Geometric properties of shipbuilding structural profiles in probabilistic terms to be used in elastic bending strength calculations

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Geometric Properties of Shipbuilding Structural Profiles in Probabilistic Terms to Be

Used in Elastic Bending Strength Calculations

Lyuben D. Ivanov

American Bureau of Shipping, USA. Corresponding author: LIvanov@eagle.org

Abstract

As a result of the corrosion wear, which is treated as a random phenomenon, the geometric properties of shipbuilding structural profiles (bulb plates, inverted angles/L-profiles, channels, symmetric or asymmetric Fundia T-bulbs, symmetric or asymmetric built-up T-bars, and flat bars) are presented as random functions of the ship's age. Evenly distributed corrosion wear along each cross section's dimension is assumed (different corrosion wear may be considered along each of the cross section's dimensions). Taylor series approximation method is applied to present the geometric properties used for assessment of elastic bending strength in probabilistic terms. The proposed method allows for comparison between different structural profiles using as a criterion the probabihty that these structural profiles will meet given Renewal Criteria requirements throughout the ship's lifetime. Based on its calculation, one can select the most reliable structural profile. The method also allows for better planning of ship repair and maintenance work.

The method can also be used to reveal the already implemented risk in the existing empirical Renewal Criteria through analysis of the structural components, their location, ship's type and size. This is one of the first steps towards development of Reliability-based Rules for building and classing of steel ships.

Keywords

Shipbuilding structural profiles, probabilistic geometric properties, corrosion of ship structures

1 Introduction

The application of the time-variant rehability theory into ship structures design, repair and maintenance requires presentation of all factors influencing the reliability ofthe hull structure as random functions of the time. These are the external and internal loads, geometric properties of the hull girder and its components, mechanical properties of the parent material and the welding joints. The report deals with probabilistic presentation of the geometric properties of shipbuilding structural profiles.

The randomness of the geometric properties is considered as a result of the probabiHstic nature of the corrosion wear. Only uniformly distributed corrosion wear (i.e. overaU corrosion) within each cross section's dimension is taken into consideration. The paper deals with geometric properties that are used for assessing elastic bending strength and is a continuation of the previous work done on deterministic calculation of the geometric properties of shipbuilding structural profiles [11] These geometric properties are presented in probabilistic terms using first order Taylor series approximation method. I f the second and third term in the equations for the mean value and the variance are kept, their contribution is neghgible. In the extreme cases, their contribution barely reaches 0.05-0.10% of the value of the first term [10].

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Geometric Properties of Shipbuilding Structural Profiles in Probabilistic Terms to Be Used in Elastic Bending Strength Calculations Lyuben D. Ivanov

The apphcation of the Taylor method for probabilistic presentation of the other geometric properties used for calculation of the shear and torsion strength and the plastic section modulus becomes too cumbersome. Therefore, they can be determined by an approximate method [12].

The foUowing structural profiles are covered by the model: Symmetric or asymmetric Fundia T-bulbs (Fig. 1 and Fig. 2); symmetric or asymmetric built-up T-bars (Fig. 3 and Fig. 4); Bulb plates (Fig. 5 and Fig. 6); Inverted angles (Fig. 7); L-profiles (Fig. 8); Channels (Fig. 9); Flat bars (Fig. 10). The geometric properties of these profiles are calculated following the method described in [11].

The essence of the Taylor series method [24] is the linear transformation of the function, Y, in the vicinity of its mean value. Applying first order Taylor series expansion for non-correlated, statistically independent variables, one can obtain:

(1.1)

where the subscript m means that the expression within the bracket should be calculated substituting Xj with its mean value Xj. The Cross Section Area, First and Second Moment of Area, Product of Inertia, and Section Modulus for symmetric bending of different types of structural profiles applied in the shipbuilding industry have been analyzed using the above given formulas.

Under these circumstances, one should make assumptions about the maximum possible initial value of the geometric property under consideration (i.e. the upper truncation value). • The geometric properties of shipbuilding structural profiles cannot decrease indefinitely during the ship's operational lifetime. Again, one should make an assumption for the minimum possible value of the geometric property under consideration (i.e. the lower truncation value). The "safest" assumption is to set this lower truncation value to zero. This assumption is not realistic because the structure's dimensions wiU shrink but wiU not disappear just because of corrosion wear (the assumed minimum possible value of the geometric properties is given further in this Section).

The procedure presented in the paper is a general one and allows for change of any of the assumed boundaries depending on the peculiarities of any specific case.

Let's continue with the parameter y presented in the foUowing non-dimensional form:

y = Y / Y _ (2.1) where Y„„,„ is the nominal value of Y

Let's mark the boundaries of the random parameter y as follows:

b„ = maximum possible value of the corresponding geometric property; b, = minimum possible value of the corresponding geometric property.

2 Type ofthe probabilistic distribution

It was shown in [10] that the above-mentioned geometric properties of shipbuUding structural profiles obey the Gaussian (normal) distribution. The normal distribution should be truncated because of the following reasons:

• The geometric properties cannot increase due to corrosion wear, i.e. they cannot be greater than their corresponding i n i t i a l values. However, what is the i n i t i a l value? A l l manufacturers of shipbuUding structural profiles specify the tolerances within which they guarantee the dimensions of the corresponding structural profile. The reason for the deviations f r o m the specified nominal values of the dimensions is the unavoidable inaccuracies of the manufacturing process. In other words, the initial dimensions are of a probabUistic nature.

Modern technology allows for very high accuracy of the dimensions, but it is questionable to target entire elimination of the existing tolerances. High accuracy is not an end in itself There is no sense increasing the accuracy i f the effect on the huU structure strength is insignificant whfle the production cost may substantially increase [3].

The starting point for finding the equation of the truncated normal distribution function of any geometric property under consideration is the premise that the area below the truncated normal distribution with boundaries b„ and b; should be equal to unity (as for a normal distribution with boundaries - oo, -F oo). The difference between the ordinates of the t\vo probability density functions wiU be a constant C that could be determined the following way [21]:

ƒ p, (y)dy = C ƒ p(y)dy = c [ p ( b „ ) - p ( b , ) ] = I (2.2)

C = l / [ p ( b „ ) - P ( b , ) ; (2.3)

Thus, the truncated normal distribution function of y.pXy). ivUI be: 0 b u - y - 0 b / - y 0 b u - y - 0 a,, ^ ' J > J a,, ^ ' J P(y)

where p(y) = - _exp y - y

(2.4)

(2.5)

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y = mean value of y, a,= standard deviation o f y, ï> = Laplace integral of probability.

The procedure for probabilistic calculation of the geometric properties of shipbuilding contains the following phases: 1. The mean values and the standard deviations of the

corrosion wear are determined for the attached plate, web plate, and the bulb head/flange for each year of the ship's operational lifetime.

2. The mean values and standard deviations of all dimensions of the structural profile's cross section are determined for each year of the ship's life span. The most accurate source of information might be the statistical data collected by the manufacturers. If they are not accessible, one could perform statistical analysis of the "as-rolled" or "as-built" structural profiles delivered in the shipyard [ 10]. Another way is to use the tolerances given by the manufacturer. As a first approximation, the mean value can be taken as equal to the middle of the tolerance range, and the standard deviation as equal to 1/5 of the tolerance range.

3. The mean values of any geometric property are determined with eq. (1.1) for each year of the ship's lifetime.

4. The first derivatives of any geometric property relatively to each dimension of the structural profile's cross section are determined. The equations of the first derivatives were substantially simplified ivithout loosing their accuracy. It was done by finding the relationship between the derived formulas for the first derivatives and previously obtained formulas for some geometric properties.

