Probabilistic analysis of historic masonry bridges to random ground motion by Monte Carlo Simulation using Response Surface Method

Construction and Building Materials 134 (2017) 199–209

Contents lists available at ScienceDirect

Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat

Probabilistic analysis of historic masonry bridges to random ground motion by Monte Carlo Simulation using Response Surface Method Kemal Hacıefendiog˘lu a,⇑, Hasan Basri Basßag˘a b, Swagata Banerjee c a

Private Sector Engineering Firms, Samsun, Turkey Karadeniz Technical University, Department of Civil Engineering, 61080 Trabzon, Turkey c Indian Institute of Technology Bombay, Department of Civil Engineering, Powai, Mumbai 400 076, India b

h i g h l i g h t s
We combine two probabilistic methods namely Monte Carlo Simulation and Response Surface Methods.
We investigate influence of uncertain material parameters on stochastic response of historic bridge.
Material parameters are modeled using normal, log-normal and Gumbel distributions.
COV (standard deviation/mean) and uncertainties in material parameters are investigated.

a r t i c l e

i n f o

Article history: Received 17 May 2016 Received in revised form 19 October 2016 Accepted 21 December 2016

Keywords: Monte Carlo Simulation Probabilistic analysis Response Surface Method Historic masonry bridge Random ground motion

a b s t r a c t The parametric investigation discloses influence of uncertain material parameters on stochastic earthquake response of historic masonry bridges subjected to random ground motion. Kurt Bridge, located in Samsun, Turkey is selected for numerical calculations. The east-west component of the Kocaeli earthquake in 1999, Turkey is selected as ground motion. For random ground motion, the power spectral density function is applied to each support point of three-dimensional finite element model of the historic masonry bridge. The uncertain material parameters of interest are the Elastic modulus, Poisson’s ratio and mass density, which are modeled using normal, log-normal and Gumbel distributions. ANSYS finite element program is used to present the probabilistic analysis of the historic masonry bridge according to Monte Carlo Simulation results which are obtained through Response Surface Method. In this study, the central composite design for three variables is chosen as a sampling method to obtain unknown coefficients. In order to estimate the probabilistic response of the bridge, obtaining the simulation number with the exact solution through Monte Carlo Simulation, determining the type of statistical distribution are carried out. Finally, the effects of the coefficient of variation and uncertainties in material parameters on the stochastic dynamic response of the bridge are investigated. Probabilistic one standard deviation of the displacement and Von Misses stress responses are presented and compared with those of deterministic method in terms of material parameter. Ó 2016 Published by Elsevier Ltd.

1. Introduction Historic structures are wealth, which transfer the cultural values from the past to the future and provide a way for the humanity’s common values. The protection and safety transfer of these structures to the future generations are possible only when necessary maintenance, repair and strengthening measures are taken. Structural behaviors that reflect the characteristics of existing

⇑ Corresponding author at: Private Sector Engineering Firms, Samsun, Turkey. E-mail addresses: [email protected] (K. Hacıefendiog˘lu), [email protected] (H.B. Basßag˘a), [email protected] (S. Banerjee). http://dx.doi.org/10.1016/j.conbuildmat.2016.12.101 0950-0618/Ó 2016 Published by Elsevier Ltd.

structures must be identified as a whole in order to accomplish these processes successfully and accurately. Undoubtedly, historical masonry bridges widely constructed in Anatolia, Turkey are among the most important historical monuments. The first of these bridges were built during the Hittite period, and followed by the construction during the Ottoman period. These historic bridges, especially the ones constructed in the 19th century during the Ottoman period, are usually single span stone arch bridges. Earthquakes that cause damage to historic bridges are one of the most important external factors for the existence of such structures over decades. Necessary precautions must be taken to protect structural integrity against earthquakes and to minimize structural

200

K. Hacıefendiog˘lu et al. / Construction and Building Materials 134 (2017) 199–209

damage and determination. In recent years, a number of studies related to dynamic analyses of masonry bridges have been carried out. These studies essentially evaluated dynamic behaviors of historic structures under seismic loading conditions by using three dimensional finite element method [1–13]. Knowing that historic structures are safe, capacity and serviceability of these structures are the most valuable information to engineers before and after structural restoration. However, it is extremely difficult to obtain such information because the determination of structural parameters is generally uncertain in nature. Experimental studies are performed by many researchers in past to identify the uncertain parameters which are primarily the mechanical properties of structures. The field and laboratory experiments performed on this topic were able to bring out the results for a wide range of mechanical properties such as compressive strength, flexural tensile strength, modulus of elasticity etc. Therefore, the use of mean (or, deterministic) values of these mechanical properties of masonry structures in numerical analysis will not provide any information about the natural variability of results due to material uncertainty. Insufficient knowledge on mechanical properties of materials used in numerical models leads to a search for alternative analyses which are capable of working directly with uncertain data considering them to be random variables, described by appropriate probability distribution functions. However, only a few studies related to probabilistic response of masonry structures are available in current literature [14–21]. Uncertainties in seismicity must be considered as well in the dynamic analyses of masonry historic structures. Behaviors of historic masonry bridges under seismic ground motion are rather unpredictable because of soil conditions and properties. Therefore, generalization of dynamic behaviors of historic bridges from one site to another site is hardly satisfactory. It is expected that each strong earthquake that causes damage to historic bridges can bring up new features of the seismic behavior of historic bridges, which may have been overlooked initially or considered to be of lesser importance. Consequently, accurate and sufficient numerical techniques must be employed to obtain necessary information for maintenance, repair and strengthening of historic bridges safely and economically against earthquakes. One of the most important of these numerical techniques is stochastic dynamic method. This technique can provide the response of historic masonry structures subjected to unpredictable seismic excitations [22,23]. With the uncertainties of external excitations (such as wind loading, seismic waves etc.) and structural parameters, it is almost impossible to have a precise prediction of seismic responses of masonry historic bridges. Therefore, the variability of material parameters as well as uncertainties in earthquake ground motions should be considered in dynamic response analysis of historic bridges. Various studies are found in the literature that combine the random vibration theory with the uncertainty of structural parameters for different structures [24–34]. Studies are also available on the stochastic dynamic analysis of masonry structures by separately considering the randomness of ground excitations and uncertainty in material properties. However, an attempt to combine these two for seismic performance evaluation of historic masonry bridges is lacking from the existing literature. In this context, it is required to calculate the stochastic response of historic masonry bridges exposed various loads from a probabilistic viewpoint. Monte Carlo Simulation (MCS) technique is capable of producing exact result based on uncertain parameters if an appropriate number of simulations are performed. Because the required number of simulation is generally large, approximate methods are used in association with MCS to obtain the structural response. Response Surface Method (RSM) is a helpful tool to carry out the probabilistic analysis [35]. The RSM is one of the designs of experiments methods used to approximate an unknown function

