Eggert

R. J. Eggert Mechanical Engineering Department,

Union College, Schenectady, NY 12308

Design Variation Simulation of Tliicl(-walled Cylinders Thick-walled cylinders exposed to high, static internal pressures may experience both elastic and plastic deformation. Primary design considerations include loads, geometry and material properties. However, variations in geometry and material properties due to conventional manufacturing processes, and variations of internal pressure due to actual usage patterns, propagate through the system resulting in off-design stresses and strains which may cause failure. These variations can be evaluated using probabilistic methods which are discussed in this paper Von Mises-distortion energy yield theory is presented to predict elastic, plastic and residual stresses in thick-walled cylinders. The design variation simulation method using Monte Carlo simulation and available statistical information is used to design a pressure vessel for servo-hydraulic experiments. The use of autofrettage to induce favorable compressive stresses at the inner bore, thereby improving the margin of safety and overall reliability, is also presented.

Introduction The design of a product requires knowledge of how the

product will be used, how it will be manufactured, and of what materials it is made. Successful design practices also attempt to compensate for aging processes such as wear and corrosion. The factor-of-safety method of design is one such design practice, where allowable stresses or factors of safety are specified on the basis of experience or recognized design codes. While such design factors may comfort a design engineer by providing "rules of thumb," these arbitrary limits often lead to overly conservative designs. Probabilistic methods, however, can provide more quantitative and qualitative design evaluations, leading to higher product functionality, quality and overall value.

Probablistic methods are deeply rooted in the fertile fields of worst case analysis, random variable algebra and Monte Carlo simulation. Worst case analysis is a simple, but effective approach to analyzing alternatives with respect to performance criteria. The design engineer merely poses the appropriate what-if questions, such as what if the load is doubled, will the machine element fail? What if the actual yield strength is only 90 percent of the nominal strength? And so on. Assuming that the right what-if questions are asked, the resulting worst cases reveal the design's weaknesses. A more systematic approach couples the worst-case of all the design variables, thereby describing the impact of simultaneous variations (Balling et al., 1986; Parkinson et al., 1991; Teng, 1992).

The random variable algebra approach incorporates statistics such as the mean value and standard deviation of the design variables and parameters. The statistics characterize

Contributed by the Reliability, Stress Analysis, and Failure Prevention Committee for publication in the JOURNAL OF MECHANICAL DESIGN.

Manuscript received July 1993; revised June 1994. Associate Technical Editor; T. H. Service.

the input random variations which propagate in a cause-and-effect fashion according to a functional relationship. For example, the margin of safety is a function of the strength and stress. Variations in strength and or stress propagate through to the margin of safety. The mean value and standard deviation of a function of independent random variables can be estimated by calculating the expected values of a Taylor series expansion of the function (Haugen, 1980; Ka-pur and Lamberson, 1977; Mischke, 1980; Rao, 1992). The mean value of the margin of safety and its standard deviation, can therefore be estimated, and thus provide quantitative measures of central tendency and dispersion of the margin of safety. Note that these estimates only depend upon the means and standard deviations of the input random variables and are valid regardless of the type of input probability distribution. The method does however, require the assumption that the input random variables be independent (i.e., not correlated). Unfortunately correlations sometimes exist, which reduces the generic value of this technique.

The likelihood of an event can be determined by matching the estimated mean and standard deviation to the mean and variance parameters of a normal (or log-normal) probability distribution. For example, the probability that the margin of safety is less than zero, assesses the likelihood that the machine element fails, and can be calculated from a standard normal distribution table. The random variable approach has been applied to cylindrical, ellipsoidal and toroidal pressure vessel analysis (Smith, 1984; Zibdeh, 1990); fatigue design (Eggert, 1992; Mischke, 1987); factors of safety (Mischke, 1986, 1970); stochastic mechanical design (Eggert, 1991, Mischke, 1989a, 1989b, 1989c, 1989d); optimal design and reliability of airplane wing structures (Rao, 1986a, 1985, 1984a, 1984b); gearboxes (Rao, 1986b); geartrains (Rao, 1984c, 1979); machine tool structures (Rao, 1977) and function generating mechanisms (Rao, 1977). A thorough treatise

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of random variable algebra and general applications in mechanical design has also been presented by Haugen (1980).

