3-D simulation of tunnel structures under blast loading

archives of civil and mechanical engineering 13 (2013) 128–134

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journal homepage: www.elsevier.com/locate/acme

Original Research Article

3-D simulation of tunnel structures under blast loading M. Buonsantia, G. Leonardib,n a

Department of Mechanics and Materials, Mediterranea University of Reggio Calabria, Italy Via Graziella, Feo di Vito 89100, Reggio Calabria, Italy b Department of Information, Electronics and Transport Engineering, Mediterranea University of Reggio Calabria, Italy, Via Graziella, Feo di Vito 89100, Reggio Calabria, Italy

ar t ic l e in f o

abs tra ct

Article history:

In this paper we want to study the behaviour of an underground structure subject to blast

Received 14 June 2012

action. Other than the normal operating loads we considered the actions generated by a

Accepted 26 September 2012

thermal gradient simulating the action of a fire load and by a pressure wave simulating an

Available online 1 October 2012

explosion. Initially the theoretical aspects of the problem were exanimated, and then a

Keywords:

model was developed in numerical form and implemented through a finite element

Tunnel

analysis. This modelling allows the simulation of a real scenario, e.g., railway tunnels with

Explosion

reinforced concrete structure that is subject to a fire generation and a subsequent

Blast

explosion. The simulation involved aspects of thermal analysis, and therefore the

Shock wave

structural problem was tackled analysing the tensions in the structure generated by the

Finite element analysis

effect of temperature–pressure generated by the fire and by the overpressure generated by the blast. Only following this approach the most important factors influencing the dynamic response and damage of structure can be identified and the appropriate preventive measures can be designated. & 2012 Politechnika Wrocławska. Published by Elsevier Urban & Partner Sp. z o.o. All rights reserved.

1.

Introduction

The tunnel and underground structures are widely used in civil engineering; thus, the possibility of explosions in tunnel caused by fire accidents or terrorist attack is also increasing with the development of subway and underground structures. Consequently, the analysis of blast wave propagation inside tunnels has great significance. The major risks are associated with the transportation of hazardous materials like LPG. LPG is a major hazardous material transported in bulk by road and railway. Because LPG is a liquefied gas, that is a superheated liquid, it is stored and transported under pressure, the vapour pressure at the

prevailing liquid temperature. In the proposed safety study, the consequences of a train accident with the rupture of a 50 m3 LPG transport vessel were analysed. Vessel rupture may occur directly by some heavy mechanical impact [1,2] or indirectly as a consequence of a fire. A fire further pressurises the vessel by increasing the LPG’s vapour pressure and at the same time affects the vessel’s structural strength. The consequence of a catastrophic rupture of a pressure vessel of LPG is a boiling liquid expanding vapour explosion (BLEVE), which may produce a substantial blast effect. The blast wave reflects repeatedly because of the limit of tunnel wall when there is an explosion in a tunnel, its close-in effect

n

Corresponding author. Tel. þ39 965 875237. E-mail addresses: [email protected] (M. Buonsanti), [email protected] (G. Leonardi).

1644-9665/$ - see front matter & 2012 Politechnika Wrocławska. Published by Elsevier Urban & Partner Sp. z o.o. All rights reserved. http://dx.doi.org/10.1016/j.acme.2012.09.002

archives of civil and mechanical engineering 13 (2013) 128–134

makes the overpressure of tunnel blast wave increase and continuance time of the blast wave longer [3]. In the past decades, a lot of experiments and simulations were conducted to study the consequences of explosive charges detonation in underground ammunition storage chambers and access tunnels [4,5]. Nevertheless, propagation and decaying of nonreactive blast waves in simple geometry structures, such as tunnels, tubes or plates assemblies, is still object of investigation. Only limited related studies can be found in literature. Chill et al. [6] investigated the dynamic response of underground electric plant subject to internal explosive loading using three-dimensional numerical method. Coupled fluid–solid interaction was considered in their study; however, the nonlinearity and failure of rock and concrete as well as the interaction between different solid media were not simulated. For traffic tunnels, Choi et al. [7] used the three-dimensional finite element method to study the blast pressure and resulted deformation in concrete lining. Lu et al. [8] and Gui and Chien [9], using FE method, looked into the blast-resistance of tunnels in soft soil subject to external explosive loadings. Van den Berg and Weerheijm [10] investigated to what extent an open space may effective mitigating blast effects from explosion in an urban tunnel system. There exist very few numerical studies investigating the dynamic and nonlinear response of underground structures subject to the simultaneous actions of fire and blast loading. To this end the dynamic response and damage of a railway tunnel structure under fire and explosion load is the subject of this paper.

