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Expected value analysis for the quasi-static response of offshore structures
Expected value analysis for the quasi-static response of offshore structures R. Burrows Department of Civil Engineering, L69 3BX, UK (Received March 1983)
University
of Liverpool,
PO Box 147, Liverpool
A probabilistic description of the quasi-static response of offshore structures to random wave loading has been developed recently in terms of the cross-covariance statistics of the mean-zero and Gaussian distributed particle kinematics at an array of locations beneath the water surface. Implementation of this model was initially restricted by the computational requirements associated with evaluation of certain high order expectations. However, approximations for these expectations have now been developed which enable some relaxation in these restrictions. The procedure employed makes use of standard properties of mean-zero Gaussian random variables and, in particular, application of Price’s theorem. The paper presents, for the first time, the complete description of the expected value analysis upon which the model is based. Key words: mathematical model, offshore structures, expected value analysis, quasi-static response
where:
Introduction The instantaneous wave loading exerted upon the submerged element i, of an offshore structure can be expressed in the form:‘!’ Fi=X2i-~IXzi_~l+Xzi
(1)
where XZi_,, XZi are linearly related to the instantaneous values of water particle velocity and acceleration, respectively, at location i and are functions also of member shape and fluid density. In real random sea conditions, the water particle kinematics can often be considered to behave as mean-zero Gaussian distributed processes. The probability density function (‘pdf) of wave load Fi is, therefore, a zero-memory nonlinear transformation of the bi-variate Gaussian process involving Xzi __Iand Xzi which may be expressed in general form as:
P(Y) =
&
f L
exp--(f(-$+-$))di~
2--m
0307-904X/83/05317-12/$03.00 0 1983 Butterworth &Co (Publishers)
1
Ltd
(2)
2
M2, M4 are the second and fourth order statistical moments
of y and the pdf is mean zero and symmetrical. In the case of wave load: y E Fi, M2 E E{Ff)
and
M4 = E{F;).
For offshore structures of the steel lattice type whose behaviour is quasi-static and linear, nodal response variables Y (internal stresses or deformation) may be expressed in terms of the elemental forces: Y = SIF, -t’ S,F,
. . . S,,F,,
(3) where S is a flexibility coefficient. It has been shown1-5 that in such circumstances the pdf of Y is of the form of equation (2), although no formal proof exists. Consequently, the probability structure of Y
Appl. Math. Modelling,
1983,
Vol. 7, October
317
Expected
value analysis R. Burrows
Solution
may be defined in terms of its second and fourth order moments. The second moment of Y is given by:
El to E23
of expectations
From equation
(Al .l), Appendix
El = E{XF} = 3R;i j=l
i=l
Similarly, the fourth moment
1:
(6)
..
and
j=i+l
of Y is:
E2 = E{Xi } = 1OSRg
(7)
The b&variate moment is the basic result: E(Y2> = i
$E{Ff}
+ 4 i i=l
i=l
i
$SiE(FfFi}
E3 = E{XiXj}
j=l
ifi n-l
= Rij
Other bi-variate moments require application of Price’s theorem6T7 (equation (Al .6), Appendix 1) for solution:
n
+ 6 c c SfStE(FfF;) i=l j=i+l n
n-1
n
+ 12 C
1
C
i=l
j=l
k=j+l kfi
j?i n-3 +
i=l
Rij
E4 = E{Xi2xi2> = 4
1
C
SiSjSkSlE(FiFjFkFl)
j=i+ll=k+l
2R; + RiiRji
(9)
and (5)
Evaluation of the joint moments of force given on the right-hand-side of these equations may be achieved by expansion of the arguments in terms of the X variables using equation (1). The resulting expectations of X must then be determined in terms of the cross-covariances of X, Rij, which can tIlen be computed directly from the surface elevation spectrum for the sea state.2 Following this procedure it can be easily shown that expressions for a total of 23 expectations are required as follows:
(10)
.
ES = E{XfXj> = 3RiiRij
El = E(X;}
dRii + E{Xi2> E{Xi2}
0
= n
E{XiXj} s
SfSjSkEf.FfFjFk)
n-2
24 1
(8)
E6 = E{X:Xf}
= 3Rii(4Rfj
+ RiiRjj)
E7 = E(X?X/}
= 24R:j + 9RiiRjj(8R~
(11) + RiiRjj)
Moments E8 to El0 are solved using Appendix E8 = E{XilXilXj> = 4 G
(12)
1.
= E{XilXilE{Xj(Xi)}
R;;2Rij
(13) 96 Fm Rf/‘Rij
E9 = E{XfJXilXj}=
(14)
E2 = E{X,S) 4
Es = E{XJj)
EIU = E(XilXilXT}
E4 = E{Xizxi’}
Rij(3RiiRjj
=G
+ Rzj)/R;/2 (15)
Es = E{X+?Xj)
Solution of the remaining b&variate moments is more complex than for the above and it is, therefore, instructive to develop the solution for El1 in detail here. Applying Price’s theorem; equation (Al .6):
E6 = E{X?X;) ET = E{X;X,+>
Rij
El1 =E{XiIXilXjlXjl}=
4
s
E{lXiXjl>dRij
0
+E{xilxil>Ef.XjIXjI)
(16)
and from the mean-zero symmetrical E(XilXil)
= E{XjlXjl)
nature of Xi and Xi:
=0
Applying Price’s theorem to the argument in equation (16):
of the integral
Rij E(lXiXjl}
=
E{sign(Xi)
sign(Xj)> dRij
i
0
+E( IXil> E{ IXjl)
(17)
and E{ IXil}, E{ IXjl) are obtained from equation (Al .2), also (a/aXi){sign(Xi)} = 2F(Xi) where F(.) is the dirac delta function. Thus applying Price’s theorem to the argu-
318
Appl. Math. Modelling,
1983,
Vol.
7, October
Expected
El4 = E{XfXjXJ
sign(Xj)>
Rij =
analysis
R.
Burrows
Similarly:
ment in the integral: E{sign(Xi)
value
El5
= ISaiRfiRij + Uj(E6)
=E{XilXilX/X,}
=gRH2
E{6(Xi) S(Xj)} dRij
4 r -0
X
+ E{Sign(Xi)} E{sign(Xj)}
(18)
(3R$ f RiiRjj)
(El
(26)
EZ6
and E{sign(Xi)}
= 0
also : E{6 (Xi) 6(Xj)> m =
m
.l 1): S(XJS(Xi)p(XiXi> dXi dXj
ss
El7 =
-cc -cc
EHilXilXjlX~lI
= P(Xi = 0, Xj = 0) = 1/[2?r(RiiRjj)1’2(1 -12)1’2] (19) where :
+ 2aiajE{IXzXtI}
+~fE{XiIXilX~lXjl)
Y = Rij/(RiiRjj)lJ2 Hence, substituting yields : El1
=-
RiiRij
(20)
successively from equations
[(4r2+ 2)arc sinr+
(19) to (16)
6~(l-r~)l’~]
(21)
71 a
+a~E{X~IXilXjlXjl)
2Ri”iRjj 115
2R~iRjj = [(12r2+3)arcsinr+r(l-r2)“2
(29)
R?!2R?!2 I1 [6r arc sinr(2r2+ =L
E{IX;X/l}
3)
71
+ (1 - r2)1’2(22r2 f S)] arc sin r(l+
(30)
6r’) E(XilXjlX~lXjl}
77
+ r( 1 - r2)1’2(8 1 + 28r2 - 2r4)}
(22)
The remaining expectations involve more than two variables and must, therefore, be reduced, if possible, to expressions involving only two variables, by substitution using Appendix I, to enable the application of Price’s theorem as applied above. (Note: in its general form,6 Price’s theorem covers multi-variate expectations of any order. However, attempts to apply this directly to expectations in three and four variables result in third and fourth order differential equations, whose solutions proved to be more difficult to obtain than the solutions to the bivariate problem which results from the above substitution).
is obtained by comparison
with equation
(29). Expectation El8 requires an expression for the conditional fourth moment, equation (A1.13): EZ8 = E(XilXilXjlXjlX,‘> =E{XilXjlXjlXjlE{X~IXiXj}}
=~3[R~~-(~~R~~+~jRj~)12~~~~l~~l~jlXjl~ +6[Rkk-(aiRik+ajRjk)][aiZE{Xi3IXilXjlXjl} + 2aiajE(IXi3Xi3I}+ai2E{XilXilXi”IXjl}] +U~E’E(Xi5IXiIXjlXjl}+4ai3aiE{IXi5Xil} +Qi~i”EtlXi”Xi”J}+6ai2ai”EcXi”lXilXi”IXjl}
EZ3 = E{X,ZXjXk> = E{XfXjE{XkIXiXj)I
and using equation
~J.~~l~il~jl~jlI
x (2r2 + 13)]
= E{X:lXilXjlXjl}
_
is El1 and applying Price’s theorem:
where E{XilXilXjlXjl}
‘II
result previously obtained by Borgman. Following the same procedure: El2
(28)
+ai4E{XiIXiIXisIXjl}}
(31)
(Al .8):
E{Xk IXiXj> = aiXi + UjXj
(23)
where : ai = (RjjRik - RijRjk)l(RiiRjj
- R$)
ai = (RiiRjk - RijRik)/(RiiRjj
- R$)
where E{XilXilX-lX.l} is EZI, E{XflXilXjlXjl> is given by equation (29), E{lX/‘X,?]} is given by equation (30), E{X: IXjlXjlXjl} is El2 and applying Price’s theorem: E{X~lXilX/lXjl1 2R?.R?. = --K! [3 arc sin r(8r4 + 24r2 + 3) 71
Hence : El3 = E{XFXjXJ
= ai(E5)
+ aj(E4)
(24)
+ r(l
Appl.
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r2)1’2(50r2
Math.
Modelling,
+
55)]
1983,
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Vol. 7, October
319
Expected
value analysis R. Burrows E21 = E(XilXilXjlXjlXkXl)
and
= biE{lX:lXjlXjlXk>
E{IXiSX$ _
+bkE{XiIXiIXjIXjIX,$)
2R?!2R?!2 11
II
[lSrarc
sinr(4r2+
3)
The final expectation first two expectations expanded as follows:
77 +5(1--2)“2(4r2+
17)-3(1--2)3’2(2y2+23)]
The remaining expectations, El9 to E23, involve four variables and, therefore, require a double substitution using equation (A1.7). So far, for conditional statistics, coefficients ai and ai have been used for the description of moments of Xk conditional on Xi and Xi. In the following coefficients bi, bi and bk are required for description of the moments of X, conditional on Xi, Xi, Xk and, with reference to Section 3, Appendix 1, are obtained as solution to : [RI@) = {Ro,)
(34)
where [R] is the matrix of cross-covariances of variables Xi, Xi, Xk;{b)=(bibjbk)T; and {R(l))=(RilRjrRkl)T EZ9 = E(XiXjXkXl>
=
aiE{X~IXilXjlXjl}
+ ajE{IX:X;l)
The expectations on the right-hand-side equations (29) and (30): E22
being given by
=E{XilXilXjlXjlX,IXklX,}
=biE~lXflXjlX~lX~IX~II + bjE{XiIXilIXi3IXkIXkI} + bkE{XiIXiIXjIXjIIXZI)
(44)
In order to expand the expectations on the right-hand-side it is necessary to determine the conditional expectations Of the form E{XklXkllXiXj} or alternatively E{IXiIIXiXj). It can be shown that the former expectation may be expressed as:
= E{X,X,X,E(X,lX,X,X,>>
= biE(XfXiXk)
(43)
on the right-hand-side is El 7 and the are of the same form and may be
E{lx?lxjlxjlxk>
(33)
E(XkIXkIIXiXjP
+ biE{XiXfXkI
+ bkE(XiXiX$ where expectations
+ bjE(XiIXiIIXTIXkI
= [Rkk-(UiRik (35)
are of the form E13.
