Risk Analysis of Industrial Structures Under Extreme Transient Loads

Risk analysis of industrial structures under extreme transient loads

D.G. Talaslidisa,*, G.D. Manolisa, E. Paraskevopoulosa, C. Panagiotopoulosa,N. Pelekasisb, J.A. Tsamopoulosc

aLaboratory of Applied Statics, Department of Civil Engineering, Aristotle University, Thessaloniki GR-54124, GreecebLaboratory of Fluid Mechanics, Department of Mechanical Engineering, University of Thessaly, Volos GR-38334, Greece

cLaboratory of Computational Fluid Dynamics, Department of Chemical Engineering, University of Patras, Patras GR-26500, Greece

Accepted 20 February 2004

Abstract

A modular analysis package is assembled for assessing risk in typical industrial structural units such as steel storage tanks, due to extreme

transient loads that are produced either as a result of chemical explosions in the form of atmospheric blasts or because of seismic activity in

the form of ground motions. The main components of the methodology developed for this purpose are as follows: (i) description of blast

overpressure and ground seismicity, (ii) transient non-linear finite element analysis of the industrial structure, (iii) development of 3D-

equivalent, continuous beam multi-degree-of-freedom structural models, (iv) introduction of soil–structure interaction effects,

(v) probabilistic description of the loading process and the stiffness/mass characteristics of the structure, and (vi) generation of fragility

curves for estimation of structural damage levels by using the Latin hypercube statistical sampling method. These fragility curves can then be

used within the context of the engineering analysis–design cycle, so as to minimize structural failure probability under both man-made

hazards such as blasts and natural hazards such as earthquake-induced transient loads.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: Chemical explosions; Industrial structures; Finite elements; Fragility curves; Latin hypercube method; Risk analysis; Seismic loads; Structural

dynamics

1. Introduction

The development of methods for risk analysis based on

probabilistic concepts in the presence of hazards such as

seismically induced ground motions or chemically induced

atmospheric blasts is a prerequisite for damage assessment

in key structures (storage tanks, piping systems, industrial

buildings, loading docks, etc.) that comprise large industrial

complexes [1–8]. Current design codes and guidelines

cannot completely cover such issues because much

uncertainty is involved, primarily with respect to the nature

of the various accident scenarios that are possible [9,10].

Further investigations are, therefore, required in order to

better understand the risk involved and to obtain results that

can be used in formulating comprehensive intervention

strategies. Since the aforementioned structural units form

an integral part of larger installations, improvement in

safety helps to ensure a reliable and continuous operation of

the entire industrial complex, which in turn implies better

environmental protection [11].

Current methods of structural analysis that assume a

deterministic environment selectively utilize part of all

available information regarding loading, material

parameters and structural configuration and are thus capable

of producing only representative (e.g. maximum or

minimum) values for the structural response [3].

The introduction of probabilistic models, especially in the

description of the structure itself, requires a different level

of analysis in order to provide quantitative information on

structural reliability and on risk of failure. Furthermore, it

is essential that any increase in sophistication implied for

the loading is not compromised by simplification in the

description of the structural model, i.e. the level of accuracy

in all steps of the analysis must be adjusted so as to produce

an all-around consistent mechanical model [2].

Recent work in this area includes use of extensive Monte

Carlo simulations (MCS) for the seismic finite element

0267-7261/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.soildyn.2004.02.003

Soil Dynamics and Earthquake Engineering 24 (2004) 435–448

www.elsevier.com/locate/soildyn

* Corresponding author. Tel.: þ30-2310-99-5707; fax: 30-2310-99-

5769.

E-mail addresses: [email protected] (D.G. Talaslidis), tsamos@

chemeng.upatras.gr (J.A. Tsamopoulos).

http://www.elsevier.com/locate/soildyn

method (FEM) analysis of nuclear reactor facilities,

including soil–structure interaction (SSI) effects [12].

Uncertainty is assumed to exist in the free-field motion,

the local site conditions and the structural parameters, which

are modelled by uniform, Gaussian and log-normal

distributions. These random processes are in turn

represented by their respective Karhunen–Loeve expan-

sions and results are cast in terms of probabilistic

in-structure spectra. Next, the non-linear stochastic dynamic

analysis of an SSI system consisting of an oscillator plus

foundation resting on a 2D soil deposit was performed using

the FEM with MCS in order to determine risk of damage

due to liquefaction under earthquakes [13]. Soil properties

were measured from cone penetration tests from a real site

and were found to match a beta-distribution. The seismic

input was modelled as a non-stationary random process and

the analysis results were presented in terms of fragility

curves. Fragility curves have also been used in conjunction

with bridges, and their computation is based on either

standard time history analyses or on simplified methods

such as the capacity spectrum approach, which is essentially

a static non-linear procedure. Specifically, 10 nominally

identical, but statistically different highway bridges were

analysed under 80 records of ground motion histories for

this purpose [14]. Also, the seismic behaviour of on-grade,

steel liquid storage cylindrical tanks subjected to

ground shaking hazard was examined and fragility curves

were developed by analysing the reported performance

of over 400 such tanks under nine separate earthquake

events [15].

As far as explosions are concerned, we mention the

construction of pressure– impulse diagrams based on

single degree-of-freedom (SDOF) representations of struc-

tural systems, where the blast load is a simplified

descending pressure pulse [16]. These diagrams are used

in identifying damage regimes in the SDOF, much like a

response spectrum and are defined in terms of the ratio

between loading time and response period of the structure.

Finally, more refined statistical sampling techniques such

as the Latin hypercube method [17] have been recently

introduced for generating variations of a typical structural

configuration for stochastic FEM analysis purposes.