2.

Theoretical background

The blast action can be decomposed in two components, thermal and shock wave’s loads. In this paragraph theoretical assumptions are developed in both cases, and furthermore some structural considerations on the thick shell behaviour are given in premises. From a structural point of view the tunnel can be considered as a half thickness-walled cylinder subject to internal and external pressures. Assuming the zaxis as the revolution axis, the deformation becomes symmetrical in respect to the z-axis. Consequently it is convenient to use cylindrical coordinates r, y, z. So we can consider a half cylinder of inner radius a and outer radius b (Fig. 1) and subject to an internal pressure pa

129

and an external pressure pb, other than the radial temperature field T(r). Due to the axial symmetry of the problem, we assume that the displacement field is radial only, namely ur ¼u(r), uy ¼0, and assuming the plane strain, we have uz ¼ 0. According to [11–13] the stresses satisfy the equilibrium conditions; without body forces, the expression becomes: @sr sr sy þ ¼0 @r r

ð1Þ

while the components of the deformation field e assume the form:   ur 1 @uy @ur 1 @uy uy 1 @ur þ ; er ¼ ; ery ¼  þ ð2Þ ey ¼ r @y 2 @r r @y r @r r where the function u(r,y) represents the displacement field over the shell. Introducing the Lame’s constitutive equations (withn and E, respectively, Poisson’s and Young’s modulus) after some simple calculations we get the basic equations governing the thick-walled half-cylinder. ur ¼

1n pa a2 pb b2 1 þ n a2 b2 pa pb rþ 2 2 E E r b2 a2 b a

b2 a2 pa pb r2 b2 a2 b2 a2 pa a2 pb b2 b2 a2 pa pb sy ¼ þ 2 2 r b a2 b2 a2 sr ¼

pa a2 pb b2



ð3Þ

Under these conditions, we recall the specific conditions of the internal and of the external pressure loads. In the first case (internal pressure) the above equations become: ! p a2 b2 1 2 sr ¼ 2 a r b a2 ! p a2 b2 1þ 2 sy ¼ 2a ð4Þ r b a2 From the equations above, a consideration can be drawn about the circumferential stress (sy tensile stress), which is at its greatest on the inner surface and is always greater than pa. In the second case (external pressure) the general Eq. (3) assume the form.   p a2 1 2 sr ¼  2 b r b a2   pb b2 a2 1þ 2 ð5Þ sy ¼ 2 r b a2 The stress paths, when no inner holes were present, are uniformly distributed in the cylinder. From now we will be

y r r

pb

a b

pa

internal pressure



x Fig. 1 – Tunnel theoretical model (a); tunnel-section with load conditions (b).

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able to describe the coupling actions over the thick-walled halfcylinder shell, and therefore we recall some basic thermoelasticity assumption. There is a significant presence in literature over the question, but we prefer to refer to [13–15]. We focus the consistence of thermal stresses induced in thick-walled half-cylinder when the temperature field is symmetrical to the z-axis. In this case we suppose the temperature T as radius function only and independent from z then ez ¼ 0. With analogous considerations to those above, the basic equations, for the coupled problem, can be written as in Eq. (6) where the a term represents the thermal expansion coefficient. ( ) ð1 þ nÞa R r ð12nÞr2 þ a2 R b ur ¼ a Trdr þ a Trdr ð1nÞr ðb2 a2 Þ sr ¼

aE 1 ð1nÞ r2

(

r2 a2

Z

b

Trdr

Z

r

) Trdr

b2 a2 a a ( ) Z r Z aE 1 r2 a2 b 2 sy ¼ Trdr þ TrdrTr ð1nÞ r2 b2 a2 a a ( ) Z b aE 2 sz ¼ TrdrTr 2 2 ð1nÞ b a a

