Static and Dynamic Analysis of U-Shaped Building - …ijetmas.com/admin/resources/project/paper/f201605251464188000.pdf · Static and Dynamic Analysis of U-Shaped Building Deepak

International Journal of Engineering Technology, Management and Applied Sciences

www.ijetmas.com May 2016, Volume 4, Issue 5, ISSN 2349-4476

183 Deepak Anand, Yogesh Kaushik and Vipin Mann

Static and Dynamic Analysis of U-Shaped Building

Deepak Ananda, Yogesh Kaushik

a and Vipin Mann

b

aDepartment of Civil Engineering, Amity University, Uttar Pradesh, Noida, INDIA

b Border Security Force, Ministry of Home Affairs, New Delhi, INDIA

Abstract

The present study deals with the static and dynamic analysis of U-Shaped buildings. Basic geometries in arrangement

performs well in amid solid tremors then those with re-contestant corners like L, Y, U and + formed in arrangement as

they tend to bend amid a quake shaking. The present study considers a building intended in Zone IV with multi-story

inflexible jointed plane edge. The primary concern of the study is to minimize torsion because of tremor in a four story

U-shaped formed uneven building by situating the shear divider at most favorablearea in the structure. Static and

element examination are done using STAAD.Pro.2008. The manual figuring is done for diverse sorts of loads, materials

and measurements of segments i.e. column, beam, slab and wall is chosen. The self-weight for various components and

the seismic weight for every floor is computed. For assessing the outline seismic power, static examination is finished by

figuring the base shear, nodal and seismic powers at every level and torsional shear in both the sidelong bearings. To

calculate the characteristic recurrence and mode shape, dynamic investigation is finished by ascertaining modular mass,

modular interest component and parallel burdens for both horizontal headings. Toward the end, examination of base

shear of both static and element investigation for all levels in both courses is contemplated and at last the aftereffects of

manual and programming figuring are analyzed.

Keywords- Asymmetrical shape, Static analysis, Dynamic analysis, Torsional shear, Base shear,Lateral load

distribution

1. Introduction

Buildings are considered asymmetric in plan due to heavy mass at eccentric position, re-entrant corner, and

diaphragm irregularity and/or in elevation due to mass irregularity and geometric irregularity. Structures with

a deviated distribution of stiffness and posture in plan arrangement experience coupled sidelong and torsional

movements amid tremors. In numerous structures the center of resistance does not correspond with the center

of mass. In this study, by placing the shear wall at adequate positions, torsional impacts ought to be minimized.

The stiffness attributes control the dynamic reaction of the building structure.Lateral loads are live loads

whose principle component is a horizontal force acting on the structure.These lateral loads differ in intensity

of earthquake depending on the building's geographic location, structural materials, height and shape. The

horizontal components of this movement are usually considered critical in a structural analysis, so the lateral

load is calculated in both X and Y directions by modal analysis based on different mode shapes to study the

distribution on these loads.Seismic forces accumulate downward in a building known as Base shear which are

greatest at the base of the building and need to be evaluated for strengthening of the structure

A review of previous studies on asymmetric building systems reveals many aspects and conclusions. The

effects of asymmetry on the response of multistorey buildings to earthquakes using a computer based

procedure which accounts for lateral load analysis of asymmetric buildingsis presented by Thambiratnam and

Corderoy [14]. A method of seismic analysis of three-dimensional asymmetric multistorey buildings with

flexible foundations is given by Sivakumaran and Balendra [13]. The behavior of asymmetric building

systems under monotonic load considering that the coupled inelastic lateral and torsional deformations could

be the governing factors in their design when buildings are subjected to a large intensity ground motion is

studied by Ferhi and Truman [8]. The dynamic responses of a multi-storey irregular building under actual

earthquake records have been investigated and the variation of the building height on the structural response

of the shear wall building is analyzed by Bagheri et.al. [1]. The seismic performance assessment of a 21-story

building having an asymmetrical plan by using in-situ measurements is done by Bui et.al. [3]. Behavior of

frame-wall irregularity on established existing reinforced concrete (RC) structures subjected to the earthquake

force through nonlinear static analysis and nonlinear dynamic analysis is studied by Kocak et.al. [10]. The




seismic demands of two-way asymmetric-plan 15-storey building subjected to bi-directional seismic ground

motions with the effects of higher modes and torsion through Consecutive Modal Pushover procedure is

estimated by Poursha et.al. [11]. Statistical analysis of the time-history earthquake response of asymmetric

building model in terms of the lateral shear and storey torque developed due to earthquake base loading is

presented by Chandler and Hutchinson [5].

