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Point estimation technique for the reliability analysis of ship structures
Computers & Structures Vol. 48, No. 5, pp. 835--841, 1993 Printed in Great Britain.
0045-7949/93 $6.110+ 0.00 ~ 1993 Pergamon Press Ltd
POINT ESTIMATION TECHNIQUE FOR THE RELIABILITY ANALYSIS OF SHIP STRUCTURES K.
RAJAGOPALAN
Ocean Engineering Centre, Indian Institute of Technology, Madras-600 036, India
(Received 14 July 1992) Abstract--A point estimation technique for the reliability analysis of ship structures has been described. The method does not require the evaluation of derivatives of the failure equation as do the traditional FOSM (first order, second moment) and AFOSM (advanced FOSM) techniques. A computer program has been presented for use with the point estimation method (PEM). This contains a subroutine that automatically generates the sign factors (+ or - ) using the points to obtain the function evaluations. Examples of reliability calculation of ship structures using PEM have been described and compared with the traditional methods such as FOSM and AFOSM.
conditions. Changes in ballast conditions and consumables as well as changes in temperature impart considerable uncertainty to the values of Ms,,.. It has been generally found that a normal distribution can be used for the still water bending moment [1]. The wave bending moment is the wave induced response of the ship moving in a sea state. All probable sea states that the ship could encounter during its life on the design route could be assessed. A typical sea state is associated with certain energy defined by the wave height spectrum. The irregular sea can be considered as the superposition of a large number of regular waves. The analysis of the ship motion in a regular wave determines the peak vertical bending moment per unit wave height at the wave frequency. The response amplitude operator, (RAO), which gives the distribution of peak vertical bending moments per unit wave height over various frequencies can thus be determined. The wave bending moment spectrum is simply
INTRODUCTION
The objective of structural reliability is to develop verification procedures aimed at ensuring that ships built according to specifications will perform acceptably. The reliability of a ship structure must be computed from the probabilistic descriptions of the loads and of the physical properties of the structure. The calculated reliability must be more than a target reliability. Target values are fixed by comparing the reliabilities of exemplary designs. The estimation of failure probabilities therefore forms the core of structural reliability. Failure probabilities can be estimated by a variety of methods such as FOSM, (first order, second moment), AFOSM (advanced FOSM), etc. These methods require the evaluation of the derivatives of the failure equation which is difficult and even impossible in many situations. In this paper a point estimation technique (PEM) is described by which the failure probability could be obtained using the values of the failure function alone evaluated at a number of preselected points. Depending on the number of variables in the problem, the number of points for the function evaluation could be large. A computer program is therefore needed for the rapid estimation of reliability.
SM, (O9) = IRAOI2SH(co)
PROBABILISTIC DESCRIPTION OF LOADS
The computation of reliability of the ship structure when the failure mode is full plastic yielding has been considered in this paper as a vehicle for illustrating the PEM. The vertical bending moments at the midship section to be considered for this mode of failure are the vertical still water and wave bending moments, Ms., and M., respectively. The still water bending moment distribution can be determined knowing all anticipated loading
(1)
in which Sn(o9) is the spectrum of wave heights. The statistics of M,, can then be calculated. The statistical quantities are obtained for each probable sea state and the most probable value of the wave bending moment can then be obtained. In the study described in this paper, the wave bending moment spectrum is determined using the NSRDC program [2] and the expected value of the vertical wave bending moment is obtained using the analysis suggested by Mansour [3]. PROBABILISTIC DESCRIPTION OF PLASTIC M O M E N T
The full plastic moment of resistance of the midship section, Mp, is subjected to uncertainties in the
835
836
K. RAJAGOPALAN Hence - -
1
LD
,.l 'I
to -f
dl
-Z-
gp=~
iI I
t2o-J
I
gp
1 t
d2
f
--Lk +tkLk
+ (tsg] + tsg2+ tog)d2-- toLo -- t2DL2o].
(2)
The plastic moment is given by
',,,..Is
_k
t~D+t2s~+tn
Mp = ay [2tDLogp + 2t2DLD(gp -- dl )
t2B
tsgl -~1 tsg2 - ~
" - tog
-~-t
_e_
k
+ t~g2 + Bt2B(D - g, - d2) + 2(D - gp)
t-
2
Fig. 1. Typical midship halfcross-section.
(,3) geometry of the scantlings and in the yield stress. In ordinary steel, the yield stress can vary as much as 10 to 15% and is also dependent on the rate of loading and the effects of welding. The thickness of scantlings is also affected by quality control in the manufacture. Quality control and other factors may also introduce correlations between variations in plate thickness and in yield stress, i.e., the plate thickness and the yield stress cannot be treated as independent random variables.
