Lateral-Torsional Response of

Structures Subjected to Seismic

Waves

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Washington, DC 20234

June 1983

! .1156

i 83-2

1933

I Cj2

. DEPARTMENT OF COMMERCE

riONAL BUREAU OF STANDARDS

NATIONAL BUREAU

OF STANDARDS

LIBRARY AUb 1 iBB'i

c&k \CO

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NBSIR 83-2727

LATERAL-TORSIONAL RESPONSE OF

STRUCTURES SUBJECTED TO SEISMIC

WAVES

Spencer T. Wu

Edgar V. Leyendecker

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Washington, DC 20234

June 1983

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS. Ernest Ambler. Dinctor

ABSTRACT

The behavior of coupled lateral-torsional systems subjected to seismic waves is

investigated analytically. This report presents the numerical results of a parametric study for structures subjected to seismic waves. Case studies are provided to show the contribution of each of the selected parameters to the

rotational response of the systems. These parameters are: geometric eccentricity, aspect-ratio of the foundation mat, damping ratio, and the ratio of the

rotational to translational frequencies. Dynamic eccentricity is selected as an index to represent the level of the response. The sensitivity due to the

deviation of the input spectrum is investigated. Accidental, eccentricities

due to seismic waves are also evaluated. Design concerns are given on how the

design eccentricity should be considered based on this study*

Key words: accidental eccentricity; building codes and standards; design

eccentricity; dynamic eccentricity; parametric study; seismic waves structural response.

iii
TABLE OF CONTENTS

Page

ABSTRACT ill

LIST OF FIGURES v

1. INTRODUCTION 1

2. THE APPROACH 2

3. ANALYTICAL RESULTS OF THE PARAMETRIC STUDY 5

4. DESIGN CONCERNS 19

5. CONCLUSION 21

6. ACKNOWLEDGMENTS 22

7. REFERENCES 23

APPENDIX. NOTATION 25

iv
LIST OF FIGURES

Page

Figure 1. A sketch of a one-story building system subjected to

earthquake excitations at the foundation mat 3

Figure 2. The impedance functions for a foundation mat with

dimensions equal to 85 ft x 85 ft 6

Figure 3. The input motions for a foundation mat with dimensions equal

to 85 ft x 85 ft 7

Figure 4. The impedance functions for a foundation mat with dimensions

equal to 168 ft x 42 ft 8

Figure 5. The input motions for a foundation mat with dimensions equal

to 168 ft x 42 ft 9

Figure 6. Spectra assumed as the free-field motions 10

Figure 7. Dynamic eccentricities for a case with foundation mat equal

to 85 ft x 85 ft for e/r = 0.1, 0.2, and 0.32, respectively .. 11

Figure 8. Accidental eccentricities for the selected cases 12

Figure 9. Dynamic eccentricities for a case with the foundation mat

equal to 168 ft x 42 ft for e/r = 0.1, 0.2, and 0.32,

respectively 13

Figure 10. Dynamic eccentricities for cases with different damping

ratios 14

Figure 11. Dynamic eccentricities for cases with different ratios of

rotational to translational frequencies ...» 16

Figure 12. Dynamic eccentricities for a case with the same conditions

as the case represented in figure 7 except the input spectrum

is represented by case b in figure 6 17

Figure 13. Comparison of the bases shown in figures 7 and 12 for

e/r = 0.32 18

Figure 14. M(t) and V(t) for the case shown in figure 7 with

e/r = 0.20 20

v

1. INTRODUCTION

There has been evidence of serious damage In buildings due to torsional effects

In earthquakes [1-2]. Therefore, It Is necessary that adequate torsional resistance be provided for building systems. Current design provisions in building

codes or standards related to torsional effects [3-5] are mainly based on engineering judgment. These provisions are not necessarily theoretically sound,

e.g., it is known that by taking accidental eccentricity as five percent of

the building dimension in design may not be conservative [5-9] but this figure

is widely used as the accidental eccentricity in present building codes. To

avoid structural failures in earthquakes, research is needed to simulate the

real behavior of structural systems subjected to seismic waves.

