Ship structural design for extreme loads

v. Scheepsbouwkunde

IN TiTUE OF

rtWehnische Hogeschool

Li

-; V .

Yos

Delft

SHIP STRUCTURAL DESlisi FOR EXTREME. LOADS

by

Edward V. LeT,jJ

Prepared for Special Issue

of Schiffstechnik in Honor of the

Birthday of Professor O. Grim

7 September 1976

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1 6 NOV. 1976

ARCHIEF

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A

Introduction

Dr.

Grim has always been interested in the extreme behavior of ships at

sea and this interest has led him into valuable investigations of non-linear

responses. In addition to such important studies the full understanding of

.extreme behavior requires the application of tools

obtained from probability

theory.

It is the latter subject that will be discussed briefly here.

The idea persists in some quarters that the problem of extreme conditions

is a deterministic one. If we knew the worst wave a ship or structure will

ever encounter, we could calculate the re:,:imum loads and design accordingly. But in reality the "worst wave" may not give the worst load. Furthermore, we cannot ever claim to know the worst wave or the worst load without some

prob-abilistic qualification -- such as the worst in 100 years (i.e.,

"the

100-year

wave"). Finally, the design of the structure cannot be so

precise

that the

strength just barely exceeds the maximum load. Variability in the strength of

the material, unfairness and misalignment of structure, material flaws and weld defects, inadequacies in the design theories, etc., all lead to variabil-ity in the strength of the structure.

The above difficulties can be overcome either by including a generous em-pirical factor of safety or by utilizing a probabilistic approach, i.e. the

use of reliability theory. Let us limit

ourselves to

the problem of designing

a ship with adequate but not excessive longitudinal hull strength. This

prob-lem takes on

an entirely different aspect when we ask simply, has the

proba-bility of

failure been reduced

to

an acceptable value? To answer it we need to know the distribution functions of load (or demand) and of strength

(capa-bility)

expressed in the same units. Then we can determine the probability of

failure by integration. Finally, we must decide

what is

an acceptable value

for this failure probability. Probability of Failure

(

If y is demand and z is capability, the probability of failure may be

de-fined as the probability that the ratio zly is greater than 1.0. Since :riy is

an instantaneous

value

of safety factor, we can also say that the failure

probability is the probability that the safety factor always exceeds unity.

It can be shown that

the

cumulative distribution of the safety factor is given

by (1) p(z I [1 - Q(y)]r(z) dz Y J.

--Where Q(y) is the cumulative distribution of demnvd, r(z) is the density function of capability.

The integral

can be

evaluated numerically to determine its value at

z0/y. =

1.0, which

is the probability of failure per cycle of loading.

NordenstrOm has investigated the probability of failure when both demand and capability can be expressed as specific probability functions, such as

normal, Weibull, and Frechet (2). Mansour has also investigated this problem

(3)

Loads

The development of the cumulative probability of demand or loads has been developed elsewhere on the basis of considering the entire population of indi-vidual peak-to-trough or peak-to-mean amplitudes of stress or bending moment

in a ship's lifetime (4). It can be estimated on the basis of a summation of

.many Rayleigh distributions (5). It need not be expressed as a discrete

dis-tribution function, although it has been shown that it can be represented by a

so-called Weibull distribution with suitable parameters (2). A log-normal or

exponential distribution is not acceptable, however (6).

The significance of such a long-term distribution as that shown in Fig. 1 is that it shows the probability, P. (per cycle of load) of exceeding

differ-ent levels of stress or bending momdiffer-ent. For design purposes it is convenient

to interpret the curve in terms of a large data sample of N cycles, where N =

1/P. Hence, an alternate scale of N is shown In the figure. For example,

-probability of exceedance of 10-6. means that in a data sample of 106 cycles of

load we would expect that one value would exceed the indicated level. Of

par-ticular interest is the bending moment at 108 which corresponds to a typical oceangoing Ship's lifetime of 25 years.

However, the above lifetime bending moment is only an average value. If

there were 100 similar ships operating in the same service for which data were

available, we would expect a

variation in

this bending

moment-to-be-exceeded-once. Karst has shown on the basis of a binomial model (approximated by the

Poisson density function) that one ship out of 100 would be expected to exceed

a stress corresponding to N = 302 x 108 = 1010 (7). This -- or the value

cor-responding to 1000 ships (1011) -- could be used as a design load. However,

as shown above, if we proceed to the concept of probability of failure the distribution itself will be utilized, rather than a single value.

Structural Response

Less is known about the distribution of structural response or

capabili-ty. Perhaps the best treatment is that of Mausour and Faulkner (8), who

con-sider the effects of a great many uncertainties. They assume that the

proba-bility density function is normal, with

a

mean value for each potential mode

of failure determined by standard structural analysis techniques. The

stand-ard deviation is determined by estimating the coefficients of variation (ra-tios of standard deviation to mean value) for each factor of uncertainty and

then combining them. Although a great deal more research is needed to

estab-lish reliable values of all coefficients, the paper gives a good idea of the order of magnitude of the variability of capability.