5. The variance/standard deviation of any geometric property is determined with eq.(l.l).

6. Kno^ving the mean values and the standard deviations of any geometric property, its truncated normal distribution function is built for any ship's age (see Fig. 11). The assumed upper and lower truncation values are given further on. 7. The envelope o f the truncated normal distribution

functions is built by putting together the probability density functions calculated with eq. (2.4) for each year of the ship's lifetime. It is a smooth surface in the 3D space (see Fig. 12).

It is worth drawing the attention to the following:

The stiffeners in ship hull structures are simply replaced when they do not meet the requirements of the Classification Societies' Renewal Criteria. It is hard to believe that they will be repaired to increase their strength (especially, rolled profiles). The hull girder is repaired, but not the individual stiffeners. The smoothness of the 3D surface is breached (sudden elevation occurs) when the corroded stiffener is replaced by a new one to meet the Renewal criteria. Then, the 3D surface becomes smooth again. This property of the envelope facilitates its calculation and allows for several practical applications. For

example, one could formulate the following problems with practical importance:

• For given time T what is the probability of the geometric property, y, under consideration being between y„and y,:

P ( y , < y ^ y „ ;T) = CT ƒ P T ( y ) d y _(2.6)

where (y) is the probabihty density function of the given geometric property, y, under consideration at time T; C^ is the coefficient determined with eq.(2.3).

• What is the probability of the geometric property y being between any y,, and y, during the time interval (T„ , T,)? I.e. jf C , J p T ( y ) d y P ( y , < y < y „ ; T , < T < T „ ) = i L ^ An dT J CT ƒ PT(>')dy d T (2.7)

where y„ = upper boundary of y; y, = lower boundary of y; T„ = upper boundary of the time T; T, = lower boundary of the time T; b„ = maximum possible boundary of y (i.e. the upper truncation of y ) ; b, = minimum possible lower boundary of y (i.e. the lower truncation of y ) ; pT(y) = probability density function of y at any time T;

Eq. (2.7) represents a kind of "average" probability thaty will not exceed given limits within a specified time-interval. The integral in the denominator of eq. (2.7) wthin the brackets multiplied with the coefficient C, should always be equal to unity. Consequendy, the denominator in eq. (2.7) is equal to

T„ - T,. Thus, the denominator's geometric interpretation is the volume below the envelope w t h i n the given time interval AT =T„ - T,. This volume can be likened to the volume of a parallelepiped with area of the base equal to unity and height equal to T„ - T,. The nominator's geometric interpretation is the volume below the envelope within given boundaries T„, T,

and y u , y,. I t may be called volume o f a "deformed" parallelepiped. Thus, the geometric interpretation of eq. (2.7) is the ratio between two volumes: the volume of the "deformed" parallelepiped and the volume of the parallelepiped mth base equal to unity and height equal to T„ - T,. I n other words, the "average" probability is equal to the volume of the "deformed" parallelepiped (see Fig. 13) divided by T„ - T,. Consequently, eq.(2.7) can be rewritten as

P ( y , < y < y ,

;T,<T<T„) =

T,. f

— - j C , | p , ( y ) d y dT (2.8)

As to the geometric interpretation of eq. (2.6), it is equal to

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Geometric Properties of Stiipbuilding Structural Profiles in Probabilistic Terms to Be Used in Elastic Bending Strength Calculations Lyuben D. Ivanov

the area below the probability density function of y for time T within the given boundaries y„ and y,(seethecrosshatched area in Fig. 13).

Using eq. (2.8) as a criterion, a parametric study was carried out on the effect of different maximum and minimum possible values of y (i.e. upper and lower truncation limits b„ and b,) on the "average" probability. Their effect was calculated for the most fi-equently used geometric properties: Total Cross Section Area, Cross Section Area of the stiffener alone. Moment of Inertia relatively to the Centroidal Axis X,, and the Section Modulus for symmetric and asymmetric bending. The maximum possible value of y, b„, was varied between 1.00 and 1.10 while the minimum possible value of y, b,, was varied between 0.5 and zero. In all cases the differences between the "average" probabilities were less than 1%. Based on this analysis, the author recommends the following values for b„ and b,:

b„ = 1.10 b, = 0 (2.9)

The author found that for the corrosion wear reported in different publications, such as [4], [16], [17], the coefficient Cjis practically equal to unity. In other words, the difference between the truncated normal distribution and the traditional normal distribution is neghgible. Then, eq. (2.7) could further be simplified and rewritten as:

P(y, < y < y„ ;T, < T < T„) = - 1 - j j" p , ( y ) d y dT

(2.10)

Basic and auxilliary

non-dimensional functions.

First derivatives

The basic and auxiliary non-dimensional functions have been derived in two formats: accurate and approximate [11]). The approximate format substantially simplifies the calculations for the first derivatives of the geometric properties. The analysis performed i n [ I I ] showed that the difference between the accurate and approximate equations does not exceed 2.7%, which is quite acceptable in this case bearing in mind the existing uncertainties. The first derivatives of the auxiliary non-dimensional functions and the auxihary parameters are given in Appendix A. As said above, only the first derivatives of the approximate auxiliary functions are given in Appendix A and used i n this work because of their simplicity and sufficient accuracy. In addition, the following auxiliary parameter, T|, is introduced to simplify the writing of the equations:

4 Fi rst derivatives of th e

geometric properties of

shipbuilding structural profiles

Bulb Plates

The components of the first derivatives of all geometric properties of bulb plates relatively to axis Xj = h, b, t j , t„, R j , RJ, R4, R5, and P are given i n Appendix B. Using these components, one can find:

First derivatives of the Cross Section Area:

g A s _ a A B , L ^ ^ R 2

^A

R

3

5AR4

^

SA

RS

~ 8x- dx-, 8x. öXj ^^'^^ First derivatives of the Static Moment relatively to axis X:

dx. dx. ' S.Xj " dx. ' ÖXj 3x, First derivatives of the Static Moment relatively to axis Y:

dSy s

_^S

Y_B

,I.

^^Y,R2

^^Y,

R

3 ^^Y.R4

^ ^ ^ Y ,

R5

SXj dX; ' 5Xj ' SXj ' SXj dx, ^^'^^

First derivatives of the Moment of Inertia relatively to axis X:

^Jx.S_^lxB.L ^Ix

.R2

^hR3 ^lx

.R4

^^Ix

.R5

SXj 5Xj dx, dx, dx, dx^ ^ ' '

First derivatives of the Moment of Inertia relatively to axis Y: 8Iy s_9lY.B.L

^

I

Y

,R2 ^I

Y

,R3

^^Y. R 4 _^ R 5

dX: dX: dX: dX; dx: dx, (4.5)

First derivatives of the Product of Inertia relatively to axes X andY:

^'xY.S _ ^IxY,B,L

^Ixy

,R2

^^XY.RS ^^XY,R4 ^ ^^XY.RS

7t-2P

(3.1)

SXj 9Xj öXj dX; 9Xj (4.6) Inverted Angles or L-Profiles

The geometric properties of inverted angles/L-profiles and their first derivatives are determined with the equations for bulb plates, substituting Rj = R j = 0.

