynamic response of multiple-span beams has been studied by Ayre et al. [6] and Honda et al. [7], where the effects of span length and the number of spans on the dynamic response were examined. Many researchers have used the finite element method to 149 0022–460X/96/470149 + 21 $25.00/0
7 1996 Academic Press Limited
150
. .
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 governing equations of motion. Dynamic analysis of beams on an elastic foundation (BEF) under moving loads has received less attention, even though the results could be readily applied to the analysis and design of railway tracks. Timoshenko et al. [4] analytically solved for the free vibration of beams on elastic foundations. Ono and Yamada [10] presented a classic method for free and forced vibration analysis of BEF. Trochanis et al. [11] presented a method for analyzing beams on elastic foundations under moving loads and applied the results to the analysis of railway tracks. They used the fast Fourier transform (FFT) technique in their analysis. The beam on an elastic foundation under a moving load has also been treated by Kenny [12], Fryba [13] and Fryba et al. [14]. Thambiratnam and Zhuge [2, 15] have developed a simple finite element model to analyze simply supported beams on an elastic foundation (BEF) of any length. At first this model was employed in the static analysis of BEF and the results were applied to railway track structures, and then the model was used in free vibration analysis of BEF. In the present 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, free vibration and dynamic analyses of beams on an elastic foundation. Axial effects and variations in foundation properties along the length of the beam can be easily accommodated in the method, which will yet have a significant reduction in complexity in comparison to any of the presently available methods. In the finite element model, the elastic foundation is represented by springs with known stiffness. The moving concentrated load is assumed to travel along the beam with constant velocity. Newmark’s method is used in the numerical integration of the equations of motion. The length of the simply supported beam, the speed of the moving load and the magnitude of the foundation stiffness are the main parameters in the study. Time histories of deflections and stresses can be obtained, from which the resulting dynamic amplifications can be calculated. The influence of the suspension stiffness on the dynamic amplifications is also investigated. The effect of the length of the beam on the response was particularly studied in order to extend the application of the method to railway tracks. Finally, the effects of two moving loads travelling on a beam were studied. It is evident that by choosing an appropriate value for the length of the beam, and the number of elements to model the beam, a simple but efficient method can be obtained for the dynamic analysis of beams on 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 Figure 1, having a uniform width b and a linearly varying thickness h(x). It will be a simple matter to consider an element having a linearly varying width if the need arises. Neglecting axial deformations, this beam on an elastic foundation element has two-degrees-of-freedom per node; 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 of the element is obtained by adding the (4 × 4) stiffness matrices kB and kF pertaining to the usual beam bending and foundation stiffness respectively. Since there are four end
151
displacements (or degrees of freedom), a cubic variation in displacement is assumed, in the form v = Aa, (1) where A = (1 x x 2 x 3) and aT = (a1 a2 a3 a4 ). The four degrees of freedom corresponding to the displacements v1 , v3 and the rotations v2 , v4 at the longitudinal nodes are given by q = Ca, (2) where q T = (v1 v2 v3 v4 ) and C is the connectivity matrix for an element ij between x = 0 and x = L (Figure 1). From equations (1) and (2), (3) v = AC−1q. If E is the Young’s modulus, and I = bh(x)3/12 is the second moment of area of the beam cross-section about an axis normal to the plane of the paper, the bending moment M in the element is given by (4) M = D d2v/dx 2 = DBC−1q, where D = EI(x) and B = d2A/dx 2 = (0 0 2 6x). The potential energy UB due to bending is UB =
g
1 2
l
d 2v M dx dx 2
0
(5)
which, upon using equations (3) and (4), becomes
6g
l
UB = 12 q T(C−1)T
7
B TDB dx C−1q.
0
(6)
Upon using Lagrange’s equations, the (4 × 4) element stiffness matrix kB is obtained from the potential energy of the element as kB = (C−1)Tk B C−1,
(7)
where k b =
g
l
B TDB dx.