Objective uncertainties, based on Ang's subdivision (9), are random

var-iations that can be measured experimentally. They include material

proper-ties, scantlings and imperfections in construction. Under the first category

-- material properties Young's modulus (E) and yield strength were

consid 2 consid

-

-ered, the first being assumed to be small enough to be ignored. The coeffi-'cient of variation for yield strength was believed to be 6-82: and a value of

87 was assumed for a tanker structure in either tension or compression.

Scantling variations include both variability in plete thickness and in areas

and inertias of stiffeners. An overall value of was selected on the basis

of an ISSC study (10). Under imperfections

in construction,

cracks and flaws

were arbitrarily excluded on the grounds that they represented non-ductile

be-havior,

but they should obviously be studied further. Shape imperfections may

be favorable or unfavorable, and they depend on mode of failure. It was

as-sumed that they can be included in the subjective uncertainties. Finally

re-sidual welding stresses were taken to be a type of imperfection that must be allowed for directly in the case of compression, and a value of 12% was as-sumed for a typical tanker (compression only).

Subjective uncertainties (9) are those that have not been measured and

therefore

must be estimated on the basis of engineering judgement. They de-..d on mode of failure -- such as deck or bottom yielding, strut-panel com-i,ression failure, grillage instability, etc. -- excluding fatigue.

Estimated

mean

values and coefficients of variation given in (8) for a

anker are summarized in Table 1.

Table 1

3

-The above figures are quoted as examples of variability in structural

capability, but are not endorsed for design use. Such data for ship

struc-tures are extremely scarce, and it Is felt that a great deal more research

needs to be done on the subject. Mention may be made of the fact that a

pro-ject in this area is expected to be undertaken soon by

the

Ship Structure

Committee.

Acceptable Failure Probability

With probability data on both structural demand and capability available,

the probability of failure can be calculated, as previously shown. The final

step in a rational approach to designing a ship for extreme loads is then the

determination of an acceptable level of probability. J.F. Dalzell, in an

in-formal memo to the Ship Research Committee (12 May 1970), gave a valuable analysis of Some published Lloyd's Register data on merchant ship losses (11), which covered 13 years (1949-1966) and 390,000 ship-years of service

experi-Estimated Uncertainties for

, nip

77;

ree.ngth

/IL

c, Displacement, Mean Stress Tons/in.2 Tanker (8)

n1 ,nn

_LAJV Coefficients Objective of Variability, Subjective % Total n,

/Jo.eee

a,

Deck yield

Bottom beam-column 16.5 15.0 3.6 8.3 3.0 7.0 9.1 11.3 (middle tank empty)

e,:;iae Bottom yield 16.5 8.6 3.0 9.1

Deck strut-panel 15.4 8.8 7.0 11.3 Deck grillage instability 16.5 8.6 7.0 11.2

--tons

ee. He assumed that the losses desienated "Foundered" (31% of all losses)

Te. cases

of

complete structural fnauie, although

there were no doubt

nu-Jous exceptions. Assuming a 25-ycer average ship life, he concluded that a

figure of

about 0.006

would be a

rease:elbie value foo the

probability of

failure that has been tacitly accepted over the past 20 years for large

oceangoing ships. In proposing a specific figure for a new design criterion,

however, we

feel that a more

conservative figure should

be adopted, and Dal-.-11's suggestion of 0.001 is tentativcAy proposed (12).

A basic approach to determining the probability level to be used in a design criterion is that of "expected loss", which has been summarized in conventient form by Freudenthal for application to maritime structures in

general (13). It is based on the principle that the best design is the one

that

minimizes

the expected total cost, where the latter consists of the sum

of initial cost and failure cost, as is explained below. It is very

diffi-cult, if not impossible, to assign a dollar value to passengers and crew, but it will be assumed -- perhaps over-optimistically -- that in this day of

ef-. ient communication and life-saving technology a ship may founder without

the crew being lost.

Expressed as an equation, the total:expected cost to

be minimized is,

L ==. I P

1

I = initial cost of the ship (or structure)

P1 = probability

of

failure (in a lifetime)

F = anticipated total cost of failure (replacement cost + cargo loss ±

temporary charter of replacement ship + loss of business from cus-tomer reactions + cost of pollution or other environmental effects, etc.)

Some approximate calculations (14) confirm that the above tentative

fig-tire of 0.001 for lifetime failure probability, is reasonable. However, more

work is needed on this subject, based on extensive, routine reliability data

on actual ship hull

girder

failures.

