Dynamic analysis of structures with closely spaced modes using the response spectrum method

o!xs-794986 $3.00 + 00 Per&mm Press Ltd.

DYNAMIC ANALYSIS OF STRUCTURES WITH CLOSELY SPACED MODES USING THE RESPONSE SPECTRUM METHOD CORNEL~U MANU Control Data Canada

Ltd., Multiple Access Division, 50 Hallcrown Place. Willowdale, Ontario. Canada (Received

24 September

1984)

evaluation is made of some of the modal maxima superposition rules currently used to estimate the response of systems with closely spaced modes. Closely spaced modes arise in structures from geometrical effects-such as symmetry and torsional unbalances-and because of a iight ap pendage with a frequency close to one of the natural frequencies of the structure. The proper use of the rules intended to account for closely spaced modes is reviewed. A slightIy torsionally unbalanced structure is analyzed using response spectrum and time history methods. The results obtained show that only adequate modal maxima superposition can avoid too much conservatism. This does not mean that conservatism is bad. but only that in safety design, a rational evaluation of the margin of safety is the essential part of the work. _

Abstract-An

1. I>TRODUCTION When making an analysis to determine the eanhquake forces for design, different approaches can

be used. The quickest .and most inexpensive approach is to use the earthquake provisions of the Building Codefi]. This is a simplified approach that does not take into account all of the important dynamic properties of structures. Special structures, such as nuclear power plant, offshore oil drilling platforms, long bridges and tall buildings, should be given a more detailed dynamic analysis. The response spectrum method has proved to be a reliable alternative to time history analysis. Use of a design spectrum curve based on a class of rep resentative earthquakes for a specific site significantly reduces the cost and uncertainties associated with a single time history analysis. The only difficulty and occasionally a source of criticism for the response spectrum method is the calculation of the total response from modal maxima, not all of which will occur at the same time. A large number of modal maxima superposition methods are currently in use. They are based on probability considerations set up mainly by Rosenblueth[2] and Rosenblueth and Elorduy[3]. It is not the goal of this paper to review all these methods. Only six of them are considered herein with the main intent of stressing the correct use of those rules developed to account for closely spaced modes. RESPONSE SPECTRUM METHOD OF ANALYSlS

The dynamic equilibrium equations for a structure subjected to a ground motion are J%@(r)+ Q(t) + Ky(tf = -M%(t),

(1)

where M, C and K are the mass, damping and stiff405

ness matrices, respectively. The time dependent relative displacement, velocity and acceleration vectors are represented by y(r), i(t) and ji(t), respectively. P is a vector with components of unity in the directions parallel to support movement and zero otherwise. The time dependent ground acceleration is represented by Z(r). In a mode superposition solution the displacement vector y(t) is expressed in terms of the modal matrix, @, and the generalized coordinates vector, 4, as y (t)

= @Q.

(2)

Substitution of eqn (2) into eqn (1) and premultiplication by a’ yields

= -cDTMPJ(r).

(3)

Using the orthogonality properties of the mode shapes, eqn (3) can be reduced to a set of uncoupled modal equations of the form (4) where {i is the modal damping ratio, wi is the natural frequency in mode i arid yr is the modal participation factor. In the response spectrum method of analysis, the ground motion is represented in terms of response spectrum curves. These curves, calculated from the recorded ground acceleration, are plots of the maximum response of a single degree of freedom oscillator over a range of its natural period and damping. Using a displacement response spectrum curve, the maximum value of the generatized displacement

406

CORNELIU

in mode i is given by

MAW

ficients expressed

(4i)max

=

vx

S,/h.

by

(5)

where (S,,)i is the displacement ordinate of the response spectrum curve for damping
w; =

lo/

r;;p

(I -

(I I)

and (Yihnax

=

cbiYi(Sd)i*

(6)

