Damage evaluation of the steel tubular column subjected to explosion and post-explosion fire condition

Engineering Structures 55 (2013) 44–55

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Damage evaluation of the steel tubular column subjected to explosion and post-explosion fire condition Yang Ding a,⇑, Ming Wang a, Zhong-Xian Li a, Hong Hao b a b

School of Civil Engineering, Tianjin University, Tianjin 300072, China School of Civil and Resource Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

a r t i c l e

i n f o

Article history: Available online 23 February 2012 Keywords: Medium-scaled blast Post-explosion fire Integrative damage Mechanical damage Geometrical damage Pressure–Impulse diagram

a b s t r a c t Fires following explosion are considered one of the most serious secondary disasters in the blast events, which are frequently occurring in the terrorist attacks and other blast accidents. In the recent decades, a significant amount of research work has been carried out to study the structural response under blast loads, and it is found that due to the superior mechanical properties of steel material, steel structures are adequate to withstand a medium-scaled blast loading. However, since the strength of steel is sensitive to temperature, the post-explosion fire action should be involved in the evaluation of structural damage in the blast events. This paper is devoted to introduce a numerical method for predicting the integrative damage of steel tubular column subjected to blast load and the following fire action. The damage caused by blast load would significantly reduce the fire resistance of steel column, which includes two aspects: mechanical damage and geometrical damage. In order to describe the mechanical damage under the blast loading, a damage scalar is defined in the constitutive model, and it is used to represent the reduction of material strength. The geometrical deformation induced by blast load is treated as the initial condition in the fire analysis. Pressure–Impulse diagram is employed to describe the damage of the steel tubular column under blast loading. In the second step, fire analysis for the explosion-survived column is carried out. Obviously, the residual vertical capacity of steel column is related with the fire exposure time. In order to clarify the interaction between the explosion and the post-explosion fire action, a more inclusive function, characterized by three variables: pressure, impulse and fire exposure time, is presented in this paper, and it can be used to predict the residual capacity of the steel column subjected to blast load and exposed to fire for a specific time. In the last section, parametric studies are conducted to observe the effects of geometric size on the failure evolution of steel columns. The main objective of this research work is to provide guidance for assessing damage level of the steel tubular columns that have survived blast loading and expose to the following fire condition. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Explosion incidents have become a serious worldwide problem. Civil engineers are responsible to strengthen the survivability of the structures which are potentially subjected to blast load and the post-explosion actions, such as fire. In the recent decades, a significant amount of research work has been carried out to study the structural response under blast load. Nurick et al. [1–3] conducted a large number of explosion tests to investigate the deformation and damage patterns of steel plates under different scaled blast loads. Morrill et al. [4] published their blast experiments, in which the damages of I-shaped steel columns and the beam–column connections subjected to blast loads are observed. Since the Second

⇑ Corresponding author. Tel.: +86 22 27405498. E-mail address: [email protected] (Y. Ding). 0141-0296/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2012.01.013

World War, lots of explosive experiments have been carried out by military departments, but quite limited results were published in the open literature. Nonetheless, some technical manuals (e.g. TM5-1300 [5–7]) have been established for security designing of civil structures. In the recent years, with the ever-accelerated updating of computer technology, several commercial numerical codes, e.g. AUTODYN, LS-DYNA, ABAQUS have been used extensively in the analysis of structures subjected to impulsive loads. Krauthammer and Ku [8] used finite difference approach to study the role of reinforcing bars in structural concrete knee-joints under open blast loads, and employed DYNA3D to obtain the stress and strains at the boundary. This approach was extended to the study of structural steel connection by Krauthammer [9]. Sabuwala et al. used ABAQUS to study the behavior of fully restrained steel connections subjected to blast loads, and some recommendations for modifications to TM5-1300 were presented in [10]. Jama et al. [11] conducted a numerical simulation of square hollow steel

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Y. Ding et al. / Engineering Structures 55 (2013) 44–55

