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Earthquake response of nonlinear plates
Nuclear Engineering and Design 54 (l 979) 407-417 O North-Holland Publishing Company
EARTHQUAKE RESPONSE OF NONLINEAR PLATES Goodarz AHMADI * Department of Mechanical Engineering, Shiraz University, Shiraz, Iran Received 28 March 1979
The finite amplitude vibration of a plate subjected to earthquake support motion is considered. Several bounds on the maximum responses of the plate are established which are based on the knowledge of the response spectra of the seismic motion. The stochastic models of the earthquake ground motion is then briefly discussed. The single mode of vibration of the plate is then considered and a perturbation series expansion is developed for the vibration amplitude. Mean square responses of the plate are then calculated for both stationary as well as nonstationary seismic excitations. It is observed that the variance of the deflection field of the nonlinear plate is less than that of the corresponding linear one. The reliability of design is also considered and the probability of no barrier crossing is briefly discussed.
1. Introduction
method the maximum responses of a single degree o f freedom structure to several past earthquakes are calculated and a smooth curve is fitted to the mean plus one standard deviation o f the maximum responses. Furthermore, some physical reasoning is used to correct the limiting range of high and low frequencies. The result is then called the "design response spectrum" and this will be used in the actual design. For more detail see [1,2,5]. Here, we are concerned with the probabilistic approach in which the ground acceleration is being modelled b y a stochastic process and the methods o f random vibration are employed. A number of stochastic models of earthquake ground motion were proposed in the past, as summarized b y Clough and Penzien [ 1], Newmark and Rosenblueth [5 ], and more recently b y Ahmadi [6]. Among the most well-known models are the white noise representations o f Housner [7] and Bycroft [8], the finite duration white noise models o f Rosenblueth et al. [5,9,10], the filtered white noise representation o f Kanai [11] and Tajimi [12] and the nonstationary models o f Amin and Ang [ 13] among others [1,2]. The random vibration o f elastic systems have attracted considerable attention in the past two decades b o t h in the field o f mechanical vibration as well as structural dynamics. The response o f lumped mass systems subjected to stationary excitations were
Analysis of the response o f structures subjected to earthquake ground acceleration is o f fundamental importance for their aseismic design. For such an analysis to be able to predict the realistic behaviour o f a structure during an actual seismic motion, certain information is necessary. First, the nature of the ground motion must be specified, and secondly the mechanical behaviour o f the structure should be modelled within a reasonable accuracy. If the time history o f the ground acceleration is given, there exist numerous analytical and numerical techniques [ 1 - 5 7 ] for the determination o f the time history of the response o f various linear and nonlinear structures. However, only the time histories o f a number o f past earthquakes are available and there is no method for predicting the time histories o f the future earthquakes for a particular site. Furthermore, the available accelerograms vary to a considerable extent in their time duration, frequency content, and maximum amplitude. To overcome this difficulty two methods are available. These are the use o f design response spectra and the probabilistic approach. In the former
* Presently academic visitor at the Department of Mechanical Engineering, University of Newcastle Upon Tyne, U.K. 407
408
G. A hmadi /Earthquake response of nonlinear plates
considered by several investigators. (See, for instance [1-5,14-17] .) Recently, the mean-square response of structures to random excitation was studied by Gersch [18], Zeman and Bogdanoff [19] and Ahmadi [20]. The response of single degree of freedom systems to nonstationary random excitation was investigated by Barnoski and Maurer [21,22], Bucciarelli and Kuo [23], Caughey and Stumpf [24], Corotis et al. [25, 26] and Bogdanoff et al. [27]. The response of beams and plates to nonstationary random loads were studied by Ahmadi and Satter [28-30]. The response of structures to nonstationary random excitation was considered by Holman and Hart [31 ]. An analysis of the earthquake response of linear continuous structures by the probabilistic approach was carried out by Ahmadi [32]. Although analysis of the linear response of elastic structures subjected to random forces is useful for understanding their behaviour during small earthquakes, in some structures, such as nuclear power plants, various components must be designed to withstand a maximum possible earthquake intensity, termed the "safe shut down earthquake", at the site of the plant. For such cases the linear response of structures is not sufficiently accurate and finite amplitude vibration of the various elements of structures must be considered. Therefore, a nonlinear stochastic system analysis must be used for an accurate design. The response of nonlinear systems to stationary random excitation was considered by Caughey [33, 34], Ariaratnam [35], Crandall [36] and Lyon [37, 38], among others, as summarized in survey articles by Caughey [39-41] and Crandall [42]. Stationary random vibration of a nonlinear plate was studied by Ahmadi, Tadjbakhsh and Farshad [43]. In the present study the response of a nonlinear plate subjected to earthquake support motion is considered. Several bounds on the maximum responses of the plate are obtained which are related to the response spectra of the seismic motion. A probabilistic approach for the analysis of the response of the plate is then undertaken. A single mode Galerkin expansion on space is considered and a perturbation method is employed for the vibration amplitude. Stationary as well as nonstationary models of the earthquake ground motion are considered and the
mean-square responses of the deflection field for both cases are obtained. It is observed that the variances of the lateral displacement of the nonlinear plate are less than those of the corresponding linearized plate for stationary as well as nonstationary loadings. The probability of no barrier crossing during a time interval is also studied.
2. Basic equations The equations governing the f'mite amplitude vibration of plates subjected to earthquake ground acceleration are given by
pH ~2w
yaw + . . . . D 3t 2
a2F
h.
H [~2w ~2F ~ Wg(t) + - D [~x ~3y2
a2w a2F]
23x~y 3x~y + ~y2 aX2 J,
(1)
and
//__it/2wX2 a2w
7
(2)
In these equations x and y are cartesian coordinates, w is the lateral deflection of the plate relative to the ground and Wg is the lateral ground acceleration; p and H are the mass density and thickness of the plate, respectively. E is Young's modulus of elasticity, v is Poisson's ratio, and D is the bending stiffness defined by
D =EH3/12(1 - / ) 2 ) .
(3)
F is Airy's stress function. The in-plane forces Nx, Ny and Nxy are given by ~2F N x = Oy2 ;
~2 F N y = Ox 2 ;
~2F
Nxy -
Ox
ay
(4)
We should mention here that in the derivation of eqs. (1) and (2) both extensional and rotary inertia effects have been neglected. Furthermore, only the effect of lateral ground acceleration is being considered while in general both lateral and axial ground accelerations are present. However, for simplicity the more restricted form of the vibration equations (1) and (2) are being considered for the present analysis. For a simply supported rectangular plate of sides
G. A hmadi /Earthquake response of nonlinear plates a and b the boundary conditions on w are a2w a2w w = 0 and ~-7~- + v - = 0 , o y 2 a t x = 0 , a,
409
For a simply supported plate the solution is of the fo rm
wOc, y, t) = HA(t) sin 02W
w=0
mrrx
~2 W
and a ~ - + v ~ - - ~ - = 0 ,
a t y = 0 , b.
(5)
a
nrry
sin.
(10)
b '
The boundary conditions for the Airy stress function for the case of a plate with stiess free edges becomes
where A (t) is a dimensionless function of time, and m and n are integer numbers. The function, eq. (10), obviously satisfies the boundary conditions eq. (5). Inserting eq. (10) into eq. (2), it follows that
32F Oy2 =0
a t x = 0 , a,
V4F = ~ - ( ~ - - - ) A
aty = 0, b.
Furthermore, the following solution for the stress function is assumed:
O2F 3x3y-0'
and
E nmlr2H 2 2
(t) [cos 2m----~+ cos ~b-~-] (.1 a
02F 3x 2 - 0
O2F 3x Oy - 0,
and
(6)
For a plate whose edges are immovably constrained, instead of eq. (6) we have and
o=0
and
axay
-0,
atx =0,a,
-0,
a t y = 0 , b,
02F
3x 0y
(7)
where u and v, the midplane displacements in the x and y directions, are given by 0
(E \3y 2
o =-flY ,[11321;-I~x2
v ~X2] -- 2 \ 3 X ]
~2F~3y 2 ] - 2\-~y]l(3w '~2 } dy.
w=0
and
aw
~x-x=0,
a t x = 0 , a,
aw 7 - =0,
a t y = 0 , b.
oy
(1 - c°s 2rnrrx a ) (1 - c°s 2 - ~ )
F* =
dx,
(g)
For a clamped plate the deflection boundary conditions become and
- cos - - b ~ ] . u 2 )
and integrating over the surface of the plate, it is found that
0
w=0
27)(,
The assumed solution, eq. (12), clearly satisfies the stress free edge boundary conditions, eq. (6), for the Airy stress function. Employing eq. (12) in eq. (11), multiplying by
a2F
u=0
cos
1)
3. Approximate analysis Exact analytical solutions of eqs. (1) and (2) are not available in the general cases. In this section an approximate single mode assumption, which is a generalization of [44], is employed in order to derive the decoupled equations for the dynamics of the time dependent amplitudes.