CONCEPT OF RELIABILITY
The supply (resistance) and demand (load) are modelled as random variables, R and S, respectively. The probability density functions of these variables are shown in Fig. 2. It is clear that the probability of failure is defined by
pf= P(R < S) =
P(R < x)P(x)
EXPRESSION FOR PLASTIC MOMENT
The midship half section of the ship considered in this paper is shown in Fig. 1. The plastic neutral axis is located at depth gp. Since it is an axis of equal areas we have B
Loto + L2ot2o + tsgp = ts(D - gp) + t28~
\ +ta B _ Lk} + Lktk + tseld2 + tsg2d2 + tcgd2. /
= f : FR(x)f~(x ) dx,
(4)
where FR(x ) is the cumulative distribution function of strength. The problem of computation of the probability of failure thus requires the knowledge of the probability distributions of load and strength. In practice, this information is difficult to obtain. However the mean and standard deviation (and perhaps the coefficients of correlation between the variables) can be obtained fairly easily. With this
-'2- X
P(R
= FR
(x)
Fig. 2. Concept of probability of failure.
Reliability analysis of ship structures t
837
and the probability of failure
Xz
p i = q~(- fl), F
i Xl D
Fig. 3. Concept of reliability index.
second moment approach the problem is formulated as follows. The uncorrelated variables ( X I , X 2. . . . . X,) are transformed into standard normal variables, X~ (which have zero mean and unit standard deviation)
x;
x,-
,,
(5)
O'xj
where #x~and ax, are the mean and standard deviation of variable X~. The failure equation can be described in the space of the standard normal variables (a typical description is shown in Fig. 3 for a problem with only two variables). The reliability index fl is obtained as the minimum distance of the failure surface from the origin. The failure point F can be located by the minimization problem
(9)
where $ is the cumulative normal distribution function. The most probable failure point F is thus assumed to coincide with the mean value of the variables in the FOSM method. (The distance fl is the tangential distance from the origin at the failure point.) The standard deviations of functions are evaluated by linearizing the functions about the mean point. This requires the evaluation of derivatives of various functions at the failure point. In the AFOSM method, the failure point is actually located by the minimization process defined in (6). This method is therefore iterative. Also the evaluation of the derivatives of the constraint function is needed in each iteration. POINT ESTIMATION TECHNIQUE
The PEM works on the analogy between the probability density function of a random variable and the distributed loading function on a beam with two supports as shown in Fig. 4. The mean of the random variable is E(x) =
xf(x)dx.
(10)
The total load is equal to unity since ~ b , f ( x ) d x is unity. The reactions at the two support points x and x+ are P and P+, respectively, where x _ = E ( x ) - a~
x+ = Ex+~r ~. minimize ~ / { x ' } r { x '} These reactions are called the point estimates. It is clear from the beam analogy that
subject to g(X) = 0.
P+ + P
(6)
P+x++P
METHODS OF EVALUATING RELIABILITY
Two methods which are currently in vogue for the evaluation of reliability are the FOSM and AFOSM techniques. Several other approaches have however been suggested [4]. Nevertheless, all these approaches require the evaluation of a number of derivatives. In the FOSM method, the mean and standard deviation of Z, where z = R - S
x
P + xZ+ + P _ x :
=1 =E(x) = E(x:),
and so on.
1
I
(7)
are evaluated at the mean point of the variables itself. Thus
3 = ~'~ O"Z
(8)
P
P.
Fig. 4. Point estimation technique.
(11)
838
K. RAJAGOPALAN
The concept of the PEM can be extended to a problem with many variables. For a problem with three random variables we can write 1 P+4-+ = P _ _ _ = ~ ( 1 +PI2+P23+P31) A~
1 P++_ = P _ _ + = ~ ( 1 +P12--Pz3--P31)
of points and their signum multipliers ( + or - ) . The point estimates are generated using an equation similar to (12) where the coefficients on the righthand side are 1/2" and the signs of the correlation coefficients are determined by multiplying the corresponding sign multipliers. The mean and standard deviation of Z can then be given by equations similar to (11). The reliability index follows from (8). It must be noted that
1 P+_+ = P_+_ = ~ ( 1 - PI2 -- P23 + P31)
#z = E(z)
(12) EXAMPLE
where Po is the correlation coefficient of the random variables x~ and x/. The sign of Po is given by the sign of the multiplication of ~, i.e. if i = - and j = + , then/j = ( - ) ( + ) = - . In general, the stencil for the generation of the + and - for any number of variables is given in Fig. 5. For a problem involving n variables, there will be 2" terms and n ( n - 1)/2 correlation coefficients. The formulation of the reliability problem using the PEM technique is thus simple. We have for the ship structure problem with full plastic yielding mode
Z = M p - M~,,- M,,.
(13)
er
VARIABLES 2
4.
-
-
4.
