Presenting a quick method for estimation of MRF dynamic ... · no certainty of the correctness of the data entry or matching ... are equivalent to m of the single frame ... ¦k u

INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING

Volume 6, No 3, 2016

© Copyright by the authors - Licensee IPA- Under Creative Commons license 3.0

Research article ISSN 0976 – 4399

Received on December, 2015 Published on February 2016 195

Presenting a quick method for estimation of MRF dynamic characteristics

using 3D modeling

Peyman Shadman Heidari1, Ali Golara2

1- F. A. Department of Civil Engineering, East Tehran Branch, Islamic Azad University,

Tehran, IRAN

2- Responsible with Strategic Planning Studies at National Iranian Gas Company

(NIGC).

[email protected]

doi:10.6088/ijcser.6018

ABSTRACT

The current research presents a quick method for estimating the lateral stiffness and torsional

stiffness of 3D MRF (moment-resistant frame) structures, considering irregular moment

frames. This study also provides a method for calculating story displacement and rotation and

natural frequencies with respect to different lateral load patterns. This study proposes a

method for calculating lateral and torsional stiffness for each frame in two directions, and

then converting the stiffness of all frames to one frame to obtain the deformation and natural

frequency for two directions. The basic idea of the proposed innovative method was

developed through the force method to obtain the lateral deformation and stiffness of 2D

building structures. Then, the mentioned procedure was expanded into 3D building structures.

Some examples have been made to compare the latter method with linear analysis. The

results showed that the suggested method can capture 3D dynamic characteristics with

accuracy compared with linear analysis.

Keywords: Lateral stiffness, torsional stiffness, natural frequency, force method, 3D building

structure.

1. Introduction

All computer programs need initial values for calculating the cross-section and material

properties of a building’s structural elements. These initial values are obtained through some

preliminarily calculations. Cross-section and material properties will be the values most

referred for optimizing in analysis and design procedures. On the other hand, there is usually

no certainty of the correctness of the data entry or matching of the data entered by the user

with reality, especially in the case of inexperienced engineers working with complicated

software. In this case, a control or final checking tool is very useful or even necessary. Most

methods used to analyze building frames, such as the cantilever, portal, factor, Spurr,

Bowman, and Witmer methods (Utku S, 1991), are limited to only regular geometric 2D

moment frames. They cannot calculate the lateral displacement or the lateral and torsional

stiffness of the 3D frame systems in the building.

The Kan and PCA methods have been presented and used for many years for regular

moment 2D frames with shear walls. However, these methods are not matched with 3D

frames and they can result in errors of even more than 50% in the calculation of lateral

displacements. Moreover, they have employed some complex formulations that are very

time-consuming when they are calculated by hand. Grigorian presented a method for

calculating the lateral response of regular 2D frames based on an analogy between the frame

Presenting a quick method for estimation of MRF dynamic characteristics using 3D modeling

Peyman Shadman Heidari, Ali Golara

International Journal of Civil and Structural Engineering 196

Volume 6 Issue 3 2016

and discretized Timoshenko beam-columns with similar boundaries. Implementation of this

method, however, is not easy enough for use in calculating the 3D irregular frame response

with respect to lateral load. Moreover, the calculation of lateral load pattern distribution to

determine lateral displacement is more necessary and important when there are eccentricities

between the mass center and the rigid center of the 3D frame system. The main aim of the

proposed method in this paper is to deal with irregular 3D frames.

Miranda and Taghavi [3] used the HS73 model to acquire the approximate structural

behavior up to 3 modes. As a follow-up study, Miranda and Akkar (Miranda E., 2006)

extended the use of HS73 (Heidebrecht A.C, 1973) to compute generalized drift spectra with

higher mode effects. In this study, the continuum model is also used to estimate the

fundamental periods of high-rise buildings More recently, Gengshu et al., studied second

order and buckling effects on buildings through the closed form solutions of continuous

systems. Eroglu and Akkar proposed lateral stiffness estimation in frames and its

implementation in continuum models for linear and nonlinear static analyses. Mohsen

Shahrouzi M 2004, suggested a quick method for estimating the eigenvalues of multistory

buildings. His proposed method was based on the developed mapping between the chain

structure and an equivalent beam model; thus, it led to a dimensionless frequency equation.