Because (S,l)i is, by definition, positive for all modes. the maximum modal displacements given by eqn (6) have signs uniquely determined by the mode shape +i and the modal participation factor 7;. Therefore, the maximum internal modal forces have unique signs, because they are consistently computed from the maximum modal displacements. METHODS OF MODAL MAXIMA SUPERPOSITION

With the response spectrum technique, an approximation of the maximum response is obtained by combining the modal maxima. Six modal maxima superposition methods are presented. I. Absolute sum of modal maxima superposition (ABS) This method, described by many authors (e.g. Clough[4]), always provides an upper bound of the total response, which can be calculated using the formula R, =

R’(sign R),

(7)

where R, is the total response, RT is the transposition of the column vector of the modal responses and sign R is the unit column vector containing the signs of the modal responses.

5; = 51 +

4. Complete

quadratic

combination

of modal maxima

t CQO This method was proposed by Der Kiureghianl7. 81 in another attempt to account for closely spaced modes in the modal superposition approach. The total response in this method is given by R, = (R’&)“‘,

This method, introduced by Rosenblueth[f], is the most widely used, and the total response is calculated as (8)

where I is the identity matrix. Other variables have the same meaning as for the ABS method.

(12)

In expression (12). trl is the earthquake duration. Examination of eqn (10) shows that for independent modes, E,I = 0 for K # 1 and e,, = I, DS rule reduces to the SRSS. This condition is satisfied when modes K and I are well separated, damping is light and earthquake duration is long. It can be seen from eqn (9) that this method suggests a very realistic way to consider the influence of closely spaced modes. Thus, cross terms in eqn (9) can assume positive or negative values depending on the particular sign of the modal responses making up these terms. However, this method was incorporated by the U.S. Nuclear Regulatory Commission in Regulatory Guide 1.92[6] with the specification that cross terms must always be taken with positive signs. This results in overly conservative results in many applications.

2. Square root of the sum of modal maxima squared superposition (SRSS)

R, = (RTfR)“‘,

&,.

(13)

where E is the matrix of the modal coupling coefficients, which for a smooth spectrum over a wide range of frequencies and a long earthquake duration is determined using the formula

8G&)“2(5K + rb)?,’ ‘.’ =

(1 - 92)’ + 4~&r(

1 + r’) + 4(5; + [/?)2’ (14)

3. Double sum modal maxima superposition (DS) Singh, Chu, and Singhi presented this method as an improvement to the SRSS, to account for closely spaced modes. According to this rule the total response is given by R, = (RTfi)“‘,

(9

where r is the matrix of the modal coupling coef-

where r = oJ0,.

(15)

It is noted that for zero damping the CQC reduces to the SRSS. This method, as the DS method, also presents a realistic approach to consider the influence of closely spaced modes.

Structures with closely spaced modes 5. Ten percent modrd moximo superposition

(TEN%) This method is also recommended by the U.S. Nuclear Regulatory Commission in Regulatory Guide 1.92 [6]. It is another way to improve the SRSS to account for closely spaced modes. Calculation of the total response is made according to the equation

407

fact always occur to which some expected portion of remaining modal responses should be added. The total response is calculated using the relationship R, =

Max 1R / + [R’ZR-

(Max 1R / )21’i2.

(181

NUMERICAL RESULTS

where the modal given as II f,/

=

I

0

coupling

for

K =

1

for

’ OK

-

01 otherwise

coeffkients

w

I

10.1

matrix,

Kfi.

c, is

(17)

Because it ignores the signs of cross terms, this method results in overly conservative results in many applications. 6. Nuvaf Research Laboratory modal maxima superposition ( NRL )

O’Hara and CunnifIll] suggested this rule, using the rationale that the largest modal response will in

A response spectrum analysis of the building structure shown in Fig. I was performed using the STRESS 3.0 [lo] computer program in which the author has recently implemented dynamic capabilities. The eigenvalue problem is solved using the determinant search algorithm]1 I]. Table 1 presents the first I5 frequencies of this structure. The response spectrum curve used in the analysis is shown in Fig, 2. This curve was generated for a damping ratio of 3% for the TAFT, July 1952N2 I E earthquake portion, given in Fig. 3, using the STARDYNE/DYNRES]lZ] computer program. Results obtained with the six methods of modal maxima combination, described in the previous section. are presented in Tables 2 and 3 in an element local system of coordinates. In these tables force values of less 0.005 kN and moments less than 0.005 kNm

Fig. I. Simple three-dimensional

building problem.

