Stochastic dynamic response of masonry minarets subjected to random blast and earthquake-induced ground motions

Stochastic dynamic response of masonry minarets subjected to random blast and earthquake-induced ground motions

Kemal Haciefendioglu*,† and Fahri Birinci

Department of Civil Engineering, Ondokuz Mayıs University, Samsun, Turkey

SUMMARY

This paper presents the stochastic seismic response analysis of masonry minarets subjected to random underground blast and earthquake-induced ground motions by using a three-dimensional fi nite element model. The random blast and earthquake-induced ground motions are represented by the power spectral density function and applied to each support point of the three-dimensional fi nite element model of the masonry minaret system. This research conducted a parametric study to estimate the effects of the blast-induced ground motion on the stochastic response of the minaret. Therefore, the analyses were carried out for the different values of the charge weight and the distance from the charge centre. In addition, in order to investigate the effect of earthquake-induced ground motion on the stochastic response of the masonry minaret, three different soil conditions—soft, medium and fi rm soils—are considered in the analyses. Finally, it is noted that underground blast and earthquake effects cause the stochastic behaviour of minaret to change considerably. Copyright © 2009 John Wiley & Sons, Ltd.

1. INTRODUCTION

Minarets are the tall, slender towers built at the side of mosques from which the call to prayer is given for Muslims. Minarets have become an integral part of Islamic faith and culture. They are deemed as the lighthouses of faith. Minarets built during the Ottoman era were sometimes more than 70-m tall. Sometimes they were constructed as stepped storeys. Staircases were built inside in order for the caller to reach the balcony. Staircases were either made of stones or bricks and were reinforced with arches under each landing. The cut stone blocks were used more widely for the construction of historical masonry minarets in Turkey.

Many masonry minarets of the villages, towns and cities in the seismic region with single and double balcony had damages. These minarets are of similar type. The ones with single balconies are 20–25 m high. These are relatively long periods of vibration and correspond to the fi rst mod vibration of 10- to 12-storey-high reinforced concrete frame building.

Many of the masonry minarets resisted a variety of potential threats, such as environmental, wind, earthquake and blast, are partly or completely damaged. The potential threat of blast and earthquake to masonry structures has not been extensively studied in recent years (Hao et al., 2002; Wu et al., 2005; Pallarés et al., 2006; Ivorra and Pallarés, 2006; Wu and Hao, 2007; Betti and Vignoli, 2008). Seismic ground motions and blast loadings are inherently nondeterministic since there are always considerable uncertainties about the intensity and frequency of their future content. The most impor-tant function of the stochastic method for both earthquake and blast loading is the power spectral density function. The power spectrum of blast-induced and earthquake ground motions can also be represented by the Kanai (1957) and Tajimi (1960) functions.

Available knowledge on the seismic behaviour of masonry minaret structures is also very limited (El Attar et al., 2005, Higazy, 2004; Menon et al., 2004). In addition, a study investigating the dynamic response of masonry minarets against the blast-induced ground motion is not available. This paper

THE STRUCTURAL DESIGN OF TALL AND SPECIAL BUILDINGSStruct. Design Tall Spec. Build. 20, 669–678 (2011)Published online 28 October 2009 in Wiley Online Library (wileyonlinelibrary.com/journal/tal). DOI: 10.1002/tal.552

Copyright © 2009 John Wiley & Sons, Ltd.

* Correspondence to: Kemal Hacıefendioglu, Department of Civil Engineering, Ondokuz Mayıs University, 55139, Kurupelit, Samsun, Turkey

† E-mail: [email protected]

670 K. HACIEFENDIOGLU AND F. BIRINCI

Copyright © 2009 John Wiley & Sons, Ltd. Struct. Design Tall Spec. Build. 20, 669–678 (2011) DOI: 10.1002/tal

carried out a three-dimensional stochastic dynamic analysis due to the random blast and earthquake-induced ground motions. All the numerical applications are performed using ANSYS (2003).

