The use of finite element techniques for calculating the dynamic response of structures to moving loads

Computers and Structures 78 (2000) 789±799

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The use of ®nite element techniques for calculating the dynamic response of structures to moving loads Jia-Jang Wu, A.R. Whittaker *, M.P. Cartmell Department of Mechanical Engineering, James Watt Building, University of Glasgow, Glasgow G12 8QQ, UK Received 19 March 1999; accepted 18 February 2000

Abstract This paper presents a technique for using standard ®nite element packages for analysing the dynamic response of structures to time-variant moving loads. To illustrate the method and for validation purposes, the technique is ®rst applied to a simply supported beam subject to a single load moving along the beam. Finally, it is applied to the problem that initiated the work: calculation of the e€ects of two-dimensional motion of the trolley on the response of the base structure of a mobile gantry crane model. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Concentrated point load; Moving loads; Finite element; Dynamic responses

1. Introduction The motivation for the work described in this paper is the improved control of mobile gantry cranes, as used in ports and rail-head freight yards all over the world. In order to simulate the dynamic response characteristics of the full-sized crane in the laboratory, a 1/10 scale crane model, shown schematically in Fig. 1(a), has already been built [1]. Cartmell et al. [2] developed a mathematical model of this laboratory rig and used it in the design of a control system. The original model assumed that the structural members were rigid. The current work aims to improve the mathematical model by including ¯exible members. The approach used is to divide the whole structure into two sections: the static framework, as shown in Fig. 1(b) and the moving substructure, as shown in Fig. 1(c). The idea is to try to simplify the relationship between the static framework and the

* Corresponding author. Tel.: +141-330-5044; fax: +141-3304343. E-mail address: [email protected] (A.R. Whittaker).

moving substructure as four time-variant moving point loads. 1 The problem can then be solved, provided a ®nite element package is capable of analysing the static structure, subjected to these four moving loads. It is the use of ®nite element packages for dynamic analysis of three-dimensional structures subject to time-variant moving loads which is the subject of this paper. Standard ®nite element packages are not usually set up to easily apply moving, time-variant loads. This is certainly the case for the I - D E A S package available for this work [3]. The techniques that we describe here will therefore have general applicability. The literature concerning the forced vibration analysis of a three-dimensional framework subjected to moving loads is rare and the authors are not aware of any work on the dynamic responses of a three-dimensional framework subjected to loads moving in two dimensions. Hence, the purpose of this paper is to address this problem.

1

In fact, due to the partial symmetry in the current application, it is found that there are only two di€erent load values.

0045-7949/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 0 ) 0 0 0 5 5 - 9

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J.-J. Wu et al. / Computers and Structures 78 (2000) 789±799

Fig. 1. The 1/10 scale crane model.

To simulate the moving load, the basic principle is to apply forces and moments to all the nodes of the ®nite element model, making these forces and moments functions of time. Nodes near to the instantaneous force application point can then be given relatively large force values, whereas nodes away from the instantaneous force application point will have relatively small (zero) force values. To develop techniques for deriving appropriate time±force and time±moment functions for all the nodes on a structure, a beam subjected to a single concentrated force will ®rst be studied. It is found that a lot of researchers have devoted themselves to studies in this ®eld [4±15]. Some form of beam (uniform beam/ non-uniform beam/linear beam/non-linear beam/multispan beam) is the subject of all the above studies. The beam is subjected to one constant moving concentrated force but not multiple time-variant moving forces. The current work starts with the single moving force model and extends it to deal with a pair of beams, each subjected to two time-variant moving concentrated forces, as is required for the analysis of the mobile crane.

