 Open Access
 Authors : Ali S. Ngab, Nada J. Attia, Md Saifuddin Shahin
 Paper ID : IJERTV10IS110110
 Volume & Issue : Volume 10, Issue 11 (November 2021)
 Published (First Online): 30112021
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Effects of Reinforced Concrete Building Characteristics on the Fundamental Natural Period Using the Rayleigh Method
Ali S. Ngab*
Civil Engineering Department University of Tripoli
Tripoli, Libya
Nada J. Attia
Civil Engineering Department University of Tripoli
Tripoli, Libya
Md Saifuddin Shahin
Civil Engineering Department University of Tripoli
Tripoli, Libya
AbstractThis study focuses on the most important single and unique property controlling the seismic response of any structure, namely; the "Fundamental Natural Period" of oscillation. The effects of the main parameters influencing the fundamental period of vibration of reinforced concrete buildings such as mass and stiffness as related to other parameters including; building height, number of storeys, type of framing system, the effect of infill brick walls, type of flooring systems and the compressive strength of concrete were investigated in this study. The work is carried out using the Rayleigh method using the STAADPro Software computer program. 140 building cases were investigated to determine the natural period for different types of buildings.
Linear Equations were suggested to calculate the fundamental natural period of reinforced concrete for bare frames with a solid slab or flat plate and for structures where infill masonry walls are included. The findings indicate that the fundamental natural period of the structure is linear with the height of the building and the number of storeys. The period is reduced significantly by more than 50% when infill walls were present in both solid and flat plate construction. As the mass increases, the period increases as long as the stiffness remains constant. When the mass remains constant and the stiffness is increased, the period decreases for all building heights and for all structural flooring systems considered in this study. When the cracking effects in beams and columns were considered, the structure became more flexible and the natural period increased significantly. This behavior was apparent in both flooring systems with flat plate and solid slabs. An increase in compressive strength of concrete in columns did not have any noticeable effect on the fundamental period of the buildings under consideration.
Keywords: Fundamental natural period, Natural frequency, Earthquake, Seismic performance, Structural dynamics, Reinforced concrete buildings, Rayleigh method.

INTRODUCTION
The most important step in the earthquakeresistance building design is the determination of the fundamental natural period of free vibration of the structure. This property is a unique parameter related to the response characteristics of the structure to seismic loading. The reciprocal of the fundamental period is called the fundamental frequency. The shape associated with this natural frequency is the mode shape of the vibrating structure.
In order to use any building code effectively, the local building response characteristics, as well as the anticipated loads such as wind and earthquake loadings, should be
established. The type of structure, its structural form, materials, and construction details play an important role in assessing its behavior under different gravity and lateral loadings. The fundamental natural period of vibration (Tn) of the building is an important parameter for the evaluation of seismic base shear [1]. It depends on basic parameters such as building height or the number of storeys.
The current study focuses on one of the basic dynamic characteristics of structures under lateral loadings. The natural period of oscillation of any system is a unique property of that system under free vibration. When a structure is subjected to an earthquake motion it vibrates at a certain frequency termed as the natural frequency of the building. The natural frequency is the number of cycles in one second of vibration. The natural period of the system is the reciprocal of its natural frequency. It is the time required for the system to complete one cycle.
The objective of this study is to investigate the effects of the main parameters influencing the fundamental natural periods of vibrations of reinforced concrete buildings such as mass and stiffness as related to other parameters including; building height, number of storeys, type of framing system, the effect of infill brick walls, type of flooring systems and the compressive strength of concrete. The work is to be carried out using the Rayleigh method of dynamic analysis using the STAADPro Software computer program. The results will be compared with the corresponding values obtained from the empirical formula recommended by different international codes for seismic design and with other similar work in the literature. 140 building cases will be investigated to determine the natural period for different types of buildings with different characteristics.