The procedure presented in this study introduces a new definition for plan irregular structure

but regular in height, especially for the mass/stiffness of the last story with respect to the

others. Hosseini and Imagh-e-naiini presented a quick method for estimating the lateral

stiffness of building structures, including regular and irregular moments and braced 2D

frames. The present paper extends the method of their study from 2D to 3D analytical

modeling. By using proposed method, the dynamic specification of a 3D model of moment

frame, including the lateral stiffness, torsional stiffness, displacement and rotation of story-

subjected lateral load, and natural frequency of a 3D system, can be calculated with good

precision. In other words, this paper provides a more accurate estimation of lateral

deformation profiles of discrete systems through the simplified continuum model. Finally,

the results obtained from this study’s proposed method and numerical modeling results were

compared to validate the accuracy of the proposed method.

2. Methodology

The main concept of this study is based on the simplification of 2D modeling. In this

method, a 2D frame with definite mechanical specifications and multi-bays is converted to

many one-bay 2D frames combined by hinges. These 2D frames can be summarized by a

one-bay, one-story 2D frame, hereinafter referred to as a module of the simplified system.

Figure 1 shows a schematic view of the proposed method to simplify a 2D frame system.

The simplified equivalent 2D system was initially proposed by Hosseini and Imagh-e-naiini

. The basic ideas of the proposed method are based on the following facts: A) In a ordinary

moment frame subjected to a lateral load, beam and columns in all bays deform similarly. B)

The lateral stiffness of a multi-story frame at each floor is mainly due to columns and beams

just below and above that floor.





Figure 1: The simple multi-bay frame subjected lateral load, and its simplified equivalent

systems: (a) the main 2D system, (b) the simplified equivalent 2D system, (c) the basic

module of simplified

In a moment 2D frame with regular geometric planes that are connected to each other by

hinges as shown in Figure. 1, the value moment of inertia for column cI and the moment of

inertia for beam gI are given by:

m

I

I

m

j

cj

c2

1

(1)

m

I

I

L

I

m

j j

gi

g

1

(2)

Where L and m are span length value and number of spans, respectively. It is obvious that the

frames shown in Figure 1(b) are equivalent to m of the single frame shown in Figure 1(c). A

similar idea was used for the n-story 2D frame shown in Figure 2. The values of cjI and giI

in Figures 2(b) and 2(c) are given by:

m

j

cijcj II12

1 (3)

m

j j

gijgi

I

I

L

I

1

(4)

In fact, each of the sub-frames in Figure 2(c) is a simple frame module, like that shown in

Figure 3, which has lateral stiffness of k:





Figure 2: Model of m-bay, n-story 2D moment frame, and its simplified equivalent systems:

(a) the main 2D system, (b) the one-step-simplified equivalent 2D system, (c) the final

simplified equivalent system

)3)(2

6)()(

12(

22

ududcc

ududccfm

kkkkkk

kkkkk

h

kk

(5)

h

EIk c

c (6)

Where:

h

EIk

gd

d (7)

h

EIk

gu

u (8)

Where

h , cI , gdI and guI are the dimension and the cross-sectional properties of the frame

module, respectively, as shown in Figure 3, and E is the modulus of elasticity of the frame

material.





Figure 3: The main frame module of the simplified 2D system for regular moment frames

2.1 Main concept of approximation lateral and torsional stiffness for 3D systems

The lateral stiffness and torsional stiffness in a 3D moment frame for X and Y directions were

obtained with the approximating method for the 2D moment frame. Then, the simplified

stiffness in the X and Y directions were summed in each direction, and the 3D system was

exchanged for two 2D moment frames in each direction of X and Y. With this method, it can

be given by:

n

iifmxfm kk

1

(9)

m

jjfmyfm kk

1

(10)

Where n is the number of moment frames in the X direction and m is the number of moment

frames in the Y direction. The coordinate of the center of stiffness (CR) for irregular 3D

building systems can be given by:

fmy

fmy

CRk

xkX (11)

fmx

fmx

CRk

ykY (12)

The value of torsional stiffness of each story can be obtained from the stiffness of each

moment frame in the X and Y direction, and then the sum of these values in each direction.

2

1

.. i

n

iifmyix xkk

(13)

2

1

.. i

n

iifmxiy ykk

(14)

The sum of torsional stiffness of columns for each story of a building system can be given by:

n

i

ich

GJk

1

.. (15)





Where G is the torsion constant for the section and J is polar inertia of column, that:

ycxc IIJ (16)

The model is composed of torsional stiffness contributions to overall lateral stiffness.

