Numerical evaluation of the dynamic response of ...

Advances in Environmental Vibration Fifth International Symposium on Environmental Vibration, Chengdu, China, October 20-22, 2011

Numerical evaluation of the dynamic response of pipelines to vibrations induced by the operation of a pavement breaker 1

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Zuhal OZDEMIR , Mohammad Amin LAK , Pieter COULIER , Stijn FRANÇOIS , Geert 1 1 LOMBAERT and Geert DEGRANDE 1. Department of Civil Engineering, K.U.Leuven, Kasteelpark Arenberg 40, B-3001 Leuven, Belgium

Abstract: This paper presents a two-and-a-half dimensional (2.5D) coupled finite element-boundary element (FE-BE) method to predict the response of lifelines subjected to ground-borne vibrations. The dynamic soil-structure interaction (SSI) problem is solved with a subdomain formulation. The presented methodology is used to assess the risk of damage to a buried high pressure steel natural gas pipeline due the operation of a pavement breaker during the rehabilitation of concrete pavements. It is observed that the stresses in the pipeline remain much lower than those induced by the operating internal pressure, which indicates that there should be no fear of damage. Keywords: Lifelines; dynamic soil-structure interaction; 2.5D FE-BE method; ground-borne vibrations; pavement breaker.

1

Introduction

Lifelines transmit resources such as water, fuel, energy and information which are of vital importance to industrial facilities and communities. Ground vibrations induced by earthquakes, construction activities, traffic, explosions or industrial activities can be potentially damaging to these structures. One particular example are vibrations generated during the rehabilitation of roads. Deteriorated jointed plain concrete pavements (JPCP) are often rehabilitated by overlaying bituminous layers on the road. Prior to the installation of the bituminous layer, the concrete pavement is cracked into smaller pieces by a pavement breaker which repeatedly drops large weights on the slabs. These pieces are subsequently seated by passes of a roller to form a stabilized foundation. The method is often not applied as it is feared that vibrations generated by the pavement breaker during the cracking phase may cause damage to pipelines close to the road. The objective of this paper is to present an efficient numerical model in order to evaluate the response of lifelines nearby a road to vibrations generated by a pavement breaker. In

this work, the incident wave field is incorporated in a 2.5D coupled FE-BE methodology developed by François et al. (2010), where the invariance of the geometry in the longitudinal direction is exploited to formulate an efficient solution procedure in the frequencywavenumber domain. The dynamic soil-structure interaction (SSI) problem is solved by means of a subdomain formulation. The presented methodology is subsequently applied to evaluate the response of a steel natural gas pipeline during the rehabilitation of concrete pavements, and the risk of damage is assessed.

2 2.1

Numerical model Subdomain formulation for dynamic SSI

In the subdomain formulation (Aubry and Clouteau, 1992), the soil-structure system is decomposed into two subdomains: the bounded structure (the pipe) and the unbounded semi-infinite layered soil (Fig. 1). Continuity of displacements and equilibrium of stresses are enforced on the interface  ps between the pipe and soil:

uˆ p  uˆ s  0 on  ps tˆ

np p

uˆ   tˆ uˆ   0 p

ns s

s

on ps

(1)

2.2

The 2.5D coupled FE-BE method

(2)

The equilibrium equation for the dynamic SSI problem is formulated in a variational form. For any virtual displacement field vˆ p imposed on the structure  p , the sum of the virtual work of the internal and inertial forces is equal to the virtual work of the external loads:

 ˆ  vˆ  : ˆ uˆ  d     vˆ 2

p

p

p

p

p

(a)



(b)

uˆ p and uˆ s represent the displacement vectors of the structure and the soil, respectively. n tˆ pp uˆ p  ˆ p uˆ p  np is the traction vector on the interface  ps with a unit outward normal vector n p and ˆ p uˆ p is the Cauchy stress tensor. A hat above a variable denotes its representation in the frequency domain. The displacement field uˆ s in the soil is decomposed into the wave field uˆ 0 and the scattered wave field uˆ sc :

 

 vˆ

p

 

 

 

uˆ s  uˆ 0  uˆ sc uˆ p

(3)