These sampling procedures strongly improve the statistical

representation of stochastic design parameters as compared

to standard MCS and also help reduce spurious correlations

in the processed data.

In this work, an integrated methodology is developed for

assessing risk to a typical industrial structure with respect to

accident scenarios involving either a chemical explosion or

an earthquake. The methodology first focuses on generation

and subsequent propagation of overpressure due to chemical

explosions [18]. In parallel, we give a description of the

seismically induced ground motions [19]. As far as

the structural model is concerned, the starting point is

non-linear, transient FEM analysis of cylindrical, thin-

walled shell structures. This implies use of efficient

triangular and quadrilateral shell finite elements based on

the generalized Hu-Washizu principle within the context

of elastoplastic material response [20]. Based on the results

of the 3D finite element analyses, a simplified model in the

form of a multi-degree-of-freedom (MDOF) continuous

beam with variable thickness is developed for capturing all

salient aspects of the tank’s response. SSI effects are

accounted for by introducing simplified, yet realistic soil

impedance coefficients [21]. Next, this MDOF system with

material, geometric and loading parameters that are

stochastic variables is used for assessing risk in the presence

of random loads [22,23]. In particular, the basic failure

mode considered is material yielding with local buckling

(e.g. ‘elephant foot’) ignored. Furthermore, two basic limit

states are distinguished, namely serviceability and ultimate

strength. Earlier work along these lines [24,25] employed

MCS with SDOF structural models, which have been

replaced here by the more efficient Latin hypercube method

[17] and by a continuous beam model, respectively. The final

results of the risk analysis is generation of fragility curves

plotting limit state probability as a function of blast load or

peak ground acceleration (PGA), which allow for an

estimation of damage levels in the structure under

investigation.

Briefly, the material presented in this paper is organized

as follows. In Section 2, the load sustained by the industrial

structure shown in Fig. 1(a) is described for two cases,

namely chemical explosions and seismic motions.

Both overpressure duration and ground acceleration filter

parameters are treated as random variables, so that

realizations of the pressure and acceleration fields can be

computed within the framework of the Latin hypercube

method. Next, the dynamic behaviour of the structure is

examined by a 3D FEM model, which is used in turn to

define a simplified MDOF model with SSI. The key part is

Section 4, where all information is integrated for the

purpose of assessing risk, which is carried out within the

framework of the Latin hypercube method and conveniently

described by a set of fragility curves. Finally, some

conclusions are drawn regarding probability for the

structure to exhibit damage of a certain level at a given

external load intensity.

2. Load description

As previously mentioned, the methodology developed

herein is modular in form; each component is now

separately analysed, starting with the loading scenarios

shown in Figs. 2 and 3.

2.1. Chemical explosion model

The majority of data on blast effects are scaled in practice

with respect to the blast pressure output of a spherical

charge of TNT explosive [7]. Of interest in the context of

D.G. Talaslidis et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 435–448436

industrial plants is explosion of a cloud of flammable

vapour. Such explosions are usually the result of massive

spills of combustible hydrocarbons in the atmosphere,

followed by ignition and acceleration of the flame

propagating through the cloud so as to produce a destructive

shock wave. Depending on the speed of the flame front,

two different mechanisms are at work, namely deflagration

and detonation. Existing guidelines for estimating blast

damage are based on the TNT equivalent yield concept that,

in conjunction with the Rankine–Hugoniot relations [26]

allows for computation of all necessary quantities

(overpressure, reflected pressure, shock velocity, stagnation

pressure, etc.) with relative ease.

The basic assumptions made for evaluating air blast

loading on an industrial structure [18] are as follows: (i) an

ideal shock is produced in which the peak overpressure is

reached instantaneously; (ii) the structure is in the Mach

reflection region where the air blast front is propagating

parallel to the ground; (iii) the shock load can be decoupled

from the structural response, and (iv) the structure is treated

as a rigid body. The loading scenario develops as the air

Fig. 1. Steel storage tank: (a) geometry, (b) vertical cross-section, (c) finite element mesh, (d) equivalent MDOF model and (e) equivalent MDOF model

with SSI.

Fig. 2. Blast load: spatial variation of the load (a) across the horizontal

plane, (b) along the height and (c) temporal variation of the net

overpressure.

D.G. Talaslidis et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 435–448 437

pressure reaches the front face (first point impact) of the

circular cylindrical tank. We have an instantaneous rise

from ambient air pressure to overpressure pðtÞ felt in

the immediate surroundings of the structure, which quickly

increases in value because of the emergence of a reflected

pressure. This is followed by rapid decay as air flows around

the tank. The total pressure, which corresponds to pðtÞ þ

cdqðtÞ (with qðtÞ the dynamic pressure and cd the drag

coefficient) reaches the stagnation point at clearing time

t1 ¼ 3d=U; with d being the tank diameter and U the shock

velocity. The pressure’s spatial variation over the cylind-

rical tank’s circumference is simply a function of the angle

that the normal to a given point on the surface traces with

respect to an axis parallel to the direction of blast

propagation. Subsequently, the rear face is completely

engulfed and a maximum pressure registers there at blast

time t2 ¼ ðd þ 3hÞ=U; where h is the height of the tank.

Also, the average roof loading can be idealized as triangular

in time with peak pressure equal to cdqðtÞ: Finally, when this

overall pressure is averaged, i.e. integrated over the entire

structure, a net horizontal force PðtÞ acting on the structure

is produced, which is uniform along the tank’s height and in

the direction of the propagating front. For simplicity, a

linear drop in time is assumed for load PðtÞ that lasts for tdseconds, with maximum intensity P0 corresponding to 2/3

of the value obtained from integrating the total pressure on

the tank in order to account for suction in the rear surfaces.