Recently, analytical models have been developed specifically for determining

dynamic response of coupled lateral-torsional structural systems. For example,

the effects of response due to geometric eccentricity between center of rigidity

and center of mass have been investigated extensively by Kan and Chopra for the

linear and non-linear systems [7-8], The response of torsionally coupled

elastic systems have been studied by Kung and Pecknold [9] based on a probabilistic ground motion model. Dynamic eccentricity has been estimated by Tso and

Dempsey in terms of geometric eccentricity [10]. However, soil-structure interaction (SSI) effects were not considered in these studies. To include SSI

effects, a simple approach [11] was proposed that made use of the impedance

functions and input motions as computed in references 12 and 13.

In this report, an analytical result will be presented to illustrate the effects

of a few parameters pertinent to the characteristics of a structural system.

The formulation presented in reference 11 will be briefly described. The parameters selected in the study are: geometric eccentricity, aspect-ratio of the

foundation mat, damping ratio, and the ratio of the rotational to translational

frequencies. The sensitivity of the response to variation in the ground motion

spectrum is also investigated. Based on these analtyical results, discussions

are given on how the design eccentricity should be considered in the related

seismic design provisions.

1
2. THE APPROACH

A one-story structural system with geometric eccentricity equal to e at the

first floor is shown in figure 1. The structure is subjected to earthquake

excitations at the ground floor (foundation). The equations of motion may be

written as,

m t V + e “t V + KyUyd + C

y

Uyd . 0 (1)

“t6 Uyt

+ V V + s

u*d + C

* - 0 (2)

• • •• m b

uyb + mt uyt

+ mte " f

y

(t) (3)

mce Uyt + V + V “tb

“ ** (t) (4)

where m, C, K, and U are equal to the mass, damping, stiffness, and displacement,

respectively. Subscript t or b denotes that the term is related to the first or the ground floor; subscript y or ^ denotes that the term is related to the translational or rotational movements; subscript d denotes tht the term is equal to

the difference between the related terms of first floor and the ground floor;

e.g., Uyd Uyt - U yb ; I^ t and are the rotational mass moments of inertia

taken with respect to the Z-axis located at the center of rigidity, as shown in

figure 1 for the first and ground floors, respectively; f yand f^ are the

earthquake excitation forces at the foundation.

"U" and ”f" may be transformed into the frequency domain, i.e.,

{u} " 1 { U S J exP U ws t) (5a)

SO

and

{ f } 2 ff s ) exp (i tOg t) (5b)

SO

where w s is the frequency. Let the excitation forces between the soil and

foundation, f(s) be expressed as [12-13]:

{f (s)} " [K

f

J [(u

f

l - {U*}] (6)

where (Kf) represents the impedance matrix, (Uf) represents the foundation

motion, and (Uf) represents the input motion. The subscript f denotes terms

related to foundation mat. Equations 1-4 can be rearranged and written in

matrix form for each w s as:

2
first

floor 3

Figure

1.A

sketch

ofa

one-story

building

system

subjected

to

earthquake

excitations

at

the

foundation

mat
- m

t <4 - 2 em_ a) t 8 ‘ m

b“8'K

fy 0

•

rr

i “ K

fy ufy — em^w2

t 8 - v* 0 V

‘ K

f*u^

- Kf*

- ni tu2 +k

t s y - eu)2m

t - K -c io)2

y y s 0 Uyb

m 0

+ Cyi U)g

- 2 em_u)

t 8 - I. f w

2 +Kx

s * 0 -% U

4>b 0

_ + C^iaig “ C^iojg

m m m m

( 7 )

The response of the structure can be calculated once the impedance functions

and the input motions of the foundation are known. With the Fast Fourier

Transform technique, the dynamic response of the system may be found in the time

domain (e.g., ref. 14). For the problem under investigation, the dynamic eccentricity, E(t), is considered to be equal to the torsional moment, M(t), divided

by the transverse shear, V(t).

