Modeling the Natural Hearbeat of Reinforced

7/30/2019 Modeling the Natural Hearbeat of Reinforced

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MODELING THE NATURAL HEARBEAT OF REINFORCED

CONCRETE BUILDINGS IN METRO MANILA

Andres Winston C. Oreta

ABSTRACT: The natural heartbeat or period of vibration is an important

dynamic property of a building since it characterises the behavior and

performance of the structure to external forces. An estimate of the fundamental

period of a building is useful to a structural engineer, civil engineer or urban

disaster manager. The present study illustrates the use of neural networks in

estimating the period of reinforced concrete (RC) buildings. Data from ambient

vibration measurements conducted in medium-rise and high-rise buildings in

Metro Manila were used to train a neural network. A model for estimating the

period of RC moment-resisting space frame buildings and RC dual buildings,

using global building parameters - type of structural system and height of the

building - was developed and its performance was evaluated and compared with

existing empirical formulas.

KEYWORDS: period, building, reinforced concrete, ambient vibration, neural network

1. INTRODUCTION

The health of a human heart can be monitored by an electrocardiogram (ECG or

EKG).

Through the tracings of an ECG (Figure 1), the physician can identify important

parameters of the heart such as the heartbeat rate and other heart rhythms. Like

the

human being, a building also has its own rhythms. A building vibrates under

ambient

conditions or under severe

loading conditions such as

earthquakes. The building


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vibration can also be monitored

similar to an ECG and important

parameter may be extracted from

the these measurements. One

important parameter which can

be obtained from vibration

measurements of a building is

the period or the natural

Figure 1. An ECG Tracing heartbeat The fundamental period is a dynamic property

of a building which characterises the

behavior and performance of the structure to external forces. The natural period,

sometimes referred to as the natural heartbeat (Pacheco 1999) of the building,

can be

determined experimentally from vibration data recorded in instrumented buildings.

Depending on the type of vibration on the building, the natural period of a building

may

be classified as either (a) period at ambient condition; or (b) period at large-

amplitude

motions or seismic condition.

At ambient condition, the vibrations of the building are usually very small in

amplitude

and quite random in waveform. The vibrations are typically induced at the base or

foundation and some may be caused by wind which acts at the surrounding walls.

Pacheco (2001) describes that the underlying principle used to evaluate the

natural

period of a building using ambient vibration testing is that the prevailing ground-

borne

and wind-induced excitations in the building are composed of an almost infinite

number


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of harmonics with different periods of vibration, and every harmonic component

whose

period corresponds to a natural period of the building is amplified in the building

response, due to resonance. Ambient vibration tests generally have to be repeated

several times, preferably at different times of the day, to check the consistency of

the

results and to avoid recording unintentionally any occasional dominant forced

excitations. The natural period at ambient condition can not include the effect of

partial

cracking in reinforced concrete which usually occurs under large-amplitude

vibrations

during an earthquake.

The period at large-amplitude vibrations, on the other hand, are obtained in

buildings that

are shaken strongly but not deformed into the inelastic range (Goel and Chopra

1997)

during past earthquakes. Such data are scarce because relatively few buildings

have

accelerometers permanently installed and earthquakes are not that frequent. For a

building with the same height, the effective natural period of a building during an

earthquake is usually longer than at ambient condition due to the reduction of the

stiffness of the building caused by partial cracking of some structural elements such

as

beams and columns and cracking of non-structural members such as in-filled walls.

The natural period can be estimated from vibration time history by converting the

data

into the frequency domain as a spectrum. In most cases whether under ambient or

seismic

condition, a predominant peak which corresponds to a natural period of the building

can

be identified from the spectrum of the vibration history of a point in a building.


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The fundamental period of a building is useful to a structural engineer, civil

engineer or

urban disaster manager. The natural period is an important parameter in the

computation

of the design base shear and lateral forces due to earthquakes. Building codes

recommend

methods for estimating the period through an empirical equation or by a refined

dynamic

modeling and analysis (NSCP 2001). Improved formulas for possible code

applications

to estimate the fundamental periods of RC and steel moment-resisting frame

buildings

using data from more recent earthquakes in the Unites States have been developed

(Goel

and Chopra 1997). Alternative period formulas for estimating seismic displacements

have

also been recommended (Chopra and Goel 2003). The building period may be used

in

classifying a building population in an urban metropolis for the purpose of

earthquake disaster preparedness and mitigation. For example, the natural period

of buildings are

compared with a similar survey of ground natural periods to identify buildings with

periods very close to the ground which may result to resonance. The natural period

of a

building may be used also as a parameter in developing seismic capacity curves

that can

be used in building damage assessments based on a scenario earthquake in

densely

populated cities (Pacheco, Tanzo and Peckley 2003).