(13)
Now employing eqs. (10) and (12) in eq. (1), multiplying the resulting equation by the corresponding mode shape of lateral displacement, and integrating over the surface of the plate, the result is +
(9)
E(nm/ab )2 8 [(m/a) 2 + (n/b)2] 2"
+
pH
A
4¢og
=-mnHTr2 [ 1 - ( - 1 ) m ] [ 1 - ( - 1 ) n]
p
\ab!
+
-b
"
(14)
The procedure which was followed here is the wellknown Galerkin method for Finding approximate solutions to nonlinear equations. It should be mentioned here that for other types of boundary conditions such as clamped and, or immovably constrained boundaries, a similar method
41 o
G. Ahmadi /Earthquake response of nonlinear plates
could be applied with a proper mode shape. For instance, for a clamped plate the form of the mode of deflection could be taken to be
w =HA(t)sin 2 rnTrx sin2 a
HTrO
b
(15) '
which satisfies the boundary conditions (9). Employing eq. (15) in eqs. (1) and (2) yields an equation for time development of A (t) similar to (14), but with different coefficients. Introducing new parameters
remains small in comparison to the linear part. Employing eq. (19) in eq. (20) and equating the coefficient of like powers of e equal to zero, yields a set of linear equations for Ai, i = 0, 1,2, .... These are A'o + 2~o-Zio +cogAo = -~(~g(t), A'l + 2~'eOoA1
+
co2A1 = -cooA2 03,
A'2 + 2~'6Oo,zi2 + topAz = -3eo~A 1A~.
(20)
The general solutions of the first two equations of eq. (20) are given by t
=7#
+
'
4 - mnHTr2 [1 - ( - 1 ) m] [1 - ( - 1 ) n ] ,
Ao(t) = - ~ f
h(t - r) (~g(r) dr,
(21)
to t
(16)
A , (t) = -W2o f h(t - r ) A 3 ( r ) dr, e=,
EH 3
(nmTr/ab) 4
D
[(m/a) 2 + (n/b)2] 3'
where h(t) is the impulsive response of the corresponding linear system and is given by
eq. (14) becomes
A" + togA + ecogA 3 = -~Wg(t).
1
(17)
For the first mode approximation m = n = 1 and the parameters defined in eq. (16) simplify correspondingly. If a damping term is also present in the plate vibration equation (1), then eq. (17) generalizes to the following equation: A' + 2~'co0d + w~A + ew2A 3 = -~Wg(t),
(22)
to
(18)
where ~"is the damping coefficient.
h(t) = ~oo exp(-c°°~'t} sin f2ot,
(23)
with g22 = co2(1 _ ~-2).
(24)
For an initially motionless plate the value to = 0, and for a stationary response the value to = _oo must be considered in eqs. (21) and (22). A two term perturbation approximation for the lateral vibration of the plate thus becomes
w(x,y, t ) = H [ A o ( t ) + earn (t)] sin mwx a sin nay --b---, (25) 4. Perturbation s o l u t i o n
The classical method for solving nonlinear differential equations is the perturbation method. (See, for instance, Stoker [45] .) Applications of the perturbation method to random vibration has been carried out by, among others, Crandall [42], Crandall and Mark [46], Caughey [41] and more recently by Ahmadi and co-workers [43,47]. The basic assumption in the perturbation method is that the solution to eq. (18) has an expansion in powers of e, i.e.
A(t) = Ao(t) + eA l(t) + J A 2 (t) + ....
(19)
with Ao (t) and A l (t) being given by eqs. (21) and (22), respectively.
5. Bounds o n m o t i o n When the plate is subjected to arbitrary seismic support motion with known response spectra, the maximum response of the lateral deflection field may be estimated from eq. (25). For instance, max iw(x, y, t)i ~< H(max tAo(t)t + e max IA1 (t)l) t
Assumption (19) is expected to be valid for small e, i.e. when the nonlinear part of the restoring force
t
t
X sin m n x sin nny a
b
.
(26)
G. Ahmadi / Earthquake response of nonlinear plates velocity response spectrum given by
The maximum absolute value o f A o (t) is given by max I Ao(t)t = c~S~(cOo, ~') t
(27)
where the displacement response spectrum is defined by t Sd(O.)0, ~') = max I J h(t - r) t 0
drt.
(28)
The maximum o f A 1 (t) may be estimated from eq. (22), i.e. max IA 1 (t)l • t
6o20ct3S3(~0, ~')h m ,
(29)
where tm hm = f o
Ih(r)l dr
411
(30)
with t m being the time duration of the earthquake ground motion. Employing eqs. (27) and (29) in eq. (26), a bound on the maximum relative deflection of the plate is obtained, i.e. max lie(x, y, t)t ~< altSa(~o, ~)
Sv(cao, ~') = caoSd(~O, ~').
(36)
The inequality (32) thus becomes max I~b(x, y, t)l ~< ¢d-/Sv(cao, ~') t,x,y
X [1 + ea2hmS2v(COo, ~')].
(37)
Evaluating the second time derivative of eq. (25) and noting that h(0) = 0;
/~(0) = 1,
(38)
the bound on the maximum absolute acceleration becomes max lff,(x, y, t) + ¢d-/lCg(t)l t,x,y (39) ~< crHSa(6Oo, ~')[1 + ect2S~(¢Oo, ~X1 + w2hm)], where the acceleration response spectrum is defined by t Sa(cOo, ~') = mtax f J~(t - r) I~g(r) dr, 0
(40)
and is approximated by
t,x,y
X [1 + eco2a2hmS2(wo, ~')1.
(31)
Sa(wo, ~') = cooSv(wo, ~) ~-- cO2Sd(WO, ~).
(41)
Similarly, the maximum absolute relative velocity is bounded by
Furthermore, in the derivation of inequality (41) the approximation
max Iw(x, y, t)[ ~< cd-/Sv(cOo, ~')
hm = f o
t
t,x,y
+ d--Ico2t~3t~mS3(cdo, ~'),
(32)
where tm ]~m = f o
I]~(r)i dr,
(33)
and the velocity response spectrum is defined by t Sv(cao, ~') = max I f / ~ ( t - r) t 0
drl.
(42)
is also employed. It should be noted here that the left hand side of eq. (39) is not exactly the total absolute acceleration, and if for instance the first mode of vibration is being considered, i.e. m = n = 1 from eq. (16), it follows that +~/l~'g =~,
16 .. + n-~- leg.
(43)
(34)
For a small damping coefficient ]~m ~-- ¢°ohm
Ih(r)l dr--~ ¢.o2hm
(35)
and Sv becomes approximately equal to the pseudo-
The difference arises from the single mode assumption of eq. (10) instead of a full Galerkin series. Therefore, whenever the response spectra of seismic motion are known either from direct calculations for actual earthquakes or from an assumed design response spactra [1,5], eqs. (31), (37) and (39) give
412
G. A hmadi / Earthquake response of nonlinear plates
upper bounds on the relative displacement, relative velocity, and absolute acceleration of the plate. However, it should be mentioned here that the bound provided by inequalities (31), (37) and (39) are quite conservative. In particular, the estimate of the maximums of the first order perturbation term A1 (t) and its time derivatives are highly conservative.
X sin 2 m n x sin2 tory a
Therefore, variance ofA o (t) and covariance ofA o (t) and A 1(t) are needed for the calculation of the mean square of the deflection field. From eq. (21) it follows that t
t
f d72
{ A 2 ( t ) ) = - 2 f drl 6. Response to stationary earthquake models
to
According to Housner [7], Bycroft [8] and Rosenblueth et al. [5,9,10] earthquake ground acceleration could be represented by a simple stationary white noise, with a constant power spectrum and a delta function autocorrelation, i.e., R ~g(Z) = 27rSo6(7),
(44)
where So is a constant power spectrum that was estimated [1,8] to be equal to 0.375 ft2/sec 4- cps for the N - S component of the E1 Centro (1940) earthquake. For Housner's standard response spectra So is found to be equal to 0.0063 ft2/sec 3. There have also been several stationary filtered white noise models of earthquakes such as those suggested by Kanai [11] and Tajimi [12] and more recently by Liu and Jhavri [48]. The autocorrelation of the deflection of the plate is defined by R w ( x , y , tl, t2) = (w(x,y, t l ) W ( x , y ,
t2))
(45)
where angular brackets, ( ) , denote the expected value (ensemble average). For the stationary response the autocorrelation function depends only on the time difference, t I - t 2 . Employing the general solution (25) in eq. (45), up to the first order in e, we find
X h(t
to
71)h(t
-
72)RWg(71,72).
-
t
(AS(t)) : 2rro~2So /
Making a change of variables in eq. (49), it follows that oe
(A 8) = 27ra2So f
h2(7) dT
which is independent of time as is expected for stationary processes. Employing the expression for the impulsive response as given by eq. (23), eq. (50) becomes (Ag) = n a 2 S o / 2 f w 3 .