0
Figure 1. A beam on an elastic foundation element.
(8)
152
. .
The above expression can be evaluated explicitly. If the element has linearly varying thickness, 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 =
g
1 2
l
v Tkf v dx,
(9)
0
where kf is the stiffness of the foundation. Use of equation (3) in equation (9) will yield UF = 12 q T(C−1)T
6g
7
l
A Tkf A dx C−1q.
0
(10)
Using Lagrange’s equations, the (4 × 4) stiffness matrix kF pertaining to the foundation stiffness is given by kF = (C−1)Tk F C−1,
(11)
where k F =
g
l
A Tkf A dx.
(12)
0
The above expression can also be evaluated explicitly and, finally, the complete stiffness matrix for the element is k = kB + kF . (13) 2.2. For dynamic analysis, it is also necessary to derive the element mass matrix. The element mass matrix is a matrix of equivalent nodal masses that dynamically represent the actual distributed mass of the element. In this investigation, the mass matrix is derived by considering the kinetic energy due to lateral velocity. This is consistent with the derivation of the stiffness matrix where axial effects were ignored. The kinetic energy of the element shown in Figure 1 is given by T=
1 2
g
l
(v˙ )Tr dVv˙ ,
(14)
0
where the lateral velocity v˙ is given by the time derivative of the displacement v and r is the mass density. Further simplification gives
6g
r T = (q˙ )T(C−1)T 2
l
7
A Th(x)A dx C−1q˙ .
0
(15)
Using Lagrange’s equations, on the kinetic energy term above, the mass matrix m is given by m = (C−1)Tm¯C−1, (16) where m¯ = r
g
l
0
A Th(x)A dx.
(17)
153
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 moving concentrated force travelling along the beam. The beam has been discretized into a number of finite elements. Following the usual procedure for stiffness analysis of structures, the governing 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 structure stiffness matrix, [N ]T is the transpose of the shape functions for the beam element [6] which are 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 vectors respectively. In equation (18), damping effects are neglected and the shape functions can be represented as [N] = [0 0 0 . . . N1i N2i N3i N4i 0 0 0 . . . ],
(19)
where
01 01
N1i = 1 − 3
x l
2
+2
01 01
N3i = 3
x l
2
−2
3
x , l
N2i = x
3
x , l
N4i = x
0 1
2
x −1 , l
$0 1 1% x l
2
−
x l
,
(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 of non-zero entries within the n × 1 vector will be four. This 4 × 1 ‘‘sub-vector’’ is time dependent as the load moves from one position to another within an element. As the load moves to the next element, this sub-vector will shift in position corresponding to the degrees 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 study to solve the governing equation (18) and the numerical procedure is implemented in a FORTRAN program. The program has been run on a mainframe computer by using the CONVEX-220 system. 3. NUMERICAL EXAMPLES AND DISCUSSION
In this section numerical examples are treated to illustrate the procedure, and the effects of some parameters are investigated.
. .