The wave field uˆ 0 is the wave field in the soil generated by the incident wave field, when zero displacement boundary conditions are assumed on the interface  ps . The wave field uˆ sc uˆ p corresponds to the displacement field radiated in the soil due to the structural displacements uˆ p on the interface  ps . The scattered wave field uˆ sc uˆ p is computed by means of a BE method, as it is posed in terms of displacements on the interface  ps . However, the wave field uˆ 0 cannot straightforwardly be computed by means of a BE formulation, since the assumption of zero volume forces does not hold in the case of the incident wave field uˆ inc . Therefore, the wave field uˆ 0 is further decomposed into the incident wave field uˆ inc and a locally diffracted wave field uˆ d0 : uˆ 0  uˆ inc  uˆ d0 (4) ˆ The wave field u d0 is computed by means of the BE method. The total soil displacement uˆ s follows from Eq. (3) and Eq. (4): uˆ s  uˆ inc  uˆ d0  uˆ sc uˆ p (5)

 

 

 

 

n  tˆ pp uˆ p d  

ps

Fig. 1 The soil-structure interaction problem: (a) the coupled soil-lifeline system and (b) the lifeline subdomain.

p

 puˆ p d 

p

 vˆ

p

ˆn  tp p d 

(6)

p

     

where ˆ p vˆ p is the strain tensor corresponding to the virtual displacement vˆ p and ˆ p vˆ p : ˆ p uˆ p represents the tensor product of the strain and stress tensors. Accounting for the stress equilibrium at the soil-structure interface  ps defined in Eq. (2), the displacement decomposition given in Eq. (5), and the continuity of displacements defined in Eq. (1), the virtual work equation (6) is elaborated with a finite element formulation for the interpolation of the displacement field with respect to the coordinates x and z. The latter can be justified due to the fact that a longitudinally invariant structure is considered. The equilibrium equation is transformed to an algebraic system of equations by a forward Fourier transform of the longitudinal coordinate y to the horizontal wavenumber ky:

K 0pp  ik y K1pp  k y2K 2pp  ik y3K 3pp  k y4K 4pp   2M pp



 











K spp k y ,   up k y ,   fp k y ,   fps k y ,   (7) where the finite element stiffness matrices K 0pp , K1pp , K 2pp , K 3pp , and K 4pp and the mass matrix M pp are independent of the wavenumber ky and the frequency  and therefore only assembled once. K spp represents the dynamic soil stiffness matrix. The force vector fp results from the external forces on the structure. The force vector fps denotes the dynamic SSI forces at the soil-structure interface S ps due to the incident wave field u inc and the locally diffracted wave field u d0 . A tilde above a variable denotes its representation in the frequency-wavenumber domain. The interface S ps between the structure and the soil is discretized into 2.5D boundary elements that match the finite element mesh of the structure. Therefore, the boundary element interpolation functions Ns  x, z  correspond to the finite element shape functions Np  x, z  on

the soil-structure interface. Discretizing the displacements and tractions, the force vector fps in Eq. (7) is written as: (8) fps k y ,   Tq tsns uˆ inc k y ,   uˆ d0 k y , 



 









where the boundary element stress transfer matrix Tq   NTp N p dS is independent of the wavenumberSps ky and the frequency . The dynamic stiffness matrix of the soil K spp is written as: (9) K spp k y ,   Tq tsns uˆ sc Np k y , 



   





The unknown tractions in Eqs. (8) and (9) are evaluated by means of the 2.5D BE formulation. The boundary element system of equations for an unbounded domain is expressed as follows (François et al., 2010):











 



T k y ,   I  us k y ,   U k y ,  ts k y ,  (10)  









where U k y ,  and T k y ,  denote the fully populated unsymmetric boundary element system matrices for an unbounded domain and represents a unity matrix, which corresponds to the integral free term in the boundary integral equation. The boundary element system matrices U k y ,  and T k y ,  are based on Green's displacements and tractions for a horizontally layered (visco-)elastic halfspace in the frequency-wavenumber domain (Schevenels et al., 2009). The solution of the system of equations (7) provides the displacement vector of the structure u p . Eq. (1) and (10) are subsequently used to evaluate the soil tractions ts at the interface S ps . The discretized form of the boundary integral equation (François et al., 2010) is finally employed to evaluate the radiated wave field u in the soil from the tractions ts and displacements u s at the interface S ps :











 





 

transformations are performed from the horizontal wavenumber ky to the coordinate y along the pipe and from the frequency domain to the time domain, respectively, to formulate the solution of the coupled system in the time-space domain. The 2.5D coupled FE-BE method has been verified by means of results available in the literature for a buried pipeline subjected to plane harmonic P- and SV-waves (Coulier et al.).