)(22

1

.h

GJykxkk iifmxi

n

iifmyi

(17)

Or:

. . .

1i

n

i x y i

i

GJk k k

h

(18)

where )1(2

E

G

Where i, n, ix , iy , j and G are the counters of each story, number of stories, distance

from each frame to stiffness center of each story in two directions, polar moment

inertia, and shear modulus of a material computed by adding moment of inertia in two

directions, respectively. Therefore, the displacement in each direction of X and Y can be

given by:

fmx

xixi

k

V (19)

fmy

yi

yik

V (20)

Where xi , yi , xiV , yiV , fmyk and fmxk are the story lateral displacement values in X and Y

directions, story shear force in X and Y directions, and estimated lateral stiffness in two

directions X and Y, respectively. Accumulated lateral displacement values of stories versus

base shear in each story have a relationship with lateral stiffness.Considering the computed

torsional stiffness values for each story, torsional moment of rigidity center, and Hooke's law,

the total in-plan rotation of each story can be computed as follows:

k

Tii (21)

3. Evaluation of proposed method for 3D frame behavior

To show the high efficiency of the proposed methods for calculating the lateral displacement,

torsion of each story, and main period of structures, some numerical examples are presented.

The examples presented here are 4, 6, 12, 18 and 24-story steel frames with irregularity in-

plan. In all models, the plan view is the same (Figure 4) and the height of the stories and the

length of beams are 3m and 5m, respectively. The modulus of elasticity is supposed to be

2100000 2kgfcm . For calculating natural frequency and base shear, a mass of 448tons has

been considered for all floors. Lateral loading of frames is defined as static loads and

calculated based on the regulations of Iranian seismic design code (IS 2800-05) with the

design basis acceleration of 0.35 for South Pars region and soil period of 0.5 second for soil





type II. The IS 2800-05 [12] is derived from UBC 1994 and BOCA 1978 and have undergone

major changes over the years. For summarizing the content of present work and avoiding

time-consuming, the analysis of a 6-story frame will be described in detail. The specification

of 3D modeled 6-story frame with irregular structure is shown in Figure 4. The stiffness

center in X and Y direction and the values of stiffness based on the proposed method are

illustrated in Table 1 and Table 2 to 4, respectively. For this frame and also other models, the

error of the proposed method in calculating the lateral displacement, rotation and the main

period of structure as the analytical parameter values will be compared with numerical result

from an analysing program.

Evaluation of the methodology presented here requires structural model to be accurate. To

analytically predict the response of structural system under seismic loads, the building

structure should be accurately described. Using reliable analytical software and definition of

strength of materials and yielding behavior of elements are necessary. The models were

analyzed herein by employing OpenSees software. All of the beams end connections within

the structure are assumed to be pinned. Therefore, the beams are modeled as elastic elements

with steel02 material. These models are built with nonlinear beam column element for

columns as well as the P-delta effects are taken into account. Fiber elements were used in all

of the nonlinear elements. The masses are lumped at floor levels, whereas the horizontal

degrees of freedom are defined. The Rayleigh damping with a specified ratio of ξ = 0.05 was

assigned at the first-vibration-mode, and the effect of nonstructural elements was not

considered.

Figure 4a: 2D plan view of studied structure with asymmetric distribution of stiffness





Figure 4b: 2D view of AF-1and AF-6 frames of studied structure

Based on the presented methodology in this paper, in order to obtain torsional stiffness for

each story, the tortional stiffness of each frame should be computed in each direction then

total value of computed stiffness for both directions should be considered as a total stiffness

of the frame. In other words, the torsional stiffness in each direction is computed by

multiplying lateral stiffness of each frame and square distance of each frame to stiffness

center which result in Equation 18. The shear modulus of steel material was given as2

784cm

ton.

Based on Equation 18, torsional stiffness in each story can be obtained by adding the lateral

stiffness of columns (ixk .. ). It is worth

h

GJstiffness of the columns () and torsional iyk .. ،

mentioning that the torsional stiffness of columns in each story should be computed, and its

value should be included separately with the torsional stiffness of each frame in the total

torsional stiffness of a frame (Table 5). With respect to the calculation of the above-

mentioned components for torsional stiffness, the torsional stiffness value would be very

small compared with the torsional stiffness in each frame; thus, it can be neglected.