Channels

As in [11], a horizontal section of the profile though h/2 is drawn. This section is denoted with X„ (see Fig. 1.6). Thus, two profiles are created - one above the new axis X„ (its geometric properties are marked with the subscript o) and one below the horizontal section (its geometric properties are marked with u). For the "above-the-horizontal-section" profiie

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Geometric Properties of Shipbuilding Structural Profiles in Probabilistic Terms to Be Used in Elastic Bending Strength Calculati Lyuben D. Ivanov

the formulas for the "Basic" Profile in [11] are used with the following input parameters: R,= R j = R j = 0, h„ = h/2, b, t„, s = 2 t„, t j . p , R2R5

Notes:

1. h„ is the height above the middle horizontal secfion 2. h is the height of the real channel profile.

3. The components of the first derivative of ali geometric properties of channels relatively to axis X , = h, b, ti, t,„ R j , R5, and P are given in Appendix C.

4. In some of the manufacturers' standards the thickness of the flange, t,, is given at section through (b-t„)/2. To obtain the parameter t j that is used:

90-B b - t „ .

t j = t f - R 2 t a n ^ -tm^ (4.7)

5. If the thickness ofthe flange, tf, is given at section through b/2, eq. (4.7) is transformed to

= I f - R 2 tan.^^-Ê-^tanp

^ = 2 ^ = 2

5x J dx. dx; dX:

(4.8)

(4.9)

ThederivativesSAg L / 5 h and 9A „ ^ / ^ ( x j T^h) are given i n Appendix C while derivatives ö A g ^ j / ^ - ^ i ^"^^ Ö A j j j / S x are given in Appendix B.

First derivatives of the Static Moment relatively to axis X: 9 Sv , 9 A . , f o r x ; = h — ^ = A „ + h ^ (4.IO) Bh for X; h 9Sv dx; dh dX: Dx. dX; (4.11)

First derivatives of the Static Moment relatively to axis Y:

9S _Y,S

9X; = 2 dx, dX: dx, (4.12)

The derivatives 9 S ^ ' ^ ^ i given in Appendix C while derivatives 9 S Y ^ 2 ' ' ^ ' ' i and dSyj^^/dx^ are given i n Appendix B.

First derivatives of the Moment of Inertia relatively to axis X: whenx, T^h 91 x.s _2 dX; when X, 91 _{,.a^Îx,R5 Îx,R2} dX; 9X; dh. h f d l dh Xa,h,a , '^^X.RS 9 x , ÎX,R: 9 A , L " 2 dX; (4.13)

V

Sh + h A 9h 9h (4.14) 2 9h The derivatives dU dh 9 ( x i ^ h ) dL. dh • and 91v

Appendix B while derivatives

9 ( x i ^ h ) „ 91. are given in Xa.L.a S A ^ L 9h dx, Dh and 9 A ,

• are given in Appendix C. 3 ( x j ^ h ) '

First derivatives of the Moment of Inertia relatively to axis Y: 91

dX; dX; dX: 9 X ;

h^ 9 A „ , 2 dx: (4.15)

The derivatives 9Iy jj2 / 9 x j and Sly j ^ j / 9 x j are given in AppendixB;9lY „ L / 9 X j derivatives - in Appendix C. First derivatives of the Product of Inertia relatively to axes X andY: whenXj = h "JxY.s _ 1 dh 2 when X ; *h SI S y . s + h — Sh (4.16) XY,S _ h ~ 2 ÖX 9S _Y,R2 dx-. dx, SX: (4.17)

Asymmetric Fundia T-Bulbs

Based on [ 11 ], a vertical section of the profile through the middle of the web plate is drawn. Thus, two profiles are created - one from the right-hand side ofthe section and anodier one from the left-hand side of the section. The characteristics of the right-of-the-section profile are marked with the subscript r while those of the left-of-the-section - with /. The width of the "right" seaion is b, = c while that of the "left" section is b, = b-c. The web plate thickness of each of these two structural profiles is:

(4.18) The input parameters are: x ^ = h, b, c, t, s, t„ , R|, R j , R„ and P The components of the first derivatives of all geometric

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properties of asymmetric Fundia T-bulbs relative to h, b, c, t, s, t,v, Rl, R2, R3, and P are given in Appendix D.

5x: dx, dx.

1(3

dX; dX; (4.19)

First derivatives of the Static Moment relatively to axis X:

dX: dX; dX;

dX; dX;

SSX.R2,, 5Sx,R2,; (4.20)

dX; dX:

First derivatives of the Static Moment relatively to axis Y:

dx, dx; dX;

aSy,R2.r gSy.R3,, ^ 8Sy,,,2,, ^ aSy „3 , (4.21)

5Xj 3Xj S x j 9 x j

First derivatives of the Moment of Inertia relatively to axis X: 3Ix,S _ SIx.L,r ^ aix,L.;

Sx; dx r a i . _{I.r ^ aix,R3.r}

5X; dx.

dl _{. L2,r aix,R2./ (4.22)}

dx. dx,

First derivatives of the Moment of Inertia relatively to axis Y:

aiy.s S l y !, , ^ Sly.L,; ^aiy.Ri.r ax, dX; dX;

^^Y,R2,T a^Y,R3,r a^^Y.R2,; a-'Y,R3,; (4.23)

dX: dx, dx,

a^XY.S _ SIxY,L,r ^^XY.L.I a^XY.R2,r 8x j dx

dl

dx. dx,

i,i ^ a^XY.R2.1 ^ ^^XY.R3,l (4.24)

dx. dx, dx,

Symmetric Fundia T-Bulbs

The same approach is used as for asymmetric Fundia T-bulbs. Based on [ 11 ], if the properties ofthe "right-hand side" of the corresponding profile are taken as a basis, all geometric properties of the whole profile could be easily determined as

shown hereafter. The components of the first derivative of all geometric properties of symmetric Fundia T-bulbs relatively to axis X , = h, b, c, t, s, t„, R„ Rj, R j , and P are given in Appendix D.

5As ^ a A , dx, dx,

(4.25)

(4.26)

d x ,

dx-First derivatives of the Moment of Inertia relatively to axis X:

dx, dxi (4.28)

Asymmetric Built-up T-Bars

The same approach as for asymmetric Fundia T-bulb is applied. Based on [11], a vertical section through the middle of the web plate is drawn that divides the structural profile into two parts - "right" and "left". Thus, the geometric properties of the whole profile could be determined by a combination of the geometric properties of both parts. The components of the first derivative of all geometric properties of asymmetric built-up bars relatively to h, b, c, t j , and t„ are given in Appendix E.

a A s _ a A , a A ,

dx, dx, ox. (4.29)

First derivatives of the Static Moment relative to axis X:

öXj dx, a x j

First derivatives of the Static Moment relatively to axis Y: (4.30)

as y s SSy , SSy ,

dx, dx, dx, (4.31)

(4.32) S I x . s S i x . , , S i x . ,

a x j a x j a x j

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Geometric Properties of Sliipbuilding Structural Profiles In Probabilistic Terms to Be Used in Elastic Bending Strength Calculations Lyuben D. Ivanov

3X: SX; 3X: (4.33)

SIxY.S _ SIxY.r SIj-Y ,

3X : Ö X: 5X: (4.34)

Symmetric Built-up T-Bars

As for symmetric Fundia T-bulbs, the properties of the "right-hand side" of the corresponding profiie are taken as a basis for calculation of the geometric properties of the whole profile. The components of the first derivative of all geom. properties of symmetric built-up T-bars are given in Appendix E.

9x: dx, (4.35)

(4.36) SSx.s^2_SSx^

9x. dx,

(4.37) dX: dx,

(4.38) dx, dx.