+ fZjRjk)]
l
x [( 1+ a”)( 1 - 2Q(a))
+ 2CLZ(Cl)]
(45)
where:
E20 = E(XilXilXjXkXl) = biE{IX~lX~X~} + biE(XilXilX/X,I + bkE{XilXilXjXi}
1 Z(o) = 3;
(36)
where :
ff2 I
and m
EtlX:lXi&, =~iE{XflXilXj)
+ ajE(IXflX/I
(37)
s cy
=aiE(IXi3lXi2}+UjE{XilXiIXi3}
(38)
E{XilXiIXjXz> = E{XjIXilXk2E{XjIXiXk)) = c,E{lX,3lX;}
+ c,E{XjlXjlX:)
(39)
with ci and ck the solutions to: (40)
where [R] is the matrix of covariances of variables Xi, Xk: {cl = (ci, ck)
T
and
Clearly, the presence of functions of the form of Z(a) and Q(s2) prevent any further expansion of equation (45) along the lines of the method adopted so far. Similar difficulties are also likely to be experienced in an expansion Of E(lX:llXiXj). E23
=E(XjIXiIXjIXjIXkIXkIX,lx,l)
Expansion
of this requires the conditional expectation, which will have a similar form to equation (45) and direct solution is, therefore, also prevented.
E(XllXlllXiXjXk}
{R(j)> = (RijRjk)T
The expectations on the right-hand-side (39) are of the form: E{XilXjlX;}
Z(X) dX
Q(a)=
E{XjlXjlXi”X,,
of equations
(37)-
Numerical computation (E22) =
= El0
JJD
of E22 and E23
{xilxilxjlxjlxklxklx~}
-m
and E{lX:1Xf}
4R!!2 = -J& (3R$ f RiiRjj)
X p(xjxjxkx,)
(E23) = 16R?!2R..
E{X,31XilXi> =
Appl.
Math.
&
Modelling,
II
1983,
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Vol.
dXi dXi dXk dX1
(41)
and
320
1 exp -2 (
7, October
IIJ -Cc
{X,lXilXjlXjlXklX,lX,Ix,l)
XP(XiXjXkXl)
dXi dXj dXk dXl
Expected
These are plotted in Figure I and it is evident that the linear approximation offers a reasonable solution only to about two standard deviations whilst the cubic is very accurate to more than three and the quintic is good to about four standard deviations. Using substitution approximations to E22 and E23 may, therefore, be obtained as follows.
and for mean-zero Gaussian variables:
where {X}r = (X,XjXkXl):
‘=
Rii
Rii
Rik
Ril
Rii
Rii
Rjk
Rjl
Rik
Rjk
Rkk
R,
Ril
Rjl
Rkl
RI1
‘Linear’approximations
Substituting
for E22 and E23
for X,lX,l
from equation
(46):
E22L =
In previous applications of the model these equations have been solved2 by numerical integration over the fourdimensional Gaussian probability spaces using constant step rectangular rule summations. For each cycle of integration the relevant conditional pdf is Gaussian in form. Therefore, to centre the computations over the significant region of this probability mass, the integration was performed over a range of k5 standard deviations of each conditional distribution about its conditional mean value, these statistics being obtained from equations (Al .S) and (Al .lO). Table 1 illustrates the sensitivity of the numerical integrations to the number of steps chosen for the computation of E23. Use of 35 steps in computation of the moments of response from equations (4) and (5) was restricted to systems of up to only four load components whilst reduction to 11 steps enabled application of the model for systems of up to 12 load components with a loss of accuracy of only about 0.1%. This later condition was therefore adopted for general use.
Approximation
value analysis R. Burrows
(49)
for XklXkl and XllXll from equation (46):
Substituting
8 E23L = - R,!$R/1/2(E21) 71
IE Best
cubic
fit \/
lf
1;
1c
for E22 and E23
An alternative approach to the use of numerical integration for the solution of expectations E22 and E23 is to employ a polynomial approximation for some of the XIX1 terms enabling evaluation in the manner applied for the other expectations. Such approximations, resulting from minimization of the ‘mean-square’ error in the statistical sense, are given by Borgman as follows:
E
6
8
XjlXjl =
J
--Ri’i/‘Xi
(Linear approx.)
71
(Cubic approx.)
(47) v
1
I
I
I
I
I
z
3
4
5
6
7
X c
(Quintic approx.)
Table 7
Sensitivity
of numerically
No. steps = N
computed 7
(48)
Figure
7
Polynomial
approximation
to XIX1
value of E23 to the number of data points considered 9
11
21
31
35
Total probability N
.) dXj..
z:zzz,:~(. 1 (E23)N o,,
lo/,’
. dX/
1.00658
95.63
1.00003
100.36
1 .ooooo
1 .ooooo
100.04
100.10
Appl.
Math.
Modelling,
0.999999
100.00
1983,
0.999999
99.99
Vol. 7, October
321
Expected value analysis ‘Cubic ‘approximation
Substituting
R. Burrows
for E22
for X,/X,)
X E~XilxilxjIxjIx~x~>I
from equation
(47): where the second expectation (52).
1
E22c =
+3R,+p X E{X~lXilXjlXjlX~X~>
(51)
1
and using equation
(Al .7):
E{XilX~lXjlXjlX~X[>=
E(XilXilXj
= biE~lX~IXjIXjIX~}+
‘Cubic ‘approximation Substituting (47):
is of the form of equation
for E23
for both X,/X,1
and X,lX,l from equation
IXj IX; E{XrIXiXjX,>>
bjE{XilXillX~lX~)
+ b~E{XilX~lXjlXjlX~4)
X E(XiIX~IXjIXjIX~XfJ
(52)
where the b’s are defined in equation (34); E{XilXilXjlXjlX,$) is El8 and the remaining expectations are of the same form, which may be expanded as follows: +
~~l~,3l~jl~jl~~> = 3a:,aiE{Xi”IXjlXjlXjl>+ +a~E(XfIXilXjlXjl}+
3&aiE{IX~X~I}
(53)
where a,& = Rkk - (ajRjk + ajRjk) from equation (Al .lO), and the expectations of the r.h.s. have been derived previously for E18. ‘Linear/cubic ‘approximation for E23 Substituting for Xk IXk 1from equation XllXrl from equation (47):
Tab/e 2
Typical
results from approximations
9Rg;Rfi’”
E~X~lX~lXjl~jl~~~~>]
(55)
where the middle two expectations are of the form of equation (52) and E{XilXilXjlXilX~Xr”> can be expanded using expressions for the conditional moments. However, this expansion yields some expectations the solution of which have not been ascertained previously and which would demand considerable effort.2 Two procedures using the above results have been examined previously,3 one including the linear approximations for E22 and E23 and the other including the cubic approximation for E22 and the linear/cubic approximation for E23. Some typical results from these approaches are given in Table 2. It is seen that the cubic approximation for E22 offers a considerable improvement in accuracy over the linear approximator and for many terms is in very
3fZffZjE{IXfX~()
+3ajai”E{Xi”iXilXi”IXil)+ai3E{IXi3X~I)
1
(46) and for
of ,622 and 623
Response Y=F,+F,tF,+F, where: F;=X~~-~IX~~-II+X~; = force on unit length member i Method
of computation Num.
Expectation
10.71
E23 E22 i c d e f
XlIX,IX,
X,IX,IX,IX,I
XlIX,IX,
X,IX,IX,IX,I
XlIX,IX,
x,Ix,Ix,Ix,I
X,IX,IX, X,lX,IX, X,IX,IX,
x,Ix,Ix,Ix,I X,IX,IX,IX,I x,Ix,Ix,Ix,I
X,IX,IX, X,IX,IX, X,lX,IX,
x,Ix,Ix,Ix,I X,IX,IX,IX,I
X,IX,IX, X,IX,IX, X,IX,IX,
X,IX,IX,IX,I x,Ix,Ix,Ix,I X,IX,IX,IX,I
-7.181 - 1.760 1 .J80
3.297 -1.875 -8.353
X,IX,IX,IX,I
Appl.
Math.
Modelling,
1983,
Vol.
7, October
Linear approx.
Linear/cubic approx.
x
lo9
7.453
7.197
-5.788 -1.818 6.139
-7.173 - 1.745 2.420
10.36
3.183 - 1.856 -8.372
x109 x109
-1.937 -6.527
x109
x109
X108 X108
2.297 1.812 3.034
1.358 1.467 4.449
2.393 1.823 -1.758
XlO’O XlO’O X106
7.784 11.88 3.341
-11.97 8.787 1.977
9.198 11.99 3.485
x10* x109 x10’”
3.238 -7.235 5.009 - 14.63
2.743 -7.468 3.790 -9.319
3.237 -7.210 5.052 - 14.67
x1o’o x109 x109 x109
4.1913
E(Y4)
322
int. (15 steps)
4.1097
4.1881
x 10L3
Expected
close agreement with the numerically computed values. However, for certain terms, notably (c), (i) and (j) in the table, the error is substantial. The reason for this behaviour is not clear but, fortunately, it appears to apply to the expectations with lower numerical values, reflecting a relatively low degree of mutual correlation, and hence the their effect on the final moment E(Y4} is small. The approximators for E23 are evidently deficient, demonstrating the need for a better approximator. Alternative linear/cubic approximator
PWJXI --AIX)~I
[I
aA
dX
--m
1
= 0
yielding AI = E{IX3(}/E{X2} which, gives the result in equation (46). There is no reason to suppose that this value of AI should minimize the errors in the application of a linear approximation of one of the variables in E22 and E23. For example, in E23 an alternative linear approximator AZ3 might be considered as that which again minimizes the ‘mean-square’ error:
&
[E{X,?XtX;(XtIX,(
E23 = E(Xf)
-A23Xt)2>]
= 105Rfi
using equation
(Al.l):
using equation
(Al .2). Similarly:
E23Lc = 63.8R:i
1
=-a m(XIX1 -A,X)2p(X)
i.e. consider Xi = Xi E Xk E Xl
then:
for E23
The polynomial expressions considered so far represent the estimators which minimize the ‘mean-square’ error in a statistical sense. The linear estimator AI, is the solution to: ;
Complete dependence:
value analysis R. Burrows
= 0
23
and E23ALc = Azs(E22~)
= 152RA
For further comparison, a complete cubic approximation, using equation (47) for all four variables, would yield a value of 164.4R:j under these conditions. This approximation, and its more precise counterpart E23~ given in an earlier section, have not been derived explicitly since they involve expectations of high order and consequently solution requires considerable effort both in algebraic expansion of the expectations and in the multi-stage integration associated with Price’s theorem. This degree of complexity, as illustrated in a partial derivation of E23c presented in reference 2, is likely to be comparable with that experienced in the derivation of (A,,)summarized in Appendix 2. From the above results, it would appear that no single approximator is likely to yield satisfactory estimates of E23. However, an ‘optimum’ linear/cubic approximator, combining with equal weighting a linear approximation equivalent to A, above with the alternative linear approximation AZ3:
hence :
A
_Erx,“xj”X:lx:l,
(57)
23 - E{X;X;XzX;} and E~~ALC
(56)
=A23@%)
The solution for A23 is summarized Comparison of the approximators
in Appendix 2. for E23
Consider firstly the two extreme conditions dependence between the variables:
of mutual
Zero correlation: E23 ‘E{XjlXjl}E{XilXjl}E(XI,IX,I)E{X~IXtI)
a condition also satisfied by all the approximators reference to equation (Al .4).
