Reliability assessment due to wear Текст научной статьи по специальности «Строительство и архитектура»

RELIABILITY ASSESSMENT DUE TO WEAR

V. Raizer •

13513 Zinnia Hills Place #28, San Diego,CA 92130 ,USA e-mail: vdraizer@hotmail.com

Evaluation of structural reliability under processes of deterioration presents very important problem in design. The structure's wear shows a reduction of bearing capacity in time that for one's turn leads to increasing the probability of failure. The reasons for long duration and irreversible change of structural features can be corrosion in steel structures, decomposition in wood structures, ageing in polymer structures, and processes of abrasion or erosion also. The problem of defects accumulation should be mentioned too, when reduction of the bearing capacity connects with load's value and its duration.

The models and peculiarities of corrosion wear and its influence on bearing capacity are discussed in this paper.

1. MODELS OF CORROSION WEAR

Corrosion is an important factor in reducing of reliability and durability due to different kinds of structures or equipments. From 10% to 12% of fabricated and used steel is lost annually due to destructive effects of corrosion. In spite of widely used protection methods, the quantity of steel destroyed is growing almost proportionally to the accumulated stores of steel. Losses from corrosion average are between 2% to 4% of GDP in almost every country. About 30% of structural steel is subjected to atmospheric corrosion, and 75% is subjected to atmospheric and aggressive corrosion simultaneously [1]. Under corrosion's influence the initial cross-section of a structural element is decreased, and consequently so its bearing capacity. Fig.1 presents the types of corrosion for structural steel.

cO b)

c) d)

Fig.1 Types of corrosion of a structural steel. a) Uniformly distributed wear. b) Irregular distributed wear. c) Corrosion with spots. d) Corrosion with ulcers. e) Corrosion with points. f) Corrosion with cracks.

The speed of a corrosion process depends upon degree of aggressive environment and is changing with 0.05mm/year to 1.6mm/year. The damage of structural steel in soil depends on the duration of an exposure, as shown in Fig.2. Data are based on 16 types of soil. Similarly, the damage of the steel from atmospheric corrosion is shown in Fig.3. Distribution of corrosion speed (measured at the inner reservoir surface along its height) for different products is presented in Fig.4.

0,05

Fig.2 Corrosion of structural steel in soil over years. Y-axis shows the mean depth of corrosion in mm.; X-axis shows the years of duration.

Fig. 3 Corrosion of structural steel in open air. 7-axis shows the average depth of corrosion (mm). X-axis shows the years of duration.

Fig. 4 Variation of corrosion's speed 1. Gasoline. 2. Kerosene. 3. Diesel.

The evaluation of structural durability depends essentially on the choice of the model that is capable to reflect the influence of an aggressive environment. When modeling corrosion processes, there are important damage characteristics to consider, such as depth of defect (5) and corrosion speed (v=d5/dt). Classification of mathematical models of corrosion (based on empirical approach)

presents in Table 1 [1,2,3]. The kinetics of the corrosion process in different metals for different aggressive environments looks very similar, and this fact presents the opportunity to use these models in design.

In general, processes of wear can be presented as time-dependent random functions of time. Type of processes depends on maintenance conditions, methods of structure's fabrications, steel's composition and others.

Table 1

# Models of corrosion Functional relationship

1 5 = Vot Linear

2 vt = kt - " Power

3 5 = a + b lg t Logarithmic

4 5 = ln(kt ) Logarithmic

5 vt = v0 exp(-ot ) Exponential

6 vt = mt2 exp(-tIt) Exponential

7 5 = 50[1 - exp(-t It)] Exponential

8 5= a 1 + b exp(-ct ) Exponential

9 t at + bt + c Fractionally linear

10 5= 5 1 + at Fractionally linear

Models of long-term processes presents as random time processes, but its uncertainty defines, due to random, independent from time parameters. Such kind of random processes were called "deterministic random processes "[4].

In the case that all loads Fi presents independent random values, probability of no failure during working life can be expressed as:

P(n)=P [ R > F R2 > F2,..., Rn > Fn ], (1)

where R1,R2,...,Rn- values of bearing capacity in considered time intervals. If designate Rn = Rp(n),then n=t -term of maintenance in years; R0-initial (random) value of bearing capacity; p(n) - monotonically decreasing nonnegative function (i=1,2,3,n.), satisfying to the conditions: p(0) = 1; p(^) = 0; dp/dt < 0. It should be mentioned also the additive property of 9 (t) function, independence of wear's process in the subsequent time interval ti from previous process's value in time ti-1, i.e. p(t1)p(t2) = p(t1 +12).

