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ScienceDirect Procedia Engineering 190 (2017) 547 – 553

Structural and Physical Aspects of Construction Engineering

Mutual Comparison of two Pavement Computing Models Veronika Valaškováa,*, Gabriela Lajčákováa a

University of Žilina, Univerzitná 8215/1, 010 26 Žilina, Slovak Republic

Abstract Moving load effect on pavements is actual engineering problem. For the numerical simulation of moving load effect on pavements the various computing models were created. In the proposed paper two computing models of pavement are described and mutually compared. One computing model is created in the sense of the beam on elastic foundation and the second in the sense of Finite Element method. The plane computing model of vehicle is adopted. ©2017 2017The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license © Published by Elsevier Ltd. This (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the issue editors. Peer-review under responsibility of the organizing committee of SPACE 2016 Keywords: Moving load; pavement; computing model; vehicle; beam on elastic foundation; Finite Element Methods.

1. Introduction The pavements are the structures directly exposed to dynamic effect of moving vehicles. To know the stress and strain states in dynamic regime of load is necessary for assessment of many engineering problems as fatigue, life time, reliability and so on. There are two basic approaches how to obtain the required data – numerical or experimental. With regard to various constructive variants of pavements the numerical approach is very effective. There are many possibilities how to create the computing model of the pavement and computing model of the load. In this paper two computing models of the pavement are introduced and mutually compared. One model comes from the solution of equation of motion of the beam on elastic foundation and the second model is created in the sense of Finite Element Method. The results of numerical simulations in time and in frequency domain can be employed for the solution of many engineering tasks [1, 2].

* Corresponding author. Tel.: +421-41-513 5613; fax: +421-41-513 5510. E-mail address: [email protected]

1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of SPACE 2016

doi:10.1016/j.proeng.2017.05.378

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Veronika Valašková and Gabriela Lajčáková / Procedia Engineering 190 (2017) 547 – 553

2. Vehicle computing model The plane computing model of vehicle is adopted for the solution of this task, Fig. 1, [3]. It is discrete computing model with 8 degrees of freedom, 5 mass and 3 mass-less degrees of freedom. The mass-less degrees of freedom correspond to vertical movements of contact points of the model with the pavement.

Fig. 1. Plane computing model of vehicle.

Vertical vibration of mass objects is described by 5 functions of time ri(t), (i = 1,2,3,4,5).

r1 t {k1 d1 t b1 d1 t k2 d2 t b2 d2 t f 2 d2(t)/dc } / m1 , r2 t {a k1 d1 t a b1 d1 t b k2 d2 t b b2 d2 t f 2 d2(t)/dc } / I y1 ,

r3 t {k1 d1 t b1 d1 t k3 d3 t b3 d3 t } / m2 , r4 t {k2 d2 t b2 d2 t f 2 d2(t)/dc k4 d4 t b4 d4 t k5 d5 t b5 d5 t } / m3 , r5 t {c k4 d4 t c b4 d4 t c k5 d5 t c b5 d5 t } / I y3 .

(1)

The contact forces Fj(t), (j = 3,4,5) corresponds to mass-less degrees of freedom

b § · F3 t G3 k3 d3 t b3 d3 t g ¨ m1 m2 ¸ k3 d3 t b3 d3 t , s © ¹ 1 a § · F4 t G4 k4 d 4 t b4 d4 t g ¨ m1 m3 ¸ k 4 d 4 t b4 d4 t , 2 s © ¹ 1 a § · F5 t G5 k5 d5 t b5 d5 t g ¨ m1 m3 ¸ k5 d5 t b5 d5 t . 2 s © ¹

(2)

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Deformations of connecting members di(t), (i = 1,2,3,4,5) are as follows

d1 (t )

r1 (t ) a r2 (t ) r3 (t ) ,

d 2 (t )

r1 (t ) b r2 (t ) r4 (t ) ,

d3 (t )

r3 (t ) u3 (t ) ,

d 4 (t )

r4 (t ) c r5 (t ) u4 (t ) ,

d5 (t )

r4 (t ) c r5 (t ) u5 (t ) .