2. THE FORMULATION OF STOCHASTIC DYNAMIC ANALYSIS

The dynamic equilibrium equation of motion for a multidegree of freedom system subjected to ground excitation can be written as:

Mu t Cu t Ku t M u tg�� ( ) + ( ) + ( ) = ( )δ (1)

where M, C and K are n × n, positive defi nite, mass, damping and stiffness matrices; u(t), u.(t) and ü(t) are the vectors of displacement, velocity and acceleration, respectively. δ is the direction vector that links the mass terms to the ground acceleration, üg(t).

A stationary stochastic model of the blast and earthquake-induced ground motions is defi ned by specifying the power spectral density function. If the power spectral density function of input process is known, the power spectral density function of output process can be determined easily. Filtered white noise model is generally used as power spectral density function for the modelling of ground motion simulation.

Since the formulation of the stochastic dynamic analysis of structural systems is given previously by many researchers, in this study, the fi nal equations will be used directly without any derivation. Detailed formulations for stochastic dynamic analysis are given in Lin (1967), Yang (1986), and Manolis and Koliopoulas (2001).

The structural response uj(t) in Equation (1) can be stated in terms of modal coordinates as:

u t Y tj jr r

r

N

( ) = ( )=

∑ψ1

(2)

where N is the number of modes that are considered to contribute to the response; ψjr is the contribu-tion of the j th mode to the uj(t); and Yr(t) is the modal coordinate. The Fourier transform of Equation (2) reveals:

U Yj jTω ψ ω( ) = ( ) (3)

where Y(ω) may be expressed as:

Y H PTω ω ϕ ω( ) = ( ) ( ) (4)

where H(ω) is the diagonal matrix of the Hj(ω) = (w2j − ω2 + 2iξjωj)−1. Here ωj and ξj are the natural

frequency and the damping ratio corresponding to the j th mode.For all mode shapes, modal forces are written as:

P M ATω ϕ δ ω( ) = ( ) (5)

where ϕ is the matrix of the mode shapes, and A(ω) is the Fourier transform of the ground acceleration.

For the response displacements ui(t) and uj(t), cross power spectral density function (Yang, 1986), Sij(ω), may be obtained as:

Sn

U U

Tnij

T

i k j k

k

n

ωω ω

( ) =( ) ( )

→ ∞→∞ =

∑lim*, ,1 2

1

(6)

If a single ground acceleration record because of blast loading and earthquake is used for the input, cross power spectral density function can be simplifi ed as follows:

RANDOM BLAST AND EARTHQUAKE-INDUCED GROUND MOTIONS 671


S S H Hij u ir js ir js

s

N

r

N

gω ω ψ ψ ω ω( ) = ( ) ( ) ( )==

∑∑�� *11

(7)

where Süg(ω) represents the power spectral density function of ground motion; ω represents the fre-

quency; H(ω) represents the frequency response function; N is the number of modes which are con-sidered to contribute to the response; ψir is the contribution of the rth mode to uj(t) displacement; and * denotes the complex conjugate. For i = j, Equation (7) gives the power spectral density function of the ith displacement. The standard deviation response of the structure can be computed from Equation (8):

σ ω ωij ijS d= ( ) ( )∞

∫0

(8)

3. GROUND MOTION MODELS

3.1. Blast-induced ground motion

An underground blast creates a complex dynamic process with the surrounding bedrock. Many studies that investigated ground motion due to underground blast have been conducted in the last two decades. Both continuum and discontinuum models have been used to simulate blast-induced stress wave in rock masses (Wu and Hao, 2005). These models are able to give a reasonable estimation of ground motions on rock surface generated by underground blasts. In this research, when analysing masonry minaret under blast-induced ground shock, the power spectral density function represented by the Kanai (1957) and Tajimi (1960) functions, which have been widely used in earthquake engineering, is used. The power spectral density function of blast ground motion may be expressed as (Wu and Hao, 2007):

S ff PF

f PF f PFSg

g

b

b

b( ) =

+−( ) +

×1 4

1 4

2 2 2

2 2 2 2 2 20

ξξ

(9)

where S0b is a scale factor depending on the intensity of the ground motion; PF describes the principal

frequency; and ξgb is a parameter governing the power spectrum shape.

