The time-dependent ral density of any linear response t stochastic process is a single ~~rner~~a~ i~tegra.t~~~. Thus depend on the efficient comput n of the frequency response function

e extended to elasto-plastic structures with certain a process which allow the dete~~nat~~n of frequency dependent envelope f~uctions associated linear system are computed in a

e sta~~nary limit.

onse of linear t~me~~nva~iant sy en random forcing functions may be in

or frequency domain. In ng ~p~~i~atio~s consideration in the frequency domain was until recently, restricte to stationary processes, nonstationary random vibrations of linearized ~echa~i~a~ or structural systems are described by ti dependent power spectral densities. Considering evolution of such a spectral density in a time-step tnanner, the characteristic features of the classical stationary case are preserv Although nonstationary system analysis of that kind d been in use in electrical and chemical engineering it was not until the famous

of Briestley” that this rocedure could compete he time-domain descripti of civil- and mechanical eering a~~~~catio~s.

n principle, the frequency domain approach is equally ~ppl~~a~~e to discrete or ~0nt~n~Q~s systems and, what is :onsi a great advantage, t e main computational etf~rt is the evaluation of (sure) frequency response bnctions of those state variables of e~g~~eeKi~g importance. Thus, weir-established numerical zedures like the Finite Element Method5 LX the

and more efficient oundary Element sed for the determin tic analysis of the li

e response to a time harmonic e d to a stationary random driving force

and, still more general, to that of a stochastic process of envelope type. ase,

nctions like th ion, e sum of two e fu

wrdely used e.g., in earthquake engineering. Using such a simple amplitude modulation of a stationary process

akes it easily possible, to integrate one of the double integrals in Priestley’s formula analytically. This was

Accepted April 1987. Discussion closes August 1987. * Based on an invited lecture presented to the 26th Polish Soiid Mechanics Conference, Gdansk, 1986.

~266-892~/87/020~2-08~2.00 c 1987 Computational Mechanics Publications

been used in

linear damping law is possible without increasing the numerical efforts.

Since engineering structures under the action of severe wind - or earthquake loadings tend to yield, stochastic dynamic plasticity becomes applicable. There is little doubt in literature about stochastic analysis of elastic- plastic structures being awkward to model, see e.g., the review of Crandall and Zhu 15. Mainly single-degree-of- freedom (SDOF) structures have been considered. Progress has been achieved recently, when Wenl6 applied a statistical linearization technique in connection with a special material's law.

However, it has been observed, that straight-forward approximations combined with results obtained by computer simulations render reliable results. The total deformation, if small, may be decomposed into an oscillatory and zero-mean linear elastic component and into a plastic deformation which remains constant over that (long) period of time where the elastic part stays within the yield-levels (barriers). Karnopp and Scharton 17 considered the fact that the yield level crossings occur in clusters, each of 'short' time duration, and thus approximated the plastic drift as a point process with jumps at those time instances, Using these approximations Vanmarcke and Veneziano ~s and Vanmarcke ~9 obtained important plastic response measures for a structure under severe earthquake loadings. Stationary conditions of the linearly damped linear part of the response were assumed. Thus, the average amount of a simple yield increment, the variance of the accumulated plastic drift and a ductility ratio determined in those papers are reliable results only in the region of weak ptastification.

Grol3mayer 2° and GroBmayer and Iwan 21 improved the response analysis of a yielding SDOF-system at the cost of a sophisticated linear model with time-variant stiffness and damping, thereby extending the range of applicability to higher plastic deformation ratios.

Keeping the associated linear system constant in time and considering the plastic drift in the form of an additional structural loading of the linear system the two approaches were combined by Ziegler and Irschik 22 while preserving the simpler character of Ref. 18. Thus, extension to a two,degree-of-freedom flame system was analogously possible 23 and the n-storey-frame in the plastic hinge model is taken into account by Irschik 24. The total loading of the associated time-invariant structure is described by the stationary power spectral density of the earthquake loading process if modulated by a time- and frequency dependent envelope function. The crucial point in that paper is the determination of the envelope function in a time-stepping manner for each mode of vibration.

