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1 ð15þ KD-232 PROTECTION AGAINST LOCAL FAILURE In addition to demonstrating protection against plastic collapse as defined in KD-231, the local failure criteria below shall be satisfied. KD Elastic Plastic Analysis Procedure. The following procedure shall be used to evaluate protection against local failure. (a) Perform an elastic plastic stress analysis based on the local criteria of Table KD and the ratcheting criteria for a series of applied loads in KD-234. Nonlinear geometry shall be used in the analysis. (b) For a location in the component subject to evaluation, determine the principal stresses, σ 1, σ 2, σ 3,the equivalent stress, σ e,usingeq. (KD-232.1) below, and the total equivalent plastic strain, ϵ peq. ðkd 232:1Þ (c) Determine the limiting triaxial strain ϵ L,k, for the k th load step increment using the equation below, ϵ Lu, m 2, and m 5 are determined from the coefficients given in Table KD ðkd 232:2Þ ϵ peq = total equivalent plastic strain e = , approximate value of the base of the natural logarithm El = minimum specified elongation, percent ϵ L,k = maximum permitted local total equivalent plastic strain at any point at the k th load increment ϵ Lu = maximum of m 2, m 3, and m 4 ln = natural logarithm m 2 = value calculated from Table KD m 3 = value calculated from Table KD m 4 = value calculated from Table KD m 5 = value listed in Table KD R = S y /S u RA = minimum specified reduction of area, percent σ 1k = principal stress in the 1 direction at the point of σ 2k = principal stress in the 2 direction at the point of σ 3k = principal stress in the 3 direction at the point of σ ek =vonmisesequivalentstressatthepointof interest S y = yield strength at the analysis temperature (see Section II, Part D, Subpart 1, Table Y-1) S u = tensile strength at the analysis temperature (see Section II, Part D, Subpart 1, Table U) (d) Determine the strain limit damage for the k th load step increment using the following equation: ðkd 232:3Þ D ϵ,k = strain limit damage for the k th loading condition Δϵ peq,k = equivalent plastic strain range for the k th loading condition or cycle (e) Add the damage occurring during the k th load step increment, D ϵ,k, to the sum of the incremental damage occurring at each previous increment to obtain the accumulated damage, D ϵ. (f) Repeat the process in (b) through (e) for all load step increments in the analysis. (g) If the component has been cold formed without subsequent heat treatment, calculate the damage from forming, D ϵform, using the equation below. If the Table KD Tabular Values for Coefficients Material Maximum Temperature m 2 m 3 m 4 m 5 ϵ p Ferritic steel [Note (1)] 480 C (900 F) 0.60 (1.00 R) 2 In [1 + (El/100)] In [100/(100 RA)] E 5 Austenitic stainless steel and 480 C (900 F) 0.75 (1.00 R) 3 In [1 + (El/100)] In [100/(100 RA)] E 5 nickel based alloys Duplex stainless steel 480 C (900 F) 0.70 (0.95 R) 2 In [1 + (El/100)] In [100/(100 RA)] E 5 Precipitation hardening, nickel based 540 C (1,000 F) 1.90 (0.93 R) In [1 + (El/100)] In [100/(100 RA)] E 5 Aluminum 120 C (250 F) 0.52 (0.98 R) 1.3 In [1 + (El/100)] In [100/(100 RA)] E 6 Copper 65 C (150 F) 0.50 (1.00 R) 2 In [1 + (El/100)] In [100/(100 RA)] E 6 Titanium and zirconium 260 C (500 F) 0.50 (0.98 R) 1.3 In [1 + (El/100)] In [100/(100 RA)] E 5 NOTE: (1) Ferritic steel includes carbon, low alloy, and alloy steels, and ferritic, martensitic, and iron based age hardening stainless steels. 69

2 component has not been cold formed, or if heat treatment has been performed after forming, the damage from forming, D ϵform, may be assumed to be zero. ðkd 232:4Þ Dϵ form = damage occurring during forming at the location in the component under consideration ϵ cf = forming strain at the location in the component under consideration (h) Add the damage from forming to the accumulated damage during loading to obtain the total accumulated damage, D ϵ t : ðkd 232:5Þ (i) The total accumulated damage, D ϵ t, shall be calculated for the case of the local criteria in Table KD and for the sequence of applied loads defined in KD-234 that will be applied to the component. This calculated D ϵ t value shall be no greater than 1.0, indicating the local failure criteria to be specified (see KD-232). The designer is cautioned that excessive distortion in the structure of the vessel may lead to failure of the