This load model is shown in Fig. 2(c) and is a somewhat

simplified version of the one used in earlier work [24,25],

but retains all salient features necessary for a rational

analysis of the tank.

As far as the statistical properties of the blast load are

concerned, we assume that time duration td of the net

horizontal force PðtÞ is a random variable with uniform

distribution. We furthermore define six load levels, which

depend on the strength of the explosion that took place.

These load levels are calibrated to produce structural

damage that ranges from minimal to severe. It thus becomes

possible to generate a family of statistically equivalent,

time-dependent net horizontal forces for each load level,

which act uniformly across the height of the structure

(for both the 3D FEM tank model as well as the continuous

beam model derived from it), all within the context of the

Latin hypercube sampling method [17].

2.2. Seismic motion model

The simple, yet commonly used model of horizontally

polarized, elastic shear (SH) waves is adopted herein.

Soil layers over bedrock are modelled by the Kanai–Tajimi

(KT) and by the high-pass (HP) filters in the frequency v

domain. Thus, the earthquake signal at the source is viewed

as a broad-banded, white noise stochastic process with a

constant power spectral density function (PSDF) S0:

This function is subsequently filtered through soil to yield

the PSDF at the surface of the ground as

SgðvÞ ¼ HkðvÞHhðvÞS0; ð1Þ

HkðvÞ ¼ {1 þ 4z2kðv=vkÞ

2}={½1 2 ðv=vkÞ2�2 þ 4z2

kðv=vkÞ2}

and

HhðvÞ ¼ ðv=vhÞ2={½1 2 ðv=vhÞ

2�2 þ 4z2hðv=vhÞ

2}

In the above, fundamental frequency vk and damping

ratio zk for the KT filter, respectively, range from 3p to 9p

(rad/s) and from 15 to 25% so as to cover a wide range of

soil configurations. As far as the HP filter is concerned, these

values are held fixed at vh ¼ p (rad/s) and at zh ¼ 100%:

In order to capture the uncertainty involved in ground

seismicity, it is common to resort to MCS by considering

certain parameters of the problem as random variables [19].

The most rational choice here is the PGA, which is a

function of time t and of random parameter g: Since ground

accelerations eventually return to a quiescent state, there is

only a fluctuating component to contend with. A rather

standard choice for the autocovariance of the PGA is the

exponential correlation function, i.e.

covg ¼ k€xgðt;gÞ2l ¼ s2

g expð2Dt=tgÞf ðtÞ ð2Þ

In the above, k l denotes the expectation operator for the

ground accelerations €xgðtÞ; sg is their standard derivation, tg

is a correlation time interval, Dt is the time lapse and f ðtÞ is a

deterministic time envelope. It is well known that the

autocovariance and the PSDF form a Fourier transform pair

so once the latter has been determined, it is possible to

compute realizations of the PGA as

€xgðtÞ ¼ffiffi2

p XNj¼1

ffiffiffiffiffiffiffiffiffiffiffiffiSgðvjÞDv

qcosðvjt þ fÞf ðtÞ ð3Þ

Fig. 3. Seismic load: spatial variation of the load (a) across the horizontal

plane, (b) along the height and (c) spectrum-consistent synthetic ground

acceleration history.


where Dv ¼ ðv2 2 v1Þ=N is the frequency increment and fj

is a random phase angle uniformly distributed in the interval

[0,2p]. The frequency contents of the ground signal

falls between cut-off frequencies v1 ¼ 0:4p and v2 ¼ 50

p (rad/s), while the envelope function shown below is

essential in ameliorating the stationary assumption inherent

in this simulation procedure:

f ðtÞ ¼

ðt=T1Þ2; t # T1

1:0; T1 , t , T2

exp{ 2 cðt 2 T2Þ=TD}; t $ T2

8>><>>:

ð4Þ

Rise and drop times, respectively, range as

T1 ¼ 1:0–3:0 s and T2 ¼ 6:0–8:0 s, while total signal

duration is TD ¼ 15:0 s and c ¼ 5 is a dimensionless

constant. Obviously, as the number of samples N in Eq. (3)

increases, the realizations become a more realistic

representation of the original random field. Finally, Fig. 3

shows the spatial and temporal distribution of the input to

the tank from the ground motions, which are described by

synthetic acceleration signals. As with the chemical

explosion case, the Latin hypercube method will be used

to produce a family of statistically equivalent acceleration

records at a given PGA level, with the KT filter parameters

and the envelope function characteristic times assumed to

be random variables with uniform distributions. A total of

six PGA levels that produce the full range of damage,

from minor to severe, will be considered.

3. Methods of structural analysis

We now present details regarding the procedure used for

analysing the steel storage tank under the extreme transient

loads previously described.

3.1. Structural models

The specific tank examined herein [27] has clear height

h ¼ 29:5 m, mean base radius r ¼ 20:0 m and its thickness t

is a variable ranging from tb ¼ 21:9 mm at the base to

tt ¼ 9:52 mm at the top, as was previously shown in Fig. 1.

There is a stiffener beam around the tank’s perimeter at the

top, while the tank’s base is welded to a special metal alloy

ring plate, which in turn rests on a concrete foundation.

The tank material is ST 37 steel, with an elasticity modulus

E ¼ 2:1 £ 108 kN/m2, a Poisson’s ratio n ¼ 0:30 and a

mass density r ¼ 7850 kg/m3. The plastic modulus is

Epl ¼ E £ 1024; while the material yield stress is

sy ¼ 2:35 £ 105 kN/m2, and the yield strain and strain to

fracture are 1y ¼ 1:12% and 1ul ¼ 10:0%; respectively.

Additional, secondary design details that are particular to

such tanks are ignored.