ð6Þ

If the temperature T is positive and if the external temperature is equal to zero, then the radial stress is always compressive, like the other stresses in the inner surface, while the latter are tensile in external surface. After numerous disasters in the building and structures, the fire-structure question was developed by many researchers, which have reproduced a significant and numerous amount of literature [16–19]. Now we analyse the structural effects after the burst. According to Meyers [20] the interaction between a detonating explosive and the material in contact, or in close proximity, is extremely complex, since it evolves detonation waves, shock waves, expanding gases, and their interrelationships. The topic was developed principally for military use, developing the computational apparatus, for instance the Gurney equation [21]. So, we assume the following basic hypothesis. (1) A shock is a discontinuous surface and has no apparent thickness. (2) The shear modulus is assumed to be zero and so it responds to the wave as a fluid, and the theory can be restricted to higher pressures. (3) Body forces and heat conduction at the shock front are negligible. (4) There is no elastic-plastic behaviour. (5) Material does not undergo phase transformations. Now, to our aim we will consider the dynamic behaviour of thick-wall cylindrical shell under internal pressure produced by shock wave. Let pc be the collapse pressure, then the shell is subject to a symmetrical internal pressure pulse, in the interval time 0rtrt, while p¼ 0 when tZt. Again we assume rigid perfectly plastic material behaviour. Supposing the pressure load symmetric, then the yielding is controlled by force in the shell middle plane. So, let Ny be the generalized membrane forces, at the yielding point we have Ny ¼ Nc (with Nc the fully plastic membrane forces). Neglecting the elastic effects, the dynamics response consist of two phases motion with Ny ¼ Nc. For major clarify we

consider, as the second phase, the time as trtrtn where tn is the response duration time. Let o be the transverse displacement of the shell middle plane and let v1 be the spherically symmetric outwards impulsive velocity, we then find the radial displacement: o ¼ Nc  t2 =m  rv1  t

ð7Þ

After some calculations we have the associated permanent radial displacement field over the shell. of ¼ mr  v1=4Nc

ð8Þ

About the next numerical implementation we will be able to represent the preliminary considerations on the problem, in particular regarding the complex nature of the physics problem. In other words here we have a coupled problem, or better, a coupled-field analysis that is one that takes into account the interaction (coupling) between two or more fields of engineering [22,23]. There are certain pros and cons with coupled-field formulations: one the one hand they allow to find solutions to problems which otherwise would not be found by finite elements, on the other there is an increase in wave-front with an insufficient matrix reformulation. The procedure for a coupled-filed analysis varies depending on which fields are being coupled, but two distinct methods can be identified: the indirect method and the direct method. Here we select the first one which involves two sequential analyses, each belonging to a different field. The two fields are coupled by applying results from the first analysis as load for the second analysis. For example in this thermal-stress analysis the nodal temperature from the thermal analysis are applied as body force loads in the subsequent stress analysis.

3.

Finite element model

The finite element model was based on a single track railway tunnel system consisting of concrete tunnel tube with the section dimensions reported in Fig. 2. The tunnel was about 10 m below the ground surface. The model extended 150 m in the longitudinal direction of the tunnel, while the length and height of the model were of 26.8 m. The finite element model was fixed at the base and roller boundaries were imposed to the four side.

3.1.

Structural model

Solid elements have been used in the finite element model. According to Schrefler et al. [19] they are more accurate in coupled thermal and mechanical analysis. In particular the ten-node tetrahedral elements [24,25] have been implemented by ANSYS Code [26]. This is a higher order of the threedimensional linear tetrahedral element (Fig. 3). When compared to the four-node, the ten-node is fits better for a more accurate modelling problem with curved boundary. For solid problems the displacement field is represented by the indicial form: ui ¼ ui,I ð2S1 1ÞS1 þ ui,J ð2S2 1ÞS2 þ ui,K ð2S3 1ÞS3 þ ui,L ð2S4 1ÞS4 þ4ðui,M S1 S2 þ ui,N S2 S3 þ ui,O S1 S3 þ ui,P S1 S4 þ ui,Q S2 S4 þ ui,R S3 S4 Þ

ð9Þ

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archives of civil and mechanical engineering 13 (2013) 128–134

where the i-index is variable as (i¼ x, y, z) representing the components of the displacement field. Likewise the spatial distribution of temperature over an element is given by:

deducted by the determinant of the matrix form, rotating through the I, J, K, and L subscripts using the right-hand rule.

T ¼ TI ð2S1 1ÞS1 þ TJ ð2S2 1ÞS2 þ TK ð2S3 1ÞS3 þ TL ð2S4 1ÞS4 þ þ4ðTM S1 S2 þ TN S2 S3 þ TO S1 S3 þ T,P S1 S4 þ TQ S2 S4 þ TR S3 S4 Þ

3.2.