When a beam is subjected to a concentrated force P, the forces on all the nodes of the beam are equal to zero except the nodes of element s (as shown in Fig. 2) that is subjected to the concentrated force. According to Trethewey [4] and Clough and Penzien [16], the external force vector {F(t)} in Eq. (1) takes the following form: n o …s† …s† …s† …s† f F …t†g ˆ 000 . . . f1 …t†f2 …t†f3 …t†f4 …t† . . . 000 ; …2† …s†

where fi …t† (i ˆ 1±4) represent the equivalent nodal forces, kT  …s† j …s† …s† …s† …s† f …t† ˆ f1 …t† f2 …t† f3 …t† f4 …t† ˆ P fN g …3† and f N g ˆ bN1 N2 N3 N4 cT

…4†

such that the Ni …i ˆ 1±4† represent shape functions [4,16] given by N1 ˆ 1 ÿ 3n2 ‡ 2n3 ;

…5†


ÿ N2 ˆ n ÿ 2n2 ‡ n3 l;

…6†

2. A single point load moving along a beam 2.1. De®nition of equivalent nodal forces The equation of motion for a multiple degreeof-freedom structural system is represented as follows: _ ‡ ‰ K Šfug ˆ f F …t†g; ‰ M Šfug ‡ ‰C Šfug

…1†

where [M], [C], [K] are the respective overall mass, damping and sti€ness matrices of the structure; f ug; fu_ g; fug are the respective acceleration, velocity and displacement vectors for the whole structure, and {F(t)} is the external force vector.

Fig. 2. The equivalent forces of the element s subjected to a concentrated force P.

J.-J. Wu et al. / Computers and Structures 78 (2000) 789±799

N3 ˆ 3n2 ÿ 2n3 ;

…7†

N4 ˆ …ÿn2 ‡ n3 †l;

…8†

nˆ

x l

…9†

noting that l is the element length and x is the distance along the element to the point of application of P, as shown in Fig. 2. 2.2. Moving point load In order to simulate the moving load, one may apply forces and moments which are a function of time to all the nodes of the ®nite element model of the whole structure. As shown in Fig. 3, a concentrated force moves with velocity V from node 1 to node n of the beam which is composed of n nodes and n)1 elements. Considering m time steps and choosing a time interval Dt, the total time, tmax is then given by tmax ˆ m Dt:

…10†

The force and moment vectors contain the force and moment information for all the nodes on the beam at all time steps:  i i i i i FtˆDt Ftˆ2Dt . . . Ftˆm i ˆ 1 to n; b F cim‡1 ˆ Ftˆ0 Dt m‡1 ; …11† b M cim‡1

 i i i i i ˆ Mtˆ0 MtˆDt Mtˆ2Dt . . . Mtˆm Dt m‡1 ;

i ˆ 1 to n; …12†

where i represents the node number. At time t ˆ 0, the concentrated force is on node 1, as shown in Fig. 3. 1 i Ftˆ0 ˆ P ; Ftˆ0 ˆ 0 …i ˆ 2 to n† and i Mtˆ0 ˆ 0 …i ˆ 1 to n†:

…13†

At any time t ˆ r Dt (r ˆ 1 to m), the position of the moving concentrated force, relative to the left end of the beam, is given by xp …t† ˆ Vr Dt:

…14†

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Remembering that l is the element length, the element number, s, that the moving concentrated force is applied to at any time t can be found from   xp …t† ‡ 1: …15† s ˆ The integer part of l The two nodes of the sth (s ˆ 1 to n)1) beam element are s and s ‡ 1. Therefore, the following equations for nodal forces and moments are formed when the moving concentrated force, P, is on the sth beam element (s ˆ 1 to n)1) at any time t ˆ r Dt (r ˆ 1 to m). s Ftˆr Dt ˆ PN1 ;

…16†

s‡1 Ftˆr Dt ˆ PN3 ;

…17†

i Ftˆr …i ˆ 1 to n except s and s ‡ 1†; Dt ˆ 0

…18†

s Mtˆr Dt ˆ PN2 ;

…19†

s‡1 Mtˆr Dt ˆ PN4 ;