4
3. ANALYTICAL RESULTS OF THE PARAMETRIC STUDY

Dynamic eccentricity is an important variable for indicating the behavior of a coupled rotational-translational system. Therefore, it is selected as an index

to represent the structural response in the investigation. The effects on dynamic eccentricity due to various parameters are illustrated with a few case studies. The analytical results are given in this section. Unless otherwise

specified, the general properties of the system are taken as:

mt * 3.63 x 10 7 kg, mfe * 1.85 x 10 8 kg, I,

t « 3.03 x 10 9 kg-m^,

1.^ ** 1.52 x lO 1 ^ kg-m , K = 1.44 x 109 N/m, K, * 2.70 x 10 11 N-m/rad,

Cy and C^ * 2 percent of tne critical damping ratios.

The soil is assumed to have a damping ratio equal to 0.05, with Poisson’s ratio

equal to 0.33 and the shear modulus equal to 2.15 x 10® N/m^. The impedance

functions and input motions are taken from references 12 and 13. In these

figures, Us represents the free field motions. The functions are shown in

figure 2 through 5 for foundation mats with dimensions equal to 25.9m x 25.9m

(85 ft x 85 ft) and 51.2m x 12.8m (168 ft x 42 ft), respectively. These curves

are used as input functions in the analysis. The input spectrum is based on curve a as shown in figure 6 for all cases except otherwise noted. Analytical

results are discussed below for each of the parameters.

Geometric Eccentricity - Dynamic eccentricities of the first floor for cases with geometric eccentricity ratios , e/r, equal to 0.1, 0.2 and 0.32 are plotted

in figure 7, where r represents the radius of gyration. These curves are obtained by solving equation 7. The foundation mat is assumed to be 25.9m x 25.9m. It is shown in the figure that the dynamic eccentricity becomes larger

as the geometric eccentricity increases. It is also shown that the frequencies

of the system are affected by the geometric eccentricity values. Accidental

eccentricity due to seismic waves can be found by solving equation 7 with e = 0

[11]. The accidental eccentricity corresponding to this case shown in figure

7 is given in figure 8 by the dotted line.

Aspect-Ratio of the Foundation Mat - To illustrate the effects due to the

aspect-ratio of the foundation mat, the dynamic eccentricities are plotted in

figure 9 for a case with the same structural properties as the previous case except for the dimensions of the foundation mat. The foundation mat selected

here is of rectangular shape with dimensions equal to 51.2m x 12.8m. These two cases have about the same mat area, but the input functions are different as shown in figures 2 through 5. Hence, the dynamic eccentricities are different

in terms of magnitude and frequency. The accidental eccentricities for this case are also shown in figure 8. As expected, the results shown in figure 8

indciate that the eccentricity increases for cases with a larger aspect-ratio.

Damping Ratio of the Structural Systems - The rotational effects due to damping

ratios are shown in figure 10. Dynamic eccentricity for the case with e/r

equal to 0.2 as shown in figure 7 is replotted here. For this case the damping

ratio is equal to 0.02. The corresponding case with a damping ratio equal to

0.07 is also shown in figure 10. It can be seen that the difference between

these cases in terms of maximum dynamic eccentricity is not significant.

5
12.5

(SI

o

Ui

Mci • uarohMQ u o u •

6

Figure

2.

The

impedance

functions

for

a

foundation

mat

with

dimensions

equal

to

85

ft

x

85

ft
MV/Us) — <J3

— cu

Ul

r» V*

Art

c\l

Art cu

CM

us*

uou :

7

Figure

3.

The

input

motions

for

a

foundation

mat

with

dimensions

equal

to

85

ft

x

85

ft
12.5

M

t nc^ • uarohMor uou .