To account for torsional shear effect of low rise buildings having some lesser modes, static analysis gave good

results which assumes that the building responds in its fundamental mode. Buildings with torsional

irregularities, or non-orthogonal systems, a dynamic procedure is required. This method holds good if higher

modes can be considered. On the bases of work done on asymmetric building earlier by other researchers, the

project is to do the static and dynamic analysis of an unsymmetrical U-Shape building. While doing analysis,

use of different method, software, and equations which has been inculcated in the study has been discussed

later on in this paper. The design of footing and slab has also been done and all other aspects have been taken

care of which come forward while designing and analyzing the building.

2. Methodology

A four-storey reinforced concrete office building is considered having multi-storey rigid jointed planes frame

(Special RC Frame). The material used are (M25) concrete and (Fe415) reinforcement. The Storey height is

taken as 3m and the Plinth level is taken at 1.22 m. The thickness of Infill Masonry Wall is considered for

Outer Wall as 250 mm including plaster and for Inner Wall as 125 mm including plaster. For the structural

elements, the size of Column is taken as 400mm x 600mm and size of Beams as 400mm x 500mm. Depth of

Slab is assumed to be 150mm thick and Parapet height is 1m. The structure is modeled and analyzed in

STAAD. Pro in accordance with IS Code of Practice.

Fig. 1. Plan of the four-storey asymmetric building

For the calculation of self-weight, the formula is given as –

For different members i.e. column, beam, slab:-

S-Wt. = Area of member X Unit wt. of material

For Brick Wall, Inner Wall , Floor Wall , Terrace Parapet:-




S-Wt. = Major dimension of element X Unit wt. of material

The Slab load Calculation for (D.L. + L.L.) is done

The seismic weights are calculated in a manner similar to gravity loads. The seismic weight of each floor is its

full dead load in addition to appropriate amount of imposed load. The weight of columns and wall in a storey

is equally distributed to the floors above and below the storey. Reduced live loads are used for analysis: zero

on terrace and 50% on other floors as per I.S: 1893(Part-1):2002, Clause 7.4

2.1Static Analysis

This methodology characterizes a progression of strengths following up on a working to speak to the impact

of tremor ground movement, regularly characterized by a seismic outline reaction range. It expect that the

building reacts in its central mode. For this to be valid, the building must be low-ascent and must not wind

altogether when the ground moves. The reaction is perused from an outline reaction range, given the common

recurrence of the building (either ascertained or characterized by the construction regulation). The

appropriateness of this strategy is stretched out in numerous construction laws by applying variables to

represent higher structures with some higher modes, and for low levels of bending. Static calculations is done

by applying the procedures defined in the Indian Standard Code I.S: 1893(Part-1):2002 for this building. The

Base Shear (Vb) of the building is calculated by-

Vb = Ah x W

where, Ah = Design Horizontal Seismic coefficient. For the Base shear force, the Design Horizontal Seismic

Coefficient (Ah) has to be calculated which is given by:

where, Z is Zone Factor as per IS-1893 (Part- I): 2002, Table-2, I is Importance Factor as per IS-1893 (Part-

I): 2002, Table-6, R is Response Reduction Factor as per IS-1893 (Part- I): 2002, Table-7 and (Sa/g) is

Medium soil site and 5% Damping as per IS-1893 (Part- I): 2002, Figure-2. After accessing all these values,

the Nodal forces and seismic shear forces at various levels are calculated and the force (Qi) is expressed as:

Qi = (Vb x Wi x Hi2 ) / ( ∑ Wi x Hi

2 )

where, Vb is the Base Shear force, Wi is Seismic Weight at „i th‟ storey and H is the Cumulative Height at „i

th‟

storey. The degre of asymmetry of the structure is measured or defined in terms of structural eccentricity,

which are necessary to evaluate the torsional behavior, the maximum displacement and the seismic elastic

response of the structure. The procedure for evaluating the eccentricities of column EX and EY may be

arranged according to the following steps:

a) Evaluation of Moment of Inertia for fig. 3 as:

Ix = db3/12 ; Iy = bd

3/12

b) Evaluation of Stiffness by:

Kx = (12 E Ix) / L3 ; Ky = (12 E Iy) / L

3

Y

0.40 m

Fig. 2. Cross-sectional view of Column

0.60 m X




c) Assuming the origin at point (0,0), the coordinates of Center of Rigidity (geometric center of stiffness

of various vertical resistance elements) or point of rotation are computed as follows:

Xr =

∑KyX

Yr =

∑KxY

∑Ky ∑Kx

in which x, y are coordinates of the Kx and Ky in the two directions respectively.

d) Computing Center Of Mass as:

Xm =

∑mxx

Ym =

∑myy

∑mx ∑my

e) The Eccentricity is evaluated as:

ex = Xm - Xr

ey = Ym - Yr

The design eccentricity, (edi) to be used at floor I shall be taken as:

edi = (1.5 exi + 0.05 bi) or edi = esi – 0.05 bi

whichever of these has more severe effect in shear of any frame. The factor 1.5 represents dynamic

amplification factor, while the factor 0.05 represents the extent of accidental eccentricity.

For evaluating the rotational stiffness (Ixy) of the structure about the center of rotation Cr, the equation is given

by:

Ip = ∑ (KxY2 + KyX

2)

in which x and y are the distances of the elements from the center of rigidity Cr. Now, the Torsional moment

(Ti) at various floors, considering the seismic force in X & Y- direction respectively is as follows:

Tix = ediy x Vi

Tiy = edix x Vi

So therefore, the Torsional shear at each column line in X & Y direction respectively is worked out by using

equation:

Vx = (T. y. Kxx) / Ip

Vy = (T. x. Kyy) / Ip

2.2 Dynamic Analysis

Static strategies are suitable when higher mode impacts are not noteworthy. This is by and large valid for

short, customary structures. In this manner, for tall structures, structures with torsional inconsistencies, or non-

orthogonal frameworks, a dynamic methodology is required. In the straight element methodology, the

building is displayed as a multi-level of-opportunity (MDOF) framework with a direct versatile solidness grid

and a comparable thick damping network. The upside of these straight element methodology as for direct

static strategies is that higher modes can be considered. Be that as it may, they depend on straight flexible

reaction and subsequently the pertinence diminishes with expanding nonlinear conduct, which is

approximated by worldwide power lessening variables. In straight element investigation, the reaction of the

structure to ground movement is computed in the time space, and all stage data is along these lines kept up.

Just straight properties are expected. The explanatory technique can utilize modular decay as a method for




decreasing the degrees of opportunity in the investigation.

The Holzer‟s method is used to determine the natural frequencies and mode shapes of the structure. This

method falls under the determinant search technique and starts by assuming a trial frequency. Displacement of

unit amplitude is assumed at one end, and the forces and the displacements of different masses are then

calculated in a progressive manner. If the force or the displacement at the other end is compatible with the

conditions prevailing there, the assumed frequency is one of the natural frequencies of the system. The

dynamic analysis of the assumed structure is done in both X-direction and Y- direction and the method can be

easily programmed in a computer. The steps are as follows:

a) Calculation of Mass (m) and Stiffness (k) by using Equation of motion for free vibration for 5 number of

storeys as:

{Ẍ} + [K] {X} = {0}

where, [M] is the Mass matrix, {Ẍ} is Acceleration vector, [K] is Stiffness matrix and {X} is the

Displacement vector.

b) Structures have a tendency to have lower regular frequencies when they are either heavier (more mass) or

more elastic (that is less stiff). One of the primary things that influence the stiffness of a building is its

stature. Taller structures have a tendency to be more

elastic, so they have a tendency to have lower characteristic frequencies contrasted with shorter structures. So,

evaluation of the Natural frequencies (ω) of the building is done by formula:

and the fundamental natural frequency is given by:

c) A building can possibly "wave" forward and backward amid a seismic tremor (or even a serious wind

storm). This is known as the „fundamental mode‟, and is the most reduced recurrence of building reaction.

Most structures, notwithstanding, have higher methods of reaction, which are remarkably actuated amid

seismic tremors. Therefore, generating different Mode shapes from values of natural frequencies.

d) For real systems there is often mass participating in the forcing function (such as the mass of ground in an

earthquake) and mass participating in inertia effects (the mass of thestructure itself, Meq ). The modal

participation factor Γ is a comparison of these two masses. For a single degree of freedom system Γ = 1.