4"
÷
3
SHIP
STRUCTURE
RELIABILITY
x(1) x(2) x(3)
x(4) x(5) x(6) x(7) x(9) x(10) x(ll) x(12) x(13) x(14) x(15)
distance between decks ( d I ) distance between inner and outer bottoms (d2) plate thickness of upper deck (to) plate thickness of lower deck (hD) plate thickness of side shell (ts) plate thickness of inner bottom (t2n) plate thickness of outer bottom (tn) plate thickness of floors (lsg I o r tsg2) half of plate thickness of centre girder (tog) depth of ship hull girder (D) breadth of hull girder (B) length of deck (Lo or L2o) yield stress (try) still water bending moment (Ms.,) wave bending moment (M.,).
The total number of point estimates needed for this example is 2 '5 = 32768. A computer program is therefore needed. The program described in this paper written for this particular problem is however quite general and the only change needed for any other problem is in the program lines defining the function for the particular problem. In terms of the 15 variables, the function Z can be written as
2~
Ill I--"
4"
OF
The failure equation for the problem of the reliability of the ship structure under full plastic yielding of midship section can be cast in 15 variables. These are described in the following with reference to Fig. 1.
x(8) Depending on the number of variables in (13), the stencil of Fig. 5 can be used to generate the number
1
(14)
, ~ = E ( z 2) - ~ .
1
P+__=P_++=~(1--P12+P23--P31) ,
4"
Z .~. [ 2 x l 2 g p X 3 + 2x12x4(g p - x I ) + -
4-
4-
4.
4.
4.
xsg~
+ xll x~(Xlo - g, - x , ) + xll x7(Xlo -- g~) +
2 3
Fig. 5. Stencil for signum multipliers.
X5 (Xl0 --
gp)2 + 2x2 (2x ~ +
x9)
(15)
839
Reliability analysis of ship structures Table 1. Variable means and standard deviations Variable l 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Mean 75.0 45.0 0.95 0.65 0.70 0.40 0.70 0.35 0.50 390.0 873.792 264.0 22.8 4909200.0 152280.0
Table 3. Effect of correlation
Standard deviation
Correlation
/~
3.75 2.25 0.02375 0.0195 0.0175 0.016 0.0175 0.014 0.0225 0.585 1.3107 0.396 1.596 490920.0 22842.0
0 20% 40%
4.518 4.326 4.157
60%
4.006
Table 2. Reliability indices Reliability index Variation in Ms,.
No correlation
40% correlation
None 5% 10%
5.778 5.368 4.518
5.074 4.789 4.157
Reliability calculations are performed on a ship with the particulars shown in Table 1 which gives the mean values and standard deviations used for the variables. The units are inches and tons. The reliability index, fl, using the P E M with the still water bending moment considered deterministic and assuming all variables as uncorrelated turns,out to be 5.778. This can be compared with the values of 5.213 and 5.913 which can be obtained with F O S M and A F O S M , respectively. It can thus be seen that the P E M is as good as the A F O S M . Yet PEM is elegant to apply and does not bother about derivatives. The variation of reliability index with variations in still water bending moment is studied in Table 2. The table also gives the reliability indices obtained assuming a 40% correction between the yield stress and all plate thicknesses. Assuming a 10% variation in still water bending moment, the variation of reliability indices due to variations in the correlation coefficient between
the yield stress and plate thickness is given in Table 3. The computer program used in the study is listed in the Appendix. CONCLUDING REMARKS The point estimation technique (PEM) described in the paper has been shown to predict the reliability o f ship structures with fair accuracy in comparison with the A F O S M technique. Furthermore, the P E M has the advantage that it does not require the evaluation of derivatives. A program for the implementation of P E M on the computer has been described and used to predict the reliabilities of a ship structure over variations in still water bending moment. It has also been shown that correlations between the variables can be easily included in the PEM. Acknowledgements--The author thanks the personnel in the Ministry of Surface Transport and in the Indian Register of Shipping for their kind support. The study is funded by the Ministry of Surface Transport, Government of India, through the project grant OEC/22/MST/89-90/KR to the Ocean Engineering Centre, Indian Institute of Technology, Madras, India. REFERENCES
1. C. G. Soares and T. Moan, Statistical analysis of still water bending moments and shear forces in tankers, ore and bulk carriers. Norwegian Maritime Res. 10, 33-47 (1982). 2. W. G. Meyers, D. J. Sheridan and N. Salvesen, Manualship motion and sea-load computer program. Report No. 3376, NSRDC, Maryland (1975). 3. A. E. Mansour, Probabilistic design concepts in ship structural safety and reliability. Transactions, S N A M E 80, 64-97 (1972). 4. G. J. White and B. M. Ayyub, Reliability-based design format for marine structures. J. Ship Res. 31, 6(b69 (1987).