Tab/e 3
Percentage errors in approximators
=O
with
would yield an estimate in error by less than 3% compared with about 40% for the other procedures under these conditions. The above approximations have been compared with values of E23 computed numerically, using 11 steps, for more general conditions of mutual dependence between the variables. The results are summarized in Table 3, in terms of the mean and variance of the errors from application to systems of seven load components which produce sets of 3.5 different combinations of expectations of the form of E23. Results (b) correspond to more highly correlated sets of variables than results (a). It is clear from the table that the ‘optimum’ linear/cubic approximator produces the best estimates of E23. This ‘closeness of agreement’ increases as the degree of mutual correlation
for E23
Approximation
Error (%X 10-Y
Linear eqn (50)
Linear/cubic eqn (54)
(a)
Mean Variance
-0.379 0.064
-0.227 0.027
(b)
Mean Variance
-0.573 0.00070
-0.395 0.00013
Alternative linear/cubic eqn (56)
Optimum linear/cubic eqn (57)
Full cubic (numerical computation)
+0.340 0.037
+ 0.056 0.016
+0.131 0.011
+ 0.429 0.00007
+0.017 0.00002
+ 0.271 0.00148
Appl.
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323
Expected
value analysis R. Burrows
between the variables increases and it is under such conditions that the expression E23 has most influence on the moment of response, E{Y4}, through equation (5). Observation has shown that for conditions of very low correlation all approximators show poor performance and the ‘optimized’ linear/cubic approximator, or more strictly its component A as, is the least stable. Nevertheless, these effects exert little influence on the statistics of response. In applications (a) and (b) from Table 3 the errors carried through by the approximations to E{ Y4} are respectively 0.7% and 0.2%.
Borgman, L. E. ‘Statistical models for ocean waves and wave forces’. In ‘Advances in hydroscience’, Academic Press, London, 1972, ~018,139 Borgman, L. E. ‘Ocean wave simulation for engineering design’, J. Waterways Harbors Div. A.S.C.E. 1969,95 (WW4), 557
Appendix 1: Statistical properties of Gaussian random variables used in evaluation of El-E23 (1) Standard properties7 of mean-zero variables: (i)
E{lxln}=1.3
Gaussian random
. ..(r~-l)R~~~ n = 2K, K = integer
Conclusions The expected value analysis for the computation of the quasi-static response of offshore structures has been presented. Application of the model was formerly restricted to systems of about 11 load components when using a 30 min run time limit on an ICL 1906s computer. This was caused, in part, by a requirement for numerical integration associated with two expectations, termed herein E22 and E23. Approximations for these expectations have been developed and investigated. Adoption of a ‘cubic’ approximator for E22 and an ‘optimum’ linear/cubic approximator for E23 provides an acceptable degree of accuracy in the computations and relaxes the restrictions on implementation of the model, enabling analysis of systems of about 30 load components.
(ii)
2K. K! . R!jL’
E(lx I”} = m).
(iii)
E{x”} = 0
(iv)
E{x”lxl}
(Al.l)
n =2K+l,K=integer
(A1.2)
n =
(A1.3)
=E{xlx”l}
odd integer
=0 n = odd integer
(Al .4)
(2) Bivariate moments are solved by expansion using Price’s theorem6 as demonstrated by Papoulis.’ Consider two arbitrary functionsgr and g2 of Gaussian random variables, (rvs), x1 and x2 respectively. Price’s theorem gives: aE{g,
.g2>
=E
f&g i
ap
(Al S) 1
21
where : P=
Acknowledgements The work involved in the development of the probabilistic model for quasi-static response was supported by the UK Department of Energy through its Offshore Structures Fluid Loading Advisory Group (OSFLAG), Project No. 5 entitled ‘Prediction of long-term wave loading on offshore structures’ (1974-78). The subsequent work on the development of the expectation approximators forms part of research contract NW/L1 .l, entitled ‘Probabilisticspectral design method for wave loading on space frames’ (1978-83) supported by S.E.R.C. through its Marine Technology Directorate. In relation to this latter study, the author wishes to acknowledge the assistance of Mr K. H. Yim, a former research worker,
Burrows, R. ‘Quasi-static response of offshore structures using probabilistic methods’, Appl. Math. Modelling 1911, 1, 32.5 Burrows, R. ‘Wave loading on offshore structur&: a probabilistic approach’, PhD Thesis, University of Liverpool, 1982 Burrows, R. ‘Probabilistic description of the response of offshore structures to random wave loading’. In ‘Mechanics of wave-induced forces on cylinders’ (ed. Shaw, T. L.), Pitman, 1979,511 Holmes, P., Ticket& R. G. and Burrows, R. ‘Prediction of longterm wave loading on offshore structures’, University of Liverpool, Dept. of Civil Engineering, OSFLAG 5, Report B, 1978 Tickell, R. G. ‘Continuous random wave loading on structural members’, Struct. Erzg. 1977,55 (5), 209 Price, R. ‘A useful theorem for non-linear devices having Gaussian inputs’, IRE, PGIT, lT4,1958 Papoulis, A. ‘Probability, random variables and stochastic processes’, McGraw-Hill, London, 1965
Appl.
-
E{xdE{x21=
when xi and x2 are mean-zero. grated to give:
E{g,.g,I=jE
Math.
Modelling,
1983,
Vol.
7, October
Rx,,*
Equation
(Al. 5) can be inte-
[~~]dp+EighE~gz)
(Al.6)
0
(3) Expectations involving more than two random variables are, where possible, reduced to expressions involving only two random variables, for application of Price’s theorem using the following properties:7 E{grg:! . . . g,> = E{glg,
. . .gn_-1.E{gnlx+2..
where gjis an arbitrary expectation:
References
324
E{x1x2)
E{g,
1x1~2
. . . x,
function
.X,-I>>
(Al.7)
of Xi and the conditional
-11
is the expected value of g, given the values of xi to X, -1. (4) Some conditional moments: conditional mean, m,: (A1.8)
m,=E{xoIxI...x,)=alxl+...+a,x, where aj’s are given by the solution to: [RI&z>
=
(Al .9)
{Rd
in which: [R]
is the matrix of cross-covariances 1 ton
(Q}
{R,)
iS
the
V&Or
(QI,
~2,
. . . , a,)
of variables
T
is the vector (RoI, Roz, _. . , R&JT
Expected
For example, when II = 2:
E{XfX~X~>
a1 =
&,Ro~
a2
(RllRoz--R21Rol)I’(R~~R22-R~zRzl)
=
value analysis R. Burrows
-R12Roz)I(R11R22-R~zRzl)
Conditional variance or second central moment,
a::
us = E{(xe - m,)21xr . . . XJ . ..+a.Ron)
=Re,-(aIRor+
(A1.lO)
Conditional second moment: E{x;lx,
. . . x,} = 13:+ m2c
= 304bE{Xi”xi”>+ 6u;b;E{Xi7Xi5}
(Al.ll)
+ 12u;brb2E(X:X;6
Similarly :2 E{xzlxr _. . x,} = m,[3oz + mf] E{x~lx,
. . .x,}
+ b${XigXi5} + 4b:b&{X;X$’
(A1.12)
= 30: + 6uzrnz + mt
+ 6b:b;E{Xi7Xi7} + 4b&E{X;X;}
(Al .13)
+ b$E{XiSXig}
and it can be shown that: .,x,}
J%x;tx,
= 15uzm,+
lOu~m~+rn~
E{x$lx, . . . x,} = 15u~+45u~mZ,+
(A1.14)
. . _x,} = 105uzm,+
15uzmt+mZ
(A1.16)
and
E{x$x,
. . x,} = 105u~+420u~m~+
Appendix
2: Derivation
(A1.17)
of A,,
With reference to equation (56): A23 = E{X~X~X~IX~I}/E{x~X~X~X:} Applying equation (Al .l l), Appendix tor, the following expansions result:
bl = (RiiRik - RijRjk)/(RiiRii
- R$)
b2 = (RiiRjk - RijRik)/(RiiRii
- R$)
u;=R
kk
-@l&k
+
b&k)
The expectations on the r.h.s. of equations (A2.3)-(A2.5) can be solved using (Al .6). They are of the form:
210u~m~
+ 28u2m6+ma c c c
(Al .9):
and
105u,4mz
+21u,Zm~+m~
(A2.5)
where the b’s are given by equation
(A1.15) E{xzlx,
+ 6u;b;EjXi5Xi7I
E{XfX;}
= 24R$ + 9RiiR,(8R$+
E(XfX/>
= 12OR$ + 6OOR~RiiRji+
E{XfXT}
= 360RiiR;
E{XfX,f}
= 5040RfiRz
+ 5040RfiRjiRB
E{XJXj}
= 2520RiiRs
+ 6300RfiRijRk
(A2.1)
1 to the denomina-
RiiRij) 225R$R$Rii
+ 540RfiRiiRi
+ 45RfiR$ + 3 15RfiR,$
+ 1575R$R$Rii E{XfXT}
= 120R$ + 5400R;RiiRij
+ 4050R;iR$Ri
+ 225R;iR,$
= u$!?{Xi4XjX~} + a:E{XfX!X$) + a$T{XfX~X~> + 2a,a,E{X;X/Xi}
+ atE{XfXfX;}
+ 2ala2E{XfX~X~} + 2a,a,E{X:XfXz}
(A2.2)
E{XjXT>
= 45360RfiR:i
+ 75600R;iRjjR$
+ 14175R?.R?.R” II
where the a’s are given by: E{XfXT}
= 20160RiiRg
11
+ 75600RfiRjjRt
+ 37800RfiRiRi
Considering now the numerator applying (Al .13):
+ 1575RgRs
in equation (A2.1) and
E{XfX,+X;lXfI}
and u2=R a
II - (aIRi
+ a2Rjl + a3Rkl)
= 3u;E(Xi4Xi41XfI}
The expectations on the r.h.s. of equation (A2.2) may be expanded using equation (A1.13). They are of the form:
= 3u${Xi4xi4} + 6u;b;E{Xi6Xi4} + 12u&b,E{X;X;}
+ 6b:b;E{XfX;} + b;E{X;xjs}
+6u~p~E{X~X~IX~l)+6u~p2~{X~X~X~~X~I} + 12u;plp2E{Xi5Xi5Xf()
E{xi’x!x;>
+ b;lE{X;x;)
+ 6u;p:E{XfX$Xf)}
+ 6u;b;E{Xi4Xi6}
+
12~;PIP&~xi5xi4x,Ix:I)
+
12a~p,p3E{Xi4Xi5X,IX~~}+p~E{X~X~~X~I}
+p~E{Xi4XilX:l}+p~E{Xi4Xi4XPIX:I}
+ 4b:b*E{xi7xi5>
+4p1~23E{Xi”Xi’IX:l}+4p1p~E{XiSXi4X131X~I}
+ 4bIb;E{Xi5xi7} (A2.3)
+4p~pzE{Xi’xi”IX~I}+4p:p3E{Xi’xiX~IX~I}
Appl.
Math.