(51)
Employing eqs. (21) and (22), the covariance of Ao and A 1 become t
f d,f
= H 2 [(A~ (t)) + 2eGto (t)A 1 (t))]
t
tO
t
t
d,, f d,: f d,3 to
tO
t
(46)
and the variance of the lateral displacement is given by OZw(X, y, t)= Rw(x, y, t, t)
(50)
0
e(Ao(t2) A 1 (tl))]
X sin 2 mTrx sin2 nny a b '
(49)
hZ(t - 71) dTl.
to
+ {?("40 ( / 1 ) A 1 ( t 2 ) ) +
(48)
For the stationary white noise model of the earthquake ground acceleration, its autocorrelation is given by a delta flmction ofTl - r2 as defined in eq. (44). Eq. (48) then reduces to
C4o(t)A,
Rw(X, y, tl, t2) = H 2 [ ( A o ( t l ) A o ( t z ) )
(47)
b
Xf
dr 4 h ( t - v ) h ( t - 7 1 ) h ( r - r a ) h ( r -
73)
to X
h ( 7 --
T4)(WE(71)WE(T2)WE(T3)WE(T4))"(52)
When the ground acceleration is being modelled by a white noise process which is Gaussian, the fourth order correlation in eq. (52) is related to the auto-
G.Ahmadi/Earthquakeresponseof nonlinearplates correlation of
(¢g(t) and is given by
( ~lg( T1) ~]g(T 2 ) ~]g(7"3 ) ~/g(T 4 ) ) = 47r2S2 [6('/1 - 7 2 ) 6 ( 7 3
- 7"4)
(53)
+ 5('/1 - T 3 ) ~ ( T 2 -- "/'4) + 5 ( 7 1 -- 7"4)6('/'2 -- 7"3)]'
The expression for the covariance, eq. (52), becomes oo
(A o (t)A1 (t)) = -12rr2sgcoga 4
f dr h(r) 0
× ? dO h(O)h(O+r) ? h2(rl) dr1. 0
(54)
0
Employing eq. (23) in eq. (54) and evaluating the integral the result is 3~r 2 a4 S 2
=
8~.2o~~ ,
(55)
which is time independent. When eqs. (52) and (55) are substituted into eq. (47), the expression for the mean-square response of the lateral deflection of the plate is found, i.e.
O2w(x, y) -
7r°~2H2 S °
(1
3rrea2S°
)
× sin2 rnTrxsin2 tory (56) a b The variance becomes time independent as would be expected for stationary processes. From eq. (56) it is observed that the mean-square response decreases with an increase of e. That is to say, the variance of the deflection field of a nonlinear plate is smaller than that of the corresponding linear plate. A similar conclusion in connection with a hardening spring oscillator, such as the Duffing oscillator, is pointed out in [37,38,42]. In a previous investigation [43] a result similar to eq. (56) was obtained also by the method of equivalent linearization and by the exact method of the Fokker-Planck equation. The 30 level is known to be a conservative estimate of the maximum response, see for instance [1,5,32, 49]. Therefore, max
[w(x,y, t)l <~ 3Ow(X,y),
(57)
t
with
Owgiven by eq. (56). From eqs. (56) and (57) it
413
is concluded that the maximum deflection of a nonlinear plate is less than that of the corresponding linear plate when both are subjected to the same earthquake support motion. This is in contrast to the relatively crude bound given by eq. (31). From the physical side of the problem, a nonlinear plate behaves like a nonlinear hardening spring and thus the restoring force increases rapidly with the increase in vibration amplitude. Hence, it is expected that the maximum deflection of the nonlinear plate be smaller than the linear one in agreement with the prediction of the statistical analysis of eq. (57). A similar analysis for the evaluation of the variances of the relative velocity and absolute acceleration of a plate could be carried out. However, when damping is small the approximate expressions for the standard deviations become
o~,(x, y) ~- ~oOw(X, y),
(58)
o~+~14~g~- 6OgOw(X,y).
(59)
Eqs. (58) and (59) may be used for approximate evaluation of the maximum relative velocity and absolute acceleration of the plate.
7. Response to nonstationary ground motions Although the stationary stochastic models of earthquake ground motion have been quite successful in the prediction of maximum response, they could hardly represent the detailed statistical properties of seismic acceleration as pointed out by Levy, Kozin and Moorman [49]. Actual earthquakes have finite durations and hence clearly have the appearance of nonstationary random processes. Nonstationary models of ground acceleration were suggested by Amin and Ang [ 13], Shinozuka and Sato [50], Jenning, Housner and Tsai [51] and Bogdanoff, Goldberg and Bernard [27], among others. In most of the nonstationary models the ground acceleration is assumed to be g!ven by
(~gq) =eq) n(t),
(60)
where e(t) is a deterministic envelope function and n(t) is a stationary random function of time. Several possible envelope functions are discussed in [1,6,32]. n(t) could be some filtered white noise process. However, for simplicity we assume that it is simply a white
4 14
G. Ahmadi / Earthquake response o f nonlinear plates
noise process. The autocorrelation of the ground acceleration thus becomes
Rf~g(tl, t2) = 27rSoe2(tl)6(tl - t2),
(61)
where So is the constant power spectrum of the white noise process n(t). The autocorrelation and variance of the lateral displacement of the plate are given by eqs. (46) and (47), respectively. The variance ofAo (t) and covariance of Ao(t) and A 1(t) are still given by eqs. (48) and (52) with to = 0 for an initially motionless plate. Employing eq. (61) into eq. (48) and integrating over the delta function, the result is
Approximate expressions (64) and (65) are for small damping and hence gZ was taken to be approximately equal to coo. The integrals in eqs. (64) and (65) could easily be evaluated for a large class of smooth envelope functions. The following cases are of interest. (a) Envelope function is a unit step function. Eqs. (64) and (65) now become 7ra2So
(Ag(t)) =-2~'w~ (1 - e-2S'c°°t)),
(66)
37r2S2(~ 4
(Ao(t)Al(t))-
8~.2co6 ( 1 - e -2~'~°°t)
t
(,zig(t)) = 27r82So f e2(r) h 2 (t - r) d r .
(62)
0
Similarly, substituting eq. (61) into eq. (52), it follows that t
X (1 - e -2~wOt
t
~'~2 S o H 2
O2w(X,y, t) = ~
0
rl)h('r- rl)drl].
X I1
0
(63) For a given envelope function e(t), the integrals in eqs. (62) and (63) could be evaluated at least by numerical techniques. However, for a smoothly varying envelope function and for small damping, simple approximate expressions could be obtained. Following [23], [28] and [32], and neglecting the highly fluctuating terms in comparison to smoothly varying contributions to the integrals in eqs. (62) and (63), it follows that 71"0~2S °
-~Ol,,)/~ - ~ o
t
f e-2~°°(t- r)e2(r) dr,
e(t)
cog
X [ f e-2~w°(t-7-')e2(r ') dr'] 0 t
=[1'
O
O,
t >tm,
2 (r2w _ 7 r o ~
T)
(69)
2 .(e2~.COo(tm_ t) _ e-2~-coot)
2~'6oo
0 T
× --f" e -2~x°°(7--~0 e2(rl) drl] .
(68)
where tm is the time duration of the earthquake strong motion. The mean square of the deflection field for t ~ tin, the mean square response becomes
t
× [ f dr e -2~'¢°o(t-r) sin 2COo(t
~'e - 2 ~ t ° 0 t
For large time t, the exponential functions of time in eq. (68) may be dropped and the stationary/me ansquare response of eq. (56) is recovered. (b) Rosenblueth finite duration white noise model. A very common non-stationary model of earthquake is the finite duration white noise model, i.e.
37r2S2t~ 4
(Ao(t)A 1 q))-- -
3~ea2S° e -2~wOt 2~.eOo3 (1 -
X sin 2Wot)7 sin 2 m~rx sin2 nny a b "3
(64)
0
0
(1 - e - 2 ~ o t )
7-
X [fdrh(t- r)f e2(rl)h(t-
( A 2 r÷~\
(67)
where terms up to the first order in e are considered in the last integral. Employing eqs. (66) and (67) in eq. (47), the corresponding variance of the lateral displacement of the plate becomes
(Ao(t)A 1 (t)) = - 12zr2sgwgc~4 [ f e2 (r')h 2 ( t - r') dr'] 0
~'e-2¢w°t sin 2COot),
(65)
X I1
3~'ect2S° {e~W°(tm-t)[cos 2coo(t - tin)
G. A hmadi / Earthquake responseof nonlinearplates + ~"sin 2coo(t - tm) - e - 2 g ' t ° ° t [cos 2Coo(t - tm) +
sin 2coot ] )~ sin 2 m~rx sin2 tory "j a b
(70)
It is of interest to note that for the finite duration white noise model the variance of deflection field for zero damping remains finite and is given by
O:w(X,y, t)
37rea2S° 2co~ "1
(2coot - sin coot) I
7rot2SoH2tm [-
11-
t ~< t m
.J
-3neot2So
× (2coot cos 2coo
(t - tm)
+ sin 2coo(t - tin) - sin 2coot)]
t > tm
I
...i
(71) This is a significant improvement over the stationary white noise model which yields unbound variance at ~'=0. The 30 level as before provides bounds on maximum deflection. From eqs. (68), (70) and (71) it follows that the variances of the deflection of the nonlinear plate is less than that of the corresponding linear plate. Several other envelope functions have been proposed which can be employed in eqs. (64) and (65) to calculate the corresponding variances of the lateral displacement of the plate. However, examples (a) and (b) show the procedure and some of the more commonly used envelopes.
