random vibration of structures with strong non-linearities ... · random vibration of structures...

Random vibration of structures with strong non-linearities subjected to seismic excitation

H.O. Soliman, A . E . Bakry Faculty of Engineering, Zagazig University, Egypt

Abstract

The response of nonlinear structures subjected to earthquake excitation modeled by white noise random process is calculated. The non-linearity of structural elements is modeled by a first order differential equation representing a curvilinear relationship between the restoring force and the displacement. that is suitable for the dynamic analysis of structures. The spectral method of analysis is used to obtain the Root Mean Square response of the structure using the complex frequency function of the structural system. The equivalent linearization technique is used to obtain the equivalent linear parameters for stiffness and damping of the non-linear structure. The degree of nonlinearity, and the damping ratio of the structure are investigated through the parametric study.

1 Introduction Many of today's structures are subjected to excitations, which are random in nature. Examples range all the way from aircraft and missile structures subjected to acoustic and aerodynamic loads to civil engineering structures acted upon by earthquake and wind loads. In some cases. the response statistics of such structures will be strongly time-dependent, but in many applications. the response may be considered stationary. Even when the actual response is not stationary. an understanding of the stationary response of a system is often helpful in predicting its general response behavior. For this reason, considerable effort has been directed towards the development of techniques for determining the stationary response statistics of stochastically excited systems.

Stationary excited linear systems have been studied in great details and numerous analytic techniques exist for treating both stationary and nonstationary problems. In the earthquake environment, not a few structures show some nonlinear response stemming from the yielding process. For such systems. unlike linear ones. the response depends on both the input intensity of ground motion

Transactions on the Built Environment vol 57, © 2001 WIT Press, www.witpress.com, ISSN 1743-3509

14 Earthquake Resisranr Enprneerrng Structwes /[I

and the restoring force-deformation characteristic, making it difficult to get exact analytical solutions in general. Various approximate techniques have been proposed for this reason. Lutes L. D. [ l ] made a review to the methods of equivalent linearization. Lutes L. D. and Shah V. S. [2] investigated the oscillators that have a bi-linear hysteretic restoring force characteristics. Lutes L. D. and Takemiya H. [3] studied the problem of predicting the response level of a non-linear oscillator subjected to random excitation. Takemiya H. and Lotus L. D. [4] studied a quite accurate equivalent linearization technique for Masing's type hysteretic systems in random vibration. Iyengar R. N. and Dash P. K. [5] presented a new technique, called Gaussian Closure. Iwan W. D. and Gates N. C. [6] investigated the accuracy o f the various approximate methods for defining equivalent linear systems for simple non-linear structures subjected to earthquake excitation. Sues R. H., et. al., [7] identified the parameters o f degrading, hysteretic restoring force model for structures subjected to extreme dynamic loads. Spanos P. D. and Red-Horse J. R. [S] studied a class of non- linear single degree o f freedom oscillators subjected to zero mean modulated Gaussian white noise excitation.

2 Method of analysis

Figure ( 1 ) shows a single story structure, where y is the displacement of the structure. measured from the static equilibrium position. The restoring force, due to the non-linear behavior o f the structure is denoted by Q(t) = aKy+ (1 -a ) y z where cc is the post-pre yield stiffness ratio. In the corresponding linear structures, the function Q(y) = ky, where k is the linear stiffness of the structure. The equation of motion for the nonlinear structure subjected to earthqsake ground motion is given by;

? + g ( y , y ) = -U, (1

It will be convenient to decompose g(y. y ) into a linear damping and stiffness

components. p y and w i y respectively ,where P = 2 j U,, together with the non- linear component p G (z, $) , where y is a scaling factor, < is the damping ratio, W,, is the undamped natural frequency for the linear system. z is the hysteretic part of the restoring force and U, is the earthquake ground acceleration.

g ( y , y ) = P y + w i y + y ~ ( r , j i ) (2)

To apply the statistical linearization procedure the non-linear system (1 ) is replaced by an equivalent linear system

y +Peq y + Oq y = ii, , P,q = 2 jCq w C q . ( 3 )

The difference ( E ) b e t w e e n t h e non- l inear and t h e e q u i v a l e n t linear equations is given by

