Moving loads on beams on viscoelastic foundations with frictional

damping

Rita Toscano Monteiro Ritto Correarita.correa@ist.utl.pt

Instituto Superior Tecnico, Lisboa, Portugal

October 2016

Abstract

In this dissertation the dynamic behavior of Euler-Bernoulli beams on viscoelastic foundations of theWinkler type submitted to moving loads is analyzed. In particular, the usefulness of passive frictionaldamping devices (discrete model) or the consideration of friction in the foundation’s granular material(continuous model) in the context of minimizing the resonance phenomena or of improving the model-ing and behavior of high speed railway lines are analyzed. Damping systems based on friction presentecological and economic advantages with respect to viscous ones. A program in Matlab environmentbased on the finite element method (FEM) that simulates the dynamic behavior of the beam, founda-tion and frictional damping devices (discrete or continuously distributed), driven by a moving load, isdeveloped. The NSCD method (“nonsmoth contact dynamics method”) is used for the time integrationof the equations. This method is suitable for friction problems, which are governed by a non-linearand non-smooth constitutive law. The effects of frictional dissipation (for both models) on the valueof the critical velocities and dynamic amplifications are evaluated by means of numerical analyzes. Forthe discrete model, different frictional dissipation solutions are compared and the maximal friction forceoptimization is addressed. The two models are also compared in situations of similar potential frictionaldissipation. For the continuous model, the corresponding dissipated energy is compared for differentdistributed frictional forces.Keywords: frictional dissipation, NSCD method, θ method, dynamic analysis, resonance, FEM,Euler-Bernoulli beam, Winkler foundation

1. IntroductionIn this dissertation the dynamic behavior of Euler-Bernoulli beams on viscoelastic foundations of theWinkler type submitted to moving loads is ana-lyzed. In particular, the utility of using Coulombfrictional damping devices or the consideration ofthe friction in the foundation’s granular materialin the context of minimizing resonance phenomenawhich occur in high-speed railway lines are ana-lyzed. Damping systems based on friction presentecological and economic advantages with respect toviscous ones.

The study of the dynamic behavior of Euler-Bernoulli beams on elastic foundation subjected tomoving loads has been under investigation since afew decades ago. Among the initial analytical stud-ies, it is worth to mention the works of Krylov [1]and Timoshenko [2] that began to develop the studyof the dynamics of beams under moving loads inthe early twentieth century. Years later, Inglis [3],Lowan [4] and Fryba [5] studied the problem ofvibrations in simply supported beams traveled bymoving loads at constant velocity using the modal

analysis approach. Timoshenko [2] presented a so-lution for the infinite beam. In the late twentiethcentury, Thambiratnam and Zhuge [6] applied thefinite element method to the dynamic analysis of abeam on elastic foundation of Winkler type trav-eled by a moving load. In 2012, Dimitrovova andRodrigues [7] analytically determined the criticalvelocity of the moving load in the case of a finitebeam on an elastic Winkler foundation and showedthat the consideration of a finite beam with a 200m length allows a good approximation for the valueof the critical velocity of an infinite beam. Usingthe FEM, Castro Jorge [8] and Castro Jorge et al.[9, 10] numerically studied the effect on the criticalspeed of a nonlinear Winkler foundation, consider-ing either that the reaction force has a cubic depen-dence on the transverse displacement of the beam orthat the Winkler foundation has a bi-linear behav-ior with different tensile and compressive stiffness.

2. Formulation of the problem

We consider a slender Euler-Bernoulli beam (railUIC60, whose properties are listed in Table 1) with

1

length L, supported on a linear viscoelastic Winklerfoundation (stiffness kf = 250 kN/m2) in paral-lel with a set of frictional damping devices of theCoulomb type submitted to a moving downwardload (F = −83.4 kN) of constant intensity in uni-form motion (velocity v and coordinate x = vt attime instant t). The value used for the load cor-responds to half the weight per wheel axle of theThalys [11] high-speed train’s locomotive, whichhas a mass of 17.000 kg, and it is the same as usedby the authors [7, 8, 9, 10].

Two different situations corresponding to two dif-ferent frictional damping models are studied: thepresence in the foundation of a finite number offrictional damping devices (FDD) (Figure 1 andnumerical results in section 3) and the considera-tion of friction between the granular material of thefoundation represented by continuously distributedFDD (Figure 2 and numerical results in section 4).