The essential component in a FEM representation of

thin-walled, circular cylindrical tanks is development of

a simple, yet robust shell finite element that can be

incorporated into an open-end computer program capable

of transient non-linear analysis [20]. For this purpose,

we employed an in-house finite element program with pre-

and post-processing capabilities for data input/output built

around the facilities of the commercial program Nastran

[28]. Furthermore, the program employs Newmark’s

beta time integration algorithm, supplemented by the

Newton–Raphson iterative scheme for capturing non-linear

response with the tangent modulus concept.

Following standard convergence studies, the tank was

finally modelled by 288 quadrilateral shell finite elements

arranged in nine circumferential bands so as to account for

variable wall thickness and the presence of the stiffener

beam. Furthermore, this FEM model is detailed enough to

capture bending distortions near the lower edge of the tank’s

wall. This particular discretization required 320 nodal points

and the resulting mesh is shown in Fig. 1(c), where the

entire tank is modelled. Among the large volume of results

generated by the free-vibration analysis, we show

the dominant (first) mode in Fig. 4, namely combined

flexural – distortional mode at an eigenfrequency

f ¼ 2:21 Hz. When modal analysis was specified for the

two types of loadings previously described, which

are symmetric about the tank’s reference diameter and

piece-wise uniform across the height, the resulting

participation factor for this mode was very low (less than

g ¼ 0:01). Therefore, the mode with the largest

participation factor had to be identified, which turned out

to be the skew-symmetric, bending mode at f ¼ 19:99 Hz

given in Fig. 5. The two participation factors for the blast

load, one for the restoring forces (‘static’) and another for

the inertia forces (‘dynamic’), were gst ¼ 0:773 and

gdyn ¼ 0:486; respectively. As far as the seismic load was

concerned, the aforementioned participation factors

assumed values of gst ¼ 0:926 and gdyn ¼ 0:713;

respectively. It is interesting to note that both blast and

seismic loads, despite being two very different types of

events, share a similar spatial variation when applied to

this particular type of industrial structure and activate the

same mode.

Based on the results of the FEM eigenvalue analysis, a

continuous, cantilevered beam model (Fig. 1(d)) was

developed for the steel tank. Firstly, the bending stiffness

coefficients were computed by assuming a step-wise

variable thickness with unit value at the bottom and by

retaining the same thickness ratios of the original structure,

which had nine circumferential bands of constant thickness

decreasing from bottom to top. The height of the beam is the

same as that of the original tank, i.e. h ¼ 29:5 m.

Next, masses were lumped at the nine levels that demark

change in thickness, and full fixity was assumed at the base.

The value of the mass coefficient at the top node was

adjusted so as to produce the fundamental frequency of

f ¼ 19:99 Hz associated with the dominant (in terms of its

participation factors) mode and the remaining masses were


adjusted by using the actual thickness ratios. Finally, a

nominal amount of modal damping of 4.5% was prescribed,

which is representative of rather flexible steel structures.

Thus, a nine-node model was established with two DOF per

node (translational and rotational) for a total of 18 active

DOF, which is far more efficient for the ensuing stochastic

analysis than the 3D FEM model.

As far as the loading sequence on this particular MDOF

model is concerned, we use a uniformly distributed

horizontal load for the blast, whose equivalent value peq is

computed by multiplying the original reference value P0 by

the inverse of the ratio defined as total tank mass divided by

the sum of lumped masses of the MDOF model. This ratio is

about 30:1, while the time variation of the equivalent load

remains unchanged. For the seismic event, we use a

step-wise uniform load whose value at the top is

peq ¼ 2mðlevel¼9Þ €xg; while the variation of peq with height

follows the thickness ratio of the original tank.

The formulation is now in terms of relative displacements,

while the time variation of the synthetic ground

accelerations remains unchanged.

The last task that remains is to adjust the material

parameters of the MDOF model for the purposes of a non-

linear, time-stepping analysis. For this purpose, a quasi-

static (‘push-over’) analysis of the 3D FEM model for a

step-wise uniformly distributed load across the height of the

tank is performed. The constitutive law used derives from

the bilinear, strain-hardening model for ductile steel,

whose values were prescribed earlier on. At each load

increment, the total base shear Vb in the FEM model is

computed, normalized by the tank’s total mass M; and

plotted in Fig. 6 versus maximum lateral deflection ut at the

tank’s top. The process is repeated until complete yielding

of the base is reached. By curve-fitting a bilinear plot to the

l ¼ Vb=M versus ut curve, a yield displacement

uy ¼ 0:0492 m can be identified. Next, the ‘pseudo-accel-

eration’ ly ¼ 559:8 m/s2 value that corresponds to uy is

converted into a load through multiplication by the tank

mass at each level, scaled by the total tank to MDOF model

mass ratio, and applied to the beam as a distributed load

(Fig. 6(c)). The moment computed at the base of the

continuous beam is the plastic (or yield) moment Mp:

Finally, this procedure was assumed to cover both blast and

seismic load cases.

Thus, considerable economy in scale has been reached,

given that a total of 18 DOF (later increased to 20 by

introducing base rotation and translation for SSI purposes)

are now sufficient to capture the basic mechanical behaviour

of the original tank. We note in passing that further

reduction in tank modelling was achieved in earlier work

[24,25] by introducing a SDOF oscillator, whose flexibility

was adjusted through energy balance considerations, i.e. by

specifying equal kinetic energies with the continuous beam

model. This was deemed necessary, since the rather

extensive MCS used within the context of risk analysis

required the simplest structural model possible for

efficiency purposes. Use of the Latin hypercube method

[17] in the present work, which generates a much smaller

sample space for comparable accuracy, allows use of the

present MDOF model.