The present study introduces the application of neural networks in estimating the

period

of reinforced concrete (RC) buildings. Data from ambient vibration measurements


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conducted in medium-rise and high-rise buildings in Metro Manila were used to train

a

neural network. A model for estimating the period of reinforced concrete moment

resisting space frame (RC-MRSF) and reinforced concrete dual (RC-Dual) buildings,

using global building parameters : (a) type of structural system; and (b) height of

the

building, was developed and its performance was evaluated.

2. EMPIRICAL AND CODE FORMULAS

Empirical models for predicting the natural period of buildings are usually developed

by

deriving empirical equations using regression analysis. In this method, the form of

the

regression equation is assumed, e.g., T = H

, where and are determined by curve

fitting using the least squares method.

2.1 Code Period Formula

The National Structural Code of the Philippines (NSCP 2001), which has its reference

the

Uniform Building Code 1997, uses an empirical equation derived from the actual

behavior of buildings in California in past earthquakes. The equation, here referred

to as

Method A, has the following form:

T = Ct

H 0.75 (1)

where:

Ct

= 0.0853 for steel moment-resisting frames

Ct

= 0.0731 for reinforced concrete moment-resisting frames and eccentrically


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braced frames

Ct

= 0.0488 for all other buildings

H = height above the base to level n

2.2 Regression Equation from Ambient Vibration Survey

In a report on a survey of vibration characteristics of multistory buildings in Metro

Manila (NDCC-PHIVOLCS-ASEP Project Team 2001), regression equations, similar to

the NSCP equation, were derived using ambient vibration data of RC-MRSF buildings

and RC-Dual buildings. The coefficients, Ct , derived are as follows:

Ct

= 0.045 for RC-MRSF buildings, six stories and above

Ct

= 0.051 for RC-Dual Buildings

A comparison of the code and survey coefficients shows that the code equation will

result

to longer periods than the survey equation, particularly for RC-MRSF buildings.

Since the code equation coefficients were derived based on data of buildings whichwere

shaken by past earthquakes resulting to cracks in RC members reducing the

stiffness,

the code periods are usually longer. Surprisingly, the coefficient of the survey

equation

for RC-Dual buildings is larger than the code coefficient for other buildings. The

applicability of the code coefficient for other buildings may have limitations; that

is

why the code provides an alternative value for Ct

for structures with concrete or masonry

shear walls where the area of the shear wall in the first story is a parameter.

Lumping


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buildings not belonging to steel moment-resisting frames, reinforced concrete

momentresisting frames and eccentrically braced frames to one group as other

buildings may

not be practical.

3. AMBIENT VIBRATION SURVEY OF BUILDINGS IN METRO MANILA

Seventy two buildings, majority of which have reinforced concrete moment resisting

space frame (RC-MRSF) and reinforced concrete dual (RC-Dual) structural systems,

were surveyed by the Philippine Institute of Volcanology and Seismology

(PHIVOLCS),

in collaboration with Japanese research groups from Kanto Gakuin University led by

Prof. Norio Abeki and Tokyo Institute of Technology lead by Prof. Saburoh

Midorikawa

from 1998 to 1999 (NDCC-PHIVOLCS-ASEP Project Team 2001). Small-amplitude

vibrations of medium-rise to high-rise buildings in Metro Manila under random,

ambient

conditions were measured using portable instruments installed at various buildings.

The

natural periods of vibration of each building along either or both the longitudinal

and

lateral direction of the floor plan of the building were extracted from spectral

analysis of

the recorded vibration histories. Although different survey equipment and

methodologies

were employed Abeki measured displacement and neglected torsional effects,

while

Midorikawa measured velocity and considered torsional vibrations both methods

yielded consistent fundamental natural periods, with similar trends, taller buildings

having longer periods. However, the longitudinal and lateral periods of the same

building

can be very different. The results of the survey of buildings for this project were


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envisioned to be used in assessing the general trend of seismic vulnerability of

buildings

around Metro Manila.