While it was possible in a first attempt to extend the widely used response spectra method to include damping through the nonlinear process of yielding in SDOF- systems the approach shown in this paper applies the time- and frequency dependent power-spectral-density method to discrete elasto-plastic structures. A further extension of the plastic hinge model to include the spreading of the plastic zones in continuous structures is in preparation and may be presented at the forthcoming IUTAM-Symposium on Nonlinear Stochastic Dynamic Engineering Systems, Igls-Innsbruck 1987 (Proc. by Springer Verlag).

Random vibrations: F. Ziegler

2. N O N S T A T I O N A R Y VIBRATIONS OF LINEAR STRUCTURES

We consider a one-dimensional nonstationary input process of envelope type with power-spectral-density given by

Sgo(t, 09) = 0 (t)S(09) (1)

where S(09) refers to a proper stationary zero mean process and O(t) is a deterministic envelope function defined for t>0. Priestley's formula 1, renders the evolutionary power spectral density of any output component w of the linear system by the scalar relation

sww(t, 09)= [~(t, 09)12s(09) (2) which is analogous to the stationary relation. The time- dependent sure frequency function was originally determined by

~9 (t, 09) = f l O(t -- z)h(z) e-io~, dz (3)

through the impulse function

h(z) = F(v) e iw dv (4) oo

where F(v) is the complex frequency response function related to the output w. Since F(v) in general is determined pointwise and numerically, integration is interchanged to perform one quadrature analytically, see Refs 2, 3 and 4 and also 9, namely

f ( t , 09, v) = 0 (t - T) e itv - o , dz (5)

to render

F ~b(t, 09) = - ~ F(v)f( t , 09, v) dv (6) oO

by a single numerical integration over a sufficiently wide frequency band, often determined by the definition of S(09) of the stationary excitation process.

In case of a Heaviside step function

f ( t , 09, v) = i{ 1 - exp[i(v- co)t]}/(v - 09) (7)

For an exponential shape function, t > 0,

we obtain

~b(t) = e ~t - e p', fl < e < 0 (8)

where

f ( t , 09, v)=f~ - f~ (9)

f~=i{exp(~t ) -exp[ i (v - -09) t ]} / (v -09+i~) (10)

and f~ is obvious. By definition the nth spectral moment at any instant

of time is given by another numerical integration

Sn(t) = 09nSww(t, 09) d09 (1 I) 03

Thus, all ingredients of a reliability analysis e.g.,

Probabilistic Engineering Mechanics, 1987, Vol. 2, No. 2 93

Random vibrations: F. Ziegler

\ ?

d 4

_ 2 _ v -

!

H~ I

~--2R •]

Fig. 1. Platform surrounded by water in distance d to a sloping beach

following Vanmarcke ~9, are determined by 'smoothing' numerical integrations. Practical applications require the most efficient determination of the frequency response function to save overall computing time on that main part.

2.1. Nonstationary vibrations of an offshore~platform Zhang 9 considered the simple model of a vertical

cantilever beam of circular cross-section with rigid end.- mass surrounded by linear compressible water. Earthquake excitation by the horizontal component of base acceleration is considered and the two cases corresponding to bedrock support and support on a layered soil are treated. The proper assumptions on the input process are band-limited white noise 0 < co ~< e)o and coloured noise represented by the output of a Kanai- Tajimi-filter, respectively, switched on at time t = 0.

The r.m.s, value of the base acceleration in bedrock is chosen cg with c=0 .2 and hence, the constant power spectral density

S=So, 0<c0~< c%, S = 0 , co>~o o (12)

renders the variance of base acceleration

2 _ ~ S de) = Solo = (cg) 2 (I 3 ) Cr gg - -

Ao

The soil response is assumed to be

S(m) = So[1 + 4L'~ (m/~g)2] / { [1 - (co/c%)2] z + 4[~ (co/c%) = }

(14)

when ~g = 0.6 and o)g = 2.7 r.p.s. Height H of the concrete platform is 16-times the

cross-sectional radius R and compressional normal force is 5 % of the buckling value. Damping is 2 % of the critical coefficient. Eigenfrequencies in vacuum are co~= 2.74r.p.s. and o)2=600r.p.s. Hence, the second and higher modes will have negligible contributions to the forced vibrations.