154
3.1. 3.1.1. A simply supported beam without foundation stiffness To validate the method, a simply supported beam without an elastic foundation subjected to a concentrated force moving with constant velocity, is analyzed and the results are compared with those from the existing analytical solution of Warburton [5] and finite element analysis of Lin and Trethewey [3], where the set of second order differential equations have been solved by the Runge–Kutta numerical integration scheme. Results for the dynamic amplification factors fD , defined as the ratio of the maximum dynamic and static deflections at the center of the beam, are computed and compared in Table 1, for different values of t/t, where t denotes the travelling time of the force moving from the left end of the beam to the right end, while t denotes the time after the moving load enters the beam from the left end. It can be seen that the present results, obtained with only four elements modelling the beam, compare quite well with the results of the others and thereby confirm the validity of the proposed numerical procedure. The maximum value of the dynamic amplification is about 1·7 and occurs at t/t = 1·234. The maximum discrepancy between the present results and those of others is less than 5%, which is acceptable, especially as the results in reference [3], obtained from a numerical model, and those in reference [5], obtained by considering only the first mode, may not be exact. With a finer mesh, results obtained from the present procedure, match more closely the results from the numerical approach in reference [3]. This problem is not directly relevant to the scope of the present study, but it was treated as a test case to validate the numerical procedure. The technique is extended to beams on an elastic foundation in the next section. 3.1.2. The effect of foundation stiffness A 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, the Poisson ratio n = 0·3 and the second moment of area I = 1·844e–4 m4. The moving load has a constant velocity V = 16·7 m/s (60 km/h), and the foundation stiffness kf is varied from 0 to 1·14e8 N/m2, and includes values typical for railway tracks. Results for the dynamic amplification of mid-span deflections are shown in Figure 3. It can be seen that the dynamic amplifications fD initially increase as kf increases. However, when kf is increased beyond the value of 1·14e6 N/m2, the dynamic amplifications fD decrease. The dynamic amplification is largest for kf = 1·14 e6 N/m2, and this value is about three times the value for kf = 0. T 1 Dynamic amplifications, fD
t/t 0·1 0·5 1·0 1·234 1·5 2·0
fD ZXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXV This study Warburton [5] Lin and Trethewey [3] 1·040 1·330 1·710 1·723 1·630 1·500
1·040 1·250 1·710 1·740 1·710 1·550
1·053 1·252 1·705 1·730 1·704 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, as shown in Figure 4, where the dynamic amplifications at the beam center are plotted with respect to the foundation stiffness. These results indicate that when a concentrated force travels along a beam, the dynamic effects are greatly influenced by the foundation stiffness.
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, 60 and 100 km/h; kf = 1·14e7.
3.1.3. The effect of travelling speed The 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 varied from 4·1 m/s (15 km/h) to 27·8 m/s (100 km/h). The results for the dynamic amplifications in mid-span deflections are shown in Figure 5. It is interesting to note that the dynamic amplifications are reasonably constant for various travelling speeds, within the range considered. However, at smaller values of the foundation stiffness, the dynamic amplification increases slightly with travelling speed, as shown in Figure 6, which also shows a slight decrease in the dynamic amplification for the largest value of the foundation stiffness treated. Hence, when the foundation stiffness is relatively large, as in the case of foundations of railway tracks, the influence of the travelling speed, in the range 15–100 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 beam In 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. The moving load has a velocity V = 8·1 m/s (29 km/h). The length of the beam is varied from 5 m to 40 m and, for each case, the dynamic amplifications on the mid-span deflections are shown in Figure 7. It can be seen that when the span length L e 10 m, these dynamic amplifications on the mid-span deflection remain constant. However, at lower values of kf this is not the case, as shown in Figure 8. When the foundation stiffness kf is reduced to 1·14e2 N/m2, the peak dynamic amplification increases with the span, as shown in the figure, and no convergence is observed for the range of L considered here. Practical values of kf are greater than 1·14e5 N/m2, and for this range the dynamic amplifications converge (to a value of approximately 3·6) for L e 20 m, and the 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 important considerations are the allowable bending stress and allowable vertical deflection. In the previous section, dynamic amplifications in deflections of beams on an elastic foundation subjected 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 bending stresses. For this purpose, the dynamic amplifications in stress fS can be defined as the ratio of the maximum dynamic stress to the maximum static stress at the center of the beam. The maximum bending stress sS in a beam may be calculated using the simple applied 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 beam A 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. The length of the beam is varied from 5 m to 40 m and the resulting dynamic amplifications in stresses fS are shown in Figure 9. It can be seen that the values of the dynamic amplifications 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 dynamic amplifications increase with the span length of the beam and no convergence is noticed for the range of span lengths considered. When kf = 1·14e5 N/m2, the dynamic amplifications sensibly converge when L e 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.