Monte Carlo simulation produces a large sample of possible what-if scenarios based on historical or a priori statistical knowledge of the input design variables and parameters. The margin of safety for example, can be assumed to be a stochastic function of the two random variables, strength and stress, each characterized by its own mean value and standard deviation. One simulation of the margin of safety is generated by subtracting an instance of stress from an instance of strength. Instances of stress and strength are computer-generated according to their respective parent probability distribution functions which are known (historically) or which can be assumed (a priori). After many instances of the dependent function are simulated, appropriate sample statistics are calculated. A distinct advantage of this approach is that the engineer need not worry about selecting the right combination of values, as compared to the worst case method. Since historical data are used to characterize the input random variables, the resulting sample of simulations is a naturally-occurring collection of the possible instances. Most of the what-if answers, if not all, are represented in the data collection, naturally or automatically. Design variation simulation methods have been applied to a number of engineering problems including mechanism design (Crawford and Rao, 1989), manufacturing and assembly (Early and Thompson, 1989), brake design (Doepker and Nies, 1989), fatigue design (Eggert, 1992), structural wind loading (Shah and Chamis, 1991); optimization (Rao, 1978; Siddall, 1983), and finite element analysis (Thacker et al., 1991; Weber and Penny, 1991). A hybrid approach using simulation to "proof-test" optimal designs obtained from random variable algebra solutions has also been presented (Eggert and Mayne, 1990; Eggert, 1989).

Design Variation Simulation The design variation simulation method goes beyond mere

variation simulation analysis to include the design engineer in an iterative feedback loop to successively synthesize, analyze and optimize the product under consideration. Computer programs can be integrated into the loop as well, such as in synthesis, to automate the generation of new candidate designs, or in the optimization phase, to determine the best feasible design. The resulting, improved design process thereby makes effective use of historical or assumed statistical knowledge, and broadens the engineer's envelope of understanding.

As a candidate design is synthesized, specific values for the controllable design variables and parameters are determined. In the case of a thick-walled cylindrical pressure vessel, the inner and outer radii, internal operating pressure, and material properties are determined. The candidate design is simulated. The simulation results are analyzed, design variables

modified, and a revised candidate design is simulated. The simulate, analyze, modify, and re-simulate iterations are continued until the design engineer (or computer program) has optimized the controllable design variables.

The simulations result in a wealth of statistical measures such as: (7) minimums and maximums of the response functions, (2) sample average x, (5) sample standard deviation (T , {4) coefficient of variation C^ = &^/x, (5) coefficient of skew ^/S], (6) coefficient of kurtosis ^2, (7) and probability of a "failure" event. The average characterizes the sample's central tendency, while the standard deviation characterizes dispersion about the average. The coefficients of skew and kurtosis measure the symmetry and peakedness of the probability density function. The coefficient of variation measures dispersion relative to the mean value or average. The probability of an event, such as strength/stress failure, is directly obtained by comparing the number of simulated instances that "fail" to meet a prescribed specification, to the total number of simulations.

The random variable algebra approach is relatively simple and fast,;/ the required partial derivatives are readily obtainable. However, it usually provides first order estimates and will be invalid if the random variables are correlated. The design variation simulation method, on the other hand, is relatively easy to program, capably deals with correlation and can provide somewhat more accurate results. However, it does require historical data or a priori assumptions and it is computationally intensive. These drawbacks may not be that problematic however. First, more and more information is becoming available regarding the statistical nature of materials, manufacturing methods, load applications and usages. And secondly, as will be shown in this paper, existing personal computers have sufficient computing memory and processing speed. The next sections discuss the elastic breakdown margin of safety for thick-walled cylinders under high internal pressures, and the application of the design variation simulation method.