for which only a few values are computed. The purpose of the present paper is to perform classical random vibration analysis (stochastic earthquake analysis) using power spectral density function that includes uncertainty in material properties simulated through MCS in combination with RSM. To achieve this goal, parametric studies are carried out to determine the optimum simulation number required for MCS. Following this, the finite element model of a historic masonry bridge under random ground motion is developed in ANSYS [36] and probabilistic analyses are performed according to the MCS using RSM through the selected parameters. Displacements and stresses at different locations within the bridge are obtained from ANSYS. 2. Random vibration method The dynamic equilibrium equation of motion for a multi-degree of freedom system subjected to ground excitation can be written as

€ ðtÞ þ CuðtÞ _ € g ðtÞ Mu þ KuðtÞ ¼ Mdu

ð1Þ

where M, C, K are n  n, positive definite, mass, damping and stiff_ € ðtÞ are the vectors of displacement, velocness matrices; u(t), uðtÞ, u ity and acceleration, respectively. d is the direction vector that links € g ðtÞ. the mass terms to the ground acceleration, u A stationary stochastic model of the earthquake-induced ground motions is defined 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 ground motion simulation. Since the formulation of the stochastic earthquake analysis of structural systems is given previously by many researchers, the current study does not provide any derivation and directly uses the final equations. For detailed formulations for stochastic earthquake analysis, the readers are referred to previous studies [37– 39]. The structural response uj ðtÞ in Eq. (1) can be stated in terms of modal coordinates as,

uj ðtÞ ¼

Nm X wjr Yr ðtÞ

ð2Þ

r¼1

where Nm is the number of modes which are considered to contribute to the response, wjr is the contribution of the jth mode to the uj ðtÞ, and Yr ðtÞ is the modal coordinate. The Fourier transform of Eq. (2) reveals,

Uj ðxÞ ¼ wTj YðxÞ

ð3Þ

where YðxÞ may be expressed as,

YðxÞ ¼ HðxÞ/T PðxÞ

ð4Þ

where HðxÞ is the diagonal matrix of the Hj ðxÞ ¼ ðx  x2 þ 2 j

2inj xj Þ1 . Here xj and nj are the natural frequency and the damping ratio corresponding to the jth mode. For all mode shapes, modal forces are written as,

PðxÞ ¼ /T MdAðxÞ

ð5Þ

where / is the matrix of the mode shapes, AðxÞ is the Fourier transform of the ground acceleration. If a ground acceleration record from an earthquake is used for the input, cross power spectral density function Sij ðxÞ can be simplified as follows Nm X Nm X Sij ðxÞ ¼ Su€ g ðxÞ wir wjs Hir ðxÞHjs ðxÞ r¼1 s¼1

ð6Þ

K. Hacıefendiog˘lu et al. / Construction and Building Materials 134 (2017) 199–209

where Su€ g ðxÞ represents the power spectral density function of the ground motion, x represents the frequency, HðxÞ represents the frequency response function, N is the number of modes which are considered to contribute to the response, wir is the contribution of the rth mode to uj(t) displacement and * denotes the complex conjugate. For i = j, Eq. (6) gives the power spectral density function of the ith displacement. The standard deviation of the structure response can be computed from Eq. (7).

Z

rij ¼

1

0

Sij ðxÞdðxÞ

The aim of the current research is to investigate the effect of uncertainties of the material parameters on the stochastic earthquake response of historic masonry bridge to random seismic excitation. In all solutions in the study, the randomness of the ground excitation is utilized as external load affecting on the bridge for each simulation number of MCS in combination with RSM. This section presents probabilistic analysis method according to Monte Carlo Simulation (MCS) results being obtained through Response Surface Method (RSM). For procedure, firstly, structural parameters (Elastic modulus, Poisson’s ratio and mass density) are selected. Then, RSM is used to obtain the approximate function through selected parameters. Finally, a historic bridge finite element model is constituted and probabilistic analyses are performed with MCS using selected parameters and response surface function. So, reduced Monte Carlo Simulations are carried out while using the random vibration method in the frequency domain for the stochastic earthquake solutions of the bridge under random seismic excitation. 3.1. Response Surface Method Response Surface Methodology (RSM) is essentially a particular set of mathematical and statistical methods used by researchers to aid the solution of certain types of problems which are pertinent to scientific or engineering processes. It is assumed that the

2

1

6 61 6 6 61 6 6 W ¼ 61 6 6 61 6 61 4

l1

^ðXÞ. General form of this function is a quadmial type of function g ratic polynomial with cross terms. In this study, the second order polynomial without mixed terms (as given in Eq. (9)) is used to ^ðXÞ. While this equation can include a large numbers of express g random variables, it is still computationally efficient and mostly used in practice for easy implementation.

