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Block random rocking and seismic vulnerability estimation
Block random rocking and seismic vulnerability estimation Luca F a c c h i n i Department of Civil Engineering, University of Florence, Florence, Italy
Vittorio Gusella Institute of Energetic& University of Perugia, Perugia, Italy
Paolo Spineili Department of Civil Engineering, University of Florence, Florence, Italy (Received September 1992; revised version accepted September 1993)
An analysis is undertaken of the rocking motion of a rigid block resting on a rigid foundation when subjected to a random ground acceleration by using an equivalent linear system. The method predicts the cumulative distribution function of the extreme values of the tilt angle and the results are compared with those obtained by other authors using a step-by-step numerical integration of the equation of motion. With the introduction of the analysis of the seismic risk, by means of the probabilistic law of the yearly maxima of the peak ground acceleration, a method of estimating the vulnerability is proposed which takes into account the maximum permissible rotation for the block and the strength of the block material. In the latter case, an approximate relation between block rotation and maximum stress at the block base corner is introduced. This method is used to estimate the seismic vulnerability of some of the ancient monuments of the Imperial Forum in Rome. Keywords: block rocking, random vibrations, statistical lineariz-
ation, seismic vulnerability
Since the end of the last century, several researchers have been involved with the dynamics of a rigid block under seismic excitations 1. This problem was first systematically investigated by Housner2. He found a relation between the time duration of a single horizontal pulse (either rectangular or sinusoidal) and its minimal intensity in order for the block to overturn. With greater intensities, the block will topple without any rocking motion, while lower ones will make the block rock without overturning. On the other hand, if the excitation does not consist of a single pulse, but of a series of pulses, then the block may topple with lower peak intensities than those predicted by Housner2. Aslam et al. 3 performed several experimental tests with a shaking table. They developed a computer program which evaluated the rocking motion of the block, either prestressed or unconnected to the floor, where the system was prevented from sliding. The coefficient of restitution, defined as the ratio between the angular velocities of the block immediately after and immediately before the impact of the block with its foundation, was assumed to be constant. The bottom surface of the block was plane or, better, slightly concave and the impact such that there was no 0141-0296/94/060412-13 © 1994 Butterworth-Heinemann Ltd
412
EngngStruct. 1994, Volume 16, Number 6
bouncing, so that the block, when impact occurred, began rotating around the opposite comer of the base. The obtained results showed the sensitivity of the response to the aspect ratio HIB, the block size R (Figure 1) and the coefficient of restitution: the stability was observed to be greater for lower coefficients of restitution, smaller aspect
~[y x
, F
Figure 7
Consideredsystem
Block random rocking and seismic vulnerability estimation: L. Facchini et al. ratio and larger block size, but the computed results also showed some exceptions to this general trend. Yim et al. 1 have performed a very extensive and deep parametric analysis of the block dynamics under a seismic excitation. A numerical procedure was developed to solve the nonlinear equations governing the motion of a rigid block on a rigid foundation under either horizontal or vertical excitations. It was noticed that the overturning of the block under ground motion of a particular intensity does not imply, in a deterministic sense, that the block will necessarily topple under the action of a more intense excitation. Housner2 developed an approximate analysis of the dynamic behaviour of a slender block under an ideal white noise pseudovelocity spectrum excitation by means of the piecewise linear equations of motion. In addition, he determined the critical angle 0or for a block to have a 50% toppling probability. This relation shows that the intensity of the ground motion required for the block to overturn is proportional to the block size R and the critical angle 0or. Also once the characteristics of the block are given, its toppling probability increases for higher ground motion intensities. These conclusions, which also hold for the single pulse excitation, should be considered from a probabilistic point of view. Yim et al.' took into account several seismic excitations, and considered evidence that systematic trends can be observed when the problem is studied from a probabilistic point of view, with the ground motion modelled as a random process. The probability of a block response exceeding a given level, as well as the probability of its overturning, increases for higher ground motion intensities, higher slenderness ratios and lower sizes. Spanos and Koh4, using a piecewise linear equation, determined the safe and unsafe regions in the excitation frequency-amplitude plane. Their study of the possible classes of the stationary response highlighted the possibility of the block vibrating at a lower frequency than the excitation (the so-called subharmonic oscillations). Moreover, these responses can have time average zero (the so-called symmetric modes) or nonzero (unsymmetric modes). It was generally assumed that the block and its foundation were infinitely rigid, but, during recent years, the model of a rigid block resting upon a flexible foundation which allows uplift has also been analysed. Generally speaking, uplift leads to a softer vibrating system which behaves nonlinearly, even though the response may be composed of a sequence of linear responses. Psycharis and Jennings5 took into consideration a continuous elastic foundation with viscous damping (i.e. a monolateral Winkler model) and a simpler model made of two couples of spring and damper, and established an equivalence criterion between the two models. They also determined the response to a single pulse and a relation between the maximum tilt angle versus impulse amplitude. The apparent rocking period and the apparent ratio of critical damping were determined by means of a parabolic approximation of a generic half cycle of the response. Also an equivalent linear system, whose parameters depended on the normalized impulse, was defined. As the vertical oscillations of the centre of mass of the system turn out to be generally small with respect to the rocking motion, Blasi and Spinelli6 developed a simplified model for the Winkler foundation, taking into account only the degree-of-freedom 'rotation of block'. The analytical
model was then applied to the study of the behaviour of multiblock columns. In order to predict the rocking response amplitude when the foundation is excited horizontally due to a harmonic acceleration, Koh et al. 7 proposed an approximate analytical method which combined smile condensation and the method of averaging. In addition, they proposed a method of predicting the maximum tilt angle under an earthquake excitation incorporating the linear response spectra. Spanos and Koh s also studied the response statistics of the block supported by a flexible foundation when the excitation is a stationary white noise modulated by a deterministic function of time. An analytical procedure, which made use of both static condensation and equivalent linearization to approximate the evolutionary process of the joint probability density function for tilt and angular velocity, was proposed. The present study was prompted by the great interest in the behaviour of monumental structures, which is related to the problem of conserving them from seismic activity.
Notation ap apy Ce e
g k
ko Pr(Y) t,
x 2,,a~ B
G(.) H
n~ n.e loJo, M
P~o,1 P~y) So Srr T str
E(.) 6
Ai
mean peak acceleration of soil motion yearly maximum of mean peak acceleration damping coefficient of equivalent linear system coefficient of restitution of angular velocity after impact acceleration due to gravity: g = 9.81 m s-2 modulus of Winkler foundation lower limitation on k to use rigid block model elastic coefficient of equivalent linear system probability density function (PDF) for random variable Y time when impact between block and its foundation occurs ground motion acceleration peak of ground motion acceleration base of block constitutive relation between maximum compressive stress at base of block and nondimensional rotation height of block amount of energy dissipated by block amount of energy dissipated by equivalent linear system moments of inertia about block base comers mass of block collapse probability cumulative distribution function (cdf) for random variable Y intensity of ideal white noise spectral density of random stationary process
time lag when envelope X attains its maximum value = 1 structural life error function dimensionless tilt angle: th = 0/0or upper bound for safety region S dimensionless tilt angle of equivalent linear system requested rotation during system life ith order spectral moment
Engng Struct. 1994, Volume 16, Number 6 413
Block random rocking and seismic vulnerability estimation: L. Facchini et al. characteristic frequency of rigid block: IX = (MgR/Io)'/2 expected frequency of forcing process expected frequency of equivalent linear system critical damping factor compressive stress at base block requested resistance structural resistance material resistance difference between structural and requested resistances standard deviation tilt angle critical tilt angle: Ocr -~ tan-' (H/B) dimensionless time variable: r = / , t dimensionless time when impact between block and its foundation occurs dimensionless frequency: to = [1//~ dimensional frequency expected value operator
tx V
or
orr or. 7/ 0
0. T
to
12
~[.]
Analytical model and equivalent linear system The mechanical system taken into consideration is shown in Figure 1. Friction is assumed to be sufficient to prevent the block sliding. The only Lagrangian coordinate is the tilt angle 0 about the base comers 0 and 0". If the horizontal ground motion acceleration is denoted by ~(t), then the equations of motion about each of the base comers are
since d2/dF = Ix2(d2/d~-2), the piecewise linear equations of motion can be rewritten in the following nondimensional form d2S - S + sign(S) =f(~'/ix) d~-2
when coupled to the initial conditions for each half cycle this gives lim ~(z) = e lim ~(T); + ~r~'r.
Ocr--O ) +
~¢,) = 0
(5)
-r--.'r.