Flat Bars and Attached Plate

S e y ^ 1 S x j . SSy 5A Sy — dx; [ dxi 1 ( dS Y,s S S y p 6x1 _{S x j}

_J

dA^^dAp) 9 x j S x j (5.1)

Ordinate of the centroid relative to axis X:

a x : a x j ^ dXi

a x j S x j

Centroidal Moments of Inertia: aA -ex — aix . g s x axi dx: 2 ^5Sx, s _{— 1 -} SSx,p , 5x, axi SAs dAp a x i (5.2) -ex f 8 A s ^ a A p ^ axi a x j (5.3) S l y i a k ax j dx a x j a x j - e y SSy, s ^ SSy p dx\ dx: ^dAs dAp^ - e y + £-dx; 5 x : J (5.4)

Product of Inertia relatively to centroidal axes X I , Y l

a i X l Y l Six Y a x j a x j aA asv - + e x e Y - e x ^ - e aSv a x j dxi ' dxj s i x Y, S s i x Y,p -+—:——+exe a x j dx; (dAg dArj

^

-X SSy, s ^ SSy^ p dx; _Sxi

_j

-ey dx; dx; ax; a x j (5.5)

The first derivatives of all geometric properties of flat bars are given in Appendix F and those for the attached plate-in Appendix G.

5 Other geometric properties

First derivatives of the foUowing geometric properties: Abscissa of the centroid relative to axis Y:

Maximum Moment of Inertia

91,,, S I . „ _ 1 f S I x i a x j 2 | a x i dx, 1 ^ IY, - IX , ^ ' 21. 91v dx, IYI -1X1 Slyi Six, dx, dx, (5.6)

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Minimum Moment of Inertia 5 U 1 fSIx,_^3Iv, 2 [ S X i Sx; I + 1x1 Yl IYI -1X1 21 + ^'XIYI 5X : Sx ; (5.7)

Angle of the Principal axis X j relatively to axis X,

S a x 2 S X | Sx,

-(^Yl ^Xl) IxiYl SIYI SIxi V ^ X j SXj (IYI-IX.) +«^IYI

Angle of the Principal axis relatively to axis X, (5.8)

S«Y2

Sorx2 SXj SX;

Moment of Inertia relatively to the Principal axis X2 (5.9) Sx j Sx Sl Sx, + - ^ c o s ax2 + - ^ L( ' Y I Ixi)sin2orx2 --2Ix,YiCOs2ax2;

Moment of Inertia relatively to the Principal axis Yj

din S/y , , d l y . y , ^- ^-sm^ay,+ 5" ' s i n 2 a y 2 + (5.10) Sx, S.T S/r, S a , S.Y, + - ^ c o s a^. 2 +-^[{^x 1 - / r i ) s i n 2 « ^ 2 + + 2/A-iriCOs2cjr^.2' (5.11)

Radius of Gyration of the Area relatively to axes X, and X j ^ g ' x i J SA

S r x , _ ' " S X j ' ^ ' S X j Srx2_ Sx, ^ ^ S X j (5.12)

Radius of Gyration of the Area relatively to axes Y, and Y^ SIY2 . SA A ^ ^ - I Sry, SXj Sx,

ST

V

, S

X

;

SX Y2 _ " " i SXi 2fA%~, ' Sx, 2 , / a % '-i- (5.13)

Perimeter of the stiffener

The formulae for the first derivatives of L are given in Appendix H. Section Modulus for symmetric bending

First derivatives of the Section Modulus relatively to axis X;.

Sx ;

Sh Scy Sx: Sx,

(5.14)

where S I ^ i / S x j is determined with eq. (5.3) , Sex / S x ; is determined with eq. (5.2), while S I ^ , / S x j is calculated with the formula:

Sh SX:

1 when X j = h

0 when x, 5^= h (5.15)

Section Modulus for asymmetric bending The first derivatives of Z.asym are:

_ •'-'asjTn [ I 0 1 x 2 S y 2 , m SXj

_1X2

[Zasj™ SXj Sx, Sa -y2 _Sx. s m ( - « x 2 )

(-«X2)-

5(-«X2)-' r 2 S I x 2 SIY2 1 x 2 + Iv Sx, Sx, SX, IY 2 ; 1 x 2 S a x 2 X , „ + Sx, , cos

(-«X2)

(5.16) IY2 SX;

The first derivatives of-x^.^and y2_„ are given in Appendix H.

6 Corrosion wear

There are a lot of published data on corrosion wear. For the sake of brevity, only the basic trends would be mentioned. According to [7], [16], [17], and [18] there is no tendency of wastage acceleration with aging because the corrosion wear rate is represented as an average corrosion rate per year. Hence, the corrosion wear is a linear function of the ship's age. It follows from [2] that after ship's age of 12-14 years, an acceleration of the corrosion wear occurs. Thus, the corrosion wear could be represented by two Unear functions with tsvo different slopes vs. ship's age. The data in [ 13] leads to a conclusion that corrosion wear stabUizes with ship's aging and even decreases (this conclusion is drawn on the basis of statistical analysis of ships in operation up to 45-46 years).

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Geometric Properties of Stiipbuilding Structural Profiles in Probabilistic Terms to Be Used in Elastic Bending Strengtti Calculations Lyuben D. Ivanov

All above given references are based on statistical analysis of thousands of data performed by qualified and respected researchers. Despite this fact, it is hard to accept that this issue is finally clarified. There is still a lot to be done before one could have greater confidence i n the prediction methods for corrosion wear rate. One of the important tasks is to present the corrosion wear rate as age dependent [ 19]. It is a hard task but worth spending time and effort, especially when Time-dependent Reliability Theory is appUed.

Probably better agreement between different statistical analyses could be achieved i f time dependent corrosion degradation is separated into three phases as proposed in [23]. The first one is the time when there is no corrosion because the protection of the metal surface works properly. The second one is initiated when corrosion starts destroying the protection. The third phase reflects the corrosion process as a result of the protection provided by the oxidized metal.

The rate of corrosion wear depends on many factors like ship's operational life, quality of steel, methods of corrosion protection, operation conditions, etc. The combination ofall these factors can vary substantially from case to case. More accurate prediction of the corrosion wear can be achieved only through incorporating more detailed theoretical understanding of the various processes which affect corrosion, i.e. building up a phenomenological model for the probabilistic description of the corrosion [14], [15]. However, even then considerable uncertainty in the probabilistic model of the corrosion wear will stiU remain

Under these circumstances, the only practical approach is to select one mathematical model to start with the calculations (for example - presenting the corrosion wear as a linear function with one slope vs. ship's age) and adjust it after each thickness gauging. Although the existing uncertainties in its prediction are too many, this simplified approach could still contribute to a better understanding of an aging ship's strength.

A significant number of detail records exist showing that the overaU corrosion wear prevails in most cases. One illustration of this fact is Fig. 14,originallymadebyMr.RongHuang [9]. Unfortunately, even the most detailed records do not include data for the change of the radii in the corners of the structural profile. To deal with this situation, several assumptions are made to ensure contmuity of the cross section due to shrinkage ofthe structural profile as a result of the corrosion wear (see further in this Section).