ð10Þ Particularly, the shape functions having the form: Si ¼

1 ðai þ bi X þ ci Y þ di ZÞ 6V

ð11Þ

In the last equation V represents the volume tetrahedral element, while the i-index vary from 1 to 4, and i -index is varying as I, J, K, L-node. The coefficients ai, bi, ci, di can be

Modelling materials

The modelled tunnel structure is surrounded by soil, the unit weight of which was 18.9 kN/m3, and this load represents the starting state of stress. Drucker–Prager elasto-plastic model was used to model the soil (Table 1). For the characterization of the reinforced concrete of the tunnel structure it was considered a C50/60 class concrete. The mechanical characteristics of the reinforced concrete (at a temperature of 20 1C) are reported in Table 2. The effects of high temperature on the mechanical and physical characteristics of reinforced concrete have been considered in the model according with the indications of the Eurocode 2 [27], in particular, the values of the elasticity modulus in function of temperature are reported in Table 3.

3.3.

Modelling of loads

A fundamental aspect in the study of fire resistance in underground structures is the definition beforehand of the fire scenario taken in the analysis, therefore choosing the best fit standard curve. A standard curve used when testing the temperature exposure is the cellulose curve defined in several standards, e.g., ISO 834 [28]: yg ¼ y0 þ 345  log10 ð8t þ 1Þ

ð12Þ

where: t is time (min); yg is the temperature (1C) at time t; y0 is the initial temperature (1C). This curve applies to materials found in typical buildings. This has been used for many years, also for tunnels, but it is clear that this curve does not represent all materials, e.g., petrol, chemicals, etc., and therefore a special curve, the hydrocarbon curve (the HC curve) [29], which was developed in the 1970s for use in the petrochemical and off-shore industries, has been applied to tunnels:

5.00

Fig. 2 – Rail tunnel section.

Table 2 – Mechanical characteristics of concrete.

4 R

P Q

2

Y

3

O X

I

Density (kg/m3)

Poisson’s Ratio n

30,000

2300

0.18

K Table 3 – Temperature dependence of concrete modulus of elasticity [37].

N

M

Z

Young’s modulus E (MPa)

Temperature (1C) Young’s modulus E (MPa)

J

20 3000

50 3000

200 1500

400 450

600 150

Fig. 3 – 3-D 10-node tetrahedral structural solid element.

Table 1 – Drucker–Prager model parameters for soil. Young’s modulus E (MPa)

Density (kg/m3)

Poisson’s Ratio n

Yield stress (MPa)

Friction angle F (1)

Specific heat (J/kg1K)

Thermal conductivity (W/m1C)

146.72

1890

0.3

0.056

43

1840

2.00

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archives of civil and mechanical engineering 13 (2013) 128–134

yg ¼ y0 þ 1080  ð120325  e0167t 20675  e25t Þ

ð13Þ

The main difference between these two curves is that the HC curve exhibits a much faster fire development and consequently is associated with a faster temperature increase than the standard ISO 834 fire curve and has traditionally been seen to be more relevant for a petroleum fire. Specific temperature curves have been developed in some countries to simulate hydrocarbon fires in tunnels. Examples of such curves are the RABT/ZTV Tunnel Curve in Germany [30] and the Rijkswaterstaat Tunnel Curve (RWS curve) in The Netherlands (based on laboratory scale tunnel tests performed by TNO in 1979 [31]). Fig. 4 displays the above mentioned temperature–time curves. In the considered model the HC curve was used to simulate the fire action, the numerical-time temperature values of the first two hours are reported in Table 4. At this fire scenario load we have to add the tensional stress caused by the vertical load of 10 m of above-ground soil and the explosion over-pressure. The blast overpressure was generated from an instantaneous release of 50 m3 LPG rail tanker at 326 K. The pressure–time curve was assumed to be of triangular shape, the duration of which was obtained from CONWEB reflected pressure diagram [32]. To calculate the decay of blast overpressure during the longitudinal direction of the tunnel the Energy Concentration Factor (ECF) method was used [33]. The Energy Concentration Factor (ECF) is the ratio between volume of the hemisphere with radius r0 and volume of the confined region included in the distance r0 from the epicentre of blast. With reference to a tunnel in which the charge is placed at the middle of a tunnel: ECF ¼

2 VHSph pr3 1p 2 r ¼ 3 0 ¼ 3A 0 VTun 2Ar0

ð14Þ

where A is the tunnel cross section area. 1350

1400

1300

1260 1300

2

Temperature ϑ −ϑ0 [K]

1200 986

1000

925

1014

4

822

co

761

800 658

co

600 566

400

oli

ng

1 RABT/ZTV Tunnel Curve

ph

1080 1029 oli

ng

ph

as

e

as e

2 Rijkswaterstaat Tunnel Curve

1

3 Hydrocarbon Curve

200

1200

3

1080

1068 1078

4 Standard Temperature Curve (STC)

0 35 10

20

30

40

50

60

70

80

90

100 110 120 130

Time [min]