F1,F2,...,Fn - Loads, corresponding to considered time intervals.

2. UNIFORMLY DISTRIBUTED CORROSION WEAR

This problem is illustrating in considering a steel pipeline's section (cylindrical tube), subjected to inner pressure, changes of the temperature and corrosion. The inner pressure F and steel yield stress Ry are random values with given distributions. The corrosion process considers deterministic. The limit state condition is taken in the form: Si < Ry. Here - intensiveness of

stresses in considered cylindrical shell. In accordance with Guber-Mises condition [5], general case looks as:

Si =-^#1 - S2)2 + (S2 - S3)2 + (Si - S3)2 (2)

In discussed situation S2 =0, and the radial and the tangential stresses reads:

S1 = —)i-, S3 = FDi--aEA0.

1 2h 4h

Here F is the inner pressure, and its maximum value is random for some time intervals; Di-inner diameter of the pipe; a-parameter of linear extension; E-modulus of elasticity; A0 -temperature drop (difference between temperature of the pipeline during use and assembly).

The reliability condition expresses as follows:

3F2 D2

-+ a2E2A02 < Rl (3)

16h2 y

As temperature's drop presents an uncertain value with unknown distribution, then temperature's stresses are given as some part of the yield stress.

aEA9 = Ry sin x (4)

X is a value of angle in the given interval [0,n/2]. The condition (3) presents now in the form:

4h

F <13DiRy cosx (5)

Corrosion wear causes a reduction of tube thickness as h = h0p(t),where h0 is the

initial thickness. In accordance with the Table 1 one can takes:

p(t) = exp(-t /t) (6)

From (6) comes:

h = h0 -S[1 - exp(-t / t)] (7)

where 5 is the depth of corrosion bubble. It is assumed that the corrosion process in interval t2 is independent of the preceding values in interval t1 , so that p(0, t +11) = p(o, t) + p(t, t +11). It is assumed also that time t takes only discrete values: t=n, where n is number of years or months. An assumption is made for pressure F supposing that statistic data belong to some period of time, a month, for example. From all observations, maximum values selects only. If the time interval is large in comparison with correlation zone, then Fisher-Tippet distribution (second type) of maximum values can be used [6].

__ P (x) = exp [-(x /£)-n] (8)

If vF = sF / F, F are correspondingly the coefficient of variation and the mean value, then parameters ^ and n are determinated from the solution of two equations, which includes gamma functions.

1 + vf 2 =r(a)r(6)

£ = F / T(a)

Gamma functions are:

œ

r(a) = J e - z za—1dz,

0

œ

(10)

r(b) = J e - zzb-1dz.

0

The case when b=0 and n=2 is excluded.

For yield stress Weibull distribution is applied.

P (x) = 1- exp[-(x l®)u] (11)

Form's parameter p is expressed through coefficient of variationvR = sR lR :

Vf(1 + 2j) - [r(1 + 1l u)]2

vR =~--(12)

R r(1 + 1l u)

Values vR and p define scale parameter ro.

Taking into account (8) and (11) the reliability function is written in the form:

P (n) = - Jexp[-(4ho*coseX) * ] ^<P V(t)d[exp{-(xl®)u}] 0 yllD^ i=o

Example. After statistic data processing of pressure in pipelines and yield stress the following values of the distribution parameters were defined: = 73.5; n = 65; ro = 42.5; p = 23.5.Coefficients of variations are: vF = 0.0201;vR = 0.0522. Temperature stresses (9.4) show essential influence on pipeline's reliability. When x = n l 3, P (n) is close to zero. P (n) values for different n are presented in the Table (2).