8. Reliability The probability that the lateral deflection does not exceed a given value during the time interval (0, t) is given by t
L(t) = Lo exp ~-[" cffr) dr} ,
+oo
k f(Wo, w; t) d&,
(73)
_oo
sin:(mnx /a) sin2(mry/b ) If=o na2S°HEt I1 co2
is equal to the expected rate that w will exceed a given value w o. Eq. (72) is based on the assumption that the barrier crossing process is a Poisson process as was obtained by Cramer and Leadbetter [52] and discussed in [25,26,33]. The hazard function is given by
a(Wo, t) -- f
I
415
(72)
where f(w, ~; t) is the joint probability density of w and &. The response of a linear system to Gaussian excitation (such as white noise) is Gaussian and therefore knowledge of the means and variances is sufficient for its complete specification. However, the probability density of the present nonlinear system is unknown and is also non-Gaussian. But for small e we assume that it is not far from Gaussian and employing a normal density function the approximate expression for the hazard function becomes
a(Wo, t ) ~ o¢o e x p f - Y ! / . 7to w
tzo~)
(74)
(i) Stationary white noise excitation. Employing eqs. (56) and (58) in eq. (74) and using the results in eq. (72), the probability that the lateral deflection of the plate does not exceed
Wo = Kow
(75)
during the time interval (0, t) becomes
L(t) = exp
/
}
To e-KZ/2 '
(76)
where To is the natural period (coo = 2~rTo) and the probability of no instantaneous crossing Lo is taken to be equal to one. (ii) Finite duration white noise excitation. Employing an approximation of eq. (58) for the standard deviation of the velocity, the hazard function becomes
2 (
/
a(Wo, t) = ~oo exp - 20w~-2),
(77)
with o2 being given by eq. (68) or eq. (70) for finite damping and by eq. (71) for zero damping. The probability of no barrier crossing then is given by.
0
where Lo is the probability of no instantaneous crossing at t = 0 and c~(t) is the hazard function which
{ 2 i { w2 }aT} L (t) = exp - ~o o exp - 202 (x, y, 7") "
(78)
416
G. A hmadi / Earthquake response of nonlinear plates
The integration in eq. (78) must be carried out numerically. 9. Further remarks In this study the response of a nonlinear plate subjected to earthquake support motion is studied. The stationary as well as nonstationary models of strong motion accelerations are considered. A single mode Galerkin method together with a perturbation expansion of vibration amplitude are employed and the mean square of the lateral displacement field is derived. It is observed that the variances of the deflection of the nonlinear plate is less than that of the corresponding linear one in both cases of stationary and nonstationary excitation. The possible use of the results for estimation of maximum responses by the consideration of 3o levels is pointed out and the probability of no barrier crossing during an interval is discussed. In the analysis of the response of the plate to a nonstationary model of earthquake ground motion, the nonstationary random excitation of the Duffing equation is treated as part of the calculation which may find interest in the areas of mechanical vibration. Acknowledgments The author would like to thank the Department of Mechanical Engineering of the University of Newcastle upon Tyne for the hospitality extended to him. The financial support of the Atomic Energy Organization of Iran is gratefully acknowledged. References [ 1] R.W. Clough and J. Penzien, Dynamics of Structures (McGraw-Hill, 1975). [2] S. Okamoto, Introduction to Earthquake Engineering (University of Tokyo Press, 1973). [3] L. Meirovitch, Analytical Methods in Vibrations (MacmiUan, 1967). [4] W.C. Hurty and M.F. Rubinstein, Dynamics of Structures (Prentice-Hall, 1964). [5] N.M. Newmark and E. Rosenblueth, Fundamentals of Earthquake Engineering (Prentice-Hall, 1971). [6] G. Ahmadi, Shiraz University, Dep. Mech. Eng., Research Report (1977). [7] G.W. Housner, Bull. Seismol. Soc. Amer. 45 (1955) 197-218.
[8] G.N. Bycroft, J. Eng. Mech. Div., ASCE 86 EM2 (1960) 1-16. [9] E. Rosenblueth and J. Bustamante, 2I. Eng. Mech. Div., ASCE 88 EM3 (1962) 75-106. [10] E. Rosenblueth, J. Eng. Mech. Div., ASCE 40, EM5 (1964) 189-220. [11] K. Kanai, Univ. Tokyo Bull. Earthquake Res. Int. 35 (1957) 309-325. [12] H. Tajimi, Proc. 2nd World Conf. Earthquake Eng. Tokyo and Kyoto, vol. II (1960) pp. 781-798. [13] M. Amin and A.H.S. Ang, J. Eng. Mech. Div., ASCE 94, EM2 (1968) 559-583. [14] S.H. CrandaU and W.D. Mark, Random Vibration in Mechanical Systems (Academic Press, 1963). [15] H. Parkus, Random Processes in Mechanical Systems (UDINE, Springer-Verlag, 1969). [16] J. Robson, C.J. Dodds, D.B. Macvean and V.R. Paling, Random Vibrations (UDINE, Springer-Verlag, 1971). [171 Y.K. Lin, Probabilistic Theory of Structural Dynamics (McGraw-Hill, 1967). [18] W. Gersch, J. Acoustic. Soc. Amer. 48 (1970) 403-413. [19] J.L. Zeman and J.L. Bogdanoff, AIAA 7 (1969) 12251231. [20] G. Ahmadi, Earthquake response of multi-degree-offreedom systems (to appear). [21 ] R.L. Barnoski and J.R. Maurer, J. Appl. Mech. Trans. ASME 36 (1969) 221-227. [22] R.L. Barnoski and J.R. Maurer, J. Appl. Mech. Trans. ASME 40 (1973) 73-77. [23] L.L. BucciareUi,Jr. and C. Kuo, J. Appl. Mech. Trans. ASME 37 (1970) 612-616. [24] T.K. Caughey and H.J. Stumpf, J. Appl. Mech. Trans. ASME 28 (1961) 563-556. [25] R.B. Corotis, E.H. Vanmarcke and C.A. Cornell, J. Eng. Mech. Div., ASCE 98 EM2 (1972) 401-414. [26] R. Corotis and T.A. Marshall, J. Eng. Mech. Div., ASCE 103 (1977) 501-513. [27] J.L. Bogdanoff, J.E. Goldberg and M.C. Bernard, Bull. Seismol. Soc. Amer. 54 (1961) 292-310. [28] G. Ahmadi and M.A. Satter, AIAA 13 (1975) 10971100. [29] M.A. Satter and G. Ahmadi, J. Sound Vib. 51 (19oe) 577-581. [30] G. Ahmadi and M.A. Satter, J. Acoust. Soc. Amer. (in press). [31] R. Holman and G.C. Hart, J. Eng. Mech. Div., ASCE 100, EM2 (1974) 415-431. [32] G. Ahmadi, Nucl. Eng. Des. 50 (1978) 327-345. [33] T.K. Caughey, J. Appl. Mech. Trans. ASME 26 (1959) 341-344. [34] T.K. Caughey, J. Appl. Mech. Trans. ASME 27 (1960) 575 -578. [35] S.T. Ariaratnam, J. Mech. Eng. Sci. 2 (1960) 195-201. [36] S.H. CrandaU,J. Appl. Mech. Trans. ASME 29 (1962) 477-482. [37] R.H. Lyon, J. Acoust. Soc. Amer. 32 (1960) 716-719. [38] R.H. Lyon, J. Acoust. Soc. Amer. 33 (1961) 1395-1403.
G. A hmadi /Earthquake response of nonlinear plates [39] T.K. Caughey, J. Acoust. Soc. Amer. 35 (1963) 16831692. [40] T.K. Caughey, J. Acoust. Soc. Amer. 35 (1963) 17061711. [41] T.K. Caughey, in: C.H. Yih, ed., Advances in Applied Mechanics, vol. II (Academic Press, New York, 1971). [42] S.H. CrandaU, J. Acoust. Soc. Amer. 35 (1963) 17001705. [43] G. Ahmadi, I. Tadjbakhsh and M. Farshad, Acoustica 40 (1978) 316-322. [44] H.F. Bower, J. Appl. Mech. Trans. ASME 35 (1968) 4 7 52. [45] J.J. Stoker, Nonlinear Vibrations (Interscience, New York, 1950). [46] S.H. Crandall and W.D. Mark, Random Vibration of
417
Mechanical Systems (Academic Press, New York, 1963). [47] G. Ahmadi and J. Hashemi, Vehicle Syst. Dyn. 2 (1973) 225-233. [48] S.C. Liu and P.P. Jhaveri, J. Eng. Mech. Div., ASCE 95 (1969) 1145-1168. [49] R. Levy, F. Kozih and B.B. Moorman, J. Eng. Mech. Div., ASCE 97 (1971) 495-517. [50] M. Shinozuka and Y. Sato, J. Eng. Mech. Div., ASCE 93 (1967) 11-40. [51] P.C. Jennings, G.W. Housner and N.C. Tsai, Report Earthquake Eng. Res. Lab. Calif., Inst. Tech. Berkeley (Apr. 1968). [52] H. Cramer and M.R. Leadbetter, Stationary and Related Stochastic Processes (John Wiley & Sons, New York, 1967).