Modelling,
1983,
Vol. 7, October
325
Expected value analysis R. Burrows +6q~q:E{Xi5X:lX~l,+q~E(Xi9X~lX:I}
+4q:q~E(Xi8X:IX:I}+6q~qZ,E{X~X~lX~1) +4qtq;E{X;X;lX:lI
+ q;E{X~X:lX~o (A2.11)
E(XfXfyX~
I)
= 3o;E{XjyXflj
+ 6u;q:E{xyJX13)}
+ 12o;q1qzwi9x,lx~l)
1
Rjk
=
2
t- 6u;q:E{Xi”X&YfI>
Rik
4q;E(X/21XfI}
+4q:qzE&%lX,3I~+
Rkl
6q:q:E~Xi’“X,“lXz?~
+4q,q~E~X~X~IX:I}+q~E{XiSXPIX:O
and
(A2.12)
0; =Rkk blRik + PzRjk + P3Rkl) The expectations on the r.h.s. of equation (A2.6) may be expanded using equations (A1.13)-(A1.15). They are of the form : E{X,?Xi’lX,“l> = 3o;E(Xi4lX/I>
Ecx,“x,“xPlx:l, = ~u~,E{X~~XPIX,~~} + 6u;q;E(XfXfIX?\} + 12~~qlqz~cXi”x:lX:l~
-t6a~q~E{X~X~IX:()+q~E{X,sXI”IX:I} + 6o&:E(X;(X:II
+ 120;q,qzE{X;X,lX:()
+4q:q,E{Xi7X15JX131}+6q~q~EIXi6XPIX:l}
+64q’:E{X4X:lX:I)
+4q~q,3E~XiSX:IX13l}+q~E(Xi4X18IX~I)
+q~E{X,slX:t>+4q:q,E:Xi7XlX:I,
(A2.13) E(Xi”xi4X,” )xf I>
+6q~q~E~X~X~IX~I}+49~q~E(Xi5X:lX:I~
(A2.7)
+q;E{X;X:IX:II
E{xpx;
Ix: 1t
= 3u;E{Xi5X13(X:l)
+ 6u;q;E’(Xi7X13 IX:l}
+ 12u;qIqzE{X~X;IX~l)
= 3o~E{X~IX:l)+6a~q:E(XSIX~l}
~6o~q~E{Xi5X:IX:~)+q~E(Xi9X:JX131~
+ 12o~4,q,E~X~x~IX:l~
+4q:q;E{Xi8XPIX13])+6q~q~E(Xi7X:IX:I)
+ 6a;q;E{X;X;IX:I)
+q;E{Xf”lX:l)
+4q1q~E(Xi6XPIX131}+q~E{XiSX:(X:I)
+4q:q~E{Xi”XlX~I}+6q:q~E(Xi8X12lX~I}
(A2.14)
+4q,q:E{Xi7X:IX~l~+q42E{Xi4YPlX:l)
(A2.8)
E(X;x~x~Ix,3l)
E{x;x;]x,“I> = lso;q,E{X~(X;I~
= 3@{Xi4X:lX,3(}+
6u~q~E’{X~X~IX~l~
+ 3ou~q,q;E{Xi”X:IX~I>
+ ~G~~~E{X~~XP(X:I)+~‘:E{X~~X:IX:I}
+ 10u~q~E{Xi7X,31X:I>+q:E‘(Xi121X,”I}
+ 6q:q;E~X~x~lX:l~
+ 5q;q~E{x~lX,IX,31}+ (A2.9)
+4qlq~E{X~X~IX:l)+q,4E(X~X~lX~I)
E{Xi”xi” 1x; I > = 15CJ;q~E{x~lx:l:
3Ou;q:q;E{X;X/IX:l>
+ ~OU&;E~X;~~X,~~)+
+ 124q*4~E{Xi5X:lX:l)
+ 4q:4&{Xi7X13lX:I~
+ 15a~q2E~Xi7X,IX~lI
10q:q~E{X~“x~lX:I~
+ lOqfq;E{X;X~lX;I>
+ 5q,q;E{Xi8X14lX:I1
+&‘{X%IX%
+ 15a;q,E{X~x~lX~l~
(A2.15)
E{Xi7Xi”X,jX~j> + 100~q~E{XjyX:O
+ 30qfq,o~EfXi’X,rXr”l>
= 3u4,E{X~XzIX:I> + Ga;q;E’{X~X~JX:J}
+ 30q&~EIx~X;rX:l, + IOa;q;E{XiSX:IX:I} +
+ 12u~qlq2E{XigX~lX:I)
fq:E{Xi’“lX,“I>
sq;2q&TXi9X,IX,3l>-t 10q~q~ECXi8X:lX:l~
+q;1E{Xi”XrlX:l)
+4q:q;E{X~“X,‘IX~I>+6q:q:E~X~X~lX~~~
+ loq~q;E{X~x~IXr”l~+ 5q&?Xfx.?lX:I1 (A2.10)
+q:E~Xi5X:lX~l~
+ 6a;q;E{xJX~(X~I}
+4qlq~E(XiSXPIX2”I)+q,4E{Xi7X:IX~l} (A2.16)
E{x,sxi”x,jx:lj = 34E{Xisx,~Xr”l}
+ 64q:E{Xi7XIIX131~
= lsu~E{xpIx,3~)
-t 12o~qIqZmi6X:IX:lI
326
Appl. Math. Modelling,
E{x~x/w:()
1983, Vol. 7, October
+ 45u4,q:E{x~lx:I~
Expected + 9Oc&lq,~{X~X~IX:I~ + 45a~q~E{X~X~~X13I}
+
+~4m&v + 15a~4~E‘{Xi’“lXr”l)
E{X;XI”IX:I)
E{xfIX:o
+ 150~(Y4~{Ix:(>+01~E{Ix19~} E{x,sx,lx~l>
= lSu~aE{IX:I}+
2oq;q;E{x~x~Ix:I)
+
E{X,4X:lX:l>
+ 154~q~E(X~X~~X~~}+6q~q~E{X~X~~X~~} (A2.17)
+&wi6XPIX:I~ E{Xi6Xi4X,2lX:I> = 3@{XfXf
= 15u~E{IX~~}+45u~~~~{IX~~}
15u;qf:E{X~X~IX~II
+ 4;E‘(Xi’21X,31} + 6q;q&{X/‘X~IX~I} + 154~4;E{Xi’“Xl”jX~I}+
lX,‘3j}
+ (Y4E{Ix;SI)
+90o~q:q:E{x~x~IX13I)
+
= ~u~,E{IX/~~} + 6u;a2E{
604&,~~&94&7>
+60u~q,q~E{X~X~lX~l~+
value analysis R. Burrows
~5~w$l~
3u4,E{IX;I)
+ 6u~a2E{Ix;~}
+(r4E{Ix~lI E{X/lX~l,
IX,3 I>+ 6u;q~E’(xi”x,
=
10&&{~X~~>
= 105u;E{IX;I}+
402u~cy*E{ 1X,51)
+ 21ou~a4E{~x~~)
IX,3 I}
124qIq,mi7X:Im
+28u2a6E{~X~~}+a8E{~X~‘~} CY E{Xi7x,lX:I)
+~u~~~E{X~X~IX~I}+~;~E~X~~X~IX~I)
105&3E{~x~~}
+ 2lu~a%{IX~I}
+4q:4;E{Xi9X:lX,‘I}+64~4~E{XisXPlX~I} E{Xi6X:IX:I}
+44~4~E(X~x~IX:l}+4~E(X~x~IX~I}
E{XiSX:IX:I)
E{xfx;xfIx:I>
+a%(
Ix:‘l)
= 15u~E{~X~~}+45u~(u*E{~X~~} + 15u~a4E{IX~~}+a6E{IX:‘I}
(A2.18)
= 15u~4rE‘{X~x~IX:l)+
= 105u~orE{IX:l~+
= lsu~olE{~x:~>+lou~clI3E(IxpI) +a%{IX:‘I)
15u~4,E{Xi5X:IX:II
E{Xi’“lX_?l, = 945u~“E{~X~~)+4725u~a2E{~X~~}
+ lOu~4:E{X~x~IX:I}+30u;f4q4,E~X~x~lX:I}
+ 315ou~(w4E{Ix:I~
+ 30u~qr4;E{x~x~IX:l)
+ 630u~&{IX~~)
+45u:a8E{IXI”I)
+ 10u~4:E{X~x~IX:I}+4:E‘(X~~x~IX:l} +cr’0E{Ix:31) + Sq;q;E{XigX:IX:
I> + 104:4,2E{X~x~IX~
I) E{X,gx,lX;l}
+ 104~4~E{Xi7X:IX~l}+54~4~E{X~x~IX:l)
E{X;x;xJX~I)
+ 36u~cy’E{IX,“I} +(w~E{IX,‘~I} E{Xi”x,“lX;l>
= 15cJ~4,E~X~x~IX:I)+ + 100~4:E~xigx,Ix:l>
15u4,42E{X;x~lX:I} + 30&4~4~E{x~x~IX:I}
+~~u~~~~E{IX:‘~}+~~E{IX:~I} E{Xi7X:IX:I)
+ 10u~4~E{X~x~IX:I}+4:E‘(X~~x~IX:I}
+ 104~423E{x~x~lX~I)+
= 105u~olE(~X~~}+105u~~3~{~x~~} +2lu~cu5E{IX~~~}+(W~E{IX~~~}
+ 104:4:E{Xi9X:IX~
I>
E{Xi6X14lX:o = 15u~E{~X~~}+45u~a~E{IX~I}
Sqrq;E{Xi7X:IX:I} (A2.20)
+452~{Xi6XPIX:I~
= 105u~E{~X~~}+420u~c2E{~X~~} + 210u~CI4E{~X~~>
+30u~4,4::E{Xi7X:~X~~}
+ 54;lq,E{X;“x:Ix~I}
1260u;a3E{IX~I}
+ 378u~c&{~X~~} (A2.19)
+4:E~Xi5X:IX:o
= 945u~cS{IX~I)+
+ 15u~a4~{~X~~~}+a~E{IX1’3~} E{X;X!lX;I)
where, in the above:
= 3u~E{~X~~}+6u~ar2E{~X~~} +a4E{Ix:r1}
41 = (Rl,Rij - RilRjl)/(RiiRll
- R;l)
42 = (RiiRjl -RilRij)/(RiiRll
- R;l)
and
E{X,5x:lX&?l> = l5u”,olE{~X,g~>+lou~(y3Eo + cYsE{Ix;sI) E{Xi”xplX:l>
= 3u~E{~X~~}+6u~a2E{~X,“~} + (Y4E(Ix;31}
The expectations on the r.h.s. of equations (A2.7)-(A2.20) may be expanded using equations (Al .13)-(A1.17), or alternatively using equation (Al .6). They are of the form: E{X?IXfl)
= 304,E(lX?l} + 6u:02E{ IX;I}
E{X;‘IX:()
= 10395u;%{Ix:I> + 623700;~(~~E{IXfI} + 51975&4E{IX:I> + 13860u~c~%{lX~l>
Appl. Math. Modelling,
1983,
Vol.
7, October
327
Expected
value
analysis
R. Burrows
E{Xi8xflX,3/} = 105u~E{JX~~}+402u~(u2E(~Xf’~}
+ 14850;ar8E{ IX;‘l}
+ 21Ou~cr4E{IX~‘I}
+ 66o~a’OE{~X:“I} +CY’%{1x,151> E{X;‘X~IX:l)
+28~~a~E{IX~~~}+a8E(IX:51)
= 10395u~001E{Ix:I) E{x,7x:lx:l)
+ 17325a~a3E{IX~~}
= 105u~olE{~x~~>+105u~~~~{~x~~~~ + 21u~a5E(IX:3~}
+ 6930u~c~%{~X~~}
E{XfxfIX:I)
+ 990cJ&7E{IX:11> + 55u~(u%(~X~3~} + cm{ E{X/“Xr”lX/I}
= 15u~E{IX~~}+45u~a~E{IX:~I)
Ix:51)
= 945u~“E{~X~~}+4725~~~2E{~X~~}
+15u~(Y4~{~x~3~>+a~E(Ix:5~) E{XiSX:IX:I)
where, in the above:
+45u~cy~E{Ix~3I}+(Y~~E{Ix~sI~ E{X:XjalXfI)
= 945u;olE(IX;I}
+ 126Ou~(~~E{IX,9~>
+ ~~~u$Y~E{IX,“I} + ~~u~(Y~E{IX;~~}+(Y~E{IX~~~}
328
Appl.
Math.
Modelling,
1983,
Vol.