3.2. Equations of motion

The MDOF model exhibiting elasto-plastic material

behaviour that was calibrated above will now be incorpor-

ated within the context of an efficient probabilistic analysis

for risk assessment. The basic parameters of this model are

mass matrix M; elastic tangent stiffness matrix K; and

damping matrix C built by specifying a damping coefficient

z ¼ 4:5% in each mode. In addition, it is necessary to

introduce ductility m as the ratio of maximum transient

response to yield displacement uy: The dynamic equilibrium

Fig. 4. FEM modal analysis of the steel tank: dominant first mode at

f ¼ 2:21 Hz with nearly zero participation factors for the two types of

transient loads.


equations of the MDOF structural system read as

M€yðtÞ þ C_yðtÞ þ RðtÞ ¼ FðtÞ ð5Þ

with yðtÞ denoting the vector of horizontal displacement and

rotational DOF and R representing the restoring force

vector. Specifically, in the elastic range, R ¼ KyðtÞ; while

beyond it, the system is plastic with much reduced stiffness

(the new coefficients are four orders of magnitude less than

the elastic ones). Lastly, vector FðtÞ corresponds to external

loads imposed on the structure. These are given by either

an equivalent, uniformly distributed load PðtÞ along the

height with ‘triangular’ time variation for chemical

explosions or by an inertia term 2MI€xgðtÞ applied along

the height of the structure for earthquake motions

(where IT ¼ ½1; 0; 1;…; 0�). In the latter case, the horizontal

displacements uiðtÞ are defined as relative to the ground

motion xgðtÞ: We note here that both loading cases are

viewed as random processes, so as to allow for statistical

generation of families of load records.

Eq. (5) constitute an initial value problem. The resulting

system of non-linear, second-order differential equations are

numerically integrated by the generalized energy

momentum algorithm [29], which in turn is based on the

generalized alpha method [30] for linear systems. This new

algorithm adopts both linear and angular momentum

conservation criteria during the current time-step to avoid

Fig. 5. FEM modal analysis of the steel tank: non-symmetric bending mode at f ¼ 19:99 Hz with the highest participation factor for the two types of transient

loads.

Fig. 6. Nonlinear FEM analysis of the steel tank: (a) monotonically

increasing lateral piece-wise uniform load, (b) computation of the plastic

moment for the continuous beam model and (c) normalized base shear

versus top horizontal deflection curves.


loss of unconditional stability that is commonly associated

with elastic systems, along with desirable numerical

dissipation characteristics. As an illustration of the type

of results that were recovered, Fig. 7 shows the evolution of

crown displacement u9ðtÞ and base moment MbðtÞ for the

empty steel tank with total mass M ¼ 403:34 t

(that corresponds to an equivalent MDOF model mass of

meq ¼ 12:324 t), which is subjected to a ‘reference’

explosive load (corresponding to load level ‘4’ in a scale

from 1–6) of magnitude P0 ¼ 246; 453 kN with linear time

decay and time duration td ¼ 0:1453 s, which exceeds the

mean duration of the event by a factor of 1.60. Also, Fig. 8

plots the same quantities, but for the synthetic earthquake

acceleration record with ‘reference’ PGA of €xg ¼ 0:38 g

and total duration TD ¼ 15:0 s, which also corresponds to a

load level of ‘4’. In both cases, 750 time steps were used in

the time integration algorithm and the criterion for yielding

was MbðtÞ $ Mp: Note that plots for SSI effects, which

will be discussed in Section 3.3, are also included in

these figures. It is interesting to note that a typical

computer run for the MDOF model consumed a CPU time

of less than 1.0 s.

Although the MDOF model cannot provide all the details

that a 3D FEM model does, it gives a good overall picture of

the dynamic response, which is on the conservative side.

Also, it is well suited for intense sudden loads that produce

non-linear response, since in this case mostly lower modes

of vibration contribute to the gross displacement picture. If

additional effects such as instability need to be studied, then

recourse must be made to the FEM model. In principle,

risk analysis is still possible, but becomes extremely

time-consuming because a very large number of fully

Fig. 7. Time evolution of the MDOF steel tank response for a chemical blast yielding horizontal load P0 ¼ 246; 453 kN : (a) top horizontal displacement u9 and

(b) base moment Mb:


non-linear time-stepping analyses need to be performed at

each load level.

3.3. Soil–structure interaction effects

SSI effects are manifested because of ground flexibility

and are important for the purposes of this work, given the

large mass (when the tank is full) and rather heavy

foundation design of most cylindrical steel storage tanks.

Since we seek the simplest model possible, the influence of

the explosion on the ground itself is neglected in the

chemical blast case, while kinematic interaction effects that

have to do with the influence of the foundation shape on the

earthquake signal that develops at the base of structure

were not taken into account. Thus, we use a set of

frequency-independent soil impedances, dampers and

masses [21], which for the case of a circular rigid foundation

of radius r resting on the ground surface are:

Kh ¼ 8Gr=ð2 2 nÞ; Ch ¼ 1:08

ffiffiffiffiffiffiffiffiKhrr3

q;

Mh ¼ 0:28rr3

ð6Þ

Ku ¼ 8Gr3=ð3ð1 2 nÞÞ; Cu ¼ 0:47

ffiffiffiffiffiffiffiffiKurr3

q;

Iu ¼ 0:49rr5

In the above formulas, subscripts h, u; respectively, denote

horizontal and rotational DOF, as shown in Fig. 1(e). We

assume relatively soft soil [21] with the following numerical

values for its properties: Poisson’s ratio n ¼ 0:25; shear

modulus G ¼ 72:0 £ 106 kN/m2, density r ¼ 1800 kg/m3

and shear wave-speed cs ¼ffiffiffiffiffiffiffiðG=rÞ

p¼ 200 m/s. Basically,

the influence of soil on the response of the overlying structure

Fig. 8. Time evolution of the MDOF steel tank response for seismic motions with a peak ground acceleration €xg ¼ 0:38g : (a) top horizontal displacement u9

and (b) base moment Mb:


is elongation of the fundamental period of vibration, an effect

that becomes more pronounced as the ratio of stiffness of

structure to that of soil increases. SSI is not necessarily

detrimental and, in fact, may lead to stress reduction in the

structure at the expense of a more pronounced kinematic

state. For stochastic loads, it usually leads an increase in the

probability for a certain type of damage to occur at a given

load level [24,25].