~ = g ( y > j ) - P , ~ y - ~ : ~ y (4)


Earthquake Resistant Engineering Strucrures III 15

To obtain the expressions for p,, . m,,, it is necessary to minimize the expected

value o f E in a least square sense. Evidently these values must satisfv the following relationships

Applying conditions (5) to equation (4): the following relationships are obtained and simplified as follow;

E { Z ~ ( Z , Y ) ) - P , , E { Z Y } - W ~ ~ E { Y ' } = O (7)

Since y (t) and z are stationary processes with zero mean, the two equations (6) and (7) can be reduced by considering the following relations,

E { z ~ ) = o E { Z ~ ) = C Y ~ E { ~ ; ' } = c Y ? Y

(8)

where 0, and o, are the standard deviation of y ( t ) and 5; ( t ) respectively. Using

equations (S), equations (6) and (7) are simplified to the following

Using Kazakov formula. Robert and Spanos [9]. which states that

~ { f ' 7 '7) = ~ ( ' 7 qT) E{V f ('7) } (10)

where V is the gradient operator defined by

p,, ando:, can be simplified by:

Using equation (2) representing g ( r , y ) , equations P,, ando:, are given by:


3 Structures with strong nonlinear stiffness

The specific case of structures with a curvilinear hysteresis characteristic will be considered. Defining a state vector q where q , = y , q: = z , the hysteretic non-

linear behavior o f the structure is represented by the following first order differential relationship , Soliman [10]:

~ ( i l l > q , ) = - y I i l l I q 2 - v i l , 1 % / + A % (14)

where cc, v, and A are parameters controlling the non-linear hysteretic behavior of the structure as shown in figure (2).

where f is the Gaussian joint probability function of ql and q2. After

long integration operations, the following expressions are obtained;

To evaluate p,, and of, according to the above equations, ~t I S requlred a

knowledge of o, and o, . Further relatlonshlps between o, and o, and P,, and

wi,may be obtained by using the equivalent linear system to compute

G,, a n d o , . It will be assumed that the input process f (t) is Gaussian with a

power spectral density function Sf (W).

-- -m

Where the appropriate frequency response function H (W) is given by

Therefore, four simultaneous equations can be found for the four unknowns P,, ,

W:,. o, and o, . A numerical solution of these equations will lead to expressions

of the four quantities.


Earthquake Resistant Engmeerzng Structures 111 17

4 Discussion of results The structure shown if Fig. (1) is used for the numerical study. Figure (2) shows the non-linear behavior of the structure, where Q is the hysteretic restoring force of the structure, and a is the post-pre yield stiffness ratio. The mass, and the stiffness of the structure are characterized by the natural period of the structure T in seconds, and the damping is characterized by the damping ratio 5. The results are obtained in terms of Root Mean Square (RMS) response and the RMS ratio which represents the ratio of RMS response of non-linear structure to that of linear structure. Fig. (3) shows the RMS displacement ratio of structures having different time period (T) and with different damping ratio where the post- pre yield ratio a equals 0.1. It is shown that the inelastic displacement response relative to the elastic one is increased with the increase of damping ratio. As a result, the damping decreases the displacement of non-linear structure less than that of linear structure. This is clarified again in Fig. (4) where the increase of damping ratio decreases the RMS displacement. The effect of damping on the RMS velocity of the structure is shown in Fig.(5). It is shown that the increase of damping ratio decreases the RMS velocity. Figure (6) shows the variation of RMS velocity ratio with damping ratio for different structures. The effect of damping on RMS velocity is the same as in the case of RMS displacement.

Fig. (7) shows the RMS ratio of restoring force for different structures. It is shown that the increase of damping ratio increases the RMS ratio of restoring force because the high values of damping ratio increases the dissipated energy in non-linear structures. It is shown also that the RMS ratio of restoring force is decreased with the increase of structure time period. Figures (8) and (9) show the RMS displacement ratio of structures having different time period (T) and with different damping ratio where the post-pre yield ratio a equals 0.3 and 0.7 respectively. It is shown that the increase of a decreases the RMS displacement ratio because the high values of a tends to represent linear structures. This is shown also in figures (10). where the lower limit of RMS ratio of velocity tends to unit value in case of high values of a . Figures (1 1) and (12) show the RMS ratio of restoring force of structures having different time period (T) with different damping ratio where a equals 0.3 and 0.7. It is shown also that the RMS ratio of restoring force approaches unit value for high values of a .