The equivalent linearization method was applied to the rigid block under harmonic excitation by Spanos and Koh4's to obtain an approximate steady-state response aproximation with the same excitation frequency. In the equivalent lineafization method, in its more general formulation9, a linear system is characterized by two constant parameters k, and c,, in such a way as to minimize the expected square value of a proper error function. If the linear system t~, + c,~e + (k,-1)S, = F(~'); F(T) =f(zll*)
(6)
is considered, subtracting equation (6) from equation (4) gives -- ~)e -- Ce+e -- ~) +
Io0 - M R X C o S (
(4)
sign(S)
-
(k,-1)S, = 0
(7)
M g R sin( Oc~O) = 0; 0 --> O
Io,0 - M R X cos(0c~+0) - M g R s i n ( 0 ~ + 0 ) = 0;
0 <-- 0
Assuming that S , - S, an error function can be defined such that
(1) E(~') = sign(S) - keS where M is the mass of the block, Io and Io, are the moments of inertia about the comers of the base; g is the acceleration due to gravity, R = 0.5 (H2+B2)lz2 is the block size; O~r = tan-'(HIB) is the critical angle of overturning and is also a measure of the block slenderness. During the impact between the foundation and the block a certain quantity of energy is dissipated and hence the angular velocity is sharply reduced; the assumption is introduced that the dissipation can be modelled by an impulse mechanism described by a coefficient of restitution e of the angular velocity. Hence the motion of the block is described, during each half cycle of its oscillation, by each part of equation (1), which are linked together by the initial conditions for a generic half cycle
(8)
The forcing process F(¢) is assumed to be a stationary Gaussian filtered white noise, which is a common assumption in earthquake engineering. Later it will be explained why this assumption is valid. The proposed procedure takes into account the energy dissipation during impact in order for the linear damping coefficient to be evaluated, since this must be computed by means of an energy balance. The basic procedure is to evaluate the expected amount of energy to be dissipated during impacts between the block and its foundation, and then tune c, accordingly. The amount of energy dissipated during the ith impact is
H.,=-~--~72; l -e 2
lim 0(t)= e lim 0(t) t~t +
where t, is the instant when the impact occurs and e is the coefficient of restitution of the angular velocity. When the block is slender (HIB >-- 2) equation (1) can be linearized, giving Io0
- M R X + M g R ( O~r-O) = 0
Io,0 - g
~r.i) =0
(9)
R X - M g R(O~,+O) = 0
When the linear system is considered, its angular velocity will be a continuous function of time, i.e. there is no discontinuity in the angular velocity.when S, = 0. The angular velocity of the linear system S, can therefore sensibly approximate the average of the two limits of the actual angular velocity
(2) ~ e ( T * ) = Si- + S ~
Defining
(MgRI '/2 ~(t) S=0/0c'; I x = \ Io ,/ ; * = l z t ; f ( t ) - O c . g
414
~7 = lim - ~(r);
t~t:
(3)
Engng Struct. 1994, V o l u me 16, N u m b e r 6
- ~1 -+ e S;•
(10)
so that the expected amount of energy dissipation during is
impact
Block random rocking and seismic vulnerability estimation: L. Facchini et al. 1-e
F[H.,] -- 2 ~
IF[~,(r*)]
(11)
where E[.] denotesflhe expected value operator. Hence the expected amount of energy dissipated during a unit time period will be 1-e
IF[H.] = 2 v, F[H.,] = 4 ~
v, IF[4~'~[~b,= 01
(12)
where v. is the expected frequency of the linear system and the vertical bar [ denotes conditional probability; in other words, F[H.] is assumed to be proportional to the expected frequency of the response process. The expected amount of energy dissipated by the linear system during the same period will be
F[H,~] = F
c, ~ dr =
the fact that a random forcing process is now dealt with, the assumption can be considered suitable. It can be found that
1 rb~e ; ~:[~l~be = 0] = E [ ~ ] = r/~, 27r ~/%
v, = - -
(18)
so that the linear system parameters can be evaluated as follows k,=
(2)1t2 1 -"1/% -;
1-e*l*~ c , = 21 + e -TI% -
(19)
The variances ~l2, and rf% can .~ computed from
~ =f~S%%(to)dto
t~] dr = c, a~,
°
(13) so that c, becomes
where
1- e IF[4~'.lth~= O] c. = 4 ~ ve .~.
(14)
Once c, has been evaluated, k~ may be computed. This can be done by minimizing the value of the square of the error function E2 defined by relation (8)
0 Ke0 E[~] = 0 --7
(15)
which yields9
ke =
(16)
2
In equations (14) and (16) the expected value operators, and hence the variances of the two processes */% and *1%, depend on the two parameters k. and c. so the equations are not linear. From the Gaussian assumption and the noncorrelation of the two processes, the joint probability density of ~be and ~ of the equivalent linear system is found to be 1 e x p ( - t/r~. p..~. (th.; &.) - 2rr r/% ,/.. 2 -tf%
(20)
~/~,e= f ~ to2 S%% (to) dto
SFF (to) S % % - (ar~ _ to~,)z + 4~ar2~to 2 ; to~, = k , -
1;
c. ~e -- 2toe
(21)
where SFF(to) is the power spectral density of the dimensionless excitation F(r) and to is the dimensionless frequency to = f~/gt. Some numerical algorithms have been develope dm to compute ke and ce, since their evaluation can be extremely difficult. The computed, values will be affected by the variances of ~b, and ~b, rather than the shapes of p%(the) and p,t,,(q~e): the proposed method can give good approximations to these parameters, so that satisfactory results can be obtained even in cases, such as the one discussed, in which the physical system exhibits a strong nonlinearity. The actual distribution for t~(r) and t~(r) will not be Gaussian, but the equivalent linearization method cannot give any information about its shape 9. Moreover, it is implicitly assumed that the solving equations have a unique solution over the whole domain of k~ and c,. This seems generally true for dynamic systems with only one stable equilibrium position excited by a wide-band random process.
~----~-/
2 '}~e] (17)
where null averages were assumed for both the processes. When deterministic excitations (e.g. harmonic accelerations) are considered, the dynamic system can exhibit steady-state responses which have nonzero time averages. These responses were defined by Spanos and Koh4 as 'unsymmetric modes' in order to distinguish them from those whose time average was zero, these being called the 'symmetric modes'. They observed that the predominant mode is symmetric and nearly harmonic, with the same frequency of excitation. From these observations, and from
Comparison of theoretical layout with numerical results The reliability of the proposed method, is evaluated by using the numerical results of Yim et aL 1 They examined the behaviour of a block with size parameter R = l0 ft and aspect ratio H / B = 5 subjected to 20 numerically evaluated acceleration histories satisfying the main characteristics of the following earthquakes: E1 Centro, California, 30 December 1934 and 18 May 1940, Olympia, Washington, 13 April 1949 and Taft, California, 21 July 1952. The method employed by Yim et al.~ consisted of prefiltering a generated ideal white noise according to the Kanal-Tajimi
E n g n g Struct. 1994, V o l u m e
16, N u m b e r
6
415
Block random rocking and seismic vulnerability estimation: L. Facchini et al. spectrum model and multiplying the response of the prefilter by a deterministic function of time X(t). The power spectral density of the prefilter response was then
where q=
1 + 4~'2(1)/1~o)2 SX~I~) = So (l -- (~-~/1"~0)2)2 + 4~2(~/~'~0)2
(22)
with 12o = 27rfo; fo = 2.5 Hz.; ~o = 60% and where So was the intensity of an ideal white noise. The time dependent envelope )((t) was t_<4s
'16
4s
15s
(23)
t ->- 15s
The coefficient of restitution e changed from 0.90 to 0.95. A constant multiplier was used to obtain the desired mean for the peak acceleration of the soil: E[X,,=] = 0.4 g. In order to evaluate the parameter So, the following distribution 1~ is assumed for the peak acceleration
P~,,,~ ( 0 = vT ~2 e x p ( - 2 ~ ) (24)
exp{-vT exp(- 2~)} It can be shown that 0.5772 / E[X~x] = r/X (2 ln(vT)) 1/2 + (2 l n - ~ ) l / 2 J
(25)
where v = l]o/2"tr = 2.5 Hz and T is the seismic event duration. Assuming that E[g,,,~] = 0.4 g and T ~ 11 s, then r/x - 1.4020 m s-2. The variance ~xz can be computed from integration of equation (22), so that So = 1.9589 x 10-L It can be shown that the power spectral density for the dimensionless excitation F('r) can be expressed as follows So/x 1 + 4~2(02/0)o) 2 Sv~oJ) = (g ~ 0~) (1 - (~o/0.~o)2)2 + 4~o2(O9/0h~)2
(26)
where OJo= 1)o/~. In order to compare the theoretical and numerical results, by means of equations (19)-(21) the two parameters k,, Ce and the power spectral density S%% were estimated for the various coefficients of restitution for a linear system equivalent to a block with size parameter R = 10 ft and slenderness ratio HIB = 10 (also considered by Yim et aL~). Assuming the model proposed by Vanmarcke ~2, the cumulative distribution function (cdf) of the extremes of tilt angle is
F~emax (05emax) [ [ffr~112q {#emax~ (27) = exp - v ,
exP(2 .q2 ,,o~) _ 1
1-hohe//
;Ai=2
(28)
Figure 2 shows the comparisons between the function (27) (continuous line) and numerical results of Yim et al. j (dots) for different values of the coefficient of restitution e. The dashed line denotes the confidence interval for the Kolmogorov-Smirnov test of hypothesis for a confidence level of 0.1%. Since the coefficients of restitution were similar, the numerical results were considered together in order to obtain 120 samples with a high statistical reliability. The theoretical analysis was repeated with a damping coefficient which was the average of those formerly obtained. The proposed method provides a reliable estimation of the maximum tilt angle as shown in Figure 3. Larger differences between theoretical and numerical results are observed as ~b approaches its critical value qbcr= 1. It will later be seen how in the vulnerability estimation these differences can be surpassed assuming, on the safe side, an admissible value ~-< 1, which also corresponds to the imposition of a limiting (deterministic) value for the resistance of the block material.
Relation between rotation of the block and compressive stresses at its base The compressive stresses at the corner of a base of a rigid block resting on a rigid foundation and subject to oscillations, are infinite. If collapse is only assumed to take place when 4~cr>- 1, there is an implicit assumption that the block material is infinitely strong. This approach is unreliable when applied to the estimation of the seismic vulnerability of monumental structures. The model of the rigid block resting on a Winkler foundation can be taken into account in order to establish an approximate relation between the block rotation and the internal stresses so that a random material resistance can be considered in the estimation of seismic vulnerability. In most of the observations of the block motion, vertical oscillations are very small and can be neglected. This hypothesis leads to the Blasi-Spinelli 6 model, where the foundation reaction balances the weight of the block, since the variation of the vertical momentum can be neglected. The block rotation thus becomes the only Lagrangian coordinate of the system. The period of the free oscillations of the block can be evaluated as the spring constant k of the foundation changes. It follows that there is a value ~ such that, if k - £, the period can be satisfactorily approximated by Housner's relation for a rigid block on a rigid foundation (it can be seen6 that/~ ~ 1000 N cm -3 in most cases of real columns). It can therefore be deduced that the dynamic response of a rigid block on a Winkler foundation whose constant k is greater than ~ can be compared to that of a rigid block on a rigid foundation, while its internal stress configuration can be evaluated from the Winkler model. The relation between the rotation, 0, and the maximum compressive stress at the base of the block, cr,~, generally depends upon the shape of the base itself (see the Appendix); the two functions (which are the inverse of each other) can thus be evaluated
l o',~x = G-1(05) = G-~(OlOcr);
416
~oiS%%(o9) dw
Engng Struct. 1994, Volume 16, Number 6
05 = G(orm,~)
(29)
Block random rocking and seismic vulnerabilityestimation: L. Facchini et al. e-(~90 /"
1 .....................
~4 •d
e=0.91 ,/
/ ,
0.5-
./ . . . . . . . .
1(~
~
.
.,.--"
~4
jpse~
"d
10o
0.5
l0 t
../
"
1"04
l0 o
÷.u
l0 t
÷.,= effi 0.92
Ii
,
,
~
,'/'i" /"" ,~ ,/
'~ 0.5 ~,,
•
....
/
1"04 "
/
~
/"
I
effi 0 . 9 3
.... .. ~ ........................ ' ' '~ . . . . . . . / ~ , .:. . . . . . .