pressure boundary. This could be in the form of buckling or bellmouthing (see KD-631.5). KD-233 PROTECTION AGAINST BUCKLING COLLAPSE In addition to evaluating protection against plastic collapse as defined in KD-231, a design factor for protection against collapse from buckling shall be satisfied to avoid buckling of components with a compressive stress field under applied design loads. KD Buckling Design Factors. The design factor to be used in structural stability assessment is based on the type of buckling analysis performed. If a collapse analysis is performed in accordance with KD-231, and imperfections are explicitly considered in the analysis model geometry, the design factor is accounted for in the factored load combinations in Table KD KD Buckling Numerical Analysis. When a numerical analysis is performed to determine the buckling load for a component, all possible buckling mode shapes shall be considered in determining the minimum buckling load for the component. Care should be taken to ensure that simplification of the model does not result in exclusion of a critical buckling mode shape. For example, when determining the minimum buckling load for a ringstiffened cylindrical shell, both axisymmetric and nonaxisymmetric buckling modes shall be considered in determination of the minimum buckling load. KD-234 RATCHETING ASSESSMENT ELASTIC PLASTIC STRESS ANALYSIS Nonintegral connections such as screwed on caps, screwed in plugs, shear ring closures, and breech lock closures are subject to failure by bell mouthing or other types of progressive deformation. If any combination of applied loads produces yielding, such joints are subject to ratcheting behavior. Stresses that produce slippage between the parts of a nonintegral connection in which disengagement could occur as a result of progressive distortion, shall be limited to the minimum specified yield strength at temperature, S y, or evaluated using the procedure in KD To evaluate protection against ratcheting using elastic plastic analysis, an assessment is performed by application, removal, and reapplication of the applied loadings. If protection against ratcheting is satisfied, it may be assumed that progression of the stress strain hysteresis loop along the strain axis cannot be sustained with cycles and that the hysteresis loop will stabilize. A separate check for plastic shakedown to alternating plasticity is not required. The following assessment procedure can be used to evaluate protection against ratcheting using elastic plastic analysis. KD Assessment Procedure. Step 1. Develop a numerical model of the component including all relevant geometry characteristics. The model used for analysis shall be selected to accurately represent the component geometry, boundary conditions, and applied loads. Step 2. Define all relevant loads and applicable load cases (see Table KD-230.1). Step 3. An elastic perfectly plastic material model shall be used in the analysis. The von Mises yield function and associated flow rule should be utilized. The yield strength defining the plastic limit shall be the minimum specified yield strength at temperature from Section II, Part D, Subpart 1, Table Y-1. The effects of nonlinear geometry shall be considered in the analysis. Step 4. Perform an elastic plastic analysis for the applicable loading from Step 2 for a number of repetitions of a loading event, or, if more than one event is applied, of two events that are selected so as to produce the highest likelihood of ratcheting. Step 5. The ratcheting criteria below shall be evaluated after application of a minimum of three complete repetitions of the loading cycle following the hydrotest. Additional cycles may need to be applied to demonstrate convergence. If any one of the following conditions is met, the ratcheting criteria are satisfied. If the criteria shown below are not satisfied, the component configuration (i.e., thickness) shall be modified or applied loads reduced and the analysis repeated. (a) There is no plastic action (i.e., zero plastic strains incurred) in the component. 70