CORNELIU MAW

408

Table I. The first 15 modes of the structure shown in Fig. I MODE

FREQUENCY

(CYCLEWSX.

1

2.186

2

3.584

3

5.961

4

6.269

5

7.133

6

7.240

7

7.378

8

9.124

9

9.299

10

9.594

11

10.551

12

11.745

13

11.893

14

12.656

15

13.833

)

Fig. 3. Part of the Taft, July 1952 N2IE history.

earthquake time

Eu

I.0

4 ,a

.6

.4

are omitted. The solution is compared against the “exact” time history analysis for the accelerogram of Fig. 3. The time history analysis was performed using the STARDYNE/DYNREl [ 121 computer program. The results presented in Table 2 were obtained using the DS and TEN% rules of modal maxima combination as indicated by the U.S. Nuclear Regulatory Commission[6]. For consistency purposes, the CQC method was used in the same idea. The ABS method yields results with a degree of conservatism varying from a factor of 1.08 to 5.95. It is the author’s opinion that this range of maximum response approximation cannot be acceptable in a rational design. A somewhat lower degree of conservatism (I .033 to 3.497) is shown by the NRL method. Besides the ABS and NRL rules all other methods

i; 8

.2

0.0

Fig. 4. Variation of the modal coupling coefficients versus the ratio of periods for DS and CQC rules of modal maxima superposition (0% damping, earthquake duration 10.0 seconds).

Ekl

1.0 .0

.b

I 4.

0.0

Fig. 2. Acceleration response spectrum curve for earthquake of Fig. 3 and 3% damptrig.

Fig. 5. Variation of the modal coupling coefficients versus the ratio of periods for DS and CQC rules of modal maxima superposition (2% damping, earthquake duration 10.0 seconds).

Structures

with closely spaced modes

409

Table 2. Response spectrum versus time history responses in the ground level support columns at the fixed end. Absolute values of modal responses are used in CQC. DS and TEN% methods of modal combination MEMBER

COUPONENT

TIKE

HISTORY

kN,kN-m

RESFCWSE AD.9

SRSS

SPECTRUWTI.ME

HISTORY DS

Cpc eddo.

1

Fx F

2

125.25

1.133

1.000

1.002

1.003

TEN \

NP.L

td-1.411

1.009

1.001

1.063

3.68

2.270

1.272

1.352

1.364

1.449

1.350

1.738

FE

49.81

1.118

0.976

0.977

0.978

0.983

0.976

1.066

M

Y

78.37

1.117

0.977

0.978

0.978

0.984

0.977

1.065

ML

74.97

2.286

1.272

1.353

1.365

1.452

1.351

1.738

Fx

129.44

1.084

1.020

1.021

1.021

1.023

1.202

1.072

Y

F

4.92

2.280

1.271

1.352

1.364

1.451

1.350

1.737

48.11

1.082

0.971

0.972

0.972

0.975

0.971

1.049

75.70

1.081

0.972

0.972

0.973

0.975

0.972

1.049

a.77

2.206

1.271

1.353

1.365

1.452

1.351

1.737

Y F M

z Y

M"

3

F:

130.66

1.162

1.039

1.041

1.041

1.047

1.040

1.098

F

3.69

2.274

1.271

1.350

1.362

1.448

1.349

1.735

Y F.