3.2.2. The effects of foundation stiffness A simply supported beam of length L = 10 m is considered, on which the load travels with a velocity of V = 16·7 m/s (60 km/h). The stiffness of the foundation is varied from 0 to 1·14e8 N/m2. This range covers the typical foundations of railway tracks. For each case the dynamic amplifications fS were calculated and are presented in Figure 11. It can be seen that initially, the dynamic amplification increase with kf . When kf is increased
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.
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 . These results are similar to those obtained for deflections. However, it is evident that the dynamic amplifications in the stresses are less than those in the deflections. The maximum values of 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 study of dynamic amplifications in bridges due to moving loads [21], and could be explained as follows. Stresses considered in these studies are those due to bending effects and are obtained from the curvatures or second derivatives of the vertical deflections. It is not necessary for a function (deflection) and its second derivative (stress) to have the same dynamic amplification and in the present case amplifications in the second derivatives of the deflection are smaller than those in the deflection. Beams on an elastic foundation with other support (boundary) conditions can be treated with equal ease by the numerical procedure established in this paper. In order to illustrate the versatility of the present model, the problem treated above is re-analyzed with the beam having fixed supports. Dynamic amplifications in the bending stresses fS are shown in Figure 12 for various values of foundation stiffness. It can be seen that the trends are similar to those for a simply supported beam, but with the maximum value occurring for kf = 1·14e6 kN/m2. 3.2.3. The effect of travelling speed A 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 from 60 km/h to 120 km/h. The results for the dynamic amplifications in stresses fS are shown in Figure 13. The values of the dynamic amplifications in the mid-span stress remains more or less constant at the different speeds. It is evident that these results are similar to those for the deflections shown in Figure 5. The maximum dynamic amplification can be observed at a speed of 100 km/h. Again, the dynamic effects on the stresses are smaller than those on the deflections, the average peak dynamic amplification being about 1·6 for stresses and 3·7 for deflections.
161
In Figure 14 is shown the effect of the travelling speed on the dynamic amplifications in the mid-span stress for different values of foundation stiffness. It can be seen that when kf is reduced to 1·14e2 N/m2, dynamic amplifications increase with travelling speed, up to about 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 usually depends to a large extent upon the engineering experience of the designer [17, 18]. The available design expressions are at best empirical. This is because the methods of analyzing railway tracks are complicated, as the tracks are infinitely long and supported on beds the properties of which can vary along the track length. Moreover, the dynamic response characteristics of the tracks are not well enough understood to form a rational design method. Although research in seeking simpler and/or improved methods of track analysis has been going on, rail selection procedures and design have remained relatively static for over 100 years [19]. This is exemplified by the continuous common adoption of the quasi-static design 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 in severity of operating conditions. Tew et al. [19] have given a comprehensive coverage of some of the more important quasi-static methods. Kerr [20] has presented an interesting compilation 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 satisfying several criteria for strength of the individual components [18, 19]. The important criteria for 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, over 100 years ago. Using the BEF model, Hetenyi [1] analyzed railway track under static loads by solving the governing differential equation. Hetenyi’s classic solution is still used, often in the form of a computer program, to analyze and design railway tracks. The approach is quasi-static where the design load is obtained by multiplying the static wheel load by one of many available impact factors [17, 19] to account for dynamic effects. There are several such formulae, each with its merits and/or limitations. For want of a more comprehensive dynamic analysis procedure, this is the method used at present. There are, however, some shortcomings in the present method of analyzing and designing railway tracks: the analysis is tedious and not well understood; and the effects of moving loads cannot be fully accounted for by using impact factors in a quasi-static approach and variations in track and/or foundation properties along the length cannot be accounted for. In an earlier paper by Thambiratnam and Zhuge [2], a special technique was proposed to handle infinitely long beams (or tracks) by using an equivalent finite beam for static analysis. The length of this beam was chosen so that when it is subjected to a concentrated load the response curves for the deflection, the bending moment and the shear force are rapidly damped away from the load [1]. This equivalent beam of finite length has been tested to give converging results. The stiffness of the spring represents the stiffness of the rail 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 to identify the required span length of the beam on an elastic foundation. The rapid damping of the beam response away from the load enables the track to be modelled as a finite beam on an elastic foundation. The point load moving at a constant speed is representative of the 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 sections of the paper can be used to obtain this finite span length, with the appropriate value for the foundation stiffness kf . For rail foundations, the range of values for this stiffness is from 5·2e6 N/m2 to 3·54e7 N/m2. From the results in the earlier sections, it can be seen that, for this range of kf , dynamic amplifications in both deflections and stresses converge for span lengths L e 10 m. The corresponding value of the dynamic amplification in stress is about 1·6, in the absence of damping.