8

Figure

4.

The

impedance

functions

for

a

foundation

mat

with

dimensions

equal

to

168

ft

x

42

ft
1.00

_ M

O

Ul

<3

Ul

COHhOS u o u

9

Figure

5.

The

input

motions

for

a

foundation

mat

with

dimensions

equal

to

168

ft

x

42

ft
aas/uj ‘A1I0013A

10

Figure

6.

Spectra

assumed

as

the

free-field

motions
W

in

cu £ in

in

cu

Id

HI I 1 1 1 1 1

1

I II 1 11 1 I i 1

1 1 1

1 1 1 H4

m

in

cu

©

in

in

©

in

cu

(*/ 3} OHM! -KD03

11

Figure

7.

Dynamic

eccentricities

for

a

case

with

foundation

mat

equal

to

85

ft

x

85

ft

for

e/r

-

0.1,

0.2,

and

0.32,

respectively
( JAI *0*331 IDBippsy

r

12

Accidental

eccentricities

for

the

selected

cases
1A

OJ IA

in

cu

1 1 1

1 1 m+-

|

1 1 ii

1

i i h

(V a) ouva *ioxa *»wa

13

Figure

9.

Dynamic

eccentricities

for

a

case

with

the

foundation

mat

equal

to

168

ft

x

42

ft

for

e/r

0.1,

0.2,

and

0.32,

respectively
I I 1 1 1 II 1 I I I 1 1 u>

at tn Ul

(V *a) ollyk 'Kboa 'tua

14

Figure

10.

Dynamic

eccentricities

for

cases

with

different

damping

ratios
Ratio of the Uncoupled Rotational Frequency to Translational Frequency - Dynamic eccentricities are shown in figure 11 as a function of the ratio between

the uncoupled rotational frequency to translational frequency. The curve with

a solid line in the figure is a replot of the case with e/r 0.2 in figure 7.

The ratio is approximately 1.5 for this case. The curve with a dotted line

represents a case with the same system except with a higher I^ t

. For this

case, I^ t is selected such that the rotational frequency is one-third higher

than the case with a solid line. As shown in figure 11, the magnitude of the

eccentricity is lower for a larger ratio of the rotational to translational

frequencies, but the frequency of the dynamic eccentricity is higher if the

ratio is higher. In reference 8 and 9, similar effects are found despite the

fact that the SSI effect was not considered in those studies.

Ground Motion Spectrum - The sensitivity of dynamic eccentricity due to a change

in the spectrum as shown in figure 6 describing ground motion are shown in

figures 12 and 13. Figure 12 shows dynamic eccentricities for e/r =0.1, 0.2,

and 0. 32 , respectively , based on the spectrum shown in case b in figure 6. A

replot of the corresponding case for e/r = 0.32 in figures 7 and 12 is shown in

the figure 13 in which, it can be seen that the difference in terms of dynamic

eccentricity is negligible due to the change in input spectrum as represented

by curves a and b in figure 6.

15
*

e

ou

ti CO

16

Figure

11.

Dynamic

eccentricities

for

cases

with

different

ratios

of

rotational

to

translational

frequencies
ft* IA

•oc

o

uV

(A

bJ

W 3) OLLVH *NTO3 *fWfl

17

Figure

12.

Dynamic

eccentricities

for

a

case

with

the

same

conditions

as

the

case

represented

in

figure

7

except

the

input

spectrum

is

represented

by

case

b

in

figure

6
T3e

o

uV

(•*/ a) ouv* 'Nroa -nm

18

Figure

13.