44.04

1.412

0.972

0.973

0.973

1.271

0.972

1.033

H

70.59

1.086

0.972

0.973

0.974

0.979

0.972

1.033

7.52

2.283

1.271

1.352

1.364

1.451

1.350

1.736

Y nz

4

Fx F

5

6

2.505

3.989

4.094

4.569

3.988

3.480

0.16

5.917

2.512

4.015

4.119

4.589

4.015

3.497

54.06

1.114

0.978

0.978

0.979

0.985

0.978

1.064

H' Y

82.69

1.114

0.978

0.979

0.979

0.985

0.978

1.064

ML

0.32

5.915

2.511

4.013

4.118

4.507

4.013

3.946

FY

0.21

5.916

2.511

4.013

4.118

4.587

4.014

3.497

FZ

52.23

1.078

0.972

0.973

0.973

0.976

0.972

1.048

M

79.89

1.078

0.972

0.973

0.973

0.976

0.972

1.047

Hz

0.38

5.915

2.511

4.012

4.117

4.586

4.013

3.496

Fx

0.47

5.678

2.402

3.953

4.057

4.528

3.952

3.448

F

0.16

5.917

2.512

4.014

4.119

4.589

4.015

3.497

F.

40.72

1.084

0.972

0.973

0.974

0.979

0.972

1.032

n

74.52

1.084

0.972

0.973

0.974

0.979

0.972

1.032

0.32

5.915

2.511

4.013

4.117

4.587

4.013

3.496

Y

Y

Y

Y

F*

124.81

1.139

1.002

1.004

1.005

1.012

1.003

1.071

3.77

2.200

1.265

1.321

1.332

1.418

1.316

1.727

Fz

49.82

1.117

0.976

0.977

0.978

0.983

0.976

1.066

M

78.38

1.119

0.977

0.970

0.978

0.984

0.977

1.065

7.69

2.191

1.265

1.321

1.333

1.418

1.317

1.727

F

Y

Y

n_

8

5.950

F;

% 7

0.46

px FY

129.45

1.084

1.020

1.021

1.021

1.023

1.020

1.072

5.05

2.190

1.265

1.321

1.332

1.417

1.316

I.727

410

C~RNELIU

MANU

Table 2. (canrind) 8

9

F

48.11

1.082

x;

0.305

2.041

1.054

1.092

1.104

n Y nz

75.70

I.061

0.972

0.972

0.973

a.99

2.185

1.265

1.321

1.332

Fx

131.21

1.147

1.036

1.037

F Y FL

3.79

2.196

1.264

44.84

1.088

70.59 7.70

z

n Y K

Note: The starred (*) components for all members in this structure.

0.971

0.972

0.972

0.971

1.049

1.189

1.077

1.442

0.975

0.972

1.049

1.416

1.116

1.726

1.038

1.044

1.037

k.089

1.319

1.331

1.416

1.314

i.724

0.972

0.973

0.973

0.979

0.972

i.033

1.086

0.972

0.973

0.974

0.979

0.972

1.033

2.188

1.265

1.320

1.332

1.417

1.316

:.726

designate maximum

0.975

values given by response spectrum analysis

Table 3. Response spectrum versus time history responses in the ground level support columns at the fixed end. Modal responses with their signs are used

MX%ER

CmFoNLlm

DS

CW y10.0

4

Fx

1.000

1.001

1.005

1.000

P Y F.

I.257

1.250

1.181

1.258

0.976

0.975

0.971

0.976

n Y HZ

0.976

0.976

0.971

0.977

1.257

1.249

1.101

1.257

5

1.020

1.021

1.022

1.020

F Y Fz

1.256

i.249

1.181

1.257

0.971

0.971

0.970

0.971

M Y HZ

0.972

0.971

0.972

0.972

1.256

1.249

1.161

1.257

F;