3 Response of pipelines vibrations generated by pavement breaker

to a

3.1 Problem definition In this section, the response of a pipeline to vibrations generated by the operation of a multihead pavement breaker (MHB) during the rehabilitation of concrete pavements is evaluated with the 2.5D coupled FE-BE method. The MHB uses 4 independently controlled cylindrical drop hammers with a mass of about 600kg to break concrete slabs (Fig. 2). The steel cylinders are dropped from a height of 1.30m about 130 times per minute. In situ vibration measurements during the operation of the MHB were carried out along the N9 motorway in Waarschoot, Belgium (Lak et al., 2009) during 8 individual impacts of the MHB at 4 points on the slab. A shock accelerometer was installed on the drop hammer of the MHB to estimate the force applied on the concrete slab. The response of the concrete slab and the free field was measured by means of seismic accelerometers.



u  Ur k y ,  ts k y ,   Tr k y ,  us k y ,  (11) where the displacements at receiver locations (xr,zr) are collected in a vector u . The matrices Ur k y , and Tr k y ,  follow from the introduction of the boundary element discretization in the integral representation theorem. The wave field u inc , u d0 and u sc at the receiver locations are evaluated by means of Eq. (11). The total displacement field is obtained by the displacement decomposition defined in Eq. (5). Two successive backward Fourier









Fig. 2 3.2

The multihead pavement breaker.

Coupled road-soil system

The impact footprint of the drop hammers of the MHB is very small with a diameter of about 0.13m, and it is therefore considered as a point load. As the first natural frequency of the drop

hammer is 1100Hz, the drop hammer is considered as a rigid mass. This reduces the problem to the evaluation of the road-soil transfer functions between a point load on the road and the vibration velocity in the free field. In the numerical model, the road is represented by a concrete plate with a thickness d=0.20m, a width B=2.85m, a Young's modulus E=430  108 N/m2, a material density =2400kg/m3 and a Poisson's ratio =0.20. The impact point of the drop hammer on the concrete slab is located at e =-0.75m (Fig. 3).

the source x' on the road and a receiver located at a point x in the free field are computed. Table 1 Dynamic soil characteristics at the site in Waarschoot. Layer 1 2 3 4 5 6 7 8 9 10

Thickness [m] 0.9 2.7 2.3 2.2 4.1 4.0 2.7 7.4 12.5 ∞

Cs [m/s] 126 130 138 224 240 250 255 328 400 500

Cp [m/s] 444 460 1270 1373 1579 1829 2000 2000 2000 2000



s=p

[−] 0.46 0.46 0.49 0.49 0.49 0.49 0.49 0.49 0.48 0.47

[−] 0.103 0.091 0.082 0.065 0.040 0.040 0.040 0.040 0.040 0.040

3.3 Free field response due to the operation of the MHB Fig. 3

Geometry of the problem.

The dynamic small-strain soil characteristics at the site were determined by means of a Spectral Analysis of Surface Waves (SASW) test (Badsar et al., 2009). A soil density of s=1900kg/m3 is assumed for all layers. The small-strain material damping ratio for the shear waves so is estimated from the experimental attenuation curve using the half-power bandwidth method, while the same values are used for the material damping ratio po for the dilatational waves. The behaviour of the soil and the slab in the vicinity of the impact of the drop hammer is highly non-linear. In order to account for the non-linear effects in the soil, an equivalent linear analysis has been performed, where the soil characteristics are iteratively modified in function of the effective strain levels in the soil until convergence is reached (Coulier et al.). This analysis resulted in a soil profile where the dynamic characteristics of the top four layers are modified. The resulting soil characteristics used throughout the remainder of this paper are summarized in Table 1. The coupled road-soil system is modelled by means of the 2.5D coupled FE-BE method presented in Section 2.2. The road is modelled with 40 2.5D shell elements. Using the coupled road-soil model, the transfer functions hˆ  x',x,   due to a vertical unit impact load at

Fig. 6a shows a zoom of the time history of the acceleration of the drop hammer recorded during in situ vibration measurements (Lak et al., 2009). The response reaches a peak value of 5350m/s2 at t  0.1975s and the impact duration ti is about 0.002s. Fig. 6b shows the corresponding frequency content of the acceleration of the drop hammer, which is mainly situated below 900Hz. This measured force is combined with the road-soil transfer functions hˆ  x',x,   to determine the free field response at several receiver locations.