EARTHQUAKE RESPONSE OF NONLINEAR PLATES Goodarz AHMADI * Department of Mechanical Engineering, Shiraz University, Shiraz, Iran Received 28 March 1979
The finite amplitude vibration of a plate subjected to earthquake support motion is considered. Several bounds on the maximum responses of the plate are established which are based on the knowledge of the response spectra of the seismic motion. The stochastic models of the earthquake ground motion is then briefly discussed. The single mode of vibration of the plate is then considered and a perturbation series expansion is developed for the vibration amplitude. Mean square responses of the plate are then calculated for both stationary as well as nonstationary seismic excitations. It is observed that the variance of the deflection field of the nonlinear plate is less than that of the corresponding linear one. The reliability of design is also considered and the probability of no barrier crossing is briefly discussed.
1. Introduction
method the maximum responses of a single degree o f freedom structure to several past earthquakes are calculated and a smooth curve is fitted to the mean plus one standard deviation o f the maximum responses. Furthermore, some physical reasoning is used to correct the limiting range of high and low frequencies. The result is then called the "design response spectrum" and this will be used in the actual design. For more detail see [1,2,5]. Here, we are concerned with the probabilistic approach in which the ground acceleration is being modelled b y a stochastic process and the methods o f random vibration are employed. A number of stochastic models of earthquake ground motion were proposed in the past, as summarized b y Clough and Penzien [ 1], Newmark and Rosenblueth [5 ], and more recently b y Ahmadi [6]. Among the most well-known models are the white noise representations o f Housner [7] and Bycroft [8], the finite duration white noise models o f Rosenblueth et al. [5,9,10], the filtered white noise representation o f Kanai [11] and Tajimi [12] and the nonstationary models o f Amin and Ang [ 13] among others [1,2]. The random vibration o f elastic systems have attracted considerable attention in the past two decades b o t h in the field o f mechanical vibration as well as structural dynamics. The response o f lumped mass systems subjected to stationary excitations were
Analysis of the response o f structures subjected to earthquake ground acceleration is o f fundamental importance for their aseismic design. For such an analysis to be able to predict the realistic behaviour o f a structure during an actual seismic motion, certain information is necessary. First, the nature of the ground motion must be specified, and secondly the mechanical behaviour o f the structure should be modelled within a reasonable accuracy. If the time history o f the ground acceleration is given, there exist numerous analytical and numerical techniques [ 1 - 5 7 ] for the determination o f the time history of the response o f various linear and nonlinear structures. However, only the time histories o f a number o f past earthquakes are available and there is no method for predicting the time histories o f the future earthquakes for a particular site. Furthermore, the available accelerograms vary to a considerable extent in their time duration, frequency content, and maximum amplitude. To overcome this difficulty two methods are available. These are the use o f design response spectra and the probabilistic approach. In the former
* Presently academic visitor at the Department of Mechanical Engineering, University of Newcastle Upon Tyne, U.K. 407
408
G. A hmadi /Earthquake response of nonlinear plates
considered by several investigators. (See, for instance [1-5,14-17] .) Recently, the mean-square response of structures to random excitation was studied by Gersch [18], Zeman and Bogdanoff [19] and Ahmadi [20]. The response of single degree of freedom systems to nonstationary random excitation was investigated by Barnoski and Maurer [21,22], Bucciarelli and Kuo [23], Caughey and Stumpf [24], Corotis et al. [25, 26] and Bogdanoff et al. [27]. The response of beams and plates to nonstationary random loads were studied by Ahmadi and Satter [28-30]. The response of structures to nonstationary random excitation was considered by Holman and Hart [31 ]. An analysis of the earthquake response of linear continuous structures by the probabilistic approach was carried out by Ahmadi [32]. Although analysis of the linear response of elastic structures subjected to random forces is useful for understanding their behaviour during small earthquakes, in some structures, such as nuclear power plants, various components must be designed to withstand a maximum possible earthquake intensity, termed the "safe shut down earthquake", at the site of the plant. For such cases the linear response of structures is not sufficiently accurate and finite amplitude vibration of the various elements of structures must be considered. Therefore, a nonlinear stochastic system analysis must be used for an accurate design. The response of nonlinear systems to stationary random excitation was considered by Caughey [33, 34], Ariaratnam [35], Crandall [36] and Lyon [37, 38], among others, as summarized in survey articles by Caughey [39-41] and Crandall [42]. Stationary random vibration of a nonlinear plate was studied by Ahmadi, Tadjbakhsh and Farshad [43]. In the present study the response of a nonlinear plate subjected to earthquake support motion is considered. Several bounds on the maximum responses of the plate are obtained which are related to the response spectra of the seismic motion. A probabilistic approach for the analysis of the response of the plate is then undertaken. A single mode Galerkin expansion on space is considered and a perturbation method is employed for the vibration amplitude. Stationary as well as nonstationary models of the earthquake ground motion are considered and the
mean-square responses of the deflection field for both cases are obtained. It is observed that the variances of the lateral displacement of the nonlinear plate are less than those of the corresponding linearized plate for stationary as well as nonstationary loadings. The probability of no barrier crossing during a time interval is also studied.
2. Basic equations The equations governing the f'mite amplitude vibration of plates subjected to earthquake ground acceleration are given by
pH ~2w
yaw + . . . . D 3t 2
a2F
h.
H [~2w ~2F ~ Wg(t) + - D [~x ~3y2
a2w a2F]
23x~y 3x~y + ~y2 aX2 J,
(1)
and
//__it/2wX2 a2w
7
(2)
In these equations x and y are cartesian coordinates, w is the lateral deflection of the plate relative to the ground and Wg is the lateral ground acceleration; p and H are the mass density and thickness of the plate, respectively. E is Young's modulus of elasticity, v is Poisson's ratio, and D is the bending stiffness defined by
D =EH3/12(1 - / ) 2 ) .
(3)
F is Airy's stress function. The in-plane forces Nx, Ny and Nxy are given by ~2F N x = Oy2 ;
~2 F N y = Ox 2 ;
~2F
Nxy -
Ox
ay
(4)
We should mention here that in the derivation of eqs. (1) and (2) both extensional and rotary inertia effects have been neglected. Furthermore, only the effect of lateral ground acceleration is being considered while in general both lateral and axial ground accelerations are present. However, for simplicity the more restricted form of the vibration equations (1) and (2) are being considered for the present analysis. For a simply supported rectangular plate of sides
G. A hmadi /Earthquake response of nonlinear plates a and b the boundary conditions on w are a2w a2w w = 0 and ~-7~- + v - = 0 , o y 2 a t x = 0 , a,
409
For a simply supported plate the solution is of the fo rm
wOc, y, t) = HA(t) sin 02W
w=0
mrrx
~2 W
and a ~ - + v ~ - - ~ - = 0 ,
a t y = 0 , b.
(5)
a
nrry
sin.
(10)
b '
The boundary conditions for the Airy stress function for the case of a plate with stiess free edges becomes
where A (t) is a dimensionless function of time, and m and n are integer numbers. The function, eq. (10), obviously satisfies the boundary conditions eq. (5). Inserting eq. (10) into eq. (2), it follows that
32F Oy2 =0
a t x = 0 , a,
V4F = ~ - ( ~ - - - ) A
aty = 0, b.
Furthermore, the following solution for the stress function is assumed:
O2F 3x3y-0'
and
E nmlr2H 2 2
(t) [cos 2m----~+ cos ~b-~-] (.1 a
02F 3x 2 - 0
O2F 3x Oy - 0,
and
(6)
For a plate whose edges are immovably constrained, instead of eq. (6) we have and
o=0
and
axay
-0,
atx =0,a,
-0,
a t y = 0 , b,
02F
3x 0y
(7)
where u and v, the midplane displacements in the x and y directions, are given by 0
(E \3y 2
o =-flY ,[11321;-I~x2
v ~X2] -- 2 \ 3 X ]
~2F~3y 2 ] - 2\-~y]l(3w '~2 } dy.
w=0
and
aw
~x-x=0,
a t x = 0 , a,
aw 7 - =0,
a t y = 0 , b.
oy
(1 - c°s 2rnrrx a ) (1 - c°s 2 - ~ )
F* =
dx,
(g)
For a clamped plate the deflection boundary conditions become and
- cos - - b ~ ] . u 2 )
and integrating over the surface of the plate, it is found that
0
w=0
27)(,
The assumed solution, eq. (12), clearly satisfies the stress free edge boundary conditions, eq. (6), for the Airy stress function. Employing eq. (12) in eq. (11), multiplying by
a2F
u=0
cos
1)
3. Approximate analysis Exact analytical solutions of eqs. (1) and (2) are not available in the general cases. In this section an approximate single mode assumption, which is a generalization of [44], is employed in order to derive the decoupled equations for the dynamics of the time dependent amplitudes.
(13)
Now employing eqs. (10) and (12) in eq. (1), multiplying the resulting equation by the corresponding mode shape of lateral displacement, and integrating over the surface of the plate, the result is +
(9)
E(nm/ab )2 8 [(m/a) 2 + (n/b)2] 2"
+
pH
A
4¢og
=-mnHTr2 [ 1 - ( - 1 ) m ] [ 1 - ( - 1 ) n]
p
\ab!