APPENDIX: PROGRAM FOR RELIABILITY ANALYSIS BY PEM
10 15
665 666
IMPLICIT REAL~8tA-H,O-Z) COMM~ N,M,St33000,15),WEIGHT(330UO),R(15,15) DIMENSION XC15),I(15),SIG(15) READ(25,10) N,KTYPE READ(25,15) tX(1),I=I,N),(SIG(1),I=I,N) FORMAT(215) FORMAT(SFIU.O) DO 665 I=I,N DO 665 J=IBN R(I,J)=O,O IF(KTYPE.EQ.I) GO TO 9999 DO 655 I=I,N READt25,15) ( R t I , J ) , J : I , N )
840
K. P-~AGOPALAN
9999 CONTIqUE DO 8888 l=1,N DO 8888 J=I,N 8888 R(J,I)=R(I,J) M=I DO I0~ I=I,N 105 M=M*2 SUMI=~.O
SUM2=~.O CALL SIGNUM
DO 193 K:I,M DO 205 KK=I,N 2~5 Z(KK)=X(KK)+S(K,KK)*SIG(KK)
GP=O-5*Z(II)*z(6)+O.5*Z(I])*I(7)e2.*Z(2),Z(8)+Z(2).Z(9) GP=GP~Z(JU)*Z(5)-2.*Z(J2)*Z(3) GP=Q.5*GPIZ(5) S1=2-*Z(12)*GP*Z(5)÷Z.*Z(12)*Z(4)*(Gp-Z(1))+;p,Go,Z(5] S1=S1tZ(11)*Z(6)*(Z(1~)-GP-Z(2))÷Z(11)*Z(7)*(Z(10)-Gp) Sl=S1t(Z(lO)'GP)*(Z(13)-~P)*Z(5) SI=SI~2-*Z(2)*(Z.*Z(8}+Z(9))*(Z(IO)-Gp-~.5*I(2)) S1=S1*Z(13)-Z(14)'Z(15) S2=51'S1 SUMI=SUMI+SI*NEIGHT(K) SUM2=SUM2+S2*WEIGHT(K) 100 CONTINUE SD=DSQRT(SUM2-SUMI*SUMI) WRITE[26,20) ~RITE(26,40) SD WRITE(26,20)
GP=O-5*X(11)*X(b)+O,5*X(11)*X(T)+2.*X(2)*X(8)+X(2),X(9) GP=GP~X(10)*X(5)-2.*X(12)*X(3) GP=O,5*GPIX(5) S1=2-*X(12)*GP*X(3)+Z.*X(12)IK(4)*(Gp-X(1))+$p,Gp,X(5) S1=S1+X(11)*X(6)*(X(10)-~P-X(Z))+K(11)*X(7)*(X(13)-Gp) SI=sI+(X(IO)'3P)*(X(I~)-Gp)*x(5) S1=S1+2.*X(Z)*(2.*X(8)+X(9))*(X(l~)-;p-O.5,X(2)) S1=S1*X(13)-X(14)-X(15) SH:S1 WRITE(26,20) WRITE(Z6,40) W~ITE(26,20) BETA:SH/SD WRITE(26,20)
20 FORMAT(5X,m 30 ~,0 50 777 C C C C C 444 C C C 555
SM
i)
WRITE(26,30) N,(X(Z),I=I,N) WRITE(26,20) FORMAT(SK,IS,4(D2U.6e2X)) WRITE(26,40) (SIG(1),I=IeN) FORMAT(5X,4(D20.b,5X]) FORMAT(SX,2015) WRITE(26,2U) DO 777 I=1,N WRITE(26,40) (R(I,J),J=I,N) WRITE(26,40) (WEIGHT(I),I=I,M) WRZTE(26o20) DO 44~ I=I,M DO 4~G J=I,N NS(I,J)=S(I,J) ~RITE(26,20) DO 555 I=I,fi WRITE(26,50) I,(NS(I,J),J=I,N] WRITE(26,20) WRITE(26,40) 3ETA WRITE(26,20) STOP END S ~ R O J T I N E SIGNUM IMPLICIT REAL*8(A-H,O'Z) COMMO~ N,M- S(33UOO,15),WEIGHr (330UO),R (15,15)
DIMENSION IJ ( 1 1 0 ) , I K ( 1 1 0 ) DO I0~ I=I,M
I1=2.I-1 I2=IlC.1 S(II,1)=-1.0 106 S(12,1 )=1.0 IF(N.LT.2) GO TO 340 DO 33] K=2,.N
Reliability analysis of ship structures LL=K-I L=2**LL DO 1 0 t I = | , M , L
I1=2hi-1 108
109 107 330 340
365 360
375 370
[2=I1-1+L DO 109 J=II,I2 S(J,K)=-I.0 13=12+I 14=I3-I+l DO I ~ J=I3,14 S(J,K)=I.0 CONTINUE CONTIWUE CONTINUE LLL=2~*N FL=LLL
IC=fl*(N-1)12 KK=I KKK=N'I DO 36) I=I,KKK II=I+1 DO 365 K=II,N IJ(KK)=I IK(KK)=K KK=KK+I CONTINUE CONTINUE DO 37] I=I,M SUM=~.O DO 375 J=l,IC LJ=IJ(J) LK=IK(J) $UM=SJM+R(LJ,LK)*S(I,LJ)*S(I,LK) SUM=I.÷SUM ~EIGHF(I)=SUMIFL CONTINUE RETUR~ END
841
0045-7949/93 $6.110+ 0.00 ~ 1993 Pergamon Press Ltd
POINT ESTIMATION TECHNIQUE FOR THE RELIABILITY ANALYSIS OF SHIP STRUCTURES K.