Table 2

(13)

T X Values of function P (n)

Time in years

1 5 10 15 20 25 30

100 0 0.9989 0.9989 0.9989 0.9989 0.9987 0.9962 0.9860

100 6 0.9989 0.0087 0.9968 0.9880 0.9590 0.8600 0.6000

100 4 0.9560 0.8500 0.5800 0.1800 - - -

120 0 0.9989 0.9989 0.9989 0.9989 0.9989 0.9975 0.9872

120 6 0.9989 0.9941. 0.9941 0.9750 0.9600 0.8990 0.8060

120 4 0.9560 0.8790 0.6870 0.3790 - - -

150 0 0.9989 0.9989 0.9989 0.9989 0.9988 0.9985 0.9900

150 6 0.9989 0.9988 0.9980 0.9900 0.9760 0.9570 0.3200

150 4 0.9989 0.8820 0.7500 0.5200 0.3800 - -

From (13) the member responsible for corrosion process's influence is picked out:

X =

n—1

n

n+M

(14)

where X characterizes decreasing of reliability in regard of corrosion's development. Parameter t in (6) and in Table 2 defines intensiveness of uniform corrosion. Physical sense of this value consists in decreasing of initial tube's thickness. This essential decreasing is possible under large values of t = 100.. .150.

Results of many experiments and real observations demonstrated [1,.3] the influence of stresses in structures to the speed of corrosion. Especially large is this influence in places of concentrations of stresses. Dependence between corrosion's speed and increasing level of stresses

M

1=0

can be as linear as nonlinear. If to take dependence between the intensiveness of stresses and the depth of the corrosion's penetration such as£ = atp exp(kSt), and substituting it in the formula for

FD

the circular stresses in cylindrical shell S1 = ~h ' then the condition of the failure reads:

FD' > R

l[h0 -atp exp(kSi)] > y (15)

After decomposition into the row exp(kS' ) = 1 + kSi, expression (15) performs to:

F< 2[ho -atp(l + yf3kRy/2)]/Dt (16)

Here the expression in brackets takes into account influence of stress state at speed of corrosion. If to take the same distribution for inner pressure (8) and for yield stress (11), and to consider process of corrosion as a function of discrete argument then the expression for reliability function can be written in the form:

P (n) = _ jnexp(- 2[4./2I"'dexp

-

x

a

. . , , (17)

0 '=1

Expression (17) allows to evaluate the reliability of pipelines, subjected to continuous corrosion and to take into account influence of stress state to the corrosion's depth penetration or corrosion's speed.

3. IRREGULAR DISTRIBUTED CORROSION WEAR

A problem of structural durability and the protection from a local corrosion turns out to be very important as well. Local corrosion leads to some local destruction seen on the surface of the structure in the form of spots, ulcers, points or cracks (Fig.1). Appearance of this destruction in time is random too.

Corrosion cavities' ensemble is based on the following assumptions:

• Events, which have to do with the appearance of various numbers of cavities at disjoint time intervals are independent.

• Probability of corrosion's cavity appearance in the arbitrary time interval t is proportional to the length of this interval with the factor of proportionality equal to p,.

• Probability of the two or more events appearance through an extremely small time interval presents an infinitely small value of more high order.

The simultaneous realization of all these assumptions should be present and have an existence of the primary flow of events - a uniform Poisson process. Such process can be described by the system of differential equations:

df

.............. (18)

^ = P( Pn-1 - Pn) dt

Initial conditions for this system of equations are:

Pn (t) = 1, when n=0

Pn (t) = 0, when n=1,2, (19)

There will be only one solution for the system (9.18) and together with the conditions (9.19) it can be presented as the Poisson distribution:

p w = _MzML- (20)

From (20) probability of the fact follows that in the moment t > t0 the system is in the

state n (n = 1,2,3,). If the number of cavities appearing in some time interval submits to Poisson distribution, then the amount time before appearance of the next cavity possesses exponential distribution [7].

P(t) = exp(-/t). (21)

The number of experimental data that connects with investigations of kinetic due to cavity growth or an increase of cavities number is very small. Experimental dependences were received in [8]:

/ = / (1 - e) (22)

Here /, P are empiric coefficients. Value / varies in wide limits and measures as

number of defects to unit of structural surface.

Important parameters for considered type of irregular corrosion are - maximum depth of a cavity, its diameter and square of a cavity.

The random value of a cavities depth Sk (k-random point on structural surface) is distributing in the final interval [0,h0], where h0 is the thickness of structural element. It is considered that this value had uniform distribution, i.e.

0 x < 0

Ps(x) = x / h0 0 < x < h0 (23)

1 x > h0

Distribution of the maximum depth for n cavities, i.e. Sn = max {x1, x2, x3,...,xn} is well known from theory of extreme values [9] and can be taken as exponential.