4. NEURAL NETWORK MODELING

4.1 Neural Network Architecture and Implementation

An ANN is a collection of simple processing units or neurons connected through

links

called connections. The topology or architecture of a three-layer feed-forward

neural

network may be presented schematically, as in Figure 2. The neural network is

represented in the form of a directed graph, where the nodes represent the neuron

or

processing unit, the arcs represent the connections with the normal direction of

signal

flow is from left to right. The processing units may be grouped into layers of input,

hidden and output neurons. The neural network in the figure consists of two input

neurons, two hidden neurons and one output neuron. The main tasks of neurons are

to

receive input from its neighboring units which provide incoming activations,

compute an

output, and send that output to its neighbors receiving its output. The strength of

the

connections among the processing units is provided by a set of weights that affect

the magnitude of the input that will be received by the neighboring units. These set

of

weights are determined by presenting the network a set of training data and using a

training algorithm such as the back-propagation neural network (BPNN) algorithm

(see

Freeman and Skapura 1991), the weights are updated until a stable set of weights

are

obtained.


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4.2 Experimental Data

The present study uses the results of the survey for RC-MRSF buildings and RC-Dual

buildings. The information available for each building in the report by

NDCCPHIVOLCS-ASEP Project Team (2001) are general location, structural system,

number

of floors, height of building and the fundamental natural periods in the lateral and

longitudinal directions. The building height was estimated from the number of floor

levels multiplied by an average floor height which is assumed here as equal to 3.5

m. The

estimated periods for the lateral and longitudinal vibrations are not always the

same since

the stiffness of the building in any direction is affected by various factors such as

dimension of building, existence of shear walls or CHB walls. The difference

between

the longitudinal and lateral periods ranges from 0.01 s to 0.85 s. Since no

information

about the lateral and longitudinal dimensions of the buildings is available, then the

input

parameters used in the ANN modeling to estimate the period are limited to thefollowing

available data; (a) type of structural system; and (b) height of the building. It is

assumed

that the building dimension is not a

significant factor in the period of

the building. This may be true for

buildings which are fairly regular

in plan and for those with rigid

floor diaphragms. To eliminate the

possible effect of building

dimension in the estimation of the


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period, the data for buildings with

the difference between

longitudinal and lateral periods

greater than 0.2 s were not

included. Hence the number of

data of buildings used was reduced

from 72 to 47.


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0

2

4

6

8

10

12

5 15 25 35 45 55 65 75 85 95 105

RC-Dual RC-MRSF

COUNT

HEIGHT (m)

Figure. 3. Histogram of Building Data


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Bldg. Height

Bldg. Type

Bldg. Period

Hidden

Layer

Input

Layer

Output

Layer

Figure 2. A three-layer feed-forward neural network Using the lateral andlongitudinal periods separately of the 47 buildings results to 94 sets

of data with 68 for RC-MRSF buildings and 26 for RC-Dual buildings. A histogram of

the building data used with respect to height is shown in Figure 3. The RC-MRSF

buildings were generally 70 m or about 20 stories or lower, while the RC-Dual

systems

ranges from 60 m to 110 m in height or about 20 to 30 stories. The data were

divided

randomly into two subsets a training set of 70 data and a testing set of 24 data.

4.3 Neural Network Architecture

A three-layered feed-forward neural network model (Figure 2) is used. The neural

network considered in the study consists of two inputs: (a) height of the building;

and (b)

type of structural system. The building height is in meters, while the type of

structural

system has a value of either 1 for RC-MRSF or 2 for RC-Dual. The output is the

natural

period at ambient conditions in seconds. The number of hidden nodes is varied and

the

simplest model which is found to be acceptable is selected.


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4.4 Network Data Preparation

The raw experimental data need to be pre-processed or normalized by scaling to

improve the training of the neural network. To avoid the slow rate of learning near

the

end points specifically of the output range due to the property of the sigmoid

function

which is asymptotic to values 0 and 1, the input and output data were scaled

between the

interval 0.1 and 0.9. The linear scaling equation : y =( 0.8 / ) x + ( 0.9 0.8 xmax /

)

was used in this study for a variable limited to minimum (x

min) and maximum (x

max )

values. The minimum and maximum values of the building height are 7 m and 119

m,

respectively while the period varies from minimum of 0.1 s to a maximum of 2.5 s.