The calculation of Modal mass (Mk) and Modal participation factor (Γ) is followed up by using equation

as:

where, (Wi) is Seismic weight of floor, (ϕik) is the Mode shape coefficient at „i th‟ floor in „k

th‟ mode and (g)

as gravity. Now to evaluate how much mass has influenced in the modes, the percentage of total mass is

calculated as




and the Modal Participation factor as:

e) For illustration, the all five modes of vibration are considered and the lateral load (Qik) acting at ith floor in

the kth mode is computed by

Qik = Ahk x Ǿik x Pk x Wi

which is in accordance with clause 7.8.4.5 c of IS: 1893 (Part 1) and the value of (Ahk) for different modes is

obtained from clause 6.4.2. Since all of the modes are well

separated (clause 3.2), the contribution of different modes is combined by the SRSS method (square root of

the sum of the square) as per clause 7.8.4.4a of IS:1893 (Part 1) as

Vi = √∑ (Vik) 2

f) Now the Lateral forces at each floors due to all modes considered or the externally applied design loads

are then obtained as is calculated as

Q roof = Vroof

Qi = Vi – Vi+1

According to clause 7.8.2 requires that the base shear obtained by dynamic analysis be compared with that

obtained from empirical fundamental period as per Clause 7.6. If VB is less than that from empirical value,

which is true in this case, the response quantities are to be scaled up.

3. Results and Discussion

From the static approach, the nodal and its seismic forces are evaluated as the base for comparative study.

From Table 1, it is noticed that the maximum nodal force is at 5th floor and maximum seismic shear force is at

ground floor. After calculating the nodal and shear forces, also evaluating the eccentricity, center of rigidity

and center of mass the Torsional shear at each column line in X-direction is worked out in table 2. and

similarly, in Y-direction in table 3.

Table 1: Nodal forces and seismic shear forces at various levels

FLOOR Wi(kN) Hi(m) Wi x Hi x Hi Qi (kN) Vi (kN)

5 8233.5 13.22 1438955.621 1679.850 1679.850

4 9192 10.22 960089.693 1120.817 2800.667

3 9192 7.22 479164.253 559.381 3360.048

2 9192 4.22 163694.813 191.099 3551.147

1 3719.54 1.22 5536.163 6.463 3557.61

3047440.543 3557.61




Table 2: Torsional shear in various storeys in X – direction (in kN)

Column Line 1st 2

nd 3

rd 4

th 5

th

1 27.863 1.867 1.770 1.477 0.885

2 20.452 1.369 1.298 1.083 0.649

3 13.374 0.896 0.849 0.709 0.425

4 11.145 0.747 0.708 0.591 0.354

5 0.000 0.000 0.000 0.000 0.000

6 -11.145 -0.747 -0.708 -0.591 -0.354

7 -13.374 -0.896 -0.849 -0.709 -0.425

8 -20.432 -1.369 -1.298 -1.835 -0.649

9 -27.863 -1.867 -1.770 -1.477 -0.885

.

Table 3: Torsional shear in various storeys in Y – direction (in kN)

Column Line 1st 2

nd 3

rd 4

th 5

th

A 147.537 9.901 9.370 7.811 4.685

B 86.872 5.830 5.517 4.600 2.759

C 40.385 2.710 2.565 2.138 1.282

D -5.304 -0.356 -0.337 -0.281 -0.168

E -91.379 -6.132 -5.803 -4.838 -2.902

F -178.043 -11.948 -11.307 -9.426 -5.654

From dynamic analysis, vibration properties such as mass and stiffness of different storeys in X-direction are




find out as shown in table 4. The natural frequencies are evaluated with reference to different amplitudes as

demonstrated in table 5. On the bases of natural frequencies evaluated above, different mode shapes are

obtained (table 6). On studying these mode shapes, the 5th mode shows maximum deflection of storey. From

table 7, it is seen that the first two modes excites 97.04% of the total mass. Hence, in this case, codal

requirements on number of modes to be considered such that at least 90% of the total mass is excited will be

satisfied by considering the first two modes of vibration only. From lateral load calculation in table 8, the

unequal transfer of load between the structural elements in different storeys due to excitations is evaluated.

