Presentation on Effect of height and number of floors to natural time period of a multi-storeyed building

““EffEct of hEight and numbEr of floorsEffEct of hEight and numbEr of floors
to natural timE pEriod of a multi-storEyto natural timE pEriod of a multi-storEy
buildingbuilding””
PRESENTED BY MENTOR
NILESH V. PRAJAPATI SHRI S.K. PATEL
ASSISTANT ENGINEER SUPERINTENDING ENGINEER
R&B DESIGN CIRCLE, GANDHINAGAR
DATE:21-09-2015
GUIDED BY
SHRI PROF. A.N.DESAI
B.V.M.ENGINEERING COLLEGE,V.V.NAGAR,ANAND
1

FLOW OF PRESENTATION
Introduction.
 Literature review.
Objective of work and mode of working.
Problem Definition.
Problem formulation.
Software verification.
Results and Discussion.
Conclusion.
Nilesh prajapati-A.E. R& B Design Circle,Gandhinagar. 2

Introduction

 As per IS 1893:2002 The approximate fundamental natural
period of vibration (T ), in seconds, of a moment-resisting
frame building without brick infill panels may be estimated by
the empirical expression:
 Ta = 0.075 h0.75
for RC frame building
= 0.085 h0.75
for steel frame building
Where
h = Height of building, in m.
This excludes the basement storeys, where basement
walls are connected with the ground floor deck or fitted
between the building columns. But it includes the basement
storeys, when they are not so connected.

 The approximate fundamental natural period of vibration
( T, ), in seconds, of all other buildings, including moment-
resisting fame buildings with brick infill panels, may be
estimated by the empirical expression:
 Ta = 0.09h/√ d
Where
h= Height of building, in m
d=Base dimension of the building at the plinth level, in m,
along the considered direction of the lateral force.

 In this work attempt is been made to show that
Natural time period is also a function of Number of
floors and not only of height of building , as it is not
mentioned in above formula .

LITERATURE REVIEW

Literature survey is essential to review the work done in
the area of performance based Engineering.
To take up the specific need to perform the analysis, the
literature like technical papers, journals and books need to
be referred.
The literature review concentrates on a range of
earthquake engineering topics and structural modeling
aspects.
 For the understanding of seismic capacity, a review of
literature is required in experimental testing, current
design practice, theoretical strength evaluation and
modeling techniques such as finite element modeling.

The literature review begins with a coverage of general
earthquake engineering topics, which serves to set the
context of the research.

Ms A Cinitha, Dr G M Samuel Knight, Dr V Ramamurthi [P12]
The dynamic relationships between the model and
prototype structure depend on the geometric and
material properties of the structure and on the type of
loading.
Parametric studies have been carried out on
extended numerical model to study the effect of height
of the building, height of storeys, number of storeys
and size of beams and columns and bracings on the
fundamental frequency.

The conclusions drawn as per their work are:
Fundamental frequency against height of the building
behavior showed a non-linear decreasing trend with
increase in height of the building irrespective of the size of
beams and columns.
Increase in height of the building from 10 m to 30 m
decreases the fundamental frequency to one third,
whereas if the cross-section of beams and columns are of
smaller sections, the fundamental frequency decrease by
more than 50% irrespective of the plan of the building.

For a building with constant height, increase in
height of the storeys decrease the fundamental
frequency by more than 20%. the fundamental
frequency of the frames with smaller span increases
by 26% as compared to frames with larger span.

L.Govinda Raju ,G.V. Ramana,C.HanumanthaRao and T.G.Sitharam [P8]
They performed study on Reinforced Cement Concrete
multistory building plane frames for different configurations
of number of bays (each of 4 m span) and storey height (3
m each) were analyzed for their natural frequencies using
Finite Element Analysis package (NISA) without considering
the infill effect.
Figure 1 shows the variation of natural frequencies for
different storey heights and bays corresponding to first and
second modes respectively.

Figure 1. Variation of natural frequency with number of bays and storeys in (a) First mode (b) Second mode.

Figures 1a and 1b represent the effect of number of
bays and storey height on the natural frequency of the
structure for first and second modes respectively.
It is evident that the magnitude of natural frequencies
is not much influenced by the number of bays.
Further the natural frequency of the structure
decreases as the number of storeys is increased.
It can be noticed from the figures that for 4 to 10
storied buildings, the natural frequency ranges between
1.5 Hz to 3.0 Hz for the first mode and 2.5 Hz to 8 Hz for
the second mode.