RAJAGOPALAN
Ocean Engineering Centre, Indian Institute of Technology, Madras-600 036, India
(Received 14 July 1992) Abstract--A point estimation technique for the reliability analysis of ship structures has been described. The method does not require the evaluation of derivatives of the failure equation as do the traditional FOSM (first order, second moment) and AFOSM (advanced FOSM) techniques. A computer program has been presented for use with the point estimation method (PEM). This contains a subroutine that automatically generates the sign factors (+ or - ) using the points to obtain the function evaluations. Examples of reliability calculation of ship structures using PEM have been described and compared with the traditional methods such as FOSM and AFOSM.
conditions. Changes in ballast conditions and consumables as well as changes in temperature impart considerable uncertainty to the values of Ms,,.. It has been generally found that a normal distribution can be used for the still water bending moment [1]. The wave bending moment is the wave induced response of the ship moving in a sea state. All probable sea states that the ship could encounter during its life on the design route could be assessed. A typical sea state is associated with certain energy defined by the wave height spectrum. The irregular sea can be considered as the superposition of a large number of regular waves. The analysis of the ship motion in a regular wave determines the peak vertical bending moment per unit wave height at the wave frequency. The response amplitude operator, (RAO), which gives the distribution of peak vertical bending moments per unit wave height over various frequencies can thus be determined. The wave bending moment spectrum is simply
INTRODUCTION
The objective of structural reliability is to develop verification procedures aimed at ensuring that ships built according to specifications will perform acceptably. The reliability of a ship structure must be computed from the probabilistic descriptions of the loads and of the physical properties of the structure. The calculated reliability must be more than a target reliability. Target values are fixed by comparing the reliabilities of exemplary designs. The estimation of failure probabilities therefore forms the core of structural reliability. Failure probabilities can be estimated by a variety of methods such as FOSM, (first order, second moment), AFOSM (advanced FOSM), etc. These methods require the evaluation of the derivatives of the failure equation which is difficult and even impossible in many situations. In this paper a point estimation technique (PEM) is described by which the failure probability could be obtained using the values of the failure function alone evaluated at a number of preselected points. Depending on the number of variables in the problem, the number of points for the function evaluation could be large. A computer program is therefore needed for the rapid estimation of reliability.
SM, (O9) = IRAOI2SH(co)
PROBABILISTIC DESCRIPTION OF LOADS
The computation of reliability of the ship structure when the failure mode is full plastic yielding has been considered in this paper as a vehicle for illustrating the PEM. The vertical bending moments at the midship section to be considered for this mode of failure are the vertical still water and wave bending moments, Ms., and M., respectively. The still water bending moment distribution can be determined knowing all anticipated loading
(1)
in which Sn(o9) is the spectrum of wave heights. The statistics of M,, can then be calculated. The statistical quantities are obtained for each probable sea state and the most probable value of the wave bending moment can then be obtained. In the study described in this paper, the wave bending moment spectrum is determined using the NSRDC program [2] and the expected value of the vertical wave bending moment is obtained using the analysis suggested by Mansour [3]. PROBABILISTIC DESCRIPTION OF PLASTIC M O M E N T
The full plastic moment of resistance of the midship section, Mp, is subjected to uncertainties in the
835
836
K. RAJAGOPALAN Hence - -
1
LD
,.l 'I
to -f
dl
-Z-
gp=~
iI I
t2o-J
I
gp
1 t
d2
f
--Lk +tkLk
+ (tsg] + tsg2+ tog)d2-- toLo -- t2DL2o].
(2)
The plastic moment is given by
',,,..Is
_k
t~D+t2s~+tn
Mp = ay [2tDLogp + 2t2DLD(gp -- dl )
t2B
tsgl -~1 tsg2 - ~
" - tog
-~-t
_e_
k
+ t~g2 + Bt2B(D - g, - d2) + 2(D - gp)
t-
2
Fig. 1. Typical midship halfcross-section.
(,3) geometry of the scantlings and in the yield stress. In ordinary steel, the yield stress can vary as much as 10 to 15% and is also dependent on the rate of loading and the effects of welding. The thickness of scantlings is also affected by quality control in the manufacture. Quality control and other factors may also introduce correlations between variations in plate thickness and in yield stress, i.e., the plate thickness and the yield stress cannot be treated as independent random variables.