7, October
= 15u4,cS{JX~‘~}+ lou;a3E{IX:31} + CY5E{Ix~5)}
+ 3150u~a4E{lx~l> + 630u~a%{~X~‘I}
+(w7_E{Ix:51}
(Y=-
Bil
u2cy=B..-_a.B. 21
II
41
and the expectations equation (Al .2).
on the r.h.s. can be solved using
University
of Liverpool,
PO Box 147, Liverpool
A probabilistic description of the quasi-static response of offshore structures to random wave loading has been developed recently in terms of the cross-covariance statistics of the mean-zero and Gaussian distributed particle kinematics at an array of locations beneath the water surface. Implementation of this model was initially restricted by the computational requirements associated with evaluation of certain high order expectations. However, approximations for these expectations have now been developed which enable some relaxation in these restrictions. The procedure employed makes use of standard properties of mean-zero Gaussian random variables and, in particular, application of Price’s theorem. The paper presents, for the first time, the complete description of the expected value analysis upon which the model is based. Key words: mathematical model, offshore structures, expected value analysis, quasi-static response
where:
Introduction The instantaneous wave loading exerted upon the submerged element i, of an offshore structure can be expressed in the form:‘!’ Fi=X2i-~IXzi_~l+Xzi
(1)
where XZi_,, XZi are linearly related to the instantaneous values of water particle velocity and acceleration, respectively, at location i and are functions also of member shape and fluid density. In real random sea conditions, the water particle kinematics can often be considered to behave as mean-zero Gaussian distributed processes. The probability density function (‘pdf) of wave load Fi is, therefore, a zero-memory nonlinear transformation of the bi-variate Gaussian process involving Xzi __Iand Xzi which may be expressed in general form as:
P(Y) =
&
f L
exp--(f(-$+-$))di~
2--m
0307-904X/83/05317-12/$03.00 0 1983 Butterworth &Co (Publishers)
1
Ltd
(2)
2
M2, M4 are the second and fourth order statistical moments
of y and the pdf is mean zero and symmetrical. In the case of wave load: y E Fi, M2 E E{Ff)
and
M4 = E{F;).
For offshore structures of the steel lattice type whose behaviour is quasi-static and linear, nodal response variables Y (internal stresses or deformation) may be expressed in terms of the elemental forces: Y = SIF, -t’ S,F,
. . . S,,F,,
(3) where S is a flexibility coefficient. It has been shown1-5 that in such circumstances the pdf of Y is of the form of equation (2), although no formal proof exists. Consequently, the probability structure of Y
Appl. Math. Modelling,
1983,
Vol. 7, October
317
Expected
value analysis R. Burrows
Solution
may be defined in terms of its second and fourth order moments. The second moment of Y is given by:
El to E23
of expectations
From equation
(Al .l), Appendix
El = E{XF} = 3R;i j=l
i=l
Similarly, the fourth moment
1:
(6)
..
and
j=i+l
of Y is:
E2 = E{Xi } = 1OSRg
(7)
The b&variate moment is the basic result: E(Y2> = i
$E{Ff}
+ 4 i i=l
i=l
i
$SiE(FfFi}
E3 = E{XiXj}
j=l
ifi n-l
= Rij
Other bi-variate moments require application of Price’s theorem6T7 (equation (Al .6), Appendix 1) for solution:
n
+ 6 c c SfStE(FfF;) i=l j=i+l n
n-1
n
+ 12 C
1
C
i=l
j=l
k=j+l kfi
j?i n-3 +
i=l
Rij
E4 = E{Xi2xi2> = 4
1
C
SiSjSkSlE(FiFjFkFl)
j=i+ll=k+l
2R; + RiiRji
(9)
and (5)
Evaluation of the joint moments of force given on the right-hand-side of these equations may be achieved by expansion of the arguments in terms of the X variables using equation (1). The resulting expectations of X must then be determined in terms of the cross-covariances of X, Rij, which can tIlen be computed directly from the surface elevation spectrum for the sea state.2 Following this procedure it can be easily shown that expressions for a total of 23 expectations are required as follows:
(10)
.
ES = E{XfXj> = 3RiiRij
El = E(X;}
dRii + E{Xi2> E{Xi2}
0
= n
E{XiXj} s
SfSjSkEf.FfFjFk)
n-2
24 1
(8)
E6 = E{X:Xf}
= 3Rii(4Rfj
+ RiiRjj)
E7 = E(X?X/}
= 24R:j + 9RiiRjj(8R~
(11) + RiiRjj)
Moments E8 to El0 are solved using Appendix E8 = E{XilXilXj> = 4 G
(12)
1.
= E{XilXilE{Xj(Xi)}
R;;2Rij
(13) 96 Fm Rf/‘Rij
E9 = E{XfJXilXj}=
(14)
E2 = E{X,S) 4
Es = E{XJj)
EIU = E(XilXilXT}
E4 = E{Xizxi’}
Rij(3RiiRjj
=G
+ Rzj)/R;/2 (15)
Es = E{X+?Xj)
Solution of the remaining b&variate moments is more complex than for the above and it is, therefore, instructive to develop the solution for El1 in detail here. Applying Price’s theorem; equation (Al .6):
E6 = E{X?X;) ET = E{X;X,+>
Rij
El1 =E{XiIXilXjlXjl}=
4
s
E{lXiXjl>dRij
0
+E{xilxil>Ef.XjIXjI)
(16)
and from the mean-zero symmetrical E(XilXil)
= E{XjlXjl)
nature of Xi and Xi:
=0
Applying Price’s theorem to the argument in equation (16):
of the integral
Rij E(lXiXjl}
=
E{sign(Xi)
sign(Xj)> dRij
i
0
+E( IXil> E{ IXjl)
(17)
and E{ IXil}, E{ IXjl) are obtained from equation (Al .2), also (a/aXi){sign(Xi)} = 2F(Xi) where F(.) is the dirac delta function. Thus applying Price’s theorem to the argu-
318
Appl. Math. Modelling,
1983,
Vol.
7, October
Expected
El4 = E{XfXjXJ
sign(Xj)>
Rij =
analysis
R.
Burrows
Similarly:
ment in the integral: E{sign(Xi)
value
El5
= ISaiRfiRij + Uj(E6)
=E{XilXilX/X,}
=gRH2
E{6(Xi) S(Xj)} dRij
4 r -0
X
+ E{Sign(Xi)} E{sign(Xj)}
(18)
(3R$ f RiiRjj)
(El
(26)
EZ6
and E{sign(Xi)}
= 0
also : E{6 (Xi) 6(Xj)> m =
m
.l 1): S(XJS(Xi)p(XiXi> dXi dXj
ss
El7 =
-cc -cc
EHilXilXjlX~lI
= P(Xi = 0, Xj = 0) = 1/[2?r(RiiRjj)1’2(1 -12)1’2] (19) where :
+ 2aiajE{IXzXtI}
+~fE{XiIXilX~lXjl)
Y = Rij/(RiiRjj)lJ2 Hence, substituting yields : El1
=-
RiiRij
(20)
successively from equations
[(4r2+ 2)arc sinr+
(19) to (16)
6~(l-r~)l’~]
(21)
71 a
+a~E{X~IXilXjlXjl)
2Ri”iRjj 115
2R~iRjj = [(12r2+3)arcsinr+r(l-r2)“2
(29)
R?!2R?!2 I1 [6r arc sinr(2r2+ =L
E{IX;X/l}
3)
71
+ (1 - r2)1’2(22r2 f S)] arc sin r(l+
(30)
6r’) E(XilXjlX~lXjl}
77
+ r( 1 - r2)1’2(8 1 + 28r2 - 2r4)}
(22)
The remaining expectations involve more than two variables and must, therefore, be reduced, if possible, to expressions involving only two variables, by substitution using Appendix I, to enable the application of Price’s theorem as applied above. (Note: in its general form,6 Price’s theorem covers multi-variate expectations of any order. However, attempts to apply this directly to expectations in three and four variables result in third and fourth order differential equations, whose solutions proved to be more difficult to obtain than the solutions to the bivariate problem which results from the above substitution).
is obtained by comparison
with equation
(29). Expectation El8 requires an expression for the conditional fourth moment, equation (A1.13): EZ8 = E(XilXilXjlXjlX,‘> =E{XilXjlXjlXjlE{X~IXiXj}}
=~3[R~~-(~~R~~+~jRj~)12~~~~l~~l~jlXjl~ +6[Rkk-(aiRik+ajRjk)][aiZE{Xi3IXilXjlXjl} + 2aiajE(IXi3Xi3I}+ai2E{XilXilXi”IXjl}] +U~E’E(Xi5IXiIXjlXjl}+4ai3aiE{IXi5Xil} +Qi~i”EtlXi”Xi”J}+6ai2ai”EcXi”lXilXi”IXjl}
EZ3 = E{X,ZXjXk> = E{XfXjE{XkIXiXj)I
and using equation
~J.~~l~il~jl~jlI
x (2r2 + 13)]
= E{X:lXilXjlXjl}
_
is El1 and applying Price’s theorem:
where E{XilXilXjlXjl}
‘II
result previously obtained by Borgman. Following the same procedure: El2
(28)
+ai4E{XiIXiIXisIXjl}}
(31)
(Al .8):
E{Xk IXiXj> = aiXi + UjXj
(23)
where : ai = (RjjRik - RijRjk)l(RiiRjj
- R$)
ai = (RiiRjk - RijRik)/(RiiRjj
- R$)
where E{XilXilX-lX.l} is EZI, E{XflXilXjlXjl> is given by equation (29), E{lX/‘X,?]} is given by equation (30), E{X: IXjlXjlXjl} is El2 and applying Price’s theorem: E{X~lXilX/lXjl1 2R?.R?. = --K! [3 arc sin r(8r4 + 24r2 + 3) 71
Hence : El3 = E{XFXjXJ
= ai(E5)
+ aj(E4)
(24)
+ r(l
Appl.
-
r2)1’2(50r2
Math.
Modelling,
+
55)]
1983,
(32)
Vol. 7, October
319
Expected
value analysis R. Burrows E21 = E(XilXilXjlXjlXkXl)
and
= biE{lX:lXjlXjlXk>
E{IXiSX$ _
+bkE{XiIXiIXjIXjIX,$)
2R?!2R?!2 11
II
[lSrarc
sinr(4r2+
3)
The final expectation first two expectations expanded as follows:
77 +5(1--2)“2(4r2+
17)-3(1--2)3’2(2y2+23)]
The remaining expectations, El9 to E23, involve four variables and, therefore, require a double substitution using equation (A1.7). So far, for conditional statistics, coefficients ai and ai have been used for the description of moments of Xk conditional on Xi and Xi. In the following coefficients bi, bi and bk are required for description of the moments of X, conditional on Xi, Xi, Xk and, with reference to Section 3, Appendix 1, are obtained as solution to : [RI@) = {Ro,)
(34)
where [R] is the matrix of cross-covariances of variables Xi, Xi, Xk;{b)=(bibjbk)T; and {R(l))=(RilRjrRkl)T EZ9 = E(XiXjXkXl>
=
aiE{X~IXilXjlXjl}
+ ajE{IX:X;l)
The expectations on the right-hand-side equations (29) and (30): E22
being given by
=E{XilXilXjlXjlX,IXklX,}
=biE~lXflXjlX~lX~IX~II + bjE{XiIXilIXi3IXkIXkI} + bkE{XiIXiIXjIXjIIXZI)
(44)
In order to expand the expectations on the right-hand-side it is necessary to determine the conditional expectations Of the form E{XklXkllXiXj} or alternatively E{IXiIIXiXj). It can be shown that the former expectation may be expressed as:
= E{X,X,X,E(X,lX,X,X,>>
= biE(XfXiXk)
(43)
on the right-hand-side is El 7 and the are of the same form and may be
E{lx?lxjlxjlxk>
(33)
E(XkIXkIIXiXjP
+ biE{XiXfXkI
+ bkE(XiXiX$ where expectations
+ bjE(XiIXiIIXTIXkI
= [Rkk-(UiRik (35)
are of the form E13.
