NOTES ON 3D MULTI MASSES DYNAMIC ANALYSIS

HADI RUSJANTO TANUWIDJAJA*

1. Introduction

Today’s engineers are familiar very well in operating and using “user-friendly” commercial computer programs; keep and believe for granted that everything had been solved perfectly and correctly from the printed computer outputs. Prof. Wilson (2004) remarks specifically that “do not use a structural analysis program unless you fully understand the theory and approximations used within the program” and “do not create a computer model until the loading, material properties and boundary conditions are clearly defined”. The author personal observation shows that most of engineers in structural design offices have not enough back up knowledge and understanding in dynamic analysis theory of structures.

3D dynamic analysis is becoming a routine structural analysis which is currently required for buildings design subjected to earthquake loads in this era of modern existence of inexpensive personal computers. Unfortunately not many textbooks and course subjects are available to be taught in the university in emphasizing the background and practical application of the theory into the reality of daily structural design office practices.

It is interesting to note what Prof Powel (2010) felt that “most of young engineers use computer programs blindly, without understanding what they are doing, This is probably true, and it is unfortunate. However, my experience tells me that young engineers are not to blame”. The paper is written for the purpose of filling the niche of many members of engineering profession who used to work with the assistance of commercial structural analysis and design computer programs but loss of confident about their doing.

Many design aspects to be mentioned for examples; present seismic code weakness that it does not state specifically how to define the principal directions for a 3D-structure of arbitrary geometric shape. Unawareness by most engineers in predefining earthquake directions may produce base shears that underestimate the appropriate design values since it is not a unique design base shear that associated with the fundamental modes of vibration in the major principal direction. In most of mix-used development buildings project, it is not uncommon to have several building towers which significant differences in building masses and number of stories then these to be united and integrated into one large several lower stories for podiums or basement floors. This particular 3D-structural frames may produce large lateral torsional moments associated to the higher modes induced by larger masses contribution at lower stories of podiums or basement floors movements.

A well designed structure should be capable of equally resisting earthquake motion from all possible directions and also should have minimum amount of lateral torsional moments in the mode shapes associated with the lower frequencies of the structure.

*Associate Professor Trisakti University & President Director Haerte Widya Consulting Engineers

2. Fundamental Assumptions One of the most important applications of the theory of structural dynamics is an analyzing the response of structures to ground motion caused by an earthquake. In general, structural response is expressed in terms of the displacement of the structure, through the solution of the dynamic force of equilibrium or equations of motion. Typical standard equations of motion due to three components of free field ground displacements can be set in the form of Ns second order differential equations :

gzgzygyxgx utumtumtumtututu..

r mkcm )()()()()()(....

..

.....

……...……(1)

The number of degrees of freedom is equal to the number of lumped masses in the system. In building type of structures, in which the floor system can have any number of columns and beams connecting to it; at the floor level intersection end of each member, six degrees of freedom exist for 3D structure and the masses are lumped at the nodes.. The in-plane deformations in the floor systems are small compared to the interstory horizontal displacements, then, it has become common practice to assume the in plane motion of all points on the floor diaphragm move as a rigid body. The in plane displacements of each floor diaphragm can be expressed in terms of two lateral displacements ux

(m) , uy(m) , and a lateral rotation about the z- axis , uzΘ

(m).

For relatively small displacements of each structural member, the materials property could be reasonably assumed as linearly elastic isotropic material where the stress-strain relationships of the materials is linear and its have equal properties in all directions.

There are several different classical methods that can be used for the solution of Eq. (1). The most common and effective approach for seismic analysis is the mode superposition method. For the purpose of dynamic response analysis, it is often advantages to express the displaced position u(t) in terms of the free-vibration mode shapes =

N..... , , , 3 21 (separation variables) ;

)()( tYtu …………………………………………….(2)

where is an Nd x N matrix containing N spatial vectors that are not a function of

time, and )(tY is a vector containing N function of time. It was noted that vibration mode

amplitudes obtained from the eigen problem solution are arbitrary, in the analysis process the amplitude (the first, actually) has been set to unity, and the other displacements have been deterimined relative to this reference value (normalizing the mode shapes with respect to the specified reference coordinate) :

Nnnn

kn

Nnnn

T

n uuuu

..........11

.......... 3231 , , , , , , 2n ……………..(3)

Because of the orthogonality property with respect to mass, i.e. 0m

T

n m ,

0m

T

n k , 0m

T

n c , for m n , therefore, IT m and

2 k

T, where I

is a diagonal unit matrix and 2 is a diagonal matrix in which the diagonal terms are

2

n

and n may or may not of free vibration frequency in radians per second. The use of

normal modes coordinates serve to transform the equations of motion, Eq. (1) from a set

of N simultaneous differential equations, which are coupled by the off-diagonal terms in mass , stiffness and damping matrices, to a set of N independent (uncoupled) normal-coordinate equations :

)()()()(

...

tPtYKtYCtYM nnnnnnn ……………………………………..(4)

where :

n

T

nnM m , n

T

nnK k , n

T

nnC c , )( )(r mp tututtP gngT

n

T

nn

....