P5n = exp [- n (h0-x)] 0 < x < h0

Psn= 1 x > h0 (24)

The next important parameter is the diameter of considered cavity, due to an assumption that this cavity has cylindrical form (Fig.5). Let the depth of the cavity is equal to x. Then the possible

region for variation of diameter is the chord AB with the length 2V2 rx - x , and r is the external radius. An assumption is taken that the random value of diameter yi has uniform distribution in the

interval [0, 2V2 rx - x ].

Pd(y)=

y

2V2

2

rx - x

Pd(y)= 1

if 0 < y <2>/2 rx - x if y > 2V2T

(25)

rx - x

Distribution of the maximum diameter for n cavities dn = max (yi, y2, y3, .. .yn) is:

Pdn = exp[-n(2^2rx - x2, 0 < y < 2^2rx - x

2

Pdn = 1,

y >

2yJ2rx - x~

2

(26)

Third parameter of this cavity is its square Ak. The knowledge of the maximum square value is important in solution of the considered problem. There are some difficulties, however, unclear even in the theory of order statistics. The point is that the maximum 5n value doesn't always correspond to the maximum value of dn. If to agree with this position then the solution will be received in safety margin. Two kinds of versions can be offered, distribution of maximum depth's value 5n and distribution of diameter's value dk for k-cavity in the first case, and otherwise: distribution of maximum diameter's value dn, and distribution of depth's value 5k in the second case.

Types of PA(x) distributions are written for three cases: Case 1:

PSn (x) = exp[-n(^0 - x)L x e [0, h0 I

Pn ( y) =

y

2yl2rx - x'

.2

0 < y < 2^2 rx - x'

2

The square of the cavity Ak is equal to the square of the segment at Fig.5:

A 2 • y y A, = r arcsin---..

2r 2 V

2y

r -— + [x - r + 4

2y

r

(27)

(28)

The maximum possible value of the cavity square Ak will be when x = h0 and

y =

2^2rh() - h02.

In pipes of large diameter x / r, y / 2r values are highly small numbers and possible reasonable approximation will be Ak = xy, and it follows:

PAk (A) =

A™ iexp[-n(h0 - x)]x 2dx 2yl2r 0

(29)

Ak is here uniformly distributed at interval [0,A*] random value. Case2:

x

Psn(x) = —, x e [0, h0] h

Pdn(y) = exp Distribution of Ak is:

n{(2^2rx - x2 - y)][ y e 0,2^2rx -

P

Ak

1 A)

(A) = Y i exp

n0 0

- n

2V2

rx - x2 -

A

rx - x

dx

x

(30)

(31)

Case 3:

It follows:

P5n (x) = exp[- n(h0 - x)], x e [0, h0 ] Pdn (y) = exp[- n(2V2

rx - x - y i y e

0.2^2rx - x2

(32)

PAk (A) = f expl - 2^2rx - x2--Id[exp(- n(h0 - x))] (33)

o I x J

The last case, as it was written before, leads to safety margin.

Example 1. Reliability of pipeline subjected to one-sided irregular corrosion.

Dimensions of the resulting cavity-depth and diameter are increasing in time in such degree that the failure of pipe will occur i.e. formation of a reach-through hole will take place. Time, tn before this hole will appear calculates from the expression:

tn

f v(t)dt = ho -Sn (34)

o

Here Sn - maximum depth from an ensemble of n cavities; v(t) = v0exp(-at) - corrosion's speed (Table 1,5). From (34) we get:

tn = -InT^V <35)

a h0 -Vn

Time distribution P(tn < t) to reach -through hole can be written as:

Pn(t) = P\Sn >

h0 -—(1 - exp(- at)) a

= 1 - exp

- n—(l - exp(- at )) a

(36)

After averaging on "n" it follows:

P (t) = E4- exp{l - exp[l - exp(- at )]} (37)

n=o n!

Example 2. Design of structural members under central tension.