A BREIF LITERATURE REVIEW
Several investigations have studied the effects of building characteristics on the natural frequencies of vibration. Bhuskade et al. [1] has studied the effects of various parameters of buildings on the natural period. They concluded that the fundamental natural period of vibration of buildings is the most important parameter in the evaluation of seismic base shear and depends on basic building characteristics such as mass, stiffness, building height and number of stories. Yogalakshmi [2] investigated the effect of building height on the natural frequency of medium to high rise buildings. His findings indicate that the height of the building was a major factor that affects the time period of vibration. The story
height and bay width also play a greater part on the time period than the effect of column sizes and number of bays. According to Sudhir et al. [3], the natural period of a building can be related to the number of stories. Whereas, the number of stories increases, the natural period also increases, although the height of the building remains constant.
M.C. Griffith and Vaculik [4] carried out an experimental program on 8 fullscale unreinforced brick masonry walls and concluded that the experimental results agree with the theoretical analytical prediction for the natural periods of the buildings investigated. H. Crowley and R. Pinho [5] studied the possibility of estimating of natural period by simplified equations that estimate the period of vibration.
They presented a simple equation which relates natural period to building height for assessment of existing of R/C building. The equation in the form of T=0.055h takes the effect of infill walls on the natural period of the building. L. Chung and T. Park [6] investigated the evaluation of natural periods using several equations. They conducted that, empirical equations applied to evaluate natural periods of R/C buildings gave various results.
Thus, the seismic loads calculated based on such empirical equations may differ greatly according to the equation applied. The values calculated based on the RayleighRitz method were twice the actual measurements. In this study, the purpose is to study the effect of the important building characteristics on the natural period of R/C buildings with and without infill walls. The study will be limited by using the Rayleigh method for free vibration.

STATEMENT OF THE PROBLEM AND OBJECTIVES
The objective of this work is to perform a parametric study of the reinforced concrete buildings and determine the fundamental natural period of vibration under free vibration. The work will focus on finding the fundamental period fo different structural characteristics of reinforced concrete structures. These include building height, stiffness and cracking effect of infill brick walls, number of storeys, and the influence of increasing the compressive strength of concrete in columns of highrise buildings. The Rayleigh method and STAADpro software program of dynamic analysis will be used in this investigation. The Rayleigh method is one of the methods recommended by the ASCE 7 standard for seismic design. The results of the work will be compared with the recommendation of the empirical equations recommend by several international codes to determine the fundamental period for base shear calculations in seismic design. These codes include UBC, IBC, Eurocode8, IS1983, Japanese standard, and others.

THE FUNDAMENTAL NATURAL PERIOD Every building has a number of natural frequencies at
which it has minimum resistance to vibration induced by external forces such as wind or earthquake. The mode of vibration with the smallest natural frequency (and largest natural period) is called the fundamental mode. The associated period is called the fundamental Natural Period and the corresponding frequency is called the Fundamental Natural
Frequency. Regular buildings held at their base have three fundamental transitional natural periods in the X, Y and Z directions.
Natural frequencies and mode shapes are functions of the structural properties and boundary conditions. If the structural properties change, the natural frequencies change, but the mode shapes may not necessarily change [7].
Dynamic systems can be characterized in terms of one or more natural frequencies. The natural frequency is the frequency at which the system would like to vibrate if it were given an initial disturbance and allowed to vibrate freely. There are many methods to evaluate the natural frequencies; the Newtons law of motion, Rayleigh method, energy methods and Lagrange equations. The ASCE7 permits the use of properly substantial analysis (e.g. Rayleigh method or dynamic modal analysis) to determine the fundamental period provided that the value of the period determined using these methods, T rational may not exceed the value of Ta recommended by the ASCE7 code. Thus, T rational CuTa. Rayleigh's method directly yields the natural frequency. Rayleigh's method, however, requires an assumed displacement function. This method reduces the dynamic system to a single degree of freedom. For a multidegree of freedom system, the Rayleigh method can be expressed in the following form:
Where:
= Fundamental angular natural frequency (radians/s).
[K] = Stiffness matrix. [M] = Mass matrix. [X] = Assumed mode shape (deflected shape). [X]T = Assumed mode shape (deflected shape) transposes Tn = Fundamental natural transitional period. 
PARAMETRIC STUDY OF DIFFERENT R/C BUILDING DETAILS
In this study, over 140 building cases were considered, with a various number of floors and specifications that influence the mass and stiffness of the structures.

Details of buildings considered for the effects of building heights on the natural fundamental period.
It makes sense to study the behavior of changing the structure height, as increasing the building height would increase its mass, therefore the natural period. Table I shows the building details while Fig. 1 and Fig. 2 show the elevations and plans of the studied structures.