Figure 1: Euler-Bernoulli beam on viscoelas-tic Winkler foundation with a finite number ofCoulomb FDD.

Figure 2: Euler-Bernoulli beam on viscoelasticWinkler foundation with a continuous distributionof Coulomb FDD.

Table 1: UIC60 rail’s properties.

cross sectional area A 7684 mm2

cross sectional inertia I 3055 cm4

mass density ρ 7800 kg/m3

Young’s modulus E 210 GPa

2.1. Frictional constitutive lawEach FDD applies a force to the section of the beamwhere it is installed that depends on the velocity of

that section according to the inclusion

R ∈ −Fu Sign(w), (1)

where Fu is the maximum frictional force developedin the damping device and the multi-application

Sign(z) =

−1, z < 0

[−1,+1], z = 0

+1, z > 0

. (2)

Inclusion (1), schematically represented in Figure 3,corresponds to the frictional constitutive law whichcan be defined as: when the velocity of the sectionis zero (w = 0), the absolute value of the force inthe frictional damping device is less than the max-imum frictional force Fu; on the other hand, whenw 6= 0 frictional force opposes the movement witha constant value given by Fu.

Figure 3: Constitutive law of a frictional dampingdevice with a maximum force of Fu.

2.2. Equation of the motionUsing the finite element method (FEM) the follow-ing equation of motion is obtained

Mq + Cq + Kq ∈ P + FΨT(xc) + R(q), (3)

where M is the mass matrix, C is the viscous damp-ing matrix, K is the stiffness matrix, P is the selfweight vector, FΨT(xc) is the moving load vector(where Ψ is the shape functions vector and xc isthe position of the moving load), R(q) is the vectorof frictional forces, q is the generalized nodal dis-placements vector and () represents the derivativewith respect to time.

Due to the non-smooth and non-linear behaviorof the constitutive frictional law (see the angularpoints in Figure 3), the NSCD method [12, 13, 14]was used to perform the time integration of theequation of the motion (3).

In the discrete model (numerical results in sec-tion 3) the vector R only has non zero values inthe components associated with the existence of a

2

FDD, which always coincide with some nodes of thefinite element mesh. In the continuous model (nu-merical results in section 4) the vector R has, ingeneral, non zero values in all components, corre-sponding both to forces and moments. In order tocompute the vector R in the continuous model, eachfinite element is divided in J sub-intervals and thenthe constitutive law is applied in all J + 1 sectionsto calculate the distributed impulses. Note that inthis case the distributed impulses depend on thefrictional force per unit length fu. Integrating theproduct of the distributed impulses by the shapefunctions over the finite element we get the general-ized vector of impulses of each finite element. Afterthe assembly of the elementary vectors we obtainthe global vector R. Based on tests that we made,we adopted J = 4 in all simulations.

All the numerical simulations whose results arepresented in the next sections were obtained con-sidering L = 200 m and using a mesh composed of200 finite elements of equal length. The time step his chosen so that the corresponding load progressionis 0.1 m, i.e the time step depends on the velocity ofthe load according to the rule h = 0, 1/v. More re-fined meshes and smaller time increments were alsoconsidered in this study but we found no significantdifferences.

3. Beam with a finite number of Coulombfrictional damping devices

In this section the most relevant numeric resultsfor the discrete model, where the beam is equippedwith a finite number of Coulomb frictional dampingdevices (FDD) are presented.

3.1. Critical velocity analysis

The critical velocity of the moving load for a givensystem is the one for which resonance occurs, a phe-nomenon characterized by very large transverse dis-placements. The theoretical value of the critical ve-locity of a linear elastic Winkler foundation [5] isgiven by

vcr = 4

√4kfEI

(ρA)2, (4)

where EI is the bending stiffness of the beam, ρAis the mass per unit length of the beam and kf isthe foundation stiffness. For the problem in hand,vcr = 206 m/s = 742 km/h. This critical velocityis very high and still unattainable by present dayhigh-speed trains; however, records of high dynamicdisplacement amplifications in railway tracks existfor relatively low velocities (v = 202 km/h) dueto the presence of very soft soils [15]. In Figures4, 5 and 6 we represent the maximum upward anddownward displacements of the beam as a functionof the velocity of the load for a different number of

discrete FDD uniformly distributed in the founda-tion and for three different values of Fu. In Figures4 and 5 it is possible to see that, in a certain range(Fu < 100 kN), for a fixed value of the force Fu

the number of FDD does not change the value ofthe critical velocity (which is practically coincidentto the critical velocity of a load on a linear elasticfoundation without frictional damping) but reducesthe peak of maximum displacements. The sameoccurs when the frictional force is varied keepingthe number of FDD constant (less than 7). On theother hand, in Figure 6, it is possible to see that theconsideration of a large number of FDD with highfrictional forces (Fu = 100 kN) decreases the valueof the critical velocity; however the correspondingpeak displacements are much lower than the onesoccurring for smaller frictional forces.