The

numerical values, 1 and 2, corresponding to the type of structural system were also

normalized to 0.1 (RC-MRSF) and 0.9 (RC-Dual).

4.5 A Neural Network Model for Predicting Natural Periods of RC Buildings

4.5.1 Selecting the Model

ANN simulations were

conducted by varying the

number of hidden layer

nodes (2 to 4 nodes) and the

BPNN learning algorithm

parameters such as the

learning parameter (0.5 1.0), momentum parameter (0.005 0.05) and number of


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epochs or cycles (1,000 4,000 cycles). A noise of 0.01% was also added to the

data.

The different networks were compared with respect to the error metrics such as

mean

average error (MAE), root mean square error (RMSE), average percent error and the

Pearson product moment correlation coefficient, R. Based on the comparison of

networks with different number of hidden layer nodes, the neural network with two

hidden layer nodes, which will be referred to as the T221 model, is the simplest

model

and has the best MAE, RMSE and R for the test data of the combined RC-MRSF and

RC-Dual buildings. Table 1 presents the connection weights for this model.

Table 1. Connection Weights of T221 Model

H idden Ou tpu t

No des Bldg . He ig ht B ldg . T yp e N ode

1 -4.451696 1.074989 -5.239752

2 -3.587910 4.178850 1.130453

Input NodesUsing eqn. (2), the neural network output, y, can be computed using

normalized inputs,

xi

, the computed weights, wji (connection weights between input nodes and hidden

layer

nodes) and wkj(connection weights between hidden layer nodes and output node).

(2)

The activation function, f [ ], in this case, is the sigmoid function f(z) = 1/ (1 + e-z ).

The

output of the network is a value between 0.1 and 0.9 and can be converted to

period in

seconds using the linear scaling equation in Sect. 4.4.

4.5.2 Performance of the T221 Model


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The performance of the T221 model in predicting the natural period at ambient

conditions is evaluated with respect to the error metrics as shown in Table 2. The

error

metrics, MAE, RMSE and percent error are shown for various cases such as

predictions

for RC-MRSF buildings or RC-Dual Buildings for the training data only, test data only

or

combined data. The mean percentage error for RC-MRSF buildings and RC-Dual

buildings for the combined training and test data are about 39% and 23%,

respectively.

The error metrics for the combined data for RC-MRSF and RC-Dual buildings for the

training data only, test data only or combined data are also shown in the last

column of

Table 2.

Table 2. Summary of Prediction Errors of T221 Model

RC-MRSF Bldg Data RC-DUAL Bldg. Data Combined Data

Training Data Only

MAE (s) 0.19 0.30 0.22

RMSE (s) 0.22 0.33 0.26

Average Percent Error 37.43 23.83 33.74

Percent < 30% Error 60.78 68.42 62.86

Test Data Only

MAE (s) 0.15 0.35 0.21

RMSE (s) 0.17 0.39 0.26


Percent < 30% Error 58.82 85.71 66.67

Combined Training and Test Data

MAE (s) 0.18 0.31 0.22


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RMSE (s) 0.21 0.35 0.27


Percent < 30% Error 60.30 73.08 63.83

Figure 4 shows the histogram of the percentage error of the predictions of the T221

model for RC-MRSF buildings and RC-Dual buildings. More than 60% of the

predictions

for RC-MRSF buildings have a percentage error less than 30%; while more than 70%

of

the predictions for RC-Dual buildings have a percentage error less than 30%.