Conclusions

The stochastic response of structures with strong non-linear behavior is obtained in terms of Root Mean Square (RMS) values and its relative values to those for linear structures. The effect of damping ratio and the post-pre yield ratio which defines the degree of nonlinearity are studied. It is found that the RMS displacement of non-linear structure is always higher than that of linear structure. The damping ratio has less effect on decreasing the displacement response of non-linear structures than that of linear structures. The high values o f post-pre yield stiffness ratio decrease the effect of non-linearity on the different response values.


18 Earthquake Resistant Engineering Strztctures III

References Lutes L.D.: "Equivalent linearization for random vibration", Journal of the Engineering mechanics division, ASCE, Vol. 96, No. EM3, pp. 227-242, June 1970. Lutes L.D. and Shah V.S.: "Transient random response of bilinear oscillators", Journal of the Engineering mechanics division, ASCE, Vol. 99. NO. EM3, pp. 715-734, August 1973. Lutes L.D. and Takemiya H.: "Random vibration of yielding oscillator", Journal of the Engineering mechanics division, ASCE, Vol. 100, No. EM2, pp. 343-358, April 1974. Takemiya H. and Lotus L.D.: "Stationary random vibration of hysteretic systems", Journal of the Engineering mechanics division, ASCE, Vol. 103, No. EM4, pp. 673-687, August 1977. Iyengar R.N. and Dash P.K.: "Study of the random vibration of nonlinear systems by the Gaussian Closure technique", Journal of Applied Mechanics, ASME, Vol. 45, pp. 393-398, June 1978. Iwan W.D. and Gates N.C.: "Estimating earthquake response of simple hysteretic structures", Journal of the Engineering mechanics division, ASCE, Vol. 105, No. EM3, pp. 39 1-405, June 1979. Sues R.H., Mau S.T., and Wen Y.K.: "Systems identification of degrading hysteretic restoring forces", Journal of Engineering Mechanics, ASCE, Vol. 114, No. 5, pp. 833-848, May 1998. Spanos P.D. and Red-Horse J.R.: "Nonstationary Solution in nonlinear random vibration", Journal of Engineering Mechanics, ASCE, Vol. 114, No. 11, pp. 1929-1943, November 1998. Robert J .B. , and Spanos P.D., "Random vibration and statistical linearization". John Wiley & Sons, New York. 1990.

[l01 Soliman: H.O., "Study of Differential hysteretic models for the stochastic analysis of structures", The Eighth Arab Structural Engineering Conference, Cairo, Oct. 2000.

- - -.-h - U 0

Figure 1 : Single story building Figure 2: Restoring force- displacement relationship


Earthquake Reslstatir Englneerlng Str.~rctwes 111 19

0.5 1 .O 1.5 Time period of structure (T)

Figure 3 : RMS ratio of displacement (a = 0.1)


Figure 4 : RMS displacement of different structures ( a = 0.1)


20 Earthquake Resistant Engineering Strzlctzwes III

0.0 0.5 1 .O 1.5 2.0 Time period of structure (T)

Figure 5 : RMS velocity of different structures (a = 0.1)


Figure 6 : RMS velocity ratio of different structures (U = 0.1)


Earthqzlake Resrstnnt Engineerzng Strzlctwes 111 2 1


Figure 7 : RMS ratio of restoring force for different structures (a = 0.1)


Figure 8 : RMS ratio of displacement for different structures (a = 0.3)


22 Earthquake Resistant Engineering Structures I11


Figure 9 : RMS ratio of displacement for different structures (a = 0.7)


Figure l 0 : RMS ratio of velocity for different structures (a = 0.7)


Eartizquake Keslstant Engneer~ng Str.zlctwes I11

Time period of structure (T)

Figure 11 : RMS ratio of restoring force for different structures (a = 0.3)


Figure 12 : RMS ratio of restoring force for different structures ( a = 0.7)


random vibration of structures with strong non-linearities ... · random vibration of structures...

Documents