Thus, the additional equations of dynamic equilibrium

for the horizontal displacement u0 and angle of rotation u0 at

the foundation (level 0) are added to Eq. (5) as follows:

The above is now a 20 DOF system, where Iu is the mass

moment of inertia corresponding to lumped mass Mh, and in

both augmented damping and stiffness matrices various

coupling terms (with mixed indices) appear. Numerical

results from computations involving the MDOF system with

SSI are also plotted in Figs. 7 and 8 for the blast and

earthquake loads, respectively. Two types of soil are

considered whose lateral stiffness varies by two orders of

magnitude, namely the soft soil given above

(with Kh ¼ 2:1 £ 109 kN/m) and a stiff soil (with

Kh ¼ 210 £ 109 kN/m). Comparisons for the blast load

reveal an increase (of roughly 20%) in the tank’s kinematic

response as time evolves. At the same time, this phenom-

enon is accompanied by a slight increase (of about 2–5%) in

the base moment. As far as the seismic loads go, the

presence (or absence) of SSI is of minor importance since all

three curves (fixed, soft soil, stiff soil) lie close together

through-out the time duration of the ground accelerations.

4. Risk analysis

The risk analysis procedure is broken down into the

following two basic steps:

4.1. Generation of structure and load samples

Risk is broadly defined as the convolution of hazard with

consequence, i.e. the probability of occurrence of an event

that can lead to structural (or other) type of failure times the

economic (or other) loss associated with this event. Risk

reduction is essentially a stochastic optimization problem

[1], and additional aspects that should be considered is the

influence of various structural components (sensitivity

analysis), along with changes that can be implemented so

as to reduce risk. On the hazard side of the equation, it is

necessary to reduce existing failure probability, which is

directly dependent on the randomness exhibited in the

resistance of the structure as well as in the loading. In

general, when performing reliability analyses for non-linear

systems, the capacity as well as the demand in conventional

structures is assumed to be log-normally distributed [31,32].

4.1.1. Blast load case

In order to model blast-induced loading for risk analysis,

we define a sequence of blasts with increasing strength

levels. Then, we integrate pressure along the entire circular

cylindrical tank configuration to produce a net, uniformly

distributed horizontal load with a simplified triangular time

variation PðtÞ ¼ P0ð1 2 t=tdÞ; where P0 is the value

registered at each load level and td is its time duration, as

was discussed in Section 2.1. For a cylindrical steel tank

with overall dimensions h ¼ 29:5 m, r ¼ 21:0 m, the

maximum load resulting from an intermediate blast giving

80.0 kPa of overpressure (load level ‘4’ in a scale from 1 to

6) is p ¼ 8354 kN/m, based on a doubling caused by the

reflected shock. The maximum equivalent horizontal

uniform load is therefore P ¼ 369; 680 kN at that level,

but in order to be consistent with negative pressure and other

effects that are produced as the blast engulfs the entire tank,

the total impulsive load is two-thirds of this value, i.e.

P0 ¼ 246; 453 kN. Finally, mean time duration of the blast

td is given by ratio 2r=U ¼ 0:0908 s; where U is the pressure

wave-speed from the Rankine–Hugoniot relations.

4.1.2. Structural configuration

As far as the structural configuration is concerned, we

consider certain key parameters to be random variables.

These are the yield (or plastic) moment Mp whose mean

value is 6300 kN m and its standard deviation is 200 kN m,

the damping coefficient z whose range is 3–6%, and the

fluid mass Mfl stored in the tank. We assume the tank stores

hydrocarbon fluids with mass density rfl ¼ 0:5 kg=m3; and

the mean value corresponds to a half-empty configuration

that gives Mfl ¼ 9227 ton (the equivalent MDOF

model fluid mass is mfl ¼ 300 ton). This fluid mass is

lumped at the MDOF model node that is at a distance h=3

from the bottom of the tank. By prescribing a standard

deviation of 50t; the samples that will be generated cover

the range from nearly empty to nearly full. Also, the bilinear

elastoplastic material model uses an elastic bending stiffness

of EI and a corresponding plastic stiffness EIpl ¼ 1024EI:

By specifying the above range and assuming a normal

distribution for these key structural parameters, the Latin

hypercube method commences generation of a family of

storage tanks. These tank samples are then matched with a

family of blast records, which come from perturbing the

time duration td of the horizontal load. This last task is done

M 0

0Mh 0

0 Iu

26664

37775

€y

€u0

€u0

8>>><>>>:

9>>>=>>>;þ

C Cij

Cji

Chh Chu

Cuh Cuu

26664

37775

_y

_u0

_u0

8>>><>>>:

9>>>=>>>;

þ

K Kij

Kji

Khh Khu

Kuh Kuu

26664

37775

y

u0

u0

8>>><>>>:

9>>>=>>>;¼

FðtÞ

F0

0

8>>><>>>:

9>>>=>>>;

ð7Þ


by considering the range of 0:10td –2:0td; which like all load

parameters follows a uniform distribution. At this point, the

user may specify a minimum number of structure–load

pairs (or samples), which in our case was taken as N ¼ 100;

following parametric studies in which the same results were

recovered for larger values of N: Finally, we specify six load

levels that range from 0:25P0 to 1:75P0; computed so as to

produce structural behaviour starting from mild elastic

response up to pronounced non-linearities. Thus, at each

load level, the MDOF model is solved at least N times and

an equal number of ductility values m are obtained. These

values are statistically processed to yield mean value mL and

second-order reliability index bL (Table 1).