Similarly, the analysis is also performed for the excitation in Y- direction. The vibration properties is stated in

table 9. When the mode shapes of X-direction (table 6) is compared with the mode shapes of Y-direction

(table 11), it can be noticed that these mode shapes are almost same except 5th mode shape. Due to large

frequency for the 5th mode in X-direction, it shows unusual behavior.

Table 4: Free Vibration Properties of the building for vibration in the X-Direction

m1 839.2966361 839.2966361 839.2966361 839.2966361 839.2966361

m2=m3=m4 937.0030581 937.0030581 937.0030581 937.0030581 937.0030581

m5 379.158002 379.158002 379.158002 379.158002 379.158002

k1=k2=k3= k4 35555.56 35555.56 35555.56 35555.56 35555.56

k5 528678.61 528678.61 528678.61 528678.61 528678.61

Table 5: Natural Frequencies of the building for vibration in the X-Direction

ω2 3.8841 33.88232 84.36862 132.17851 1969.880745

a5 1 1 1 1 1

Fi5=m5*ω2*a5 1472.687596 12846.75275 31989.03739 50116.53976 746896.0475

Fk5=Fi5 1472.687596 12846.75275 31989.03739 50116.53976 746896.0475

k5=(Fk5/k5) 0.002785601 0.02429974 0.060507531 0.094795853 1.412760103

a4=a5- k5 0.997214399 0.97570026 0.939492469 0.905204147 -0.412760103

Fi4=m4*ω2*a4 3629.275624 30976.37326 74270.31347 112111.0436 -761866.1102

Fk4=Fi4+Fk5 5101.96322 43823.12601 106259.3509 162227.5833 -14970.06274

K4=(Fk4/k4) 0.143492698 1.232525265 2.988543869 4.562650211 -0.421032962




a3=a4- k4 0.853721702 -0.256825005 -2.0490514 -3.657446064 0.008272859

Fi3=m3*ω2*a3 3107.046352 -8153.638516 -161985.0024 -452980.7959 15269.91319

Fk3=Fi3+Fk4 8209.009572 35669.4875 -55725.65152 -290753.2126 299.8504511

k3=(Fk3/k3) 0.230878365 1.003204211 -1.567283753 -8.177433082 0.008433293

a2=a3- k3 0.622843336 -1.260029216 -0.481767647 4.519987018 -0.000160434

Fi2=m2*ω2*a2 2266.784495 -40003.20273 -38085.49336 559807.932 -296.1262596

Fk2=Fi2+Fk3 10475.79407 -4333.715227 -93811.14488 269054.7194 3.724191486

k2=(Fk2/k2) 0.294631671 -0.121885726 -2.63843812 7.567163036 0.000104743

a1=a2

- k2 0.328211665 -1.13814349 2.156670473 -3.047176018 -0.000265177

Fi1=m1*ω2*a1 1194.49799 -36133.59452 170492.6834 -377397.8327 -489.4590073

Fk1=Fi1+Fk2 11670.29206 -40467.30975 76681.53849 -108343.1134 -485.7348158

k1=(Fk1/k1) 0.328226923 -1.138142944 2.156668 -3.047149682 -0.01366129

a0=a1

- k1 -1.52582E-05 -5.45606E-07 2.47205E-06 -2.63351E-05 0.013396113

Table 6: Mode Shapes of the building for vibration in the X-Direction

a1 1 1 1 1 1

a2 1.897687995 1.107091704 -0.223384914 -1.483336372 0.605007274

a3 2.601131504 0.225652571 -0.950099436 1.200273973 -31.1975323

a4 3.03832711 -0.857273506 0.435621705 -0.297063295 1556.547329

a5 3.046814319 -0.878623837 0.463677698 -0.328172706 -3771.070212




Table 7: Modal Mass and Modal Participation factor in X -Direction (Clause: 7.8.4.5.)

Storey Weigh

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Level t (kN)

Roof 8233.5 1

8233.

8233.5 1

8233.

8233.5 1

8233.

8233.5 1

8233.

8233.5 1

8233.5 8233.5

5

5

5

5

4th

17443

33102.

10176 11266.

-

- 20225.

5561.2

9192 1.89

1.10

-0.22

2053. 458.6 -1.48

0.60

3364.58

Floor

.5

4

.4 2

13635 8

3

4

3rd

23909

2074.

-

11032 13242.