Figure 4: Maximum displacement as a function ofthe velocity of the load for Fu = 10 kN and for fourdifferent numbers of frictional damping devices.

To test the efficiency of a frictional damping sys-tem, the product of the number of FDD by the max-imum force developed in each device Fu was used asa measure of the cost of implementation of each sys-tem. Three groups of systems were tested in whichthe cost measure is invariant, according to Table 2.

In Figures 7 to 9 we represent the maximum up-ward and the maximum downward displacement ofthe beam as a function of the velocity of the load forthe three groups where the cost is invariant. Fromthese figures it can be concluded that the higherthe product of the number of FDD by the fric-tional force (higher cost), more the solutions differfrom each other. Another observation is that, atleast for the higher invariant cost (3100 kN), thegreater the number of FDD the smaller the value ofthe critical velocity and of the corresponding max-imum displacements. It is better to distribute a

3

Figure 5: Maximum displacement as a function ofthe velocity of the load for Fu = 50 kN and for fourdifferent numbers of frictional damping devices.

Figure 6: Maximum displacement as a function ofthe velocity of the load for Fu = 100 kN and for fourdifferent numbers of frictional damping devices.

larger number of FDD along the beam with a lowerfrictional force rather than to distribute a smallernumber of FDD with higher frictional force. Thisadvantage is more relevant the higher the cost of thefrictional damping solution. However, it should benoted that, although generally it is better to havea larger number of FDD with lower frictional force,locally (for a particular velocity) it may be worse(see Figure 7 at v = 175 m/s).

3.2. Optimization of the frictional force

For a fixed number of FDD, the greater the fric-tional force Fu the more expensive the system is.It is then interesting to find, for a fixed number ofFDD, what is the corresponding optimum value ofFu according to the maximum displacement (wmax)minimization criterion. In Figure 10 we represent

Table 2: Tested solutions with invariable cost.

number of FDD Fu (kN) cost31 100 310015 206.(6) 31007 442.857 31003 1033.(3) 310031 50 155015 103.(3) 15507 221.429 15503 516.(6) 155031 10 31015 20.(6) 3107 44.286 3103 103.(3) 310

Figure 7: Maximum displacement as a function ofthe velocity of the load for a number of FDD timesFu = 3100 kN.

the maximum upward and downward displacementsof the beam as a function of the frictional force forv = 206 m/s (theoretical critical velocity) and dif-ferent number of FDD. Optimal values for the fric-tional force lie between Fu = 90 kN and Fu = 315kN and decrease when the number of FDD in-creases.

4. Beam on a Winkler-Coulomb foundation

In this section the most relevant numerical resultsfor the continuous model, where the internal fric-tion in the granular material of the foundation isconsidered, are presented.

4.1. Critical velocity analysis

In Figure 11 we represent the maximum upward anddownward displacements of the beam as a functionof the velocity of the load in the case of a linearelastic Winkler foundation and in the case of a fric-tional damped foundation with four different valuesof the frictional parameter fu. When the frictional

4

Figure 8: Maximum displacement as a function ofthe velocity of the load for a number of FDD timesFu = 1550 kN.

Figure 9: Maximum displacement as a function ofthe velocity of the load for a number of FDD timesFu = 310 kN.

parameter is small (fu = 1 and 2 kN/m), the fric-tional damping does not affect the value of the crit-ical velocity but reduces the value of the peak ofmaximum displacements. For higher values of thefrictional parameter (fu = 5 and 10 kN/m), thereis a reduction of the value of the peak of maximumdisplacements and the value of the critical velocityincreases. The higher the frictional parameter, themore rigid is the beam (because more sections areblocked) and the natural frequency of the systemincreases resulting in a higher value of the criticalvelocity. In Figure 12 we represent a global normof the displacement w, defined by

W =

∫ ton0

∫ L

0|w(x, t)|dxdtton

, (5)