beams under a transverse blast load, and it was concluded that the consideration of steel strain-rate hardening significantly affects the accuracy of numerical analysis. Lee et al. [12] used LS-DYNA to simulate the complex interaction between blast wave and a steel wide-flange section column. All these researches above vary in their methods or applicability, but only limited to the explosive response. The post-explosion actions are not taken into account. The lessons learned from the events of September 11, 2001 reflected that the catastrophic damage of steel structure is most likely to occur during the fire process after explosion. Due to the temperature-sensitive property of steel material, a large number of researches on fire resistance of steel structures have been carried out [13–23]. However, fairly limited work published in the open literature is about the combined effects of blast and fire, which could lead to the progressive collapse of structure as in the case of the terrorist attack on the World Trade Center Buildings. Izzuddin et al. [24,25] proposed an integrated analysis method using beam element for explosion and fire analysis of steel structures. The employment of beam element can improve the calculation efficiency significantly, but this method is deficient in simulating the local buckling and shear damage induced by blast load. Liew and Chen [26,27] adopted fiber element to model the frame element that were not subjected to blast load directly, and model the frame element impacted by blast load directly with more complex element. Additionally, LS-DYNA was adopted by Liew [28] to simulate the survivability of steel column subjected to blast and fire, and the interaction between blast load and fire action for the damage of steel column was discussed. By these methods, the geometric deformation and distortion caused by blast load can be involved in the fire resistant analysis. However, the mechanical damage of steel constitutive under blast load was neglected in most of the existing research work, and it would be demonstrated in this paper. In order to obtain a comprehensive damage assessment of the steel structure subjected to blast load and the post-explosion fire action, the physical state of realistic structure after explosion should be clarified, and it would be the initial condition for the subsequent fire action. Actually, because of too many uncertainties of both structural properties and blast loads, it is difficult to get sufficiently detailed information of the structure damage after explosion. However, the most common explosion-induced damage can be categorized into three forms [29], which are stated as follows: 1. Geometrical damage. The irreversible deformation results from plastic strain that commonly exists in the explosion-survived steel structures. The damage pattern is identified by loading conditions. In general, the most common damage patterns of steel members are bending deformation, shear deformation and local buckling. 2. Mechanical damage. Blast load induces irreparable distortion of material micro-mechanism, and due to the energy dissipation caused by plastic deformation during blast loading, the degradation of material properties is not negligible for the following fire analysis. 3. Damage of fire-resistant coating. It has been found that most of the existing fire-resistant materials are mechanically weak, and would crack or fall down when subjected to impact, blast, or large-scaled deformation [30]. This paper is devoted to introduce a numerical approach for evaluating the integrative damage of unprotected steel tubular columns subjected to both blast load and the following post-explosion fire action. The constitutive model of steel material is described by Johnson–Cook strength criterion, in which the strain rate effect and stress hardening caused by blast load are considered. Moreover, a

damage scalar is employed to record the mechanical damage induced during blast loading, and it is set to be an initial material property in the fire analysis. The reliability of the material model is validated through numerical simulations of several published test cases. Both blast analysis and fire analysis are conducted in the commercial code LS-DYNA, and the material models including strength criterion, failure model, damage criterion, and softening property during fire action are compiled in the subroutines. All three kinds of blast damage presented above have been addressed in this method. With the application of this numerical method, the damage of unprotected steel tubular column subjected to blast load is demonstrated by Pressure–Impulse diagram. Similarly, in the integrative analysis of blast and fire action, a more inclusive function, characterized by three variables: peak pressure, impulse and fire exposure time, is given to identify the interaction between blast and fire action. Additionally, a parametric study of geometrical size of steel tubular column is carried out, and more sufficiently detailed information is presented to provide a general guidance for the engineering application of this method. 2. The steel material model 2.1. Johnson–Cook strength model Johnson–Cook (J–C) strength model is a phenomenological model, which is based on large numbers of experiments [31]. It has been vastly proved that Johnson–Cook model is accurate and efficient to describe the mechanical properties of metal materials that experience high-rated deformation or melting process [32]. Johnson–Cook strength model can be expressed as:

ry ¼ ½A þ Bðepeff ÞN ð1 þ C ln e_ Þ½1  ðT H ÞM 

ð1Þ

The equation of J–C strength model is composed of threes parts. The first bracket in Eq. (1) denotes the strain hardening model, in which the yield stress ry is determined by effective plastic strain epeff . A is the elastic yield stress, and B and N are two parameters controlling the development of stress hardening. The second bracket concerns the strain-rate hardening, and the last set in Eq. (1) represents the stress softening caused by high temperature, which is significantly important in the simulation of those cases as metal forming, penetration, etc. TH is a homologous temperature, and M is a material parameter. Jama et al. [11] conducted a detailed research for studying the thermal softening of steel beams subjected to blast load, it is found that the adiabatic temperature rise due to blast loading is less than or around 200 °C. According to the temperature-dependent property of steel material, the strength of steel under 220 °C almost keeps constant. So in this study, the thermal softening caused by blast load is not considered. 2.2. Johnson–Cook failure criterion Based on the strength model, a failure criterion was additionally defined by Johnson and Cook [33]. In numerical simulations, the failure criterion provides a critical mechanical state in which the corresponding element or basic particle could be mathematically eroded with a comparatively small physical error. The Johnson– Cook failure model is expressed as





eF ¼ D1 þ D2 exp D3 DF ¼

X Depeff

eF

P

reff

 ð1 þ D4 ln e_ Þð1 þ D5 T H Þ

ð2Þ

ð3Þ

where D1, D2, D3, D4, D5 are another five material constants, reff is the effective stress, Depeff is the increase of effective plastic strain

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Y. Ding et al. / Engineering Structures 55 (2013) 44–55

and original yield stress ry is calculated by Johnson–Cook strength model as Eq. (1). The cumulated damage scalar D is determined by Eqs. (4) and (5).

and P is the hydrostatic pressure, DF is the cumulative failure parameter, and the failure occurs when DF = 1.0. It should be noted that the cumulative failure parameter DF only determines the critical state of the catastrophic failure of metal material, but not affects the reduction of material strength or modulus.