+ fZjRjk)]
l
x [( 1+ a”)( 1 - 2Q(a))
+ 2CLZ(Cl)]
(45)
where:
E20 = E(XilXilXjXkXl) = biE{IX~lX~X~} + biE(XilXilX/X,I + bkE{XilXilXjXi}
1 Z(o) = 3;
(36)
where :
ff2 I
and m
EtlX:lXi&, =~iE{XflXilXj)
+ ajE(IXflX/I
(37)
s cy
=aiE(IXi3lXi2}+UjE{XilXiIXi3}
(38)
E{XilXiIXjXz> = E{XjIXilXk2E{XjIXiXk)) = c,E{lX,3lX;}
+ c,E{XjlXjlX:)
(39)
with ci and ck the solutions to: (40)
where [R] is the matrix of covariances of variables Xi, Xk: {cl = (ci, ck)
T
and
Clearly, the presence of functions of the form of Z(a) and Q(s2) prevent any further expansion of equation (45) along the lines of the method adopted so far. Similar difficulties are also likely to be experienced in an expansion Of E(lX:llXiXj). E23
=E(XjIXiIXjIXjIXkIXkIX,lx,l)
Expansion
of this requires the conditional expectation, which will have a similar form to equation (45) and direct solution is, therefore, also prevented.
E(XllXlllXiXjXk}
{R(j)> = (RijRjk)T
The expectations on the right-hand-side (39) are of the form: E{XilXjlX;}
Z(X) dX
Q(a)=
E{XjlXjlXi”X,,
of equations
(37)-
Numerical computation (E22) =
= El0
JJD
of E22 and E23
{xilxilxjlxjlxklxklx~}
-m
and E{lX:1Xf}
4R!!2 = -J& (3R$ f RiiRjj)
X p(xjxjxkx,)
(E23) = 16R?!2R..
E{X,31XilXi> =
Appl.
Math.
&
Modelling,
II
1983,
(42)
Vol.
dXi dXi dXk dX1
(41)
and
320
1 exp -2 (
7, October
IIJ -Cc
{X,lXilXjlXjlXklX,lX,Ix,l)
XP(XiXjXkXl)
dXi dXj dXk dXl
Expected
These are plotted in Figure I and it is evident that the linear approximation offers a reasonable solution only to about two standard deviations whilst the cubic is very accurate to more than three and the quintic is good to about four standard deviations. Using substitution approximations to E22 and E23 may, therefore, be obtained as follows.
and for mean-zero Gaussian variables:
where {X}r = (X,XjXkXl):
‘=
Rii
Rii
Rik
Ril
Rii
Rii
Rjk
Rjl
Rik
Rjk
Rkk
R,
Ril
Rjl
Rkl
RI1
‘Linear’approximations
Substituting
for E22 and E23
for X,lX,l
from equation
(46):
E22L =
In previous applications of the model these equations have been solved2 by numerical integration over the fourdimensional Gaussian probability spaces using constant step rectangular rule summations. For each cycle of integration the relevant conditional pdf is Gaussian in form. Therefore, to centre the computations over the significant region of this probability mass, the integration was performed over a range of k5 standard deviations of each conditional distribution about its conditional mean value, these statistics being obtained from equations (Al .S) and (Al .lO). Table 1 illustrates the sensitivity of the numerical integrations to the number of steps chosen for the computation of E23. Use of 35 steps in computation of the moments of response from equations (4) and (5) was restricted to systems of up to only four load components whilst reduction to 11 steps enabled application of the model for systems of up to 12 load components with a loss of accuracy of only about 0.1%. This later condition was therefore adopted for general use.
Approximation
value analysis R. Burrows
(49)
for XklXkl and XllXll from equation (46):
Substituting
8 E23L = - R,!$R/1/2(E21) 71
IE Best
cubic
fit \/
lf
1;
1c
for E22 and E23
An alternative approach to the use of numerical integration for the solution of expectations E22 and E23 is to employ a polynomial approximation for some of the XIX1 terms enabling evaluation in the manner applied for the other expectations. Such approximations, resulting from minimization of the ‘mean-square’ error in the statistical sense, are given by Borgman as follows:
E
6
8
XjlXjl =
J
--Ri’i/‘Xi
(Linear approx.)
71
(Cubic approx.)
(47) v
1
I
I
I
I
I
z
3
4
5
6
7
X c
(Quintic approx.)
Table 7
Sensitivity
of numerically
No. steps = N
computed 7
(48)
Figure
7
Polynomial
approximation
to XIX1
value of E23 to the number of data points considered 9
11
21
31
35
Total probability N
.) dXj..
z:zzz,:~(. 1 (E23)N o,,
lo/,’
. dX/
1.00658
95.63
1.00003
100.36
1 .ooooo
1 .ooooo
100.04
100.10
Appl.
Math.
Modelling,
0.999999
100.00
1983,
0.999999
99.99
Vol. 7, October
321
Expected value analysis ‘Cubic ‘approximation
Substituting
R. Burrows
for E22
for X,/X,)
X E~XilxilxjIxjIx~x~>I
from equation
(47): where the second expectation (52).
1
E22c =
+3R,+p X E{X~lXilXjlXjlX~X~>
(51)
1
and using equation
(Al .7):
E{XilX~lXjlXjlX~X[>=
E(XilXilXj
= biE~lX~IXjIXjIX~}+
‘Cubic ‘approximation Substituting (47):
is of the form of equation
for E23
for both X,/X,1
and X,lX,l from equation
IXj IX; E{XrIXiXjX,>>
bjE{XilXillX~lX~)
+ b~E{XilX~lXjlXjlX~4)
X E(XiIX~IXjIXjIX~XfJ
(52)
where the b’s are defined in equation (34); E{XilXilXjlXjlX,$) is El8 and the remaining expectations are of the same form, which may be expanded as follows: +
~~l~,3l~jl~jl~~> = 3a:,aiE{Xi”IXjlXjlXjl>+ +a~E(XfIXilXjlXjl}+
3&aiE{IX~X~I}
(53)
where a,& = Rkk - (ajRjk + ajRjk) from equation (Al .lO), and the expectations of the r.h.s. have been derived previously for E18. ‘Linear/cubic ‘approximation for E23 Substituting for Xk IXk 1from equation XllXrl from equation (47):
Tab/e 2
Typical
results from approximations
9Rg;Rfi’”
E~X~lX~lXjl~jl~~~~>]
(55)
where the middle two expectations are of the form of equation (52) and E{XilXilXjlXilX~Xr”> can be expanded using expressions for the conditional moments. However, this expansion yields some expectations the solution of which have not been ascertained previously and which would demand considerable effort.2 Two procedures using the above results have been examined previously,3 one including the linear approximations for E22 and E23 and the other including the cubic approximation for E22 and the linear/cubic approximation for E23. Some typical results from these approaches are given in Table 2. It is seen that the cubic approximation for E22 offers a considerable improvement in accuracy over the linear approximator and for many terms is in very
3fZffZjE{IXfX~()
+3ajai”E{Xi”iXilXi”IXil)+ai3E{IXi3X~I)
1
(46) and for
of ,622 and 623
Response Y=F,+F,tF,+F, where: F;=X~~-~IX~~-II+X~; = force on unit length member i Method
of computation Num.
Expectation
10.71
E23 E22 i c d e f
XlIX,IX,
X,IX,IX,IX,I
XlIX,IX,
X,IX,IX,IX,I
XlIX,IX,
x,Ix,Ix,Ix,I
X,IX,IX, X,lX,IX, X,IX,IX,
x,Ix,Ix,Ix,I X,IX,IX,IX,I x,Ix,Ix,Ix,I
X,IX,IX, X,IX,IX, X,lX,IX,
x,Ix,Ix,Ix,I X,IX,IX,IX,I
X,IX,IX, X,IX,IX, X,IX,IX,
X,IX,IX,IX,I x,Ix,Ix,Ix,I X,IX,IX,IX,I
-7.181 - 1.760 1 .J80
3.297 -1.875 -8.353
X,IX,IX,IX,I
Appl.
Math.
Modelling,
1983,
Vol.
7, October
Linear approx.
Linear/cubic approx.
x
lo9
7.453
7.197
-5.788 -1.818 6.139
-7.173 - 1.745 2.420
10.36
3.183 - 1.856 -8.372
x109 x109
-1.937 -6.527
x109
x109
X108 X108
2.297 1.812 3.034
1.358 1.467 4.449
2.393 1.823 -1.758
XlO’O XlO’O X106
7.784 11.88 3.341
-11.97 8.787 1.977
9.198 11.99 3.485
x10* x109 x10’”
3.238 -7.235 5.009 - 14.63
2.743 -7.468 3.790 -9.319
3.237 -7.210 5.052 - 14.67
x1o’o x109 x109 x109
4.1913
E(Y4)
322
int. (15 steps)
4.1097
4.1881
x 10L3
Expected
close agreement with the numerically computed values. However, for certain terms, notably (c), (i) and (j) in the table, the error is substantial. The reason for this behaviour is not clear but, fortunately, it appears to apply to the expectations with lower numerical values, reflecting a relatively low degree of mutual correlation, and hence the their effect on the final moment E(Y4} is small. The approximators for E23 are evidently deficient, demonstrating the need for a better approximator. Alternative linear/cubic approximator
PWJXI --AIX)~I
[I
aA
dX
--m
1
= 0
yielding AI = E{IX3(}/E{X2} which, gives the result in equation (46). There is no reason to suppose that this value of AI should minimize the errors in the application of a linear approximation of one of the variables in E22 and E23. For example, in E23 an alternative linear approximator AZ3 might be considered as that which again minimizes the ‘mean-square’ error:
&
[E{X,?XtX;(XtIX,(
E23 = E(Xf)
-A23Xt)2>]
= 105Rfi
using equation
(Al.l):
using equation
(Al .2). Similarly:
E23Lc = 63.8R:i
1
=-a m(XIX1 -A,X)2p(X)
i.e. consider Xi = Xi E Xk E Xl
then:
for E23
The polynomial expressions considered so far represent the estimators which minimize the ‘mean-square’ error in a statistical sense. The linear estimator AI, is the solution to: ;
Complete dependence:
value analysis R. Burrows
= 0
23
and E23ALc = Azs(E22~)
= 152RA
For further comparison, a complete cubic approximation, using equation (47) for all four variables, would yield a value of 164.4R:j under these conditions. This approximation, and its more precise counterpart E23~ given in an earlier section, have not been derived explicitly since they involve expectations of high order and consequently solution requires considerable effort both in algebraic expansion of the expectations and in the multi-stage integration associated with Price’s theorem. This degree of complexity, as illustrated in a partial derivation of E23c presented in reference 2, is likely to be comparable with that experienced in the derivation of (A,,)summarized in Appendix 2. From the above results, it would appear that no single approximator is likely to yield satisfactory estimates of E23. However, an ‘optimum’ linear/cubic approximator, combining with equal weighting a linear approximation equivalent to A, above with the alternative linear approximation AZ3:
hence :
A
_Erx,“xj”X:lx:l,
(57)
23 - E{X;X;XzX;} and E~~ALC
(56)
=A23@%)
The solution for A23 is summarized Comparison of the approximators
in Appendix 2. for E23
Consider firstly the two extreme conditions dependence between the variables:
of mutual
Zero correlation: E23 ‘E{XjlXjl}E{XilXjl}E(XI,IX,I)E{X~IXtI)
a condition also satisfied by all the approximators reference to equation (Al .4).
