Chapter 6 Reliability-based design and code developments

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1 Chapter 6 Reliability-based design and code developments 6. General Reliability technology has become a powerul tool or the design engineer and is widely employed in practice. Structural reliability analysis documents reliability design and saety assessment o structure. According to the current development o reliability techniques in engineering, structural reliability analysis can be used or the ollowing applications: ) Estimate reliability in existing designs. hat is the level o risk (probability o ailure) o the structures designed by using codiied actors o saety in traditional designs? Reliability analysis given in this report can be ollowed to estimate the risk or existing ships. ) Develop probability based design code requirements. In this aspect, according to the present development achievement o structural reliability technology, structural reliability technology is mainly used to calibrate the partial coeicients in structure design codes. This has been a major application o reliability methods. 3) Perorm ailure analysis. A structure component ails. In the subsequent investigation, the question is asked, what was the cause o ailure and the probability o occurrence o this event? and is the probability o this occurrence small enough that we can avoid making a signiicant investment to make a design change? Reliability analysis and assessment can help to resolve these questions. 4) Compare alternative designs. Reliability as an index o structural perormance has been used very successully by the oshore industry or the purpose o comparing competing design concepts. 5) Support economic value (cost~beneit) analysis. The tradeo between cost and risk is analysed in order to gain insights on the decision making process, i.e., in theory, decisions could be made to minimize total expected lie cycle costs. 6) Develop a strategy or design and maintenance o ageing structures. Risks are reduced when a structure is periodically inspected and repaired or replaced i necessary. Economic value analysis can be also employed to develop optimal maintenance strategies that lead to minimum cost without reducing the reliability below a speciied level. 7) Quantitative Risk Assessment with Structure Reliability Analysis The ollowing will give a simple guidance about reliability assessment and calibration o partial saety coeicients. Quantitative risk assessment with structure reliability analysis will be introduced in Chapter Reliability assessment o structure saety 8

2 Reliability assessment o structural saety is on the basis o reliability calculation o the structure. The ollowing is the basic procedure o structural reliability assessment: a) Establish a target reliability (saety standard). b) Calculate the structural reliability. c) Assess whether the calculated reliability is suicient and modiy the design i necessary. d) Evaluate the results o the reliability analysis with respect to parametric sensitivity considerations. See section 6.4 or a simple example. 6.3 Calibration o partial saety coeicients 6.3. General method Standard deterministic design methods result in designs against speciied limit-states by application o design values o the governing basic variables, where each design value is ormed by the product o a partial coeicient and the characteristic value or the variable in question. The saety redundancy in traditional deterministic design is ensured by the deterministic saety actors. The drawbacks o these kinds o saety actors are usually shortage o strict theory derivations and they are determined with more objective ingredients. ith a reliability method available, a set o partial coeicients can be derived which will result in designs with a given target reliability. The general steps o determining reliability partial saety coeicients o a structure in question are: a) Calculate the structural reliability o the initial design structure. This part includes establishing physical modelling, limit state unction, identiication o random variables and their distribution types and parameters, o course, includes reliability calculation, etc. b) Adjust the design and calculate the structure reliability again until the given target reliability is met. c) Determine partial saety actors: these coeicients shall be determined on basis o the relationships between the characteristic values o the design variables and the corresponding values in the design points. This step includes that the characteristics o the governing stochastic variables shall be speciied as quantiles o their respective probability distributions. In general, partial coeicients are introduced in pairs, one partial coeicient associated with a load variable and another associated with a resistance variable. In general, a partial coeicient can be represented as ollow: x * x (6.3.) xc where is partial saety coeicient about stochastic variable x. x * x is the value o design point o stochastic variable x. x is the characteristic value o stochastic variable x. c 9