+
-b
"
(14)
The procedure which was followed here is the wellknown Galerkin method for Finding approximate solutions to nonlinear equations. It should be mentioned here that for other types of boundary conditions such as clamped and, or immovably constrained boundaries, a similar method
41 o
G. Ahmadi /Earthquake response of nonlinear plates
could be applied with a proper mode shape. For instance, for a clamped plate the form of the mode of deflection could be taken to be
w =HA(t)sin 2 rnTrx sin2 a
HTrO
b
(15) '
which satisfies the boundary conditions (9). Employing eq. (15) in eqs. (1) and (2) yields an equation for time development of A (t) similar to (14), but with different coefficients. Introducing new parameters
remains small in comparison to the linear part. Employing eq. (19) in eq. (20) and equating the coefficient of like powers of e equal to zero, yields a set of linear equations for Ai, i = 0, 1,2, .... These are A'o + 2~o-Zio +cogAo = -~(~g(t), A'l + 2~'eOoA1
+
co2A1 = -cooA2 03,
A'2 + 2~'6Oo,zi2 + topAz = -3eo~A 1A~.
(20)
The general solutions of the first two equations of eq. (20) are given by t
=7#
+
'
4 - mnHTr2 [1 - ( - 1 ) m] [1 - ( - 1 ) n ] ,
Ao(t) = - ~ f
h(t - r) (~g(r) dr,
(21)
to t
(16)
A , (t) = -W2o f h(t - r ) A 3 ( r ) dr, e=,
EH 3
(nmTr/ab) 4
D
[(m/a) 2 + (n/b)2] 3'
where h(t) is the impulsive response of the corresponding linear system and is given by
eq. (14) becomes
A" + togA + ecogA 3 = -~Wg(t).
1
(17)
For the first mode approximation m = n = 1 and the parameters defined in eq. (16) simplify correspondingly. If a damping term is also present in the plate vibration equation (1), then eq. (17) generalizes to the following equation: A' + 2~'co0d + w~A + ew2A 3 = -~Wg(t),
(22)
to
(18)
where ~"is the damping coefficient.
h(t) = ~oo exp(-c°°~'t} sin f2ot,
(23)
with g22 = co2(1 _ ~-2).
(24)
For an initially motionless plate the value to = 0, and for a stationary response the value to = _oo must be considered in eqs. (21) and (22). A two term perturbation approximation for the lateral vibration of the plate thus becomes
w(x,y, t ) = H [ A o ( t ) + earn (t)] sin mwx a sin nay --b---, (25) 4. Perturbation s o l u t i o n
The classical method for solving nonlinear differential equations is the perturbation method. (See, for instance, Stoker [45] .) Applications of the perturbation method to random vibration has been carried out by, among others, Crandall [42], Crandall and Mark [46], Caughey [41] and more recently by Ahmadi and co-workers [43,47]. The basic assumption in the perturbation method is that the solution to eq. (18) has an expansion in powers of e, i.e.
A(t) = Ao(t) + eA l(t) + J A 2 (t) + ....
(19)
with Ao (t) and A l (t) being given by eqs. (21) and (22), respectively.
5. Bounds o n m o t i o n When the plate is subjected to arbitrary seismic support motion with known response spectra, the maximum response of the lateral deflection field may be estimated from eq. (25). For instance, max iw(x, y, t)i ~< H(max tAo(t)t + e max IA1 (t)l) t
Assumption (19) is expected to be valid for small e, i.e. when the nonlinear part of the restoring force
t
t
X sin m n x sin nny a
b
.
(26)
G. Ahmadi / Earthquake response of nonlinear plates velocity response spectrum given by
The maximum absolute value o f A o (t) is given by max I Ao(t)t = c~S~(cOo, ~') t
(27)
where the displacement response spectrum is defined by t Sd(O.)0, ~') = max I J h(t - r) t 0
drt.
(28)
The maximum o f A 1 (t) may be estimated from eq. (22), i.e. max IA 1 (t)l • t
6o20ct3S3(~0, ~')h m ,
(29)
where tm hm = f o
Ih(r)l dr
411
(30)
with t m being the time duration of the earthquake ground motion. Employing eqs. (27) and (29) in eq. (26), a bound on the maximum relative deflection of the plate is obtained, i.e. max lie(x, y, t)t ~< altSa(~o, ~)
Sv(cao, ~') = caoSd(~O, ~').
(36)
The inequality (32) thus becomes max I~b(x, y, t)l ~< ¢d-/Sv(cao, ~') t,x,y
X [1 + ea2hmS2v(COo, ~')].
(37)
Evaluating the second time derivative of eq. (25) and noting that h(0) = 0;
/~(0) = 1,
(38)
the bound on the maximum absolute acceleration becomes max lff,(x, y, t) + ¢d-/lCg(t)l t,x,y (39) ~< crHSa(6Oo, ~')[1 + ect2S~(¢Oo, ~X1 + w2hm)], where the acceleration response spectrum is defined by t Sa(cOo, ~') = mtax f J~(t - r) I~g(r) dr, 0
(40)
and is approximated by
t,x,y
X [1 + eco2a2hmS2(wo, ~')1.
(31)
Sa(wo, ~') = cooSv(wo, ~) ~-- cO2Sd(WO, ~).
(41)
Similarly, the maximum absolute relative velocity is bounded by
Furthermore, in the derivation of inequality (41) the approximation
max Iw(x, y, t)[ ~< cd-/Sv(cOo, ~')
hm = f o
t
t,x,y
+ d--Ico2t~3t~mS3(cdo, ~'),
(32)
where tm ]~m = f o
I]~(r)i dr,
(33)
and the velocity response spectrum is defined by t Sv(cao, ~') = max I f / ~ ( t - r) t 0
drl.
(42)
is also employed. It should be noted here that the left hand side of eq. (39) is not exactly the total absolute acceleration, and if for instance the first mode of vibration is being considered, i.e. m = n = 1 from eq. (16), it follows that +~/l~'g =~,
16 .. + n-~- leg.
(43)
(34)
For a small damping coefficient ]~m ~-- ¢°ohm
Ih(r)l dr--~ ¢.o2hm
(35)
and Sv becomes approximately equal to the pseudo-
The difference arises from the single mode assumption of eq. (10) instead of a full Galerkin series. Therefore, whenever the response spectra of seismic motion are known either from direct calculations for actual earthquakes or from an assumed design response spactra [1,5], eqs. (31), (37) and (39) give
412
G. A hmadi / Earthquake response of nonlinear plates
upper bounds on the relative displacement, relative velocity, and absolute acceleration of the plate. However, it should be mentioned here that the bound provided by inequalities (31), (37) and (39) are quite conservative. In particular, the estimate of the maximums of the first order perturbation term A1 (t) and its time derivatives are highly conservative.
X sin 2 m n x sin2 tory a
Therefore, variance ofA o (t) and covariance ofA o (t) and A 1(t) are needed for the calculation of the mean square of the deflection field. From eq. (21) it follows that t
t
f d72
{ A 2 ( t ) ) = - 2 f drl 6. Response to stationary earthquake models
to
According to Housner [7], Bycroft [8] and Rosenblueth et al. [5,9,10] earthquake ground acceleration could be represented by a simple stationary white noise, with a constant power spectrum and a delta function autocorrelation, i.e., R ~g(Z) = 27rSo6(7),
(44)
where So is a constant power spectrum that was estimated [1,8] to be equal to 0.375 ft2/sec 4- cps for the N - S component of the E1 Centro (1940) earthquake. For Housner's standard response spectra So is found to be equal to 0.0063 ft2/sec 3. There have also been several stationary filtered white noise models of earthquakes such as those suggested by Kanai [11] and Tajimi [12] and more recently by Liu and Jhavri [48]. The autocorrelation of the deflection of the plate is defined by R w ( x , y , tl, t2) = (w(x,y, t l ) W ( x , y ,
t2))
(45)
where angular brackets, ( ) , denote the expected value (ensemble average). For the stationary response the autocorrelation function depends only on the time difference, t I - t 2 . Employing the general solution (25) in eq. (45), up to the first order in e, we find
X h(t
to
71)h(t
-
72)RWg(71,72).
-
t
(AS(t)) : 2rro~2So /
Making a change of variables in eq. (49), it follows that oe
(A 8) = 27ra2So f
h2(7) dT
which is independent of time as is expected for stationary processes. Employing the expression for the impulsive response as given by eq. (23), eq. (50) becomes (Ag) = n a 2 S o / 2 f w 3 .