)()( L ,

nL is defined as modal participation factors or in this case earthquake excitation

factors., r is a vector of ones for the structure which represents the displacements resulting from a unit ground displacement excitation either in x, y (both translations), or z

(rotational), and n is the mode number ; so that 12

1

in

N

i

in

T

n mm

Note : For rotational excitation the Mass Moment of Inertia (MMI) = mr 2

= OI which

shall be used in the above equations, and no

T

nnM I

( 12

1

in

N

iino

T

n I I o )

3. Dynamic Analysis Using Design Response Spectra In designing structures to perform satisfactorily under earthquake excitations, the engineer needs a much more precise characterization of the ground shaking of the specific site under consideration. For this purpose, study and observation of the response of a single oscillator freedom (SDOF) induced by ground motion has proved to be invaluable. The graphs showing the absolute maximum values of the structural

response, ),(paS , plotted as function of period )2

(T for discrete values of

damping ratio, , are called pseudo acceleration response spectra. The response

spectral values and shapes , paS for earthquake ground motions depend upon many

independent variables respectively such as as source mechanism (SM), epicentral distance (ED), focal depth (FD), geological conditions (GC), Richter magnitude (RM), soil condition (SC), damping ratio and period. Due to the lack of knowledge as to their influences, in the modern design response spectrum curves when normalized and averaged to a fixed intensity level , currently are specified in terms of only two

parameters SC and . Using the direct statistical approach as similarly developed by

Seeds (1976) the average pseudo-acceleration spectra for different types of site-soil conditions and correlated to the numerous recorded past earthquakes expressed in terms of g had been normalized with respect to peak ground accelerations as shown in Fig 2, of the Indonesian seismic design code SNI 1726-2002 (2010 ?), the typical design spectra for Jakarta was copied and shown in Fig.1.

0.20

0.13

0.10

0.08

0.050.04

0 0.5 1.0 2.0 3.00.60.2

lunak) (TanahT

0.20C

sedang) (TanahT

0.08C

keras) (TanahT

0.05C

0.38

0.30

0.20

0.15

0.12

0 0.5 1.0 2.0 3.00.60.2

lunak) (TanahT

0.50C

sedang) (TanahT

0.23C

keras) (TanahT

0.15C

0.50

0.75

0.55

0.45

0.30

0.23

0.18

0 0.5 1.0 2.0 3.00.60.2

lunak) (TanahT

0.75C

sedang) (TanahT

0.33C

keras) (TanahT

0.23C

0.60

0.34

0.28

0.24

0 0.5 1.0 2.0 3.00.60.2

lunak) (TanahT

0.85C

sedang) (TanahT

0.42C

keras) (TanahT

0.30C

0.85

0.70

0.90

0.83

0.70

0.36

0.32

0.28

0 0.5 1.0 2.0 3.00.60.2

(Tanah lunak)T

0.90C

(Tanah sedang)T

0.50C

(Tanah keras)T

0.35C

0.95

0.90

0.83

0.380.360.33

0 0.5 1.0 2.0 3.00.60.2

(Tanah lunak)T

0.95C

(Tanah sedang)T

0.54C

(Tanah keras)T

0.42C

T

Wilayah Gempa 1

C

T

Wilayah Gempa 2

C

T

Wilayah Gempa 3

C

T

Wilayah Gempa 5

C

T

Wilayah Gempa 4

C

T

Wilayah Gempa 6

C

Fig.1 – Design Spectra for Jakarta (SNI 1726-2002)

The standard design procedures for dynamic analysis will be described as follows :

1. The first step in the dynamic analysis of a structural model is the calculation of the 3D

mode shapes and natural frequencies of vibration (or natural periods), n or nT .

2. The value of maximum response for the modal spectral acceleration, namS

, nn T , =

Cn , is found from the seismic design code which is usually expressed in units of gravitational acceleration g.