D.E. Allen and G. Pernica [P6]
They performed study for Repetitive forces
produced by human activities for the floor acceleration
due to cyclic force and stated that the taller the
columns supporting the floor on which the rhythmic
activity takes place the lower the natural frequency of
the floor.
An example of this occurred when aerobics on the
top storey of a 26-storey building caused second
harmonic resonance due to the axial flexibility of the
columns.

This resonance produced annoying vibrations of
approximately 1% g (gravity) in the offices below.
If the vibration is very large (more than 20% g), and
occurs frequently (e.g., in a health club), then fatigue
failure of the floor can occur.
To prevent collapse due to fatigue or overloading, the
National Building Code (NBC) requires a dynamic
analysis of a floor structure if it has a natural frequency of
less than 6 Hz.

Objective and Mode of working
 In this work attempt is been made to show that
Natural time period is also a function of Number of
floors and not only of height of building , as it is not
mentioned in formula as per IS1893:2002.
 And also to find out the effect of variation of bays (in
plan) on natural time period, for same height of
building and same storey height.
 For This STAAD Pro. Software is used for relevant
analysis

The specific objectives were as follows:
•To prepare various R.C.C. models in STADD-Pro .
•To assess the change in natural time period with
respect to variation of height and number of floors of
R.C.C. building.
•To assess the change in natural time period with
respect to variation of number of bays (in plan) for the
same height of R.C.C. building.

PROBLEM IDENTIFICATION

 In this research work STADD-Pro software has been used. In
it, various model of R.C.C. framed building were prepared.
 The height of RCC building varies from 60m to 90m with
respect to increase in number of floors from 20 to 30
numbers for the constant storey height of 3 m.
 The plan dimension of all models are 70 m × 70 m. All
columns are of same size and also all beams are of same
size in each model.
 In each sub sequent model there has been a variation in
number of floors. Suppose in first model, number of floors
are 20. In next model, the number of floors would be 21.
Thus, the variation of each number of floors has been
conducted in each sub-sequent model.

Using STADD-Pro software, ANALYSIS of each model
has been carried out.
Then, concrete design is been carried out manually for
maximum axial load for column and for maximum BM for
beam.
With this actual design the MODAL CALCULATION has
been carried out using STAAD-Pro software.
As the number of floors increases, height of building will be
increased and due to this, variation in natural Time-period
has been obtained.

•For the further analysis, for same height of building
and same storey height, variation in bays (in plan) was
made to find out the effect of number of bays on
natural time period.
•For this variation of bays was made from 11 to 14 in
numbers.
•Hence plan dimension varies from 50m ×50m to
70m× 70m and keeping other dimensions as constant.

Problem formulation:
 Plan dimension : 70 m × 70 m
 Height of building : 90 m for sample model (varies from 60 m to 90 m)
 Height of each storey : 3m (constant)
 Number of bays along X-direction: 14 nos.
 Number of bays along Y-direction: 14 nos.
 Length of each bay(in X-direction): 5m
 Length of each bay(in Y-direction): 5m
 Number of floors varies as :20,21,22,23,24,25,26,27,28,29,30.
 Column size: 450 mm × 300 mm (may be changed as per actual design)
 Beam size: 300 mm × 600 mm (may be changed as per actual design)
 Modules of elasticity of concrete: 2 × 10^5
 Grade of concrete: M-20
 Grade of steel: Fe-415
 Density of concrete: 25 KN/m3
 Density of wall: 20 KN/m3
 Live load: 4 KN/m2
 Slab thickness: 120 mm
 Wall thickness: 230 mm (periphery wall)
 115 mm (internal wall)
 230 mm (parapet wall)

Fig STAAD-Pro software display

Plan of a building

Front view

3D view of a model

Load calculation
Various loads were calculated as below:
Slab thickness is 120 mm
Density =25 KN/m3
(for M25)
Floor load:
DL = 0.12×25
=3.00 KN/m2
LL=3.00 KN/m2
Wall load:
For typical floor
Height of storey =3 m
Height of wall =3.0-0.6
=2.4 m
Density =18 KN/m3
Load from external wall =18×.23×2.4
=9.936 KN/m
Load from Internal wall =18×0.115×2.4
=4.968 KN/m
For top floor
Height of parapet wall = 1.00 m
Thickness =0.23 m
Load from parapet wall=18×0.23×1
=4.160 KN/m

As per this revised design, sizes carried out for all
columns as 1000*1000 mm and all the beams as
300*600 mm.