~'"
~
/,
~j
........ i "
,,'" ~ /:/
: .~.~ / O. " ,,I(" ~ "/ , .~/
"d
~
I0 e
¢/
/¢ ",~ /,,"
/
lo-°',"
10~
. . . . . . .
÷-,--
. . . . . .
÷.,.,
Figure 2 Comparisons between results obtained by statistical linearization ( equation (27)) and numerical results of Yim et aL ~ (*), for different values of coefficient of restitution e ( - - - confidence interval for Kolmogorov-Smirnov test of hypothesis for a
confidence level of 0.1%)
ce = 0.0225
0.9
//
................................................... - ....
/
0.8
.
/
0.7
J "1=
//
.i
''i. "
/
//'"
//
0.6 :,
0.5
~
-
•
" ....
.
//
.
i
.
. .
. .
. .
.
.
-
i_
0.4
0.3I~~i//
/......... . . . . . . . . . . .
0.20.i i~f//'I/'/ n~
- ,,"
i
-'i
i
i
1"04
t
t
i
i
t
i
10o
t
i
t
i
l0 t
Figure 3 Cumulative comparison between numerical results (*) of Yim e t aL 1 (neglecting restitution coefficient variation) and statistical linearization ( equation (27)) with a mean damping coefficient for equivalent linear system ( - - - confidence interval for Kolmogorov-Smirnov test of hypothesis for a confidence level of 0.1%)
In the previous method the stress is only affected by the block weight. More precisely, the vertical reaction of the comer of the base of a rigid block resting on a rigid foundation can be obtained once the vertical acceleration of the centre of mass of the block (d2ys/df) is given. Thus
Ry= M(g+d2ys/dI2)= M g ( l + d2yg/dr2)
(30)
where yg is the vertical coordinate of the centre of mass and
Ys = R cos(0c:-0);
dys/dt = R sin(0c:0)d0/dt
d2yJdt 2 = R sin( O~:-O)d201dt2-R cos(Oc:-O)(dO/dt) 2 (31)
Lineadzing the expression for d2y,/dt2 gives
d2ys/df ~- R[d20/dt2 ( Ocr-O)-(d~/dt) 2] =/j2R~.cr[~(l__t~ )
_
(~2] =
_l.t2R~cr [(1_~)2 + fi~2] (32)
where the equation of motion for free oscillations ~ - ~ + 1 = 0 of the system is employed. If the expressions of potential and kinetic energies are introduced denoted by W and T, respectively, then
W=d~-th2/2;
T=~/2;
T=W,~-W
d2ysldt 2 --- -/z2R~'c,[1 + 2(W,~,-2W)]
Engng Struct. 1994, Volume 16, N u m b e r 6
(33) (34)
417
Block random rocking and seismic vulnerability estimation: L. Facchini et al. Now, as W~,~ ~ W ~ 0 and 1/2 -> Wm,~ -> 0, it can be shown that 0--< 1 +2(Wm,~-2W)-----2
less than an arbitrarily fixed value $ during the system life 7",r. Equation (37) defines the CDF for the extreme rotation during the whole structural life T~,~. Thus
(35)
For the structures which will be examined later, the leading term ~2R~r lies in the range 7 x 10-2-3 x 10-~ m s-2, which can be neglected when compared with gravity (g -~ 9.81
P%(~b,)=
exp{-L,rf~[1-F.~(~s,apy)]pAey (apy)daey
}
= Prob[~b < ~b,]
(39)
m s-2).
The corresponding PDF can be obtained from this last equation as follows
Estimation of seismic vulnerability
Admissible maximum rotation for the block
p%(~b~) = ~
In the previous section the starting point was the shape of the power spectral density of the forcing process, coupled to the mean of its peak acceleration. If an upper bound is imposed for the maximum rotation of the block, for. instance the toppling condition <-- $-- 1, a safety region can be determined and the curve which expresses the collapse probability to the given event can be plotted. In order for the seismic vulnerability to be estimated, the collapse probability to a given event must be combined with the probability that this event takes place in the structural life. The upper bound for the rotation will be denoted by and S =- [-~;~] C [-!;1] E R will stand for the safety region of the dynamic system. Let ~(~[ap) be the safety probability and ~3~(t~lap) the collapse probability for the block subjected to a seismic motion whose mean peak acceleration is ap. Then
~3,($1a~) - 1 - ~ (¢la~) -- Prob[l¢t < $1a~l
S(T~,r)=exp{-T~,r f ~ ~3~((blapy)PA,y(apy) dapy}
~3c(~lap) = 1 - F..=(~b,ap)
f2 °
(38)
where ap is the mean peak acceleration of the earthquake. The critical value t~ can be introduced via a collapse criterion. If overturning is considered~- then (b = 1. Otherwise if a constitutive equation between the rotation and the stress at the base of the block is developed, a deterministic critical value for stresses can be fixed and then a critical value for rotation can be obtained.
~( dps,apy) =
418
Engng Struct. 1994, Volume 16, Number 6
~(~b,;a,,) clap,
(40)
[1
- F.m~x ( (]ls,apy)]PApy ( apy)
(41)
Hence using the second of equation (29), the PDF for the maximum stress of the block, during the whole structural life T,,~, is as follows
pz~s(CL)= p% [G(cL)] ~
OG
¢,.ox='~,
(42)
This PDF must eventually be combined with the PDF for the structural resistance, which comprises two distinct components. The former comes from the material strength for which the following standard Gaussian distribution can be conveniently employed (43)
Pr-v((r.) = N(/~u, ~i~u)
The latter comes from the overturning condition, that is, if the rotation equals its critical value (~b = ~bcr= 1 determined by the toppling condition). Then the compressive stress at the base of the block will attain its maximum value. Because the constitutive equation is a monotonically increasing function, greater compressive stresses will correspond to greater rotations, which are impossible to obtain, as the block will topple. In order to obtain a comprehensive description of an equivalent strength of the structure which takes into account the effective material strength and the toppling phenomenum, a Dirac delta function will be combined with equation (43) to obtain the structural strength PDF
pER(O.r)= P~R (1) (err) + Kr~(ffr 0~4,= &)
(44)
where
(1) tN(~u , "qzu) if trr < crl~ $) PI~R(t~r)= [0
Introduction of a random strength for the block material Once the probability density function (PDF) of the yearly extremes for ground peak acceleration apy is given, equation (37) may be viewed as the probability of the rotation being
~(¢~s,apy)dapy}
where
(37)
where pApr .is the probability distribution function of the yearly maximum mean peak acceleration, apy, of seismic motion for the considered site. Equation (27) gives the probability that ~b,~o~-< ~b, where th,,~ is the maximum rotation and th is a fixed value. When ~b= t~; equation (27) gives the safety probability and the collapse probability is
P%(~b~)
=- Ltrexp{- L,r f2
(36)
It will be assumed that the time series of the seismic events can be modelled as a Poisson process and that the structural system does not change with time. Then the safety probability of the system during its life T~,~ becomes ~2
0
(45)
otherwise
and Kr is a normalizing constant. Consider the random variable
Block random rocking and seismic vulnerability estimation: L. Facchini et al. ~
(46)
= ~;. - ~ s
difference between equivalent strength and maximum stress of the block during the whole structural life T,,~, whose PDF can be evaluated from the convolution of equations (42) and (45), which can be simplified in the expression
p:~z(~bz) =
f : 6=4,)-~rspr.R(or~ + tr,) p:~,(cL) dcr,
(47)
where
M* = "Zimi; R* - ~mi M ~Ri ,
I~ = ~i R*2 ~ lGi
The characteristics of the equivalent block depend not only on the sizes Ri of the columns but also on their aspect ratios HJBi through Ri and Ion. With regard to the seismic risk, a Fisher-Typpet II type CDF for the yearly peak acceleration was assumed to fit the available seismic data, so that
The collapse probability can be evaluated from
[ \apy/ j P : a , = f ° p x z ( ~ z ) dcr~
(48)
Application of proposed method to some ancient slender monuments The proposed method has been employed in the very first estimation of the seismic vulnerability of some of the most ancient structures in Rome, which have a dynamic behaviour that can be described, as a first approximation, by means of the rigid block model. Foca's column (Figure 4(a)), Vespasiano's temple (Figure (4b)), Dioscuri's temple (Figure 4(c)) and Saturn's temple (Figure 4(d)) have been analysed. These last three structures have a series of columns linked by an overhanging architrave, so that an equivalent block was defined (Figure 5). The following assumptions were made • The components of the structures and their foundations are infinitely stiff • The friction between the foundation and the columns and between the columns and the architrave is sufficient to prevent sliding • The geometry of the architraves are such that they are unlikely to uplift from the top of the columns • There are no relative displacements among the superimposed elements ('rocchi' of marble or granite) which constitute the columns Hence the motion of the system can be described by only one Lagrangian coordinate, 4~The mechanical characteristics (e.g. mass, momentum of inertia etc.) of the fictitious block can be evaluated by equating its total energy to that of the actual s t r u c t u r e 1° (Figure 5). Letting the subscript tot indicate the characteristics of the fictitious block, the superscript * those of an intermediate system, the subscript tr those of the architrave and the subscript i those of the single colunms, then
Mrot = M* + mtr IG tot =
tl-2[/~ +
(4mtr + M*)R .2] - Mot RE,
Rtot = 2mtr + M* R* + 1 m t r htr M,o, 2 M ,o, a = 1
1 mtr htr
2 M,o, R,o,
(49)
(50)
(51)
where u~ = 6.6552 x 10-3 and u2 = 1.9313.
The analysis of Saturn's temple and other monuments Saturn's temple is made of eight granite columns which bear an architrave; the mean diameter of the columns is 64 cm and they are 13.00 m high. The architrave is approximated to two prisms whose centres of mass are 1.89 m above the top of the columns. Dynamic tests were carried out in order to study the dynamic behaviour of this monument. The results confirmed that the temple could be idealized as a rigid block model. Also it was possible to compute a reliable approximation of the Winkler constant k of 1500 N cm -3 which proved high enough to consider the system as a rigid block on a rigid foundation. Several equivalent linear systems were defined to calculate F . , ~ (~b;ap) in which the mean ground acceleration ap ranged from 0.1 g to 1.3 g (it should be noted that the characteristics of the linear system change when the ground motion intensity increases) (see Figure 6). In order to define the statistical properties of the material resistance Et,, the experimental results of Blasi et al. 14 were utilized so that a Gaussian PDF was assumed with/x~u = 55.73 N mm -2 and r/~U = 7.62 N mm -2. The structural life was assumed to be T,,r = 50 years, giving the collapse probability from (equation (48)) as
If,, ~- 0.00706 In order for the influence of the strength to be considered, several average values were taken into account, with the same coefficient of variation. The results are reported in Table 1 and plotted in Figure 7. If a best fit cubic spline curve is plotted for these data, it asymptotically approaches the collapse probability p f a1) il ~
0.00638
(52)
which is obtained when a critical (deterministic) threshold t~ = 1 is considered for the structural collapse (i.e. the toppling condition). This shows that, as the average value for the material strength increases, only a small fraction of P:ait is due to material failure, and the greater part is due to overturning. Owing to this fact, when other ancient structures were considered, the analysis did not take into account the material strength, and only the toppling condition ~ = 1 was imposed to evaluate the collapse probability. In Figure 8 the overturning probability is plotted versus the mean peak
Engng Struct. 1994, Volume 16, Number 6 419
Block random rocking and seismic vulnerability estimation: L. Facchini et al.