Figure 10: Maximum displacement as a functionof frictional force for v = 206 m/s and differentnumber of FDD.

given by the double integration of the transversedisplacement over the beam and over time, dividedby the simulation time. Near the critical velocitythis parameter is lower for the highest values of thefrictional parameter (fu = 5 and 10 kN/m) but forsub-critical velocities this parameter is higher forthe same curves. This occurs because, after the pas-sage of the load, the beam has significant permanentdisplacements. The total energy loss is representedin Figure 13 as a function of the velocity of the load.It is curious that the curve obtained for the highestvalue of the frictional parameter (fu = 10 kN/m)is not the curve with the higher total energy loss,such as it is not the curve with the lower globalnorm of the displacement, although it is the curvewith lower maximum displacements. When the fric-tional parameter increases too much, the maximumdisplacements decrease but the displacements thatoccur on the beam in general are higher and thesystem dissipates less energy.

4.2. Dynamic amplificationsThe dynamic amplifications of displacement andbending moment at the mid span section was anal-ysed. The dynamic amplification factors consideredare defined by

φw(τ) =w(τ)

wsta

φM (τ) =M(τ)

Msta,

where wsta and Msta are, respectively, the displace-ment and bending moment of the mid span sectionwhen the load acts there statically (v = 0 m/s)without any frictional dissipation (fu = 0 kN/m).The effect of the velocity of the load (v = 50, 100,150 and 200 m/s) on the dynamic amplification fac-

5

Figure 11: Maximum displacement as a function ofthe velocity of the load in the case of a linear elasticfoundation and four frictional damped foundations.

Figure 12: W as a function of the velocity of theload in the case of a linear elastic foundation andfour frictional damped foundations.

tors is represented in Figure 14. It may be con-cluded that dynamic amplifications are higher fora velocity near the critical velocity. It may be alsoobserved that higher frictional forces have a benefi-cial effect for velocities close to the critical one butthey are almost irrelevant for non-critical velocities.

5. Comparison between the two modelsIn this section the comparison between the twomodels, discrete and continuous friction, used forthe simulation of the frictional behavior is pre-sented. Two identical friction solutions were sim-ulated:

• discrete case: 199 FDD, each with Fu = 1 kN;

• continuous case: fu = 0.995 kN/m with L =

Figure 13: Total energy loss as a function of thevelocity of the load in the case of a linear elasticfoundation and four frictional damped foundations.

200 m.

Note that the potential capacity for dissipation ofenergy is the same in both cases: 0.995 × 200 =199× 1 kN. In Figure 15 we represent the displace-ment at the mid span section for both models whereit is possible to see that they are equal. In Figure16 we represent the impulses at the mid span sec-tion for both models. It can be observed that theyare practically equal in the region where the dis-placements (and the velocities) are more relevantbut are different in the side regions where the wavehas almost vanishing displacements (and velocities).Nevertheless, note that, as long as the force devel-oped in the FDD does not attain its maximum valuethat corresponds to a rest position, i.e. there is nomotion in the section.

6. Conclusions and further developmentsThis study focused on the effect of frictional dissi-pation in the critical velocity, maximum displace-ments, energy loss and dynamic amplifications of abeam on a foundation under a moving load. Twosituations were modeled: a finite number of fric-tional damping devices uniformly distributed alongthe beam (discrete model) and continuously dis-tributed frictional damping in the foundation (con-tinuous model).

In the discrete model it was observed that, in acertain range (Fu < 100 kN and for less than 7FDD), the frictional dissipation doesn’t change thecritical velocity but reduces the peak of maximumdisplacements. When comparing different solutionswith the same cost (the number of FDD times thefrictional force), the higher the cost more the so-lutions differ from each other. Moreover, it is bet-ter to distribute a larger number of FDD along thebeam with a lower maximal frictional force rather

6

(a) φw for fu = 1 kN/m.

(b) φM for fu = 1 kN/m.

(c) φw for fu = 2 kN/m.

(d) φM for fu = 2 kN/m.

Figure 14: Dynamic amplifications at mid span asa function of τ .

(a) Result of the discrete friction algorithm.

(b) Result of the continuous friction algo-rithm.