To calculate natural time period for each model,
MODAL MASS CALCULATION has been performed
using Eigen value extraction method in STAAD.pro.,
for the variation of each number of floors and height
for sub-sequent model.
 The number of floor varies from 20 to 30 and height
of building varies from 60m to 90m respectively for the
constant storey height of 3m.
For the further analysis the variation in numbers of
bays (in plan) has been made for each building height,
keeping other dimensions (i.e. number of floors and
storey height)as constant and plan dimension varies
as 50m,55m,60m,65m,and 70m keeping storey
height(3m) and total height of building as constant for
that particular model.

For example, If the first model is of 20 storey building and
total height of a building is 60 m and the number of bays
varies as from 10 to 14 each of 5m. hence plan dimension
varies as 50m,55m,60m,65m, and 70m keeping storey
height(3m) and total height of building as constant(60m) for
that particular model.
For these models also the MODAL CALCULATION has
been carried out using Eigen value extraction method using
STAAD-Pro software to calculate natural time period of a
building.

SOFTWARE VARIFICATION

To verify the software, mass and stiffness has been
manually calculated.
 After calculating mass and stiffness natural time
period and natural frequency has been calculated for
a single model and natural time period has been
calculated.

Sample calculations to calculate natural time period using Eigen value constant.
Slab = 0.12×25×70×70 = 14700 KN
Beam = 0.3×0.6×25×70×70= 9450 KN
Live load = 70×70×1.5 =7350 KN
Column = 1×1×25×225×3 =16875 KN
Ex. Wall= 0.23×20×70×4×2.4 = 3091.2 KN
Int. Wall=0.115×20×70×26×2.4 = 10046.4 KN
Total mass (m) = 61512.6 KN
= 6151260 kg
E=2×105
N/mm2
I=B×D3
/12
=8.33× 1010
N/mm
K = ∑ ωn
=
= 2.0889×1012
N/m
= 589.067Hz
h= height of column
ω = = 589.067× 0.077*
= 45.358 Hz
T =
= 0.234 Sec
= 0.13845 Sec
Indicates the Eigen-value constant (for20storey) which is calculated from iteration method for different lumped mass systems.

Sr. No
No of
floors
Storey height
(m)
Height of
building
Mass (kg)
Stiffness, *1012
(N/m)
Natural frequency
ωn (Hz)
Constant Frequency ω (Hz)
Natural time period
T (sec)
1 20 3 60 6019884 2.0889 589.067 0.077 45.358 0.13845
Table 1 Results for natural time period for different models using the Eigen
value constant.

RESULTS AND DISCUSSION

Sr no No. of floors storey height
(m)
Height of
building (m)
Natural time
period T(second)
(as per
IS1893:2002)
Natural frequency
(cycle/second)
(ω=2π/T)
1 20 3.00 60.00 0.6454 9.7301
2 21 3.00 63.00 0.6777 9.2668
2 22 3.00 66.00 0.7100 8.8454
4 23 3.00 69.00 0.7422 8.4609
5 24 3.00 72.00 0.7745 8.1084
6 25 3.00 75.00 0.8068 7.7840
7 26 3.00 78.00 0.8391 7.4847
8 27 3.00 81.00 0.8713 7.2075
9 28 3.00 84.00 0.9036 6.9501
10 29 3.00 87.00 0.9359 6.7104
11 30 3.00 90.00 0.9681 6.4867
Table 2 Results for Natural Time period for Different models as per IS1893:2002.

Table 3 Results for Natural time period calculated based on MODALCALCULATION
using Eigan value extraction method in STAAD.Pro. Software.
Sr. no. No of floors storey height
(m)
Height of
building (m)
Time Period
(T)
(seconds)
Natural frequency
ω (cycle/second )
1 20 3 60 0.13462 7.43
2 21 3 63 0.14116 7.08
3 22 3 66 0.14768 6.77
4 23 3 69 0.15420 6.49
5 24 3 72 0.16070 6.22
6 25 3 75 0.16720 5.98
7 26 3 78 0.17368 5.76
8 27 3 81 0.18016 5.55
9 28 3 84 0.18663 5.36
10 29 3 87 0.19309 5.18
11 30 3 90 0.19954 5.01