I
Ii i
a)
~"
b)
"..'~ II I
i i
I
i
~f;v i I
I
'
i
Figure 4 Structural schemes of analysed monuments: (a) Foca's column; (b) Vespasiano's temple; (c) Dioscuri's temple; and (d) Saturn's temple (measures in centimetres)
420
Engng Struct. 1994, Volume 16, Number 6
Block random rocking and seismic vulnerability estimation: L. Facchini et
el.
A
~ 24.00 II
~ 20.00
numerically evaluated data
%
best fit cubic spline
Figure5 Equivalence criterion for evaluation of mechanical characteristics of fictitious block
16.00 >, 4-,
o 12.00
acceleration ap; this result is obtained from the CDF of the maximum tilt angle versus ap (see for example Figure 6) for ~ = 1. The characteristics of the equivalent block, the modulus of the elastic foundation and the collapse probability are shown in Table 2.
_Q 0
O_
o 03 00
-6 o
Conclusions
8.00
O
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
O
asymptotic value (toppling condition only) 4.00
0.00
.......
.......
................. :so
.......
.......
material resistance mean, N / m m s
With regard to the block random rocking, them is a satisfactory agreement between the results obtained from the equivalent linear system and those obtained from numerical integrations of the equation of motion, so that the proposed procedure can be considered accurate enough for a first estimation of the stochastic characteristics of the response. When the cumulative distribution function for the extremes of the response is considered, a greater difference between the two methods can be observed when $ approaches So,In order to estimate the seismic vulnerability, the influence of these differences can be safely catered for by means of a reduction of the upper bound for the block rotation. An extended model allowing for a random material strength is proposed by means of approximate relations between the block rotation and the compressive, stresses at the comer of the base. If this approach is followed, the reduction of the upper bound for the rotation corresponds to the introduction of a deterministic value for the strength of the material. Moreover, taking into account an equivalent
Figure7 Seismic vulnerability versus average value for material resistance (Saturn's temple)
Table 1 Change of 50 year collapse probability of Saturn's temple according to different values of mean /~ and standard deviation ~/of material resistance P'Xu (Nmm -2)
~-u (Nmm -2)
Pro. (T.= = 50 years)
14.0 16.8 19.6 22.4 25.2 28.0 42.0 55.7
1.9 2.28 2.66 3.04 3.42 3.8 5.7 7.6
20.50279 14.18266 10.44586 8.38832 7,49777 7,19037 7,05766 7.05718
x x x x x x x x
10~ 10-3 10-~ 10-~ 104 10-s 10-~ 10-3
SATURN TEMPLE 1
i
==
0.9
. . . .
0.8
•
~ :i_
i ¸ iii-
0.7
.
0.6
. . . .
0.5
. . . . . . . . .
0.4
i.._ i_
:
0.3
i
0.2 0.1 100-2
10-1
100
101
==
Figure 6 Cumulative distribution functions for extreme tilt angle when block is subjected to different forcing processes whose mean peak acceleration ranges from 0.1 g to 1.3 g (duration of stationary portion of seismic event ~ 10 a)
Engng Struct. 1994, V o l u m e
16, N u m b e r 6
421
Block random rocking and seismic vulnerability estimation: L. Facchini et al. ROMAN FORUM MONUMENTS
0.6 . . . . . 0.5
..'"o fzo •
0.4
.:-f"
•
0.3 ~
0.2 /,:" :f/./'/,,~
O
-c • • d .................
temialcof Vespasian temiale of Sattirn
0.1
°o
0.2
024
016
028
i
114
122
Figure 8 Overturning probability versus mean peak acceleration for examined structures considering only toppling criterion ~ = 1 (duration of stationary portion of seismic event - 10 s) Table 2 Characteristics of equivalent block, frequency of free oscillation, modulus of elastic foundation of block and collapse probability for a structural life of 50 years
Foca's column Temple of Castors Temple of Vespasian Temple of Saturn
Mass (kg)
Io (kg m 2)
R (m)
Ocr (rad)
/~ (Hz)
k
Pf..
(N cm -a)
(Ts, = 50 years)
43540 204300 197820 694950
2410000 2860000 8820000 31170000
6.45 7.72 8.35 11.04
0.098 0.119 0.120 0.107
1.07 1.24 1.36 1.55
1000 1900 1000 1500
0.01030 0.00889 0.00670 0.00638
structural strength (due to the random material strength and overturning) the discrepancies between the two methods are reduced. The procedure has been used to estimate the seismic Vulnerability for historically valuable structures, which can be idealized as a single rigid block. The results for the examined monuments indicated that material strength has negligible influence upon the probability of structural collapse: this is because of the high strength of marble and the relatively low stresses at the base in the toppling condition. For this reason a rigid block on a rigid foundation appears to be an adequate model for initial estimation of seismic vulnerability. The results show that the structures have a high vulnerability, markedly higher than the admissible levels for new buildings. Concern over these results are sufficient to suggest that a research programme involving comparisons between several methods of analysis and a repair intervention programme should also be planned.
References 1 Yim, C. S., Chopra, A. K. and Penzien, J. 'Rocking response of rigid blocks to earthquakes', Earthquake Engng Struct. Dyn. 1980, 8, 565-587 2 Housner, G. W. 'The behavior of inverted pendulum structures during earthquakes', Bull. Seism. Soc. Am. 1963, 53, 403-417 3 Aslam, M., Godden, W. G. and Scalise, D. T. 'Earthquake rocking response of rigid bodies', J.sStruct. Div. ASCE 1980, 106, 377-392
422
Engng Struct. 1994, Volume 16, Number 6
4
Spanos, P. D. and Koh, A. S. 'Rocking of rigid blocks due to harmonic shaking', J. Eng. Mech. ASCE 1984, 110, 1627-1642 5 Psycharis, I. N. and Jennings, P. C. 'Rocking of slender rigid bodies allowed to uplift', Earthquake Eng. Struct. Dyn. 1983, 11, 57-76 6 Blasi, C. and Spinelli, P. 'Un metodo di calcolo dinamico per sistemi formati da blocchi rigidi sovrapposti', lng. Sismica 1986, l, 12-21 7 Koh, A.S., Spanos, P. D. and Roesset, J. M. 'Harmonic rocking of rigid blocks on flexible foundation', J. Eng. Mech. ASCE 1986, 112, 1165-1180
8 Spanos, P. D. and Koh, A. S. 'Analysis of block random rocking', Soil Dyn. Earth. Engng 1986, 5, 178-183 9 Roberts, J. B. and Spanos, P. D. Random vibration and statistical linearization, John Wiley, Chichester, 1990 10 Facchini, L. 'Strutture monolitiche sottoposte ad azioni aleatorie: applicazioni al comportamento sismico di opere monumentali', Graduate Thesis, Faculty of Engineering, University of Florence, Italy, 1990 11 Cramer, H. 'Mathematical analysis of statistics' Princeton University Press, 1946 12 Vanmarcke, E. H. 'On the distribution of the first-passage time for normal stationary random processes', J. AppL Mech. ASME 1975, 42 13 Gusella, V. 'Structural failure and stochastic discrete process of random events: An application to seismic vulnerability analysis of a historic building', Struct Safety 1991, 11, 13-28 14 Biasi, C., Sorace, S. and Coil, M. 'Decay of the mechanical properties of marble due to long term small stress configuration', Die Pep. ST/9, Department of Civil Engineering, University of Florence, Italy, 1990
Appendix The relation between the tilt angle 0 and the maximum stress at the base of the block (i~,~ (equation (29)) depends
Block random rocking and seismic vulnerability estimation: L. Facchini e t al.
!i
1 1
°l°t
I.
3,
1
'
"--~--
Ii
1
1
i
i
I
A
a)
/
.
(' .... .
.
.
,
.
l . - . & .
"
•
I
[
~----
I
-
I
i
.