The principal frequency derived from empirical observations may be expressed as (Wu and Hao, 2007):

PF R Q R Q= ( ) ≤ ≤ ( )−465 62 0 3 101 3 0 13 1 3. , .

.Hz (10)

where R and Q indicate the distance in metres measured from the charge centre and the TNT charge weight in kilograms, respectively. The scale factor of the spectrum may be determined from follow equation (Wu and Hao, 2007):

S R Q04 2 18 2 891 49 10= × ( )− −. . . m s2 3 (11)

Figure 1(a,b) illustrates the power spectral density functions for different values of charge weight and distance from charge centre, respectively.

3.2. Earthquake-induced ground motion

The power spectral density function of ground acceleration is assumed to be of the form of fi ltered white noise ground motion model originally developed by Kanai (1957) and Tajimi (1960), and modi-fi ed by Clough and Penzien (1993).



S Sug g g

g g g fg�� ω

ω ξ ω ωω ω ξ ω ω

ωω ω ξ

( ) =+

−( ) +×

−( ) +0

4 2 2 2

2 2 2 2 2 2

4

2 2 2

4

4 4 ff f2 2 2ω ω

(12)

where ωg and ξg are the resonant frequency and damping ratio of the fi rst fi lter; ωf and ξf are those of the second fi lter; and S0 is the spectrum of the white-noise bedrock acceleration.

The fi lter parameters depend on the soil type (Der Kiureghian and Nevenhofer, 1991). In this study, soft (S), medium (M) and fi rm (F) soil types are chosen for offshore wind turbine example, and fi lter parameters for these soil types, proposed by Der Kiureghian and Nevenhofer (1991), are utilized (see Table 1).

In this study, the east–west component record of the Kocaeli earthquake of 1999 has been applied to masonry minaret in direction x. S0 can be estimated for each soil type by equating the variance of the power spectral density function of acceleration to the variance of the Kocaeli earthquake accelera-tion. The calculated values of the intensity parameter for each soil type are: S0(soft) = 0.00214 (m2/s3); S0(medium) = 0.00153 (m2/s3); and S0(fi rm) = 0.00103 (m2/s3). The power spectral density func-tions of the Kocaeli earthquake for each soil type appear in Figure 2.

0 200 400 600 800

Frequency (Hz)

0.000

1.000

2.000

3.000

4.000

Pow

er S

pect

ral

Den

sity

Fun

ctio

n (m

²/s³

) Charge weightQ=100 kg

Q=200 kg

Q=300 kg

(a)

1000 0 200 400 600 800 1000Frequency (Hz)

0.000

1.000

2.000

3.000

4.000

Pow

er S

pect

ral

Den

sity

Fun

ctio

n (m

²/s³

)

DistanceR=25 m

R=50 m

R=100 m

(b)

Figure 1. The power spectral density functions for different (a) charge weights and (b) distances from charge centre.

Table 1. Filter parameters for different soil types.

Soil type ωg (rad/s) ξg ωf (rad/s) ξf

Firm 15.0 0.6 1.5 0.6Medium 10.0 0.4 1.0 0.6Soft 5.0 0.2 0.5 0.6

0 10 20 30 40 50

Frequency (rad/s)

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

Pow

er S

pect

ral

Den

sity

Fun

ctio

n (m

²/s³

) Firm soil

Medium soil

Soft soil

Figure 2. Power spectral density functions of the Kocaeli earthquake for each soil type.