Table 4 Results for Natural Time period for Different models calculated based on MODALCALCULATION using Eigan value
extraction method in STAAD.Pro. Software for the variation of bays (in plan).
No of floors storey height Height of building No. of bays plan Dimension Time Period (T)
x y x y
20 3 60 10 10 50 50 0.13356
3 60 11 11 55 55 0.13392
3 60 12 12 60 60 0.13431
3 60 13 13 65 65 0.13444
3 60 14 14 70 70 0.13462
21 3 63 10 10 50 50 0.13998
3 63 11 11 55 55 0.14038
3 63 12 12 60 60 0.14070
3 63 13 13 65 65 0.14095
3 63 14 14 70 70 0.14116
22 3 66 10 10 50 50 0.14637
3 66 11 11 55 55 0.14682
3 66 12 12 60 60 0.14717
3 66 13 13 65 65 0.14745
3 66 14 14 70 70 0.14768
23 3 69 10 10 50 50 0.15276
3 69 11 11 55 55 0.15324
3 69 12 12 60 60 0.15363
3 69 13 13 65 65 0.15394
3 69 14 14 70 70 0.15420
24 3 72 10 10 50 50 0.15913
3 72 11 11 55 55 0.15966
3 72 12 12 60 60 0.16008
3 72 13 13 65 65 0.16042
3 72 14 14 70 70 0.16070
25 3 75 10 10 50 50 0.16549
3 75 11 11 55 55 0.16606
3 75 12 12 60 60 0.16652
3 75 13 13 65 65 0.16689
3 75 14 14 70 70 0.16720
26 3 78 10 10 50 50 0.17184
3 78 11 11 55 55 0.17245
3 78 12 12 60 60 0.17295
3 78 13 13 65 65 0.17335
3 78 14 14 70 70 0.17368

No of floors storey height
Height of
building
No. of bays plan Dimension
Time
Period (T)
27 3 81 10 10 50 50 0.17817
3 81 11 11 55 55 0.17883
3 81 12 12 60 60 0.17937
3 81 13 13 65 65 0.17980
3 81 14 14 70 70 0.18016
28 3 84 10 10 50 50 0.18450
3 84 11 11 55 55 0.18521
3 84 12 12 60 60 0.18578
3 84 13 13 65 65 0.18628
3 84 14 14 70 70 0.18663
29 3 87 10 10 50 50 0.19082
3 87 11 11 55 55 0.19157
3 87 12 12 60 60 0.19218
3 87 13 13 65 65 0.19267
3 87 14 14 70 70 0.19309
30 3 90 10 10 50 50 0.19713
3 90 11 11 55 55 0.19793
3 90 12 12 60 60 0.19857
3 90 13 13 65 65 0.19910
3 90 14 14 70 70 0.19954

Keeping constant Building Height i.e. 90 m, and same plan dimension now variation
is made for the Numbers of floors for the same building and with same member
properties, following results for the natural time period were obtained from Eagan
value factor method as shown in table no 5 and table no 6.

heigh
t
Natural
frequency ωn
(Hz)
Constant Frequency ω
(Hz)
Natural
time
period T
(sec)
90 529.42 0.05149 26.797 0.2
90 451.41 0.0551 24.87 0.256
90 395.69 0.05926 23.451 0.268
90 348.877 0.0641 22.363 0.281
90 311.154 0.06979 21.71 0.289
90 278.8 0.07304 20.35 0.308
90 251.14 0.0766 19.23 0.327
90 227.53 0.08053 18.32 0.343
90 207.37 0.08488 17.6 0.357
Sr.
No
Storey
height
(m)
Mass (kg) Stiffness,1012
(N/m)
1 3 6151260 1.667
2 3.25 6428700 1.31
3 3.5 6706100 1.05
4 3.75 6983500 0.85
5 4 7261100 0.703
6 4.25 7538500 0.586
7 4.5 7816000 0.493
8 4.75 8093500 0.419
9 5 8371000 0.36
TABLE 5. TABLE 6.

DISCUSSION :
From the results obtained for natural time period using STAAD.Pro. and
formula given in IS1893:2002 it can be stated that values obtained from
STAAD.Pro. using Eigen value extraction method are different than the
values obtained from formula given in code, in which natural time period
is only function of total height of the building and plan dimension of a
building. And as the number of floor increases, Natural time period
also increases.
From this, it can be stated that natural time period is also changes with
variation in number of floors.