CONCEPT OF RELIABILITY
The supply (resistance) and demand (load) are modelled as random variables, R and S, respectively. The probability density functions of these variables are shown in Fig. 2. It is clear that the probability of failure is defined by
pf= P(R < S) =
P(R < x)P(x)
EXPRESSION FOR PLASTIC MOMENT
The midship half section of the ship considered in this paper is shown in Fig. 1. The plastic neutral axis is located at depth gp. Since it is an axis of equal areas we have B
Loto + L2ot2o + tsgp = ts(D - gp) + t28~
\ +ta B _ Lk} + Lktk + tseld2 + tsg2d2 + tcgd2. /
= f : FR(x)f~(x ) dx,
(4)
where FR(x ) is the cumulative distribution function of strength. The problem of computation of the probability of failure thus requires the knowledge of the probability distributions of load and strength. In practice, this information is difficult to obtain. However the mean and standard deviation (and perhaps the coefficients of correlation between the variables) can be obtained fairly easily. With this
-'2- X
P(R
= FR
(x)
Fig. 2. Concept of probability of failure.
Reliability analysis of ship structures t
837
and the probability of failure
Xz
p i = q~(- fl), F
i Xl D
Fig. 3. Concept of reliability index.
second moment approach the problem is formulated as follows. The uncorrelated variables ( X I , X 2. . . . . X,) are transformed into standard normal variables, X~ (which have zero mean and unit standard deviation)
x;
x,-
,,
(5)
O'xj
where #x~and ax, are the mean and standard deviation of variable X~. The failure equation can be described in the space of the standard normal variables (a typical description is shown in Fig. 3 for a problem with only two variables). The reliability index fl is obtained as the minimum distance of the failure surface from the origin. The failure point F can be located by the minimization problem
(9)
where $ is the cumulative normal distribution function. The most probable failure point F is thus assumed to coincide with the mean value of the variables in the FOSM method. (The distance fl is the tangential distance from the origin at the failure point.) The standard deviations of functions are evaluated by linearizing the functions about the mean point. This requires the evaluation of derivatives of various functions at the failure point. In the AFOSM method, the failure point is actually located by the minimization process defined in (6). This method is therefore iterative. Also the evaluation of the derivatives of the constraint function is needed in each iteration. POINT ESTIMATION TECHNIQUE
The PEM works on the analogy between the probability density function of a random variable and the distributed loading function on a beam with two supports as shown in Fig. 4. The mean of the random variable is E(x) =
xf(x)dx.
(10)
The total load is equal to unity since ~ b , f ( x ) d x is unity. The reactions at the two support points x and x+ are P and P+, respectively, where x _ = E ( x ) - a~
x+ = Ex+~r ~. minimize ~ / { x ' } r { x '} These reactions are called the point estimates. It is clear from the beam analogy that
subject to g(X) = 0.
P+ + P
(6)
P+x++P
METHODS OF EVALUATING RELIABILITY
Two methods which are currently in vogue for the evaluation of reliability are the FOSM and AFOSM techniques. Several other approaches have however been suggested [4]. Nevertheless, all these approaches require the evaluation of a number of derivatives. In the FOSM method, the mean and standard deviation of Z, where z = R - S
x
P + xZ+ + P _ x :
=1 =E(x) = E(x:),
and so on.
1
I
(7)
are evaluated at the mean point of the variables itself. Thus
3 = ~'~ O"Z
(8)
P
P.
Fig. 4. Point estimation technique.
(11)
838
K. RAJAGOPALAN
The concept of the PEM can be extended to a problem with many variables. For a problem with three random variables we can write 1 P+4-+ = P _ _ _ = ~ ( 1 +PI2+P23+P31) A~
1 P++_ = P _ _ + = ~ ( 1 +P12--Pz3--P31)
of points and their signum multipliers ( + or - ) . The point estimates are generated using an equation similar to (12) where the coefficients on the righthand side are 1/2" and the signs of the correlation coefficients are determined by multiplying the corresponding sign multipliers. The mean and standard deviation of Z can then be given by equations similar to (11). The reliability index follows from (8). It must be noted that
1 P+_+ = P_+_ = ~ ( 1 - PI2 -- P23 + P31)
#z = E(z)
(12) EXAMPLE
where Po is the correlation coefficient of the random variables x~ and x/. The sign of Po is given by the sign of the multiplication of ~, i.e. if i = - and j = + , then/j = ( - ) ( + ) = - . In general, the stencil for the generation of the + and - for any number of variables is given in Fig. 5. For a problem involving n variables, there will be 2" terms and n ( n - 1)/2 correlation coefficients. The formulation of the reliability problem using the PEM technique is thus simple. We have for the ship structure problem with full plastic yielding mode
Z = M p - M~,,- M,,.
(13)
er
VARIABLES 2
4.
-
-
4.
4"
÷
3
SHIP
STRUCTURE
RELIABILITY
x(1) x(2) x(3)
x(4) x(5) x(6) x(7) x(9) x(10) x(ll) x(12) x(13) x(14) x(15)
distance between decks ( d I ) distance between inner and outer bottoms (d2) plate thickness of upper deck (to) plate thickness of lower deck (hD) plate thickness of side shell (ts) plate thickness of inner bottom (t2n) plate thickness of outer bottom (tn) plate thickness of floors (lsg I o r tsg2) half of plate thickness of centre girder (tog) depth of ship hull girder (D) breadth of hull girder (B) length of deck (Lo or L2o) yield stress (try) still water bending moment (Ms.,) wave bending moment (M.,).