-

9192 2.60

62192 0.22

468.04 -0.95

8733. 8297.5 1.20

-31.19

28676 8946445

Floor

.6

2

.9 5

3

8

2nd

27928

84855.

-

6755.3

4004.

-

1.4E+0

9192 3.03

-0.85

7880. 0.43

1744.3 -0.29

2730. 811.16 1556.55

2.2E+10

Floor

.3

3

7

2

7

1

6

1st

11332

34528.

-

2871.4

1724.

-

-

3719.5 3.04

-0.87

3268. 0.46

799.6 -0.32

1220. 400.58 -3771.1

5.3E+10

Floor

.7

8

1

6

1E+07

1

7

Total

88847 22291 9335. 29594. 3175. 19533. 1680. 42913. 8163.5

7.5E+10

39529

.7

2

9 5

73 7

08 6

6

Mk 3541268.658 294513.9408 51630.12997 6577.582979 0.088651269

% of Total Mass 89.586508 7.450571549 1.306131643 0.166398753 2.24269E-06

Pk 0.398577418 0.315462182 0.162576982 0.039150375 1.08594E-07

Table 8: Lateral load calculation by modal analysis method (EQ in X –Direction)

Floor Weigh Mode1 Mode 2 Mode3 Mode4 Mode5

Level t W

Ǿi1 Qi1 Vi1 Ǿi2 Qi2 Vi2 Ǿi3 Qi3 Vi3 Ǿi4 Qi4 Vi4 Ǿi5 Qi5 Vi5

i (kN)

5 8233.5 1 50.40 50.40 1 117.81 117.81 1 95.80 95.80 1 29.01 29.01 1 8.04E-05 8E-05

4 9192 1.89 106.79 157.19 1.10 145.61 263.43 -0.22 -23.89 71.90 -1.48 -48.04 -19.03 0.60 5.43E-05 0.0001

3 9192 2.60 146.37 303.57 0.22 29.68 293.11 -0.95 -101.62 -29.70 1.20 38.87 19.84 -31.19 -0.002 -0.002

2 9192 3.03 170.98 474.55 -0.85 -112.76 180.35 0.43 46.59 16.88 -0.29 -9.62 10.22 1556.55 0.139 0.1371

1 3719.5 3.04 69.38 543.93 -0.87 -46.76 133.59 0.46 20.06 36.95 -0.32 -4.30 5.91 -3771.1 -0.137 8E-05




Table 9: Free Vibration Properties of the building for vibration in the Y-Direction