4. NUMERICAL APPLICATION

A masonry mosque, which was named as the Ulu Mosque, located in Samsun, Turkey, is selected as an application. The Ulu Mosque located in a large yard has dual minarets made of hewn stone. It was completed in 9 September 1884 by Hacý Ali of Batumi and repaired by Sultan Abdulaziz’s mother. The dual minarets have single balconies. A picture of the mosque and its masonry minaret are shown in Figure 3(a). Figure 3(b) shows in detail the approximate geometrical properties of the minaret. The minaret is 32.07 m high with 99 stairs from the fl oor to the balcony; each step is 0.25 m high. From inside, a cylindrical shaft of constant inner diameter extends from the bottom square base up to the level of the balcony. A helical staircase, constructed from stones interlocked with the stones of the inner cylindrical shaft and resting at its centre on a common stone column, exists at the centre of the minaret. The minaret consists of a 3.13 × 3.13-m square base.

ANSYS (2008) is used to compute the stochastic behaviour of the masonry minaret by using a fi nite element model. Three-dimensional SOLID186 elements, which exhibit quadratic displacement behaviour, were used for modelling both the minaret body and the internal helical stair. The element has 20 nodes and three degrees of freedom per node, translations in the nodal x, y and z directions. The fi nite element model of the minaret and the stone block with stairs are shown in Figure 3(c).

The modulus of elasticity, Poisson’s ratio and mass density of masonry material are taken as 2000 MPa, 0.2 and 16 kN/m3, respectively. In the model, linear elastic material behaviour is assumed and the stiffness degradation is neglected.

(b) (c)

A-A Section

B-B Section

3.13 m

3.13

m

1.86 m

1.45 m

C-C Section

1.68 m

1.45 m

1.45 m

TNT

R (Distance)

Rock

-1.35 m

7.79 m

10.75 m

24.45 m

30.72 m

BB

AA

CC

(a)

Figure 3. (a) Photograph, (b) geometrical and cross-sectional properties and (c) fi nite element model of the Ulu Mosque’s minaret.



5. NUMERICAL RESULTS AND DISCUSSIONS

5.1. Blast-induced ground motion

In order to evaluate the effect of the random blast-induced ground motion on the stochastic response of the masonry minaret, a parametric study is performed for the different values of the charge weight and the distance from the charge centre. The TNT charge weights used in simulations are, respectively, 100, 200 and 300 kg. In addition, the distances from the structure to blast centre used in the analysis are, respectively, 25, 50 and 100 m.

For the TNT charge weights of 100, 200 and 300 kg, the x-direction displacement power spectral density (PSD) values, depending on the frequency ranging from 0.0 to 2.0 Hz, at top of the minaret are shown in Figure 4. In this section, the distance from the structure to blast centre was chosen as 25 m. It can be seen from the fi gure that the variation in the TNT charge weight causes signifi cant change in the displacement at the top of the minaret. The displacement PSD values clearly increase with increasing the TNT charge weight value.

The shaded image contours of the one standard deviation (1σ) of the Von Misses stress responses (N/m2) on the minaret for the different values of the TNT charge weight are shown in Figure 5. It can be observed from this fi gure that the maximum value of 1σ Von Misses stress contour is clearly increasing depending on the increase in the TNT charge weight. The maximum values for all three cases occur in the same region on the minaret. The maximum stress accumulation occurred at around the door opened to the balcony and the region between transition segments. It should be noted that the stiffness and strength of the minaret are reduced over the height of the transition segment with polygonal shape near its bottom. As expected from previous studies related to the minarets, the maximum stress accumulations and damages will occur at this transition segments.

The effects of the distances of 25, 50 and 100 m from the structure to blast centre used in the analysis on the stochastic response of the minaret are illustrated in Figures 6 and 7. In this section, the TNT charge weight was chosen as 300 kg. For this purpose, the x-direction displacement PSD values for the different distances from the structure to blast centre, at the top of the minaret, depend-ing on the frequency ranging from 0.0 to 2.0 Hz, are shown in Figure 7. It is concluded from the fi gure that the displacement values decrease as the distance from the structure to blast centre increases. The same comments can be made for the one standard deviation (1σ) of the Von Misses stress responses (N/m2) due to the different distances as illustrated in Figure 7. The maximum stress accu-mulations for the distances of 25, 50 and 100 m occur in the same regions on the minaret. The maximum stress accumulation occurred at around the door opened to the balcony and the region of between transition segment.