Elastic Cylinder Wall Stresses The thick-walled cylinder shown in Fig. 1 has an inner

bore radius a, outer wall radius c, is subjected to a static internal pressure pi and zero external pressure. In the closed case, the cylinder is modelled as being capped at both ends. The resulting axial forces, sustained by the cylinder wall, thereby create tensile axial stresses. In contrast, the open case refers to the situation where the axial forces are sustained by external supports such as bolts, thereby resulting in no axial stresses. Solution of the governing equations is credited to Lame, however a more contemporary derivation is available (Faupel and Fisher, 1981). The resulting nominal stresses are defined as the tangential or hoop stress o- ,, the radial stress o- , and the axial or longitudinal stress a^^ (open case), (T^ (closed case). The stresses are given below as

Nomenclature

a = inner bore radius b = elastic-plastic boundary

b/c = autofrettage ratio COV = coefficient of variation

C„

m.

= coefficient of variation of pressure

= outer surface radius = margin of safety-elastic

breakdown = internal pressure

R = radius or diameter ratio (c/a) r = radius

Sy = yield strength S„ = ultimate strength X = average of a random variable

a-/, = hoop stress o- = radial stress

o- j. = axial stress (open case) a^o = axial stress (closed case) a' = Von Mises equivalent stress

&^ = standard deviation ^/P^ = coefficient of skew jSj = coefficient of kurtosis

Subscripts o = open case c = closed case h = hoop r = radial z = axial

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/

- ^ ^ / <?, + da,

y J

Fig. 1 Thick-walled cylinder under Internal pressure

functions of the radius r, the nondimensional index of wall thickness R, defined as the ratio of outer to inner wall radii (or diameters), R = c/a, and the internal pressure p,:

Pi

R^ - 1

R' - 1

c 1 + -^

c l - - r

' R^-\

(T,„ = 0

(1)

(2)

(3)

(4)

These nominal stresses are the principal stresses and can be combined into one equivalent Vori Mises stress cr' defined as:

/ I \ ' ^ ^' = \l\i^k - o>)' + (^. - o-z)' + (o> - o-/,)1 j (5)

Substituting the Lame solution for CT/,, (T„ a^^, and cr. . into the above equation, the equivalent Von Mises stress distributions are obtained for a function of the dimensionless wall location parameter r/c:

Pi

R^-l

Pi

R^ -

3

1

{r/c)

3

[ir/c)'\

4

1/2

1/2

(6)

(7)

Figure 2 illustrates the nominal and Von Mises stresses for a closed cylinder of wall thickness ratio R = 3. The stresses have been normalized by /?, resulting in a convenient multipurpose graph. As the radius r increases, a^ and cr,. decrease in magnitude while the axial stress o- . does not vary with location. The inner bore undergoes the largest hoop and radial stresses thereby producing the largest Von Mises stress, approximately twice the value of the internal pressure. Figure 3 shows the effects of changing the diameter ratio R on the open case Von Mises stress as a function of wall location r/c for R values 1.5, 2, 3, and 5. Each stress curve is bounded on the left by the minimum value of r/c which is a/c or \/R.

U

Von Mises

Hoop, \ \ \

Axial ^ „

Radial/

1 2 3 Radial Location r

Fig. 2 Normalized nominal and Von Mises stresses for thick-walled cylinder; (R = 3; a = 1; c = 3; closed case)

0.2 1.2 0.4 0.6 0.8 Location in Wall r/o

Fig. 3 Von Mises stresses for thick-wailed cylinders; (R = 1.5, 2, 3, and 5; open case)

Table 1 Normalized Von Mises stresses for open and closed thick-walled cylinders for various diameter ratios

[o'lpi.

i.o'lp^.