(51)
Employing eqs. (21) and (22), the covariance of Ao and A 1 become t
f d,f
= H 2 [(A~ (t)) + 2eGto (t)A 1 (t))]
t
tO
t
t
d,, f d,: f d,3 to
tO
t
(46)
and the variance of the lateral displacement is given by OZw(X, y, t)= Rw(x, y, t, t)
(50)
0
e(Ao(t2) A 1 (tl))]
X sin 2 mTrx sin2 nny a b '
(49)
hZ(t - 71) dTl.
to
+ {?("40 ( / 1 ) A 1 ( t 2 ) ) +
(48)
For the stationary white noise model of the earthquake ground acceleration, its autocorrelation is given by a delta flmction ofTl - r2 as defined in eq. (44). Eq. (48) then reduces to
C4o(t)A,
Rw(X, y, tl, t2) = H 2 [ ( A o ( t l ) A o ( t z ) )
(47)
b
Xf
dr 4 h ( t - v ) h ( t - 7 1 ) h ( r - r a ) h ( r -
73)
to X
h ( 7 --
T4)(WE(71)WE(T2)WE(T3)WE(T4))"(52)
When the ground acceleration is being modelled by a white noise process which is Gaussian, the fourth order correlation in eq. (52) is related to the auto-
G.Ahmadi/Earthquakeresponseof nonlinearplates correlation of
(¢g(t) and is given by
( ~lg( T1) ~]g(T 2 ) ~]g(7"3 ) ~/g(T 4 ) ) = 47r2S2 [6('/1 - 7 2 ) 6 ( 7 3
- 7"4)
(53)
+ 5('/1 - T 3 ) ~ ( T 2 -- "/'4) + 5 ( 7 1 -- 7"4)6('/'2 -- 7"3)]'
The expression for the covariance, eq. (52), becomes oo
(A o (t)A1 (t)) = -12rr2sgcoga 4
f dr h(r) 0
× ? dO h(O)h(O+r) ? h2(rl) dr1. 0
(54)
0
Employing eq. (23) in eq. (54) and evaluating the integral the result is 3~r 2 a4 S 2
8~.2o~~ ,
(55)
which is time independent. When eqs. (52) and (55) are substituted into eq. (47), the expression for the mean-square response of the lateral deflection of the plate is found, i.e.
O2w(x, y) -
7r°~2H2 S °
(1
3rrea2S°
)
× sin2 rnTrxsin2 tory (56) a b The variance becomes time independent as would be expected for stationary processes. From eq. (56) it is observed that the mean-square response decreases with an increase of e. That is to say, the variance of the deflection field of a nonlinear plate is smaller than that of the corresponding linear plate. A similar conclusion in connection with a hardening spring oscillator, such as the Duffing oscillator, is pointed out in [37,38,42]. In a previous investigation [43] a result similar to eq. (56) was obtained also by the method of equivalent linearization and by the exact method of the Fokker-Planck equation. The 30 level is known to be a conservative estimate of the maximum response, see for instance [1,5,32, 49]. Therefore, max
[w(x,y, t)l <~ 3Ow(X,y),
(57)
t
with
Owgiven by eq. (56). From eqs. (56) and (57) it
413
is concluded that the maximum deflection of a nonlinear plate is less than that of the corresponding linear plate when both are subjected to the same earthquake support motion. This is in contrast to the relatively crude bound given by eq. (31). From the physical side of the problem, a nonlinear plate behaves like a nonlinear hardening spring and thus the restoring force increases rapidly with the increase in vibration amplitude. Hence, it is expected that the maximum deflection of the nonlinear plate be smaller than the linear one in agreement with the prediction of the statistical analysis of eq. (57). A similar analysis for the evaluation of the variances of the relative velocity and absolute acceleration of a plate could be carried out. However, when damping is small the approximate expressions for the standard deviations become
o~,(x, y) ~- ~oOw(X, y),
(58)
o~+~14~g~- 6OgOw(X,y).
(59)
Eqs. (58) and (59) may be used for approximate evaluation of the maximum relative velocity and absolute acceleration of the plate.
7. Response to nonstationary ground motions Although the stationary stochastic models of earthquake ground motion have been quite successful in the prediction of maximum response, they could hardly represent the detailed statistical properties of seismic acceleration as pointed out by Levy, Kozin and Moorman [49]. Actual earthquakes have finite durations and hence clearly have the appearance of nonstationary random processes. Nonstationary models of ground acceleration were suggested by Amin and Ang [ 13], Shinozuka and Sato [50], Jenning, Housner and Tsai [51] and Bogdanoff, Goldberg and Bernard [27], among others. In most of the nonstationary models the ground acceleration is assumed to be g!ven by
(~gq) =eq) n(t),
(60)
where e(t) is a deterministic envelope function and n(t) is a stationary random function of time. Several possible envelope functions are discussed in [1,6,32]. n(t) could be some filtered white noise process. However, for simplicity we assume that it is simply a white
4 14
G. Ahmadi / Earthquake response o f nonlinear plates
noise process. The autocorrelation of the ground acceleration thus becomes
Rf~g(tl, t2) = 27rSoe2(tl)6(tl - t2),
(61)
where So is the constant power spectrum of the white noise process n(t). The autocorrelation and variance of the lateral displacement of the plate are given by eqs. (46) and (47), respectively. The variance ofAo (t) and covariance of Ao(t) and A 1(t) are still given by eqs. (48) and (52) with to = 0 for an initially motionless plate. Employing eq. (61) into eq. (48) and integrating over the delta function, the result is
Approximate expressions (64) and (65) are for small damping and hence gZ was taken to be approximately equal to coo. The integrals in eqs. (64) and (65) could easily be evaluated for a large class of smooth envelope functions. The following cases are of interest. (a) Envelope function is a unit step function. Eqs. (64) and (65) now become 7ra2So
(Ag(t)) =-2~'w~ (1 - e-2S'c°°t)),
(66)
37r2S2(~ 4
(Ao(t)Al(t))-
8~.2co6 ( 1 - e -2~'~°°t)
t
(,zig(t)) = 27r82So f e2(r) h 2 (t - r) d r .
(62)
0
Similarly, substituting eq. (61) into eq. (52), it follows that t
X (1 - e -2~wOt
t
~'~2 S o H 2
O2w(X,y, t) = ~
0
rl)h('r- rl)drl].
X I1
0
(63) For a given envelope function e(t), the integrals in eqs. (62) and (63) could be evaluated at least by numerical techniques. However, for a smoothly varying envelope function and for small damping, simple approximate expressions could be obtained. Following [23], [28] and [32], and neglecting the highly fluctuating terms in comparison to smoothly varying contributions to the integrals in eqs. (62) and (63), it follows that 71"0~2S °
-~Ol,,)/~ - ~ o
t
f e-2~°°(t- r)e2(r) dr,
e(t)
cog
X [ f e-2~w°(t-7-')e2(r ') dr'] 0 t
=[1'
O
O,
t >tm,
2 (r2w _ 7 r o ~
T)
(69)
2 .(e2~.COo(tm_ t) _ e-2~-coot)
2~'6oo
0 T
× --f" e -2~x°°(7--~0 e2(rl) drl] .
(68)
where tm is the time duration of the earthquake strong motion. The mean square of the deflection field for t ~
t
× [ f dr e -2~'¢°o(t-r) sin 2COo(t
~'e - 2 ~ t ° 0 t
For large time t, the exponential functions of time in eq. (68) may be dropped and the stationary/me ansquare response of eq. (56) is recovered. (b) Rosenblueth finite duration white noise model. A very common non-stationary model of earthquake is the finite duration white noise model, i.e.
37r2S2t~ 4
(Ao(t)A 1 q))-- -
3~ea2S° e -2~wOt 2~.eOo3 (1 -
X sin 2Wot)7 sin 2 m~rx sin2 nny a b "3
(64)
0
0
(1 - e - 2 ~ o t )
7-
X [fdrh(t- r)f e2(rl)h(t-
( A 2 r÷~\
(67)
where terms up to the first order in e are considered in the last integral. Employing eqs. (66) and (67) in eq. (47), the corresponding variance of the lateral displacement of the plate becomes
(Ao(t)A 1 (t)) = - 12zr2sgwgc~4 [ f e2 (r')h 2 ( t - r') dr'] 0
~'e-2¢w°t sin 2COot),
(65)
X I1
3~'ect2S° {e~W°(tm-t)[cos 2coo(t - tin)
G. A hmadi / Earthquake responseof nonlinearplates + ~"sin 2coo(t - tm) - e - 2 g ' t ° ° t [cos 2Coo(t - tm) +
sin 2coot ] )~ sin 2 m~rx sin2 tory "j a b
(70)
It is of interest to note that for the finite duration white noise model the variance of deflection field for zero damping remains finite and is given by
O:w(X,y, t)
37rea2S° 2co~ "1
(2coot - sin coot) I
7rot2SoH2tm [-
11-
t ~< t m
.J
-3neot2So
× (2coot cos 2coo
(t - tm)
+ sin 2coo(t - tin) - sin 2coot)]
t > tm
I
...i
(71) This is a significant improvement over the stationary white noise model which yields unbound variance at ~'=0. The 30 level as before provides bounds on maximum deflection. From eqs. (68), (70) and (71) it follows that the variances of the deflection of the nonlinear plate is less than that of the corresponding linear plate. Several other envelope functions have been proposed which can be employed in eqs. (64) and (65) to calculate the corresponding variances of the lateral displacement of the plate. However, examples (a) and (b) show the procedure and some of the more commonly used envelopes.