…20†

i Mtˆr …i ˆ 1 to n except s and s ‡ 1†; Dt ˆ 0

…21†

where N1 ; N2 ; N3 ; N4 are given by Eqs. (5)±(8), and Eq. (9) can be re-written in terms of the global xp (t) instead of the local x: nˆ

xp …t† ÿ …s ÿ 1†l : l

…22†

It is noted that when n is an integer, there is only one nodal force, as would be expected, since an integer value for n means that P is coincident with a node. Hence, the time±force and time±moment functions are determined for all the nodes of the beam when it is subjected to a moving concentrated force. The procedure described above has so far only been developed for beam elements. The same principle could be applied if other element types were used, but the corresponding shape functions would need to be used. By ignoring moments at each end of each element (neglecting f2 …s† (t) and f4 …s† (t) in Fig. 2), a simple linear interpolation for the forces would allow the whole procedure to be generalised much more easily:

Fig. 3. A beam subjected to a concentrated force, P, moving with velocity, V.

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 x …s† f1 …t† ˆ P 1 ÿ ; l x …s† f3 …t† ˆ P : l

J.-J. Wu et al. / Computers and Structures 78 (2000) 789±799

…23† …24†

This is referred to as the ``simple method''. A further analysis could be performed, setting N2 and N4 in Eq. (4) to zero. This is referred to as the ``no moment'' method, as opposed to the ``full'' method where N1 to N4 are given by Eqs. (5)±(8). To illustrate the principles involved, consider a simply supported beam of length 1 m with 11 nodes equally spaced along the beam. Let a force of constant amplitude of 1 N travel with a constant velocity of 0.1 m/s

from one end to the other. Figs. 4(a) and 5(a) show the force/time and moment/time graphs for nodes 3±5 of the beam, covering the time period when their values are non-zero. Figs. 4(b) and 5(b) show the force/time and moment/time graphs for each node of the beam. All these ®gures illustrate a feature of this modelling technique: that both the force and moment on each node are zero for all times other than while the force is travelling from the previous node to the next node. To try to assess the suitability of the di€erent calculation methods, the quantity Px can be re-calculated from the equivalent nodal force and moment values. Considering Fig. 2 and performing a static analysis, taking moments about the left end of an element yields

Fig. 4. (a) Force±time graphs for nodes 3±5, (b) force±time graph for each node.

J.-J. Wu et al. / Computers and Structures 78 (2000) 789±799

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Fig. 5. (a) Moment±time graphs for nodes 3±5, (b) moment±time graph for each node.

…s†

…s†

…s†

Px ˆ f2 …t† ‡ f4 …t† ‡ lf3 …t†:

…25†

Eq. (25) can be applied three times, once with the values of f2 , f3 and f4 from the full model, once with f2 and f4 being zero, but the value of f3 calculated from the full model (the no moment case) and ®nally with f2 and f4 still zero, but the value of f3 calculated from the simple model (Eq. (24)). Fig. 6 shows the results of this procedure, just for the time when the force is on element 3 (P is travelling from node 3 to node 4). It is interesting to note that the re-generated values of Px, when calculated

by either the simple or full method have no discernible di€erence, indicating that for the particular example chosen, either method would be suitable. It is also clear that the simple method is more accurate than the no moment method, which is explained by the foregoing analysis. The values of f1 and f3 as calculated by the simple analysis are not as accurate estimates of the true f1 , f3 values as those calculated by the full analysis, but they produce a more realistic loading e€ect on the beam element than would be obtained by using the true values of f1 and f3 , but ignoring the moments, f2 and f4 .

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J.-J. Wu et al. / Computers and Structures 78 (2000) 789±799

The results of the above procedures are shown in Fig. 7. Only a slight di€erence can be detected between these curves, so the ®nite element method has been shown to be viable. The time±force functions were re-calculated using the ``simple'' and ``no moment'' methods, and the results of the three di€erent analyses appear very similar to Fig. 7 when looking at the full plot. Fig. 8 shows just the results for the ®rst 1.2 s of the simulation and it can be seen that the full method and the simple method are still in good agreement, but the ``no moment'' method shows a slight discrepancy. This is to be expected, since it has been shown in Section 2.3 that the ``no moment''

Fig. 6. Px on node 3 using di€erent calculation methods.