I
,
I
,
1
i 4 L
i.a, Figure9
I
iI - d - d-_Jr_J F . * .?. . . . . d
k
L
Schemes to obtain relations beWveen block rotation 0 and maximum compressive stress at base of block for either (a)
circular; or (b) cave square shape on the shape of the base and the rocking condition. The block is supposed to topple after the uplift of the base. With reference to a circular shape (Figure (9a)), for no uplift condition P
O <- O <- k--~,
P
t r , ~ = - ~ - ~ + kOr
for the uplift condition P
P r-dl
kwr3 <- 0 <- Oc,. tr.= - ~ d _ d l where P is the weight of the block, A is the area of the
portion of the base in contact with the ground, k is the modulus of the Winkler foundation, and P k A(d - dl)' A = RE(a + sin ct cos or), 2 sin3a d = ~ R a - sin a cos ct' dl = R cos cz 0=
With reference to the cave square base (Figure (9b)), for the no uplift condition P P 0 <-- 0 --< ~-~, A = 4b 2 -- 4 ( b - s ) 2, O',nax = ~ + k b 0
Engng Struct. 1994, V o l u m e 16, N u m b e r 6
423
Block random rocking and seismic vulnerability estimation: L. Facchini et al. for the uplift condition
P
- kAb
<-- 0 <-- 0~,, ( r , . a ~ -
Pb-dl A b-d
-
d I --< b
--
A, = 2b(b-dj), A~ = 2(b-s)(b--s-dO, dl = (b+dl)/2,
where
O-
dv = (b-s+dO/2,
P
d = (A, d, - Av dv)l(A, - A v), b - s <- d, < b,
k A (d-dO -b<-d~
424
b(b2-~) d = 2b(b-dO - 4 ( b - s ) 2 ' s - b
<-s-b,
A = 2 b ( b - - d 0 - 4 ( b - s ) z,
Engng Struct. 1994, Volume 16, Number 6
A = 2b(b-dO,
d = (b+dO/2
S,
Vittorio Gusella Institute of Energetic& University of Perugia, Perugia, Italy
Paolo Spineili Department of Civil Engineering, University of Florence, Florence, Italy (Received September 1992; revised version accepted September 1993)
An analysis is undertaken of the rocking motion of a rigid block resting on a rigid foundation when subjected to a random ground acceleration by using an equivalent linear system. The method predicts the cumulative distribution function of the extreme values of the tilt angle and the results are compared with those obtained by other authors using a step-by-step numerical integration of the equation of motion. With the introduction of the analysis of the seismic risk, by means of the probabilistic law of the yearly maxima of the peak ground acceleration, a method of estimating the vulnerability is proposed which takes into account the maximum permissible rotation for the block and the strength of the block material. In the latter case, an approximate relation between block rotation and maximum stress at the block base corner is introduced. This method is used to estimate the seismic vulnerability of some of the ancient monuments of the Imperial Forum in Rome. Keywords: block rocking, random vibrations, statistical lineariz-
ation, seismic vulnerability
Since the end of the last century, several researchers have been involved with the dynamics of a rigid block under seismic excitations 1. This problem was first systematically investigated by Housner2. He found a relation between the time duration of a single horizontal pulse (either rectangular or sinusoidal) and its minimal intensity in order for the block to overturn. With greater intensities, the block will topple without any rocking motion, while lower ones will make the block rock without overturning. On the other hand, if the excitation does not consist of a single pulse, but of a series of pulses, then the block may topple with lower peak intensities than those predicted by Housner2. Aslam et al. 3 performed several experimental tests with a shaking table. They developed a computer program which evaluated the rocking motion of the block, either prestressed or unconnected to the floor, where the system was prevented from sliding. The coefficient of restitution, defined as the ratio between the angular velocities of the block immediately after and immediately before the impact of the block with its foundation, was assumed to be constant. The bottom surface of the block was plane or, better, slightly concave and the impact such that there was no 0141-0296/94/060412-13 © 1994 Butterworth-Heinemann Ltd
412
EngngStruct. 1994, Volume 16, Number 6
bouncing, so that the block, when impact occurred, began rotating around the opposite comer of the base. The obtained results showed the sensitivity of the response to the aspect ratio HIB, the block size R (Figure 1) and the coefficient of restitution: the stability was observed to be greater for lower coefficients of restitution, smaller aspect
~[y x
, F
Figure 7
Consideredsystem
Block random rocking and seismic vulnerability estimation: L. Facchini et al. ratio and larger block size, but the computed results also showed some exceptions to this general trend. Yim et al. 1 have performed a very extensive and deep parametric analysis of the block dynamics under a seismic excitation. A numerical procedure was developed to solve the nonlinear equations governing the motion of a rigid block on a rigid foundation under either horizontal or vertical excitations. It was noticed that the overturning of the block under ground motion of a particular intensity does not imply, in a deterministic sense, that the block will necessarily topple under the action of a more intense excitation. Housner2 developed an approximate analysis of the dynamic behaviour of a slender block under an ideal white noise pseudovelocity spectrum excitation by means of the piecewise linear equations of motion. In addition, he determined the critical angle 0or for a block to have a 50% toppling probability. This relation shows that the intensity of the ground motion required for the block to overturn is proportional to the block size R and the critical angle 0or. Also once the characteristics of the block are given, its toppling probability increases for higher ground motion intensities. These conclusions, which also hold for the single pulse excitation, should be considered from a probabilistic point of view. Yim et al.' took into account several seismic excitations, and considered evidence that systematic trends can be observed when the problem is studied from a probabilistic point of view, with the ground motion modelled as a random process. The probability of a block response exceeding a given level, as well as the probability of its overturning, increases for higher ground motion intensities, higher slenderness ratios and lower sizes. Spanos and Koh4, using a piecewise linear equation, determined the safe and unsafe regions in the excitation frequency-amplitude plane. Their study of the possible classes of the stationary response highlighted the possibility of the block vibrating at a lower frequency than the excitation (the so-called subharmonic oscillations). Moreover, these responses can have time average zero (the so-called symmetric modes) or nonzero (unsymmetric modes). It was generally assumed that the block and its foundation were infinitely rigid, but, during recent years, the model of a rigid block resting upon a flexible foundation which allows uplift has also been analysed. Generally speaking, uplift leads to a softer vibrating system which behaves nonlinearly, even though the response may be composed of a sequence of linear responses. Psycharis and Jennings5 took into consideration a continuous elastic foundation with viscous damping (i.e. a monolateral Winkler model) and a simpler model made of two couples of spring and damper, and established an equivalence criterion between the two models. They also determined the response to a single pulse and a relation between the maximum tilt angle versus impulse amplitude. The apparent rocking period and the apparent ratio of critical damping were determined by means of a parabolic approximation of a generic half cycle of the response. Also an equivalent linear system, whose parameters depended on the normalized impulse, was defined. As the vertical oscillations of the centre of mass of the system turn out to be generally small with respect to the rocking motion, Blasi and Spinelli6 developed a simplified model for the Winkler foundation, taking into account only the degree-of-freedom 'rotation of block'. The analytical
model was then applied to the study of the behaviour of multiblock columns. In order to predict the rocking response amplitude when the foundation is excited horizontally due to a harmonic acceleration, Koh et al. 7 proposed an approximate analytical method which combined smile condensation and the method of averaging. In addition, they proposed a method of predicting the maximum tilt angle under an earthquake excitation incorporating the linear response spectra. Spanos and Koh s also studied the response statistics of the block supported by a flexible foundation when the excitation is a stationary white noise modulated by a deterministic function of time. An analytical procedure, which made use of both static condensation and equivalent linearization to approximate the evolutionary process of the joint probability density function for tilt and angular velocity, was proposed. The present study was prompted by the great interest in the behaviour of monumental structures, which is related to the problem of conserving them from seismic activity.
Notation ap apy Ce e
g k
ko Pr(Y) t,
x 2,,a~ B
G(.) H
n~ n.e loJo, M
P~o,1 P~y) So Srr T str
E(.) 6
Ai
mean peak acceleration of soil motion yearly maximum of mean peak acceleration damping coefficient of equivalent linear system coefficient of restitution of angular velocity after impact acceleration due to gravity: g = 9.81 m s-2 modulus of Winkler foundation lower limitation on k to use rigid block model elastic coefficient of equivalent linear system probability density function (PDF) for random variable Y time when impact between block and its foundation occurs ground motion acceleration peak of ground motion acceleration base of block constitutive relation between maximum compressive stress at base of block and nondimensional rotation height of block amount of energy dissipated by block amount of energy dissipated by equivalent linear system moments of inertia about block base comers mass of block collapse probability cumulative distribution function (cdf) for random variable Y intensity of ideal white noise spectral density of random stationary process
time lag when envelope X attains its maximum value = 1 structural life error function dimensionless tilt angle: th = 0/0or upper bound for safety region S dimensionless tilt angle of equivalent linear system requested rotation during system life ith order spectral moment
Engng Struct. 1994, Volume 16, Number 6 413
Block random rocking and seismic vulnerability estimation: L. Facchini et al. characteristic frequency of rigid block: IX = (MgR/Io)'/2 expected frequency of forcing process expected frequency of equivalent linear system critical damping factor compressive stress at base block requested resistance structural resistance material resistance difference between structural and requested resistances standard deviation tilt angle critical tilt angle: Ocr -~ tan-' (H/B) dimensionless time variable: r = / , t dimensionless time when impact between block and its foundation occurs dimensionless frequency: to = [1//~ dimensional frequency expected value operator
tx V
or
orr or. 7/ 0
0. T
to
12
~[.]
Analytical model and equivalent linear system The mechanical system taken into consideration is shown in Figure 1. Friction is assumed to be sufficient to prevent the block sliding. The only Lagrangian coordinate is the tilt angle 0 about the base comers 0 and 0". If the horizontal ground motion acceleration is denoted by ~(t), then the equations of motion about each of the base comers are
since d2/dF = Ix2(d2/d~-2), the piecewise linear equations of motion can be rewritten in the following nondimensional form d2S - S + sign(S) =f(~'/ix) d~-2
when coupled to the initial conditions for each half cycle this gives lim ~(z) = e lim ~(T); + ~r~'r.
Ocr--O ) +
~¢,) = 0
(5)
-r--.'r.