1.039

1.039

1.038

1.039

F Y Fz

1.256

1.249

1.180

1.257

0.972

0.972

0.974

0.972

M Y %

0.972

0.972

0.974

0.972

1.256

1.249

1.180

1.257

Fx

2.144

2.117

2.116

1.787

F

2.16l

2.135

2.136

1.811

0.977

0.976

0.972

0.977

0.977

0.977

0.972

0.976

2.160

2.134

2.135

i.810

Y F' Z M* Y %

5

TEN \ ed-1.411

F Y FZ n Y XL

2.161

2.135

2.135

1.811

0.972

0.972

0.971

0.972

0.972

0.972

0.971

0.972

2.160

2.134

2.135

1.810

Structures

with closely Table

6

spaced

modes

411

3. (conlinlted)

PX

2.125

2.098

2.096

1.170

F

2.161

2.135

2.136

1.811

Fz

0.972

0.970

0.975

0.972

I4 Y

0.972

0.972

0.975

0.972

%

2.160

2.134

2.135

1.810

Fx

1.003

1.003

1.008

1.002

F

1.253

1.245

1.171

1.257

Fz

0.976

0.975

0.971

0.976

n

Y

7

Y

0.976

0.976

0.971

0.977

"z

1.253

1.246

1.172

1.257

Fx

1.021

1.021

1.022

1.020

1.253

1.246

1.172

1.257

0.971

0.971

0.970

0.971

Y

8

F Y Fz n;

1.033

1.030

0.976

1.039

"

0.972

0.971

0.970

0.972

%

1.253

1.245

1.172

1.257

Fx

1.036

1.036

1.034

1.036

F

1.252

1.244

1.170

1.256

Fz

0.972

0.972

0.974

0.972

n

0.972

0.972

0.974

0.972

1.252

1.245

1.171

1.256

Y

9

Y

Y %

Note: The starred (*I comoonents for all members in this &&re.

slightly

designate -

maximum

underestimate the shear forces in the di: rection of motion and grossly overestimate them in the direction normal to the motion. Figures 4, 5 and 6 present graphical variations of the modal coupling coefficients for the CQC and DS methods of modal maxima superposition. The higher values of these coefficients for the DS method than those for the CQC rule should be interpreted in the sense that the DS asks for a larger degree of correlation than the CQC. The limiting case of the correlation is however given by the TEN% rule. Therefore, use of these methods as recommended by the U.S. Nuclear Regulatory Commission does nothing else than to increase the response given otherwise by the SRSS rule. Obviously, a rational superposition of modal maxima should account for the actual effect of closely spaced modes. With the current rules of total response estimation, it appears that the best solution is obtained only if the actual sign of modal maxima is retained in those methods where it makes a difference (i.e. DS, TEN% and CQC). Table 3 presents results obtained in this idea. Although the time history analysis was performed for the earthquake portion shown in Fig. 3 (with a duration of 1.411 seconds), and the response spectrum curve of Fig. 2 is based on this duration, Tables 2 and 3

values given by response

spectrum

analysis

present results of the DS method for r* = 10.0 seconds, as well. These calculations were done because some of the general purpose computer programs (e.g. ANSYS[l3]) have hard coded the DS rule for an earthquake duration of 10.0 seconds. Figure 7 shows graphical plots of the correlation coefficients for different earthquake durations. It can be seen that the correlation decreases as the motion duration increases (DS). A correct use of the DS method (refer to Table 2, tcl = 1.411 seconds) shows that this method (DS) yields the best estimation of the total response with the exception of the central columns on a line normal to the ground motion. Otherwise, the debate for this particular problem would be between TEN% and DS if the actual signs of modal maxima is carried in the superposition calculations. The CQC results seem to envelope the maxima of the DS and TEN% (refer to Table 3). Definitely, if more sophisticated expressions for the correlation coefficients (as presented by Der Kiureghian[‘l, 81) would be used in the CQC rule different results are expected. Only this definition of the modal coupling coefficients was implemented in STRESS’3.0 [IO]. because, to the author’s knowledge, this is the only way the CQC rule is currently available in other

412

CORNELIU MANU

The slight unconservatism shown by some of the modal maxima combinations should not be a point of concern. It can be noticed that in the present study an unsmoothed response spectrum curve was used. Different rules of spectrum smoothing and “peak widening” are always applied to the raw response spectrum curve. All these rules are actually intended to account for certain uncertainties and they end up making the final results conservative.