3 As in the traditional deterministic design, or a variable that has been assigned a partial coeicient, the design value shall be taken as the product o its characteristic value and its partial coeicient. The design values o the basic random variables are denoted by q, r and l, respectively. The design values may be written as id id kd q r l id jd kd φ ξ in which q ic, ric and lkc are characteristic values o load, resistance and geometric parameters, i are load actors, φ j are resistance actors, and ξ k are geometrical actors. Veriication o the structure with respect to the limit state is done in standard deterministic design by inserting the design values and the dimension parameters into the limit state unction to ascertain that G ( q, r, l ) 0 id jd where G ( Qi, R j, Lk ) is a limit state unction. Q i, Rj, Lk are the relevant load parameters, strength parameters and nominal geometric parameters. or all other variables, the design value shall be taken directly as the characteristic value or speciied value. rom equation (6.3.), it can be concluded that the design values are equal to the values in design point and at the ailure surace. It also means that the limit state unction shall equal zero i the design values are substituted or the stochastic variables in the expression or the limit state unction. In addition, according to current design practice, a load coeicient is used as a actor on the characteristic load to give the design load, and a strength coeicient is used as a divisor on the characteristic strength to give the design strength. Thereore, the strength coeicient shall become x c sx (6.3.) * x In the design process, a set o partial coeicients is introduced. The number o partial coeicients can be equal to or less than the number o stochastic variables. I less than the number o stochastic variables, the partial coeicients which are introduced should be assigned to those o the stochastic variables that are important as determined by the reliability analysis. A common choice is to introduce one partial coeicient associated with a load variable and one partial coeicient associated with a resistance variable. All stochastic variables shall be represented by characteristic values. or a variable which has been assigned a partial coeicient, the design value should be taken as the product o its partial coeicient and its characteristic value. or all other variables the design value shall be taken directly as the characteristic or design value Characteristic values i j k q r l kd x ic jc kc 0

4 To determine the partial coeicients, the characteristic value or each variable should be given irst. In this section, simple guidance to determine characteristic values will be introduced. Characteristic values or the governing stochastic variables should be speciied as quantiles o their respective probability distributions. The characteristic values o loads and strength parameter are reerence values to be used in the design process. Characteristic values given in rules should normally be used to the extent such are given. The characteristic values o load and resistance basic variables are to be based on a prescribed statistical conidence level Characteristic values or strength parameters The characteristic values or basic variables o stochastic resistance (strength) are to be based on a prescribed statistical conidence level. Characteristic resistance is to be determined on the basis o reliable data and appropriate statistical techniques based on recognized methods or testing. The ollowing guidance may normally be applied in the choice o the characteristic resistance o steel (DNV, Classiication Notes No.30.6). here high structural resistance reduces structural saety, the characteristic resistance is to be determined such that it has a low probability o being exceeded. The characteristic value o strength (or resistance) is normally to be calculated based on net scantlings, i.e. with any corrosion excluded. In cases where the eect o the corrosion addition reduces the structural saety, the corrosion addition is to be included. The characteristic value o strength (or resistance) is normally to be based on the 5 th or 95 th percentile o the test result, whichever is the most unavourable. The characteristic atigue strength (or resistance) is normally to be based on the.5 th percentile o the test results. The characteristic value o resistance or design yield strength is to be taken as the smaller o: a) minimum upper yield stress; b) yield strength at 0.% oset; c) 0.85 times o minimum tensile strength o the material. The ollowing guidance may normally be applied in the choice o the characteristic resistance o aluminium. or welded aluminium alloy structures the characteristic material yield stress o the weld region is normally to be based upon either: a) 0.% proo stress in the annealed condition, b) appropriate test results. The ollowing guidance may, normally, be applied in the choice o the characteristic resistance o concrete. The characteristic value o strength is normally based on the 8 th or the 9 nd percentile o the test result, whichever is the most unavourable. The characteristic atigue strength is normally based on the.5 th percentile o the test results. here high structural