Fig. 1 shows the sloping beach in distance d which is considered 24-times and 3-times the cross-sectional radius R, respectively, and R = 1.

The time-harmonic hydrodynamic pressure in the water body (sound velocity c) is a solution of the three- dimensional Helmholtz-equation

A3p+kZp=O, k=og/c (!5)

with boundary conditions:

@ = 0 at fixed part of boundary Fz 0n

0p On -pmZwcosO at surface of vibrating column Y2

p = 0 at free surface F3 fsurface waves are neglected)

To build up the interaction problem the pressure distribution on F 2 is needed° Hence, we use the associated boundary integral equation

i]7}

with the real part of the half-space Green's ~nct~on

[-cos(ktri) cos(k,'H

By that imaging technique the boundary condition on F3 is built in. r' is the point vector to the image source, Using a proper discretizadon net of the remaining surface F I u F 2 a !inear set of equations resuh

[Z]{ p} = {q} '!9~

The pressure is constant in each surface element in this simple version of BEM, and

Z q = - grj on ;']

Substructure synthesis method renders the ume reduced equation of motion of the platform surrounded by water

([K] - + Z) = {<} + {i%} + [G]{ n

where common notation has been used and

Psi= Li(x) Ps(R, 8, x)R cos 8 dO I dx (22)

is the generalized force due the rigid motion of the platform in water projected on the ith mode of vibration in vacuum (shape function L3. The matrix [P~] of hydrodynamic forces is associated to the modal matrb. [qh] in vacuum. Each column is the force excited at the platform when vibrating in the corresponding free mode of vibration at a fixed excitation frequency c~. The force {Pc} is the modal loading due to support motion %..

Orthogonalization, i.e., multiplication by the transposed modal matrix [4L-] r renders the linear equations

{ u]{ Y} = { v}

coupled by the hydrodynamic forces. Variations of the

94 Probabilistic Engineering Mechanics, I987, Vot. 2, No. 2

Random vibrations: F. Ziegler

X

2 7

8

8

4

2

O. I

X g

/

n/12

8

I ] l l l l [ ] [ l l l l l t l l ] l l i l

1.0

.8

.F~

.4

.2

P

' ] mt~ O. '''

0 .01 .02 .03 .04 .05 0 .01 .02

1

7) I~IIiI Ill I

d- 24~

d= 4R

d= 2E P

- - -7 I I I i * I I t t I [ mUg

• O3 . 0 4 . O5

Fig. 2. Distribution o f hydrodynamic pressure at forcing frequency oo = 1.2 r.p.s. Radius R = 1, H = 16R. Distance d = 2R, sloping angle ~t = O, respectively

25

20

15

10

Y

E-

- 5 J l l [ l l l i l i ~ i l l l l t ' l l l t l l l ' l t l l l l l [ i l l l l l l t l [ l l l l l l l f r t l l l E i l ' J J J ' ~ l l l l l l f [ l l l L I } l l l [ J i l i l l l l l l l l l l l , , ~

O .2 .4 .6 .8 1.0 1.2 1.4 1.S 1.8 2.0

Fig. 3. Frequency response function o f tip-deflection. 1 in vacuum, R = 1, H -= 16R, ~o 1 = 2.74 r.p.s., 2 in water, d = 24R, 3 in water, d = 3R

pressure distribution with distance and slope of the beach are studied in Ref. 25, an example is shown in Fig. 2.

Frequency response function of the tip-deflection is shown in Fig. 3. Evolution of the corresponding power spectral density is shown in Figs 4 and 5. Probability of first passage of that deformation through a critical level Yc is easily determined assuming a normal distribution and shown in Fig. 6.