Based on computed lateral stiffness values, the story shear force, and Hooke's law, story

lateral displacement can be computed by Equations 19 and 20 and are presented in Table 6.

The total in-plan rotation of each story can be computed by using Equation 21 and Table 7

presents estimated rotation values calculated by stiffness center and in-plan torsional moment

provided by determined lateral load distributions in the X and Y direction for considered

structure. In order to calculate the main period of structure, lateral and torsional stiffness

values should be computed based on the above-mentioned equations 15 to 18. Thus, the





rotation and transition component of mass moment inertia for each story is calculated by the

following equation:

222

2

1mdbamm (22)

Where a, b, m, m and d are rotational mass moment inertia, total mass of each story, length

and width of panel, and distance between mass center of panel and total mass center of each

story. The results are shown in Table 8.

3.1 Numerical result from analysing program

At this stage, the exact values of lateral and torsional displacement were calculated by

OpenSees software. These results consist of lateral displacement values for each story and the

torsion of each story in the X and Y directions. Table 9 presents the exact values of

displacement in 2 directions and the rotations of each story. Figure 5a and 5b show schematic

views of the lateral and torsional displacement of the stories.

Figure 5a: Schematic views of stories lateral and Torsional displacements for sixth stories of

studied building

Figure 5b: schematic views of stories lateral and tortional displacements for sixth stories of

studied building





3.2. Calculation of the error of proposed method

Based on the method presented in this study, the analytical parameter values used in this

study were compared with the corresponding exact values obtained by OpenSees software;

the calculated errors can also be computed as follows:

100%

exact

exactestimate

ntdisplacemee (23)

100%

exact

exactestimate

rotatione

(24)

100%

exact

exactestimate

priodT

TTe (25)

Figure 6: The error in displacement calculation in X and Y direction

Figure 7: The absolute error in rotation calculation in X and Y direction

Where , , and T are lateral displacement of each story, torsion of each story, and main

period of structures obtained by the presented method and exact salutation, respectively.





Table 10 to 12 show the calculated errors of lateral displacement, torsion and the main period

values of 6-story structure for each stories computed by the exact solution and the proposed

method. Combining all the error, from Figure 6 to 8 it can be seen that the error of proposed

method in calculating the lateral displacements and rotation of stories and main period of

irregular buildings are less than 12%. Therefore, this shows a good agreement between the

proposed method with analysing program.

Figure 8: The error in main period of models

Table 1: Specifications of selected structure

Story Story height

(metre)

Center of Rigidity

(metre)

X Y

story 6 3 10.177 10.061

story 5 3 10.180 10.056

story 4 3 10.196 10.065

story 3 3 10.203 10.054

story 2 3 10.203 10.054

story 1 3 10.265 10.071

Table 2: The obtained values of stiffness based on simplified model in presented

methodology

Frame 1

Story )(cmh h

EIk c

c L

EIk

gu

u L

EIk

gd

d )3)(2

6)()(

12(

22

ududcc

ududcc

xfmkkkkkk

kkkkk

h

kk

story

6 37.6 188648 191992 544685 300

story

5 38.9 188648 188648 649795 300





story 4

44.9 275183 188648 649795 300

story

3 55.2 275183 275183 837213 300

story

2 55.2 275183 275183 837213 300

story

1 111.1 1.00E+20 275183 837213 300

Table 3: Calculation of the stiffness center based on simplified model in presented

methodology for the structure in Y direction

Kfmy Calculate of CRX

Story Frame

A

Frame

B

Frame

C

Frame

D

Frame

E

Frame

F

Kfmy

Kfmy

* x )(mXCR

story

6 22.9 26.8 25.9 13.2 13.2 12.4 114.4 1164.6 10.177

story

5 23.6 27.5 21.6 13.5 13.5 12.8 112.5 1144.8 10.180

story

4 25.5 30.3 23.3 14.8 14.8 13.9 122.5 1249.2 10.196

story

3 28.6 34.6 26.2 16.8 16.8 15.7 138.7 1414.9 10.203

story

2 28.6 34.6 26.2 16.8 16.8 15.7 138.7 1414.9 10.203

story

1 36.4 47.1 32.4 22.6 22.6 20.6 181.8 1866.2 10.265

Table 4: Calculation of the stiffness center based on simplified model in presented