Figure 15: Displacement at the mid span section asa function of τ .

than to distribute a smaller number of FDD withhigher maximal frictional force. This advantage ismore relevant the higher the cost of the frictionaldamping solution. For a particular number of FDDthere is an optimal value of the corresponding fric-tional force (that is, of the force that minimizes thepeak displacements) and its value decreases whenthe number of FDD increases.

In the continuous model, as in the discrete model,it was noticed that when the frictional parameter issmall (fu = 1 and 2 kN/m), the frictional dissipa-tion does not change the value of the critical ve-locity but reduces the peak of maximum displace-ments. For higher values of the frictional parameter(fu = 5 and 10 kN/m) there is a reduction of thevalue of the peak of maximum displacements andthe value of the critical velocity increases. The en-ergy dissipation is larger in the case of a foundationof higher frictional force but only up to a certainlimit (< 10 kN/m), from which friction becomes dis-advantageous and less energy is dissipated. The dy-namic amplification of the displacement and bend-ing moment at the mid span are higher when thevelocity is close to the critical value.

In the future, it would be interesting to improvethis study by considering some improvements in

7

(a) Result of the discrete friction algorithm.

(b) Result of the continuous friction algo-rithm.

Figure 16: Impulse at the mid span section as afunction of τ .

modeling and analysis: (i) obtaining the form of thestationary wave in an infinite beam in the case of afoundation with distributed friction, which requiresthe consideration of special boundary conditions toavoid reflections at the beam ends; (ii) the consid-eration of a distributed moving load simultaneouslywith concentrated loads (HSLM - high speed loadmodel - defined in Eurocode); (iii) the considera-tion of a moving mass; (iv) the consideration of amoving oscillator with the eventual consideration ofthe rail’s corrugation; (v) the improvement of thetime integration algorithm in order to reduce thenumber of equilibrium iterations.

Acknowledgements

The author would like to thank professors AntonioPinto da Costa and Fernando Simoes for the excel-lent orientation provided and family and friends forcompany and support.

References

[1] A.N. Krylov. Uber die erzwungenenschwingungen von gleichformigen elastichenstaben (about forced vibrations of prismaticelastic beams). Mathematische Annalen,

61:211–234, 1905.

[2] S.P. Timoshenko, W. Weaver, and D.H. Young.Vibration Problems In Engineering. Jonh Wi-ley, 1974.

[3] C.E. Inglis. A mathematical treatise on vibra-tion in railway bridges. Cambridge UniversityPress, 1934.

[4] A.N. Lowan. On transverse oscillations ofbeams under the action of moving variableloads. Philosophical Magazine, 19(127):708–715, 1935.

[5] L. Fryba. Vibration os solids and structuresunder moving loads. Research Institute ofTransport, 1972.

[6] D. Thambiratnam and Y. Zhuge. Dynamicanalysis of beams on an elastic foundation sub-jected to moving loads. Journal of Sound andVibration, 198(2):149–169, 1996.

[7] Z. Dimitrovova and A.F.S. Rodrigues. Criticalvelocity of a uniformly moving load. Advancesin Engineering Software, 50:44–56, 2012.

[8] P. Castro Jorge. Analise dinamica de vigasfinitas sobre fundacao elastica sujeitas a cargasmoveis. Dissertacao para obtencao do grau deMestre em Engenharia Civil, Instituto Supe-rior Tecnico, 2013.

[9] Pedro Castro Jorge, Antonio Pinto da Costa,and Fernando Manuel Fernandes Simoes. Dy-namics of beams on non-uniform nonlinearfoundations subjected to moving loads. Com-puters and Structures, 148:26–34, 2015.

[10] P. Castro Jorge, A. Pinto da Costa, and F.M.F.Simoes. Finite element dynamic analysis of fi-nite beams on a bilinear foundation under amoving load. Jornal of Sound and Vibration,346:328–344, 2015.

[11] Thalys. https://www.thalys.com.

[12] J.J. Moreau. Some numerical methods inmultibody dynamics: application to granularmaterials. European Journal of Mechanics,13(4):93–114, 1994.

[13] M. Jean. The non-smooth contact dynamicsmethod. Computer Methods in Applied Me-chanics and Engineering, 177:235–257, 1999.

[14] V. Acary and B. Brogliato. Numerical Methodsfor Nonsmooth Dynamical Systems. Springer,2008.

8

[15] C. Madshus and A.M. Kaynia. High-speed rail-way lines on soft ground: Dynamic behaviourat critical train speed. Journal of Sound andVibration, 231(3):689–701, 2000.

9