Ogl O^

fl = 5.0

1.8057

1.8042

1.0003

fl=3.0

1.9526

1.9486

1.0021

fl=2.0

2.3333

2.3094

1.0104

fl=1.5

3.2167

3.1177

1.0324

fl=l.l

11.058

9.980

1.108

Similarly, all curves are bounded on the right by the maximum value of r/c which is c/c = 1. Note that little decrease in peak stress is obtained for R values larger than 3. For example, the benefit of increasing the wall thickness from i? = 3 to 5, is to decrease the peak stress from about 2 p, to 1.8 pi or about ten percent. The maximum Von Mises stresses at the bore, are obtained by substituting the minimum value of the location parameter into Eqs. (6) and (7):

(37?" + 1) 1/2

R^ 1 Pi

^/3R^

R^ 1 Pi

(8)

(9)

The open and closed Von Mises stresses in (8) and (9) are similar for values of R > 2.0 as seen in Table 1. At R - 2.0 the open condition stresses are about 1.04 percent larger than the closed. At R = 3 and R = 5 the open condition stresses are 0.21 and 0.03 percent larger, respectively.



Table 2 Simulation results assessing variations in static pressure (p = 350 MPa; a = 25 mm; c = 50 mm; S„ = 950.3 MPa)

s,

"o

m.

c,

all

0% 5%

10% 15%

0% 5%

10% 15%

Min.

840.46

784.39 726.95 658.97 590.99

7.46 -49.20

-119.95 -190.70

Avg.

950.30

816.85 816.53 816.22 815.91

133.45 133.76 134.07 134.38

Max.

1219.28

850.74 920.10 990.89

1062.00

412.57 446.40 502.59 570.01

Std. Dev.

48.91

8.21 41.77 82.20

122.91

49.57 64.27 95.61

132.24

Skew

.392

.150

.019

.013

.011

.370

.168

.045

.011

Kurt.

2.859

3.071 1.894 1.831 1.819

2.859 2.745 2.351 2.119

The elastic breakdown failure criterion for thick-walled cylinders compares the yield strength Sy to the maximum equivalent Von Mises stress a' at the inner bore. This elastic breakdown margin of safety m^, can be restated as:

m„ > 0

Sy- 0-' >0

(10)

(11)

Note that the yield strength Sy embodies material properties and that the Von Mises stress depends upon the geometry and the internal pressure. In the typical design situation, after the operating pressure is characterized, the appropriate material and geometry are determined to guarantee a satisfactory margin of safety. The next section discusses how design variations can be simulated and the design modified to successively converge to a satisfactory design.

Design Example An experimental program has been undertaken by the

author and a group of graduate and undergraduate students to investigate the viability of using an Instron servo-hydraulic testing machine and specially designed testing fixtures to assess thick-walled high-pressure cylinder fatigue, fatigue of reciprocating seals subjected to high pressure, computer-aided fatigue testing and cylinder fatigue resistance improvement gained by autofrettage processes. The Instron 245 kN (55 kip) load frame will axially load a 20 mm diameter piston-fluid-cylinder fixture to hydraulically generate internal pressures up to 350 MPa (50.8 ksi). To corroborate preliminary design calculations, the author prepared a design variation simulation for the candidate thick-walled cylinder of inner radius 25 mm, outer radius 50 mm, internal pressure 350 MPa, external pressure 0 MPa (gage), made of 4340 steel whose average yield strength and standard deviation have been determined as 950 MPa (138 ksi), and 48.9 MPa (7.1 ksi) respectively (Mischke, 1989a). A Fortran computer program was written which includes subroutines to generate pseudorandom numbers; generate uniform and Weibull vari-ates using the inverse cumulative distribution method; generate normal and log-normal variates using a rational function routine; simulate instances of the inner and outer radii, internal pressure, yield strength, resulting Von Mises stress and margin of safety; and then calculate desired statistical measures.