methodology for the structure in X direction

Kfmx Calculate of CRY

Story Frame

1

Frame

2

Frame

3

Frame

4

Frame

5

Frame

6 Kfmx

Kfmx

* y )(mYCR

story

6 37.6 42.9 41 20.9 20.9 19.2 182.5 1836 10.061

story

5 39 44.3 39.9 21.6 21.6 19.9 186.1 1871.4 10.056

story

4 45 52.4 46.2 25.4 25.4 23 217.3 2187.3 10.065

story

3 55.2 63.9 57 31 31 28 266.2 2676.4 10.054

story

2 55.2 63.9 57 31 31 28 266.2 2676.5 10.054

story

1 111.1 190.8 121.4 82.9 82.9 57.9 646.9 6514.9 10.071





Table 5: Computed tortional stiffness value for each story based on presented methodology

Story 2

1

.. i

n

iifmyix xkk

2

1

.. i

n

iifmxiy ykk

h

GJ

rad

cmtonk

story 6 73991812.6 117637485.9 3951203 195580501

story 5 75975688.8 121735502.4 4849043 202560234

story 4 82826611.8 141587081.3 4849043 229262736

story 3 93518500.3 173058973.2 6216127 272793600

story 2 93518500.3 173058973.2 6216127 272793600

story 1 122636329.5 392564687.4 6216127 521417144

Table 6: Estimation of displacement values of rigidity center for each story

)(cmyi )(cmyi )(cmxi )(cmxi )(tonVyi )(tonVxi cmtonk fmy cmtonk fmx Story

8.42 0.649 4.27 0.407 74.3 74.3 114.4 182.5 story

6

7.77 1.235 3.86 0.746 138.8 138.8 112.5 186.1 story

5

6.54 1.555 3.12 0.877 190.6 190.6 122.5 217.3 story

4

4.98 1.657 2.24 0.863 229.8 229.8 138.7 266.2 story

3

3.33 1.846 1.38 0.962 256.1 256.1 138.7 266.2 story

2

1.48 1.481 0.42 0.416 269.2 269.2 181.8 646.9 story

1

Table 7: Estimated in-plan rotation values of each story provided by torsional moment for X

and Y direction

Story

rad

cmtonk

yx , cmtonTyxi

, radyx, radyx,

story 6 195580501 72050875.1 0.000368 0.00407

story 5 202560234 134715356 0.000665 0.003702

story 4 229262736 184942076 0.000807 0.003037

story 3 272793600 223060778 0.000818 0.00223

story 2 272793600 248533176 0.000911 0.001412

story 1 521417144 261269375 0.000501 0.000501





Table 8: Calculated lateral and tortional stiffness values, transformed mass moment inertial

and tortional mass moment inertial

Mass

2sec.. cmton

MassY

2sec

cmton

MassX

2sec

cmton

rad

cmtonk

cmtonk fmy

cmtonk fmx

Stor

y

1230858 0.4382 0.4382 1955805

01 114 182

stor

y 6

1283569 0.4570 0.4570 2025602

34 112 186

stor

y 5

1285954 0.4578 0.4578 2292627

36 123 217

stor

y 4

1301065 0.4632 0.4632 2727936

00 139 266

stor

y 3

1304104 0.4643 0.4643 2727936

00 139 266

stor

y 2

1304104 0.4643 0.4643 5214171

44 182 647

stor

y 1

Table 9: Exact values of displacements in 2 directions and rotations of each story

Story Diaphragm Load )(cmUX )(cmUY )(radRZ x

story 6 D1 EX 4.235 0.179 0.004103

story 5 D1 EX 3.761 0.156 0.003626

story 4 D1 EX 3.097 0.126 0.003003

story 3 D1 EX 2.234 0.089 0.002294

story 2 D1 EX 1.351 0.052 0.001436

story 1 D1 EX 0.402 0.014 0.00047

BASE D1 EX 0 0 0

story 6 D1 EY 0.301 8.195 0.00408

story 5 D1 EY 0.216 7.428 0.003638

story 4 D1 EY

0.176 6.327 0.003034

story 3 D1 EY

0.125 4.794 0.002229

story 2 D1 EY

0.075 3.229 0.001434

story 1 D1 EY

0.027 1.44 0.000528

BASE D1 EY 0 0 0





Table 10: Calculated errors of lateral displacement values for each stories computed by the