3. Calculate the modal mass participating factor or modal earthquake excitation factor.

By definition the modal participation factor and the generalized mass nM

Ln = r m T

n = in

N

i

im1

=N

i

iniWg 1

1 …………………………………………(5)

N321 LLLL ...... , , , m 1 ………………………………………………(6)

4. Define the quantity of nM

2nL as effective modal mass. Since total mass

1 m 1TM = N

nM

2nL (for rotational excitation then 1 I o1oTI =

N

n nM1

2nL

);

the number of significant modes of vibration to be considered in design for most of building codes require that at least 90% of the effective modal participating mass should be included in the calculation of response for each principal direction.

5. Calculate respectively the modal story shears ( sf ), base shears (V ), overturning

(OTM) and lateral torsional moments (ts, torques) , displacements (u) or drifts :

gCM

SM

n

n

n, nam

n

n m mf sin

nn LL ………………………………………(7)

s

N

i

sin fV f 1

1 = gCM

SM

n

N

n n

nam

N

n n

1,

1

2n

2n LL

……………………………(8)

N

i

in hfOTM1

sin ……………………………………………………………..(9)

gCM

SM

n

n

nnam

n

nosn m It ,

nn LL…………………………………(10)

2

n

2

, m u

gC

M

S

M

n

n

n

n

nam

n

nn

nn LL…………….……………………….(11)

4. Practical Examples

EXAMPLE 1 - Regular Symmetrical Plan

A three story with regular plan ETABS OUTPUT

Mode Period Mode UX UY RZ SumRX SumRY SumRZ

1 0.521735 1 -0.0354 0 0 0 99.1767 0

2 0.496507 1 -0.0296 0 0 98.8866 99.1767 0

3 0.488673 1 -0.0195 0 0 98.8866 99.1767 93.6149

4 0.144574 2 0 -0.0349 0 98.8866 99.9515 93.6149

5 0.138088 2 0 -0.0297 0 99.9569 99.9515 93.6149

6 0.134828 2 0 -0.0202 0 99.9569 99.9515 99.3592

7 0.071017 3 0 0 0.00349 99.9569 100 99.3592

8 0.069642 3 0 0 0.00285 100 100 99.3592

9 0.063753 3 0 0 0.0018 100 100 100

4 -0.0321 0 0

4 0.0117 0 0

4 0.0361 0 0

5 0 -0.0324 0

5 0 0.0106 0

5 0 0.0363 0

6 0 0 -0.00309

6 0 0 0.00126

6 0 0 0.00345

7 -0.0177 0 0

7 0.038 0 0

7 -0.0264 0 0

8 0 0.0182 0

8 0 -0.0382 0

8 0 0.0257 0

9 0 0 -0.00168

9 0 0 0.00362

9 0 0 -0.00261

BACKWARD ANALYSIS FROM ETABS

m 1 2 3 4 5 6 7 8 9

1 384.9984 0 0 0 0 0 0 0 0

2 0 384.9984 0 0 0 0 0 0 0

3 0 0 40712.54 0 0 0 0 0 0

4 0 0 0 407.1168 0 0 0 0 0

5 0 0 0 0 407.1168 0 0 0 0

6 0 0 0 0 0 43836.77 0 0 0

7 0 0 0 0 0 0 419.5584 0 0

8 0 0 0 0 0 0 0 419.5584 0

9 0 0 0 0 0 0 0 0 45594.14

story 3 story 2 story 1

3

2

1

mode shapes Ф

1 2 3 4 5 6 7 8 9

-0.0354 0 0 -0.0321 0 0 -0.0177 0 0

0 -0.0349 0 0 -0.0324 0 0 0.0182 0

0 0 0.00349 0 0 -0.00309 0 0 -0.00168

-0.0296 0 0 0.0117 0 0 0.038 0 0

0 -0.0297 0 0 0.0106 0 0 -0.0382 0

0 0 0.00285 0 0 0.00126 0 0 0.00362

-0.0195 0 0 0.0361 0 0 -0.0264 0 0

0 -0.0202 0 0 0.0363 0 0 0.0257 0

0 0 0.0018 0 0 0.00345 0 0 -0.00261

3

2

1

Lnx Lny Lnz Lnx2 Lny

2 Lnz2

-33.861 0 0 1146.567 0 0

0 -34.0029 0 0 1156.197 0

0 0 349.0910266 0 0 121864.5

7.550876 0 0 57.01573 0 0

0 7.07146 0 0 50.00554 0

0 0 86.73236352 0 0 7522.503

-2.42038 0 0 5.858215 0 0

0 2.23776 0 0 5.00757 0

0 0 -28.7086896 0 0 824.1889

∑ = 1209.441 1211.21 130211.2

total 1211.674 1211.674 130143.5

accuracy (%) 99.82 99.96 100.05

Base Shear V = Lnx2

S am

630.6116 6179.994 0 0

0 0 635.9082 6231.9

0 0 0 0

13.11362 128.5135 0 0

0 0 11.50128 112.7125

0 0 0 0

1.34739 13.20442 0 0

0 0 1.151741 11.28706

0 0 0 0

Vbx Vby

fs1 fs2 fs3 fs4 fs5 fs6 fs7 fs8 fs9

2487.428 0 0 -210.337 0 0 37.17652 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