The first aspect investigated was the impact of internal pressure variation on the margin of safety. A steady, or static pressure variation results from using the vessel at a pressure different from the design pressure. A malfunctioning control valve or deliberate constant overload pressure are typical examples. Fluctuating or fatigue loading is different and will be considered in a future paper. The internal pressure was simulated from a uniform distribution, arbitrarily chosen, with mean and standard deviation P ~ U(350,35) MPa. The inner and outer radii were simulated as normally distributed

c 0.20

0.15

0.10

0.05

0 100 200 300 100 500 600 Margin of Safety (MPa)

• 1 0 % • •5% • 0 %

Fig. 4 Relative frequency diagram of margin of safety {Cp = 0, 5,10, and 15 percent; C^ = 1 percent)

Table 3 Reliability estimates for thicl(-walled cylinder for increasing Cp

c. 0% 5%

10% 15%

Normal

.9965

.9813

.9196

.8452

Log-Normal

.9979

.9818

.9163

.8474

Relative Frequency

1.0000 .9884 .9099 .8089

Time (S)

38.12 38.12 38.06 38.06

variates a ~ N(25,0.25) mm, and c ~ N(50,0.5) mm, respectively. The material yield strength was simulated as a Weibull distribution (Mischke, 1989a) with mean and standard deviation Sj, ~ W(950.3, 48.9) MPa, (i.e., Weibull distribution parameters ^0 = 840 MPa, b = 2.4, Q = 965 MPa). Four different coefficients of variation of the internal pressure C were examined (0 percent, 5 percent, 10 percent, and 15 percent). Each run resulted in 25000 simulated instances of the input and output variables. The simulated arrays were then processed by the statistical subroutine. The results are shown in Table 2. When the internal pressure is deterministic (C = 0 percent), the average margin of safety w^ is 133.45 MPa and the standard deviation 49.57 MPa. This is largely due to the variation contributed by the yield strength. As C^ increases to 15 percent there is little change in m ,, however the standard deviation increases to 132.24 MPa. The total variation thus measured, is more than twice that caused by the yield strength. The visual impact of static pressure variation on the probability density function (pdf) can be approximated by a relative frequency histogram or diagram, as shown in Fig. 4. Note how the probability of failure increases as Cp increases. This is seen as the area to the left of the vertical line at w , = 0. The pdf is slightly skewed to the right and its kurtosis decreases, or flattens out with increasing C .

The cylinder reliabilities were estimated and are shown in Table 3. If little or no pressure variation exists, the reliability approximates 100 percent. As variation increases to C^ = 15 percent, the reliability decreases to 81 percent. The upper and lower limits of a uniform distribution are u,l = ;u, + 1.731 &, therefore for C^ = 15 percent, the upper and lower limits of pressure would correspond to + 26 percent of the mean pressure. This simple example reinforces the importance of using equipment at, or under, the design pressure and to guarantee that control systems do not permit positive pressure excursions. Normal and log-normal reliability estimates were also obtained using the simulated averages and standard deviations. It appears that for C = 15 percent, the normal and lognormal overestimate reliability by about 4 percentage points (85 percent versus 81 percent). Each simu-



Table 4 Simulation results for elastic-breakdown margin of safety ntg, yield strengtii Sy, and Von MIses stress o-g for different C,,. values

Sy

O^

m„

c«. all

0% 5%

10%

0% 5%

10%

Min.

840.46

675.23 605.55 564.83

-103.99 -317.33

-3364.57

Avg.

950.30

816.03 821.18 840.57

134.26 129.11 109.72

Max.

1219.28

958.12 1248.93 4273.54

510.37 511.81 542.61

Std.Dev.

48.91

81.66 93.58

145.01

95.16 105.48 152.91

Skew

0.392

0.007 0.206 2.471

0.049 -0.104 -2.111

Kurt.