In general, the corrosion wear of the bulb head in vertical and horizontal direction is not the same. The figures in the corners of corroded rolled sections cannot be described as figures with radii anymore. However, bearing in mind that the effect of these figures on the geometric properties for elastic bending is relatively smaU, one could assume that they could still be described as figures with radii (these radii may be different from those of the "as-rolled" structural profile). Thus, i t is possible to use all equations for calculation of the geometric

properties of corroded structural profiles derived in this work. As to the effect of the angle underneath the bulb head on the geometric properties for elastic bending, its effect is not neghgible. Although the available statistical data for its change due to corrosion wear are insufficient, all first derivatives of the geometric properties relatively were derived for completeness. Thus, the mathematical model f o r implementation of the Taylor series is suitable for full use in the future when more complete statistical data might be available. For "present day" use the angle is assumed as unchanged due to corrosion. To ensure continuity of the cross section when it shrinks due to overall corrosion, the assumed new cross section shape is shown in Fig. 15 that refers to bulb plates, inverted angles/L-profiles, and channels. As shown in Fig. 15, the corrosion of the bulb head, b, T , at time T along its upper surface, along its side opposite to the web plate and along its inclined surface underneath is assumed as uniform. Its magnitude has been determined from the following equation:

cos [5

l+cosf5

cosf! <^b.T =^v.T (6.1)

where is the overall corrosion wear of the bulb head in vertical direction at time T (it is defined as the difference between the ship's age and the longevity of the corrosion protection, i.e. T = T, -T, where Ts is the ship's age and T„ is the longevity of the corrosion protection. Consequendy:

. J 1 . D _1+cosP (6.2)

The same principle was appUed for Fundia T-bulbs bearing in mind that the leading parameter is t instead of t j (as is the case with bulb plates). The assumed shrinkage of built-up T-bars is shown in Fig. 16 and that of Flat bars - in Fig. 10.

The new dimensions of all structural profiles under consideration used in this work are given hereafter.

Bulb Plates

Bearing in mind eq. (6.2), one could calculate the new dimensions of the cross section of corroded bulb plate at time T with the following formulae:

, , cosp £ . , . , < ,T cosp 2 1+cosP _(6.3)

As to radii R2 j and R3 ^, the foUowing equations were appUed: J. „ cosp „ T. cosB „

^•^ ^ 1+cosP '-^ " ^ 1+cosP '-^ '

K.T = corrosion wear of the web plate at time T; 5,.j = gauged corrosion wear in vertical direction of the bulb head at time T;

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8 = assumed corrosion wear ofthe bulb head at time T used in the calculations (it ensures continuity of the cross section); v„ = "vertical" thickness of the bulb head for the "as-rolled" profde measured in the middle of b; VT = "vertical" thickness of the bulb head at time T measured in the middle of b^.

The vertical sections where v„ and v^ are measured may not be the same. This discrepancy does not have any significant effect on the results for the probabilities P(y; < y < yu", T) and P (y; < y < yu; T; < T < Tu). The radii and R j j and the angle P were assumed as unchanged, i.e.

R ,

_P

_T

_=P

(6.5)

The equations for the parameter t2 were determined the foUowing way:

If originally t j > R3:

* ^ " 1+ c°o?p'^v. T; t W.T = t „ - ; s T = s; p T = P (6.10)

R , , , ^ R , - ^ .

Asymmetric and Symmetric buUt-up T-bars h,. = h — ^ bx=b-5hor,T

^v,T (6.11)

(6.12)

(6.13) Where the subscriptT earmarks the value of the corresponding cross section's dimension at time T; 5„,T is the overall corrosion wear of the web plate at time T; 8|,„,j is the gauged horizontal overall corrosion wear of the flange at time T (i.e. corrosion in direction paraUel to axis Y)

__^!P_(,anp-t-tanii) 1+cosp'

«3

if

1

2

A . T j__cosp_, , 14-cos P^ " (6.6) Flat bars hx = h - ^ v , T Attached plates tp,T = t p - ^ p , T ' p , T = / p (6.14) (6.15) If originally t2 = R,: . _ cosp , l+cosp^"'^ (6.7)

where 5 is the corrosion wear of the thickness of the attached plate at time T.

where the subscript T earmarks the value of the corresponding cross section's dimension at time T; 5V,T is the overall corrosion wear of the web plate at time T; 5 is the gauged vertical overaU corrosion wear ofthe bulb head at time T (corrosion in direction parallel to axis Y).

Inverted angles/L-profiles

The dimensions of the cross section are calculated with the equations for bulb plates. The only difference is for L-profUes when calculating the parameter tj. In the specifications ofthe manufacturer, the thickness of the flange and radius R j are given (see Fig. 8). Then, t j can be calculated with the equation 'z.T - l f . T - R 2 , T - t f - ^ v . T - R l . T (6.8) Channels

The new dimensions of the corroded cross section are calculated with eqs. (6.3) - (6.7), bearing i n mind that R3= R, = 0.

Asymmetric and Symmetric Fundia T-bulbs t, t. cosp ^ h^. = h —S.. T ^ I-HcosP ''^ cosP ^ CT

=C

—S,, T ^ 1-t-cosp "'"^ H-cosP "'^ (6.9)

Calculation of the mean values

and the variances / standard

deviations of the cross section's

dimensions

Formulae for the general case

(i.e. for any law of the corrosion wear)

The derived set of formulae for probabilistic presentation of the geometric properties of corroded shipbuUding structural profiles is valid for any type of the probabUity density function of the corrosion wear. Consequendy, for any dimension x of the cross section one could write:

« ^ x . i l D x . T ^ D x . o + D ^ . T (7.1) where Xy = value of x at time T (x is any dimension of the cross section that is used to calculate the geometric properties of the cross section); x„ = initial value of x for roUed" or "as-built" structural profile; 5J,T = corrosion wear of the corresponding parameter x at time T; x ^ = the mean value of X at time T; x^= mean of the initial value of x (x is any dimension of the cross section). It is assumed here that it is equal to the mean of the nominal value of x. UsuaUy, it deviates

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Geometric Properties of Sliipbuilding Structural Profiles in Probabilistic Terms to Be Used in Elastic Bending Strength Calculations Lyuben D. Ivanov

from the nominal values given in the specifications for shipbuilding structural profiles up to 1 -2% [10]; , T = mean value of the corrosion wear of x at time T; y is the variance of X at time T; D^ ^ is the variance of the initial value of x. It is also assumed here that it is equal to the variance of the nominal value of x; D^.T is the variance of the corrosion wear of X at time T. The "o" subscript is omitted in the text following below for simplicity.

Linear dependency of die corrosion wear on die time As a first approach, a constant average corrosion wear is assumed throughout the ship's lifetime after failure of the corrosion protecnon. Then, any parameter x at time T can be calculated with the formula:

XT = X - 5 . r = x - T 5 ^ _ „ (7.2) where 8^ „ = average corrosion wear per unit time of any parameter x. Consequently, the mean and the variance of x will be:

x , I X 5 „ (7.3) where Dj „ is the average variance of the corrosion wear of any parameter x per unit time. Thus, the mean values and variances of all dimensions of the cross section used as input data for probabilistic presentation of the geometric properties of shipbuilding structural profiles wiU be:

Bulb plates h 7 = h - T - ^ 5 : : l+cosp • D , , = D , + f T ^ 2 £ P . •"•^ I, l+cos( (7.4) PT=P D p T = D p (7.9) (7.10) (7.11)

Mean Value and Variance of the parameter tj j when originally

^2,1 - R 3 . T = ^ 3 - T - ; - 5 . , 1+cosP D.2,T=DR3,T=DR3+ T— cos(^ -cosp (7.12)