5 resistance reduces structural saety, the characteristic resistance is to be determined such that it has a low probability o being exceeded Characteristic values or loads The characteristic values or loads are normally to be determined as the loads that cause load eects with a given probability o being exceeded. or loads without a statistical representation, the characteristic value is the speciied value that will deine operational limitations or the structure. The ollowing guidance may normally be applied in the choice o the characteristic loads. Permanent loads: The characteristic value o a permanent load is normally taken as the expected value based on accurate data o the unit, mass o the material and the volume in question. Live loads: The characteristic value o a live load is the maximum (or minimum) speciied value which produces the most unavourable load eect in the structure under consideration. The speciied value is to be determined on the basis o relevant speciications. A speciied load history is to be used in atigue limit states (LS). Deormation loads: The characteristic value o a deormation load is a value which produces the most unavourable load eects in the structure under consideration. Deormation loads due to environmental phenomena are to be determined with a similar probability o being exceeded as or environmental loads. or other deormation loads, the characteristic value is to be taken as a speciied value. A speciied or expected load history is to be used in atigue limit states (LS). Environmental loads: The characteristic value o an environmental load is normally the maximum or minimum value (whichever is the most unavourable) corresponding to a load eect with a prescribed probability o exceedence. In most cases, the characteristic value o an environmental load is taken as the most probable extreme value, with a speciied return period. Inormation on the joint probability o the various environmental loads may be taken into account i such inormation is available and can be adequately documented. or the atigue limit state, the characteristic value is normally taken as the expected load history. or the serviceability limit state, the characteristic value is a speciied value, dependent on operational requirements. or the temporary design conditions, the characteristic values may be based on speciied values, which are to be selected dependent on the measures taken such that the required saety level is obtained. The saety level in the temporary design conditions, however, is not to be inerior to the saety level required or operating design conditions. Accidental loads: The characteristic value o an accidental load is to correspond to a load eect with a prescribed annual probability o being exceeded. This annual 4 probability is normally taken to be equal, or less than 0, unless some other

6 probability o exceedence can be justiied. or temporary design conditions, the characteristic value may be a speciied value dependent on practical requirements. The saety level related to the temporary design condition is not to be inerior to the saety level required or the operating design conditions Illustration The example given here is an axially loaded steel component. In this section, irst, the procedure o structural reliability analysis is illustrated. Then, a simple calibration o partial coeicients related to the axially loaded steel component is deduced Procedure o structural reliability analysis and assessment In this example the design o the axially loaded steel component is governed by the extreme axial orce Q in its lietime. The yield strength o steel is σ and the crosssectional area o the steel component is A. The ollowing is the procedure o the structural reliability analysis a) Identiy ailure modes In this example, only yielding ailure is considered. b) Establish relevant limit-state unctions The limit-state unction is chosen as G σ A Q c) Identiy stochastic variables and parameters and speciy their probability distributions Q ollows a Gumbel distribution (i.e. Extreme value distribution type I). Its probability density and cumulative distribution unctions are, a( x b) ( x) aexp[ a( x b) e ] Q a { ( x b e ) } Q ( x) exp where a and b 79. correspond to a mean value E[ Q] 80MN and a standard deviation D[ Q] MN. The yield strength o steel σ ollows a lognormal distribution with mean value E[ σ ] 400MPa and standard deviation D[ σ ] 4MPa. Its probability density unction and the cumulative distribution unctions are: ln y λ σ ( y) exp y < π yε ε ln x λ ( u) du Φ ε σ ( y) σ The area A is used as the design parameter. y Q, σ are independent. The joint probability density distribution unction is ( x, y) ( x) ( y) where Q (x) is the probability density distribution unction o Q, X Q σ 3

7 σ (y) is the probability density distribution unction o σ. d) Calculate the structural reliability The ailure probability in question is P X Ay x 0 ( x, y) dxdy The reliability and reliability index are P, β Φ ( ) I the area A is given r P 0.57m, then P P 0 4, P , β r e) Establish a target reliability or this example, the target reliability index is taken to be β 3. 79, corresponding the ailure probability P 0 4. ) Assess the reliability o the estimated structure According to the above calculation, in order to meet a target reliability index β 3.79, the area A should not be less than 0.57m. This means that i the area A is less than 0.57m, the target reliability index β will less than 3.79, such that the designed structure will not have enough saety reliability (redundancy) Procedure o calibration o partial coeicients On the basis o the above reliability analysis and assessment, the ollowing is the procedure or determination o partial coeicients related to the axially loaded steel component. a) Determination o characteristic values related to the structure in question In this example, the characteristic value o the extreme axial orce is taken as the mean extreme value, q c E[ Q] 80MN. The characteristic value o the yield stress o steel is taken as the 5% quantile, σ σ 36. MPa., c,5% 8 b) Calculation o the coeicients One partial coeicient is introduced or the orce and another or the strength. Setting these values into the limit-state unction at the limit state surace yields σ, c A qc 0, i.e., which gives a requirement to the ratio o the partial coeicients, 0.86 The reliability analysis gave the design point values σ * MPa or the strength. The partial coeicients are thereore selected as * q.079 q c * σ σ, c 0.98 q * 86. 3MN or the orce and 4

