Nội dung

EM 1110-2-6054 1 Dec 01 Figure 6-3. Required dimensions of a discontinuity (after British Standards Institution 1980) where t = component thickness, l = effective crack length, (2a), b = effective dimension of crack in the throughthickness direction, and P = effective dimension of the distance between the edge of component and edge of crack in the through-thickness direction Direction of Principal stress Figure 6-4. Resolution of a discontinuity (after British Standards Institution 1980) (b) Check interaction with neighboring discontinuities to obtain the idealized discontinuity dimensions; idealizations for interaction of discontinuities are shown in Figures 6-5 and 6-6. (c) Check interaction with surfaces by recategorization as shown in Figure 6-7 for surface or embedded discontinuities (idealized or actual). (d) Determine final idealized effective dimensions for fracture analysis. (3) Determine the stress level by an appropriate structural analysis, assuming no crack exists. Structural loading can be divided into primary stress σp and secondary stress σs. The primary stress consists of membrane stress σm and bending stress σb, due to imposed loading. Examples of secondary stresses include stress increase due to stress concentration imposed by geometry of the detail under consideration, thermal stress, and residual stress. For discontinuities at non-heat-treated welds, the residual tensile stress should be taken as the yield stress. An estimate of the residual stress should be used for post-heat-treated weldments. The applied stress is the sum of primary σp and secondary σs stresses. If the applied stress is greater than the yield stress, EPFM must be employed. If applied stress is less than the yield stress and the plane-strain factor βIc ≤ 0.4 (Equation 2-2), LEFM should be used based on KIc. When the applied stress is less than the yield stress and βIc > 0.4, Kc (a function of plate thickness) should be used instead of KIc, if available. Otherwise, EPFM based on CTOD analysis must be employed. 6-4 EM 1110-2-6054 1 Dec 01 Figure 6-5. Interaction of coplanar discontinuities (Extracts from British Standards Institution 1980. Complete copies of the standard can be obtained by post from BSI Publications, Linford Wood, Milton Keynes, MK14 6LE) (4) Determine material properties including yield strength σys, modulus of elasticity E, KIc (based on the level of applied stress and the value of βIc), Kc, or CTOD. KIc may be estimated from Charpy V-Notch (CVN) test values by the transition method (paragraph 5-8) if direct KIc test data are not available. (5) Perform fracture assessment to determine the critical discontinuity size. (6) If the discontinuity is noncritical, determine the remaining life using a fatigue analysis as described in paragraphs 6-7 and 6-8. These steps are further discussed in the following paragraphs. 6-5 EM 1110-2-6054 1 Dec 01 Figure 6-6. Interaction of noncoplanar discontinuities (Extracts from British Standards Institution 1980. Complete copies of the standard can be obtained by post from BSI Publications, Linford Wood, Milton Keynes, MK14 6LE) Figure 6-7. Interaction of discontinuities with surfaces (Extracts from British Standards Institution 1980. Complete copies of the standard can be obtained by post from BSI Publications, Linford Wood, Milton Keynes, MK14 6LE) 6-6 EM 1110-2-6054 1 Dec 01 6-4. Linear-Elastic Fracture Mechanics a. Fundamental concepts of LEFM are described by Barsom and Rolfe (1987). LEFM is valid only under plane-strain conditions, when βIc ≤ 0.4. The basic principle of LEFM is that incipient crack growth will occur when the stress-intensity factor KI (the driving force) equals or exceeds the critical stress-intensity factor KIc (or KId for dynamic loading) (the resistance). For nonplane-strain cases, an initial evaluation based on an approximate analysis using LEFM with Kc taken as the resistance could be carried out. b. KI characterizes the stress field in front of the crack and is related to the nominal stress σ and crack dimension a for a given load rate and temperature by K I = Cσ a (6-1) where C is the dimensionless correction factor for a given geometry and loading type. If C is known, KI can be computed for any combination of σ and a. Stress-intensity factors for various types of geometries can be calculated using the information included in Figures 6-8 through 6-16 (Barsom and Rolfe 1987). Barsom and Rolfe and Tada, Paris, and Irwin (1985) contain compilations of solutions for a wide variety of configurations. c. After the stress-intensity factor is determined by Equation 6-1, it should be compared to the critical stress-intensity factor KIc (or KId for dynamic loading, or Kc for approximated nonplane-strain cases). A factor of safety (FS) = 2.0 applied to crack length is considered appropriate to prevent fracture. Therefore, the crack is considered to be acceptable if KI < Kic / 2 . d. To determine the allowable maximum crack size or nominal stress for a given KIc (or KId or Kc), substitute KIc for KI and solve for a or σ using Equation 6-1. The critical discontinuity size acr a structural member can tolerate at a given nominal stress σ and KIc with a factor of safety applied to the crack size is given by Equation 6-2: a cr = 1  K Ic    FS  Cσ  2 (6-2) e. In determining the nominal stress when stress gradients are present, an approximate method is to linearize the stress distribution, and divide it into membrane stress σm and bending stress σb. The stressintensity factor for each component of stress can be calculated separately and then added together. The total applied stress (σp and σs) can be linearized and resolved into σm and σb as shown in Figure 6-17. 