Mean Value and Variance of the parameter t2,T when originally tj > R3: if t j - T - - ^ ^ ( t a n p + t a n i i ) 1-1-cosp cosP l+cosp (lanp-t-lanii) (7.13) D . 2 , i = D . 2 + T - j _ _ c o s P _ , . UcosP^ ^ if t j - T , cosP / „ ^ _{1 (lanp-i-laiin)} 1-l-cosp^ 5v,« <R3,T: t 2 , T - R 3 , T - R 3 - T j ^ 5 y , „ D . 2 , r = D . 3 , r = D , 3 + [ T^ J (7.14) cosp , 5 „ l+cosp D b . T = D b + T ' cosp 1+cospJ D , D , ' . , r = t w - T 5 „ , , D „ _ T = D „ + T ^ D , DR2,T=DR2 + l+cosp ' cosp ^ 1+cosf D ,

5 -W r-^^I—

D,

(7.5) (7.6) (7.7) (7.8)

where the subscript T earmarks the value of the corresponding cross section's dimension at dme T; 5~^= mean value of the overall corrosion wear of the web plate at time T; 5, ^ = mean value of the overall corrosion wear of the bulb head in vertical direction at timeT; 5 „ , n = mean value ofthe overall corrosion wear of the web plate per unit time; 5y^„= mean value of the overall corrosion wear of the bulb head in vertical direction per unit time; D„ „ = average variance of the overall corrosion wear of the web plate per unit time; D^ , = average variance of the overall corrosion wear of the bulb head in vertical direction per unit time; ïï = mean of the nominal value of h; b = mean of the nominal value of b ; t „ = mean of the nominal value of t„4 R 2= *he mean of the nominal value of R2; R 3 = mean of the nominal value of R,; R 4 = mean of the nominal value of R4; R j = mean of the nominal value of R5; D|, = variance of the nominal value of h; Dt = variance of the nominal value of b; D„ = variance of the nominal value of t„.; DRJ = variance of the nominal value of R2 ; DE3 = variance of the nominal value of R3 ; DRJ = variance of the nominal value of R, ; = variance of the nominal value of R5

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Geometric Properties of Sliipbuilding Structural Profiles in Probabilistic Terms to Be Used in Elastic Bending Strength Calculations Lyuben D. Ivanov

Due to lack of information, it was assumed that the means of the nominal values of all radii are equal to their nominal value given in the manufacturer's Specifications. It was also assumed that variances / standard deviations of the nominal values of all radii are equal to zero.

^ ^ • ^ " ^ ^ l + cosp _cosP_, 1 + cosP '

(7.23)

Inverted angles /L-profiles

The formulae for bulb plates are used. As to the parameter t j , its mean value and variance are calculated with the formulae:

t 2 , T = t f - T ^ v , „ - R 2 , T ; D . 2 , T = D f + T X . + D R 2 , T (^-IS) where R 2 .^and D^^.T are determined with eq. (7.7).

1 + cosP

P

T

=P

(7.24)

(7.25)

Channels

The mean values and the variances of all cross section's dimensions are determined with the equadons for bulb plates bearing in mind that R3 = R4 = 0.

Asymmetric and Symmetric Fundia T-bulbs

h 7 = h - T ^ 6 ; : ; I > H , r = D H + f T ^ f D v . l+cosp 1+cosP ) (7.16) ^ = b - 2 T ^ 5 ; : : ; D , , = D , + 4 T - ^ l l+cosp l+cosp (7.17) ^ = c - T ^ H i P _ 5 ^ , „ ; D , , = D , + f T - H 2 i P _ l D , „ ^ l+cosp ''^ ° [ 1+cospJ (7.18)

where the subscript T earmarks the value of the correspondmg cross section's dimension at time T; c = mean of the nominal value of c; t = mean of the nominal value of t; ? = mean of the nominal value of s; R j = mean of the nominal value of the radius R,; D,, = variance of the nominal value of c; D, =

variance ofthe nominal value of t; D, = variance of the nominal value of h; DR, = variance of the nominal value of R,.

Due to lack of information, the same assumptions for the means and variances/standard deviations of the radii were made as for bulb plates. The parameters s and P were assumed as not changing with the ship's aging. The means of their nominal values were assumed as equal to the nominal values while the standard deviations of their nominal values were assumed as equal to zero.

Asymmetric and Symmetric built-up T-bars

b . = b - T 5 hor, a « = T= C- - 5 | , „ , r 7 = ï - 2 T - ^ 5 ; 7 ; D , , = D , + 4 f T ^ ^ l D _ ^ l+cosp ''^ ' 1+cospJ _ (7.19) '2,1 = t 2 - T 5 ^ , „ D , , , = D , + T ^ D „ „ , _ „ D c . T = D c + - J - D h o , , „ D . 2 , T = D . 2 + T ' D v , < , (7.26) (7.27) (7.28) (7.29) — ^ _ . ^ o s p _ g -'•^ ' l+cosp (7.20) tw.T = t „ - T 5 „ _ „ ; D ^ , T =Dw + T ' D , , , „ (7.21) (7.22) t„,T. = t , - T 5 „ , < , D „ . i = D „ +T^D„_„ (7.30)

Where the subscript T earmarks the value of given cross section's dimension at time T; 5^^,. g= mean value of the overall corrosion wear of the flange in horizontal direction per unit time; by mean value of the overall corrosion wear of the flange in vertical direction per unit time; S„ „ = mean value of the overall corrosion wear of the web plate pet unit time; D|,„„= average variance of the overall corrosion wear of the flange in horizontal direction per unit time; D^.^ = average variance ofthe overall corrosion wear of the flange in vertical direction per unit time; D„„ = average variance of the overall corrosion wear of the web plate per unit time.

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Geometric Properties of Shipbuilding Structural Profiles in Probabilistic Terms to Be Used in Elastic Bending Strength Calculations Lyuben D. ivanov

Flat bars

h 7 = h - T 5 ; ; ; d , ^ = d , + t ^ d , , „ ( 7 . 3 1 )

K^ = Ï^-^K^

D v , , i = I > w + T ' - D „ , „ (7.32)

where the subscript T earmarks the value of given cross section's dimension at dme T; 8 y „ = mean value of the overall corrosion wear of the flat bar in verdcal direcdon per unit time (if no data, one could assume 5ya = 5^v,o/2); 5„,a = mean value of the overall corrosion wear of the flat bar per imit dme; D, „ = average variance of the overall corrosion wear of the flat bar in verdcal direction per unit time (if no data, one coidd assume D,.^ = I^«/>); D»vi = average variance of the overall corrosion wear of the flat bar in horizontal direction per unit time.

Attached plates

^ = ~ , - ^ ^ D p , T = D p + T ' D p , „ (7.33)

T ^ = i; D , ^ = D , (7.34)

ë 7 = ë D e , . = D e (7.35) where the subscript T earmarks the value of given cross section's

dimension at time T ; 5 p „ = meanvalueofthe overall c orrosion wear of the attached plate per unit time; Dp„ = average variance of the overall corrosion wear of the attached plate per unit time

When the geometric properties are pjesented i n non-dimensional format, the new mean value, y , and the standard deviation, ^ D ^ , will be:

y = Y/Y„„,„ ^ = ^ / Y „ „ „ (7.36) The developed procedure has been applied to all major

shipbuilding structural profdes. For each of them about 20 geometric properties have been calculated.