^ðXÞ ¼ a þ g

ð7Þ

3. Probabilistic analysis

l2 l2 l2

l3 l3 l3 l3 l3

ðl1 Þ2

ðl2 Þ2

201

n n X X bi x i þ ci x2i þ e i¼1

ð9Þ

i¼1

Here a, bi and ci are the coefficients of the polynomial, n is the number of the variables x and e is the random error that originates due to the negligence of higher order terms. To obtain unknown coefficients of this polynomial equation, the experimental points should be chosen. There are many sampling methods namely star, full factorial, central composite and BoxBehnken designs are available to obtain the experimental points. In this study, central composite design as shown in Fig. 1 for three variables is selected as a sampling method. The experimental points are selected around the mean values of random variables according to their standard deviations;

xi ¼ li  kri

ð10Þ

where li and ri are respectively the mean value and standard deviation of random variable xi. The coefficients of the response surface function including a, bi, ci are achieved by using the least square method as given in the following equation. 1

coeff ¼ ðWT WÞ WT y

ð11Þ

where coeff is the coefficient vector, W indicates the design matrix that includes the experimental points and y represents the response vector obtained from the response surface function corresponding to the experimental points. Design matrix W for second-degree polynomial and a three-variable problem is obtained around the mean values as follows:

ðl3 Þ2

3 7

2 7 l1 þ kr1 ðl1 þ kr1 Þ ðl2 Þ2 ðl3 Þ2 7 7 2 2 2 7 l1  kr1 ðl1  kr1 Þ ðl2 Þ ðl3 Þ 7 7 2 2 2 7 l1 l2 þ kr2 ðl1 Þ ðl2 þ kr2 Þ ðl3 Þ 7 7 2 l1 l2  kr2 ðl1 Þ2 ðl2  kr2 Þ ðl3 Þ2 7 7 27 2 2 l1 l2 l3 þ kr3 ðl1 Þ ðl2 Þ ðl3 þ kr3 Þ 5 2 1 l1 l2 l3  kr3 ðl1 Þ2 ðl2 Þ2 ðl3  kr3 Þ

ð12Þ

researcher is concerned with a system involving some response y which depends on the input variables x1, x2, . . .. . ., xn [40]. So, response y can be written as

y ¼ gðx1 ; x2 ; . . . ; xn Þ

x2

ð8Þ

where g is a function representing the implicit process. In the civil engineering problems, this process generally indicates the finite element method. Finite element procedure gives the numerical solution through implicit calculations. With the Response Surface Method, the actual limit state function g(X) is replaced by a polyno-

x1 x3 Fig. 1. Central Composite Design (number of experimental points: 2n + 2n + 1).

202

K. Hacıefendiog˘lu et al. / Construction and Building Materials 134 (2017) 199–209

ments and stresses) for each iteration in the simulation are obtained as

3.2. Monte Carlo Simulation The Monte Carlo Simulation (MCS) is a special technique to numerically generate results without actually doing any physical testing. It involves ‘sampling’ at ‘random’ to simulate artificially a large number of experiments and to observe the results [41,42]. In the MCS technique, sample values are generated for each random variable xi and these are used to obtain mean values of structural response. Samples of random variable xi are generated by randomly selected zi as follows,

xi ¼ F1 X ðzi Þ

ð13Þ

where zi is a sample generated from a uniformly distributed random variable with bounds 0 and 1, and F1 X ð:Þ is the inverse CDF (cumulative distribution function) of a standard normal variate. In this study, MCS is used in combination with RSM to obtain the probabilistic response of abridge under consideration. For this purpose, the approximate functions are obtained first using RSM for one standard deviation (1r) results indicated by ^ g (). Then, MCS is applied by using these functions. The 1r results (displace-

Hi ¼^ gðXi Þ

ð14Þ

where {Xi} is the vector of sample values for random parameters. The mean values of the one standard deviation (1r) are calculated by;

Hmean ¼

N 1X Hi N i¼1

ð15Þ

The algorithm of RSM-MCS combination used in this study is shown in the flow chart in Fig. 2. 4. Numerical example 4.1. The model of Kurt Bridge A masonry bridge, Kurt Bridge, located in Samsun, Turkey is selected for numerical calculations. Kurt Bridge shown in Fig. 3 is

Fig. 2. The flow chart of the algorithm of RSM-MCS combination used in this study.

K. Hacıefendiog˘lu et al. / Construction and Building Materials 134 (2017) 199–209

203

Fig. 3. Photographs of Historic Kurt Bridge in Turkey.