2.3. Implementation and validation To check the technique, a uniform undamped simply supported beam of length 1 m and cross section 2 cm  1 cm is modelled with 10 beam elements (11 nodes) using I - D E A S . The beam is made of steel with density 7820 N/m3 and modulus of elasticity 206.8 GN/ m2 . A vertical point load of (sin 10t) N starts at the left end and travels to the right end in 10 s with constant velocity. I - D E A S is ®rst used to calculate the ®rst 10 natural frequencies and mode shapes. The ``full'' analysis of Sections 2.1 and 2.2 is coded into a F O R T A N program which builds the time±force and time±moment functions of all the nodes of the ®nite element model. This information is stored as an ASCII universal ®le which is read by I - D E A S before performing the forced vibration analysis. To check the procedure, the natural frequencies and normal modes of the beam were calculated by classical methods and the forced vibration results calculated using a series solution in terms of the normal modes. Procedures similar to this are described by Lee [17], Lee [18] and Michaltsos et al. [19] when dealing with the moving mass problem. The di€erence between moving mass and moving load is that the acceleration of the mass is ignored in the moving load formulation. It is evident from earlier works [17±19] that treating a moving mass on a beam as a moving load is a valid approximation when the ratio of the moving mass to the beam mass is suitably small. Timoshenko [20], Norris [21] and Rogers [22] each present their own analytical solution for dealing with the moving load problem. Of these themes, RogersÕ is the one with the fewest simpli®cations, so this is used for comparison with the current results.

Fig. 7. Central response (x ˆ 0.5 m) of simply supported beam subjected to sinusoidal force.

Fig. 8. Comparison of central responses calculated by three di€erent methods.

J.-J. Wu et al. / Computers and Structures 78 (2000) 789±799

method is the worst approximation of the real loading situation. 3. A beam subjected to two moving point loads For the next analysis, two concentrated forces move with velocity V from the left end to the right end of the beam, as shown in Fig. 9. The equations of Sections 2.1 and 2.2 are valid for calculating the contribution to the force and moment vectors corresponding to the ®rst moving force. To calculate the contribution of the second moving force, Eqs. (14), (15) and (22) need to be modi®ed slightly. Eq. (14) becomes x…t† ˆ Vr Dt ‡ xf

…26†

or, in words, the global x(t) value describing the position of the second force is equal to the global x(t) value of the ®rst force plus the spacing between the forces, xf . Now the element with the force on is given by   Vr Dt ‡ xf s ˆ The integer part of ‡ 1: …27† l Finally, the correct expression for the local xp (t) value is substituted into Eq. (9) to give nˆ

Vr Dt ‡ xf ÿ …s ÿ 1†l : l

…28†

The overall force and moment vectors can now be obtained by adding the contributions of the two forces together at each node. 4. The mobile crane problem By referring to Figs. 1 and 10(a), it can be seen that the model mobile crane problem can be reduced to a problem of two parallel beams, each subjected to two moving, time-variant forces. The e€ect of the moving substructure is taken into account by the position of the pairs of forces on each ®xed rail and the e€ect of the trolley movement can be taken into account by allowing the magnitude of the contact forces with the ®xed rails to vary. It should be pointed out that the physical scale model is not exactly the same as the actual crane. In the