(3)

Derivations with respect to time t are denoted by dot above the symbol. Gj, (j = 3,4,5) represent the gravity forces acting at the contact points. uj(t), (j = 3,4,5) are the components of kinematical excitation of contact points. Meaning of other symbols is clear from Fig. 1. 3. Pavement computing model as the beam on elastic foundation The computing model of pavement comes from the theory of the beam on elastic foundation (BEF) described by the equation EI

w 2vx, t wvx, t w 4vx, t μ 2 μωb kv x, t 4 wt wx wt 2

px, t

(4)

The function vx, t described the bending line of the beam at point x in time t, E is the modulus of elasticity, I is the cross section quadratic moment, μ is mass intensity per unit length, k is the coefficient of foundation compressibility, ωb is the damping angular frequency and px, t is the continuously distributed load. The equation (4) is the partial differential equation for the unknown function vx, t . To avoid the solution of partial differentia equation the assumption about the shape of bending line of the beam is adopted. The solution is carried out in the sense of Fourier method on the section of the length l. The wanted function vx, t is expressed as the product of two functions

v( x, t )

v0 ( x) q(t )

2πx · 1§ ¨1 cos ¸ q(t ) . 2© l ¹

(5)

Function v0 ( x) is known function and it is dependent on x and function q(t ) is unknown function and it is dependent on time t. By substituting the assumption (5) into equation (4) we obtain 4 1 § 2πx · 2πx · 1ª § 2πx · 2πx º § § 2π · qt μ¨1 cos » ¸ q t μωb ¨1 cos ¸ qt «k ¨1 cos ¸ EI ¨ ¸ cos 2 © l ¹ l ¹ 2 ¬« © l ¹ l ¼» © © l ¹

px, t .

(6)

It is ordinary differential equation for the function q(t ) . In case of moving vehicle the discrete load must be transformed on continuous load. We can do it by the advance proposed by Dirac [4]. p( x, t )

¦ j

f

F j (t )δ ( x x j )

¦¦ p j

n 1

n, j

n2πx 1 (t ) (1 cos ), l 2

(7)

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where l

pn, j (t )

n2πx 2 1 ε j F j (t )δ ( x x j ) (1 cos ) dx l 0 l 2

p( x, t )

¦¦ l ε F (t ) 2 (1 cos

³

n2πx j 2 1 ε j F j (t ) (1 cos ). l l 2

(8)

Then f

2

1

j

j

j

n 1

n2πx j 1 n2πx ) (1 cos ). l l 2

(9)

When we assume only the 1st member of the series the equation (9) is simplified

p( x, t )

¦ ε F (t ) 2l (1 cos 1

j

2πx j

j

l

j

Coefficient ε j

)(1 cos

2πx ). l

1 when the contact force F j (t ) occurs in observed section otherwise ε j

deflections are observed in one point only, for example x

p( x, t )

¦ ε F (t ) l (1 cos 1

j

j

(10)

j

2πx j l

0 . When the

l / 2 , equation (10) is simplified

).

(11)

Also the left hand side of equation (6) is simplified 4 ª 1 § 2π · º qt μ q t 2 μωb qt «k EI ¨ ¸ » 2 © l ¹ »¼ «¬

p( x, t ) .

(12)

4. Pavement computing model created in the sense of FEM One of the most important part of the process of numerical simulation is to create a proper computing model. For this case the plane model of TATRA 815 lorry was chosen as the most representative vehicle (Fig. 1). In this case the vehicle moved on the pavement with specific material characteristics. In the article a numerical 2D model of the vehicle as well as pavement are created using computer software ADINA. Computational model of the pavement is based on Kirchhoff theory of thin plates on a flexible substrate shown in Fig. 2.

Fig. 2. The numerical model of interaction system.

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The pavement is modeled as a beam element of 50 m length. Pavement is composed with surface for acceleration (5 m), surface for plain ride (40 m) and surface for breaking (5 m). Winkler elastic foundation is composed on springs with specific stiffness. Winkler elastic foundation is supported with boundary conditions and degrees of freedom necessary for the computation. An effective implicit time integration scheme was proposed for the finite element solution of nonlinear dynamics problems. Direct time integration is widely used in the finite element solutions of transient wave propagation problems. We consider the Bathe method and the Newmark trapezoidal rule. 5. Parameters of the pavement Numerical calculations were realized for the asphalt pavement. The composition of the pavement is shown in Fig. 3. Three upper layers are considered as the beam of the depth h h1 h2 h3 40 50 50 140 mm and the width b = 1.0 m. The equivalent modulus of elasticity and cross section moment of inertia was calculated for those three layers

E

I

E1h1 E2 h2 E3h3 h1 h2 h3 1 3 bh 12

1 1.0 0.143 12

5500 40 6000 50 3050 50 40 50 50

4803.5714 # 4800 MPa ,

2.2866667 104 m 4 .