8 Accuracy of the method

To check the accuracy of the proposed method, calculations for the major shipbuilding structural profiles were carried out in the following way:

The classification societies Rules control the web plate thickness of rolled profiles and the web plate and flange thickness of built-up T-bars [ 1 ] , [6]. It has been shown in [10] that this approach, although very simple, is quite reasonable to control the remaining strength of structural profiles when the permissible reduction due to corrosion of the web plate/flange thickness is around 25%. Using the same approach, series of calculations were performed by the Monte Carlo simulation

method to verify the proposed analytical method. As an example, the results for the probability that the web plate/ flange thickness will meet the Renewal Criteria requirements (25% permissible reduction) are given in Fig. 17. Although there are no requirements for the maximum permissible reduction of the cross sectional area and the other geometric properties, calculations were performed with both Monte Carlo simulation method and the proposed analytical method for the section modulus for symmetric and asymmetric bending. The results are shown in Fig. 18 and Fig. 19. One can observe in Fig. 17, Fig. 18, and Fig. 19 that the results are almost identical. The differences (see Fig. 20) are between plus 1% and minus 1.5%, which is a negligible difference for such type of calculations. Thus, a conclusion is made that the accuracy of the proposed analytical method is very high and it can be recommended for use.

9 Numerical examples

The calculations were performed for all aforementioned shipbuilding structural profdes assuming hnear dependency of the corrosion wear on the ship's age (i.e., constant corrosion rate in the third period of corrosion activity). To make easier the comparison between the results for each structural profile, the duration of the fu-st period (i.e, no corrosion) was assumed as equal to two years.

The two probabilities P ( y ; < y < y u ; T , < T < T „ ) and PÖ'; £ y ^ yu; T) were calculated for each structural profile. The former probability represents the probability of given geometric property being between specified limits within given time period. In the paper, 75% and 100% of the original value of the corresponding geometric property were assumed as its lower and upper limit while for the duration of the time period the time between sixteen and twenty years were selected. The latter probability represents the annual probabUity of given geometric property being between specified limits (in both cases, 75% and 100% of the corresponding geometric property were used as lower and upper limit). For the sake of brevity, only the resiUts for the Cross Sectional Area of the stiffener itself. As, Section Modulus for symmetric and asymmetric bending, SM,j„ and SMo,j„ , and the web plate/flange thickness, t„ , t f . are given in the paper (the results for the other geometric properties can be found in [10]). A stable trend in aU calculations was noticed: the mean values and the standard deviations of aU geometric properties change almost linearly with the ship's age (e.g.. Fig. 21 - Fig. 24).

Bulb Plates (Fig. 5)

The dimensions of the tested bulb plate are given in Table 1. The nominal values of some of its geometric properties are given in Table 2. The corrosion data used in the calculations are given in Table 3. An assumption was made in this particular example that the corrosion wear along the süfièner's cross section is uniform and equal to the corrosion wear of the web plate.

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Table 1 Nominal values ofthe dimensions of the bidb plate's cross section (see Fig. 5)

Parameter Dimension Notation Value Height ofthe stifleiier cm h 25.00

Width ofthe bulb head cm b 4.50

"Flat region" ofthe bulb

head cm t2 1.00

Web thickness cm tw 1.20

Radius in lower end of the

bulb head cm R 2 1.00

Radius in upper end of tlie

bulb head, right cm R3 1.00

Radius in upper end ofthe

bulb head, len cm 0.20

Radius below the bulb head cm Rs 1.00 Angle of incHned edge ofthe

bulb head degrees

P

30.00

Thickness ofthe attached

plate cm 1.20

Length ofthe attached plate cm h 70.00

Angle of the attached plate degrees e 15.00

Table 2 Nominal values of some of the geometric properties of the bulb plate *

Parameter Dimension Notation Value Total Cross Section Area

(bulb plate plus attached plate)

cm^ A 122.13 Cross Section Area ofthe

bulb plate alone cm^ As 38.13

Ordinate of the Centroid

relatively to original axis X cm ex 4.13 Abscissa ofthe Centroid

relatively to original axis Y cm ey 0.84 Moment of Inertia relatively

to Centroidal axis Xi

4

cm 1X1 11059.66 Moment of Inertia relatively

to Centroidal axis Y j cm A IYI 3 2 0 4 0 . 0 8 Radius of Gyration of Ixi cm rxi 9.52 Radius of Gyration of lyi cm rYi 16.20 Angle ofthe Principle Axes

X 2 relatively to axis Xi •* * degrees -69.96 Moment of Inertia relatively

to Principal axis X 2 cm A I.X2 = Ima\ 3 5 2 5 8 . 2 8 Moment of Inertia relatively

to Principal axis Y 2

A

cm IY2 ° Imin 7841.47 Radius of Gyration of Ixj cm rx2 16.99 Radius of Gyration of IY2 cm rY2 8.01 Section Modulus for

symmetric bending cm' SMsym 530.03 Angle relatively to Xi ofthe

N A . for asym. bending * * degrees 15.40 Section Modulus for

asymmetric bending cm' SMasym 409.94 Perimeter of the stiffener

alone cm Lb 55.68

Table 3 Means and standard deviations/variances of the corrosion wear a used as input in the calculations for the bulb plate

Bulb head

g [mm/year] 0.1000 Bulb head Dv,o [mm/year]' 0.0016 Bulb head

CTya [mm/year] 0.0400

Web plate

8 ^ [nun/year] 0.1000 Web plate Djv „ [mm/ycar]' 0.0016 Web plate

CT„_„ [mm/year] 0.0400

Attached plate

5 ^ [imn/year] 0.1070 Attached plate Dpa [mm/year]' 0.00096 Attached plate

ap_„ [mm/year] 0.0310

However, the calculations are not limited to the above mentioned geometric properties or assumptions for the type of the corrosion wear. All other geometric properties can also be calculated for any kind of the corrosion wear along the cross section's dimensions and presented in probabilistic terms with the proposed methodology.

The average probability P(0.75y„„< y < y„„„; 16< T < 20) is given in Table 4 and the annual probability P(0.75y„„„ < y < ynom; T) is given in Fig. 25

Table 4 Results for P(0.75y„„ < y < y„„; 16 < T < 20) for the bulb plate

Geometric

property tw As Atolal Average

probability 0.9585 0.9991 0.9981 0.9991 0.9880

• The effect of the welding melal is e.'^cluded

** Sign of the angle means rotation clock wise relatively to the axis for comparison The length of the attachment o f t h e bulb plate to the attached plate is excluded

I n v e r t e d angle (Fig. 7)

The example for inverted angles is based on data for inverted angle BUAIOOF described in [5]. The reason for its selection \vas the intention of having a structural profile with geometric properties close to those ofthe bulb plate. Thus, more realistic comparison beUveen the results for both profiles could be done (although this unequal inverted angle may not be manufactured anymore, it still can be used for comparison purposes).

The same input data for the corrosion wear, and the same boundaries of y and tw are used as for the bulb plate. The probability P ( y/ < y < yu ; T ) for the geometric properties (but not for t„.) of inverted angles and L-profdes drops earlier and more sharply than the corresponding probability for bulb plates.