The total number of point estimates needed for this example is 2 '5 = 32768. A computer program is therefore needed. The program described in this paper written for this particular problem is however quite general and the only change needed for any other problem is in the program lines defining the function for the particular problem. In terms of the 15 variables, the function Z can be written as
2~
Ill I--"
4"
OF
The failure equation for the problem of the reliability of the ship structure under full plastic yielding of midship section can be cast in 15 variables. These are described in the following with reference to Fig. 1.
x(8) Depending on the number of variables in (13), the stencil of Fig. 5 can be used to generate the number
1
(14)
, ~ = E ( z 2) - ~ .
1
P+__=P_++=~(1--P12+P23--P31) ,
4"
Z .~. [ 2 x l 2 g p X 3 + 2x12x4(g p - x I ) + -
4-
4-
4.
4.
4.
xsg~
+ xll x~(Xlo - g, - x , ) + xll x7(Xlo -- g~) +
2 3
Fig. 5. Stencil for signum multipliers.
X5 (Xl0 --
gp)2 + 2x2 (2x ~ +
x9)
(15)
839
Reliability analysis of ship structures Table 1. Variable means and standard deviations Variable l 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Mean 75.0 45.0 0.95 0.65 0.70 0.40 0.70 0.35 0.50 390.0 873.792 264.0 22.8 4909200.0 152280.0
Table 3. Effect of correlation
Standard deviation
Correlation
/~
3.75 2.25 0.02375 0.0195 0.0175 0.016 0.0175 0.014 0.0225 0.585 1.3107 0.396 1.596 490920.0 22842.0
0 20% 40%
4.518 4.326 4.157
60%
4.006
Table 2. Reliability indices Reliability index Variation in Ms,.
No correlation
40% correlation
None 5% 10%
5.778 5.368 4.518
5.074 4.789 4.157
Reliability calculations are performed on a ship with the particulars shown in Table 1 which gives the mean values and standard deviations used for the variables. The units are inches and tons. The reliability index, fl, using the P E M with the still water bending moment considered deterministic and assuming all variables as uncorrelated turns,out to be 5.778. This can be compared with the values of 5.213 and 5.913 which can be obtained with F O S M and A F O S M , respectively. It can thus be seen that the P E M is as good as the A F O S M . Yet PEM is elegant to apply and does not bother about derivatives. The variation of reliability index with variations in still water bending moment is studied in Table 2. The table also gives the reliability indices obtained assuming a 40% correction between the yield stress and all plate thicknesses. Assuming a 10% variation in still water bending moment, the variation of reliability indices due to variations in the correlation coefficient between
the yield stress and plate thickness is given in Table 3. The computer program used in the study is listed in the Appendix. CONCLUDING REMARKS The point estimation technique (PEM) described in the paper has been shown to predict the reliability o f ship structures with fair accuracy in comparison with the A F O S M technique. Furthermore, the P E M has the advantage that it does not require the evaluation of derivatives. A program for the implementation of P E M on the computer has been described and used to predict the reliabilities of a ship structure over variations in still water bending moment. It has also been shown that correlations between the variables can be easily included in the PEM. Acknowledgements--The author thanks the personnel in the Ministry of Surface Transport and in the Indian Register of Shipping for their kind support. The study is funded by the Ministry of Surface Transport, Government of India, through the project grant OEC/22/MST/89-90/KR to the Ocean Engineering Centre, Indian Institute of Technology, Madras, India. REFERENCES
1. C. G. Soares and T. Moan, Statistical analysis of still water bending moments and shear forces in tankers, ore and bulk carriers. Norwegian Maritime Res. 10, 33-47 (1982). 2. W. G. Meyers, D. J. Sheridan and N. Salvesen, Manualship motion and sea-load computer program. Report No. 3376, NSRDC, Maryland (1975). 3. A. E. Mansour, Probabilistic design concepts in ship structural safety and reliability. Transactions, S N A M E 80, 64-97 (1972). 4. G. J. White and B. M. Ayyub, Reliability-based design format for marine structures. J. Ship Res. 31, 6(b69 (1987).