Fig. 4. Finite element model of the historic masonry bridge system.

built on the Istavroz Brook which is drawing the borders of Vezirkopru and Havza, and connects Tahna Village (Havza) and Tekkekıran Village (Vezirkopru). The bridge stands over two high arches. The bridge has three pointed arch windows, one is located between the arches and the remaining two of them are located on

Fig. 5. Cross-section of the historic masonry bridge system.

the sides of the arches. Face stone and irregular stones are observed at the rubble bonding system of the bridge. In the construction of the bridge, grave stones and architectural pieces belonging to Byzantine and Roman periods are also used as a gathered material. The architectural style and the bond system of the bridge are confirming to the 13th–14th century architecture [43]. Probabilistic analyses of the historic masonry bridge to random ground motion are carried out by using ANSYS [36] finite element program. In the analyses, the effects of the uncertainty in material parameters on the stochastic earthquake response of a historic masonry bridge are investigated in detail. SOLID45 element is used for the three-dimensional modeling of the Kurt Bridge. This element is defined by eight nodes having three translations degrees of freedom in the nodal x, y, and z directions at each node. Figs. 4 and 5 show the finite element model and cross-section of the Kurt Bridge and specific node points. Due to the complicated geometry of the bridge model, 24,744 tetrahedral shaped elements are used

Table 1 Material properties of the bridge. Material

Side walls Stone arches Filling

Modulus of Elasticity (N/m2)

Density (kg/cm3)

Poisson’s Ratio

Mean

COV: %

Mean

COV: %

Mean

COV: %

2.5  109 3.0  109 1.5  109

5–20 5–20 5–20

0.20 0.25 0.05

5–20 5–20 5–20

2464.4 2140.7 1600.0

5–20 5–20 5–20

K. Hacıefendiog˘lu et al. / Construction and Building Materials 134 (2017) 199–209

204

for the mesh system. At the support points where the bridge contacts to the ground, fixed boundary conditions are assumed. Therefore, all the DOFs at these support points are restrained. Stone arch, side wall and timber block sections of the bridge are also taken into account in the finite element model. Material properties used for these sections are obtained from previous studies performed on similar historic bridges [4,44,45]. Firstly, in the analyses, three types of statistical distributions called as normal, lognormal, and Gumbel are considered to model the variability of Elastic modulus, Poisson’s ratio and mass density of the historic bridge. To facilitate the comparison among results obtained by using each of these distributions for all parameters, simulations are performed using the same mean and standard deviation for all distributions. The mean and COV (standard deviation/mean) of the material properties are considered as listed in Table 1.

4.2. Random ground motion model In this study, a stationary assumption is made for the stochastic earthquake analysis where the statistical parameters are independent of time. A stationary model simplifies the computations and gives satisfying results. Because earthquake ground motions occur rarely, the data that can be considered in random process are very few. So while calculating the statistics to represent the random process, like ensemble averages, some difficulties are encountered. Of course, the use of a single record is not a sufficient basis to produce general conclusions. However, ergodicity assumption is made to overcome these difficulties and only one earthquake record from local area (local acceleration) can be used in this study [46]. The east-west component of the 1999 Kocaeli earthquake in Turkey, recorded on the firm soil condition, are considered in this study. Power spectral density function (PSD) of this earthquake ground motion are calculated to obtain absolute maximum values. The average response spectra for stochastic process is more simple to reduce difficulties of calculating the stochastic earthquake

Acceleration Spectral Density Function (m²g³)

0.025 Filtered white noise model 0.020

Earthquake PSD

0.015 0.010 0.005

responses. Finally, the filtered white noise model modified by Clough and Penzien [47] is easily prefer to generalize the design PSD function of earthquake ground acceleration in local area. The power spectral density function of Kocaeli earthquake ground acceleration is assumed to be in the form of filtered white noise ground motion model originally developed by Kanai [48] and Tajimi [49], and modified by Clough and Penzien [47]. This is given as

!

Su€ g ðxÞ ¼ S0

x4g þ 4n2g x2g x2 x4  2 2 2 ðx2g  x2 Þ þ 4ng x2g x2 ðx2f  x2 Þ þ 4n2f x2f x2

!

ð16Þ The east-west component of the 1999 Kocaeli earthquake in Turkey, recorded on the firm soil condition, is chosen as ground motion to perform the study. Acceleration spectral density function of filtered white noise ground motion for firm soil type is shown in Fig. 6. The random ground motion is applied to the bridge in the X direction as it is shown in Fig. 4. The calculated values of the intensity parameter for Kocaeli earthquake is S0 = 0.00103 (m2/s3), the resonant frequency and damping ratio of the first filter are xg = 15.0 rad/s, ng = 0.6 and those of the second filter are xf = 1.5 rad/s, nf = 0.6. In all analyses, the input ground motion is considered as a random ground motion by using the Clough and Penzien Model [47] and the uncertainty of the response of the structure due to randomness of the model parameters is kept at the forefront.

4.3. Stochastic earthquake analysis with deterministic material parameters This section of the paper demonstrates the stochastic earthquake responses of the historic bridge subjected to random ground motions when deterministic values of material properties are used. Such an analysis with deterministic parameters will help to understand the displacement and stress fields within the bridge. The shaded image contours of the one standard deviation (1r) of the displacement (in m) and Von Mises stress responses (in N/m2) on the historic masonry bridge appear in Figs. 7 and 8. Notably, the maximum displacements occur at middle and the tops of the big arches as illustrated in Fig. 7. In addition, the maximum Von Mises stresses appear at bridge sections close to the base and the tops of the big arches of the bridge. 4.4. Stochastic earthquake analysis under uncertain material parameters

0.000 0

10

20

30

40

50

Frequency (rad/s) Fig. 6. Earthquake power spectral density function and filtered white noise ground model.