Tab/e 3
Percentage errors in approximators
=O
with
would yield an estimate in error by less than 3% compared with about 40% for the other procedures under these conditions. The above approximations have been compared with values of E23 computed numerically, using 11 steps, for more general conditions of mutual dependence between the variables. The results are summarized in Table 3, in terms of the mean and variance of the errors from application to systems of seven load components which produce sets of 3.5 different combinations of expectations of the form of E23. Results (b) correspond to more highly correlated sets of variables than results (a). It is clear from the table that the ‘optimum’ linear/cubic approximator produces the best estimates of E23. This ‘closeness of agreement’ increases as the degree of mutual correlation
for E23
Approximation
Error (%X 10-Y
Linear eqn (50)
Linear/cubic eqn (54)
(a)
Mean Variance
-0.379 0.064
-0.227 0.027
(b)
Mean Variance
-0.573 0.00070
-0.395 0.00013
Alternative linear/cubic eqn (56)
Optimum linear/cubic eqn (57)
Full cubic (numerical computation)
+0.340 0.037
+ 0.056 0.016
+0.131 0.011
+ 0.429 0.00007
+0.017 0.00002
+ 0.271 0.00148
Appl.
Math.
Modelling,
1983,
Vol. 7, October
323
Expected
value analysis R. Burrows
between the variables increases and it is under such conditions that the expression E23 has most influence on the moment of response, E{Y4}, through equation (5). Observation has shown that for conditions of very low correlation all approximators show poor performance and the ‘optimized’ linear/cubic approximator, or more strictly its component A as, is the least stable. Nevertheless, these effects exert little influence on the statistics of response. In applications (a) and (b) from Table 3 the errors carried through by the approximations to E{ Y4} are respectively 0.7% and 0.2%.
Borgman, L. E. ‘Statistical models for ocean waves and wave forces’. In ‘Advances in hydroscience’, Academic Press, London, 1972, ~018,139 Borgman, L. E. ‘Ocean wave simulation for engineering design’, J. Waterways Harbors Div. A.S.C.E. 1969,95 (WW4), 557
Appendix 1: Statistical properties of Gaussian random variables used in evaluation of El-E23 (1) Standard properties7 of mean-zero variables: (i)
E{lxln}=1.3
Gaussian random
. ..(r~-l)R~~~ n = 2K, K = integer
Conclusions The expected value analysis for the computation of the quasi-static response of offshore structures has been presented. Application of the model was formerly restricted to systems of about 11 load components when using a 30 min run time limit on an ICL 1906s computer. This was caused, in part, by a requirement for numerical integration associated with two expectations, termed herein E22 and E23. Approximations for these expectations have been developed and investigated. Adoption of a ‘cubic’ approximator for E22 and an ‘optimum’ linear/cubic approximator for E23 provides an acceptable degree of accuracy in the computations and relaxes the restrictions on implementation of the model, enabling analysis of systems of about 30 load components.
(ii)
2K. K! . R!jL’
E(lx I”} = m).
(iii)
E{x”} = 0
(iv)
E{x”lxl}
(Al.l)
n =2K+l,K=integer
(A1.2)
n =
(A1.3)
=E{xlx”l}
odd integer
=0 n = odd integer
(Al .4)
(2) Bivariate moments are solved by expansion using Price’s theorem6 as demonstrated by Papoulis.’ Consider two arbitrary functionsgr and g2 of Gaussian random variables, (rvs), x1 and x2 respectively. Price’s theorem gives: aE{g,
.g2>
=E
f&g i
ap
(Al S) 1
21
where : P=
Acknowledgements The work involved in the development of the probabilistic model for quasi-static response was supported by the UK Department of Energy through its Offshore Structures Fluid Loading Advisory Group (OSFLAG), Project No. 5 entitled ‘Prediction of long-term wave loading on offshore structures’ (1974-78). The subsequent work on the development of the expectation approximators forms part of research contract NW/L1 .l, entitled ‘Probabilisticspectral design method for wave loading on space frames’ (1978-83) supported by S.E.R.C. through its Marine Technology Directorate. In relation to this latter study, the author wishes to acknowledge the assistance of Mr K. H. Yim, a former research worker,
Burrows, R. ‘Quasi-static response of offshore structures using probabilistic methods’, Appl. Math. Modelling 1911, 1, 32.5 Burrows, R. ‘Wave loading on offshore structur&: a probabilistic approach’, PhD Thesis, University of Liverpool, 1982 Burrows, R. ‘Probabilistic description of the response of offshore structures to random wave loading’. In ‘Mechanics of wave-induced forces on cylinders’ (ed. Shaw, T. L.), Pitman, 1979,511 Holmes, P., Ticket& R. G. and Burrows, R. ‘Prediction of longterm wave loading on offshore structures’, University of Liverpool, Dept. of Civil Engineering, OSFLAG 5, Report B, 1978 Tickell, R. G. ‘Continuous random wave loading on structural members’, Struct. Erzg. 1977,55 (5), 209 Price, R. ‘A useful theorem for non-linear devices having Gaussian inputs’, IRE, PGIT, lT4,1958 Papoulis, A. ‘Probability, random variables and stochastic processes’, McGraw-Hill, London, 1965
Appl.
-
E{xdE{x21=
when xi and x2 are mean-zero. grated to give:
E{g,.g,I=jE
Math.
Modelling,
1983,
Vol.
7, October
Rx,,*
Equation
(Al. 5) can be inte-
[~~]dp+EighE~gz)
(Al.6)
0
(3) Expectations involving more than two random variables are, where possible, reduced to expressions involving only two random variables, for application of Price’s theorem using the following properties:7 E{grg:! . . . g,> = E{glg,
. . .gn_-1.E{gnlx+2..
where gjis an arbitrary expectation:
References
324
E{x1x2)
E{g,
1x1~2
. . . x,
function
.X,-I>>
(Al.7)
of Xi and the conditional
-11
is the expected value of g, given the values of xi to X, -1. (4) Some conditional moments: conditional mean, m,: (A1.8)
m,=E{xoIxI...x,)=alxl+...+a,x, where aj’s are given by the solution to: [RI&z>
=
(Al .9)
{Rd
in which: [R]
is the matrix of cross-covariances 1 ton
(Q}
{R,)
iS
the
V&Or
(QI,
~2,
. . . , a,)
of variables
T
is the vector (RoI, Roz, _. . , R&JT
Expected
For example, when II = 2:
E{XfX~X~>
a1 =
&,Ro~
a2
(RllRoz--R21Rol)I’(R~~R22-R~zRzl)
=
value analysis R. Burrows
-R12Roz)I(R11R22-R~zRzl)
Conditional variance or second central moment,
a::
us = E{(xe - m,)21xr . . . XJ . ..+a.Ron)
=Re,-(aIRor+
(A1.lO)
Conditional second moment: E{x;lx,
. . . x,} = 13:+ m2c
= 304bE{Xi”xi”>+ 6u;b;E{Xi7Xi5}
(Al.ll)
+ 12u;brb2E(X:X;6
Similarly :2 E{xzlxr _. . x,} = m,[3oz + mf] E{x~lx,
. . .x,}
+ b${XigXi5} + 4b:b&{X;X$’
(A1.12)
= 30: + 6uzrnz + mt
+ 6b:b;E{Xi7Xi7} + 4b&E{X;X;}
(Al .13)
+ b$E{XiSXig}
and it can be shown that: .,x,}
J%x;tx,
= 15uzm,+
lOu~m~+rn~
E{x$lx, . . . x,} = 15u~+45u~mZ,+
(A1.14)
. . _x,} = 105uzm,+
15uzmt+mZ
(A1.16)
and
E{x$x,
. . x,} = 105u~+420u~m~+
Appendix
2: Derivation
(A1.17)
of A,,
With reference to equation (56): A23 = E{X~X~X~IX~I}/E{x~X~X~X:} Applying equation (Al .l l), Appendix tor, the following expansions result:
bl = (RiiRik - RijRjk)/(RiiRii
- R$)
b2 = (RiiRjk - RijRik)/(RiiRii
- R$)
u;=R
kk
-@l&k
+
b&k)
The expectations on the r.h.s. of equations (A2.3)-(A2.5) can be solved using (Al .6). They are of the form:
210u~m~
+ 28u2m6+ma c c c
(Al .9):
and
105u,4mz
+21u,Zm~+m~
(A2.5)
where the b’s are given by equation
(A1.15) E{xzlx,
+ 6u;b;EjXi5Xi7I
E{XfX;}
= 24R$ + 9RiiR,(8R$+
E(XfX/>
= 12OR$ + 6OOR~RiiRji+
E{XfXT}
= 360RiiR;
E{XfX,f}
= 5040RfiRz
+ 5040RfiRjiRB
E{XJXj}
= 2520RiiRs
+ 6300RfiRijRk
(A2.1)
1 to the denomina-
RiiRij) 225R$R$Rii
+ 540RfiRiiRi
+ 45RfiR$ + 3 15RfiR,$
+ 1575R$R$Rii E{XfXT}
= 120R$ + 5400R;RiiRij
+ 4050R;iR$Ri
+ 225R;iR,$
= u$!?{Xi4XjX~} + a:E{XfX!X$) + a$T{XfX~X~> + 2a,a,E{X;X/Xi}
+ atE{XfXfX;}
+ 2ala2E{XfX~X~} + 2a,a,E{X:XfXz}
(A2.2)
E{XjXT>
= 45360RfiR:i
+ 75600R;iRjjR$
+ 14175R?.R?.R” II
where the a’s are given by: E{XfXT}
= 20160RiiRg
11
+ 75600RfiRjjRt
+ 37800RfiRiRi
Considering now the numerator applying (Al .13):
+ 1575RgRs
in equation (A2.1) and
E{XfX,+X;lXfI}
and u2=R a
II - (aIRi
+ a2Rjl + a3Rkl)
= 3u;E(Xi4Xi41XfI}
The expectations on the r.h.s. of equation (A2.2) may be expanded using equation (A1.13). They are of the form:
= 3u${Xi4xi4} + 6u;b;E{Xi6Xi4} + 12u&b,E{X;X;}
+ 6b:b;E{XfX;} + b;E{X;xjs}
+6u~p~E{X~X~IX~l)+6u~p2~{X~X~X~~X~I} + 12u;plp2E{Xi5Xi5Xf()
E{xi’x!x;>
+ b;lE{X;x;)
+ 6u;p:E{XfX$Xf)}
+ 6u;b;E{Xi4Xi6}
+
12~;PIP&~xi5xi4x,Ix:I)
+
12a~p,p3E{Xi4Xi5X,IX~~}+p~E{X~X~~X~I}
+p~E{Xi4XilX:l}+p~E{Xi4Xi4XPIX:I}
+ 4b:b*E{xi7xi5>
+4p1~23E{Xi”Xi’IX:l}+4p1p~E{XiSXi4X131X~I}
+ 4bIb;E{Xi5xi7} (A2.3)
+4p~pzE{Xi’xi”IX~I}+4p:p3E{Xi’xiX~IX~I}
Appl.
Math.