All beams are (250*400).

All slab thicknesses = 200mm.

fc'= 30 MPa, fy = 420 MPa.

Infill walls for all cases = 250mm.

Live load= 3KN/m2.

Bay length in Xdirection plan is 5m (center to center). (7bays) = 35m.

Bay length in Zdirection plan is 5m (center to center). (5bays) = 25m.
TABLE I. EFFECT OF HEIGHT CASE SPECIFICATIONS
No. of storeys
Building height (m)
Column size (mm)
Slab thickness (mm)
R/C frame
R/C frame
+ infill
R/C flat plate
R/C flat plate +infill
Bld.#
Bld.#
Bld.#
Bld.#
2
6
500*500
200
A2
B2
C2
D2
5
15
500*500
200
A5
B5
C5
D5
10
30
500*500
200
A10
B10
C10
D10
15
45
500*500
200
A15
B15
C15
D15
20
60
500*500
200
A20
B20
C20
D20
25
75
500*500
200
A25
B25
C25
D25
Fig. 1. Reinforced concrete buildings elevations for the effect of height case.
Fig. 2. Reinforced concrete building floor plans for the effect of height case.


Details of buildings considered for the effects of building stiffness and cracking on the natural fundamental period.
In this study, the beam elements moment of inertia (Ig) was modified to be 0.35 Ig for beams and 0.70 Ig for columns. Table II shows the specifications of the structure, Fig. 3 and Fig. 4 show the elevations and plans of cases.

All beams are (350*400).

All slab thicknesses = 200mm.

fc'= 30 MPa, fy = 420 MPa.

Infill walls for all = 250mm.

Live load= 3KN/m2.

Bay length in both Xdirection and Zdirection plan is 5m (center to center). (5bays) = 25m.
TABLE II. EFFECT OF STIFFNESS CASE SPECIFICATIONS
10
30
600*600
200
E10
F10
G10
H10
15
45
700*700
200
E15
F15
G15
H15
20
60
800*800
200
E20
F20
G20
H20
25
75
900*900
200
E25
F25
G25
H25
Fig. 3. Reinforced concrete buildings elevations for the effect of stiffness case.
Fig. 4. Reinforced concrete building floor plans Floor plans for effect of stiffness.


Details of buildings considered for the effects of building mass on the natural fundamental period.
Seismic mass plays a very important role in providing stability of structures. Table III and Table IV show the specifications of the structure for (Case A) and (Case B) while Fig. 5 and Fig. 6 show the elevations and plans of the studied structures.

All beams are (350*400).

fc'= 30 MPa, fy = 420 MPa.

Infill walls for all cases = 250mm.

Live load= 3KN/m2.

Bay length in both Xdirection and Zdirection plan is 5m (center to center). (7bays) = 35m.
No. of storeys
Building height (m)
Column size (mm)
Slab thickness (mm)
R/C frame
R/C frame + infill
R/C frame
R/C frame + infill
Bld.#
Bld.#
Bld.#
Bld.#
2
6
400*400
200
I2
J2
K2
L2
5
15
500*500
200
I5
J5
K5
L5
10
30
600*600
200
I10
J10
K10
L10
15
45
700*700
200
I15
J15
K15
L15
20
60
800*800
200
I20
J20
K20
L20
25
75
900*900
200
I25
J25
K25
L25
No. of storeys
Building height (m)
Column size (mm)
Slab thickness (mm)
R/C frame
R/C frame + infill
R/C frame
R/C frame + infill
Bld.#
Bld.#
Bld.#
Bld.#
2
6
400*400
200
I2
J2
K2
L2
5
15
500*500
200
I5
J5
K5
L5
10
30
600*600
200
I10
J10
K10
L10
15
45
700*700
200
I15
J15
K15
L15
20
60
800*800
200
I20
J20
K20
L20
25
75
900*900
200
I25
J25
K25
L25
TABLE III. EFFECT OF MASS (CASE A) SPECIFICATIONS
No. of storeys
Building height (m)
Column size (mm)
Slab thickness (mm)
Uncracked
Cracked (0.35 Ig beams 0.7 Ig columns)
R/C frame
R/C frame
+infill
R/C frame
R/C frame
+infill
Bld.#
Bld.#
Bld.#
Bld.#
2
6
400*400
200
E2
F2
G2
H2
5
15
500*500
200
E5
F5
G5
H5
TABLE IV. EFFECT OF MASS (CASE B) SPECIFICATIONS
No. of storeys
Building height (m)
Column size (mm)
Slab thickness (mm)
R/C frame
R/C frame + infill
R/C frame
R/C frame + infill
Bld.#
Bld.#
Bld.#
Bld.#
2
6
400*400
300
M2
N2
O2
P2
5
15
500*500
300
M5
N5
O5
P5
10
30
600*600
300
M 10
N10
O10
P10
15
45
700*700
300
M 15
N15
O15
P15
20
60
800*800
300
M 20
N20
O20
P20
25
75
900*900
300
M 25
N25
O25
P25
Fig. 5. Reinforced concrete building elevations of different heights and
mass.
Fig. 6. Reinforced concrete building floor plans of different heights and
mass.