. .
164
4.3. It is necessary to address the applicability of the simple model developed herein to railway track analysis. Railways track which have been modelled and analyzed as beams on an elastic foundation have stood the test of time. However, the dynamic effects on the track response have not been fully understood, as mentioned earlier, and in this paper an attempt is made to investigate the dynamic amplifications, in order to provide a more realistic design. As with other track analysis models, the model proposed in this paper also relies on the same beam on elastic foundation concept. Real loads on railway tracks are sprung masses, although they have been simplified up to now as moving point loads. In this 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 damping and the mass of the wheel, the governing equations can be derived as
$
[M] 0
%6 7 $
[N]Tm1 m1
q¨ [K] 0 + y¨ −k[N] k
%6 7 6
7
q [N]Tm1 g = , y 0
(22)
where m1 is the sprung mass, k is the spring stiffness and y is its deflection, measured from the static equilibrium position before the moving load enters the beam. Double dots above quantities denote second time derivatives of those quantities. The other notations are as before. A convergence study was carried out to determine the effective length of the simply supported beam on an elastic foundation to be used in modelling railway tracks. In Figure 15 it is shown that when L e 10 m, dynamic amplification in the mid-span bending stress converges for stiffness ratios in the range 0 Q k/kf Q 0·20, where k is the stiffness of the sprung mass. Analogous results were obtained for fD , the dynamic amplification in deflection. 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 resting on an elastic foundation with kf = 1·14e7 N/m2 is considered, on which the mass moves with a velocity of 16·6 m/s (60 km/h). Dynamic amplifications in the mid-span deflections fD and the mid-span stresses fS are shown in Figures 16 and 17 for different values of the stiffness ratio k/kf . It can be seen that the effect of the spring stiffness is a reduction in 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 to superpose bending moments and deflections caused by adjacent wheel loads, as shown in Figure 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 is subjected to two wheel loads, P0 and P1 , at a constant distance apart of X1 = 3·2 m and moving 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 can be 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 f=1+
India
V 58·14(kf )0·5 V2 3 × 10 4
Germany (for speeds up to 100 km/h)
f=1+
South Africa
f = 1 + 4·92
Clarke
f=1+
AREA
f = 1 + 5·21
V D
10·65V D(kf )0·5 V D
f = (1 + 3·86 × 10−5V 2) 2/3
WMATA
either one or two wheel loads. However, the duration of the peak response, especially the deflection, is increased with two wheel loads acting on the beam. Therefore, in a real railway track structure, the duration of the peak response will depend on the speed of the train and its length, which in turn will determine the number of axles contributing to the maximum response. Similar results were obtained when the travelling speed was reduced to V = 60 km/h. 4.5. Dynamic loading on railway tracks has been subject of extensive investigation by railway authorities throughout the world. The main method adopted to cater for dynamic effects in 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 a dimensionless impact factor (q1). The expressions used for the calculation of the impact factor 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 train speed, the wheel diameter, the vehicle unsprung mass, the track condition (including track stiffness, geometry and joint condition), the track irregularities, the track construction, the static 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 speeds and foundation stiffnesses used in the examples treated in this paper. In all of the above formulae, 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 present study on dynamic analysis, it has been found that the foundation stiffness kf has a significant effect on the response of the beam. Impact factors calculated from the various formulae give an average value of about 1·5. This value is close to the maximum dynamic amplification in stress fS obtained in this paper. However, for vertical deflection, the dynamic analysis indicated a higher amplification, at about 3. When damping effects are considered, 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 spring stiffness of the wheel load is included in the analysis.