0.0 2.01.51.00.5Frequency (Hz)

0.0x10 0

4.0x10 -1

8.0x10 -1

1.2x10 0

1.6x10 0

Dis

plac

emen

t (m

2/H

z)

Charge weightQ=100 kg

Q=200 kg

Q=300 kg

Figure 4. The x-direction displacement power spectral densities for the different values of TNT charge weight.



(a) (b) (c)

Figure 5. One standard deviation (1σ) Von Misses stress contours for (a) TNT = 100 kg, (b) TNT = 200 kg and (c) TNT = 300 kg.

0.0 0.5 1.0 1.5 2.0

Frequency (Hz)

0.0x10 0

4.0x10 -1

8.0x10 -1

1.2x10 0

1.6x10 0

Dis

plac

emen

t (m

2/H

z)

R(distance) =25 m

R(distance) =50 m

R(distance) =100 m

Figure 6. The x-direction displacement power spectral densities for the different distances of the blast centre.

5.2. Earthquake-induced ground motion

In this section, to understand the effect of earthquake-induced ground motion on the stochastic response of the masonry minaret, the different soil conditions were considered for the support site. For homogeneous soil condition, the soft, medium and fi rm soil conditions were selected in the analyses.

The displacement PSD values for the soft, medium and fi rm soil conditions at the top of the minaret are compared with each other in Figure 8. As can be seen from the fi gure, the x-direction displacement PSD values, within the effective frequency range from 0.0 to 2.0 Hz, obtained for the fi rm and medium soil conditions are close to each other, and the soft soil condition induces the largest displacements. In addition, the shaded image contours of the one standard deviation (1σ) of the Von Misses stress responses (N/m2) on the minaret are shown in Figure 9. It can be seen from the fi gure that the



(a) (b) (c)

Figure 7. One standard deviation (1σ) Von Misses stress contours for the distances of (a) 25 m, (b) 50 m and (c) 100 m from the structure to blast centre.

0.0 0.5 1.0 1.5 2.0Frequency (Hz)

0.0x10 0

3.0x10 -3

6.0x10 -3

9.0x10 -3

1.2x10 -2

Dis

plac

emen

t (m

2/H

z)

Soft soil

Medium soil

Firm soil

Figure 8. The x-direction displacement power spectral densities for the different soil conditions.

maximum Von Misses stress value for the soft soil condition is the largest value. The maximum stress accumulations for the different soil conditions occur in the same regions on the minaret. It is also concluded from the fi gure that the Von Misses stress accumulation values on the region of between transition segment decrease as the soil becomes harder.

6. CONCLUSIONS

The aim of this study is to show the effect of the blast in underground and earthquake-induced ground motions on the stochastic responses of masonry minarets. The results for the fi nite element structural system have been modelled by using the computer software ANSYS (2003).

In this study, the blast-induced ground motion effect was separately investigated for the different values of the charge weight and the distance from the charge centre. The analyses for the blast-induced



Figure 9. One standard deviation (1σ) Von Misses stress contours for the (a) soft, (b) medium and (c) fi rm soil conditions.

ground motion shows that the stochastic dynamic response values increase with the increase of the blast charge weight, but decrease with the increase in the distance between the structure and blast centre. In addition, the results obtained from the earthquake-induced ground motion show that the values of the stochastic dynamic response of the masonry minaret system increase generally as soil gets softer. The maximum stress accumulations occur at around the door opened to the balcony and the region of between transition segment of the masonry minaret. The numerical results presented in this paper demonstrated the importance of blast and earthquake-induced ground motions for masonry minarets.

As a result, in order to make masonry buildings safe against blast-induced ground motions and earthquakes, and repair the damaged by them, the parametric studies performed above must be taken into account in the analyses. Although the results obtained from this study belong to a specifi c example, the observations here have applicability to many situations.

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Stochastic dynamic response of masonry minarets subjected to random blast and earthquake-induced ground motions