2.4. Verification of material model

2.3. Damage evolution of steel material under blast load

Yuen et al. [3] carried out several field tests for studying the deformation of mild steel plates subjected to large-scaled explosions. In this study, in order to verify the accuracy of Johnson–Cook strength model and the damage model for simulating steel structure under blast load, several explosion cases are numerically simulated, and the results are compared. The plates tested in the experiments had a quadrangle size of 700 mm  700 mm, and the boundary was fully fixed. A tetragonal area with a dimension of 500 mm  500 mm was exposed to blast pressure, as illustrated in Fig. 1 [3]. In Yuen et al.’s experiments, the peak pressure and duration time of the blast load were measured, so the loading data is directly employed in these numerical simulations. The blast pressure is uniformly applied on the exposed area of the steel plates, and it is assumed to linearly diminish during the positive loading phase. Under the Momentum Energy Transfer Study (METS) project, the Naval Explosion Ordnance Disposal Technology Division (NAVEODTECHDIV) acquired the material characterization data for ASTM A36, and the parameters of Johnson–Cook strength model are presented in the reference paper [38,39]. As far as authors’ survey, there’s no available document for helping to determine the parameters of Johnson–Cook failure model of the mild carbon steel in the open literatures. However, the values of D1–D5 for Weldox 460 E have been verified and provided in some published papers [40], In this paper, the parameters D1–D5 for Weldox 460 E are adopted. Bonora presented the damage parameters for several different materials in the reference paper [41], and the mild carbon steel is included. According to the documents presented above, the values of the material parameters are summarized in Table 1. The numerical simulations of the steel plates subjected to blast load are carried out in the nonlinear explicit solver, LS-DYNA, and the algorithms of material models including Johnson–Cook strength model, failure model and damage scalar are compiled as subroutines. The dynamic responses of steel plate under blast

It is well known that material damage substantially stems from some irreversible phenomena in material microstructure such as micro-cracking and micro-void formation. Much research work on the steel damage evolution under impact load has been published in the open literature [34–38], and most of those damage criterions are established on the development of the micro-void existing in raw material. In this study, a plasticity damage model based on continuum damage mechanics (CMD) proposed by Bonora [41] is employed to denote damage occurring in mild carbon steel. The damage model can be generally expressed as

DD ¼ a  f

ðDcr  D0 Þ1=a f lnðecr Þ  lnðeth Þ





p

Deeff rH ðD  DÞða1Þ=a p req cr eeff





rH 2 rH ¼ ð1 þ mÞ þ 3ð1  2mÞ 3 req req

ð4Þ

2 ð5Þ

in which, DD is the increase of damage, and D is the cumulated damage, D0 is the initial amount of damage, and Dcr and ecr are critical damage and the corresponding strain respectively, eth is the threshold strain that is the beginning strain of plasticity damage. In Eq. (5), m is the Poisson’s ratio, rH is the hydrostatic stress, and req is the equivalent von Mises stress. The presence of damage affects both the yield strength and the modulus. In this study, the reduction related with the damage scalar is expressed as [36]:

  15ð1  mÞ GD ¼ G0 1  D 7  5m

ð6Þ

rDy ¼ ry ð1  4DÞ

ð7Þ

where GD is the shear modulus of damaged material, and G0 is the initial shear modulus. rDy is the yield strength of damaged material,

Fig. 1. The modeling of steel plate subjected to blast load [3].

Table 1 Determination of material parameters. G0 (GPa)

m

A (ksi)

B (ksi)

N

C

TM (°C)

TR (°C)

D1

78.85

0.3

41.50

72.54

0.228

0.0171

1500

20

0.0705

D2

D3

D4

D5

eth

ecr

Dcr

D0

a

1.732

0.54

0.015

0.0

0.202

1.0

0.1

0.0

0.198

47

Y. Ding et al. / Engineering Structures 55 (2013) 44–55 Table 2 Comparison between the test measured results and the numerically predicted results. Blast no.