(13)

(14)

The considered length of the beam was l = 6.0 m. This length resulted from the analysis of in situ experimental measurements [5]. The other layers 4 – 6 are introduced into calculation as Winkler elastic foundation. Modulus of compressibility K [MN/m3] was calculated by the program LAYMED [6]. The mass intensity of the beam μ [kg/m] is calculated as

μ

ρbh

2200 1.0 0.14 308.0 # 310.0 kg/m .

(15)

The damping is considered via damping angular frequency ωb model is mV = 13 000 kg.

MGMA; 40 mm; E = 5500 MPa; Q = 0.33 AC;

150 mm; E = 2000 MPa; Q = 0.22

GS;

150 mm; E = 400 MPa; Q = 0.30 f;

E = 4800 MPa

E = 3050 MPa; Q = 0.33

SC;

S;

h = 140 mm

50 mm; E = 6000 MPa; Q = 0.30

CG; 50 mm;

E=

0.1 rad/s . The total mass of vehicle computing

I = 2.2866· 10-4 m4

{

60 MPa; Q = 0.35 k # 156.04549 MPa

Fig. 3. Pavement computing model, MGMA – medium grained mastic asphalt, AC – asphalt concrete, CG – coated gravel, SC – soil cement, GS – gravel sand, S – subgrade.

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6. Mutual comparison of obtain results At the 1st step the dependence of pavement dynamic deflections at various speeds of vehicle motion were calculated for the pavement computing model as the beam on elastic foundation. The maxima pavement deflections wmax versus speed of vehicle motion V are plotted on Fig. 4. The absolute maximum of deflection occurs at the speed 60 km/h. At the 2nd step the time history of pavement dynamic deflection at selected point for both pavement computing models (BEF and FEM) were calculated and mutually compared, Fig. 5. The obtained results are very similar. Maximal difference occurs at the time 11.34 s. The BEF computing model gives the value of deflection wBEF = 0.1778 mm. The FEM computing model gives the value of deflection wFEM = 0.1884 mm. The difference D = wFEM – wBEF = 0.0106 mm. The difference in percentage is 5.96 %. The value wBEF is assumed as 100 %. The FEM computing model gives major values. Maximal pavement deflection wmax

0.1778 0.1776 0.1774 0.1772

0

20

40

60 V [km/h]

80

100

120

Fig. 4. Maxima pavement deflections versus speed of vehicle motion.

Comparison of BEF and FEM models, V=60 km/h 0

w [mm]

wmax [mm]

0.178

-0.1 BEF FEM -0.2 10.7

10.8

10.9

11

11.1

11.2 t [s]

11.3

11.4

11.5

11.6

Fig. 5. Comparison of deflections for BEF and FEM pavement computing models.

11.7


Acknowledgements This work was supported by the Grant National Agency VEGA of the Slovak Republic, project number 1/0005/16. References [1] K. Kotrasová, E. Kormaníková, Dynamic Analysis of Liquid Storage Cylindrical Tanks Due to Earthquake. In: Advanced Materials Research. No. 969 (2014), p. 119-124. ISSN 1662-8985. [2] J. Králik, O. Ivánková, Deterministic and probability analysis of ventilating chimney seismic reliability, 12-internationa symposium, Srtga, Macedonia, Sept. 27-29, 2007, 2 (2007) 403-410. [3] J. Melcer, et al., Dynamics of Transport Structures (in Slovak), first ed., EDIS, Žilina, 2016. [4] L. Frýba, Vibration of Solids and Structures under Moving Load, first ed., ACADEMIA, Praha, Noordhoff International Publishing, Groningen, 1972. [5] Final Report VTP 0402840504 Reengineering of Highway Network in Slovak Republic (in Slovak). VUIS-CESTY, Bratislava, 2000. [6] B. Novotný, A. Hanuška, Theory of layered half-space (in Slovak), first ed., VEDA, SAV, Bratislava, 1983.

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