2.859

1.809 2.448

28.580

2.346 2,650

24.137

Table 5 Reliability estimates for thiicic-waiied cylinder for increas-

c„ 0% 5%

10%

Normal

.9209

.8895

.7635

Log-Normal

.9176

.8873

.7766

Relative Frequency

.9118

.8791

.8070

Time (S)

37.73 37.95 37.95

lastly, autofrettage could be considered. It is a process which mechanically induces compressive stresses at the inner bore which offset the stresses caused by the pressure.

0,40'

0.3S

^ 0.30

% 0.26 cr 0)

£ 0.20'

'^ 0.10

~« «-!-« -800 -600 -400 -200 0 200 400 600 Margin of Safety (MPa)

Fig. 5 Relative frequency diagram of margin of safety (C, ,. = 0, 5, and 10 percent; Cp = 10 percent)

lation run required about 38 seconds on a 16 MHz 386-sx IBM compatible computer equipped with a math coprocessor.

A second aspect to be investigated was the impact of geometric variation as measured by the standard deviations for the inner and outer radii. For this set of simulations, the cylinder radii instances were generated as normal variates with 0 percent, 5 percent, and 10 percent coefficients of variation and mean values of ix^ = 25 mm, and fx.^ = 50 mm. The yield strengths were generated using the same Weibull distribution and the pressure variates were generated from a uniform distribution of 350 MPa mean and 35 MPa standard deviation. Each run resulted in 25000 simulated instances of the input and output variables. The simulated arrays were then processed by the statistical subroutine. The results are shown in Table 4. The margin of safety for C^^ = 0 percent is 134.26 MPa, or slightly better than the 1 percent COV case from Table 2, where w^ = 134.07 MPa. However, larger radii variations significantly add to the dispersion of the margin of safety. Note that for C„^ = 10 percent the standard deviation for m^ is 152.91 MPa, compared to 95.6 MPa (Cp = 10 percent, C^,,. = 1 percent case from Table 2). The relative frequency is graphed in Fig. 5. Note that as the geometric variations increase, the curve skews to the negative side, and probabilities of failure increase.

The cylinder reliability for C„^ = 0 percent is abut 91.2 percent as measured by the relative frequency and is shown in Table 5. By relaxing the geometric tolerances to C j. = 1 percent, the reliability decreases to 90.99 percent (from Table 3). If the cylinder were dented, or otherwise significantly out of tolerance, resulting in 10 percent radii COVs, the reliability would drop to 80.7 percent.

The design variation simulation results have shown that the initial candidate design concepts are unsatisfactory. One improvement can be attained by increasing the diameter ratio to R = 3; resulting in the reliability approaching 100 percent. Another change could be to restrict excessive pressure excursions with a relief valve, or other control. And

Autofrettage of Thick-walled Cylinder As the internal pressure is increased, the inner bore

stresses exceed the elastic breakdown limit. With further increases, a plastic/elastic interface at radius b, expands from the inner bore, to the outer surface, until the whole cylinder is in a state of plastic strain. The pressure required to cause plastic strain to radius b is (Faupel and Fisher, 1981):

Ml.) = f 1 - + 2 In (12)

The outer portion of the cylinder, from b to c, undergoes elastic stresses:

<"'>•• 7f I c ( ; 1 + T (13)

( 0 > ) e . = .('-\/3 \c

(o-.c)

1 - 7 (14)

(15)

(o-zo). = 0 (16)

The inner portion, from a to b, experiences plastic stresses:

((^h)p y

+ 1 -F 2 In

(-.). = ^ - I - H-2

(oic)/. - I + 2 In

(Oio)n = 0

(17)

(18)

(19)

(20)

The cylinder becomes fully plastic when b = c. Ultimately it bursts at the following pressure (Faupel and Fisher, 1981):