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actual full-size crane, the x motion is achieved by the whole structure moving on wheels, whereas the y motion of the trolley is on the rails Q. There are no moving rails P. In the scale model, the static framework (Fig. 1(b)) is ®xed to the ground (no wheels), so the moving rails were introduced to allow x motion. This actually makes the modelling problem slightly more complicated for the scale model than for the real structure. 4.1. Derivation of frame±trolley contact point loads Fig. 10 shows the top view of the crane model structure. The trolley moves on the moving substructure along the x axis and the moving substructure moves on the two ®xed rails along the y axis. There are four points of contact between the two substructures: A, B, C and D. The details of calculation are not relevant to this paper, but it is clear that analysis of the moving substructure could result in expressions for Fz1 …t†: the force at points A and B, and Fz2 …t†: the force at points C and D. These forces will all be time dependent because of the e€ect of the trolley movement. Fig. 11 shows the force±time graphs used to create the forced response results given in this paper. The forces correspond to the trolley motion as described below. The trolley accelerates from x ˆ ÿ0:49 m, y ˆ ÿ0:6 m to the maximum x and y velocities (0.3124 and 0.5236 m/s, respectively) in 1 s. The ®nal y position of 0.709 m is reached in 3.5 s, with a constant velocity period of 1.5 s followed by a deceleration period of 1 s. After 3.5 s, there is no further y movement of the trolley. The ®nal x position of 0.4472 m is reached in 4 s, with a constant velocity period of 2 s and a deceleration period of 1 s. For the remainder of the 10 s simulation time, the trolley remains stationary at x ˆ 0:4472 m, y ˆ 0:709 m. 4.2. Calculation of time-variant node position for motion along the principal ( y ) axis As shown in Fig. 10(a), the moving sub-structure moves on the two ®xed rails along the y axis. This means that the y co-ordinates of the four contact points, A, B, C and D vary with time. In order to calculate the four y co-ordinates of the contact points at time t, it is necessary to calculate Cy …t†, the position along the y axis of

Fig. 9. A beam subjected to two moving concentrated forces.

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J.-J. Wu et al. / Computers and Structures 78 (2000) 789±799

Fig. 10. (a) Top view of the crane model structure, (b) the forces diagram of the side view of the moving substructure (either of the two moving rails).

ty ˆ

Vcy max ÿ Vcy0 : acy

…31†

After the y co-ordinate of the centre of the trolley is acquired, the y co-ordinates, CyA …t† to CyD …t† of the four contact points, A to D, are easily found:

Fig. 11. The forces applied to the static framework.

the centre of the trolley at time t. If the trolley is moving along the y axis with a constant acceleration, Cy …t† ˆ Cy0 ‡ Vcy0 t ‡

1 a t2 ; 2 cy

…29†

where Cy0 and Vcy0 are the initial displacement and velocity, respectively, and acy is the acceleration. If the trolley accelerates to the highest speed and then moves at that speed along the y axis, Cy …t† ˆ Cy0 ‡ Vcy0 ty ‡ 12acy ty2 ‡ Vcy max …t ÿ ty †;

…30†

CyA …t† ˆ Cy …t† ÿ

yb ; 2

…32†

CyB …t† ˆ Cy …t† ‡

yb ; 2

…33†

CyC …t† ˆ Cy …t† ‡

yb ; 2

…34†

CyD …t† ˆ Cy …t† ÿ

yb ; 2

…35†

where yb is the separation of the two moving rails as shown in Fig. 10(a). 4.3. A pair of beams, each subjected to two moving point loads As mentioned in Sections 2.2 and 3, the overall force and moment vectors for a beam subjected to two moving point loads can be obtained, at each node, by adding together the contributions of the two forces. Similarly, the overall force and moment vectors for a pair of beams, each subjected to moving point loads, can now