Comparison

of

the

bases

shown

in

figures

7

and

12

for

e/r

0.32
4. DESIGN CONCERNS

In the present seismic building codes or provisions, a design eccentricity is

used to represent the level of the rotational response. To select the design

eccentricity properly, dynamic effects on structural systems should be considered. In this study, dynamic eccentricity has been found for a structure subjected to seismic waves. In general, the maximum shear and torsion will not occur at the same time. The maximum dynamic eccentricity always occurs at the

time of maximum torsion but not the maximum shear. For this reason, the use of

dynamic eccentricity as a reference index not only is a proper approach but can lead to an economical design. To illustrate this point, V(t) and M(t) for the case with e/r * 0.2 shown in figure 7 are plotted in figure 14. The vertical

axis in the figure represents the ratios of V(t)/VTT)ay or MCtl/Mn^, where Vmax

and M^iax are the maximum shear and torsion values, respectively. In this case,

the V(t) corresponding to M^y is about 88 percent of the Vmav . Thus, it would

be overly conservative to design a structure based on both Mmav as well as Vmax

as in the present practice.

The term "accidental eccentricity" is used in the literature to cover the

torsional effects due to several factors. In reference 3 it is stated that

"these factors include rotational component of ground motion about a vertical

axis; unforeseeable differences between computed and actual values of stiffness, yield strengths and dead-load masses; and unforeseeable unfavorable

distribution of live-load masses." Actually the "rotational component of

ground motion about a vertical axis" can be computed with an assumption that

the free-field seismic motion is known. For the approach presented here, the

accidental eccentricity can be determined as a special case to the general

solution of the problem. It is shown that the eccentricity is larger for a structure with a larger aspect ratio of the mat. On the other hand, the effect

of accidental eccentricity on the response of symmetric buildings cannot be

ignored. The solutions obtained here verify the concepts presented in

reference 6.

19
<0 CU

e

oo

V

C/1

Ul

( “a/a <10 asuodsaa *iuas

20

Figure

14.

M(t)

and

V(t)

for

the

case

shown

in

figure

7

with

e/r

-

0.20
5. CONCLUSION

Analytical results of a parametric study are given for a structure subjected to

seismic waves. For the accidental eccentricity due to seismic waves, the

results shown here verify the concept presented earlier in reference 7, i.e.,

the accidential eccentricities on symmetric buildings cannot be ignored.

Based on the selected index, dynamic eccentricity, it is also shown in this

study that the rotational response of a system depends greatly on geometric

eccentricity, the dimensions of the foundation mat and the ratio of rotational

frequency to translational frequency. The maximum dynamic eccentricity of the

system is not affected significantly by the damping of the system, nor by the

variation of the input spectrum as shown in figure 6. It is important that

all parameters essential to the rotational responses of the system shall be

considered in determining the design eccentricity. The related design provisions in current building codes which consider only the effects of the geometric

eccentricity should be updated. To develop consistent [15 ] seismic provisions

based on this approach, further study of various structures under specific

conditions is needed.

21
6. ACKNOWLEDGMENTS

The authors are grateful to Drs. B. R. Ellingwood, H.S. Lew, R.D. Marshall, and

J.R. Shaver for their valuable comments.

22
7.

REFERENCES

1* Bustamante, J.I. and Rosenblueth, E. , "Building Code-Provisions on Torsional Oscillation," Proc. 2nd World Conference Earthquake Engineering,

Vol. 2, Tokyo, Japan, 1960.

2. Blume, J.A. , Newmark, N.M. , and Corning, L.H. , "Design of Multistory

Reinforced Concrete Buildings for Earthquake Motions," PCA Publications,

1961.

3. Applied Technology Council, 3-06, "Tentative Provisions for the Development

of Seismic Regulations for Buildings," NBS SP 510, National Bureau of

Standards, Washington, D.C., 20234, 1978.

4. Uniform Building Code, 1973 Edition, International Conference of Building

Officals, Whittier, California.

5. SEAOC, "Recommended Lateral Force Requirements and Commentary," 1974 Edition.

6. Newmark, M.M. , "Torsion in Symmetric Buildings," Proc. 4th World Conference

Earthquake Engineering, Chile, I, A319, 1969.