( ( )) i

x

ji f w kj y = f

w

Figure 5 and Figure 6 show the comparison of the predictions of the T221 model

and

the survey regression equation with respect to the experimental values for RC-MRSF

and

RC-Dual buildings, respectively. The Pearson product moment correlationcoefficients,

R, is a measure of the linear relationship between the predicted and experimental

data

sets an R value equal to 1.0 means the predicted and experimental values are

equal. The

R values of the T221 model are slightly better than the R values for the survey

equations

(NDCC-PHIVOLCS-ASEP Project Team 2001). The R value of the T221 model for

RCMRSF buildings is about 0.7849 for the combined training and test data, as

compared to

R = 0.7793 for the survey regression equation. On the other hand, the R value of

the

T221 model for RC-Dual buildings is about 0.5023 for the combined training and test


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data, as compared to R = 0.4871 for the survey regression equation. The R value

for the

combined training and test data for both RC-MRSF and RC-Dual buildings is about

0.85

for the T221 model.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

RC-MRSF

T221 (Training Data)

T221 (Test Data)

Predicted Period (s)

Experimental Period (s)

T221 (R=0.7849)

0

0.2

0.4

0.6

0.8

1

1.2


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1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

RC-MRSF

Survey Regression Eqn. (R=0.7793)



(a) (b)

Figure 5. Predictions for RC-MRSF Buildings. (a) T221 model; (b) Survey Regression

Eqn

0

5

10


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15

20

0 20 40 60 80 100 120 140 160 180

RC-MRSF

Training Data

Test Data

Count

Percent Error (%)

T221 Model

0 2

4

6

8

10

0 10 20 30 40 50 60 70 80 90 100

RC-DUAL

Training Data

Test Data

Count

Percent Error (%)

T221 Model

Figure 4. Percentage Error of T221 Neural Network Predictions The performance of

the T221 model with respect to the parameter, building height, is


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compared with respect to the empirical models derived using regression analysis.

By

setting the input for building type as constant, either 1 or 2, and varying the input

for

building height, the neural network predictions of the natural period at ambient

conditions as a function of height can be derived as shown in Figure 7. Shown in the

figure also are the curves corresponding to the survey regression equation and the

code

equation. Plotted also in the figure are experimental periods obtained from the

survey.

The trend of the T221 predictions when compared to the regression equations is the

same

taller buildings have longer periods however there is a gradual decrease in the

slope

of the T221 curve as the height increases. The predictions of the T221 model are

also

slightly larger than the survey regression equation for most RC-MRSF buildings and

for

heights less than about 95 m for RC-Dual buildings..

0

0.5

1

1.5

2

0 10 20 30 40 50 60 70 80

RC-MRSF

NSCP (2001)

Survey Regression Eqn.

T221


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Survey Expt. D ata Natural Period (s)

Building Height (m)

0

0.5

1

1.5

2

50 60 70 80 90 100 110 120 130

RC-DUAL

NSCP (2001)

Survey Regression Eqn.

T221

Survey Expt. Data

Natural Period (s)

Building Height (m)

(a) (b)

Figure 7. Natural Period with Respect to Building Height

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5

RC-DUAL

Survey Regression Eqn. (R=0.4871) Predicted Period (s)


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0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5

RC-DUAL

T221 (Training Data)

T221 (Test Data)


Experim ental Period (s)

T221 (R=0.5023)

(a) (b)

Figure 6 Predictions for RC-Dual Buildings. (a) T221 model; (b) Survey Regression

Eqn. As expected, the code predictions for RC-MRSF buildings are greater than the

neural

network model and the survey regression equation since the code equation was

derived

from large-amplitude vibrations of buildings in the US during past earthquakes. The

effective natural periods of buildings during earthquakes become longer due to the

reduction of the stiffness of the building caused by partial cracking of some

structural

elements such as beams and columns and cracking of non-structural members such

as infilled walls. On the other hand, a comparison of the predictions for RC-Dual

buildings

using the neural network model, survey regression equation and the code equation

shows


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that the curves are nearly coincident, although the T221 predictions are slight

larger at

heights less than about 95 m. How come the periods at ambient condition and the

code

periods which are based on large-amplitude vibrations are almost similar needs

further

investigation. However, it must be noted here that the code equation used in the

Figure

7(b) is the one specified for all other buildings. Obviously the code equation for

all

other buildings has limitations and its applicability to RC-Dual buildings needs to be

clarified. Incidentally, the code provides an alternative value for Ct

for structures with

concrete or masonry shear walls where the area of the shear wall in the first story is

a

parameter.

Figure 8 shows the curves of the seismic coefficient, C, using base shear equations

of the

NSCP (2001) and the corresponding periods predicted using the T221 model and the

code equation for an RC-MRSF building with specified base shear parameters. As

expected, the seismic coefficients using ambient vibration periods (T221) are larger

than

the NSCP since the predicted periods are smaller when compared to the code

periods.