Fig. 6. Variation of the modal coupling coefficients versus the ratio of periods for DS and CQC rules of modal maxima superposition (5% damping, earthquake duration 10.0 seconds).

Acknon~ledgemenrs-The author wishes to thank Control Data Canada Ltd. for allowing him to publish this work. Implementation of the Dynamic Analysis capabilities in STRESS 3.0 was uartiallv funded bv the Natural Sciences and Engineeiing Risearch C&ncil of Canada through a grant to the author. This financial support is gratefully acknowledged. Special thanks are expressed 10 Miss Anne Jackson for typing and editing the manuscript.

REFERENCES 1. Uniform Building Code. international Conference of Building Officials, Pasadena, CA (19671. 2. E. Rosenblueth, A basis for aseismic design of structures. Ph.D. dissertation, University of Illinois. Urbana, IL (1951). 3. E. Rosenblueth and J. Elorduy, Responses of linear systems to certain transient disturbances. Proceed-

ings of the 4tk World Conference on ~arr~q~ake Engineering, Santiago, Chile (l%Q), No. A-l, pp. 18S-

Fig. 7. Variation of modal coupling coefficients versus the ratio of periods for DS and CQC rules of the modal maxima su~~osition (2% damping, ea~hquake duration 5.0, 10.0 and 20.0 seconds).

general

purpose

DYNE[lZ],

computer programs ANSYS[13], TABS[l4]).

(STAR-

CONCLUSION Although many rules of modal maxima superposition are currently in use by practitioners, only six of them, which were implemented in the STRESS 3.0 computer program, were considered in the present study. The results show that a correct account for closely spaced modes, when maximum response is evaluated, can be achieved only if the actual signs of the modal response is retained in the required calculations. It appears, that for this problem, the best methods to be used are either DS or TEN%.

196. 4. R. W. Clough, Earthquake analysis by response spectrum superposition. Bull. Seismol. Sot. Am. 52, NO. 3, 647-660 (1962). 5. A. K. Singh, S. L. Chu, and S. Singh, Influence of closely spaced modes in response spectrum method of analysis. Report SAD-126, Sargent & tundy Engineers, Chicago, Ii (19731. _ 6. U.S. Nuclear Renulatorv Commission. Reaulatorv Guide 1.92. Revis& I, combining modal responses and spatial components in seismic response analysis. Washington, D.C. (1976). 7. A. Der Kiureahian. On resnonse of structures to stationary exci&ion.’ Repo; No. UCB/EERC-79/32, Earthquake Engineering Research Center, University of California, Berkeley, CA (1979). 8. A. Der Kiureghian. A response spectrum method for random vibrations. Report No. UCB/EERC-8005, Earthquake Engineering Research Center. University of California, Berkeley, CA (19801. 9. G. J. O’Hara and P. F. Cunnif, Elements of normal mode theory. Report NRL-6002, U.S. Naval Research Laboratory, Washington, D.C. (1963). 10. STRESS 3.0, User Manual. Control Data Canada Limited. Multiole Access Division, Toronto. Canada (1984). ’ Il. K. J. Bathe and E. L. Wilson, Eigensolution of large structural systems with small bandwidth. ASCE J. Engng Me&. Div. 99,467-479 (1973). 12. STARDYNE. User Manual. Svstem Development Corporation, Santa Monica, CA-t 1979). . 13. ANSYS, Theoretical Manual. Swanson Analysis Systems, Inc., Houston, PA (1983). 14. E. L. Wilson and A. Habibullah. A program for threedimensional static and dynamic analysis of multistorey buildings, in Structural Mechanics Software Series, Vol. II. Univ. Press of Virginia, VA (19781.