2030, 2293, 698, 2118 and 2030, 2293, 698, 2118, 663 of nodal points were selected in longitudinal bridge model direction for displacements and to define the critical points for stress distribution. The selected points have been assigned according to displacement

Fig. 7. 1r displacement contours.

K. Hacıefendiog˘lu et al. / Construction and Building Materials 134 (2017) 199–209

205

Fig. 8. 1r Von Mises stress contours.

Table 2 The comparison of the results. Node

Function

ANSYS

Difference

2118 (stress) 663 (displacement)

1074434.53 0.0819509844

1073910.00 0.0818079

0.049% 0.175%

at the right end and towards middle of the bridge as shown in Fig. 4. For this purpose, the results calculated from the functions obtained from RSM are compared with those obtained from ANSYS. The functions are given as follows;

r2118 ¼ 1412874:2 þ 0:0004274224593  E sidewalls  0:001184163383  E filling  0:00003943358025  E stonearches  1:751585185E  14  E sidewalls^ 2 þ 2:479646091E  13  E filling^ 2 þ 4:344855967E  15  E stonearches^ 2 u663 ¼ 0:2147485326  1:062915506E  11  E sidewalls  2:995065844E  11  E filling  3:474671811E  11  E stonearches þ 6:590123457E  22  E sidewalls^ 2

Fig. 9. The change of 1r Von Mises stresses with simulation numbers.

and stress contours from stochastic earthquake analysis to represent the all bridge.

þ 5:166145405E  21  E filling^ 2 þ 3:022091907E  21  Estonearches ^ 2 The comparison of the results obtained from functions given above and ANSYS is demonstrated in Table 2. (b) Determination of MCS number

(a) Comparison of the results obtained from ANSYS and Response Surface Function The accuracy of the response surface functions related to Elastic modulus obtained from 1r Von Mises stresses of the selected node 2118(r2118) and 1rdisplacement of the selected node 663(u663) are checked before further analyses. These two nodes are located

Fig. 9 shows 1r Von Mises stresses of node 2118 when the number of simulation is varied for a wide range. In this, the normal distribution of Elastic modulus is considered as the only uncertain parameter in the model. Running processes are performed by the simulation numbers from 10000 to 1000000 with an increment of 10000 steps. As seen from this figure, the response does not have

Table 3 1r displacement of the bridge obtained for different statistical distributions and uncertainty ranges of Elastic modulus, Poisson’s ratio and mass density. Node

Deterministic

Normal

Difference

Lognormal

Difference

Gumbel

Difference

COV = 5% 698 2030 2293

0.037314 0.053707 0.000674

0.037402 0.053841 0.000663

0.24% 1.67% 0.25%

0.037400 0.053840 0.000663

0.23% 1.67% 0.25%

0.037396 0.053838 0.000663

0.22% 1.68% 0.24%

COV = 10% 698 2030 2293

0.037314 0.053707 0.000674

0.037499 0.053963 0.000673

0.50% 0.24% 0.48%

0.037505 0.053969 0.000673

0.51% 0.24% 0.49%

0.037501 0.053975 0.000673

0.50% 0.20% 0.50%

COV = 15% 698 2030 2293

0.037314 0.053707 0.000674

0.037657 0.054183 0.000690

0.92% 2.26% 0.89%

0.037667 0.054188 0.000690

0.95% 2.26% 0.90%

0.037663 0.054184 0.000690

0.93% 2.28% 0.89%

COV = 20% 698 2030 2293

0.037314 0.053707 0.000674

0.037913 0.054494 0.000713

1.60% 5.69% 1.47%

0.037879 0.054480 0.000713

1.52% 5.72% 1.44%

0.037909 0.054495 0.000713

1.60% 5.73% 1.47%

K. Hacıefendiog˘lu et al. / Construction and Building Materials 134 (2017) 199–209

206

Table 4 1r Von Mises stresses of the bridge obtained for different statistical distributions and uncertainty ranges of Elastic modulus, Poisson’s ratio and mass density. Node

Deterministic

Normal

Difference

Lognormal

Difference

Gumbel

Difference

COV = 5% 698 2030 2293

10364300.000 5872670.000 4974670.000

10354165.910 4968621.954 5864668.932

0.10% 0.12% 0.14%

10353911.870 4968501.154 5864407.906

0.10% 0.12% 0.14%

10354067.540 4968422.581 5864327.492

0.10% 0.13% 0.14%

COV = 10% 698 2030 2293

10364300.000 5872670.000 4974670.000

10360966.870 4976693.072 5869330.196

0.03% 0.04% 0.06%

10362650.020 4977486.996 5870346.827

0.02% 0.06% 0.04%

10362723.180 4977798.821 5869682.086

0.02% 0.06% 0.05%

COV = 15% 698 2030 2293

10364300.000 5872670.000 4974670.000

10374026.500 4992021.536 5877018.802

0.09% 0.35% 0.07%

10374909.820 4992456.424 5877978.216

0.10% 0.36% 0.09%

10376920.480 4993651.742 5879268.163

0.12% 0.38% 0.11%

COV = 20% 698 2030 2293

10364300.000 5872670.000 4974670.000

10396424.680 5014808.707 5892545.793

0.31% 0.81% 0.34%

10392661.570 5013070.619 5888884.577

0.27% 0.77% 0.28%

10394809.550 5013906.638 5891213.923

0.29% 0.79% 0.32%

any significant variation for a simulation number beyond 400000. Hence, this number is selected for MCS for the remaining part of the paper. (c) Selection of the type of statistical distribution To determine the influence of the type of statistical distribution (normal, lognormal and Gumbel) on the probabilistic response of the bridge, 1r displacements and stresses at the specific node points depending on variability of COV are illustrated in Tables 3

and 4. In these two tables, mean response (displacement and stress) of the bridge obtained by using three statistical distributions of material properties are compared with the same when deterministic values of these material properties are considered. It can be seen that the type distribution has negligible influence on the probabilistic response of the historic bridge. In all cases, bridge response measured at any particular node is almost the same when different statistical distributions of material properties are used. With the increase in COV, the variation in structural response obtained by using probabilistic and deterministic mate-

Fig. 10. Effect of COV on 1r displacements measured at (a) node698, (b) node2030 and (c) node2293.