m1 839.2966361 839.2966361 839.2966361 839.2966361 839.2966361

m2=m3=m4 937.0030581 937.0030581 937.0030581 937.0030581 937.0030581

m5 379.158002 379.158002 379.158002 379.158002 379.158002

k1=k2=k3= k4 80000 80000 80000 80000 80000

k5 1189526.87 1189526.87 1189526.87 1189526.87 1189526.87

Table 10: Natural Frequencies of the building for vibration in the Y-Direction

ω2 8.73912 73.15 189.82941 297.40176 353.10384

a5 1 1 1 1 1

Fi5=m5*ω2*a5 3313.507278 27735.40785 71975.33982 112762.2571 133882.1465

Fk5=Fi5 3313.507278 27735.40785 71975.33982 112762.2571 133882.1465

k5=(Fk5/k5) 0.002785567 0.346692598 0.060507536 0.094795889 1.673526831

a4=a5- k5 0.997214433 0.653307402 0.939492464 0.905204111 -0.673526831

Fi4=m4*ω2*a4 8165.772318 44778.8481 167108.2176 252249.9334 -222842.6683

Fk4=Fi4+Fk5 11479.2796 72514.25595 239083.5575 365012.1905 -88960.52181

k4=(Fk4/k4) 0.143490995 0.906428199 2.988544468 4.562652382 -1.112006523

a3=a4 - k4 0.853723438 -0.253120797 -2.049052004 -3.657448271 0.438479692

Fi3=m3*ω2*a3 6990.784515 -17349.34841 -364466.3915 -1019207.791 145075.118

Fk3=Fi3+Fk4 18470.06411 55164.90753 -125382.8341 -654195.6008 56114.5962




k3=(Fk3/k3) 0.230875801 0.689561344 -1.567285426 -8.17744501 0.701432453

a2=a3 - k3 0.622847636 -0.942682142 -0.481766578 4.51999674 -0.262952761

Fi2=m2*ω2*a2 5100.239046 -64613.10602 -85692.17665 1259571.032 -87000.38688

Fk2=Fi2+Fk3 23570.30316 -9448.198485 -211075.0107 605375.4315 -30885.79068

k2=(Fk2/k2) 0.294628789 -0.118102481 -2.638437634 7.567192894 -0.386072383

a1=a2 - k2 0.328218847 -0.82457966 2.156671056 -3.047196154 0.123119623

Fi1=m1*ω2*a1 2687.646995 -56518.15249 383608.6717 -849151.0563 40735.28174

Fk1=Fi1+Fk2 26257.95015 -65966.35097 172533.6609 -243775.6248 9849.491064

k1=(Fk1/k1) 0.328224377 -0.824579387 2.156670762 -3.04719531 0.123118638

a0=a1 - k1 -5.53012E-06 -2.7334E-07 2.94194E-07 -8.4473E-07 9.84301E-07

Table 11: Mode Shapes of the building for vibration in the Y-Direction

a1 1 1 1 1 1

a2 1.897659572 1.143227497 -0.223384358 -1.48332976 -2.135750219

a3 2.601079877 0.306969489 -0.950099459 1.200266765 3.561411921

a4 3.038260729 -0.792291434 0.435621585 -0.29706132 -5.470507598

a5 3.04674765 -1.212739106 0.463677573 -0.328170538 8.122182142

After studying the modal mass and modal participation factors for both the directions, it can be concluded

from the graph (fig. 4) that due to excitation generated in both directions, have almost similar quantity of mass

participation i.e. first two modes prevails maximum mass contribution (more than 90%). It is also seen that,

the mode having maximum mass participation has maximum value for modal participation factor. From table

14, which accounts for base shear, it has been noticed that even though the base shear by the static and the




dynamic analysis were comparable there was considerable difference in lateral load distribution with building

height, and therein lied the advantage of dynamic analysis. For instance, the storey moments are significantly

affected by change in load distribution.

Table 12: Modal Mass and Modal Participation factor in Y -Direction (Clause: 7.8.4.5.) Store

Weig

y

ht

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Leve

(kN)

l

Roof

8233.

1 8233.5

8233.5 1 8233.5 8233.5 1 8233.5 8233.5 1 8233.5 8233.5 1

8233.5

8233.

5

5

4th

9192 1.89

17443.

33101.4 1.14 10508.5 12013.7 -0.22 -2053.3 458.68 -1.48 -13635 20224.9 -2.13

-19632

4192

Floor

3

8.7

3rd

9192 2.60

23909.

62189.5 0.30 2821.66 866.165 -0.95 -8733.3 8297.52 1.20 11032.9 13242.4 3.56

32736. 1165

Floor

1

5

88

2nd

9192 3.03

27927.

84851.6 -0.79 -7282.7 5770.05 0.43 4004.23 1744.33 -0.29 -2730.6 811.15 -5.47

-50285

2750

Floor

7

84

1st 3719.

3.04

11332.

34527.3 -1.21 -4510.8 5470.46 0.46 1724.67 799.69 -0.32 -1220.6 400.57 8.12

30210. 2453

Floor

54 5

8

77

Total

88846.

222903

9770.14 32353.8

3175.74 19533.7

1680.35 42912.5

1264.0 6872

39529

1

6

12

Mk

3541279.476

295036.2546

51630.24694

6579.882739

232.5109212

% of Total

Mass 89.58678167 7.463784968 1.306134602 0.166456932 0.005882028

Pk

0.398585786

0.301977626

0.16257718

0.039157721

0.001839401

Table 13: Lateral load calculation by modal analysis method (EQ in Y–Direction)

Floor

Weight

Mode1 Mode 2 Mode3 Mode4 Mode5

Level

W (kN) Ǿi1 Qi1 Vi1 Ǿi2 Qi2 Vi2 Ǿi3 Qi3 Vi3 Ǿi4 Qi4 Vi4 Ǿi5 Qi5

Vi5

i

5 8233.5 1 75.59 75.59 1 165.70 165.70 1 120.47 120.47 1 29.01 29.01 1 1.36 1.36302

4 9192 1.89 160.15 235.75 1.14 211.48 377.18 -0.22 -30.04 90.42 -1.48 -48.05 -19.03 -2.13 -3.24 -1.8869