Pbur 25^

73 In R (21)

The burst pressure is higher than the pressure required to plastically strain the entire wall. Therefore, during autofrettage, as the pressure is removed the material elastically recovers and a residual compressive stress distribution is obtained. This expansion/relaxation process is called autofrettage. For diameter ratios larger than 2.22 (closed) or 2.02 (open), the full effect of autofrettage can be attained with autofrettage ratios b/c less than 100 percent. Under these conditions, higher b/c values cause an elastic recovery



-800 20 25 30 35 40 45 50 55

Radial Distance (mm)

S h • • • * - Sr ••••»••• Sz

V M o V M o

Fig. 6 Residual stress distributions of autofrettaged cyiinder (tV c = 75 percent; b = 37.50 mm; Sy = 950.3 MPa)

that exceeds the inner bore yield strength, and re-yielding occurs.

Autofrettage produces the following residual stresses in the elastic portion of the cylinder:

{'^h)re = ^

1 + (7 b^^

{<rr)r ^

R^ - 1

b

c

1 - I - + 2 1n-c j a

(22)

b\^ b 1 - I - + 2 In -

c / a

(23)

(o,,=i((^r-^fi\\c] R^ - 1

b\^ b 1 - I - + 2 /« -

c / a

{'^zo)re = 0

(24)

(25)

The residual stresses in the plastic portion of the cylinder are:

5„ ilb

( ^>- = ^ l7j -^^^ '%

R'-l 1 +

b\^ b 1 - I - +lln-

c I a (26)

Sy ilb'^ ^^r)rp=^\\-\ - l + 2 1 n -

\/3" ^\c

1 / ? 2 - l - ( 7 !

b + 21n - (27)

20 25 30 35 40 45 50 56 Radial Distonce (mm)

• Sh •

• VMo •

Sr

VMo

Fig. 7 Total stress distributions for autofrettaged cyiinder (jy c =" 67 percent; b = 33.49 mm; Sy = 950.3 MPa, p = 350 MPa)

55

>

1

1100

1000

900

300

\ \

\

\

\

'v..

\

+-''"

,.-

...-•••

y'

. > f

X ,*

. . - • ' • • " ^

/ /

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Autofrettage Ratio b/c

350 MPa • • 500 MPa

Fig. 8 Optimum autofrettage ratios

i^zc)rp-^\\-A +21n

R^ 1 - - + 2 In -

c / a

(Oio),p = 0

(28)

(29)

The residual stress distributions shown in Fig. 6 result from a 75 percent autofrettage (i.e., b/c = 75 percent). Note that the maximum Von Mises stresses occur at the inner bore where the residual stresses are maximum. Figure 7 shows the final total stress distributions for a cylinder that has been previously autofrettaged to h/c = 67 percent and now operates at a static pressure of 350 MPa. It is obtained by superposing Eqs. (l)-(4) with Eqs. (22)-(29).

How much autofrettage is enough? For a given operating pressure, as the depth of autofrettage is increased, the maximum Von Mises stress location switches from the inner bore to the autofrettage depth b. Therefore in some situations, such as when the elastic/plastic interface approaches the outer wall, yield failure begins near or at the outer surface rather than the inner bore. The optimum autofrettage therefore, minimizes the maximum Von Mises stresses, and is a function of a, b, c, S^, and the operating pressure. A coarse, modified Newton's optimization routine was used to generate Fig. 8. It shows that for an operating pressure of 200 MPa, the optimum autofrettage results at about b/c = 60 percent;



Table 6 Simulation results for autofrettaged cylinder; (tyc = 67 percent; p = 350 MPa; a = 25 mm; c = 50 mm; S = 950.3 MPa)

Sy

o'a

nig

blc

all

50.0% 67.0% 80.0%

50.0% 67.0% 80.0%

Min.

840.46

658.97 530.71 767.97

-119.95 83.45

116.14

Avg.

950.30

816.22 666.07 823.51

134.07 329.06 285.26

Max.

1219.28

990.89 870.47 893,08

502.59 559.89 511.24

Std.Dev.