Numerical Example

A very interesting numerical example of a reliability analysis, leading to an estimate of lifetime failure probability was given by Mansour and

Faulkner (8) for a frigate. Of greater interest for the present purpose

are

results given in less complete form for the 85,000-ton deadweight tanker. In

both examples the demand (or load) probabilities were determined in a

some-whnt :simplified manner. Hence, a recalculation of the tanker load

probabil-ity distribution has been made, in order to permit actual results of a com-plete failure probability calculation to be presented in numerical and graph-ical form. The starting point is the tabulation of "Expected bending moment" vs. significant wave height given in Table IV (8), obtained by means of the MIT Seakeeping program and Pierson-Moskowitz spectra assuming long-crested

irregular head seas. Expected bending moment values were converted to MS of

.record by dividing by 1.25 and to stress by dividing by the section modulus

(deck) of 152,600 in.2-ft. _{Resulting values were corrected approximately for}

the effect of short-crestedness and for an equal distribution of all ship-wave headings using a factor of 0.70 obtained from previous calculations. Data

were plotted

and read off to correspond to our usual wave height _groups,

-4--+ where

Standard deviations of rms.stresses were. c7!atimated on the basis of previous

calculations,

and

the Walden distribution of wave heights in the North

Atlan-tic was assumed. (15).

See

'Table 2 below.

Table 2

Assumptions for Long-Term 'toad Calculations

_.

'-c&

...-k, .)

The

procedure for obtaining

a

long-term stress curve for each weathet

group and then integrating to obtain a single long-term curve is described in

detail elsewhere (5) (16). Results are presented graphically in Fig. 2,

where the cumulative distribution of demand as calculated is shown on the same .scale as the density function of capability, as given by Mansour and

Faulkner (8) for the critical case of deck strpt-pauel buckling in the

sag-ging condition (mean stress = 15.4 tons per _{sq. in.,}

standard deviation =

15.4 x 0.113 = 1.74 tons per sq. in.) and also for the _{case of deck grillage}

instability. See Table 1,

-Numerical solution of the previously given equation for probability of

failure by deck strut-panel

buckling gives

_{a value of p = 1.2 x 10-12 for the}

full load condition if still water bending moment is zero. This would be en-.

tirely acceptable, for it

implies

that only one ship out of 10,000 similar

ships (with lifetimes. of 108 cycles) would fail.

However, When the constant still water bending Moment of 671,000 ft.-'tons (sag) is included, the probability decreases to P1 = 3.7 x 10-8,

com-pared to 1.3 x 10-8 in (8). The reason

for this

moderate discrepancy is

be-lieved to be differences in the methods of calculation of demand. If a ship

encounters 108 cycles of stresS in.a

_normal

_{lifetime, either figure implies}

'that the ship would be expected to fail by deck strut-panel buckling during

its lifetime. This is obviously unacceptable. Assuming

that

the ballast

'loading would give

a much lower probability of failure,

the

combined

proba-bility for loaded and ballast would be slightly reduced, but would not change

the

situation significantly.

- If it is assumed that, the design of the deck is modified to increase its.

load carrying ability to equal or exceed the grillage strength,

then

the

ea-__ 5 Sig. Wave .Height Range Ft. Frequency of Occurrence % Mean RAS Stress Tons/in .2 Standard Deviation. -RMS Stress

Tonslin.2

_

_ 0 - .3 8.75 0.0229 0.0020 3 - 6 23.75 0.0275 0.0033 6:- 9 30.70 . 0.0760 . 0.0066 91 ,-- 12 20.35 0.2294 0.1075 12 - ify 6.90. 0.3670 0.1396. 1i6 - 21 4.95 0.6147 0,1841 21_.,. 27 3.35 0.8716 0,2176 27 - 34 1.06 i.0689 0.2353 .. 34. - 42' 0.17 1.2916 0.2392 > 47.

nj19

.

1,crilA

0.2556

pability is governed by the latter. (Mean.

Strength = 16,5

tonsiin,2 and

standard deviation = 11,2%). On this basis the probability of failure

be-comes,

PI = 7.5 x 10-9. On the basis of a binomial model, assuming a ship 'lifetime of 109 cycles, this means, that there is a lifetime failure

probabil-ity of 0.528. Or 53 ships out of a, fleet

of

100 Would be expected to fail.

This Is also unacceptable, but the result, must be considered tentative for a

number of reasons:

North Atlantic sea

conditions have been assumed, which are generally

more severe than most tanker routes 7- In spite of the famous Cape

of Good Hope rollers.

Assumed standard deviations of structural capability (8)

may

be too

severe.

3, No allowance was made for lateral

longitudinal

bending in oblique.

seas, which tends to increase deck edge. stresses (14).