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Geometric Properties of Shipbuilding Structural Profiles in Probabilistic Terms to Be Used in Elastic Bending Strength Calculati Lyuben D. Ivanov

Table 5 Nominal values of the dimensions of die inverted angle's cross section (the original dimensions ivere in Imperial system)

Parameter Dimension Notadon Value Height of the stiflener cm h 25.400

Width ofthe flange cm b 10.200

"Flat region" ofthe flange cm t2 0.617

Web thicbiess cm tw 1.097

Radius in lower end of

the flange cm Rl 0.480

Radius iu upper end of

the flange, right cm R3 0.000

Radius in upper end of

the flange, left cm R. 0.000

Radius below die flange cm R5 1.370 Angle of inclined edge of

the flange degrees P 0.000

plate cm tp 1.200

Length ofthe attached plate cm _/p 70.000 Angle ofthe attached plate degrees G 15.00

The average probabihty P(0.75y„„,< y < y„„; 16 < T <20) is given in Table 7 and the annual probability P(0.75y„„„ y < yoon,; T) is given in Fig. 26.

Table 7 Results for P(0.75y„„„ < y < y„„; 16 < T < 20) for the inverted angle

Geometric

property tw As Aiolal SMsjTO Average

probability 0.9769 0.7716 0.9763 0.3065 0.2416

L-profile (Fig. 8)

The example for profiles is based on data for Fundia L-profile [22]. The same input data for the corrosion wear are used as for the bulb plate. The probability P ( y; < y < y„ ; T ) for the geometric properties (but not for t„) of L-profiles drops earher and more sharply than the corresponding probability for bulb plates.

Table 6 Nominal values of someof thegeometricproperties of the inverted angle

Parametei' Dimension Notation Value Total Cross Section Area

(inverted angle plus

attached plate) cm^ A 122.20

Cross Section Area of the

inverted angle alone cm' As 38.20

Ordinate ofthe Centroid

relatively to original axis X cm ex 4.50 Abscissa of the Centroid

relatively to original axis Y cm ey 1.07 Moment of Inertia relatively

4

cm 1x1 12243.84 Moment of Inertia relatively

to Centroidal axis Yi cm* IYI 32298.98 Radius of Gyration of Ixi cm rxi 10.01 Radius of Gyration of lyi cm TYI 16.26 Angle ofthe Principle Axes

X 2 relatively to Centroidal

axis Xl degrees «X2 -68.22

Moment of Inertia relatively

to Principal axis X 2 cm'' 1x2 = Imix 36110.92 Moment of Inertia relatively

lo Principal axis Y 2 cm* Iy2 = Imin 8431.90 Radius of Gyration of 1x2 cm 1X2 17.19 Radius of Gyration of IY2 cm ryj 8.31 Section Modulus for

symmetric bending cm' SMsym 585.73 Angle relatively to Xi ofthe

N.A. for asym. bending degrees 16.45 Section Modulus for

asymmetric bending cm' SMKym 444.25 Perimeter ofthe stiffener

alone (minus tw) cm Linv 69.31

Table 8 Nominal values of the dimensions of the L-pro£ile's cross section

Parameter Dimension Notation Value Height ofthe stiffener cm h 25.000

"Flatregion" ofthe

flange cm t2 0.750

Web tliickness cm tw 1.050

Radius in lower end of

the flange cm R 2 0.750

the flange, right cm R3 0.000

the flange, left cm R, 0.000

Radius below the flange cm Rs 1.500 Angle of inclined edge of

plate cm lo 1.200

Length ofthe attached

plate cm /p 70.000

Angle ofthe attached

plate degrees 9 15.000

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Table 9 Nominal values of some of the geometric properties of the L-profde

Parameter Dmi ension Notation Value Cross Section Area ofthe

L-profde alone cin' As 38.54

Total Cross Section Area (L-profile plus attached plate)

cm' A 122.54

relatively to original axis X cm e x 4.61

Abscissa ofthe Centroid

relatively to original axis Y cm e y 1.06

Moment of Inertia relatively to Centroidal axis Xi

4

cm 1X1 12419.42

Moment of Inertia relatively to Centroidal axis Yi

4

cm lyi 32268.12

Radius of Gyration of I x i cm rxi 10.07 Radius of Gyration of I y i cm ryi 16.23 Angle of the Principle Axes

X 2 relatively to Centroidal axis Xl

degrees AX2 -68.06

Moment of Inertia reladvely to Principal axis X 2

4

cm 1X2 ™ lim-x 36113.55 Moment of Inertia reladvely

4

cm I y 2 = Iraill 8573.99

Radius of Gyration of 1x2 cm RX2 17.17

Radius of Gyration of Iy2 cm RY2 8.36

Section Modulus for

symmetric bending cm' SMjyn, 609.15

Angle relatively to axis Xi ofthe "Elastic" N.A. for asymmetric bending

degrees 16.48

Section Modulus for

asymmetric bending cm' SMasym 463.50

Perimeter of die stiftener

alone (minus t„) cm Linv 65.98

The average probability P(0.75y„„„<y<y„„; 1 6 < T < 20)is given in Table 10 and the annual probability P(0.75y„,„ < y < ynon,; T) is given in Fig. 27.

Table 10 Results for P(0.75y,«,„ < y < y„„; 16 < T < 20) for the L-profde Geometric property tw A s Atotal SMgjm Average probability 0.9566 0,7567 0.9750 0.4257 0.3404 Channel (Fig. 9)

This structural profde was widely used in riveted structures. Although at present it is very rarely used, an example is given here for completeness. It is based on data for the hot rolled channel S 2 3 5 J R G 2 given in [8] and [ 2 0 ] . There are negligible differences between the obtained results for the geometric properties and the corresponding data given by the manufactiurer. However, they do not affect the accuracy of the calculated P (y/ < y < y u ; T ) a n d P ( y , < y < y u ; T l < T < T J .

The input data for the cross section's dimensions and the major numerical results are given in Table 11 and Table 12. To facilitate the comparison, the data for the corrosion wear were taken the same as already described in the previous sections. Table 11 Nominal values ofthe dimensions of the channel's

cross section

Pai'ameter Dimension Notation Value

Height ofthe channel cm h 40.000

"Flat region" ofthe flange cm t2 0.617

Web thickness cm tw 1.400

Radius in lower end ofthe

flange cm R 2 0.900

Radius in upper end ofthe

flange, riglit cm R3 0.000

Radius in upper end of die

flange, left cm R4 0.000

Radius below the flange cm Rs 1.800

Angle of inclined edge of

Table 12 Nominal values of some of the geometric properties of the channel

Parameter Dimension Notation Value Cross SectionArea ofthe

channel alone cm As 90.64

relatively to original axis X cm Cx 20.00

Abscissa ofthe Centroid

relatively to original axis Y cm ey 2.63 Moment of Inertia relatively

4

cm 1X1 20329.52

Moment of Inertia relatively to Centroidal axis Yi

4

cm lyi 823.28

Radius of Gyration of Ixi cm rxi 14.98

Radius of Gyration of lyi cm ryi 3.01

Angle ofthe Principle Axes X 2 relatively to Centroidal axis Xl

degrees a x 2 0.00

Moment of Inertia relatively to Principal axis X 2

4

cm 1x2 ~ IncK 20329.52 Moment of Inertia relatively

4

cm I y 2 ~ Imin 823.28

Radius of Gyration of 1x2 cm rx2 14.98

Radius of Gyration of lyi cm r y 2 3,01

Section Modulus for

symmetric bending cm' SMsym 1016.48

Angle relatively to axis Xi ofthe "Elastic" N.A. for asymmetric bending

degrees

_w

0.00

Section Modulus for

asymmetric bending cm' SMaaym 1016.48

Perimeter ofthe channel

alone cm Lch 118.05

The average probability P(0.75y„„ < y < y„„; 16 < T < 20) is given in Table 13 and the annual probability P(0,75ynom < y <y„om;T)is given Fig. 28.