K. Hacıefendiog˘lu et al. / Construction and Building Materials 134 (2017) 199–209

207

Fig. 11. Effect of COV on 1r Von Mises stresses measured at (a) node 698, (b) node 2030 and (c) node2293.

rial properties increases. However, these variations are very small. Maximum variations of 5.73% in displacement and 0.81% in stress at node 2030 are noticed. Based on these observations, normal distributions are decided to employ to define uncertainties of all material properties in the following part of the analysis.

analysis at the same node. The same level of uncertainty results in 0.50%, 0.01%, 0.06% and 0.81% change in stresses at node 2293 of the bridge from its deterministic response. With all the results presented in Figs. 10 and 11, it can be generally highlight that the uncertainty in Elastic modulus has the biggest impact on the stochastic earthquake response of the bridge.

(d) Effects of the COV and uncertainties in material parameters on the results 5. Conclusions In order to investigate the effect of uncertainties involved in material parameters (i.e., Elastic modulus, Poisson’s ratio and mass density) on the stochastic earthquake response of the historic masonry bridge subjected to random ground motion, 1r displacements and 1r Von Mises stresses recorded at selected nodal points are plotted against the coefficient of variation (COV) in Figs. 10 and 11, respectively. Three specific nodes on the bridge are selected because the responses of the bridge can be easily distinguished according to parametric changes in material parameters. The COVs are selected as 5%, 10%, 15% and 20% for the parameters. As can be seen from these two figures, increase in COV of material parameters causes significant variations in displacements and stresses. While increased uncertainties in Elastic modulus and combination of all material parameters cause higher displacements and stresses of the bridge, increased uncertainties in Poisson’s ratio and density have the opposite or no impact on the same response in all cases except for one. 20% COVs in Elastic modulus, Poisson’s ratio, density and combination of all respectively produce 5.43%, -4.24%, 0.24% and 5.69% change in bridge displacement at node 2030 when compared with the same response obtained from deterministic

This study investigated the influences of uncertainty in material parameters on the stochastic response of a historic masonry bridge subjected to random ground motion. For this purpose, probabilistic analysis of the bridge is carried out with MCS obtained through RSM. The probabilistic responses of the bridge at specific node points are compared with the same response obtained by using deterministic material properties. To demonstrate the accuracy of the response surface functions, results obtained from the functions are compared with the same obtained from the finite element analysis in ANSYS. A very good agreement is observed within these two sets of results. Hence, the approximate functions (response surface functions) obtained through the RSM can appropriately be used to simulate structural response that can otherwise be obtained from ANSYS. As a result, the analyses are carried out more efficiently than the direct MCS considering the time consumption. The effect of the type of statistical distribution (normal, lognormal and Gumbel) on the probabilistic response of the historic bridge were investigated in the paper. The results showed that

208

K. Hacıefendiog˘lu et al. / Construction and Building Materials 134 (2017) 199–209

the type distribution could be neglected for the probabilistic response of the historic bridge. In all cases, bridge response measured at selected particular nodes is almost the same when different statistical distributions of material properties are used. Based on the MCS-RSM results, the increasing the coefficient of variation values of material parameters which are Elastic modulus, Poisson’s Ratio and mass density produce greater effect on the stochastic response of the bridge. While the uncertainties in Elastic modulus and combination of all material parameters generally increase the displacements and stresses of the bridge whereas the uncertainties in Poisson’s ratio and mass density generally decrease the same responses. 20% COVs in Elastic modulus, Poisson’s ratio, density and combination of all respectively produce 5.43%, 4.24%, 0.24% and 5.69% change in bridge displacement at node 2030 when compared with the same response obtained from deterministic analysis at the same node. The same level of uncertainty results in 0.50%, 0.01%, 0.06% and 0.81% change in stresses at node 2293 of the bridge from its deterministic response. This paper found similar results from the other previous studies which indicated that the variability of the Elastic modulus has a noteworthy influence on the response of the historic bridges. However, the variability in mass density can be neglected in a stochastic dynamic analysis of the historic masonry bridge. Consequently, in order to analyze the historic bridge against random ground motion, the influence of uncertain material parameters should be taken into account. Although the results obtained from this study belong to a specific example, the observations here have applicability to many situations.