4. Since the calculation of failure probability depends on the

uncer-tain "tails" of the distributions of demand and capability, there is a possibility for appreciable error in the calculated probabilities. The still water bending moment has been considered to be

determinis-tic. However, in a more complete treatment, the variability of

still

water moffient should be taken into consideration (14).

Nevertheless, results given in this simple example are close enough to

reasonable figures to encourage further work on all aspects of this interest- 5,

ing approach to ship structural design. CONCLUSION

It is concluded that the basic calculation procedures are now available

for rational

ship longitudinal

strength

design on the basis of reliability

principles. The greatest need appears to be for data On the probability

as-pects of hull strength. Second in importance is the need for statistical

da-ta on heavy weather damage to longitudinal strength members related to number

of ships "at risk." A third area for further work is the collection of wave spectra for unusually severe situations, such as the approaches to the Eng-lish Channel (continental shelf) and the waters off South Africa.

,

ACKNOWLEDGEMENTS

This paper

would not have been possible

without

the basic

wOrk of

Man-sour and Faulkner (8) on structural capability, nor without the continuing

support of the American

Bureau of Shipping and Ship

Structure Committee for

research. at Webb Institute. The author wishes to thank various members of

a

the Webb staff for their assistance, including Professor Robert Zubaly,

Pro-fessor Bruce Stephan and Student Assistant R.P. Dallinga (of Delft Technical University).

-6

Pie

-References

.1. E. Abrahamsen

and

N. NordenstrOm, "Design

and

_{Reliability of Ship}

Struc-tures",

Trans, Spring Meeting, SNAtiE. 1970,

?, Nordenstrft, "Methods for Predicting Long-Term Distributions of Wave

Loads and Probability of Failure for

shills",

Appendix VIII. Det Norske

Veritas Research Report No. 69-23e-S, 1969.

3, A.E. Mansour, "Probabilistic

Design

Concepts in Ship Structural Safety

and Reliability", Trans. SNAME,

1972-4; E.G.U. Band, "Analysis; of Ship Data to Predict Long-Term Trends of Hull

Bending Moments", ABS Report, November 1966.

5-,

E:V. Lewis,

'Predicting Long-Term Distributions of Wave-Induced Bending

Moment on Ship Hulls", SNAZIE Spring Meeting, 1967.

6. E.V. Lewis, "Long-Term Applications of

St. Denis/Pierson

_{Technique to}

Ship Design", _{SNAME Seakeeping Symposium, October 1973.}

1.0.J. Karst,

"Statistical Techniques", Appendix A to Reference 16.

A.E. Mansour and D. Faulkner, "On Applying the Statistical Approach to

Extreme

Sea Loads and Ship Hull

Strength",

Transaction of the Royal

In-stitution of Naval Architects, 1972.

Ang, "Elements of Structural Reliability and its Implications in

Design", and, 'Extended Reliability Basis for Formulation of Design

Crite-ria", Proc. ASCE Engineering Mechanics Division Conferences,

"Probabil-istic Concepts and Methods in -Engineering", and "Probability and Random

Processes in Engineering", Purdue University, Lafayette, Indiana,

No-Vember 1969,

.Report of Committee 10 on "Design Procedure", Proc. of the Fourth ISSC

Tokyo, 1970, published by SNAJ. February 1971.

114, 14.1 Beer, "Analysis

of

World Merchant Ship Losses', Trans. RINA, 1969_

. 12. R.S. Little, F.V. Lewis, and F.C. Bailey, "A Statistical Study of

Wave-Induced Bending Moments. on Large Oceangoing Tankers and Bulk Carriers",

Trans. SNAME, 1971.

13 A-M. Freudeuthal, and W.S. Gaither,. "Probabilistic Approach tO Economic

Design of Maritime Structures", XXIInd International Navigation Congress

Section II,

Subject 5, Paris,

1969.

14. E..V. Lewis, D. Hoffman, W.M. Maclean, R. van Hoff, and R. Zubalye "Load

Criteria for Ship Structural Design", Ship Structure Committee Report

SSC-240, 1973...

15, _{IL. Walden,_ "Die Eigenschaften der Meereswellen im Nordatl}

Ozean", Deutscher Wetterdienst Seewetteramt Publ. No, 41,

16: D. Hoffman, 3. Williamson, and

E.V.

Lewis, "Analysis and

of Full-Scale Data on

Midship

Bending Stress of .Dry Cargo

Structure Committee Report, SSC-196, June 1969.

--t antischen. 1964 Interpretation Ships", Ship N. A.H-S.

-7-i!,3-' 10-5

0-5 r.-T4 VO-3

(X>Xj) TOTAL PROBABILITY OF EXCEEDING_Xj

Fig- 1.

:Typical Long-Term Probability Curves, Extrapolated

from Full-scale Data and Predicted from Model Tests.

S.S- Wolverine State in the North Atlantic

(5)

4g 11/11/1111 . LONG

,MODEL

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