2199.374 0 0 81.06925 0 0 -84.3994 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

1493.191 0 0 257.781 0 0 60.42726 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

Vbx 6179.994 0 0 128.5135 0 0 13.20442 0 0 base sehar x-x

fs1 fs2 fs3 fs4 fs5 fs6 fs7 fs8 fs9

0 0 0 0 0 0 0 0 0

0 2462.572 0 0 -198.823 0 0 35.34254 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 2216.053 0 0 68.78407 0 0 -78.4422 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 1553.275 0 0 242.7516 0 0 54.38673 0

0 0 0 0 0 0 0 0 0

Vby 0 6231.9 0 0 112.7125 0 0 11.28706 0 base shear y-y

x-x SEISMIC FORCES FOR EACH STORY

y-y SEISMIC FORCES FOR EACH STORY

Note : ETABS OUTPUT agrees very well with the theoritical results of backward-analysis

EXAMPLE 2 - Single story irregular plan

Single story irregular plan ETABS OUTPUT

Mode Period Mode UX UY SumRX SumRY SumRZ

1 0.277944 1 0.0915 0.0259 1.838 22.9208 76.0175

2 0.187345 2 -0.1654 0.0463 7.697 97.739 93.8501

3 0.170171 3 -0.0287 -0.1837 100 100 100

BACKWARD ANALYSIS

m

27.36 0 0

0 27.36 0

0 0 434.0511

Ф

0.0915 -0.1654 -0.0287

0.0259 0.0463 -0.1837

-0.04163 -0.0211 -0.01119

Lnx2 Lny

2 Lnz2 Iox=LnΘx Ioy=LnΘy

6.267212 0.502148 326.5085 -45.236 -12.8045

20.47874 1.604701 83.87772 41.44526 -11.6017

0.616589 25.261 23.59076 3.813897 24.4116

27.36254 27.36785 433.977 0.023134 0.005415

should equal to zero

Vbx Vby TORQUES Tbx Tbx

4.700409 46.06401 0.376611 3.690788 -33.927 -332.485 -9.60339 -94.1132

15.35905 150.5187 1.203526 11.79455 31.08395 304.6227 -8.70125 -85.2723

0.462442 4.531931 18.94575 185.6683 2.860423 28.03214 18.3087 179.4252

fs1 fs2 fs3

46.06401 150.5187 4.531931 Vbx

13.03888 -42.1343 29.00752

-332.485 304.6227 28.03214 Tbx

x-x seismic forces

fs1 fs2 fs3

13.03888 -42.1343 29.00752

3.690788 11.79455 185.6683 Vby

-94.1132 -85.2723 179.4252 Tby

y-y seismic forces

NOTE : ETABS OUTPUT agrees very well with the theoritical results of backward-analysis