J.-J. Wu et al. / Computers and Structures 78 (2000) 789±799

be obtained by adding the contributions of the four forces together at each node. 4.4. Results This section shows a few typical results for a representative node: node 60. Node 60 is a quarter of the way along rail Q, as shown in Fig. 10. Typical trolley movements are chosen in an attempt to assess whether the predicted responses are accurate. Fig. 12 shows the results of the simulations. The ®rst curve (marked Ô+Õ) is the result of the x; y trolley movement as described in Section 4.1. To create this curve, the damping factor f for all modes was chosen as 0.003. The curve marked ÔoÕ is for the same x; y motion, but with the damping factor increased to 0.01. Both the curves are identical. This is to be expected, since the dynamic e€ects are insigni®cant under these loading conditions. The curves merely show what would be the de¯ection if the force were applied statically at the point corresponding to the particular time. After 3.5 s, the z values remain constant, since at this time the trolley reaches the end of the speci®ed travel. The dashed line in Fig. 12 represents the de¯ections when the trolley follows the same path, but at half the speed. As expected, this curve is similar to the previous two, except that the time taken to reach the ®nal position is twice as long. Fig. 12 also shows one curve for the same x motion, but zero y motion and another curve for the same y motion, but zero x motion. It can be seen that the structure appears sti€er when the trolley moves in the x direction than when it moves in the y direction. In fact, most of the deformation corresponding to the simultaneous x; y motion is as a result of y movement. Since the loading to create the responses of Fig. 12 does not really test the dynamic model, the simulations were repeated with the same loads, but the force was

Fig. 12. z displacement of node 60.

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applied suddenly at the starting position to create an impulsive type loading. This is not an attempt to model the real loading situation for the actual crane (it actually represents the trolley suddenly being added to the static structure), but is merely a means of providing a more severe test for the mathematical model. Fig. 13 shows the results for the suddenly applied load, followed by simultaneous x; y motion of the trolley. The transient e€ects are clearly visible, but the underlying trend is the same as in Fig. 12. Fig. 14 shows the e€ect of increasing the damping factor for all modes from 0.003 to 0.01, with predictable results. As a ®nal test for the dynamic model, a simulation was performed where a vertical sinusoidal force of 50 N was applied to the moving trolley. Fig. 15 shows the vertical response of node 60 to a forcing frequency of 9.5 rad/s, chosen to be well away from any resonance of the

Fig. 13. z displacement of node 60 for simultaneous motion in the x, y direction.

Fig. 14. z displacement of node 60 for simultaneous motion in the x, y direction (n ˆ 0.01).

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Fig. 15. z displacement of node 60 when vertical sinusoidal force of amplitude 50 N and frequency 9.5 rad/s is applied to a trolley which moves in the x; y direction.

Fig. 16. z displacement of node 60 when vertical sinusoidal force of amplitude 50 N and frequency 124.4 rad/s is applied to a trolley which moves in the x, y direction.

structure. Fig. 16 shows equivalent results for a forcing frequency of 124.4 rad/s: chosen to be close to a natural frequency. Both these graphs show sensible results. Fig. 15 shows the sinusoidal response at 9.5 rad/s superimposed on the same trend as in Fig. 12. In Fig. 16, the large amplitude of vibration due to the resonance makes the relatively small static de¯ections observed under the other loading conditions insigni®cant. 5. Conclusions A technique for using standard ®nite element packages to analyse the dynamic response of structures to time-variant moving loads has been developed. A computer program has been written which calculates the

time-variant external nodal forces on a whole structure, which provide the equivalent load to point forces that move around the structure. The calculation of the equivalent nodal forces to represent the moving loads has been performed by three approximate methods. In the ®rst ``full'' method, equivalent nodal forces and moments were calculated. This required the shape functions for the element. The second method simply ignored the moments calculated using method 1. The third, ``simple'', method ignored any moment applied at the nodes at the outset and therefore did not require knowledge of the shape functions. The technique has been applied to a simply supported beam with one moving force and to a threedimensional mobile crane structure with four moving forces. Both the applications give encouraging results with either the full model or the simple model. All the observations in Sections 2.3 and 4.4 indicate that the technique is providing sensible, realistic results, but future work will involve the experimental validation of the results. This is particularly important for the mobile crane work, where the calculation methods for the contact forces between the two substructures also need to be validated. This work was performed using the I - D E A S ®nite element package, which is very accommodating because of its support of the ASCII universal ®les. However, many other packages have similar features, so the technique is potentially more generally applicable. The technique has only been applied to beam elements so far, but in principle, it is general and can be adapted to any element type. The use of the simple model, rather than the full model will make the application to ®nite element models using other element types more straightforward, but further work must be done before the simple method can be proved to be accurate enough for all applications.

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