Thickness (mm)

Scaled distance (m/ kg1/3)

Pressure peak (kPa)

Duration (ms)

Impulse (N s)

Test Measured dt (mm)

Numerically investigated results dc (mm)

Error (%)

dD c (mm)

Error (%)

1 2 3 4 5

3 3 3 6 6

0.83 0.72 0.79 0.84 1.05

1495.98 1993.45 1634.19 1443.44 908.25

3.53 5.95 8.99 3.71 8.71

660.44 1483.78 1836.86 668.75 989.11

40.70 60.60 50.90 14.70 18.10

36.81 49.82 43.97 12.50 15.27

9.56 17.79 13.61 14.97 15.64

38.45 54.04 47.18 12.95 16.52

5.53 10.83 7.31 11.90 8.73

Note: dc anddD c are the calculated displacements in the mid-point of the steel plates modeled by Johnson–Cock strength criterion without damage scalar and with damage scalar respectively.

pressures with different explosion charge mass, different stand-off distance or different plate thickness are concluded in Table 2, and they are compared with the test results given previously [3]. It is observed that the irreversible deformations of the steel plates in numerical simulation are slightly smaller than those measured in the tests. In the authors’ opinion, the error of the numerical simulation probably stems from the values of material parameters and the boundary conditions. Compared with dc, dDc is obviously much closer to the test-measured result dt. This reflects that the consideration of strength reduction and stiffness degradation would efficiently improve the accuracy of numerical prediction. Nevertheless, the results of this case study have verified the accuracy of the proposed steel material model with consideration of damage caused by blast load. 3. Development of steel temperature in fire Majority of steel members are required to be protected by fireretardant coating, which can efficiently slow down heat transfer from fire to steel components. However, it has been observed that most of the existing fire protection materials are mechanically fragile, and they are easily damaged by dynamic actions, such as impacts, earthquakes and blast loads [18,30]. The mechanics of a steel member directly exposed to fire is quite sensitive to temperature rising, and the survivability in fire condition is rather lower than protected members. The main objective of the fire analysis in this paper is to study the response of an unprotected steel column totally engulfed in the fire after explosion. One of the most intractable issues in fire analysis is the calculation of the thermal transfer between steel members and fire environment. The temperature of the air around steel member is correlated with many uncertainties, such as the property of fire source, the material of the furniture in site, the distance between fire source and target member etc. However, the ISO834 standard temperature curve is widely adopted to describe the time-dependent temperature of fire environment. A theoretic approach for calculating the thermal transfer between the target steel member and the ambient fire air was proposed by Eurocode [42,43].

The increase of temperature in time interval can be calculated by the following function:

h_ net:c ¼ ac ðhg  hm Þ

ð10Þ

in which ac is the coefficient of heat transfer by convection, hg is the gas temperature in the vicinity of exposed member, hm is the temperature of steel member. The net radiation heat flux hnet,r is determined by:

hnet;r ¼ Uem ef rst ½ðhr þ 273Þ4  ðh þ 273Þ4 

ð11Þ

where U is the configuration factor, em is the surface emissivity of steel member, ef is the emissivity of fire, rst is the Stephan Boltzman constant, and hr is the effective radiation temperature of the fire environment. In the study, the steel member is assumed to be fully engulfed by fire, so the radiation temperature hr can be represented by the gas temperature hg, which could be assumed to follow the ISO834 temperature curve, so

hr ¼ hg ¼ 20 þ 345log10 ð8t þ 1Þ

ð12Þ

where t is the fire exposure time. 3.2. Calculation of steel temperature in fire As presented in the above section, some material constants and thermodynamic constants are involved in the algorithm for determining the steel temperature. According to the guidance of Eurocode, the values of these constants are determined and listed in Table 3. The specific heat of steel ca is related with steel temperature hm, and the relation can be described as the following piecewise equations: For 20 °C 6 hm < 600 °C,

ca ¼ 425 þ 0:0773hm  1:69  103 h2m þ 2:22  106 h3m

ð13Þ

For 600 °C 6 hm < 735 °C,

13002 738  hm

ð14Þ

For 735 °C 6 hm < 900 °C,

ca ¼ 545 þ ð8Þ

where Ksh is the correcting factor for the shallow effect, Am and V are the surface area and volume of the member per unit length, which are assumed to keep constant in the analysis, ca is the specific heat of steel, qa is the density of steel, Dt is the time interval, h_ net is the net heat flux, which is composite of convective heat flux hnet,c and radiation heat flux hnet,r.