48.91

82.20 44.99 24.40

95.61 62.80 54.63

Skew

0.392

0.013 0.026 0.057

0.045 0.067 0.207

Kurt.

2.859

1.831 2.683 1.986

2.351 2.729 2.810

Table 7 Reliability estimates for autofrettage cylinder

b/c

50.0% 67.0% 80.0%

Normal

0.9196 1.0000 1.0000

Log-Normal

0.9163 1.0000 1.0000

Relative Frequency

0.9099 1.0000 1.0000

Time (S)

38.12 52.78 52.79

for 500 MPa, the optimum solution results at about b/c = 70 percent and for 350 MPa, the optimum is about b/c = 70 percent. The 350 MPa case was further optimized using a higher accuracy search resulting in b/c = 61.0 percent and maximum Von Mises stress equal to 609.4 MPa.

The design example is evolving. However, will an autofrettaged cylinder provide the reliability we require? The next section discusses the application of the design variation simulation method to an autofrettaged thick-walled cylinder.

Simulation of Autofrettaged Cylinder A computer program was written to calculate the residual

stress distribution as a function of the autofrettage radius b, according to Eqs. (22)-(29). Residual stress distributions were determined for b/c — 67 percent. Then elastic stresses as a function of operating pressure were simulated and superposed to the residual stresses. The internal pressure was simulated from a uniform distribution with mean and standard deviation P ~ U(350,35) MPa. The inner and outer radii were simulated as normally distributed variables a ~ N(25,0.25) mm, and c ~ N(50,0.5) mm, respectively. And the material yield strength was simulated as a WeibuU distribution with mean and standard deviation S ~ W(950.3, 48.9) MPa.

The smallest autofrettage ratio is 50 percent which obtains an inner bore radius of 25 mm. Similarly, the largest amount of autofrettage corresponds to the outer radius 50 mm which obtains a 100 percent autofrettage ratio. Three cases of autofrettage are examined: 50 percent, 67 percent (deterministic optimum) and 80 percent. The 50 percent values shown in Table 6 correspond to the "no-autofrettage" case. Therefore data from Table 2 is repeated here for comparison, The 67 percent and 80 percent autofrettage results were obtained, using 25000 simulations each, and assumed that the_ autofrettage depth b is characterized by a mean value of b and a 5 percent COV, normally distributed.

As shown for the optimal autofrettage case, the average margin of safety increases to 329.06 MPa as compared to the non-autofrettaged margin of 134 MPa. Note also that the minimum margin of safety is now 83.45 MPa, compared to — 119.95 MPa or failure. The 80 percent autofrettaged results indicate that the margin of safety, although better than the untreated cylinder, are somewhat worse that the optimum, as should be expected from the optimization results.

The simulated reliability increases to 100 percent from 90.99 percent as shown in Table 7. A reliability of 100 percent is expected since Table 6 shows the minimum margin to be greater than zero. Note that the required computer time was about 53 seconds.

Q B 1 e -»T"^ ' i 1 1—

ovg = 529 MPa

-1 1 1 T = ^ ^ so 100 150 200 250 300 350 400 450 500 550

Margin of Safety (MPa)

Fig. 9 Relative frequency for autofrettaged cylinder

The margin of safety relative frequency for the optimally autofrettaged cylinder is shown in Fig. 9. Its almost symmetric appearance reflects the 0.067 coefficient of skew and the almost "normal" peakedness is depicted by the 2.729 coefficient of kurtosis versus 3.0 for a normal distribution.

Summary The design of thick-walled cylinders subjected to large,

static internal pressures, includes considerations of geometry, loading and material properties. Naturally occurring variations of these items, can be evaluated using the design variation simulation method, in a reasonable amount of time on a personal computer. In addition, autofrettaged thick-walled cylinders exhibit an optimum autofrettage ratio which can be determined using a conventional optimization technique. Also, as shown in the example design, an autofrettaged cylinder may provide reliable and safe operation.

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