3 9192 2.60 219.52 455.28 0.30 56.78 433.97 -0.95 -127.79 -37.35 1.20 38.88 19.84 3.56 5.41 3.53245

2 9192 3.03 256.42 711.70 -0.79 -146.57 287.40 0.43 58.58 21.23 -0.29 -9.62 10.22 -5.47 -8.32 -4.792

1 3719.54 3.04 104.05 815.75 -1.21 -90.78 196.62 0.46 25.23 46.46 -0.32 -4.30 5.92 8.12 5.00 0.20926




Fig. 3. Graph b/w Modes and its Modal mass and Modal Participation factor

Table 14: Base Shear at Different Storeys

X Direction

Storey

Storey Shear

Shear

Floor Q(static) Q(dynamic (V) Storey Moment, Storey Moment,(M)

(V)

Level KN scaled) KN (dynamic (M) (static) KNm (Dynamic) kNm

(static)

scaled) KN

KN

5 1679.85 1030.535086 1679.85 1030.535086 5039.55 3091.605258

4 1120.817 969.9901549 2800.667 2000.525241 13441.551 9093.180981

3 559.381 683.4266057 3360.048 2683.951847 23521.695 17145.03652

2 191.099 535.901461 3551.147 3219.853308 34175.136 26804.59644

1 6.463 337.7566917 3557.61 3557.609999 38515.4202 31144.88064

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

500000

1000000

1500000

2000000

2500000

3000000

3500000

4000000

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5

MODAL MASS MODAL PARTICIPATION FACTOR




Y Direction

Storey

Storey Shear

Shear

Floor Q(static) Q(dynamic (V) Storey Moment, Storey Moment,(M)

(V)

Level KN scaled) KN (dynamic (M) (static) KNm (Dynamic) kNm

(static)

scaled) KN

KN

5 1679.85 932.5255077 1679.85 932.5255077 5039.55 2797.576523

4 1120.817 990.6059463 2800.667 1923.131454 13441.551 8566.970885

3 559.381 745.4660001 3360.048 2668.597454 23521.695 16572.76325

2 191.099 582.102525 3551.147 3250.699979 34175.136 26324.86318

1 6.463 306.9100211 3557.61 3557.61 38515.4202 30665.14738

Table 15: Comparison of various properties

Models Base Shear Cm Cr Weight C.G. Rotational Stiffness Eccentricity

1.22m

X(KN) Y(KN) X(m) Y(m) X(m) Y(m) (KN) X(m) Y(m) Z(m) 3m Column Column X(m) Y(m)

Manually 3557.61 3557.61 9 15 9.728 15 39529 319027148.5 4743641781 -0.728 0

Static

9.37 7.05 15.02

Dynamic

3425.61 3421.33 9.25 15 9.727 15 38996.54 9.37 7.05 15.02 319027040 4743641088 -0.47 0

After comparing various properties of static and dynamic behavior from table 15, it is noticed that the base

shear comes out to be almost same in both direction and by analysis. Also, the center of mass and center of

rigidity is equal. If we take weight into consideration, it is seen that there is a decrease of 1.35 % by dynamic

analysis. The important output noticed here is that there is 35.4% change in the eccentricity in X-direction.

4. Conclusions

The paper presents an investigation on the static and dynamic behavior of an asymmetric building. Dynamic

analysis shows an important difference while considering study of higher mode shapes. Following conclusions

have been made based on the study:

It was noticed that even though the base shear by the manual static and the manual dynamic analysis

were comparable, there was considerable difference in lateral load distribution with building height and

there in lied the advantage of dynamic analysis.

Difference of values between manual and STAAD Pro result for base shear in X and Y direction was




observed to be 3.71% and 3.83% respectively.

For center of mass, center of rigidity and rotational stiffness, negligible difference was observed.

Weight calculations by manual and STAAD Pro showed the difference of 1.35%.

As per the results obtained, the change in eccentricity of the building in X-direction is observed.

For future study, Consecutive Modal Pushover analysis can be done to enhance the high torsion capacity of

asymmetric building. Also Base Isolation technique may come handy for unsymmetrical building to study the

seismic behavior of the structure. Moreover, Seismic investigation between various irregular shape structures

should be possible and contrasting the consequences of seismic reaction of each shape with can help in

accessing better decision of lopsided shape depending on seismic execution.

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