ð9Þ

The net convective heat flux component should be determined by:

ca ¼ 666 þ

3.1. Methodology of temperature calculation

Am =V _ hnet Dt Dh ¼ K sh ca qa

h_ net ¼ h_ net;c þ h_ net;r

17820 hm  731

ð15Þ

Table 3 Constants for temperature calculation. Ksh

qa (kg/m3)

ac (W/m2K)

U

em

ef

rst (W/m2 K4)

0.9

7850

25

1.0

0.8

1.0

5.67  108

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Y. Ding et al. / Engineering Structures 55 (2013) 44–55

For 900 °C 6 hm < 1200 °C,

ca ¼ 650

ð16Þ

The temperature elevation of a steel column exposed in a fire testing furnace was observed by Wang and Li [18]. The cross section of the column is H140  100  6  6. According to the method presented previously, the temperature of the steel column is calculated, and the development follows the dash line in Fig. 2. The ISO834 curve in Fig. 2 represents the atmosphere temperature around the steel column. It can be found that the temperature of steel column is very close to the surrounding atmosphere when it is exposed to fire for more than one hour. The calculated temperature curve approximately agrees well with the test result, which is shown as the dot line in Fig. 2. Both the yield strength and the material modulus are sensitive to temperature rising. The reduction factor for the stress–strain relationship of carbon steel at elevated temperature can be graphically described as the polylines in Fig. 3. 4. Damage of steel column subjected to blast load 4.1. Modeling of the steel column A steel column with a rectangular section is numerically modeled, and the detailed configuration of the column is illustrated

in Fig. 4a. The initial bending deformation is considered in the model, which is characterized as an initial 0.1% deformation in the mid-span before loading. The steel column is numerically modeled by Belytschko–Wong–Chiang thin shell elements. In order to avoid the inauthentic distortion on the boundary, rigid blocks showed in Fig. 4b are connected to each end of the steel column, and both vertical load and boundary constraints are applied on the surfaces of the rigid blocks. A calculation for studying the vertical capacity of the steel column is carried out by commercial analysis tool LS-DYNA. Actually, the loading process is treated as a quasi-static problem, so in order to improve the accuracy of the numerical simulation, the load applied on the top surface of rigid foundation increases slowly as 1000 kN/s. The whole loading process lasts 10 s, and the vertical top displacements in intervals are recorded. The final history curve is derived as Fig. 5. Fig. 5 indicates that when the vertical load increases to 8000 kN, a catastrophically irreversible buckling occurs in the column, and the vertical resistance of the steel column dissipates thoroughly. In this study, the vertical capacity of the column is approximately determined as 7800 kN. 4.2. Blast load and damage index Blast loads applied on structures are usually identified by two parameters: peak pressure and positive impulse. Besides, the negative impulse should also be considered if the suction action during blast negative phase is concerned. In this study, the simplified blast pressure model as Fig. 6b is loaded on one lateral surface of the steel column, and 20% of vertical capacity is considered as the normal vertical load, which is applied on the top surface of the rigid foundation in numerical model. After subjected to blast load or exposed in fire condition, the vertical resistance of survived steel column would degrade, and the general damage level can be represented by a damage index, which is defined as:

DCOL ¼ 1:0 

Fig. 2. Temperature of steel column.

Rresi Rini

ð17Þ

where Rresi is the residual resistance of the survived column, and Rini is the initial resistance, which equals to 7800 kN for the steel column described above. According to the value of damage index DCOL, the damage degree is classified into four groups. The detailed categorization is presented in Table 4. 4.3. Damage evaluation of steel column under blast load

Fig. 3. Reduction factor of yield strength and modulus.

Two damage patterns as given in Fig. 7 have been observed during the numerical simulations of steel tubular column under blast loads. It is well known that the threat caused by explosion is defined by two equally important elements, i.e. peak pressure and positive impulse. With different values of these two elements, the damage modes are quite different. Shi et al. observed shear damage model and flexural damage model during the numerical simulation of reinforced concrete column subjected to blast load [44]. Due to the significant difference of material property, the damage patterns of steel column under blast blasts are different with RC column to some extent. Under quasi-static loads, flexural damage pattern is also observed in the steel tubular column, as Fig. 7a, and the irreversible buckling firstly occurs in the middle span located on the front face. A remarkable bending deformation has taken place before the catastrophic collapse. Subjected to impulsive load, a serious buckling on the front face and partial lateral faces occurs soon after the peak pressure is applied. In Fig. 7b,

Y. Ding et al. / Engineering Structures 55 (2013) 44–55

only a small integral displacement is observed when an irreparable damage takes place. Pressure–Impulse (P–I) diagram is commonly used in the preliminary design or assessment of protective structures to establish safe response limits for given blast-loading scenarios. Since the Second World War, P–I diagrams have been widely accepted to predict the structural damage induced by impact-driven loadings such as blast loads [44–46]. In this study, dynamic analyses of the steel column under numbers of blast scenarios differed in pressure and impulse are simulated, and according to the damage definition presented in the above section, the damage index of the steel column in each blast scenarios is calculated. Accordingly, the P–I coordinate space can be divided into four subspaces by three critical P–I curves as Fig. 8. For three critical damage levels, DCOL = 0.2, 0.5 and 0.8, the representative curves can be fitted by a general function, which can be expressed as

ðP  P 0 ÞðI  I0 Þ ¼ lðP 0 =2 þ I0 =2Þb

ð18Þ

49

Fig. 5. Ultimate vertical capacity of steel column.