168
. . 5. CONCLUSIONS
A simple finite element method for the dynamic analysis of beams on an elastic foundation, 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 moving loads in general and, as shown in the paper, can be easily extended to treat railway track structures. The effects of some important parameters, such as the foundation stiffness, the travelling 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 and 3 respectively, when the moving load was modelled without any spring stiffness. These amplifications diminish with the spring stiffness. Impact factors calculated from several formulae 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 in the amplitudes of the responses. The entire analysis has been conveniently programmed and, with few elements, gives converging results which, for a limiting case, compare well with those from existing solutions.
REFERENCES 1. M. H 1961 Beams on Elastic Foundation. Ann Arbor, MI: The University of Michigan Press. 2. D. P. T and Y. Z 1993 Journal of Microcomputers in Civil Engineering 8, 467–476. Finite element analysis of track structures. 3. Y. H. L and M. W. T 1990 Journal of Sound and Vibration 136, 323–342. Finite element analysis of elastic beams subjected to moving dynamic loads. 4. S. T, D. H. Y and W. W 1974 Vibration Problems in Engineering. New York: John Wiley; fourth edition. 5. G. B. W 1976 The Dynamic Behaviour of Structures. Oxford: Pergamon Press. 6. R. S. A, F. F and L. S. J 1950 American Society of Mechanical Engineers, Transactions of the Journal of Applied Mechanics 17, 1–12. Transverse vibration of a two-span beam under the action of a moving constant force. 7. H. H, T. K and H. Y 1986 International Association of Bridge and Structure Engineering 98, 57–75. Dynamic factors of highway steel girder bridges. 8. F. V. H 1978 Shock and Vibration Digest 10, 27–35. Finite element analysis of structures under moving loads. 9. J. H, T. Y and K. K 1986 Journal of Sound and Vibration 96, 45–53. A finite element method prediction of the vibration of a bridge subjected to a moving vehicle load. 10. K. O and M. Y 1989 Journal of Sound and Vibration 130, 269–297. Analysis of railway track vibration. 11. A. M. T, R. C and J. B 1987 Journal of Geotechnical Engineering 113, 879–895. Unified approach for beams on elastic foundations under moving load. 12. J. T. K 1954 Journal of Applied Mechanics 21, 359–364; 1955, 22, 436. Steady-state vibrations of beams on elastic foundation for moving load. 13. L. F 1972 Vibration of Solids and Structures under Moving Loads. Groningen: Noorhoff. 14. L. F, S. N and N. Y 1993 Journal of Sound and Vibration 163, 31–35. Stochastic finite elements for a random foundation with uncertain damping under a moving force. 15. D. P. T and Y. Z 1993 in Proceedings of Second Asia-Pacific Conference on Computational Mechanics, Sydney, Australia, 231–235. A simple finite element method for free vibration of beams on elastic foundation. 16. K. C. R, H. R. E, D. W. G and D. A. N 1983 The Finite Element Method. Oxford: BSP Professional Books. 17. C. E 1989 Modern Railway Track. Duisburg, Germany: MRT Productions. 18. N. F. D 1980 Railway Track Design: a Review of Current Practice. Canberra, ACT: Australian Government Publishing Serivce.
169
19. G. P. T, S. M and P. J. M 1991 A Review of Track Design Procedure. Melbourne, Victoria: BHP Research Laboratories for Railways; Volume 1, Rails. 20. A. D. K 1978 in Proceedings of a Symposium held at Princeton University, April 1975. Oxford: Pergamon Press. Railroad track mechanics and technology. 21. J. S, D. P. T and G. H. B 1996 Research Report, Physical Infrastructure Centre, Queensland University of Technology, Brisbane, Australia. Dynamic load testing of a curved bridge.