3 . N O N S T A T I O N A R Y V I B R A T I O N S O F E L A S T O -

P L A S T I C n - S T O R E Y F R A M E

Irschik 2¢ considered the shear-wall beam discretized to a chain-like n-degree-of-freedom structure, Fig. 7. Excitation by the horizontal component of an earthquake modelled by an envelope process is taken into account.

Probabilistic Engineering Mechanics , 1987, Vol. 2, No. 2 95

RavMom vibrations: F. ZiegIer

Syy. ~

F F 600 ~-

5O0

400

300

200

100

f -

E

Fig. 4. ] stationary limit, 2 time 42s, 3 time 15s

r l

i . 5 2 2 , 5 3 3 . 5 4 ~ 5

Evolution of the power spectral denszty of ~p aefiec~zon afier switching on a white noise ground acceleration process,

S y y " "

i~5s

8SS Z

800

E

200 !

Fig. 5. process.

1 1.5 2 2 .5 3 3°5 4 4°~

Evolution of the power spectral density of tip defiection after switching 9n a coloured no~se ground acceleration Kanai-Tajimi filter parameters are [g = 0.6, e)g = 2.7

The relative storey deformation

xi = wl-- wi- 1 (24)

is split into the elastic part {~ and into the nonlinear (plastic) part x~,

xi=~i+ x~i (25)

For ideal elastic-plastic material behaviour, Fig. 7d, the plastic drift of the ith storey is represented by ~ ' and

the shearing force becomes

Q~=c~(xi- ~ ) = c ~ (26)

where c~ is the local stiffness. Eliminating the shearing forces in the momentum equation of the ith storey mass m~ renders

i

h = t k = ! (27)

96 Probabilistic Engineering Mechanics, 1987, Vol. 2, No. 2

1.0

. 8

. 6

.4.

. 2

O.

Fig. 6.

Random vibrations: F. Ziegler

P

t I I l I l t l I t t t t l I 1 ! L

0 5 I t ] 15 20 25 30 35 40 4 5 58

First passage probability of tip deflection through the critical values. 1 . . . yc = 2, 2 . . . y~ = 3, 3 . . . y¢ = 4

n

Cn

¢i

1

Cl

! Q

/ / / / [

Fig. 7.

mn

Qi*t lira mi w- " • i / / / / / / / z / c - d

d) I Oi N

Q1 ~ Ci

w~ , -Fi

I _ _1

n-storey building excited by ground motion due to the horizontal component wg of an earthquake," basic model according to plastic hinge theory

The linear part of the solution is excited by the ground acceleration as well as by the drift process. The left hand side of the equations has constant coefficients and represents the associated linear system is. We transform to diagonal mass and symmetric stiffness matrices by

i

~i=11i--~]i-l' l~i= ~ ~k (28) k = l

to get

i

mii/i-Cirh-1 +(ci+ci+ x)rh-Ci+ lrh+ l = - m l w o - m l ~. ~k k = l

i= 1 . . . . . n (29)

and use modal analysis

{r/} = [q~]{q} (30)

to render the oscillator equations, modal damping has

been added,

O j + 2(~oj~ j + co]q j = f ~,

The modal forcing functions are

j = l . . . . . n (31)

m*fj = - ~ej~g + ~ ~,Y~ i = 1

(32)

where the participation factors

~={c~j } r [m]{ I } , { I}= ~, {ek} k = l

~,={~ j } r [m] { I , } , { I , }= ~, {ek} k = i

(33)

and unit v e c t o r s {ek} are introduced together with the modal mass

m* = { ~b j} r [m] { ~bj} (34)

The envelope process of ground acceleration ~g is represented by the time-dependent power spectral density

Sgg(t, 09) = ~(t)S(~o) (35) Consequently, we characterize the effective modal excitation f j including the drift by changing the given envelope function if(t) and including frequency dependence,

~(t, 09) = ~k(t)~ke(t , 09) (36)