From above Results of research work it can be seen that
natural time period is also a function of number of floors and
not only of height of building.
To calculate natural time period mathematical formula can be
revised using rigorous analysis.
CONCLUSION :

Future Scope of work:
This work is based on regular building where plan dimension of a
building in both the direction has been kept symmetrical. Further
work in this area can be achieved by considering irregular
building.
In this work column size has been kept constant throughout the
building height. One may go for the variation of sizes for the
different levels as per the actual design consideration.
In this research work there is no shear wall considered. One may
go for the with shear wall consideration.

References:
Technical Papers:
•Mills, R.S. “Small-scale modeling of the nonlinear response of steel-framed buildings to earthquakes” Design for Dynamic
Loading and Modal Analysis, Construction Press, pp.171-177.(1979)
•Krawinkler,H. and Benjamin.J. Wallace., “Small-scale model experimentation on steel assemblies” Report No.75, The John A.
Blume Earthquake Engineering Centre, Department of Civil Engineering, Stanford University, Stanford.(1985)
•Lagomarsino, S., “Forecast models for damping and vibration periods of buildings” J. of Wind Eng. and Ind Aerodyn. Vol. 48,
pp.221-239,(1993)
•Tamura, Y., Suganuma, S. , “Evaluation of amplitude-dependent damping and natural frequency of buildings during strong
winds.” J. of Wind Eng. and Ind. Aerodyn., Vol. 59, pp. 115-130.(1996)
•Goel, K.R.and Chopra, K.A. “Period formulas for moment- resisting frame Buildings”, J.of Struct.Eng., ASCE,Vol.123,pp.1454-
1461. (1997),
•D.E. Allen and G. Pernica, Control of Floor Vibration,dec (1998)
•Bhandari, N. and Sharma, B. K., Damage pattern due to January,2001 Bhuj earthquake, India: Importance of site amplification and
interference of shear waves, Abstracts of International Conference on Seismic Hazard with particular reference to Bhuj Earthquake
of 26 January 200I, NewDelhi,(2001),.
•L. Govinda Rajul, G. V. Ramana, C. HanumanthaRao and T. G. Sitharaml ,site specific ground response analysis,(2003)
•Kim, N.S., Kwak, Y.H.and Chang, S.P, “Modified similitude law for pseudo dynamic test on small-scale steel models”J.of
Earthquake Eng. Society of Korea, Vol.7, pp. 49-57. (2003)
•Tremblay, R. and Rogers, C.A. “Impact of capacity design provisions and period limitations on the seismic design of lowrise steel
buildings” Intl.J.of Steel Struct., Vol. 5, pp.1-22. (2005)
•Technical paper by Dr V Kanwar, Dr N Kwatra, Non-memberDr P Aggarwal, Dr M L Gambir, Evaluation of Dynamic Parameters
of a Three-storey RCC Building Model using Vibration Techniques , July 04, (2007)
•Thecnical paper by Ms A Cinitha, Dr G M Samuel Knight, Dr V Ramamurthi, Evaluation of Free Vibration Characteristics of
Steel Space Frames,(2007)
•Rama Raju. K, Nagesh.R. Iyer,, Ms A Cinitha, Evaluation of Free Vibration Characteristics of Steel Space Frames, oct (2008)
•Siefko Slob, Micro Seismic Hazard Analysis
•Siefko Slob, Robert Hack , Tom Scarpas, Bas van Bemmelen and Adriana duque , a methodology for seismic microzonation using
GIS & SHAKE.

Technical Journals:
1)Earthquake tips,IITK,KANPUR,INDIA,2002
IS Codes:
1)IS 1893:2002 Part-I Indian standard Criteria for earthquake resistant design of structures.
2)IS 13920:1993 Indian standard code of practice for Ductile detailing of reinforced concrete
structures subjected to seismic forces.
3)IS 456:2000 Indian standard code of practice for Plain and Reinforced concrete.

Gratitude to :
Shri Vivek. P. Kapadiya sir, Chief Engineer and MD,GWSRDC,Gandhinagar for
this motivation for this presentation and valuable guidance.
Shri S.K.Patel sir, Superintending engineer, R&B design circle, Gandhinagar for
his valuable support and guidance.
Shri Sunil.B.Prajapati my elder cousin brother and Superintending
engineer, C.D.O, irrigation dept., Gandhinagar for his valuable guidance
throughout my Career.

thank you all !!

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