Cylindrical element having a ring cross-section is considered. This element is subjected to irregular corrosion under deterministic load F. If A0 is initial value of cross-section (t = 0), Ak is square of cavity with given distribution PAk(A,) then the condition of no failure will be:

F / (Ao - Ak) < Ry or Ak < Ao - F / Ry (38)

Substituting the last expression into distribution function as an argument and carrying out an average on n and Ry probability of no failure in t moment is:

<» ( t ) n œ f F / \

P(t) = exp(-4t)X^-j-JPAk Ao - — p(Ry)dRy (39)

n=0 n! 0

V Ry y

Here p(Ry) - is density of yield stress distribution. In numerical example the following data are taken. External diameter D = 6.26in; initial thickness h0 = 0.24in; F = 127929ft; /u = [1 - exp(-pt )]and P = 0.05; Ry = 290Mpa; sRy = 25Mpa. Parameters of the cavity are

dk = Vk = 0.008in. Results of numerical realization are shown at Fig.6

P

0,9

0,8

Q7

0,6 0,5

P

Q9 03

Q7

Q6---- I 1 I I

0 5 10 tt 20 25 J*

Fig.7 Variation of the function of reliability due to number of cavities

4 CALIBRATION OF MODEL PARTIAL FACTOR

Partial factor for model uncertainties can be determined from comparison with identical structures operating in normal or in aggressive environment. Let us consider the structure under load F and with resistance equal to R. In case when random value of the load maximum for the definite period of time (one year, for example) has distribution PF (x) and year's load maximums are independent random values, the reliability function can be written as follows

Pi(H) = J Pp" (x)dPR (x) (40)

0

It is assumed that there is a structure operating in aggressive environment in the terms of uniformly corrosion. To guarantee the sufficient reliability level in the design it is necessary to go on additional expense of structural material such as increasing cross-section square, for example. The condition of no failure is

F <rDR (41)

-__ Mcy**

ytljqr-E

1 >

"

Fig.6 The reliability function

Though the corrosion process is continuous in time it is proposed to consider the function (p(t) which influences to geometric characteristics of cross-section as a function of discrete argument p(n). (41) to nth year could be rewritten as

F <yDRp(n) (42)

Reliability function will be

P2(n) = JTl P; [xp(0]R (x)

0 i=1

The equation for definition of yD arrives from the equality (40) to (43).

w n w

fn PF [yDxpi)] (x) = f P; (x)dPR (x)

0 i=1 0

If we consider tensioned non-corrosive structural element then (41) can be presented as

F < RAn

Where A0 -initial cross-section square. Function of reliability will be

Px(n) = J Pf ( xA0)dPR ( x)

(43)

(44)

(45)

(46)

For corroding structural element cross-section square is - A0yD, and yD >1. The failure condition can be expressed as

F <YdA0^)R (47)

Reliability function (43) will be

^ n

P2(n) = Jn Pf [dA xç(î)\IPr (x)

(48)

0 i=1

Equality (44) for the fast n value allows to determinate yD. Fisher-Tippet distribution (8) was chosen for PF (x). Equality (44) will be performed to

exp

From here, it follows

- n

( xA0 ^

V t J

n

dPR (X) = Jn eXP

0 i=1

(YdAox^(i)^ V

dPR (x)

yd =■

(49)

(50)

Introducing corrosion model in the form of (6) and presenting the sum in (50) in the row we

will get

Yd =

n

expf — J + exp( — J +... + exp( —

(51)

After transformation we get

so

1

i=1

1

Yd =■

expi T V1

n

expiai-,

Table 3 contains modal factor's values in accordance with (52)

(52)

Table 3

n n Yd Yd Yd

t = 100 t = 150 t = 200

10 1.0666 1.0461 1.0364

10 20 1.1304 1.0828 1.0612

30 1.2098 1.1280 1.0920

10 1.0657 1.0439 1.0337

15 20 1.1374 1.0851 1.0618

30 1.2263 1.1354 1.0958

10 1.0665 1.0433 1.0327

20 20 1.1443 1.0881 1.0632

30 1.2401 1.1424 1.0999

Aggressiveness of environment can be classified depending on parameters: heavily aggressive t = 100, middle aggressive t = 150, weakly aggressive

REFERENCES

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3. Tsikerman L.Ya. (1977), Computerized Diagnostics of Pipeline's Corrosion, Nedra, Publ. House, Moscow (in Russian), 319pp.

4. Middlton D. (1961), Introduction to a Statistical Communication Theory, Book 1, Soviet Radio Publ. House Moscow (in Russian), 782pp.

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6. Gumbel E.J. (1967), Statistics of Extremes, Columbia University Press, New York.

7. Kapur K.S. and Lamberson L.R. (1977), Reliability in Engineering Design, Wiley, New York., 604pp.

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