Modelling,
1983,
Vol. 7, October
325
Expected value analysis R. Burrows +6q~q:E{Xi5X:lX~l,+q~E(Xi9X~lX:I}
+4q:q~E(Xi8X:IX:I}+6q~qZ,E{X~X~lX~1) +4qtq;E{X;X;lX:lI
+ q;E{X~X:lX~o (A2.11)
E(XfXfyX~
I)
= 3o;E{XjyXflj
+ 6u;q:E{xyJX13)}
+ 12o;q1qzwi9x,lx~l)
1
Rjk
=
2
t- 6u;q:E{Xi”X&YfI>
Rik
4q;E(X/21XfI}
+4q:qzE&%lX,3I~+
Rkl
6q:q:E~Xi’“X,“lXz?~
+4q,q~E~X~X~IX:I}+q~E{XiSXPIX:O
and
(A2.12)
0; =Rkk blRik + PzRjk + P3Rkl) The expectations on the r.h.s. of equation (A2.6) may be expanded using equations (A1.13)-(A1.15). They are of the form : E{X,?Xi’lX,“l> = 3o;E(Xi4lX/I>
Ecx,“x,“xPlx:l, = ~u~,E{X~~XPIX,~~} + 6u;q;E(XfXfIX?\} + 12~~qlqz~cXi”x:lX:l~
-t6a~q~E{X~X~IX:()+q~E{X,sXI”IX:I} + 6o&:E(X;(X:II
+ 120;q,qzE{X;X,lX:()
+4q:q,E{Xi7X15JX131}+6q~q~EIXi6XPIX:l}
+64q’:E{X4X:lX:I)
+4q~q,3E~XiSX:IX13l}+q~E(Xi4X18IX~I)
+q~E{X,slX:t>+4q:q,E:Xi7XlX:I,
(A2.13) E(Xi”xi4X,” )xf I>
+6q~q~E~X~X~IX~I}+49~q~E(Xi5X:lX:I~
(A2.7)
+q;E{X;X:IX:II
E{xpx;
Ix: 1t
= 3u;E{Xi5X13(X:l)
+ 6u;q;E’(Xi7X13 IX:l}
+ 12u;qIqzE{X~X;IX~l)
= 3o~E{X~IX:l)+6a~q:E(XSIX~l}
~6o~q~E{Xi5X:IX:~)+q~E(Xi9X:JX131~
+ 12o~4,q,E~X~x~IX:l~
+4q:q;E{Xi8XPIX13])+6q~q~E(Xi7X:IX:I)
+ 6a;q;E{X;X;IX:I)
+q;E{Xf”lX:l)
+4q1q~E(Xi6XPIX131}+q~E{XiSX:(X:I)
+4q:q~E{Xi”XlX~I}+6q:q~E(Xi8X12lX~I}
(A2.14)
+4q,q:E{Xi7X:IX~l~+q42E{Xi4YPlX:l)
(A2.8)
E(X;x~x~Ix,3l)
E{x;x;]x,“I> = lso;q,E{X~(X;I~
= 3@{Xi4X:lX,3(}+
6u~q~E’{X~X~IX~l~
+ 3ou~q,q;E{Xi”X:IX~I>
+ ~G~~~E{X~~XP(X:I)+~‘:E{X~~X:IX:I}
+ 10u~q~E{Xi7X,31X:I>+q:E‘(Xi121X,”I}
+ 6q:q;E~X~x~lX:l~
+ 5q;q~E{x~lX,IX,31}+ (A2.9)
+4qlq~E{X~X~IX:l)+q,4E(X~X~lX~I)
E{Xi”xi” 1x; I > = 15CJ;q~E{x~lx:l:
3Ou;q:q;E{X;X/IX:l>
+ ~OU&;E~X;~~X,~~)+
+ 124q*4~E{Xi5X:lX:l)
+ 4q:4&{Xi7X13lX:I~
+ 15a~q2E~Xi7X,IX~lI
10q:q~E{X~“x~lX:I~
+ lOqfq;E{X;X~lX;I>
+ 5q,q;E{Xi8X14lX:I1
+&‘{X%IX%
+ 15a;q,E{X~x~lX~l~
(A2.15)
E{Xi7Xi”X,jX~j> + 100~q~E{XjyX:O
+ 30qfq,o~EfXi’X,rXr”l>
= 3u4,E{X~XzIX:I> + Ga;q;E’{X~X~JX:J}
+ 30q&~EIx~X;rX:l, + IOa;q;E{XiSX:IX:I} +
+ 12u~qlq2E{XigX~lX:I)
fq:E{Xi’“lX,“I>
sq;2q&TXi9X,IX,3l>-t 10q~q~ECXi8X:lX:l~
+q;1E{Xi”XrlX:l)
+4q:q;E{X~“X,‘IX~I>+6q:q:E~X~X~lX~~~
+ loq~q;E{X~x~IXr”l~+ 5q&?Xfx.?lX:I1 (A2.10)
+q:E~Xi5X:lX~l~
+ 6a;q;E{xJX~(X~I}
+4qlq~E(XiSXPIX2”I)+q,4E{Xi7X:IX~l} (A2.16)
E{x,sxi”x,jx:lj = 34E{Xisx,~Xr”l}
+ 64q:E{Xi7XIIX131~
= lsu~E{xpIx,3~)
-t 12o~qIqZmi6X:IX:lI
326
Appl. Math. Modelling,
E{x~x/w:()
1983, Vol. 7, October
+ 45u4,q:E{x~lx:I~
Expected + 9Oc&lq,~{X~X~IX:I~ + 45a~q~E{X~X~~X13I}
+
+~4m&v + 15a~4~E‘{Xi’“lXr”l)
E{X;XI”IX:I)
E{xfIX:o
+ 150~(Y4~{Ix:(>+01~E{Ix19~} E{x,sx,lx~l>
= lSu~aE{IX:I}+
2oq;q;E{x~x~Ix:I)
+
E{X,4X:lX:l>
+ 154~q~E(X~X~~X~~}+6q~q~E{X~X~~X~~} (A2.17)
+&wi6XPIX:I~ E{Xi6Xi4X,2lX:I> = 3@{XfXf
= 15u~E{IX~~}+45u~~~~{IX~~}
15u;qf:E{X~X~IX~II
+ 4;E‘(Xi’21X,31} + 6q;q&{X/‘X~IX~I} + 154~4;E{Xi’“Xl”jX~I}+
lX,‘3j}
+ (Y4E{Ix;SI)
+90o~q:q:E{x~x~IX13I)
+
= ~u~,E{IX/~~} + 6u;a2E{
604&,~~&94&7>
+60u~q,q~E{X~X~lX~l~+
value analysis R. Burrows
~5~w$l~
3u4,E{IX;I)
+ 6u~a2E{Ix;~}
+(r4E{Ix~lI E{X/lX~l,
IX,3 I>+ 6u;q~E’(xi”x,
=
10&&{~X~~>
= 105u;E{IX;I}+
402u~cy*E{ 1X,51)
+ 21ou~a4E{~x~~)
IX,3 I}
124qIq,mi7X:Im
+28u2a6E{~X~~}+a8E{~X~‘~} CY E{Xi7x,lX:I)
+~u~~~E{X~X~IX~I}+~;~E~X~~X~IX~I)
105&3E{~x~~}
+ 2lu~a%{IX~I}
+4q:4;E{Xi9X:lX,‘I}+64~4~E{XisXPlX~I} E{Xi6X:IX:I}
+44~4~E(X~x~IX:l}+4~E(X~x~IX~I}
E{XiSX:IX:I)
E{xfx;xfIx:I>
+a%(
Ix:‘l)
= 15u~E{~X~~}+45u~(u*E{~X~~} + 15u~a4E{IX~~}+a6E{IX:‘I}
(A2.18)
= 15u~4rE‘{X~x~IX:l)+
= 105u~orE{IX:l~+
= lsu~olE{~x:~>+lou~clI3E(IxpI) +a%{IX:‘I)
15u~4,E{Xi5X:IX:II
E{Xi’“lX_?l, = 945u~“E{~X~~)+4725u~a2E{~X~~}
+ lOu~4:E{X~x~IX:I}+30u;f4q4,E~X~x~lX:I}
+ 315ou~(w4E{Ix:I~
+ 30u~qr4;E{x~x~IX:l)
+ 630u~&{IX~~)
+45u:a8E{IXI”I)
+ 10u~4:E{X~x~IX:I}+4:E‘(X~~x~IX:l} +cr’0E{Ix:31) + Sq;q;E{XigX:IX:
I> + 104:4,2E{X~x~IX~
I) E{X,gx,lX;l}
+ 104~4~E{Xi7X:IX~l}+54~4~E{X~x~IX:l)
E{X;x;xJX~I)
+ 36u~cy’E{IX,“I} +(w~E{IX,‘~I} E{Xi”x,“lX;l>
= 15cJ~4,E~X~x~IX:I)+ + 100~4:E~xigx,Ix:l>
15u4,42E{X;x~lX:I} + 30&4~4~E{x~x~IX:I}
+~~u~~~~E{IX:‘~}+~~E{IX:~I} E{Xi7X:IX:I)
+ 10u~4~E{X~x~IX:I}+4:E‘(X~~x~IX:I}
+ 104~423E{x~x~lX~I)+
= 105u~olE(~X~~}+105u~~3~{~x~~} +2lu~cu5E{IX~~~}+(W~E{IX~~~}
+ 104:4:E{Xi9X:IX~
I>
E{Xi6X14lX:o = 15u~E{~X~~}+45u~a~E{IX~I}
Sqrq;E{Xi7X:IX:I} (A2.20)
+452~{Xi6XPIX:I~
= 105u~E{~X~~}+420u~c2E{~X~~} + 210u~CI4E{~X~~>
+30u~4,4::E{Xi7X:~X~~}
+ 54;lq,E{X;“x:Ix~I}
1260u;a3E{IX~I}
+ 378u~c&{~X~~} (A2.19)
+4:E~Xi5X:IX:o
= 945u~cS{IX~I)+
+ 15u~a4~{~X~~~}+a~E{IX1’3~} E{X;X!lX;I)
where, in the above:
= 3u~E{~X~~}+6u~ar2E{~X~~} +a4E{Ix:r1}
41 = (Rl,Rij - RilRjl)/(RiiRll
- R;l)
42 = (RiiRjl -RilRij)/(RiiRll
- R;l)
and
E{X,5x:lX&?l> = l5u”,olE{~X,g~>+lou~(y3Eo + cYsE{Ix;sI) E{Xi”xplX:l>
= 3u~E{~X~~}+6u~a2E{~X,“~} + (Y4E(Ix;31}
The expectations on the r.h.s. of equations (A2.7)-(A2.20) may be expanded using equations (Al .13)-(A1.17), or alternatively using equation (Al .6). They are of the form: E{X?IXfl)
= 304,E(lX?l} + 6u:02E{ IX;I}
E{X;‘IX:()
= 10395u;%{Ix:I> + 623700;~(~~E{IXfI} + 51975&4E{IX:I> + 13860u~c~%{lX~l>
Appl. Math. Modelling,
1983,
Vol.
7, October
327
Expected
value
analysis
R. Burrows
E{Xi8xflX,3/} = 105u~E{JX~~}+402u~(u2E(~Xf’~}
+ 14850;ar8E{ IX;‘l}
+ 21Ou~cr4E{IX~‘I}
+ 66o~a’OE{~X:“I} +CY’%{1x,151> E{X;‘X~IX:l)
+28~~a~E{IX~~~}+a8E(IX:51)
= 10395u~001E{Ix:I) E{x,7x:lx:l)
+ 17325a~a3E{IX~~}
= 105u~olE{~x~~>+105u~~~~{~x~~~~ + 21u~a5E(IX:3~}
+ 6930u~c~%{~X~~}
E{XfxfIX:I)
+ 990cJ&7E{IX:11> + 55u~(u%(~X~3~} + cm{ E{X/“Xr”lX/I}
= 15u~E{IX~~}+45u~a~E{IX:~I)
Ix:51)
= 945u~“E{~X~~}+4725~~~2E{~X~~}
+15u~(Y4~{~x~3~>+a~E(Ix:5~) E{XiSX:IX:I)
where, in the above:
+45u~cy~E{Ix~3I}+(Y~~E{Ix~sI~ E{X:XjalXfI)
= 945u;olE(IX;I}
+ 126Ou~(~~E{IX,9~>
+ ~~~u$Y~E{IX,“I} + ~~u~(Y~E{IX;~~}+(Y~E{IX~~~}
328
Appl.
Math.
Modelling,
1983,
Vol.
7, October
= 15u4,cS{JX~‘~}+ lou;a3E{IX:31} + CY5E{Ix~5)}
+ 3150u~a4E{lx~l> + 630u~a%{~X~‘I}
+(w7_E{Ix:51}
(Y=-
Bil
u2cy=B..-_a.B. 21
II
41
and the expectations equation (Al .2).
on the r.h.s. can be solved using