7. Kung, S.Y. and Pecknold, D.A. , "Effect of Ground Motion Characteristics on

the Seismic Response of Torsionally Coupled Elastic Systems," University

of Illinois, Report no. UILU-ENG-82-2009 , June 1982.

8. Kan, C.L. and Chopra, A.K. , "Coupled Lateral Torsional Reponse of Buildigns

to Ground Shaking," Report EERC 76-13, University of California, Berkeley,

1976.

9. Kan, C.L. and Chopra, A.K. , "Torsional Coupling and Earthquake response of

Simple Elastic and Inelastic Systems," Journal of Structural Division,

ASCE, August 1981.

10. Tso, W.K. and Dempsey, K.M. , "Seismic Torsional Provisions for Dynamic

Eccentricity," Earthquake Engineering and Structural Dynamics, EERI, Vol.

8,

1980.

11. Wu, S.T. and Leyendecker, E.V., "A Note on Seismic Design Provisions for

Torsional Effects," Journal of ASCE, Under Review.

12. Wong, H.L. and Luco, J.E., "Dynamic Response of Rectangular Foundations to

Obliquely Incident Seismic Waves,” Earthquake Engineering and Structural

Dynamics, Vol. 6, 1978.

13. Wong, H.L. and Luco, J.E., "Tables of Impedance Functions and Input Motions

for Rectangular Foundations," Report No. CE 78-15, University of Southern

Californai, December 1978.

14. Lysmer, J., Udaka, I., Tsai, Chan-Feng, and Seed, H.B., "FLUSH Program

Manual," EERC Report, No. 75-30, 1975.

23
15. Fenves, S.J. and Wright, R. N., "The Representation and Use of Design

Specifications," Technical Note 940, National Bureau of Standards,

Washington, D.C., 1977. *

4

24
APPENDIX. NOTATION

Symbols :

C - damping coefficient

E « dynamic eccentricity

Ea

* accidential eccentricity

K = stiffness

M = moment

m = mass

U = displacement

V = shear

e = geometric eccentricity

f = force

i = / - 1

t = time

a) = frequency

Subscripts

b,f = term related to foundation mat

d = term denotes the difference between the related terms at the first floor

and the foundation

t = term related to the first floor

y

= term related to y direction (translation)

<j) = term related to <|> direction (rotation)

s = index number

Subscripts:

• = time derivative

25
H BS.1 14A <wcv. ?^o)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. 3. Publication Date NBSIR 83-2727 6/30/83

4.

TITLE AND SUBTITLE

Lateral-Torsional Response of Structures Subjected to Seismic Waves

5.

AUTHOR(S)

Spencer T, Wu and Edgar V, Leyendecker

6.

PERFORMING ORGANIZATION (If joint or other than NBS. see Instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Gram No.

I. Type of Report & Period Covered

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. Stote, ZIP)

NBS (STRS)

10.

SUPPLEMENTARY NOTES

[721 Document describes a computer program; SF-185, FIPS Software Summary, Is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document Includes a significant

bibliography or literature survey, mention it here)

The behavior of coupled lateral-torsional systems subjected to seismic waves is

investigated analytically. The report presents the numerical' results of a parametric

study for structures subjected to S-H waves. Case studies are provided to show the

contribution of each of the selected parameters to the rotational response of the

systems. These parameters are: geometric eccentricity, aspect-ratio of the foundation

mat, damping ratio, and the ratio of the rotational to translational frequencies.

Dynamic eccentricity is selected as an index to represent the level of the response.

The sensativity due to the deviation of the input spectrum is investigated. Accidental

eccentricities due to seismic waves are also evaluated. Design concerns are given on how the design eccentricity should be considered based on this study.

12.

KEY WORDS ( Six to twelve entries ; alphabetical order; capitalize only proper names; and separate key words by semicolons)

accidental eccentricity; building codes and standards; design eccentricity;

dynamic eccentriciry ; parametric study; seismic waves; structural response.*

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