Pacheco (2001) suggested, for tentative comparison between code periods and

ambient

vibration periods, multiplying the ambient vibration period of about 1.3 to account

for the

increase in period due to the reduction of stiffness when partial cracking occurs in

RCMRSF under seismic conditions. The seismic coefficient for the adjusted period is

shown


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as the middle curve in Figure 8. The base shear computation using the adjusted

ambient

vibration periods of the T221 model resulted to values less than the ambient

vibration

periods but slightly larger than the base shear using code periods; hence larger

lateral

forces and a conservative design is obtained. Base shear computation using the

adjusted

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 10 20 30 40 50 60 70 80

RC-MRSF

NSCP (2001)

T221

1.3 * T221

Base Shear Parameters

Z=0.4, I=1.0, R=3.5,

Na=1.0, Nv=1.0, Ca=0.44, Cv=0.64 Seismic Coefficient (C=V/W)

Building Height (m)

Figure 8. Seismic Coefficient with Respect to Building Height ambient vibration

periods may be considered as an upper bound for RC-MRSF

buildings in this example.


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5. CONCLUSIONS

An alternative approach in estimating the natural period of RC buildings at ambient

conditions was presented using ANNs. One advantage of neural network modeling is

that

there is no need to know a priori the functional relationship among the various

variables

involved, unlike in regression analysis. The ANNs automatically construct the

relationships for a given network architecture as experimental data are processed

through

a learning algorithm. The present study illustrates the capability of neural networks

in

estimating the period of reinforced concrete (RC) buildings using data from ambient

vibration measurements conducted in medium-rise and high-rise buildings in Metro

Manila. Because of the limited information of the vibration data, only two input

parameters were considered - type of structural system and height of the building -

in

developing an ANN model for estimating the period of RC moment-resisting space

frame

(MRSF) buildings and RC dual buildings. If more information become available from

vibration survey of buildings such as building dimension, length of walls, type of

soil,

etc., increasing the number of input parameters in an ANN model for predicting the

natural period of a building can be done easily by simply modifying the neural

network

architecture.

REFERENCES

Chopra, A.K. and Goel, R. K. (2003). Building period formulas for estimating

seismic

displacements, http://ceenve.calpdy.edu.goel/research/period

%20formula/eeri_period.pdf


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Freeman, J. A. and Skapura, D.M. (1991). Neural Networks: Algorithms,

Applications, and

Programming Techniques, AddisonWesley, Reading, Mass. , USA, pp. 89-125

Goel, R. K. and Chopra A. K. (1997). Period formulas for moment-resisting frame

buildings. J.

Structural Engineering, American Society of Civil Engineers, 123 (11), 1454-1481,

New York.

National Structural Code of the Philippines (NSCP 2001) , Volume 1. 5th ed.,

Association of

Structural Engineers of the Philippines (ASEP), Manila

NDCC-PHIVOLCS-ASEP Project Team (2001). Survey of vibration characteristics of

about 100

multistory buildings, February 26, 2001, Report submitted to PHIVOLCS, Quezon

City

Pacheco, B. M. (1999). Why every building needs an electrocardiogram, Phil.

Civil Engineering,

Phil. Institute of Civil Engineers, Vol. 2, July-Dec. 1999, pp. 66-74, Quezon City

Pacheco, B. M. (2001). The natural heartbeat of 100 buildings in Metro Manila,

Proc. JapanPhilippine Workshop on Safety and Stability of Infrastructures against

Environmental Impacts,

University of the Philippines, Quezon City , Sept. 10-13, pp. 33-46

Pacheco, B. M., Tanzo, W. T. and Peckley, Jr. D.C.(2003). Survey of experts

judgement on seismic

capacity of selected building types in Metro Manila using the Delphi Technique,

Proc. 10th ASEP

International Convention -Art & Science of Structural Engineering, Vol. 2, pp. 593-

620, Quezon

City

ACKNOWLEDGEMENT

The author expresses his sincere thanks to the University Research Coordination

Office (URCO) of De La


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Salle University for supporting this research. Thanks also to PHIVOLCS for sharing

the ambient vibration

data of the project: Survey of Vibration Characteristics of about 100 Multistory

Buildings.

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Modeling the Natural Hearbeat of Reinforced