Fig. 6 Measured (a) time history (between t=0.18s and t=0.22s) and (b) frequency content of the acceleration of the drop hammer of the MHB. Fig. 7 shows the time history and frequency content of the measured free field vertical velocity at points at 5m, 11m, 15m, 21m, 33m and 41m from the center of the concrete slab during the impact of the drop hammer of the MHB. Superimposed on Fig. 7 are the computed free field responses at the considered receiver locations. A reasonable agreement between measurements and predictions is obtained.

A high pressure steel natural gas pipeline (API 5L Grade X70) is considered, which center is located at a depth of D=1.405m below the free surface of the soil and at a lateral distance A=4.0m from the center of the road (Fig. 3). The pipe has an external radius ro=0.305m and a wall thickness t=0.009m. The steel has a Young's modulus E=200GPa, a Poisson's ratio =0.30 and a density =7800kg/m3. The specified minimum yield strength of the steel is equal to fy=485MPa. The pipe is subjected to an operating internal pressure p =84bar. The pipe is modelled with 36 2.5D shell elements.

Fig. 7 Measured (solid line) and predicted (dotted line) time history (left hand side) and frequency content (right hand side) of the free field vertical velocity for receivers located at (a) 5m, (b) 11m, (c) 15m, (d) 21m, (e) 33m and (f) 41m from the center of the road during the impact of the drop hammer of the MHB. The resulting incident wave field uˆ inc is used to evaluate the dynamic response of coupled soil-pipe system due to the operation of the MHB. 3.4

Response of the structure

The response of the coupled pipeline-soil system is first evaluated by means of the 2.5D coupled FE-BE method under the incident wave field generated by a vertical unit impact load applied at a point on the concrete slab. The pipe response is subsequently combined with the measured force applied by the drop hammer of the MHB in the frequency domain (Fig. 6b). Two inverse Fourier transformations are applied to obtain the time history of the response of the pipe in the spatial domain.

Fig. 8 Axial stress yy (r =ro) (left hand side) and hoop stress (r =ri) (right hand side) on the first 10m of the steel natural gas pipeline due to the operation of the MHB at (a) t=0.2075s, (b) t=0.2225s, (c) t=0.2375s, (d) t=0.2525s, (e) t=0.2675s and (f) t=0.2825s.

Fig. 8 shows a sequence of the axial stress

yy (r =ro) and the hoop stress (r =ri) on the first 10m of the steel pipeline due to the operation of the MHB. The wave front needs 0.01s to reach the pipe, and impinges at y =0m at t =0.2075s, which is clearly apparent in Fig. 8a. Stresses are subsequently developed in the pipe, and the axial stress yy reaches extrema of -2.95MPa and 2.29MPa at t =0.2325s, while the extrema of the hoop stressare equal to -2.45MPa and 2.58MPa at t =0.23s. A maximal Von Mises stress v of 2.86MPa is obtained. At an operating internal pressure p=84bar, the longitudinal stress in the pipe is equal to yy = pri/2t =138MPa, while the hoop stress equals  pri/t =276MPa. The axial stress yy and hoop stressinduced in the pipe due to the operation of the MHB are much smaller than those generated by the operating internal pressure. The total Von Mises stress v including the effect of the operating internal pressure reaches 241MPa. The steel pipeline behaves in the linear elastic range under the combined effect of the operating internal pressure and the vibrations generated by the operation of the MHB. There should hence be no fear that the operation of the MHB causes damage to steel pipelines close to the road.

4

Conclusions

In this paper, the incident wave field generated by ground-borne vibrations is incorporated in the 2.5D coupled FE-BE methodology developed by François et al. (2010) to analyze the dynamic response of lifelines. The presented numerical model is used to assess the risk of damage to a high pressure steel natural gas pipeline due to the operation of a pavement breaker. It is observed that the stresses induced in the pipeline due to the operation of the pavement breaker are much lower than those induced by the operating internal pressure. These results indicate that it should not be feared to use a multihead pavement breaker in order to crack deteriorated concrete pavements.

Acknowledgement The results presented in this paper have been obtained within the frame of the IWT-project VIS-CO 060884, “Vibration controlled

stabilization of concrete slabs for durable asphalt overlaying with crack prevention membrane”, funded by IWT Vlaanderen (Institute for the Promotion of Innovation by Science and Technology in Flanders). The third author is a Research Assistant and the fourth author is a Postdoctoral Research Fellow of the Research Foundation-Flanders (FWO). The financial support is gratefully acknowledged.

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