Thus, the approximation applicable to slightly damped linear oscillators with intermediate eigenfrequencies excited by a broad-band process having a narrow-band output renders for equation (31) the power spectral densities of the modal driving forces in the form

Sj(t, o9) = ZP2jS(o~j)~b (t)~ke(t , ogj)/m .2 (37)

Probabilistic Engineering Mechanics, 1987, Vol. 2, No. 2 97

Random vibrations: F. Ziegler

Assuming ~e(t, co) known, the output power spectra1 density is easily determined and the variance of any modal coordinate qj is directly calculated according to a formula derived by Spanos z6

7g ff 2 (t) = ~ e x p ( - 2~jcojt)S (CO j)

L~ 2 (t x exp a ~ o exp m,--~ (2:jcofi)~(z, c@ dr (38)

Vanmarcke ~9 reports alt the equations necessary to determine the second moments of the relative deformations {~, see also Irschik a4.

The effective envelope function O~(t, co) enters nonlinearly an energy balance. Consequently, equivalence between the change of the mean power of the effective forcing function of equation (31) without and including plastic drift and the average dissipated power due to the yielding process within the storeys is required:

, . 2 * 2 2 * 2 ~mj ![5~ j ~(t)/m~ - ~ ~ O(t, cofl/mj iS(coil

- ~ ~,(2#.. ,Ffijrcm3/m~ 2 = 0 i = i

(39)

~j

[ Structure 1

0.3

0 . 2 ~

0 3 6 9 12 15

Fig. 8. Effective envelope functions Oj(t), j = L 2 of equation (43) renderin9 the effective random forcing function of the associated linear system, equations (31), (37) in the case of the structure described in Table ]

The crucial assumptions on the drift process enters the latter expressmn, and Irschik 24 gave an tmproved formula for the mean yield increment 3~- of the ith store;? which depends on ~(t, co) through the varia~ce o-{ of the relative storey displacement ~e.

a~ m,&i 2 3~ 2 F~

Corresponding yield force ~s F~, the vietd barrier :s a~ = F/c~, and &; is an average frequency of the ith storey~ Compatible with the basic assumption of treating clusters of yield level crossing as points in time. at which the plastic drift ~ ' jumps to a different value, yielding, i.e crossing of totat x~ through the barrier a; of ~, is assumed in the ith storey only.

The average rate of those piastic clumps is estimated from

f2a. i= 2Va.i/( NaA " ) :4~

where v~.~ is the average number of exceedances of ~he given level a~ by the relative storey deformation x~ ane ( N , . ) is the expected number of consecutive peaks outside the elastic barrier See Ref, 24 for detai!s of the lengthy expressions.

Assuming the average state of the system known ~t some time-instant t v _ ~ and putting

~(tp, coil = ~9~(t~_ ~, o@ + Ap~ ~ (42,

the nonlinear equation (39) is sotved for the envelope increment &p~j.

Fig. 8 shows the effective e~velope functions

~;(0 = ~ ( : , %)/mY ~ :~.3)

of a two-storey frame defined in Table i. Stationary results compare favourably well with those given by Gazetas 2v

The variance of the nonlinear part of the store? deformation is determined by the time-integral

((xff)2> = 2 I #~ ~,&*a?(z) d ~ , , o ~ 0

(441

4o CONCLUSIONS

The spectral method discussed in this paper serves as a simple means to extend the time-harmonic soiution of linearized time-invariant systems even :o the nonstationary stochastic response if the input is an envelope random process, AdditionM manipulatmns

Table 1. Mechanical properties o f a two storey example structure. Ductility ratios lq at probability Pi = 0.5, time duration s = 20s refer (ai to simulatio~ results o f Gazetas 27 and (b) to present theory, Irschik 24

(mi/14.6) x 10-3 (ci/14.6) x 10- 3 (Fi/4.45) x 10-3 ~ ( P : = 0 . 5 ) i kg N / m N a b