Details of buildings considered for the effects of number of storeys on the natural fundamental period.
Table V shows the specifications of the structure. Fig. 7 and Fig. 8 show the elevations and plan of the studied structures figures with the following parameters:

All beams are (300*600).

All slab thickness = 200mm.

fc' = 30 MPa, fy = 420 MPa.

Infill walls for all cases = 250mm.

Live load= 3KN/ m2.

Bay length in both Xdirection and Zdirection plan is 5m (center to center). (8bays) = 40m.
TABLE V. EFFECT OF NUMBER OF STOREYS CASE SPECIFICATIONS
No. of storeys
Building height (m)
Story height (m)
Slab thickness (mm)
R/C frame
R/C frame
+ infill
Bld.#
Bld.#
24
120
5
120
Q24
R24
27
120
4.44
120
Q27
R27
30
120
4
120
Q30
R30
34
120
3.53
120
Q34
R34
40
120
3
120
Q40
R40
Fig. 7. Reinforced concrete building elevations of different number of
storeys.
Fig. 8. Reinforced concrete building floor plan for different number of storeys.


Details of buildings considered for the effect of compressive strength at the natural fundamental period
Table VI shows the specifications of the structure while Fig. 9 and Fig. 10 shows the elevation and plan of the studied structures with the following parameters:

All beams are (300*600).

All slab thickness =120mm.

fc' = varies, fy = 420 MPa.

Infill walls for all cases = 250mm.

Live load= 3KN/m2.

Bay length in both Xdirection and Zdirection plan is 5m (center to center). (8bays) = 40m.

TABLE VI. EFFECT OF COMPRESSIVE STRENGTH CASE SPECIFICATIONS
No. of storeys
Height (m)
Column size (mm)
fc (MPa)
R/C frame
R/C frame
+ infill
Bld.#
Bld.#
40
120
500*500
35
Q35
R35
40
120
500*500/p>
56
Q56
R56
40
120
500*500
70
Q70
R70
40
120
500*500
84
Q84
R84
40
120
500*500
98
Q98
R98
Fig. 9. Reinforced concrete building elevation with varying compressive
strength.
Fig. 10. Reinforced concrete building floor plan with varying compressive
strength.