8. Reliability The probability that the lateral deflection does not exceed a given value during the time interval (0, t) is given by t
L(t) = Lo exp ~-[" cffr) dr} ,
+oo
k f(Wo, w; t) d&,
(73)
_oo
sin:(mnx /a) sin2(mry/b ) If=o na2S°HEt I1 co2
is equal to the expected rate that w will exceed a given value w o. Eq. (72) is based on the assumption that the barrier crossing process is a Poisson process as was obtained by Cramer and Leadbetter [52] and discussed in [25,26,33]. The hazard function is given by
a(Wo, t) -- f
I
415
(72)
where f(w, ~; t) is the joint probability density of w and &. The response of a linear system to Gaussian excitation (such as white noise) is Gaussian and therefore knowledge of the means and variances is sufficient for its complete specification. However, the probability density of the present nonlinear system is unknown and is also non-Gaussian. But for small e we assume that it is not far from Gaussian and employing a normal density function the approximate expression for the hazard function becomes
a(Wo, t ) ~ o¢o e x p f - Y ! / . 7to w
tzo~)
(74)
(i) Stationary white noise excitation. Employing eqs. (56) and (58) in eq. (74) and using the results in eq. (72), the probability that the lateral deflection of the plate does not exceed
Wo = Kow
(75)
during the time interval (0, t) becomes
L(t) = exp
/
}
To e-KZ/2 '
(76)
where To is the natural period (coo = 2~rTo) and the probability of no instantaneous crossing Lo is taken to be equal to one. (ii) Finite duration white noise excitation. Employing an approximation of eq. (58) for the standard deviation of the velocity, the hazard function becomes
2 (
/
a(Wo, t) = ~oo exp - 20w~-2),
(77)
with o2 being given by eq. (68) or eq. (70) for finite damping and by eq. (71) for zero damping. The probability of no barrier crossing then is given by.
0
where Lo is the probability of no instantaneous crossing at t = 0 and c~(t) is the hazard function which
{ 2 i { w2 }aT} L (t) = exp - ~o o exp - 202 (x, y, 7") "
(78)
416
G. A hmadi / Earthquake response of nonlinear plates
The integration in eq. (78) must be carried out numerically. 9. Further remarks In this study the response of a nonlinear plate subjected to earthquake support motion is studied. The stationary as well as nonstationary models of strong motion accelerations are considered. A single mode Galerkin method together with a perturbation expansion of vibration amplitude are employed and the mean square of the lateral displacement field is derived. It is observed that the variances of the deflection of the nonlinear plate is less than that of the corresponding linear one in both cases of stationary and nonstationary excitation. The possible use of the results for estimation of maximum responses by the consideration of 3o levels is pointed out and the probability of no barrier crossing during an interval is discussed. In the analysis of the response of the plate to a nonstationary model of earthquake ground motion, the nonstationary random excitation of the Duffing equation is treated as part of the calculation which may find interest in the areas of mechanical vibration. Acknowledgments The author would like to thank the Department of Mechanical Engineering of the University of Newcastle upon Tyne for the hospitality extended to him. The financial support of the Atomic Energy Organization of Iran is gratefully acknowledged. References [ 1] R.W. Clough and J. Penzien, Dynamics of Structures (McGraw-Hill, 1975). [2] S. Okamoto, Introduction to Earthquake Engineering (University of Tokyo Press, 1973). [3] L. Meirovitch, Analytical Methods in Vibrations (MacmiUan, 1967). [4] W.C. Hurty and M.F. Rubinstein, Dynamics of Structures (Prentice-Hall, 1964). [5] N.M. Newmark and E. Rosenblueth, Fundamentals of Earthquake Engineering (Prentice-Hall, 1971). [6] G. Ahmadi, Shiraz University, Dep. Mech. Eng., Research Report (1977). [7] G.W. Housner, Bull. Seismol. Soc. Amer. 45 (1955) 197-218.
[8] G.N. Bycroft, J. Eng. Mech. Div., ASCE 86 EM2 (1960) 1-16. [9] E. Rosenblueth and J. Bustamante, 2I. Eng. Mech. Div., ASCE 88 EM3 (1962) 75-106. [10] E. Rosenblueth, J. Eng. Mech. Div., ASCE 40, EM5 (1964) 189-220. [11] K. Kanai, Univ. Tokyo Bull. Earthquake Res. Int. 35 (1957) 309-325. [12] H. Tajimi, Proc. 2nd World Conf. Earthquake Eng. Tokyo and Kyoto, vol. II (1960) pp. 781-798. [13] M. Amin and A.H.S. Ang, J. Eng. Mech. Div., ASCE 94, EM2 (1968) 559-583. [14] S.H. CrandaU and W.D. Mark, Random Vibration in Mechanical Systems (Academic Press, 1963). [15] H. Parkus, Random Processes in Mechanical Systems (UDINE, Springer-Verlag, 1969). [16] J. Robson, C.J. Dodds, D.B. Macvean and V.R. Paling, Random Vibrations (UDINE, Springer-Verlag, 1971). [171 Y.K. Lin, Probabilistic Theory of Structural Dynamics (McGraw-Hill, 1967). [18] W. Gersch, J. Acoustic. Soc. Amer. 48 (1970) 403-413. [19] J.L. Zeman and J.L. Bogdanoff, AIAA 7 (1969) 12251231. [20] G. Ahmadi, Earthquake response of multi-degree-offreedom systems (to appear). [21 ] R.L. Barnoski and J.R. Maurer, J. Appl. Mech. Trans. ASME 36 (1969) 221-227. [22] R.L. Barnoski and J.R. Maurer, J. Appl. Mech. Trans. ASME 40 (1973) 73-77. [23] L.L. BucciareUi,Jr. and C. Kuo, J. Appl. Mech. Trans. ASME 37 (1970) 612-616. [24] T.K. Caughey and H.J. Stumpf, J. Appl. Mech. Trans. ASME 28 (1961) 563-556. [25] R.B. Corotis, E.H. Vanmarcke and C.A. Cornell, J. Eng. Mech. Div., ASCE 98 EM2 (1972) 401-414. [26] R. Corotis and T.A. Marshall, J. Eng. Mech. Div., ASCE 103 (1977) 501-513. [27] J.L. Bogdanoff, J.E. Goldberg and M.C. Bernard, Bull. Seismol. Soc. Amer. 54 (1961) 292-310. [28] G. Ahmadi and M.A. Satter, AIAA 13 (1975) 10971100. [29] M.A. Satter and G. Ahmadi, J. Sound Vib. 51 (19oe) 577-581. [30] G. Ahmadi and M.A. Satter, J. Acoust. Soc. Amer. (in press). [31] R. Holman and G.C. Hart, J. Eng. Mech. Div., ASCE 100, EM2 (1974) 415-431. [32] G. Ahmadi, Nucl. Eng. Des. 50 (1978) 327-345. [33] T.K. Caughey, J. Appl. Mech. Trans. ASME 26 (1959) 341-344. [34] T.K. Caughey, J. Appl. Mech. Trans. ASME 27 (1960) 575 -578. [35] S.T. Ariaratnam, J. Mech. Eng. Sci. 2 (1960) 195-201. [36] S.H. CrandaU,J. Appl. Mech. Trans. ASME 29 (1962) 477-482. [37] R.H. Lyon, J. Acoust. Soc. Amer. 32 (1960) 716-719. [38] R.H. Lyon, J. Acoust. Soc. Amer. 33 (1961) 1395-1403.
G. A hmadi /Earthquake response of nonlinear plates [39] T.K. Caughey, J. Acoust. Soc. Amer. 35 (1963) 16831692. [40] T.K. Caughey, J. Acoust. Soc. Amer. 35 (1963) 17061711. [41] T.K. Caughey, in: C.H. Yih, ed., Advances in Applied Mechanics, vol. II (Academic Press, New York, 1971). [42] S.H. CrandaU, J. Acoust. Soc. Amer. 35 (1963) 17001705. [43] G. Ahmadi, I. Tadjbakhsh and M. Farshad, Acoustica 40 (1978) 316-322. [44] H.F. Bower, J. Appl. Mech. Trans. ASME 35 (1968) 4 7 52. [45] J.J. Stoker, Nonlinear Vibrations (Interscience, New York, 1950). [46] S.H. Crandall and W.D. Mark, Random Vibration of
417
Mechanical Systems (Academic Press, New York, 1963). [47] G. Ahmadi and J. Hashemi, Vehicle Syst. Dyn. 2 (1973) 225-233. [48] S.C. Liu and P.P. Jhaveri, J. Eng. Mech. Div., ASCE 95 (1969) 1145-1168. [49] R. Levy, F. Kozih and B.B. Moorman, J. Eng. Mech. Div., ASCE 97 (1971) 495-517. [50] M. Shinozuka and Y. Sato, J. Eng. Mech. Div., ASCE 93 (1967) 11-40. [51] P.C. Jennings, G.W. Housner and N.C. Tsai, Report Earthquake Eng. Res. Lab. Calif., Inst. Tech. Berkeley (Apr. 1968). [52] H. Cramer and M.R. Leadbetter, Stationary and Related Stochastic Processes (John Wiley & Sons, New York, 1967).