1 ! .0 675 1 i 3.87 3A7 2 t .0 450 ~ 4.37 5 3@

(m*/14.6) x 10 .3 e)j (S,o(~3 '9.29) x 102 j kg 1/s m2/s 3" ~ P2

! 5.0 15.0 18.5 1.043 I.ff3 2 1.25 36.74 5.8 2.00 °-- •.50

98 Probabilistic Engineering Mechanics, t987, Vol. 2, No. 2

have low c o m p u t e r costs in compa r i son to the ca lcula t ions of the frequency response functions. N o restr ic t ions app ly with respect to discrete or con t inuous l inear s t ructures and to single po in t or d i s t r ibu ted r a n d o m loadings , e.g., r a n d o m wave load ings of deep pile foundat ions . Large systems are ana lysed with the help of the subs t ruc ture synthesis m e t h o d under the ac t ion of t ime-ha rmonic forces.

The concept of t ime-dependen t m o d u l a t i o n of power spectra l densi ty when general ized to t ime and frequency dependence gives a s imple a p p r o a c h to discrete elast ic- plast ic s tructures. Effective loadings of a t ime- invar ian t associa ted l inear system are de te rmined and upda t e d to the progress ing yield process. Several app rox ima t ive assumpt ions on the yield process m a k e it poss ible to calcula te the mean yield increment and by superpos i t ion the l inear and nonl inear par t of de format ion . Numer i ca l results compa re well with those der ived from c o m p u t e r s imula t ions in the s t a t ionary limit. F o r an one-s torey frame mode l tests have been per formed recent ly in the l a b o r a t o r y of the Ins t i tu te of Mechanics which conf i rmed also the nons t a t i ona ry results. F i r s t test results on a two s torey frame also confi rm the app rox ima t ions .

The next s tep in deve lopmen t is the imp lemen ta t ion of d a m a g e accumula t ion , i.e., mate r ia l stiffness deg rada t ion , recognized in cyclic plast ici ty.

A C K N O W L E D G E M E N T S

Research repor ted herein is suppo r t ed by the Aus t r ian Science F o u n d a t i o n F W F under g ran t $30-03, 1985-89. This suppor t is grateful ly acknowledged .

R E F E R E N C E S

1 Priestley, M. B. Power spectral analysis of non-stationary random processes, Journal of Sound and Vibrations, 1967, 6, 86- 97

2 HSllinger, F. Bebenerregte Schwingungen elastiseher Sperrenkonstruktionen, Doctoral Dissertation 1982, Technical University of Vienna, Austria A-1040

3 HSllinger, F. Time Harmonic and Nonstationary Stochastic Vibrations of Arch Dam-Reservoir-Systems, Acta Mechanica, 1983, 49, 153-167

4 HSllinger, F. and Ziegler, F. lnstation~ire Zufallsschwingungen einer elastischen Gewichtsmauer bei beliebig geformtem Becken, ZAMM, 1983, 63, 49-54

5 Meirovitch, L. and Hale, A. L. On the Substructure Synthesis Method, Publication of the Institute of Sound and Vibration Research 1980, University of Southampton, England

6 Chakrabarti, P. and Chopra, A. K. Earthquake Analysis of Gravity Dams Including Hydrodynamic Interaction, Earthquake Engineering and Structural Dynamics, 1973, 2, 143- 160

7 Porter, C. S. and Chopra, A. K. Dynamic Analysis of Simple Arch Dams Including Hydrodynamic Interaction, Earthquake Engineering and Structural Dynamics, 1981, 9, 573-597

8 Yeh, C.-S. and Ho, Y.-C. Earthquake Induced Water Pressures on a Gravity Dam from a Reservoir with Inclined Basin,

Random vibrations: F. Ziegler

Proceedings of the CCNAA-AIT Joint Seminar, March 1984, Taiwan

9 Zhang, Bo. Untersuchung von Bohrinseln unter Bebenbelastung, Doctoral Dissertation 1986, Technical University of Vienna, Austria A-1040

