1. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),
ISSN 0976 – 6316(Online), Volume 5, Issue 6, June (2014), pp. 10-15 © IAEME
10
INVESTIGATION OF THE CRITICAL DIRECTION OF SEISMIC FORCE
FOR THE ANALYSIS OF R.C.C FRAMES
Sohel Ahmed Quadri Mangulkar Madhuri N
P.G Student, Dept. of Structural Asst Professor Dept. of Structural
Engineering, J.N.E.C, Aurangabad Engineering, J.N.E.C, Aurangabad
(M.S).India (M.S).India
ABSTRACT
A simple method which can be applied in seismic codes to determine the critical angle of
seismic incidence is proposed in this paper. Two 4-story reinforced concrete buildings with moment
resisting frames, one with square and the other with rectangular plan, have been analysed by
Equivalent Static Method of analysis. A set of values from 0 to 90 degrees, with an increment of 10
degrees, have been used for angle of excitation. Buildings’ columns have been divided into three
main categories, including corner, side, and internal columns, and axial force and bending moment
values in different columns, have been investigated in all cases. The results show that the axial forces
of columns may exceed the ordinary cases up to 13% by varying the angle of excitation. Each
column gets its maximum axial force and moments with a specific angle of excitation, which is not 0
or 90 degree necessarily, and it varies from column to column.
Keywords: Angle of Excitation, Equivalent Static method, IS 1893:2002 (Part-1) Provisions,
Columns Axial Forces, Moments.
1. INTRODUCTION
An earthquake can be explained in the horizontal plane as two orthogonal acceleration
components of different intensities, which can excite a structure with any horizontal incidence angle.
This situation is assumed in several codes considering two orthogonal components of equal
intensities. Although in the seismic design of structures the directions of ground motion incidence
are usually applied along the fixed structural reference axis, it is known that for most world tectonic
regions the ground motion can act along any horizontal direction; therefore, this implies the
existence of a possible different direction of seismic incidence that would lead to an increase of
structural response. Critical angles are earthquake incidence angles producing critical responses.
INTERNATIONAL JOURNAL OF CIVIL ENGINEERING
AND TECHNOLOGY (IJCIET)
ISSN 0976 – 6308 (Print)
ISSN 0976 – 6316(Online)
Volume 5, Issue 6, June (2014), pp. 10-15
© IAEME: www.iaeme.com/ijciet.asp
Journal Impact Factor (2014): 7.9290 (Calculated by GISI)
www.jifactor.com
IJCIET
©IAEME

2. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),
ISSN 0976 – 6316(Online), Volume 5, Issue 6, June (2014), pp. 10-15 © IAEME
11
The maximum structural response associated to the directions of ground seismic motions has
been examined in several papers. Lopez and Torres (1997) have tried to present a simple method,
which can be applied to determine the critical angle of seismic incidence and the corresponding peak
response of structures subjected to two horizontal components applied along any arbitrary directions
and to the vertical component of earthquake ground motion. In their method the seismic components
are given in terms of response spectra that may be equal or have different spectral shapes. In that
study the structures are discrete, linear systems with viscous damping. Their method, which is based
on the response spectrum method of analysis, requires. For the general case of three arbitrary
response spectra, their method requires the solution of five seismic loading cases. These procedures
are usually identified in technical literature as complete quadratic combination rule with three
seismic components or CQC3. A more accurate structural response can be obtained with equilibrium
static method of analysis; however, for practical applications it requires the use of several ground
motions and hence enormous numerical efforts. Several examples of this procedure are presented in
technical literature. See for Fernández-Dávila (2000), Mahmood Hosseini and Ali salami (2008).
In this study two set of 4-story reinforced concrete buildings with moment resisting frames,
one with square and the other with rectangular plan, have been analysed by Equivalent Static Method
of analysis. A set of values from 0 to 90 degrees, with an increment of 10 degrees, have been used
for angle of excitation. The details of study and its result are described briefly in the following
section of paper.
2. PARAMETRIC DETAILS OF MODELS WHICH ARE STUDIED
In this work we consider the following two structures for the seismic analysis.
Figure 1: Square Structure Figure 2: Rectangular Structure
The basic specifications of above Structures are: In square structure dimensions of all the
beams from B1 to B9 are =0.23m x 0.45m and all the columns from C1 to C9 are = 0.3m x 0.3 m. In
rectangular structure dimensions of all the beams from B1 to B6 are = 0.23 m × 0.38 m; and all the
beams from B7 to B12 are = 0.23 m × 0.45 m. M20, Fe415 materials are used for both the structure.
The structures are assumed to be located in seismic Zone - V on a site with hard soil. Response
reduction factor as 5 for special moment resisting frame is considered for seismic analysis.

3. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),
ISSN 0976 – 6316(Online), Volume 5, Issue 6, June (2014), pp. 10-15 © IAEME
12
3. METHOD OF ANALYSIS
The present study undertaken deals with Linear Static Method of Analysis or Equivalent
Static Method of Analysis of 3D frames that can be used for regular structure with limited height.
For the 3D modelling and analysis of the Structure standard software package is used. Seismic force
is applied with incidence angle of 0 to 90 degrees, with an increment of 10 degrees and column
forces have been investigated in all cases. The columns have been divided into three main categories,
including corner, side, and internal columns. The natural period of the building is calculated by the
expression, T=0.09H/√D as per IS 1893:2002(part1), where H is the height and D is the base
dimension of the building in the considered direction of vibration. The lateral load calculation and its
distribution along the height are done as per B.I.S provisions. The seismic weight is calculated using
full dead load plus 25% of live load. Load combinations as per clause 6.3.1.2 of IS 1893:2002
(Part-1) are considered.
4. NUMERICAL RESULTS AND DISCUSSION
Among different internal forces following variation were observed in axial forces of columns.
Figure 3: Axial force variation for corner columns of square plan
Figure 4: Axial force variation for corner columns of rectangular plan.

4. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),
ISSN 0976 – 6316(Online), Volume 5, Issue 6, June (2014), pp. 10-15 © IAEME
13
Figure 3 and 4 represents the increase in axial force with respect to angle of seismic incidence
from 0 to 90 degrees with an increment of 10 degrees in all the corner columns of square and
rectangular structure respectively. Table 1 shows the percent variation of axial force of columns of
the first storey for various maximizing cases with respect to the base case.
Table 1: Percentage of variation of axial force in column in various cases
Building
Plane Shape
Column
Category
Critical
Angle
Variation
Percent
Square
Corner 49 12.55%
Side 10 0.34%
Middle 90 -
Rectangular
Corner 52 7.84%
Side 10 0.31
Middle 90 -
It can be seen in Table 1 that the maximum axial force in each column may occur by a
specific angle of seismic incidence, which is different from the critical angle for other columns. It
should be mentioned that the maximum variations does not necessarily belong to the same column in
each category.
The variation in moments due to the effect of critical direction of seismic force for square
structure is shown in table 2.
Table 2: variation of moments in columns for square structure
COLUMN
CATEGORY
COLUMN
NAME
Critical
angle
Moments in KN for
(0 or 90 degree)
Moments in KN for
critical angle
My Mz My Mz
Corner
C1 49 -3.193 -65.108 -49.923 -43.814
C7 49 65.108 -3.193 43.814 -49.923
C3 49 -3.193 65.108 -43.814 49.923
C9 49 65.108 3.193 49.923 43.814
SIDE
C2 10 -66.259 0 -65.292 13.087
C4 10 0 -66.259 -13.087 -65.292
C6 10 0 66.259 13.087 65.292
C8 10 66.259 0 65.292 -13.087
It is seen in table 2 that maximum bending moment in the entire corner columns were
obtained by applying seismic force at an angle of 49 degree. It is observed for column one and seven
that bending moments in one direction decreases by 32.70% but bending moments in other direction
increases 15.635 times, similarly for column three and nine the bending moments in one direction
decreases by 23.29% but bending moments in other direction increases13.712 times which makes
seismic incidence angle 49 degree critical. Maximum bending moment in the entire side columns

5. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),
ISSN 0976 – 6316(Online), Volume 5, Issue 6, June (2014), pp. 10-15 © IAEME
14
were obtained by applying seismic force at an angle of 10 degree. All the side columns which were
uni-axial become biaxial column. This makes seismic incidence angle 10 degree critical. The
bending moment in central column occurs mainly by applying seismic force at an angle of incidence
of either 0 or 90 degree.
Table 3: variation of moments in columns for rectangular structure
Column
Category
Column
Name
Critical
angle
Moments in KN for
(0 or 90 degree)
Moments in KN for critical
angle
My Mz My Mz
Corner
C1 52 -9.248 -72.562 -66.712 -57.773
C7 52 72.562 -9.248 57.773 -66.712
C3 52 -9.248 72.562 -57.773 66.712
C9 52 72.562 9.248 66.712 57.773
SIDE
C2 10 -191.932 0 -189.149 28.782
C4 10 0 -141.245 -38.618 -139.125
C6 10 0 141.245 38.618 139.125
C8 10 191.932 0 189.149 -28.782
It is seen in table 3 that maximum bending moment in the entire corner columns were
obtained by applying seismic force at an angle of 52 degree. It is observed for column one and seven
that bending moments in one direction decreases by 20.381% but bending moments in other
direction increases 7.213 times similarly for column three and nine the bending moments in one
direction decreases by 8% but bending moments in other direction increases by 6.247 times which
makes seismic incidence angle 52 degree critical. Maximum bending moment in the entire side
columns are obtained by applying seismic force at an angle of 10 degree. All the side columns which
were uni-axial become biaxial columns. This makes seismic incidence angle 10 degree critical. The
bending moment in central column occurs mainly by applying seismic force at an angle of incidence
of either 0 or 90 degree.
5. CONCLUSIONS
Based on the numerical results it can be concluded that:
• The axial forces of column may exceed the ordinary cases up to 13% in corner columns. This
critical direction is very close to the diagonal direction of the structure.
• Critical direction for columns on edge is very close to the conventional direction we normally
follow as far as Axial Forces are considered.
• There is increase in moments for columns on edge if we apply Seismic force other than the
conventional directions.
• There is no unique specific angle of incidence for each structure which increases the value of
internal forces of all structural members together; each member gets its maximum value of
internal forces by a specific angle of incidence.
• It is recommended to apply Seismic force in a set of different angle ranging between 0 to 90
degrees for safe design.

6. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),
ISSN 0976 – 6316(Online), Volume 5, Issue 6, June (2014), pp. 10-15 © IAEME
15
ACKNOWLEDGEMENTS
The authors wish to thank the Management, Principal, Head of Civil Engineering Department
and Staff of Jawaharlal Nehru Engineering College and Authorities of DR.Babasaheb Ambedkar
Marathwada University along with Mr Kareem M Pathan for their support.
REFERENCES
[1] Lopez, Oscar A. And Torres, R. (Sept. 1997). The critical angle of seismic incidence and the
maximum structural response, Earthquake Engineering & Structural Dynamics, Vol. 26,
no. 9, pp. 881-894.
[2] Fernández-Dávila, I., Cominetti, S., and Cruz, E.F. (2000). “Considering the Bi-Directional
Effects and the Seismic Angle Variations in Building Design”. 12th World Conference on
Earthquake Engineering, E.Q.C., Auckland, paper 0435.
[3] Smeby, W. And Der Kiureghian, A. (1985). “Modal Combination Rules for Multicomponent
Earthquake Excitation”. Earthquake Engineering and Structural Dynamics, 13, 1-12.
[4] Angelo Marinilli and Oscar A.Lopez Professor, Instituto de Materials y Modelos
Estrucutrales IMME Facultad de Ingenieria. Universidad Central de Venezuela.” Evaluation
of critical response and critical incidence angles obtained with RSA and RHA” The 14th
World Conference on Earthquake Engineering October 12-17, 2008, Beijing China.
[5] Mahmood Hosseini and Ali Salemi “Studying the effect of earthquake excitation angle on the
internal forces of steel building’s elements by using non-linear time history analyses”, The
14th World Conference on Earthquake Engineering October 12-17, 2008, Beijing China.
[6] Victor I. Fernandez-Devila 1 and Ernesto F. Curz “ Study of the effect of in-plan asymmetry
in multi-story buildings subjected to Uni and bidirectional seismic motions” 13th World
conference on Earthquake Engineering, Vancouver B.C., Canada August 1-6, 2004, Paper
No. 1042.
[7] Faramarz Khoshnoudian and Mehdi Poursha “Responses of three dimensional buildings
under bi-directional and unidirectional seismic excitations” 13th World conference on
Earthquake Engineering, Vancouver, B.C., Canada August 1-6, 2004 Paper No. 55.
[8] Mangulkar Madhuri N. and Gaikwad Madhukar V., “Review on Seismic Analysis of
Elevated Water Tank”, International Journal of Civil Engineering & Technology (IJCIET),
Volume 4, Issue 2, 2013, pp. 288 - 294, ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316.
[9] Mangulkar Madhuri N. and Gaikwad Madhukar V, “Comparison between Static and
Dynamic analysis of Elevated water Tank”, International Journal of Civil Engineering &
Technology (IJCIET), Volume 4, Issue 3, 2013, pp. 12 - 29, ISSN Print: 0976 – 6308,
ISSN Online: 0976 – 6316.
[10] Misam.A and Mangulkar Madhuri.N., “Structural Response of Soft Story-High Rise
Buildings under Different Shear Wall Location”, International Journal of Civil Engineering &
Technology (IJCIET), Volume 3, Issue 2, 2012, pp. 169 - 180, ISSN Print: 0976 – 6308,
ISSN Online: 0976 – 6316.
[11] Snehal D. Poojara and Dr. Paresh V. Patel, “Axial Deformation of Columns in Multi-Story
R.C. Buildings”, International Journal of Civil Engineering & Technology (IJCIET),
Volume 5, Issue 3, 2014, pp. 294 - 300, ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316.
[12] Gaikwad Madhukar V. and Prof. Mangulkar Madhuri N., “Seismic Performance of Circular
Elevated Water Tank with Framed Staging System”, International Journal of Civil
Engineering & Technology (IJCIET), Volume 4, Issue 4, 2013, pp. 159 - 167, ISSN Print:
0976 – 6308, ISSN Online: 0976 – 6316.