8 According to current design practice, a load coeicient is used as a actor on the characteristic load to give the design load, and a material coeicient m is used as a divisor on the characteristic strength to give the design strength. Hence, these coeicients become.079 m Calibration procedure or calculating partial saety actors or a ship structure As described above, partial saety actors are used in the calibration procedure to assure a speciied reliability level. or ship structure (SSC 368), M + M SM (6.4.) φ σ where M M Y y y,, φ are the partial saety actors or the characteristic values,, σ, respectively. Y The relevant nomenclature lies at the end o section 6.4. The above partial saety actors can be determined by the method o calculating design points. The ollowing procedure is another similar engineering method to determine the partial saety actors or ABS Ships (SSC 368): ) By trial and error determine,, φy in equation (6.4.) that gives the β t arget. ) ind out or dierent ratios o M M, the value o β determined rom ORM (or SORM) using the,, φy obtained in the irst step, and check i: a) the obtained β is close to the target β, and b) the obtained β range is smaller than that o ABS rules. 3) I the determined,, φy give β close to β t arget and β range is smaller, then they can be used in the new calibrated code, otherwise make changes in them to satisy the criteria a. and b. above. The procedure described above can be implemented as ollows. Equation (6.4.) can be rewritten as: SM + m (6.4.) M φ σ where m is the ratio o wave bending moment to stillwater bending moment. It is obvious that in equation (6.4.) φ Y is arbitrary, so it is set to be 0.86, i.e. a material or strength saety actor o.5. Thereore, i two ships can be ound with saety indices equal to 3.0 (here 3.0 is the target reliability index), a pair o tentative values or, can be determined. One ship can be directly chosen rom Table 6.4.; it is the ship with L 74.5m, C b 0.6, β By trial and error, another ship can be ound by changing section modulus o the ship with L 3.5m, C rom y y b 5

9 66690m cm to 66374m cm to make β equal to The values o and can be obtained by solving the resulting two equations when the values are substituted in equation The resulting and are:.03.5 L (m) C β ( L / B 5.0) β ( L / B 9.0) b Table 6.4. Saety indices o ABS Ships (see SSC365, p6) Using these partial saety actors, new set o section moduli can be calculated or which reliability analysis is perormed to determine the saety index or every ship. The result is listed in table The β s in table 6.4. are very close to each other ( < β < 3. 0 ), as compared to the range o β derived rom ABS Rules. Thereore, the calibrated model or the section modulus that gives uniorm saety or all ship sizes is given by equation 6.4. with.03 φ Y

10 L (m) C β ( L / B 5.0) b Table 6.4. Saety indices o redesigned ABS Ships (see SSC365, p7) The main beneit that accrues rom the redesign exercise, according to the new saety check ormat, is uniorm reliability and structural saety among dierent ship sizes, which in some cases could lead to weight savings. Code calibration exercises such as this can highlight sometimes large dierences in implicit saety levels or dierent ailure modes in a structure, a situation that can be rectiied in a new generation reliability based code. The ollowing is the part o the nomenclature o this section: B Ship breadth C Block coeicient b L Ship length M Stillwater bending moment M ave bending moment SM Section modulus σ Y Yield strength β Saety index Note: other symbols are deined where used 7

11 6.5 Conclusions Reliability technology has become a powerul tool or the design engineer and is widely employed in practice. Structural reliability analysis can be used in the ollowing applications : a) Estimate reliability in existing designs. b) Develop probability based design code requirements. c) Perorm ailure analysis. d) Compare alternative designs. e) Support economic value analysis. ) Develop a strategy or design and maintenance o ageingstructures. g) Quantitative Risk Assessment with Structural Reliability Analysis. In this chapter, reliability assessment and reliability based design code are introduced in detail, especially or reliability-based design code. irst, the general method o determining partial saety actors is derived. Then, a demonstration is given o how partial saety actors are determined, calibrated, and used in new designs that have uniorm saety. Last, this part provides an illustration o how probability based methods can be used to develop calibrate a code (or design criteria) in order to produce designs with uniorm saety (target saety index) over a wide range o the basic parameters involved in the design. 8