10 Ziegler, F., Irschik, H. and Heuer, R. Nonstationary Response of Polygonally Shaped Membranes to Random Excitation in Random Vibration-Status and Recent Developments. The S. H. Crandall Festschrift (eds I. Elishakoff and R. H. Lyon), Elsevier, Amsterdam, 1986, 555-565

11 Ziegler, F. and Hasslinger, H. L. Earthquake Vibrational Response of Turbo-Machines in Proceedings IUTAM/ IFToMM Symposium Dynamics of Multibody Systems (eds G. Bianchi and W. Schiehlen), Springer-Verlag, Berlin, 1986, 311- 323

12 Dasgupta, G. and Sackmann, J. L. An alternative representation of the elastic-viscoelastic correspondence principle for harmonic oscillation, Journal of Applied Mechanics, 1977, 44, 57-60

13 Ziegler, F. Erzwungene Schwingungen und ihre DS.mpfung, Sitzungsberichte der mathematische-naturwissenschaftlichen Klasse, Austrian Academy of Sciences, Springer Verlag Wien, New York, 1982, 191,233-239

14 Ziegler, F. Elastic-Viscoelastic Correspondence in Case of Numerically Determined Discrete Elastic Response Spectra, ZAMM, 1983, 63, T135-137

15 Crandall, S. H. and Zhu, W. Q. Random Vibration: A Survey of Recent Developments, Journal of Applied Mechanics, 1983, 50, 953-962

16 Wen, Y.-K. Stochastic Response and Damage Analysis of Inelastic Structures, Journal of Probabilistic Engineering Mechanics, 1986, 1, 49-57

17 Karnopp, D. and Scharton, T. D. Plastic Deformation in Random Vibration, Journal of the Acoustical Society of America, 1966, 39, 1154-1161

18 Vanmarcke, E. H. and Veneziano, D. Probabilistic Seismic Response of Simple Inelastic Systems in Proceedings of the 5th World Congress of Earthquake Engineering, Rome, Italy, 1973, 2851-2863

19 Vanmarcke, E. H. Structural Response to Earthquakes in Seismic Risk and Engineering Decision (eds C. Lomnitz and E. Rosenblueth), Elsevier, Amsterdam, 1976, 287-337

20 GroBmayer, R. Stochastic Analysis of Elasto-Plastic Systems, Proc. ASCE, Journal of the Engineering Mechanics Division, 1981, 39, 97-115

21 Grol3mayer, R. and Iwan, W. D. A Linearization Scheme for Hysteretic Systems Subjected to Random Excitation, Earthquake Engineering and Structural Dynamics, 1981, 9, 171- 185

22 Ziegler, F. and Irschik, H. Nonstationary Random Vibrations of Yielding Beams, Paper M 15/1" in Proceedings of 8th Conference on Structural Mechanics in Reactor Technology, SMiRT, Brussels, Belgium, 1985, 123-128

23 Irschik, H. and Ziegler, F. Nonstationary Random Vibrations of Yielding Frames, Nuclear Engineering and Design, 1985, 90,357- 364

24 Irschik, H. Nonstationary Random Vibrations of Yielding Multi-Degree-of-Freedom Systems: Method of Effective Envelope Functions, Acta Mechanica, 1986, 60, 265-280

25 Zhang, Bo. and Ziegler, F. Ein Randintegralgleichungsverfahren ftir die Schwingungen einer elastischen S~ule in einem Fliissigkeitsbeh~ilter, ZAMM, 1986, 66, T115-117

26 Spanos, P.-T. D. Probabilistic Earthquake Energy Spectra Equations, Proceedings ASCE, Journal of the Engineering Mechanics Division, 1980, 106, 147-159

27 Gazetas, G. Random Vibration Analysis of Inelastic Multi- Degree-of Freedom Systems Subjected to Earthquake Ground Motions, M.I.T. Report, 1976, Cambridge, Massachusetts

Probabilistic Engineering Mechanics , 1987, Vol. 2, No. 2 99