Fig. 4 – Tunnel curves according to fire course models for tunnels. Table 4 – HC curve values. Time (min) Temperature y(1C)

0 0

1 741

5 928

10 1014

20 1068

30 1078

In this perspective the Energy Concentration Factor (ECF) is a purely geometrical factor that takes into account the increase in spatial density of energy caused by the reduction of the volume available for gas expansion. When a BLEVE occurs in open environment, the peak sideon overpressure of the blast wave is normally evaluated by means of scaled overpressure curves [34,35] that provide the scaled overpressure, PS/P0, as function of the Sachs scaled distance. The Sachs scaled distance is then given by the following expression:  1=3 P0 R ¼ r0 ð15Þ 2W where r0 (m) is the distance from the blast source, W (J) is the expansion work devoted to blast-wave generation and P0 (Pa) is the atmospheric pressure. In case of a BLEVE event in a tunnel the ECF method allows to predict the blast-wave overpressure at different locations from the blast source. The use of the Energy Concentration Factor leads to a modified expression of the Sachs scaled distance on the form:  1=3 P0 ð16Þ R0 ¼ r0 2ECF  W In the considered case of railway tunnel, the numerical values of the overpressure at the different distances are reported in Table 5: During the propagation of the blast wave over the first 75 m from the BLAVE to the tunnel opening, the blast overpressure falls from 1700 kPa (vapour pressure at 326 K) down to approximately 86 kPa. This decay is solely through the intense energy dissipation in the strong leading shock of the blast wave.

3.4.

Numerical simulation and analysis

The 3-Dimensional model is representative of a tunnel section 150 m long, while the length and height of the model are 25 m. This model was implemented by quadratic tetrahedral type elements with the ANSYS FEM package [36], obtaining 95,003 elements and 147,528 nodes as shown in Fig. 5. The mesh on the 3-D computational domain was constructed after deciding for a greater refinement (0.3 m) of the tunnel walls elements as illustrated in the same Fig. 5. The finite element model was fixed at the base and elastic supports were imposed to the other sides to simulate the soil reaction. The analysis was carried out in two steps. The first step obtained the initial stress state caused by soil load and fire and the second step analysed the dynamic response under blast loading. Table 5 – Calculation of the overpressure in a tunnel of 28 m2 section area for the BLAVE of a 50 m3 LPG rail tanker. Distance (m) ECF R’ P0/Ps Ps (Pa)

50 87 0.52 1.1 111,457

100 349 0.66 0.8 81,060

150 785 0.76 0.65 65,861

archives of civil and mechanical engineering 13 (2013) 128–134

133

Fig. 5 – Meshed model.

Fig. 6 – Temperature distribution (1C) and thermal strain (m) at t ¼ 1800 s.

Fig. 7 – Deformation (m) of the tunnel at the explosion instant.

Consequently the following load conditions were considered in the FE analysis:

(1). from time t¼ 0 to time t¼ 120 min the tunnel was subjected to the surrounding soil load and to the fire thermal stress; (2). at the instant t¼ 2 s the structure was subjected to the blast over pressure.

4. Results of numerical simulation and discussion First, a thermal analysis was carried out to investigate the effects of the fire on the structure. Therefore, on the base of this analysis, the distribution of the temperature inside of the structure is known. In Fig. 6 the temperature distribution and the thermal strain after 1800 s are showed. The thermal analysis shows a distribution of temperatures which have a linear variation in time and assumes uniform values along the gallery and reaches the maximum value after 30 min (Fig. 6). Subsequently, the mechanical behaviour of the models was analysed introducing also to thermal stress, the explosion load (Figs. 7 and 8).

Fig. 8 – Mises stress (Pa) of the tunnel at the explosion instant. Analyzing the deformed model (Fig. 7), it can be seen that the gallery shows an increase of concavity in the middle of gallery, where the instantaneous explosive impact can be considered main source of the deformations. Fig. 8 shows the Mises stress of tunnel section, from meddle, where explosion is localized, to the tunnel opening.

5.

Conclusions

The proposed study was motivated by the fact that explosion in an underground structure may not only cause direct life loss, but also damage the structure and lead to further loss of lives and properties.

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Preventive measures should therefore be undertaken to protect existing subway and tunnel structures, at least the most important or most vulnerable sections, from collapse under internal blast loading; such internal blast loading should also be properly taken into consideration in the design of new structures. The importance of this issue was pointed out in numerous documents and reports [6,7]. The proposed methodology can be used to evaluate the structure integrity of existing subways and tunnels and to design new underground structures taking into account the internal blast loading still need to be developed.

references

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