Next, five limit states are defined for the structure, as

listed in Table 2. To each such state, the corresponding

ductility capacity (or resistance factor R) is assumed to obey

a logarithmic distribution with mean value mR and second

moment reliability index bR: It should be noted here that the

second-order reliability index is equal to the logarithmic

standard deviation of the structural capacity population

(the same holds true for the structural response as well). It is

subsequently used, along with the relevant mean value, to

evaluate limit state probabilities. Finally, this procedure was

repeated for the MDOF model with SSI so as to capture the

influence of ground flexibility on the ductility. The relevant

results appear in Tables 3 and 4 for the stiff and soft soils,

respectively. We mention in passing that the ground

parameters were not perturbed, which implies a fixed

foundation configuration.

4.1.3. Seismic load case

Next, we focus on seismic loads, which are characterized

by the PGA output described in Section 2.2. At a fixed

acceleration level, we again discern a number of possible

variations prior to activating the realizations given by

Eqs. (1)–(4) for producing synthetic accelerograms. These

are produced by assuming the KT filter frequency vk and

damping zk; as well as time the envelope rise time T1; are all

random variables with uniform distributions whose range is

3p–9p rad/s, 0.15–0.25% and 1.5–3.0 s, respectively.

We furthermore define six levels for the peak white noise

intensity S0 of the earthquake signal ranging from 0.005 to

0.0394 m2/s3, in equal increments. These correspond to PGA

levels of 0:15g–0:45g: Thus, at each PGA value, the

synthetic ground motion family is matched with

the structural model family of samples within the context

of the Latin hypercube method and at least N ¼ 100 pairs

are solved, yielding an equal number of ductility values m:

These values were again statistically processed, assuming

a logarithmic distribution, to give a mean value mL and

a second moment reliability index bL; as shown in Table 5.

Since soil flexibility was of little consequence to the tank’s

response to ground accelerations, no additional compu-

tations for SSI effects were performed here.

4.2. Fragility curves

The last step is to introduce the fragility of a structure

with respect to a given limit state, which is defined as

Table 1

Logarithmic distribution of structural response ðLÞ to blast loads in terms of

ductility (MDOF model)

Explosion load

Level

Mean value,

mL

Second moment

reliability index,

bL

1 0.2989 0.5261

2 0.5978 0.5261

3 1.1186 0.6799

4 2.3262 0.8924

5 4.3843 1.0394

6 7.3092 1.1396

Table 2

Logarithmic distribution of structural capacity ðRÞ in terms of ductility

Limit state Mean

value, mR

Second moment

reliability index, bR

Non-structural damage 1.0 0.5

Minor damage 2.0 0.5

Moderate damage 4.0 0.5

Severe damage 6.0 0.5

Collapse 7.5 0.5

Table 3

Logarithmic distribution of structural response ðLÞ to blast loads in terms of

ductility (MDOF model with SSI effects for Ku ¼ 210 £ 109 kN/m)

Explosion load

level

Mean value, mL Second moment

reliability index,

bL

1 0.2588 0.5350

2 0.5177 0.5350

3 0.8217 0.5838

4 1.5972 0.7884

5 2.9335 0.9458

6 4.8959 1.0621

Table 4

Logarithmic distribution of structural response to blast loads ðLÞ in terms of

ductility (MDOF model with SSI effects for Ku ¼ 2:1 £ 109 kN/m)

Explosion load

level

Mean value, mL Second moment

eliability index,

bL

1 0.2546 0.5883

2 0.5091 0.5883

3 0.8325 0.6572

4 1.7324 0.8917

5 3.2938 1.0628

6 5.6269 1.1817


the probability Pf of load L exceeding resistance R and is

given by

Pf ¼ PðR # LÞ ¼ð1

0½1 2 FsðrÞ�fRðrÞdr ð8Þ

where Fs is the cumulative probability distribution of L and

fR is the probability density function of R: For both R and L

log-normally distributed, which is common assumption

[32], Eq. (8) can be written as

Pf ¼ Fð2‘nðmR=mLÞ=

ffiffiffiffiffiffiffiffiffiffiffis2

R þ s2L

qÞ ð9Þ

where F is the standardized normal distribution function.

All results from the previous sub-section can now be

summarized in the fragility curves of Figs. 9 and 10, which

plot limit state damage probability Pf versus blast-induced

net horizontal force and normalized PGA, respectively. In

the former case, the first graph is for the standard fixed-base

MDOF model, while the second and third graphs are for SSI

with stiff and soft soils, respectively. In the later case, there

is only one graph for the fixed-base MDOF model. In

parallel, Tables 6–9 give all Pf values that were used in

constructing the aforementioned graphs. We observe that

for blast loads, the failure probability is more pronounced as

compared to failure probability for seismic motions, at least

for the range of loads that were considered here. Apparently,

the impulsive character of the blast loads falling on what is

essentially a flexible structure, especially in the presence of

the extra fluid mass it stores, is more detrimental than the

relatively slowly developing phenomenon of ground

motions. Finally, in order to look at the effect of SSI more

closely for blast loads, we compare Pf for two values of the

ground horizontal stiffness that are two orders of magnitude

apart (Kh ¼ 210 £ 109 and 2.1 £ 109 kN/m). As the soil

progressively softens, damage probability clearly increases

(with the Pf curves moving ‘upwards’), reflected by the fact

that the overall structural system has more inherent

uncertainty. It should be noted here that Pf ¼ 1:0 denotes

certain failure. Thus, it is possible for blast loads of a

fixed level of intensity to produce more damage in the

presence of foundation flexibility, as compared to the

absence of such an effect.