5. Integrated study of steel column under blast and fire action 5.1. Numerical implementation

where P0, I0, l and b are four parameters controlling the shape of P–I curves. The values of these parameters are determined by fitting analysis, and they are listed in Table 5. With the employment of the Pressure–Impulse diagram illustrated in Fig. 8, the residual capacity of survived column can be determined. However, if the column continues to be subjected to some post-explosion actions, such as fire, the mechanical damage caused by blast load should be determined. In this study, a damage scalar is involved in the material model, which has been introduced in the above section, so the mechanical damage can be recorded, including the damage level and locations. Fig. 9 presents the damage fringe of the steel column subjected to a blast load identified as P = 3300 kPa, I = 2.0E5 kPa ms. The numerical results reflect that most of mechanical damages are located in the middle and the ends. The progressive collapse in fire analysis usually begins with the local irreversible buckling. The mechanically damaged location would become the most vulnerable part in the whole steel column. The consideration of mechanical damage would improve the reliability of the following fire analysis.

Due to the small loading surface and the ductile mechanics of steel material, most steel columns can survive in a medium-scaled explosion. However, fire is one of the most common secondary actions after explosion. The mechanics of steel material is sensitive to temperature. The resistance of unprotected steel member decreases rapidly with the increase of temperature. Therefore, a reliable prediction of the sustaining time for the structural members exposed in fire is very important for evacuation and rescue of survivors. In this study, an integrated analysis of explosion and fire action for steel column is carried out. The whole analysis is divided into several sub-steps, which are introduced in the flow chart in Fig. 10. The initial temperature of the steel column is assumed to be 20 °C. The dynamic response of steel column under blast load only lasts for an extremely short time, but the fire action usually covers a comparatively long time. So it is impossible to conduct the both two analyses based on the same time unit. In this paper, the temperature of the steel column is assumed to be uniformly

Fig. 4. Details of the steel tubular column.

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Y. Ding et al. / Engineering Structures 55 (2013) 44–55

Fig. 6. Simplification of blast load.

Table 4 Categorization of damage level. DCOL

0.0–0.2

0.2–0.5

0.5–0.8

0.8–1.0

Damage level

Low damage

Medium damage

High damage

Collapse

distributed, and no temperature gradation existing along the column, so there is no thermal stress caused by temperature rising, and the thermal analysis of steel column exposed in fire is not time-depended. In this study, the fire action is assumed to be completed in a scaled-shorten time. By this approach, both the mechanical damage and geometric damage caused by blast load is taken into account for the following fire analysis. According the results of the tentative calculations, it is found that the numerical result is not sensitive to the rising rate of temperature. In the most cases of this study, the column temperature increases 20 °C per second during the action of fire.

and 12 respectively show the distributions of different damage levels when the steel column is exposed to fire for 15 min and 19 min after blast load. In Fig. 12, there are only two critical damage curves, i.e.DCOL = 0.5 and DCOL = 0.8, which means even if the steel column has not been exposed to blast load, more than 20% of vertical capacity would be lost due the fire action for 19 min. For the steel column subjected to blast load and fire action, the damage levels are controlled by three parameters: peak pressure P, impulse I and fire exposure time t. A series of numerical cases with different values of these three parameters are simulated, and it is found that for each specific fire exposure time, the P–I diagrams can all be expressed as the Eq. (18), in which l and b keep constant as the value given in Table 5, and P0 and I0 are correlated with fire exposure time t, which have been illustrated in Figs. 13–15. According to the numerical data, the relations between P0, I0 and t can be fitted by the Boltzmann Equations expressed as:

P0 ¼ 5.2. A general damage evaluation The explosion-survived column continues to lose vertical capacity with rising of temperature in the following fire condition, and the damage index DCOL can still be used to represent the residual resistance in the fire analysis. The P–I diagram can be considered to be variable with the increase of fire-exposure time. Figs. 11

I0 ¼

A1  A2 þ A2 1 þ eðtx0 Þ=dx A01  A02 0

0

1 þ eðtx0 Þ=dx

þ A02

ð19Þ

ð20Þ

In Eqs. (19) and (20), A1, A2, x0, dx, A01 , A02 , x00 , dx0 are the parameters controlling the curve shapes of Boltzmann Equation, and the values of these parameters are respectively given in Figs. 13–15.

Fig. 7. Two damage patterns of steel column.