The equivalent linearization method was applied to the rigid block under harmonic excitation by Spanos and Koh4's to obtain an approximate steady-state response aproximation with the same excitation frequency. In the equivalent lineafization method, in its more general formulation9, a linear system is characterized by two constant parameters k, and c,, in such a way as to minimize the expected square value of a proper error function. If the linear system t~, + c,~e + (k,-1)S, = F(~'); F(T) =f(zll*)
(6)
is considered, subtracting equation (6) from equation (4) gives -- ~)e -- Ce+e -- ~) +
Io0 - M R X C o S (
(4)
sign(S)
-
(k,-1)S, = 0
(7)
M g R sin( Oc~O) = 0; 0 --> O
Io,0 - M R X cos(0c~+0) - M g R s i n ( 0 ~ + 0 ) = 0;
0 <-- 0
Assuming that S , - S, an error function can be defined such that
(1) E(~') = sign(S) - keS where M is the mass of the block, Io and Io, are the moments of inertia about the comers of the base; g is the acceleration due to gravity, R = 0.5 (H2+B2)lz2 is the block size; O~r = tan-'(HIB) is the critical angle of overturning and is also a measure of the block slenderness. During the impact between the foundation and the block a certain quantity of energy is dissipated and hence the angular velocity is sharply reduced; the assumption is introduced that the dissipation can be modelled by an impulse mechanism described by a coefficient of restitution e of the angular velocity. Hence the motion of the block is described, during each half cycle of its oscillation, by each part of equation (1), which are linked together by the initial conditions for a generic half cycle
(8)
The forcing process F(¢) is assumed to be a stationary Gaussian filtered white noise, which is a common assumption in earthquake engineering. Later it will be explained why this assumption is valid. The proposed procedure takes into account the energy dissipation during impact in order for the linear damping coefficient to be evaluated, since this must be computed by means of an energy balance. The basic procedure is to evaluate the expected amount of energy to be dissipated during impacts between the block and its foundation, and then tune c, accordingly. The amount of energy dissipated during the ith impact is
H.,=-~--~72; l -e 2
lim 0(t)= e lim 0(t) t~t +
where t, is the instant when the impact occurs and e is the coefficient of restitution of the angular velocity. When the block is slender (HIB >-- 2) equation (1) can be linearized, giving Io0
- M R X + M g R ( O~r-O) = 0
Io,0 - g
~r.i) =0
(9)
R X - M g R(O~,+O) = 0
When the linear system is considered, its angular velocity will be a continuous function of time, i.e. there is no discontinuity in the angular velocity.when S, = 0. The angular velocity of the linear system S, can therefore sensibly approximate the average of the two limits of the actual angular velocity
(2) ~ e ( T * ) = Si- + S ~
Defining
(MgRI '/2 ~(t) S=0/0c'; I x = \ Io ,/ ; * = l z t ; f ( t ) - O c . g
414
~7 = lim - ~(r);
t~t:
(3)
Engng Struct. 1994, V o l u me 16, N u m b e r 6
- ~1 -+ e S;•
(10)
so that the expected amount of energy dissipation during is
impact
Block random rocking and seismic vulnerability estimation: L. Facchini et al. 1-e
F[H.,] -- 2 ~
IF[~,(r*)]
(11)
where E[.] denotesflhe expected value operator. Hence the expected amount of energy dissipated during a unit time period will be 1-e
IF[H.] = 2 v, F[H.,] = 4 ~
v, IF[4~'~[~b,= 01
(12)
where v. is the expected frequency of the linear system and the vertical bar [ denotes conditional probability; in other words, F[H.] is assumed to be proportional to the expected frequency of the response process. The expected amount of energy dissipated by the linear system during the same period will be
F[H,~] = F
c, ~ dr =
the fact that a random forcing process is now dealt with, the assumption can be considered suitable. It can be found that
1 rb~e ; ~:[~l~be = 0] = E [ ~ ] = r/~, 27r ~/%
v, = - -
(18)
so that the linear system parameters can be evaluated as follows k,=
(2)1t2 1 -"1/% -;
1-e*l*~ c , = 21 + e -TI% -
(19)
The variances ~l2, and rf% can .~ computed from
~ =f~S%%(to)dto
t~] dr = c, a~,
°
(13) so that c, becomes
where
1- e IF[4~'.lth~= O] c. = 4 ~ ve .~.
(14)
Once c, has been evaluated, k~ may be computed. This can be done by minimizing the value of the square of the error function E2 defined by relation (8)
0 Ke0 E[~] = 0 --7
(15)
which yields9
ke =
(16)
2
In equations (14) and (16) the expected value operators, and hence the variances of the two processes */% and *1%, depend on the two parameters k. and c. so the equations are not linear. From the Gaussian assumption and the noncorrelation of the two processes, the joint probability density of ~be and ~ of the equivalent linear system is found to be 1 e x p ( - t/r~. p..~. (th.; &.) - 2rr r/% ,/.. 2 -tf%
(20)
~/~,e= f ~ to2 S%% (to) dto
SFF (to) S % % - (ar~ _ to~,)z + 4~ar2~to 2 ; to~, = k , -
1;
c. ~e -- 2toe
(21)
where SFF(to) is the power spectral density of the dimensionless excitation F(r) and to is the dimensionless frequency to = f~/gt. Some numerical algorithms have been develope dm to compute ke and ce, since their evaluation can be extremely difficult. The computed, values will be affected by the variances of ~b, and ~b, rather than the shapes of p%(the) and p,t,,(q~e): the proposed method can give good approximations to these parameters, so that satisfactory results can be obtained even in cases, such as the one discussed, in which the physical system exhibits a strong nonlinearity. The actual distribution for t~(r) and t~(r) will not be Gaussian, but the equivalent linearization method cannot give any information about its shape 9. Moreover, it is implicitly assumed that the solving equations have a unique solution over the whole domain of k~ and c,. This seems generally true for dynamic systems with only one stable equilibrium position excited by a wide-band random process.
~----~-/
2 '}~e] (17)
where null averages were assumed for both the processes. When deterministic excitations (e.g. harmonic accelerations) are considered, the dynamic system can exhibit steady-state responses which have nonzero time averages. These responses were defined by Spanos and Koh4 as 'unsymmetric modes' in order to distinguish them from those whose time average was zero, these being called the 'symmetric modes'. They observed that the predominant mode is symmetric and nearly harmonic, with the same frequency of excitation. From these observations, and from
Comparison of theoretical layout with numerical results The reliability of the proposed method, is evaluated by using the numerical results of Yim et aL 1 They examined the behaviour of a block with size parameter R = l0 ft and aspect ratio H / B = 5 subjected to 20 numerically evaluated acceleration histories satisfying the main characteristics of the following earthquakes: E1 Centro, California, 30 December 1934 and 18 May 1940, Olympia, Washington, 13 April 1949 and Taft, California, 21 July 1952. The method employed by Yim et al.~ consisted of prefiltering a generated ideal white noise according to the Kanal-Tajimi
E n g n g Struct. 1994, V o l u m e
16, N u m b e r
6
415
Block random rocking and seismic vulnerability estimation: L. Facchini et al. spectrum model and multiplying the response of the prefilter by a deterministic function of time X(t). The power spectral density of the prefilter response was then
where q=
1 + 4~'2(1)/1~o)2 SX~I~) = So (l -- (~-~/1"~0)2)2 + 4~2(~/~'~0)2
(22)
with 12o = 27rfo; fo = 2.5 Hz.; ~o = 60% and where So was the intensity of an ideal white noise. The time dependent envelope )((t) was t_<4s
'16
4s
15s
(23)
t ->- 15s
The coefficient of restitution e changed from 0.90 to 0.95. A constant multiplier was used to obtain the desired mean for the peak acceleration of the soil: E[X,,=] = 0.4 g. In order to evaluate the parameter So, the following distribution 1~ is assumed for the peak acceleration
P~,,,~ ( 0 = vT ~2 e x p ( - 2 ~ ) (24)
exp{-vT exp(- 2~)} It can be shown that 0.5772 / E[X~x] = r/X (2 ln(vT)) 1/2 + (2 l n - ~ ) l / 2 J
(25)
where v = l]o/2"tr = 2.5 Hz and T is the seismic event duration. Assuming that E[g,,,~] = 0.4 g and T ~ 11 s, then r/x - 1.4020 m s-2. The variance ~xz can be computed from integration of equation (22), so that So = 1.9589 x 10-L It can be shown that the power spectral density for the dimensionless excitation F('r) can be expressed as follows So/x 1 + 4~2(02/0)o) 2 Sv~oJ) = (g ~ 0~) (1 - (~o/0.~o)2)2 + 4~o2(O9/0h~)2
(26)
where OJo= 1)o/~. In order to compare the theoretical and numerical results, by means of equations (19)-(21) the two parameters k,, Ce and the power spectral density S%% were estimated for the various coefficients of restitution for a linear system equivalent to a block with size parameter R = 10 ft and slenderness ratio HIB = 10 (also considered by Yim et aL~). Assuming the model proposed by Vanmarcke ~2, the cumulative distribution function (cdf) of the extremes of tilt angle is
F~emax (05emax) [ [ffr~112q {#emax~ (27) = exp - v ,
exP(2 .q2 ,,o~) _ 1
1-hohe//
;Ai=2
(28)
Figure 2 shows the comparisons between the function (27) (continuous line) and numerical results of Yim et al. j (dots) for different values of the coefficient of restitution e. The dashed line denotes the confidence interval for the Kolmogorov-Smirnov test of hypothesis for a confidence level of 0.1%. Since the coefficients of restitution were similar, the numerical results were considered together in order to obtain 120 samples with a high statistical reliability. The theoretical analysis was repeated with a damping coefficient which was the average of those formerly obtained. The proposed method provides a reliable estimation of the maximum tilt angle as shown in Figure 3. Larger differences between theoretical and numerical results are observed as ~b approaches its critical value qbcr= 1. It will later be seen how in the vulnerability estimation these differences can be surpassed assuming, on the safe side, an admissible value ~-< 1, which also corresponds to the imposition of a limiting (deterministic) value for the resistance of the block material.
Relation between rotation of the block and compressive stresses at its base The compressive stresses at the corner of a base of a rigid block resting on a rigid foundation and subject to oscillations, are infinite. If collapse is only assumed to take place when 4~cr>- 1, there is an implicit assumption that the block material is infinitely strong. This approach is unreliable when applied to the estimation of the seismic vulnerability of monumental structures. The model of the rigid block resting on a Winkler foundation can be taken into account in order to establish an approximate relation between the block rotation and the internal stresses so that a random material resistance can be considered in the estimation of seismic vulnerability. In most of the observations of the block motion, vertical oscillations are very small and can be neglected. This hypothesis leads to the Blasi-Spinelli 6 model, where the foundation reaction balances the weight of the block, since the variation of the vertical momentum can be neglected. The block rotation thus becomes the only Lagrangian coordinate of the system. The period of the free oscillations of the block can be evaluated as the spring constant k of the foundation changes. It follows that there is a value ~ such that, if k - £, the period can be satisfactorily approximated by Housner's relation for a rigid block on a rigid foundation (it can be seen6 that/~ ~ 1000 N cm -3 in most cases of real columns). It can therefore be deduced that the dynamic response of a rigid block on a Winkler foundation whose constant k is greater than ~ can be compared to that of a rigid block on a rigid foundation, while its internal stress configuration can be evaluated from the Winkler model. The relation between the rotation, 0, and the maximum compressive stress at the base of the block, cr,~, generally depends upon the shape of the base itself (see the Appendix); the two functions (which are the inverse of each other) can thus be evaluated
l o',~x = G-1(05) = G-~(OlOcr);
416
~oiS%%(o9) dw
Engng Struct. 1994, Volume 16, Number 6
05 = G(orm,~)
(29)
Block random rocking and seismic vulnerabilityestimation: L. Facchini et al. e-(~90 /"
1 .....................
~4 •d
e=0.91 ,/
/ ,
0.5-
./ . . . . . . . .
1(~
~
.
.,.--"
~4
jpse~
"d
10o
0.5
l0 t
../
"
1"04
l0 o
÷.u
l0 t
÷.,= effi 0.92
Ii
,
,
~
,'/'i" /"" ,~ ,/
'~ 0.5 ~,,
•
....
/
1"04 "
/
~
/"
I
effi 0 . 9 3
.... .. ~ ........................ ' ' '~ . . . . . . . / ~ , .:. . . . . . .
~'"
~
/,
~j
........ i "
,,'" ~ /:/
: .~.~ / O. " ,,I(" ~ "/ , .~/
"d
~
I0 e
¢/
/¢ ",~ /,,"
/
lo-°',"
10~
. . . . . . .