Based on these results, the following general conclusions

can be drawn: (i) for every damage level, the probability that

a certain structural configuration will exhibit the type of

damage associated with it, increases with increasing load

level; (ii) for every load level, the probability that a structure

will exhibit damage greater than a certain predefined

Table 5

Logarithmic distribution of structural response ðLÞ to seismic loads in terms

of ductility (MDOF model)

Seismic load level Mean value mL Second moment

reliability index bL

1 0.1989 0.3680

2 0.2956 0.3460

3 0.3703 0.3847

4 0.4166 0.3506

5 0.4958 0.3704

6 0.5384 0.3558

Fig. 10. Fragility curves for a steel storage tank under seismically induced

ground motions without SSI effects.

Fig. 9. Fragility curves for a steel storage tank under chemical blast loads:

(a) without SSI effects, (b) with SSI effects for stiff soil ðKh ¼ 210 £ 109

kN=mÞ and (c) with SSI effects for soft soil ðKh ¼ 2:1 £ 109 kN=mÞ:


Table 7

Fragility matrix for atmospheric blast loads (MDOF model with SSI effects for Ku ¼ 210 £ 109 kN=m)

Load level (kN) Level of damage

Non-structural

damage

Minor

damage

Moderate

damage

Severe

damage

Collapse

61613.3 0.03247 0.00262 0.00009 0.00001 0.00000

123226.6 0.18430 0.03247 0.00262 0.00041 0.00013

184839.9 0.39919 0.12359 0.01975 0.00485 0.00201

246453.2 0.69201 0.40482 0.16272 0.07815 0.04880

308066.5 0.84278 0.63984 0.38596 0.25179 0.19013

369679.8 0.91198 0.77715 0.56835 0.43124 0.35819

Table 6

Fragility matrix for atmospheric blast loads (MDOF model)


Non-structural

damage

Minor

damage

Moderate

damage

Severe

damage

Collapse

61613.3 0.04808 0.00441 0.00018 0.00002 0.00000

123226.6 0.23924 0.04808 0.00441 0.00074 0.00025

184839.9 0.55282 0.24555 0.06553 0.02328 0.01207

246453.2 0.79540 0.55871 0.29809 0.17715 0.12623

308066.5 0.89999 0.75191 0.53170 0.39281 0.32079

369679.8 0.94502 0.85115 0.68595 0.56301 0.49174

Table 8

Fragility matrix for atmospheric blast loads (MDOF model with SSI effects for Ku ¼ 2:1 £ 109 kN=m)


Non-structural

damage

Minor

damage

Moderate

damage

Severe

damage

Collapse

61613.3 0.03818 0.00379 0.00018 0.00002 0.00001

123226.6 0.19095 0.03818 0.00379 0.00070 0.00025

184839.9 0.41217 0.14427 0.02867 0.00838 0.00388

246453.2 0.70455 0.44413 0.20652 0.11215 0.07587

308066.5 0.84493 0.66450 0.43432 0.30481 0.24177

369679.8 0.91090 0.78992 0.60486 0.48005 0.41140

Table 9

Fragility matrix for seismic loads (MDOF model)

Load level (%g) Level of damage

Non-structural

damage

Minor damage Moderate damage Severe damage Collapse

0.153 0.00464 0.00010 0.00000 0.00000 0.00000

0.222 0.02251 0.00083 0.00001 0.00000 0.00000

0.281 0.05767 0.00375 0.00008 0.00001 0.00000

0.325 0.07580 0.00510 0.00011 0.00001 0.00000

0.379 0.12980 0.01250 0.00040 0.00003 0.00001

0.418 0.15650 0.01624 0.00054 0.00004 0.00002


level decreases as the level of damage increases; (iii) due to

the additional effect of foundation flexibility, the probability

for a certain damage level to occur at fixed load intensity

shows an increase with decreasing ground stiffness; and (iv)

blast loads are more critical for a tank, more so as the mass

of the fluid it stores increases, while seismic loads seem to

have a relatively minor effect on such a flexible structure,

especially when the mass of the stored fluid is low.

5. Conclusions

An integrated numerical approach has been developed for

assessing risk posed by chemical explosions and earthquake

motions to typical industrial structural facilities. The analysis

yields, as a final product, a set of fragility curves that allow

computation of failure probabilities for steel storage tanks

due to overpressure caused by a chemical explosion or due to

ground acceleration caused by an earthquake. These curves

are useful within the context of planning prevention and

intervention strategies for large industrial complexes, where

the type and intensity of the external loads are key parameters

in estimating the level of damage sustained by various

structural units. In addition, the presence of compliant soil

causes SSI phenomena that increase the probability for a

certain damage level to occur at fixed load intensity. In

closing, it should be noted that the present methodology is

generally enough to be expanded to other categories of

structures (e.g. buildings, bridges) as well as to other types of

extreme environmental and man-induced loads (e.g. high

winds, ocean waves, fire hazards).

Acknowledgements

The authors wish to thank the Greek General Secretariat

for Research and Technology for its financial support of the

project No. 1573 entitled ‘Development of Methodologies

for Evaluating Risk in Industrial Units due to Explosions’.

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Risk Analysis of Industrial Structures Under Extreme Transient Loads