EXAMPLE 3 - Second Story irregular plan

Second Story irregular plan ETABS OUTPUT

Mode UX UY RZ Mode Period SumUX SumUY SumRZ

1 -0.0897 -0.0062 0.03659 1 0.511953 23.7206 0.2562 65.7177

1 -0.0415 -0.0073 0.01792 2 0.367243 86.8642 0.4052 88.5947

2 0.1491 -0.0051 0.0224 3 0.319362 86.8769 90.2398 88.8441

2 0.0652 -0.0052 0.01052 4 0.163567 89.2762 90.2624 96.7824

3 0 0.1687 0.00363 5 0.107548 97.8402 92.7827 97.726

3 0.0029 0.0863 -0.00091 6 0.100288 100 100 100

4 0.0364 0.0269 -0.01883

4 -0.0749 -0.0294 0.03637

5 0.0632 -0.0471 0.0063

5 -0.1362 0.0861 -0.01192

6 0.0309 0.0712 0.00747

6 -0.0676 -0.1375 -0.01716

BACKWARD ANALYSIS

m

27.36 0 0 0 0 0

0 27.36 0 0 0 0

0 0 434.0511 0 0 0

0 0 0 28.8 0 0

0 0 0 0 28.8 0

0 0 0 0 0 456.102

2

1

2 1

Ф

STORY 1 2 3 4 5 6

2x -0.0897 0.1491 0 0.0364 0.0632 0.0309

2y -0.0062 -0.0051 0.1687 0.0269 -0.0471 0.0712

2z 0.03659 0.0224 0.00363 -0.01883 0.0063 0.00747

1x -0.0415 0.0652 0.0029 -0.0749 -0.1362 -0.0676

1y -0.0073 -0.0052 0.0863 -0.0294 0.0861 -0.1375

1z 0.01792 0.01052 -0.00091 0.03637 -0.01192 -0.01716

Lxn2 Lyn

2 Lzn2 Iox n Ioy n

13.31806 0.144303 578.6564 -87.7871 -9.13793

35.48747 0.083692 210.8576 86.5032 -4.20085

0.006976 50.42522 1.346883 0.096929 8.241168

1.348423 0.012262 70.81639 -9.77192 -0.93187

4.811039 1.418538 7.30196 5.927058 -3.2184

1.213205 4.048015 21.01625 5.049458 9.223563

∑ 56.18517 56.13203 889.9955 0.017588 -0.02432

∑ 56.18517 56.13203 889.9955

Lxn2

Cumm % Cumm Lyn2

Cumm % Cumm Lzn2

Cumm % Cumm

13.31806 13.31806 23.70387 0.144303 0.144303 0.257077 578.6564 578.6564 65.0179

35.48747 48.80553 86.8655 0.083692 0.227995 0.406176 210.8576 789.514 88.70989

0.006976 48.81251 86.87791 50.42522 50.65322 90.23941 1.346883 790.8609 88.86122

1.348423 50.16093 89.27788 0.012262 50.66548 90.26126 70.81639 861.6773 96.81816

4.811039 54.97197 97.8407 1.418538 52.08402 92.7884 7.30196 868.9792 97.63861

1.213205 56.18517 100 4.048015 56.13203 100 21.01625 889.9955 100

56.18517 56.13203 889.9955

BASE SHEARS V

Vbx Vby

7.324934 71.78435402 0.079367 0.777792

19.51811 191.2774596 0.046031 0.451101

0.003837 0.037598432 27.73387 271.792

0.310137 3.039344537 0.00282 0.02764

1.106539 10.84408113 0.326264 3.197385

0.279037 2.734564791 0.931044 9.124226

BASE TORQUES Tbx TbY

-48.28293 -473.1726716 -5.02586 -49.25342

47.57676 466.2522513 -2.310467 -22.64258

0.053311 0.522449246 4.532642 44.4199

-2.247542 -22.02590784 -0.21433 -2.100437

1.363223 13.35958785 -0.740232 -7.254277

1.161375 11.38147895 2.121419 20.78991

NOTE : ETABS OUTPUT agrees very well with the theoritical results of backward-analysis

5. Conclusions Theoretical 3D-dynamic analyses had been briefly elaborated and clearly applied through practical design examples by the assistance of ETABS computer program. It has been shown that for simple regular plans through irregular structural plans with only one type of rigid diaphragm, ETABS output agrees very well with the theoretical results of backward analysis.

Particular cases for the irregular structural plans with multi masses and multi rigid diaphragms as commonly found in the case of mixed use building analytical model will be separately written for the next publication.

6. References BSN (2002), “Indonesian Earthquake Resistance Design Requirements for Buildings,” SNI 03-1726-2002, 64 pp.

RSNI 03-1726-xxxx, (2010), “Draft of The Indonesian Earthquake Resistance Standard Design Requirements for Structural and Non- Structural Buildings,” 106 pp.

Clough, R., Penzien, J. (2003) , “Dynamic of Structures,” 2nd Edition (revised), Computer and Structure Inc., 739 pp.

Paz, M, (1991), “Structural Dynamics, Theory and Computation,” 3rd Edition, Van Nostrand Reinhold, New York, 626 pp.

Chopra, A.K. (1995), “Dynamics of Structures, Theory and Applications to Earthquake Engineering,” Prentice Hall , New Jersey, 729 pp.

Wilson, E. L. (2004), “Static & Dynamic Analysis of Structures, A Physical Approach With Emphasis on Earthquake Engineering,” 4th Edition, Computer and Structure Inc., 390 pp.

Seed, H.B., Ugas, C., and Lysmer, L. (1976), “Site Dependent Spectra for Earthquake Resistant Design,” Bulletin of the Seismological Society of America, Vol. 66, No.1, February.

Powell, G., H. (2010), “Modeling for Structural Analysis, Behavior and Basics,” Computer and Structure Inc., 365 pp.