÷-,--
. . . . . .
÷.,.,
Figure 2 Comparisons between results obtained by statistical linearization ( equation (27)) and numerical results of Yim et aL ~ (*), for different values of coefficient of restitution e ( - - - confidence interval for Kolmogorov-Smirnov test of hypothesis for a
confidence level of 0.1%)
ce = 0.0225
0.9
//
................................................... - ....
/
0.8
.
/
0.7
J "1=
//
.i
''i. "
/
//'"
//
0.6 :,
0.5
~
-
•
" ....
.
//
.
i
.
. .
. .
. .
.
.
-
i_
0.4
0.3I~~i//
/......... . . . . . . . . . . .
0.20.i i~f//'I/'/ n~
- ,,"
i
-'i
i
i
1"04
t
t
i
i
t
i
10o
t
i
t
i
l0 t
Figure 3 Cumulative comparison between numerical results (*) of Yim e t aL 1 (neglecting restitution coefficient variation) and statistical linearization ( equation (27)) with a mean damping coefficient for equivalent linear system ( - - - confidence interval for Kolmogorov-Smirnov test of hypothesis for a confidence level of 0.1%)
In the previous method the stress is only affected by the block weight. More precisely, the vertical reaction of the comer of the base of a rigid block resting on a rigid foundation can be obtained once the vertical acceleration of the centre of mass of the block (d2ys/df) is given. Thus
Ry= M(g+d2ys/dI2)= M g ( l + d2yg/dr2)
(30)
where yg is the vertical coordinate of the centre of mass and
Ys = R cos(0c:-0);
dys/dt = R sin(0c:0)d0/dt
d2yJdt 2 = R sin( O~:-O)d201dt2-R cos(Oc:-O)(dO/dt) 2 (31)
Lineadzing the expression for d2y,/dt2 gives
d2ys/df ~- R[d20/dt2 ( Ocr-O)-(d~/dt) 2] =/j2R~.cr[~(l__t~ )
_
(~2] =
_l.t2R~cr [(1_~)2 + fi~2] (32)
where the equation of motion for free oscillations ~ - ~ + 1 = 0 of the system is employed. If the expressions of potential and kinetic energies are introduced denoted by W and T, respectively, then
W=d~-th2/2;
T=~/2;
T=W,~-W
d2ysldt 2 --- -/z2R~'c,[1 + 2(W,~,-2W)]
Engng Struct. 1994, Volume 16, N u m b e r 6
(33) (34)
417
Block random rocking and seismic vulnerability estimation: L. Facchini et al. Now, as W~,~ ~ W ~ 0 and 1/2 -> Wm,~ -> 0, it can be shown that 0--< 1 +2(Wm,~-2W)-----2
less than an arbitrarily fixed value $ during the system life 7",r. Equation (37) defines the CDF for the extreme rotation during the whole structural life T~,~. Thus
(35)
For the structures which will be examined later, the leading term ~2R~r lies in the range 7 x 10-2-3 x 10-~ m s-2, which can be neglected when compared with gravity (g -~ 9.81
P%(~b,)=
exp{-L,rf~[1-F.~(~s,apy)]pAey (apy)daey
}
= Prob[~b < ~b,]
(39)
m s-2).
The corresponding PDF can be obtained from this last equation as follows
Estimation of seismic vulnerability
Admissible maximum rotation for the block
p%(~b~) = ~
In the previous section the starting point was the shape of the power spectral density of the forcing process, coupled to the mean of its peak acceleration. If an upper bound is imposed for the maximum rotation of the block, for. instance the toppling condition <-- $-- 1, a safety region can be determined and the curve which expresses the collapse probability to the given event can be plotted. In order for the seismic vulnerability to be estimated, the collapse probability to a given event must be combined with the probability that this event takes place in the structural life. The upper bound for the rotation will be denoted by and S =- [-~;~] C [-!;1] E R will stand for the safety region of the dynamic system. Let ~(~[ap) be the safety probability and ~3~(t~lap) the collapse probability for the block subjected to a seismic motion whose mean peak acceleration is ap. Then
~3,($1a~) - 1 - ~ (¢la~) -- Prob[l¢t < $1a~l
S(T~,r)=exp{-T~,r f ~ ~3~((blapy)PA,y(apy) dapy}
~3c(~lap) = 1 - F..=(~b,ap)
f2 °
(38)
where ap is the mean peak acceleration of the earthquake. The critical value t~ can be introduced via a collapse criterion. If overturning is considered~- then (b = 1. Otherwise if a constitutive equation between the rotation and the stress at the base of the block is developed, a deterministic critical value for stresses can be fixed and then a critical value for rotation can be obtained.
~( dps,apy) =
418
Engng Struct. 1994, Volume 16, Number 6
~(~b,;a,,) clap,
(40)
[1
- F.m~x ( (]ls,apy)]PApy ( apy)
(41)
Hence using the second of equation (29), the PDF for the maximum stress of the block, during the whole structural life T,,~, is as follows
pz~s(CL)= p% [G(cL)] ~
OG
¢,.ox='~,
(42)
This PDF must eventually be combined with the PDF for the structural resistance, which comprises two distinct components. The former comes from the material strength for which the following standard Gaussian distribution can be conveniently employed (43)
Pr-v((r.) = N(/~u, ~i~u)
The latter comes from the overturning condition, that is, if the rotation equals its critical value (~b = ~bcr= 1 determined by the toppling condition). Then the compressive stress at the base of the block will attain its maximum value. Because the constitutive equation is a monotonically increasing function, greater compressive stresses will correspond to greater rotations, which are impossible to obtain, as the block will topple. In order to obtain a comprehensive description of an equivalent strength of the structure which takes into account the effective material strength and the toppling phenomenum, a Dirac delta function will be combined with equation (43) to obtain the structural strength PDF
pER(O.r)= P~R (1) (err) + Kr~(ffr 0~4,= &)
(44)
where
(1) tN(~u , "qzu) if trr < crl~ $) PI~R(t~r)= [0
Introduction of a random strength for the block material Once the probability density function (PDF) of the yearly extremes for ground peak acceleration apy is given, equation (37) may be viewed as the probability of the rotation being
~(¢~s,apy)dapy}
where
(37)
where pApr .is the probability distribution function of the yearly maximum mean peak acceleration, apy, of seismic motion for the considered site. Equation (27) gives the probability that ~b,~o~-< ~b, where th,,~ is the maximum rotation and th is a fixed value. When ~b= t~; equation (27) gives the safety probability and the collapse probability is
P%(~b~)
=- Ltrexp{- L,r f2
(36)
It will be assumed that the time series of the seismic events can be modelled as a Poisson process and that the structural system does not change with time. Then the safety probability of the system during its life T~,~ becomes ~2
0
(45)
otherwise
and Kr is a normalizing constant. Consider the random variable
Block random rocking and seismic vulnerability estimation: L. Facchini et al. ~
(46)
= ~;. - ~ s
difference between equivalent strength and maximum stress of the block during the whole structural life T,,~, whose PDF can be evaluated from the convolution of equations (42) and (45), which can be simplified in the expression
p:~z(~bz) =
f : 6=4,)-~rspr.R(or~ + tr,) p:~,(cL) dcr,
(47)
where
M* = "Zimi; R* - ~mi M ~Ri ,
I~ = ~i R*2 ~ lGi
The characteristics of the equivalent block depend not only on the sizes Ri of the columns but also on their aspect ratios HJBi through Ri and Ion. With regard to the seismic risk, a Fisher-Typpet II type CDF for the yearly peak acceleration was assumed to fit the available seismic data, so that
The collapse probability can be evaluated from
[ \apy/ j P : a , = f ° p x z ( ~ z ) dcr~
(48)
Application of proposed method to some ancient slender monuments The proposed method has been employed in the very first estimation of the seismic vulnerability of some of the most ancient structures in Rome, which have a dynamic behaviour that can be described, as a first approximation, by means of the rigid block model. Foca's column (Figure 4(a)), Vespasiano's temple (Figure (4b)), Dioscuri's temple (Figure 4(c)) and Saturn's temple (Figure 4(d)) have been analysed. These last three structures have a series of columns linked by an overhanging architrave, so that an equivalent block was defined (Figure 5). The following assumptions were made • The components of the structures and their foundations are infinitely stiff • The friction between the foundation and the columns and between the columns and the architrave is sufficient to prevent sliding • The geometry of the architraves are such that they are unlikely to uplift from the top of the columns • There are no relative displacements among the superimposed elements ('rocchi' of marble or granite) which constitute the columns Hence the motion of the system can be described by only one Lagrangian coordinate, 4~The mechanical characteristics (e.g. mass, momentum of inertia etc.) of the fictitious block can be evaluated by equating its total energy to that of the actual s t r u c t u r e 1° (Figure 5). Letting the subscript tot indicate the characteristics of the fictitious block, the superscript * those of an intermediate system, the subscript tr those of the architrave and the subscript i those of the single colunms, then
Mrot = M* + mtr IG tot =
tl-2[/~ +
(4mtr + M*)R .2] - Mot RE,
Rtot = 2mtr + M* R* + 1 m t r htr M,o, 2 M ,o, a = 1
1 mtr htr
2 M,o, R,o,
(49)
(50)
(51)
where u~ = 6.6552 x 10-3 and u2 = 1.9313.
The analysis of Saturn's temple and other monuments Saturn's temple is made of eight granite columns which bear an architrave; the mean diameter of the columns is 64 cm and they are 13.00 m high. The architrave is approximated to two prisms whose centres of mass are 1.89 m above the top of the columns. Dynamic tests were carried out in order to study the dynamic behaviour of this monument. The results confirmed that the temple could be idealized as a rigid block model. Also it was possible to compute a reliable approximation of the Winkler constant k of 1500 N cm -3 which proved high enough to consider the system as a rigid block on a rigid foundation. Several equivalent linear systems were defined to calculate F . , ~ (~b;ap) in which the mean ground acceleration ap ranged from 0.1 g to 1.3 g (it should be noted that the characteristics of the linear system change when the ground motion intensity increases) (see Figure 6). In order to define the statistical properties of the material resistance Et,, the experimental results of Blasi et al. 14 were utilized so that a Gaussian PDF was assumed with/x~u = 55.73 N mm -2 and r/~U = 7.62 N mm -2. The structural life was assumed to be T,,r = 50 years, giving the collapse probability from (equation (48)) as
If,, ~- 0.00706 In order for the influence of the strength to be considered, several average values were taken into account, with the same coefficient of variation. The results are reported in Table 1 and plotted in Figure 7. If a best fit cubic spline curve is plotted for these data, it asymptotically approaches the collapse probability p f a1) il ~
0.00638
(52)
which is obtained when a critical (deterministic) threshold t~ = 1 is considered for the structural collapse (i.e. the toppling condition). This shows that, as the average value for the material strength increases, only a small fraction of P:ait is due to material failure, and the greater part is due to overturning. Owing to this fact, when other ancient structures were considered, the analysis did not take into account the material strength, and only the toppling condition ~ = 1 was imposed to evaluate the collapse probability. In Figure 8 the overturning probability is plotted versus the mean peak
Engng Struct. 1994, Volume 16, Number 6 419
Block random rocking and seismic vulnerability estimation: L. Facchini et al.