3 (f) High strength alloy steel bolts and studs may be evaluated for cyclic operation by the methods of Article KD-3 using the design fatigue curve of Figure KD-320.5, provided (1) the material is one of the following: (-a) SA-193, Grade B7; SA-193, B16; SA-320, L7 (-b) SA-320, L7M and SA-320, L43 (2) "V" type threads shall have a minimum thread root radius no smaller than times the pitch, and in no case smaller than in. (0.102 mm). (3) fillet radii at the end of the shank shall be such that the ratio of the fillet radius to shank diameter is not less than The bolt stress shall be determined using the root area. See KD-622 for the fatigue strength reduction factor for threads. K r may be assumed to be 1.0 when Figure KD is used. The designer should use caution in calculating bolt load from applied torque. The designer shall consider that corrosion effects on a bolted connection can reduce bolt fatigue life. (g) When the operational cycle being considered is the only one that produces significant fluctuating stresses, the calculated number of design cycles N f is determined as follows. (1) Evaluate the fatigue penalty factor, K e, using ΔS n and the following equations parameters m and n are determined from Table KD-322.1: ðkd 322:1Þ ðkd 322:2Þ ðkd 322:3Þ ΔS n is the primary-plus-secondary stress intensity range. Otherwise, the alternate method given in KD-323 may be used to calculate K e. (2) Identify the applicable fatigue curve for the material as explained in (a). (3) Enter the curve from the ordinate axis at the value: ðkd 322:4Þ E(curve) = Modulus of Elasticity given on the Design Fatigue Curve E(analysis) = Modulus of Elasticity used in the analysis (4) Read the corresponding number of cycles on the abscissa. This is the calculated number of design cycles N f. ð15þ KD-323 ALTERNATIVE METHOD FOR EVALUATING THE FATIGUE PENALTY FACTOR, K e The fatigue penalty factor, K e, may be calculated using the following operations: ðkd 323:1Þ ðkd 323:2Þ ðkd 323:3Þ 77

4 ðkd 323:4Þ and E ya = the modulus of elasticity at the point under consideration, evaluated at the mean temperature of the cycle (Δε t ) ep = the equivalent total strain range from the elastic plastic analysis for the points of interest (Δε t ) e = the equivalent total strain range from the elastic analysis for the points of interest Δσ e = the range of primary-plus-secondary-plus-peak equivalent stress However, if using this alternative method, K e = 1.0 if shakedown is shown in accordance with KD-234. KD-330 CALCULATED CUMULATIVE EFFECT NUMBER OF DESIGN CYCLES If there are two or more types of stress cycles which produce significant stresses, the alternating stress intensity and the associated mean stress shall be calculated for each type of stress cycle. The cumulative effect of all of the stress cycles shall be evaluated using a linear damage relationship as specified in (a) through (f). (a) Calculate the number of times each type of stress cycle of type 1, 2, 3, etc., will be repeated during a specific design service life period L. It is recommended that L be based on the design service L d as specified in the User s Design Specification; designate these numbers n 1, n 2, n 3, etc., or generally n i. (b) For each type of stress cycle, determine S a by the procedures given in KD Designate these quantities S a 1, S a 2, S a 3, etc., or generally S ai. (c) For each value S ai, use the applicable design fatigue curve to determine the maximum number of design repetitions N i if this type of cycle were the only one acting. Designate these as N 1, N 2, N 3, etc., or generally N i. (d) For each type of stress cycle, calculate the usage factor U i = n i /N i. (e) Calculate the cumulative usage factor from: ðkd 330:1Þ The cumulative usage factor U shall not exceed 1.0. (f) Calculate the design service L d using the equation: ðkd 330:2Þ KD-340 FATIGUE ASSESSMENT OF WELDS ELASTIC ANALYSIS AND STRUCTURAL STRESS (a) An equivalent structural stress range parameter is used to evaluate the fatigue damage for results obtained from a linear elastic stress analysis. The controlling stress for the fatigue evaluation is the structural stress that is a function of the membrane and bending stresses normal to the hypothetical crack plane. (b) Fatigue cracks at pressure vessel welds are typically located at the toe of a weld. For as welded and weld joints subject to post weld heat treatment, the expected orientation of a fatigue crack is along the weld toe in the through thickness direction, and the structural stress normal to the expected crack is the stress measure used to correlate fatigue life data. For fillet welded components, fatigue cracking may occur at the toe of the fillet weld or the weld