APPENDIX: PROGRAM FOR RELIABILITY ANALYSIS BY PEM
10 15
665 666
IMPLICIT REAL~8tA-H,O-Z) COMM~ N,M,St33000,15),WEIGHT(330UO),R(15,15) DIMENSION XC15),I(15),SIG(15) READ(25,10) N,KTYPE READ(25,15) tX(1),I=I,N),(SIG(1),I=I,N) FORMAT(215) FORMAT(SFIU.O) DO 665 I=I,N DO 665 J=IBN R(I,J)=O,O IF(KTYPE.EQ.I) GO TO 9999 DO 655 I=I,N READt25,15) ( R t I , J ) , J : I , N )
840
K. P-~AGOPALAN
9999 CONTIqUE DO 8888 l=1,N DO 8888 J=I,N 8888 R(J,I)=R(I,J) M=I DO I0~ I=I,N 105 M=M*2 SUMI=~.O
SUM2=~.O CALL SIGNUM
DO 193 K:I,M DO 205 KK=I,N 2~5 Z(KK)=X(KK)+S(K,KK)*SIG(KK)
GP=O-5*Z(II)*z(6)+O.5*Z(I])*I(7)e2.*Z(2),Z(8)+Z(2).Z(9) GP=GP~Z(JU)*Z(5)-2.*Z(J2)*Z(3) GP=Q.5*GPIZ(5) S1=2-*Z(12)*GP*Z(5)÷Z.*Z(12)*Z(4)*(Gp-Z(1))+;p,Go,Z(5] S1=S1tZ(11)*Z(6)*(Z(1~)-GP-Z(2))÷Z(11)*Z(7)*(Z(10)-Gp) Sl=S1t(Z(lO)'GP)*(Z(13)-~P)*Z(5) SI=SI~2-*Z(2)*(Z.*Z(8}+Z(9))*(Z(IO)-Gp-~.5*I(2)) S1=S1*Z(13)-Z(14)'Z(15) S2=51'S1 SUMI=SUMI+SI*NEIGHT(K) SUM2=SUM2+S2*WEIGHT(K) 100 CONTINUE SD=DSQRT(SUM2-SUMI*SUMI) WRITE[26,20) ~RITE(26,40) SD WRITE(26,20)
GP=O-5*X(11)*X(b)+O,5*X(11)*X(T)+2.*X(2)*X(8)+X(2),X(9) GP=GP~X(10)*X(5)-2.*X(12)*X(3) GP=O,5*GPIX(5) S1=2-*X(12)*GP*X(3)+Z.*X(12)IK(4)*(Gp-X(1))+$p,Gp,X(5) S1=S1+X(11)*X(6)*(X(10)-~P-X(Z))+K(11)*X(7)*(X(13)-Gp) SI=sI+(X(IO)'3P)*(X(I~)-Gp)*x(5) S1=S1+2.*X(Z)*(2.*X(8)+X(9))*(X(l~)-;p-O.5,X(2)) S1=S1*X(13)-X(14)-X(15) SH:S1 WRITE(26,20) WRITE(Z6,40) W~ITE(26,20) BETA:SH/SD WRITE(26,20)
20 FORMAT(5X,m 30 ~,0 50 777 C C C C C 444 C C C 555
SM
i)
WRITE(26,30) N,(X(Z),I=I,N) WRITE(26,20) FORMAT(SK,IS,4(D2U.6e2X)) WRITE(26,40) (SIG(1),I=IeN) FORMAT(5X,4(D20.b,5X]) FORMAT(SX,2015) WRITE(26,2U) DO 777 I=1,N WRITE(26,40) (R(I,J),J=I,N) WRITE(26,40) (WEIGHT(I),I=I,M) WRZTE(26o20) DO 44~ I=I,M DO 4~G J=I,N NS(I,J)=S(I,J) ~RITE(26,20) DO 555 I=I,fi WRITE(26,50) I,(NS(I,J),J=I,N] WRITE(26,20) WRITE(26,40) 3ETA WRITE(26,20) STOP END S ~ R O J T I N E SIGNUM IMPLICIT REAL*8(A-H,O'Z) COMMO~ N,M- S(33UOO,15),WEIGHr (330UO),R (15,15)
DIMENSION IJ ( 1 1 0 ) , I K ( 1 1 0 ) DO I0~ I=I,M
I1=2.I-1 I2=IlC.1 S(II,1)=-1.0 106 S(12,1 )=1.0 IF(N.LT.2) GO TO 340 DO 33] K=2,.N
Reliability analysis of ship structures LL=K-I L=2**LL DO 1 0 t I = | , M , L
I1=2hi-1 108
109 107 330 340
365 360
375 370
[2=I1-1+L DO 109 J=II,I2 S(J,K)=-I.0 13=12+I 14=I3-I+l DO I ~ J=I3,14 S(J,K)=I.0 CONTINUE CONTIWUE CONTINUE LLL=2~*N FL=LLL
IC=fl*(N-1)12 KK=I KKK=N'I DO 36) I=I,KKK II=I+1 DO 365 K=II,N IJ(KK)=I IK(KK)=K KK=KK+I CONTINUE CONTINUE DO 37] I=I,M SUM=~.O DO 375 J=l,IC LJ=IJ(J) LK=IK(J) $UM=SJM+R(LJ,LK)*S(I,LJ)*S(I,LK) SUM=I.÷SUM ~EIGHF(I)=SUMIFL CONTINUE RETUR~ END
841