exact solution and proposed method

Story )(cmxi )(cmUX xe% )(cmyi )(cmUY

ye%

story 6 4.27 4.235 0.9 8.42 8.195 2.8

story 5 3.86 3.761 2.7 7.77 7.428 4.7

story 4 3.12 3.097 0.7 6.54 6.327 3.4

story 3 2.24 2.234 0.3 4.98 4.794 4.0

story 2 1.38 1.351 2.0 3.33 3.229 3.0

story 1 0.42 0.402 3.6 1.48 1.44 2.8

Table 11: Calculated errors of torsions values for each stories computed by the exact solution

and proposed method

Story radx )(radRZ x xe .% rady )(radRZ y

ye .%

story 6 0.00407 0.004103 -0.8 0.00407 0.00408 -0.2

story 5 0.003702 0.003626 2.1 0.003702 0.003638 1.7

story 4 0.003037 0.003003 1.1 0.003037 0.003034 0.1

story 3 0.00223 0.002294 -2.8 0.00223 0.002229 0.0

story 2 0.001412 0.001436 -1.7 0.001412 0.001434 -1.5

story 1 0.000501 0.00047 6.6 0.000501 0.000528 -5.1

Table 12: Obtained main period of the structure based on the exact solution and the proposed

method

priode% (sec)estimateT (sec)exactT Mode

-3.6 1.7079 1.772 1

4. Conclusion

In this study, a simplified method for estimating lateral stiffness, torsional stiffness, story

displacement and main period of structure is proposed. Results from the presented method

and from the exact analysis by OpenSees software for the irregular MRF structures have been

compared to evaluate the validation of the proposed method. The results showed that there is

good agreement with insignificant error between the proposed method presented in this study

and the exact solution based on analytical modeling. Thus, the proposed methods can provide

a proper alternate for estimating initial lateral stiffness and lateral displacement and rotation

of even irregular structures and also for final checking of designs. The results showed that the

proposed method can provide the lateral displacement and rotation values with a maximum

error of less than 12%. Therefore, the proposed method can estimate numerical values with

acceptable accuracy and be applied for 3D structures.





5. References

1. Building and Housing Research Center (BHRC). Iranian Code of Practice for Seismic

Resistant Design of Buildings, Standard No. 2800-05, 3rd edition, (2005) Building

and Housing Research Center, Tehran, Iran.

2. Dym C.L., Williams H.E., (2007), Estimating fundamental frequencies of tall

buildings, Journal of Structural Engineering, 133 (10), pp 1479–1483.

3. Eroğlu T., Akkar S., (2011), Lateral stiffness estimation in frames and its

implementation to continuum models for linear and nonlinear static analysis, Bulletin

of Earthquake Engineering, 9(4), pp 1097-1114.

4. Gengshu,T.,Y-L Pi, Bradford, M.A., Tin-Loi, F., (2008), Buckling and second-order

effects in dual shear-flexural systems, Journal of Structural Engineering, 134(11), pp

1726–1732.

5. Georgian, M., (1993), On the lateral response of regular high-rise frame, (1993) Struct.

Design Tall Build, 2, pp 233–252.

6. Golara A, (2014), Probabilistic seismic hazard analysis of interconnected

infrastructure: a case of Iranian high-pressure gas supply system, Natural hazards,

73(2), pp 567-577.

7. Heidebrecht A.C., Stafford B.S., (1973), approximate analysis of open-section shear

walls subjected to torsional loading, Journal of the Structural Division, pp 2355-2373.

8. Hosseini, M. and Imagh-e-Naiini, M. R., A quick method for estimating the lateral

stiffness of building systems, (1999) Structural. Design Tall Build, 8, pp 247–260.

9. Miranda E., Taghavi S., (2005), Approximate floor acceleration demands in

multistory Building, I: formulation, Journal of Structural Engineering, ASCE, 131(2),

pp 203–211.

10. Miranda, E., Akkar, S.D., (2006), Generalized interstory drift spectrum, Journal of

Structural Engineering, ASCE, 132(6), pp 840–852.

11. Shahrouzi M.A, (2004), quick method for eigenvalue estimation of multistory, 13th

World Conference on Earthquake Engineering, Canada, Paper No. 3086.

12. Utku S., Norris, C.H., Wilbur J.B., (1991), Elementary Structural Analysis, McGraw-

Hill, New York.

Documents

Presenting a quick method for estimation of MRF dynamic ... · no certainty of the correctness of the data entry or matching ... are equivalent to m of the single frame ... ¦k u