6-5. Elastic-Plastic Fracture Assessment Rearranging Equation 2-2, the upper limit of plane-strain behavior may be determined as t K Ic = 2.5 σ ys (6-3) When this upper limit is exceeded, extensive plastic deformation occurs at the crack tip (crack tip blunting) and a nonlinear EPFM model must be used for analysis. (LEFM analysis using Kc may be used if the applied stress is less than yield stress.) Crack growth criteria for nonlinear fractures can be modeled by an R-curve, J-integral, or CTOD analysis (Barsom and Rolfe 1987). The CTOD method is the recommended method of EPFM analysis for evaluating hydraulic steel structures. The recommended procedure for cases where the applied stress (σp + σs) is greater than the yield stress is as follows (British Standards Institution 1980). 6-7 EM 1110-2-6054 1 Dec 01 Figure 6-9. Double-edge crack (Barsom and Rolfe 1987, p 40. Reprinted by permission of PrenticeHall, Inc., Englewood Cliffs, NJ.) Figure 6-8. Through-thickness crack (Barsom and Rolfe 1987, p 39. Reprinted by permission of Prentice-Hall, Inc., Englewood Cliffs, NJ.) a. Determine the effective discontinuity parameter ā. This is the equivalent through-thickness dimension that would yield the same stress intensity as the actual discontinuities under the same load. (1) For through-thickness discontinuities, ā = l/2, where l is the measured crack length. (2) For surface discontinuities, ā is determined by Figure 6-18. (3) For embedded discontinuities, ā is determined by Figure 6-19. b. Determine allowable discontinuity parameter ām that is calculated by  δ crit    ε y  am = C  6-8 (6-4) EM 1110-2-6054 1 Dec 01 Figure 6-10. Single-edge crack (Barsom and Rolfe 1987, p 40. Reprinted by permission of Prentice-Hall, Inc., Englewood Cliffs, NJ.) Figure 6-11. Cracks growing from round holes (Barsom and Rolfe 1987, p 42. Reprinted by permission of Prentice-Hall, Inc., Englewood Cliffs, NJ.) where C = values determined by Figure 6-20 δcrit = critical CTOD (paragraph 5-8c) εy = yield strain of the material In determination of C, if the sum of primary and secondary stresses, excluding residual stress, is less than 2σys, the total stress ratio (σp + σs)/σys (including residual stress) is used as the abscissa in Figure 6-20. If this sum exceeds 2σys, an elastic-plastic stress analysis should be carried out to determine the maximum equivalent plastic strain that would occur in the region containing the discontinuity if the discontinuity were not present. The value of C may then be determined using the strain ratio ε/εy as the abscissa in Figure 6-20. 6-9 EM 1110-2-6054 1 Dec 01 Figure 6-12. Cracks growing from elliptical holes (Barsom and Rolfe 1987, p 43. Reprinted by permission of PrenticeHall, Inc., Englewood Cliffs, NJ) where KT = theoretical stress concentration factor, aN = half of the long dimension of the ellipse, b = half of the short dimension of the ellipse, and f = radius at the narrow end of the ellipse Figure 6-13. Edge-notched beam in bending (Barsom and Rolfe 1987, p 45. Reprinted by permission of Prentice-Hall, Inc., Englewood Cliffs, NJ) where M = bending moment per unit thickness, B = beam width, W = beam depth, and g = function that describes effect of a/w on KI 6-10 EM 1110-2-6054 1 Dec 01 Figure 6-14. Embedded elliptical or circular crack (Barsom and Rolfe 1987, p 47. Reprinted by permission of Prentice-Hall, Inc., Englewood Cliffs, NJ.) 6-11 EM 1110-2-6054 1 Dec 01 Figure 6-15. Surface crack (Barsom and Rolfe 1987, p 48. Reprinted by permission of Prentice-Hall, Inc., Englewood Cliffs, NJ.) Figure 6-16. Cracks with wedge forces (Barsom and Rolfe 1987, p 52. Reprinted by permission of PrenticeHall, Inc., Englewood Cliffs, NJ.) 6-12 EM 1110-2-6054 1 Dec 01 Figure 6-17. Linearization of stresses (Extracts from British Standards Institution 1980. Complete copies of the standard can be obtained by post from BSI Publications, Linford Wood, Milton Keynes, MK14 6LE) c. If the effective discontinuity parameter ā is smaller than the allowable discontinuity parameter ām, then the discontinuity is acceptable. Using the procedure described in b above results in a factor of safety equal to approximately 2.0 in the determination of ā m; Figure 6-20 was developed as a design curve. Therefore, the calculated critical crack size would be equal to 2.0 ām (British Standards Institution 1980). 6-6. Fatigue Analysis a. For most lock gates and spillway gates that have vibration problems, fatigue loading is a real concern and a fatigue evaluation may be required. Fatigue analysis is used to predict when the cyclic loading will cause a crack to propagate to critical size resulting in fracture. A fatigue analysis can also provide crack growth rates that are useful in determining inspection intervals. b. The total fatigue life is the sum of the fatigue crack-initiation life and the fatigue crack-propagation life to a critical size (Barsom and Rolfe 1987). NT = Ni + Np (6-5) where NT = total fatigue life NI = initiation life Np = propagation life 6-13