PARAMETRIC STUDY OF DIFFERENT R/C BUILDING RESULTS

Results of the effects of building heights on the natural fundamental period.
Table VII and Table VIII show software analysis results in both directions, while Fig. 11 and Fig. 12 shows the relation between natural period and the number of storeys in different cases in Xdirection and Zdirection. The results show that taller reinforced concrete buildings have larger fundamental translational natural periods than short or intermediate buildings in both solid and flat plate construction as increasing the height means increasing the mass and therefore increasing the natural period.
TABLE VII. EFFECT OF BUILDING HEIGHT RESULTS TXDIRECTION
R/C frame
R/C frame + infill
R/C flat plate
R/C flat plate
+ infill
Bld.#
Tn (s)
Bld.#
Tn (s)
Bld.#
Tn (s)
Bld.#
Tn (s)
A2
0.267
B2
0.232
C2
0.272
D2
0.233
A5
0.662
B5
0.427
C5
0.687
D5
0.425
A10
1.335
B10
0.697
C10
1.394
D10
0.693
A15
2.028
B15
0.987
C15
2.118
D15
0.979
A20
2.739
B20
1.335
C20
2.857
D20
1.322
A25
3.472
B25
1.763
C25
3.623
D25
1.742
Fig. 11. Effect of building height on Tn in Txdirection.
TABLE VIII. EFFECT OF HEIGHT RESULTS TZDIRECTION
R/C frame
R/C frame + infill
R/C flat plate
R/C flat plate
+infill
Bld.#
Tn (s)
Bld.#
Tn (s)
Bld.#
Tn (s)
Bld.#
Tn (s)
A2
0.268
B2
0.233
C2
0.273
D2
0.231
A5
0.666
B5
0.431
C5
0.689
D5
0.428
A10
1.349
B10
0.719
C10
1.402
D10
0.714
A15
2.057
B15
1.005
C15
2.136
D15
1.048
A20
2.801
B20
1.453
C20
2.906
D20
1.524
A25
3.584
B25
2.061
C25
3.717
D25
2.032
Fig. 12. Effect of building height on Tn in Tzdirection.

Results of the effects of building stiffness and cracking on the natural fundamental period.
Table IX and Fig. 13 showing the relation between Tn and number of storeys in different cases, we can notice that the structure became more flexible and the natural period increased significantly. This behavior is apparent in both flooring systems with flat plate and solid slabs.
No. of storeys
Uncracked
Cracked
0.35Ig beams 0.7 Ig columns
R/C frame
R/C frame + infill
R/C flat plate
R/C flat plate
+ infill
Bld.
#
Tn (s)
Bld.
#
Tn (s)
Bld.
#
Tn (s)
Bld.
#
Tn (s)
2
E2
0.354
F2
0.333
G2
0.393
H2
0.384
5
E5
0.658
F5
0.384
G5
0.715
H5
0.425
10
E10
1.189
F10
0.590
G10
1.270
H10
0.640
15
E15
1.724
F15
0.788
G15
1.824
H15
0.847
20
E20
2.293
F20
1.050
G20
2.409
H20
1.120
25
E25
2.907
F25
1.357
G25
3.030
H25
1.451
No. of storeys
Uncracked
Cracked
0.35Ig beams 0.7 Ig columns
R/C frame
R/C frame + infill
R/C flat plate
R/C flat plate
+ infill
Bld.
#
Tn (s)
Bld.
#
Tn (s)
Bld.
#
Tn (s)
Bld.
#
Tn (s)
2
E2
0.354
F2
0.333
G2
0.393
H2
0.384
5
E5
0.658
F5
0.384
G5
0.715
H5
0.425
10
E10
1.189
F10
0.590
G10
1.270
H10
0.640
15
E15
1.724
F15
0.788
G15
1.824
H15
0.847
20
E20
2.293
F20
1.050
G20
2.409
H20
1.120
25
E25
2.907
F25
1.357
G25
3.030
H25
1.451
TABLE IX. EFFECT OF BUILDING STIFFNESS RESULTS
Fig. 13. Effect of building stiffness and cracking on the natural fundamental
period.

Results of the effects of building mass on the natural fundamental period.
Table X and Table XI shows software analysis results while Fig. 14 and Fig. 15 show the relation between Tn and number of storeys in different cases. As the building increased in height, slab thickness and the columns, it caused the mass to increase and therefore the natural period to increase, as long as the stiffness remains constant.
TABLE X. EFFECT OF BUILDING MASS ON TN (CASE A) RESULTS
R/C frame
R/C frame + infill
R/C flat plate
R/C flat plate
+ infill
Bld.#
Tn (s)
Bld.#
Tn (s)
Bld.#
Tn (s)
Bld.#
Tn (s)
I2
0.363
J2
0.388
K2
0.362
L2
0.388
I5
0.654
J5
0.757
K5
0.692
L5
0.821
I10
1.193
J10
1.422
K10
1.270
L10
1.536
I15
1.718
J15
2.057
K15
1.855
L15
2.252
I20
2.183
J20
2.717
K20
2.475
L20
2.994
I25
2.865
J25
3.401
K25
3.134
L25
3.770
Fig. 14. Effect of building mass on Tn (Case A).
TABLE XI. EFFECT OF BUILDING MASS ON TN (CASE B) RESULTS
R/C frame
R/C frame + infill
R/C flat plate
R/C flat plate + infill
Bld.#
Tn (s)
Bld.#
Tn (s)
Bld.#
Tn (s)
Bld.#
Tn (s)
M2
0.377
N2
0.400
O2
0.363
P2
0.386
M5
0.615
N5
0.695
O5
0.602
P5
0.688
M10
1.002
N10
1.158
O10
1.000
P10
1.165
M15
1.356
N15
1.577
O15
1.360
P15
1.602
M20
1.727
N20
2.008
O20
1.745
P20
2.053
M25
2.127
N25
2.463
O25
2.160
P25
2.530
Fig. 15. Effect of building mass on Tn (Case B).