References [1] J. Dulinska, Evaluation of dynamic characteristics of masonry arch bridges: linking full-scale experiment and fem modeling, Adv. Mater. Res. 133–134 (2010) 605–610, http://dx.doi.org/10.4028/www.scientific.net/AMR.133134.605. [2] S. De Santis, G. De Felice, Evaluation of the seismic response of masonry arch bridges modeled using beam elements with fiber cross section, in: B. Chen, J. Wei (Eds.), ARCH’10-6th Int. Conf. Arch Bridg., College of Civil Engineering, Fuzhou University, Fuzhou, China, 2010, pp. 919–926. [3] A. Galasco, S. Lagomarsino, A. Penna, S. Resemini, Non-linear seismic analysis of masonry structures, 13th WCEE, World Conference of Earthquake Engineering, Vancouver, B.C., Canada (2004) 15. [4] B. Sevim, A. Bayraktar, A.C. Altunisßik, S. Atamtürktür, F. Birinci, Assessment of nonlinear seismic performance of a restored historical arch bridge using ambient vibrations, Nonlinear Dyn. 63 (2010) 755–770, http://dx.doi.org/ 10.1007/s11071-010-9835-y. [5] H. Gonen, M. Dogan, M. Karacasu, H. Ozbasaran, H. Gokdemir, Structural failures in refrofit historical murat masonry arch bridge, Eng. Fail. Anal. 35 (2013) 334–342, http://dx.doi.org/10.1016/j.engfailanal.2013.02.024. [6] G. Milani, P.B. Lourenço, 3D non-linear behavior of masonry arch bridges, Comput. Struct. 110–111 (2012) 133–150, http://dx.doi.org/10.1016/ j.compstruc.2012.07.008. [7] J.M.C. Kishen, M. Asce, A. Ramaswamy, C.S. Manohar, Safety assessment of a masonry arch bridge : field testing and simulations, J. Bridge Eng. 18 (2013) 162–171, http://dx.doi.org/10.1061/(ASCE)BE.1943-5592.0000338. [8] S. Atamturktur, Calibration Under Uncertainty for Finite Element Models of Masonry Monuments, The Pennsylvania State University, 2009. [9] B. Sevim, S. Atamturktur, A.C. Altunisik, A. Bayraktar, Ambient vibration testing and seismic behavior of historical arch bridges under near and far fault ground motions, Bull. Earthquake Eng. 14 (2016) 241–259. [10] A. Bayraktar, T. Türker, A.C. Altunisik, Experimental frequencies and damping ratios for historical masonry arch bridges, Constr. Build. Mater. 75 (2015) 234– 241. [11] A.C. Altunisik, A. Bayraktar, A.F. Genç, Determination of the restoration effect on the structural behavior of masonry arch bridges, Smart Struct. Syst. 16 (2015) 101–139. [12] L. Pelà, A. Aprile, A. Benedetti, Comparison of seismic assessment procedures for masonry arch bridges, Constr. Build. Mater. 38 (2013) 381–394. [13] A. Brencich, D. Sabia, Experimental identification of a multi-span masonry bridge: the Tanaro Bridge, Constr. Build. Mater. 22 (2008) 2087–2099. [14] F. Ceroni, M. Pecce, S. Sica, A. Garofano, Assessment of seismic vulnerability of a historical masonry building, Buildings 2 (2012) 332–358, http://dx.doi.org/ 10.3390/buildings2030332.