I
Ii i
a)
~"
b)
"..'~ II I
i i
I
i
~f;v i I
I
'
i
Figure 4 Structural schemes of analysed monuments: (a) Foca's column; (b) Vespasiano's temple; (c) Dioscuri's temple; and (d) Saturn's temple (measures in centimetres)
420
Engng Struct. 1994, Volume 16, Number 6
Block random rocking and seismic vulnerability estimation: L. Facchini et
el.
A
~ 24.00 II
~ 20.00
numerically evaluated data
%
best fit cubic spline
Figure5 Equivalence criterion for evaluation of mechanical characteristics of fictitious block
16.00 >, 4-,
o 12.00
acceleration ap; this result is obtained from the CDF of the maximum tilt angle versus ap (see for example Figure 6) for ~ = 1. The characteristics of the equivalent block, the modulus of the elastic foundation and the collapse probability are shown in Table 2.
_Q 0
O_
o 03 00
-6 o
Conclusions
8.00
O
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
O
asymptotic value (toppling condition only) 4.00
0.00
.......
.......
................. :so
.......
.......
material resistance mean, N / m m s
With regard to the block random rocking, them is a satisfactory agreement between the results obtained from the equivalent linear system and those obtained from numerical integrations of the equation of motion, so that the proposed procedure can be considered accurate enough for a first estimation of the stochastic characteristics of the response. When the cumulative distribution function for the extremes of the response is considered, a greater difference between the two methods can be observed when $ approaches So,In order to estimate the seismic vulnerability, the influence of these differences can be safely catered for by means of a reduction of the upper bound for the block rotation. An extended model allowing for a random material strength is proposed by means of approximate relations between the block rotation and the compressive, stresses at the comer of the base. If this approach is followed, the reduction of the upper bound for the rotation corresponds to the introduction of a deterministic value for the strength of the material. Moreover, taking into account an equivalent
Figure7 Seismic vulnerability versus average value for material resistance (Saturn's temple)
Table 1 Change of 50 year collapse probability of Saturn's temple according to different values of mean /~ and standard deviation ~/of material resistance P'Xu (Nmm -2)
~-u (Nmm -2)
Pro. (T.= = 50 years)
14.0 16.8 19.6 22.4 25.2 28.0 42.0 55.7
1.9 2.28 2.66 3.04 3.42 3.8 5.7 7.6
20.50279 14.18266 10.44586 8.38832 7,49777 7,19037 7,05766 7.05718
x x x x x x x x
10~ 10-3 10-~ 10-~ 104 10-s 10-~ 10-3
SATURN TEMPLE 1
i
==
0.9
. . . .
0.8
•
~ :i_
i ¸ iii-
0.7
.
0.6
. . . .
0.5
. . . . . . . . .
0.4
i.._ i_
:
0.3
i
0.2 0.1 100-2
10-1
100
101
==
Figure 6 Cumulative distribution functions for extreme tilt angle when block is subjected to different forcing processes whose mean peak acceleration ranges from 0.1 g to 1.3 g (duration of stationary portion of seismic event ~ 10 a)
Engng Struct. 1994, V o l u m e
16, N u m b e r 6
421
Block random rocking and seismic vulnerability estimation: L. Facchini et al. ROMAN FORUM MONUMENTS
0.6 . . . . . 0.5
..'"o fzo •
0.4
.:-f"
•
0.3 ~
0.2 /,:" :f/./'/,,~
O
-c • • d .................
temialcof Vespasian temiale of Sattirn
0.1
°o
0.2
024
016
028
i
114
122
Figure 8 Overturning probability versus mean peak acceleration for examined structures considering only toppling criterion ~ = 1 (duration of stationary portion of seismic event - 10 s) Table 2 Characteristics of equivalent block, frequency of free oscillation, modulus of elastic foundation of block and collapse probability for a structural life of 50 years
Foca's column Temple of Castors Temple of Vespasian Temple of Saturn
Mass (kg)
Io (kg m 2)
R (m)
Ocr (rad)
/~ (Hz)
k
Pf..
(N cm -a)
(Ts, = 50 years)
43540 204300 197820 694950
2410000 2860000 8820000 31170000
6.45 7.72 8.35 11.04
0.098 0.119 0.120 0.107
1.07 1.24 1.36 1.55
1000 1900 1000 1500
0.01030 0.00889 0.00670 0.00638
structural strength (due to the random material strength and overturning) the discrepancies between the two methods are reduced. The procedure has been used to estimate the seismic Vulnerability for historically valuable structures, which can be idealized as a single rigid block. The results for the examined monuments indicated that material strength has negligible influence upon the probability of structural collapse: this is because of the high strength of marble and the relatively low stresses at the base in the toppling condition. For this reason a rigid block on a rigid foundation appears to be an adequate model for initial estimation of seismic vulnerability. The results show that the structures have a high vulnerability, markedly higher than the admissible levels for new buildings. Concern over these results are sufficient to suggest that a research programme involving comparisons between several methods of analysis and a repair intervention programme should also be planned.
References 1 Yim, C. S., Chopra, A. K. and Penzien, J. 'Rocking response of rigid blocks to earthquakes', Earthquake Engng Struct. Dyn. 1980, 8, 565-587 2 Housner, G. W. 'The behavior of inverted pendulum structures during earthquakes', Bull. Seism. Soc. Am. 1963, 53, 403-417 3 Aslam, M., Godden, W. G. and Scalise, D. T. 'Earthquake rocking response of rigid bodies', J.sStruct. Div. ASCE 1980, 106, 377-392
422
Engng Struct. 1994, Volume 16, Number 6
4
Spanos, P. D. and Koh, A. S. 'Rocking of rigid blocks due to harmonic shaking', J. Eng. Mech. ASCE 1984, 110, 1627-1642 5 Psycharis, I. N. and Jennings, P. C. 'Rocking of slender rigid bodies allowed to uplift', Earthquake Eng. Struct. Dyn. 1983, 11, 57-76 6 Blasi, C. and Spinelli, P. 'Un metodo di calcolo dinamico per sistemi formati da blocchi rigidi sovrapposti', lng. Sismica 1986, l, 12-21 7 Koh, A.S., Spanos, P. D. and Roesset, J. M. 'Harmonic rocking of rigid blocks on flexible foundation', J. Eng. Mech. ASCE 1986, 112, 1165-1180
8 Spanos, P. D. and Koh, A. S. 'Analysis of block random rocking', Soil Dyn. Earth. Engng 1986, 5, 178-183 9 Roberts, J. B. and Spanos, P. D. Random vibration and statistical linearization, John Wiley, Chichester, 1990 10 Facchini, L. 'Strutture monolitiche sottoposte ad azioni aleatorie: applicazioni al comportamento sismico di opere monumentali', Graduate Thesis, Faculty of Engineering, University of Florence, Italy, 1990 11 Cramer, H. 'Mathematical analysis of statistics' Princeton University Press, 1946 12 Vanmarcke, E. H. 'On the distribution of the first-passage time for normal stationary random processes', J. AppL Mech. ASME 1975, 42 13 Gusella, V. 'Structural failure and stochastic discrete process of random events: An application to seismic vulnerability analysis of a historic building', Struct Safety 1991, 11, 13-28 14 Biasi, C., Sorace, S. and Coil, M. 'Decay of the mechanical properties of marble due to long term small stress configuration', Die Pep. ST/9, Department of Civil Engineering, University of Florence, Italy, 1990
Appendix The relation between the tilt angle 0 and the maximum stress at the base of the block (i~,~ (equation (29)) depends
Block random rocking and seismic vulnerability estimation: L. Facchini e t al.
!i
1 1
°l°t
I.
3,
1
'
"--~--
Ii
1
1
i
i
I
A
a)
/
.
(' .... .
.
.
,
.
l . - . & .
"
•
I
[
~----
I
-
I
i
.
I
,
I
,
1
i 4 L
i.a, Figure9
I
iI - d - d-_Jr_J F . * .?. . . . . d
k
L
Schemes to obtain relations beWveen block rotation 0 and maximum compressive stress at base of block for either (a)
circular; or (b) cave square shape on the shape of the base and the rocking condition. The block is supposed to topple after the uplift of the base. With reference to a circular shape (Figure (9a)), for no uplift condition P
O <- O <- k--~,
P
t r , ~ = - ~ - ~ + kOr
for the uplift condition P
P r-dl
kwr3 <- 0 <- Oc,. tr.= - ~ d _ d l where P is the weight of the block, A is the area of the
portion of the base in contact with the ground, k is the modulus of the Winkler foundation, and P k A(d - dl)' A = RE(a + sin ct cos or), 2 sin3a d = ~ R a - sin a cos ct' dl = R cos cz 0=
With reference to the cave square base (Figure (9b)), for the no uplift condition P P 0 <-- 0 --< ~-~, A = 4b 2 -- 4 ( b - s ) 2, O',nax = ~ + k b 0
Engng Struct. 1994, V o l u m e 16, N u m b e r 6
423
Block random rocking and seismic vulnerability estimation: L. Facchini et al. for the uplift condition
P
- kAb
<-- 0 <-- 0~,, ( r , . a ~ -
Pb-dl A b-d
-
d I --< b
--
A, = 2b(b-dj), A~ = 2(b-s)(b--s-dO, dl = (b+dl)/2,
where
O-
dv = (b-s+dO/2,
P
d = (A, d, - Av dv)l(A, - A v), b - s <- d, < b,
k A (d-dO -b<-d~
424
b(b2-~) d = 2b(b-dO - 4 ( b - s ) 2 ' s - b
<-s-b,
A = 2 b ( b - - d 0 - 4 ( b - s ) z,
Engng Struct. 1994, Volume 16, Number 6
A = 2b(b-dO,
d = (b+dO/2
S,