Y. Ding et al. / Engineering Structures 55 (2013) 44–55

51

In Fig. 16, three critical surfaces divide the whole space into four subspaces, and each one represents a damage level. The integrative damage of the steel tubular column subjected to blast load and exposed in the post-explosion fire action can be evaluated by the applications of Eqs. (21) and (22) and Fig. 16, and the values of parameters have been given in Figs. 13–15. 5.3. Parametric study of geometric size Further studies are carried out to investigate the effects of different geometric parameters on the damage of steel tubular column subjected to blast load and fire. The geometric parameters involved in this study are column height H, lateral width B0, and thickness ts. The scopes of each parameter in parametric study are listed in Table 6. Based on a significant amount of numerical simulations, the parameters in Eq. (21) can be respectively fitted as functions of these interested geometric parameters. The expressions are listed as the following equations. For DCOL = 0.2

Fig. 8. Pressure–Impulse diagram for steel column.

Table 5 Determination of the parameters for P–I curves.

A1 ¼ 3000ð2:88  0:52HÞð1:05 þ 11:62B0  15:93B20 Þð0:22 þ 49:00ts Þ

DCOL

P0 (kPa)

I0 (kPa ms)

l

b

0.2 0.5 0.8

3070 3370 3380

14800 19700 23500

0.8749 0.8742 0.8610

1.8590 1.8620 1.9120

ð23Þ

A2 ¼ 100; 000ð16:70 þ 9:52H  1:28H2 Þð6:40 þ 48:37B0  79:04B20 Þ  ð1:78  50:37t s Þ

ð24Þ

x0 ¼ ð45:83 þ 33:25H  4:31H2 Þð0:16 þ 5:55B0  9:19B20 Þ  ð1:20  12:73ts Þ

ð25Þ

dx ¼ ð6:24  3:14H þ 0:43H2 Þð7:63  47:36B0 þ 84:68B20 Þ  ð0:42 þ 39:48t s Þ

ð26Þ

A01 ¼ 10000ð2:36  0:25HÞð0:81 þ 12:37B0  21:13B20 Þ  ð0:86 þ 116:02t s Þ

ð27Þ

A02 ¼ 100000ð3:64  0:67HÞð2:29 þ 21:73B0  35:83B20 Þ  ð1:96 þ 188:16t s Þ

ð28Þ

X 00 ¼ ð19:26 þ 19:42H  2:66H2 Þð0:26 þ 4:96B0  8:32B20 Þ  ð1:05  3:01t s Þ 0

dx ¼ ð12:90  6:76H þ 0:92H2 Þð8:94  52:53B0 þ 86:92B20 Þ

Fig. 9. Mechanical damage caused by blast load.

 ð0:44 þ 35:55t s Þ Substituting Eqs. (19) and (20) into Eq. (18) gives:

"

A1  A2

!#

2

0

0

0

ð30Þ

For DCOL = 0.5

13

A A  4I  @ 1 tx20 þ A02 A5 FðP; I; tÞ ¼ P  tx0 þ A2 0 1 þ e dx 1 þ e dx0 2 0 13b 1 A1  A2 A01  A02  l4 @ þ þ A2 þ A02 A5 tx0 2 1 þ etxdx0 0 1 þ e dx0

A1 ¼ 3000ð2:793  0:47HÞð0:13 þ 2:91B0 Þð0:23 þ 77:72t s Þ

ð31Þ

A2 ¼ 100000ð0:47 þ 0:23HÞð18:3  99:92B0 þ 140:79B20 Þ  ð0:69 þ 19:33ts Þ ð21Þ

ð22Þ

F(P, I, t) is an artificial function with three parameters: peak pressure P, impulse I and fire exposure time t, and F(P, I, t) = 0 represents a three dimensional surface in the P–I–t coordinate space. In Fig. 16, surface A, B and C represent three critical damage diagrams DCOL = 0.2, 0.5 and 0.8 respectively.

ð32Þ

x0 ¼ ð10:41 þ 5:20H  0:72H2 Þð1:01 þ 0:63B0  2:05B20 Þ  ð1:08  5:02ts Þ

FðP; I; tÞ ¼ 0

ð29Þ

ð33Þ

dx ¼ ð5:12  2:50H þ 0:33H2 Þð19:78  121:10B0 þ 195:00B20 Þ  ð0:059 þ 69:31t s Þ

ð34Þ

A01 ¼ 10000ð2:80  0:24HÞð0:0017 þ 6:81B0  11:61B20 Þ  ð0:71 þ 108:35t s Þ

ð35Þ

52

Y. Ding et al. / Engineering Structures 55 (2013) 44–55

Finite element modeling of steel tubular column

Program the subroutines for material model

The numerical model and subroutines are linked in LS-DYNA, and the boundary condition is defined in the model.