Results of the effects of number of storeys on the natural fundamental period
Table XII and Fig. 19 shows results, we can notice that increasing the number of storeys in a building of the same height has made the fundamental period decreased slightly in bare buildings but in buildings with infill walls there was no effect.
TABLE XII. EFFECT OF NUMBER OF STOREYS RESULTS
No. of Storeys
Height (m)
Colum size (mm)
R/C frame
R/C frame + infill
Bld.#
Tn (s)
Bld.#
Tn (s)
24
120
500*500
Q24
5.780
R24
3.164
27
120
500*500
Q27
5.618
R27
3.134
30
120
500*500
Q30
5.461
R30
3.095
34
120
500*500
Q34
5.434
R34
3.105
40
120
500*500
Q40
5.208
R40
3.164
Fig. 16. Effect of number of storeys on Tn.

Results of the effect of compressive strength on the natural fundamental period
Table XIII shows software analysis results while Fig. 17 shows the relation between Tn and compressive strength. It can be noticed that increasing the compressive strength in a building of the same height and same no. of storeys has made the fundamental period decreased slightly.
TABLE XIII. EFFECT OF COMPRESSIVES STRENGTH RESULTS
No. of Storeys
Height (m)
Column size (mm)
fc (MPa)
R/C frame
R/C frame + infill
Bld.#
Tn (s)
Bld.#
Tn (s)
40
120
500*500
35
S35
4.807
T35
2.915
40
120
500*500
56
S56
4.310
T56
2.590
40
120
500*500
70
S70
4.098
T70
2.450
40
120
500*500
84
S84
3.937
T84
2.341
40
120
500*500
98
S90
3.802
T90
2.309
Fig. 17. Effect of compressive strength on Tn.


DISCUSSION OF RESULTS
Seismic building codes generally consider the natural period as a necessary parameter to estimate the structure response coefficient. Therefore, there are empirical formulas based on general properties of the buildings the determine natural period, which could be known before a preliminary analysis, these equations can calculate the neutral period by the building characteristics such as building height H or dimensions D. The natural period formulas proposed by some of the chosen seismic codes are shown in the next Table XIV:
TABLE XIV. TA FORMULAS ON INTERNATIONAL CODES
T=0.02H
Code Name
Ta formula
Parameters
UBC 1997[8]
0.073 H^0.75
H: height of the structure.
Eurocode 8 [9]
0.075 H^0.75
H: height of the structure.
ASCE 716[10]
T=0.1N
N: number of floors.
Japan 1981[11]
H: height of the structure.
India 1984 and
others (*) [12]
T = 0.09h/
H: height of the structure. D: dimension parallel to the applied seismic force
(*): Egypt 1988, IBC2009, NEHRP 1994 Iran 1988, Indonesia 1983.
The results shown in the Fig. 18 and Fig. 19 below are the data obtained using the Rayleigh method to determine the fundamental natural period of different reinforced concrete buildings. It is clear from the figures relating Tn and building height that there is a strong linear relation between the two. Moreover, the effect of infill walls on the reduction of the fundamental period is appreciable. There is great scatter as to the recommended formulas from different codes and no definite conclusion can be made.
Studying the trends of the data from this investigation the general equations can be recommended (discussed in the following section) for reinforced concrete frames with solid and flat slabs and for reinforced concrete with infill walls.
The fundamental natural period of reinforced concrete bare frames & masonry infill with solid slabs and flat plates was predicted by excel as a linear equation as shown in the Fig. 20, 21, 22 and 23.
Fig. 18. Comparison of Ta to the current study TnX direction.
Fig. 19. Comparison of empirical formulas to the current study TnZ
direction.
Fig. 20. Excel predicted linear equation for RC flat plate with bare frames.
Fig. 21. Excel predicted linear equation for RC Frame with bare frames. From this data the fundamental natural frequency of reinforced bare frames with solid slab or flat plate can be simplified as the following linear equation:
Tn=0.045H
where H is the total height of the building in meters.
Fig. 22. Excel predicted linear equation for RC Flat Plate + Infill.
Fig. 23. Excel predicted linear equation for RC Frame + Infill.
As when the infill masonry walls are included the excel predicted equation is:
Tn=0.025H
In terms of number of floors, the recommended simplified equations from the excel data take the following format:

Reinforced concrete bare frame: Tn=0.15N

Reinforced concrete frame with infill walls: Tn=0.10N
This last equation has been recommended by many investigators and is shown in the above graphs.


CONCLUSIONS
In summary, some major and general trends related to fundamental natural periods of reinforced concrete buildings of regular geometries are:

The fundamental natural periods of reinforced concrete buildings increase with increasing of mass and decreases with increasing stiffness. However, when cracking is considered in both columns and beams, the natural periods significantly increase.

Increasing the number of storeys in a building of the same height, the fundamental period decreased slightly in R/C bare buildings but in buildings with infill walls there was no effect.

The fundamental natural periods of reinforced concrete buildings depend greatly on the unreinforced masonry infill walls.

The fundamental natural periods do not decrease significantly with increasing concrete compressive strength in columns.

Flat plate flooring systems have slightly larger natural period than buildings with solid concrete slabs in buildings with and without infill masonry walls.

The empirical formulae for natural period recommended by (ASCE716) were in close agreement with the results obtained by the Rayleigh method in this study. Other formulas recommended by different codes were significantly different.

The fundamental natural period of reinforced concrete bare frames with solid slabs or flat plates can be predicted by the following linear equation: Tn=0.045H, where H is the total height of the building in meters. When the infill masonry walls are included, the recommended equation is Tn=0.025H. In terms of number of floors (N), the recommended equations take the following format for Reinforced concrete bare frames; Tn=0.15N, while in reinforced concrete frames with infill walls Tn=0.10N.
REFERENCES

R. Bhuskade Amravati, Ram Meghe and C.Sagane ,Effects of various parameters of building on natural time period, India 2017.

N.J Yogalakshmi and S. Baskar, Effect of storey height on the natural frequency of medium to high rise building, 2014.

Patel S. K, Desai A.N and V.B Patel, Effect of number of storeys to natural time period of building, Anand, Gujarat, India, May 2011.

M.C. Griffith and J. Vaculik, Outofplane flexural strength of unreinforced clay brick masonry walls, (URM) buildings in North America, September 2007.

H. Crowley and R. Pinho, Simplified equations for estimating the period of vibration of existing buildings, First European Conference on Earthquake Engineering and Seismology, Switzerland, September 2006.

L. Chung and T. Park , Evaluation of natural frequency through measurement of ambient vibration for walltype RC structures, Dankook University, Korea May 2016.

C. V. R. Murty, Rupen Goswami, A. R. Vijayanarayanan and Vipul V. Mehta, Some Concepts in Earthquake Behaviour of Buildings Gujarat State Disaster Management Authority, India. Chapter II, September 2012.

International Conference of Building Officials 1997 Uniform Building Code (UBC97), Volume 2, Chapter 16, DIV. I, Whittier, CA, April 1997

Eurocode 8: Design of structures for earthquake resistance Part 1: General rules, seismic actions and rules for buildings, The European Union Per Regulation, July 2009.

American Society of Civil Engineers (ASCE), ASCE/SEI 716, Minimum Design Loads and Associated Criteria for Buildings and Other Structures Reston, VA, USA, 2017

Otani Shunsuke, Japanese development of earthquake resistant building design, International Symposium on Earthquake Engineering,
University of Montenegro, Podgorica, Montenegro 2000

M. Rahimian, A. Mazroi, A. Momayez and N. Jokar , Reconnaissance study of the natural period of RC buildings for Iranian seismic code revision, World Conference on Earthquake Engineering Vancouver, Canada, August 2004.