throat, and both locations shall be considered in the assessment. It is difficult to accurately predict fatigue life at the weld throat due to variability in throat dimension, which is a function of the depth of the weld penetration. It is recommended to perform sensitivity analysis the weld throat dimension is varied. KD-341 ASSESSMENT PROCEDURE The following procedure can be used to evaluate protection against failure due to cyclic loading using the equivalent structural stress range. Step 1. Determine a load history based on the information in the User s Design Specification and the histogram development methods in KD-350. The load history should include all significant operating loads and events that are applied to the component. 78

5 t b = thickness of blind end (E-110) t ess = structural stress effective thickness t H = thickness of head at joint (Figure KD-830.2) t L = thickness of layer at joint (Figure KD-830.2) t n = thickness of layer n (KD-802) = nominal thickness of nozzle wall less corrosion allowance (Figure KD-830.6) = thickness of nozzle wall (Figure KD-1130) t p = thickness of attached pipe wall (H-142) t r = minimum wall thickness without opening (H-120) t rn = required thickness of seamless nozzle wall (Figure KD-1122) t S = shell thickness (Figure KD-830.2) t w = thickness of vessel wall (E-110) U = cumulative usage factor (KD-330) V REF = longitudinal crack displacement (D-405) W = wind load (KD-230) = mass flow of any gas or vapor (KR-523.3, KR-531) = total design bolt load W A = assembly loads (e.g., shrink fit, wire winding, sealing preload) (KD-230) W a = rated air flow for relief device (KR-531) W c = total effective clamping preload on one lip W m1 = minimum operating bolt load W m2 = minimum gasket seating bolt load W pt = pressure test wind load case (KD-230) W T = theoretical mass flow (KR-523.3) w = width of the element to determine structural stresses from Finite Element Analysis X = absolute value of the range (load or stress) under consideration using the Rainflow Cycle Counting Method X b = basic clamp dimension to neutral axis X g = global X axis X i = average radial distance from bolt cutout area X L =localx axis, oriented parallel to the stress classification line X 5 = modified clamp dimension to neutral axis X 6 = modified clamp dimension to neutral axis x = diameter at any point (KD-911) = through wall thickness coordinate x 1 = any diameter of cylinder (KD-911) x 2 = any diameter of winding (KD-911) Y = wall ratio or D O /D I of a shell (KD-220, KD-502) = weld offset (Figure KD-830.2) = absolute value of the adjacent range (load or stress) to previous X using the Rainflow Cycle Counting Method Y g = global Y axis Y i = ratio of outside diameter to inside diameter of inner layer (KD-802) Y L =localy axis, oriented normal to the stress classification line Y o = ratio of outside diameter to inside diameter of outer layer (KD-802) y = radial offset in buttwelding of unequal section thicknesses (Figure KD-1121) Z = D O /D, D can be any point in the wall (KD-220) = compressibility factor (KR-531) = clamp hub taper angle α =shapefactor(kd-210, Figure , 9-100) = angle, maximum angle (Figure KE-321) = maximum rake angle (E-110) α r = thermal expansion of reinforcing metal (H-150) α v = thermal expansion of vessel wall (H-150) β = factor in equivalent alternating stress intensity (KD-312) = factor = 0.2 (KD-932) γ 1 = true strain in the micro strain region of the stress strain curve (KD-231.4) γ 2 = true strain in the macro strain region of the stress strain curve (KD-231.4) Δ = difference, increment ΔK = range of stress intensity factor (KD-430) ΔS ess,k = equivalent structural stress range parameter for the k th cycle = computed equivalent structural stress range parameter from Part 5 = average relative standard deviation of fatigue strength (KD-932.3) ΔT = operating temperature range (H-150) Δε k = local nonlinear structural strain range at the point under evaluation for the k th cycle = elastically calculated structural strain range at the point under evaluation for the k th cycle = local nonlinear structural stress range at the point under evaluation for the k th cycle = structural stress range = elastically calculated structural bending stress range at the point under evaluation for the k th cycle = elastically calculated structural stress range at the point under evaluation for the k th cycle 263