A vertical pressure applied on the top surface increases slowly from zero to 20% of the capacity, and the loading condition is kept for some time to ensure that the stress wave caused by vertical loading has propagated completely. If the column collapses,

Blast load is applied on one face of the steel column, and the induced structural response trails off after a certain time.

DCOL > 0.8

The temperature of the explosion-survived steel column begins to increase for a specified time, and the mechanical property degrades accordingly.

If the column collapses,

The thermal property of steel material is defined in subroutine

If the column survives the fire action for the specified time, calculate the residual vertical resistance.

DCOL > 0.8

According

to

the

residual

vertical

resistance, the damage index DCOL is calculated. Fig. 10. Flow chart of numerical analysis.

A02 ¼ 100000ð3:17 þ 1:63HÞð3:028  3:454B0 Þ ð0:71 þ 108:35t s Þ

ð36Þ

X 00 ¼ ð35:25 þ 31:10H  4:15H2 Þð0:22 þ 8:26B0  13:98B20 Þ  ð1:25  16:13ts Þ

ð37Þ

0

dx ¼ ð0:74 þ 0:58H  0:11H2 Þð0:84 þ 12:73B0  21:97B20 Þ  ð1:36  21:97ts Þ

ð38Þ

Fig. 12. P–I diagram with 19 min fire exposure time.

For DCOL = 0.8

A1 ¼ 3000ð2:685  0:445HÞð0:10 þ 3:05B0 Þð0:114 þ 55:91t s Þ ð39Þ A2 ¼ 100000ð11:71  2:12HÞð3:67  17:93B0 þ 30:11B20 Þ  ð0:49 þ 91:26ts Þ

ð40Þ

x0 ¼ ð52:63 þ 42:54H  5:62H2 Þð1:10  0:32B0 Þ Fig. 11. P–I diagram with 15 min fire exposure time.

 ð1:10  5:83ts Þ

ð41Þ

53

Y. Ding et al. / Engineering Structures 55 (2013) 44–55

Fig. 13. The values of parameters P0 and I0 for DCOL = 0.2.

Fig. 14. The values of parameters P0 and I0 for DCOL = 0.5.

Fig. 15. The values of parameters P0 and I0 for DCOL = 0.8.

dx ¼ ð13:34  6:69H þ 0:87H2 Þð13:66  83:30B0 þ 137:00B20 Þ

A02 ¼ 100000ð0:773 þ 1:092HÞð0:67  10:82B0 þ 17:51B20 Þ

 ð0:083 þ 60:19t s Þ

 ð1:347 þ 21:77t s Þ

A01 ¼ 10000ð3:82  0:42HÞð1:09  1:12B0 þ 2:79B20 Þð0:597 þ 101:02ts Þ

ð44Þ

X 00 ¼ð14:99 þ 26:44H  3:60H2 Þð1:23 þ 14:69B0  24:23B20 Þ ð43Þ

 ð1:17  10:90ts Þ

ð45Þ

54

Y. Ding et al. / Engineering Structures 55 (2013) 44–55

Fig. 16. Pressure–Impulse–Exposure time diagram of the steel column.

formulae for predicting the damage of any-sized steel tubular column subjected to blast load and fire action are given in the paper.

Table 6 Scope of geometric parameters. Height H (m)

Width B0 (mm)

Thickness ts (mm)

3.0 3.3 3.6 3.9 4.2

200 250 300 350 400

12 16 20 24 28

Acknowledgments The authors wish to acknowledge the financial supports from the National Science Foundation of China (No. 50638030 and No. 50528808), the National Key Technologies R&D Program of China (No. 2006BAJ13B02) and the Tianjin Municipal Application Basis and Forefront Technology Research Program of China (No. 08JCZDJC19500).

0

dx ¼ ð2:02 þ 3:22H  0:51H2 Þð4:16 þ 34:77B0  58:60B20 Þ  ð1:56  39:62ts Þ

ð46Þ References

In Eqs. (23)–(46), the units for geometric parameters H, B0, and ts are meter. Given the specific geometric sizes of a steel tubular are determined, the damage caused by blast load and the following fire action can be evaluated by Eq. (22), in which the parameters are calculated by Eqs. (23)–(46).

6. Conclusion In this paper, a numerical method for evaluating damage of steel tubular column subjected to blast load and post-explosion fire action is presented. Both geometric damage and mechanical damage caused by blast load are taken into account in the fire analysis for the explosion-survived column. A plasticity damage model based on continuum damage mechanics is employed in the material constitutive model, and its reliability is validated by the simulation of a blast test. With application of the proposed numerical method, an artificial function F(P, I, t) = 0 is derived to describe the integrated damage induced by both explosion and fire. A more inclusive damage diagram with three parameters peak pressure, impulse and fire exposure time is proposed in this study. Parametric studies are also conducted to study the effect of column height, width and thickness on the damage evolution. Based on the numerical results,

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