[15] P. Hradil, J. Zˇák, D. Novák, M. Lavicky´, Stochastic analysis of historical masonry structures, in: P.B. Lourenço, P. Roca (Eds.), Hist. Constr., November 1, Guimarães, Portugal, 2001, pp. 647–654. [16] F. Parisi, N. Augenti, Uncertainty in seismic capacity of masonry buildings, Buildings 2 (2012) 218–230, http://dx.doi.org/10.3390/ buildings2030218. [17] L. Schueremans, D. Van Gemert, Probability density functions for masonry material parameters – a way to go ?, in: P.B. Lourenço, P. Roca, C. Modena, S. Agrawal (Eds.), V Int. Conf. Struct. Anal. Hist. Constr. Possibilities Numer. Exp. Tech. Ed., New Delhi, 2006, pp. 921–928. doi:ISBN 972-8692-27-7. [18] G. Falsone, M. Lombardo, Stochastic representation of the mechanical properties of irregular masonry structures, Int. J. Solids Struct. 44 (2007) 8600–8612, http://dx.doi.org/10.1016/j.ijsolstr.2007.06.030. [19] M. Rota, a. Penna, G. Magenes, A methodology for deriving analytical fragility curves for masonry buildings based on stochastic nonlinear analyses, Eng. Struct. 32 (2010) 1312–1323, http://dx.doi.org/10.1016/j. engstruct.2010.01.009. [20] F.J. Suarez, R. Bravo, Historical and probabilistic structural analysis of the Royal ditch aqueduct in the Alhambra (Granada), J. Cult. Herit. 15 (2014) 499–510, http://dx.doi.org/10.1016/j.culher.2013.11.010. [21] A. Moropoulou, A. Bakolas, P. Moundoulas, E. Aggelakopoulou, S. Anagnostopoulou, Optimization of compatible restoration mortars for the earthquake protection of Hagia Sophia, J. Cult. Herit. 14 (2013) e147–e152, http://dx.doi.org/10.1016/j.culher.2013.01.008. [22] K. Haciefendiog˘lu, Effect of material uncertainty on stochastic response of dams, Proc. ICE Geotech. Eng. 163 (2010) 83–89, http://dx.doi.org/ 10.1680/geng.2010.163.2.83. [23] K. Haciefendioglu, F. Birinci, Stochastic dynamic response of masonry minarets subjected to random blast and earthquake-induced ground motions, Struct. Des. Tall Spec. Build. 678 (2011) 669–678, http://dx.doi.org/10.1002/tal. [24] B.H. Jensen, W.D. Iwan, Response o f systems with uncertain parameters to stochastic excitation, J. Eng. Mech. 118 (1992) 1012–1025. [25] Z. Lei, C. Qiu, Neumann dynamic stochastic finite element method of vibration for structures with stochastic parameters to random excitation, Comput. Struct. 77 (2000) 651–657. [26] W. Gao, J.J. Chen, J. Ma, Z.T. Liang, Dynamic Response Analysis of Stochastic Frame Structures, 42, 2004. [27] W. Gao, J. Chen, M. Cui, Y. Cheng, Dynamic response analysis of linear stochastic truss structures under stationary random excitation, J. Sound Vib. 281 (2005) 311–321, http://dx.doi.org/10.1016/j.jsv.2004.01.014. [28] W. Gao, Stochastically optimal active control of a smart truss structure under stationary random excitation, J. Sound Vib. 290 (2006) 1256–1268, http://dx. doi.org/10.1016/j.jsv.2005.05.019. [29] A. Chaudhuri, S. Chakraborty, Reliability of linear structures with parameter uncertainty under non-stationary earthquake, Struct. Saf. 28 (2006) 231–246, http://dx.doi.org/10.1016/j.strusafe.2005.07.001. [30] W. Gao, Random seismic response analysis of truss structures with uncertain parameters, Eng. Struct. 29 (2007) 1487–1498, http://dx.doi.org/10.1016/j. engstruct.2006.08.025. [31] K. Hacıefendiog˘lu, A. Bayraktar, H.B. Basßag˘a, Estimation of stochastic nonlinear dynamic response of rock-fill dams with uncertain material parameters for non-stationary random seismic excitation, Nonlinear Dyn. 61 (2009) 43–55, http://dx.doi.org/10.1007/s11071-009-9630-9. [32] J. Dai, W. Gao, N. Zhang, N. Liu, Seismic random vibration analysis of shear beams with random structural parameters, J. Mech. Sci. Technol. 24 (2010) 497–504, http://dx.doi.org/10.1007/s12206-009-1210-x. [33] Y. Zhao, Y.H. Zhang, J.H. Lin, W.P. Howson, F.W. Williams, Analysis of stationary random responses for non-parametric probabilistic systems, Shock Vibr. 17 (2010) 305–315, http://dx.doi.org/10.3233/SAV-2010-0514. [34] K. Hacıefendiog˘lu, Seasonally frozen soil’s effect on stochastic response of masonry minaret–soil interaction systems to random seismic excitation, Cold Reg. Sci. Technol. 60 (2010) 66–74, http://dx.doi.org/10.1016/ j.coldregions.2009.08.007. [35] M.E. Kartal, H.B. Basßag˘a, A. Bayraktar, Probabilistic nonlinear analysis of CFR dams by MCS using response surface method, Appl. Math. Model. 35 (2011) 2752–2770, http://dx.doi.org/10.1016/j.apm.2010.12.003. [36] ANSYS, Swanson Analysis System, Ansys Inc, Canonsburg, PA, 2013. [37] Y.K. Lin, G.Q. Cai, Probabilistic Structural Dynamics, McGraw-Hill, New York, 2004, p. 562. [38] C. Yang, Random vibration of structures, Earthq. Eng. Struct. Dyn., John Wiley and Sons Inc, New York, 1986, http://dx.doi.org/10.1002/eqe.4290140509. 817-817. [39] G.D. Manolis, P.K. Koliopoulos, Stochastic structural dynamics in earthquake engineering, WIT Press; Hard/CD-ROM Edition, Southampton (2001) 296. (accessed March 24, 2014). [40] R.H. Myers, Response surface methodology-current status and future directions, J. Qual. Technol. 31 (1999) 30–44. [41] G. Fishman, Monte Carlo: concepts, algorithms, and applications, Springer Series in Operations Research and Financial Engineering (2003). (accessed March 22, 2014). [42] I.M. Sobol, A Primer for the Monte Carlo Method, first ed., CRC Press, 1994. pp. 128. [43] G. Samsun, Ministry of Culture and Tourism, Samsun, 2010.

K. Hacıefendiog˘lu et al. / Construction and Building Materials 134 (2017) 199–209 [44] G. Frunzio, M. Monaco, A. Gesualdo, 3D F.E.M. analysis of a Roman arch bridge, in: P.B. Lourenço, P. Roca (Eds.), Hist. Constr., Guimarães, Portugal, 2001, pp. 591–598. [45] N. Diamanti, A. Giannopoulos, M.C. Forde, Numerical modelling and experimental verification of GPR to investigate ring separation in brick masonry arch bridges, NDT E Int. 41 (2008) 354–363, http://dx.doi.org/ 10.1016/j.ndteint.2008.01.006. [46] K. Soyluk, A.A. Dumanoglu, Comparison of asynchronous and stochastic dynamic responses of a cable-stayed bridge, Eng. Struct. 22 (2000) 435–445, http://dx.doi.org/10.1016/S0141-0296(98)00126-6.

209

[47] R.W. Clough, J. Penzien, Dynamics of Structures, second ed., McGraw-Hill Education, Singapore, 1993. (accessed April 04, 2014). [48] K. Kanai, Semi-empirical formula for the seismic characteristics of the ground, Bull. Earthquake Res. Inst. 35 (1957) 307–325. [49] H. Tajimi, A statistical method for determining the maximum response of a building structure during an earthquake, Proc. 2nd World Conf. Earthq. Eng., Tokyo and Kyoto, Japan (1960) 781–797.