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EM 1110-2-6054 1 Dec 01 Figure 7-3. Two-stage CVN-KId-KIc correlation (°C = 5/9 (°F – 32); 1 psi- in . = 1.099 kPa- m ; 1 ft-lb = 1.36 J) (3) A CVN-KIc correlation that is valid at higher temperatures in the upper shelf region is given by 2  K Ic   CVN  - 0.0098    = 0.646  σy   σy  (7-3) where KIc = MPa - m σy = static yield stress in MPa CVN = joules (For non-SI units, 2  K Ic   CVN  - 0.05    =5  σy   σy  7-3 EM 1110-2-6054 1 Dec 01 structure is framed similar to the standard tainter gate geometry as described by EM 1110-2-2702 with a 0.95-cm (3/8-in.) skin plate, C12 × 25 vertical ribs, two W30 × 118 horizontal girders, and W18 × 80 strut arm frames. All connections are riveted except for the use of bolts at the strut arm-trunnion block detail. The gates have Type J side seals and steel bottom seal details. The gates have a history of structural problems including significant gate vibrations and buckled web and flange plates on the strut arm. No extreme loads or unusual events had been reported since the last inspection. A change in operational practice was instituted to avoid gate opening settings that cause structural vibration. Because of the history of problems at this site, a thorough visual inspection was made previously on several gates. (2) Inspection. An in-depth inspection was made of the gate with particular attention to the critical areas. Weather conditions at the dam site during the inspection were sunny and warm. The examination was conducted while water was being released from the gates. The following conditions were noted: (a) Member or component deformation. Local web and flange plate buckling on the strut arms adjacent to the knee brace intersection from the upper horizontal girder was visible on several gates and is most severe on Gate 24. The condition has not deteriorated since the last inspection and was most likely caused by excessive ice loads on the structure. (b) Seal problems. Water was observed flowing through the side seals. (c) Rivet deterioration. Corrosion and a small amount of section loss were visible on some rivet heads. (d) Mechanical/electrical problems. At Gate 25, one chain hoist was out of its guide on the skin plate. This hoist was toward the Minnesota side of the gate. (e) Fabrication defects. There was no previous indication that fabrication defects existed in the structural members, and none were observed during this inspection. (f) Corrosion. Paint loss and blistering were visible along the top surface of the web on the upper horizontal girder under the diversion plate. Blistered paint was left intact during the inspection. (g) Fatigue cracking. No fatigue cracks were observed. (h) Vibration or other unusual behavior. To check for vibration, the gate was fully closed and then reopened approximately 3.0 cm (0.1 ft) when vibration began. By rough measurement, the vibration frequency was estimated at 5-10 Hz. The amplitude of vibration was maximum at midspan of the gate and was sufficient to create an audible noise and make ripples in the backwater. The vibration ceased when the gate was opened further. (i) Application of unusual loads. Except for the noted vibration, no unusual or extreme loads were reported. There was, however, an extensive accumulation of debris on the structural members in back of the skin plate, primarily large timber pieces. (3) Evaluation. Because several detrimental conditions were detected during the inspection, the structural integrity of the spillway gate must be evaluated. (a) Since an evaluation of the local buckling of the strut arms was conducted when it was first observed and the amount of buckling on the strut arms had not increased since the last inspection, it is believed that the structural capacity of the buckled members or of the gate is not in jeopardy at this time. 7-5 EM 1110-2-6054 1 Dec 01 (b) The amount of water leakage from the side seals is considered tolerable and will have no effect on normal gate operations. (c) Misalignment of the chain hoist is not severe enough to jeopardize operation of the gate but should be corrected. (d) Deterioration due to corrosion and rivet head loss are considered minor and will have no effect on normal gate operations or gate strength. (e) Flow-induced structural vibrations can cause serious damage to the spillway gate. In previous studies, stress ranges of approximately 27.6 MPa (4 ksi) have been calculated. Although this stress range is below the 41.4-MPa (6-ksi) threshold for fatigue crack growth at riveted details, the presence of groove welds to waterseal gaps between adjacent skin plates and tack welds to attach the diversion plate to the gate ribs may reduce this threshold stress range. However, since no fatigue cracks were detected and it is known how to control the gate vibrations, the structural capacity is not in jeopardy. (f) Although the accumulation of debris on the gate structure has not caused any structural or corrosion problems, it should be removed. (4) Recommendations. Based on the evaluation of conditions for the riveted tainter gates, the following recommendations are provided as steps that should be taken to ensure structural integrity for normal operations until the next regular inspection: (a) Continue operation of the spillway gates outside the range that causes vibration. (b) Schedule maintenance at Gate 25 to make repairs or adjustments to reinstall the chain hoist in the guide on the skin plate. (c) Schedule maintenance to remove large debris from all gate structures. (d) The buckled strut arm members should be occasionally monitored by lock personnel to detect any increases in deformation or distress to adjacent components. (e) Gate vibrations should be monitored by lock personnel to detect any changes. The inspection interval should be reduced to 2 to 3 years to monitor the buckled members and any future effects of the noted vibration problem more closely. b. Fatigue evaluation. (1) To illustrate fatigue strength considerations, let it be assumed that during the inspection of tainter gates a more significant mode of vibration had recently been observed. Because of this new information, a thorough inspection was made at all fatigue-sensitive details on several gates where this vibration was observed. However, no fatigue cracks were visible. (2) Based on the inspection findings in this assumed example, a field study was recommended to determine the significance of these new vibrations. The results of the field study revealed that vibrations of approximately five cycles per second or Hertz (Hz) were producing cyclic stresses of up to 55.2 MPa (8 ksi) at several details on the riveted structure. (3) The integrity of the riveted gate structure must be assessed by determining the fatigue strength of the details that are subjected to these cyclic loads. Since the measured maximum stress range is less than 7-6 EM 1110-2-6054 1 Dec 01 68.9 MPa (10 ksi), the Category C curve will be used to determine the approximate number of cycles to failure at the detail (this does not imply that the entire structure will fail). By projecting lines on the Sr-N curve shown in Figure 6-22, it can be determined that the number of cycles to failure is approximately 12.5 million. With the measured frequency of vibration equal to 5 Hz, it would take approximately 694 hours (29 days) of vibration at this stress range to exceed the fatigue strength of the riveted connection. But because this new mode of vibration has only recently been observed, it is probable that not many cycles have accumulated to date. In fact, unless the gates in this assumed example are allowed to vibrate for extended periods, it may take up to 3-1/2 years before fatigue cracks develop if vibrations are limited to 1/2 hour per day while the gates are being adjusted. (4) The recommended action to address this assumed condition would consist of three steps: • Minimize the occurrence of gate vibrations by operating outside the range causing vibration. • Reduce the inspection interval to approximately 1 year and inspect a greater number of gates to ensure that similar vibration is not occurring. • Begin engineering studies to determine solutions to reduce the stresses caused by these vibrations. c. Fracture evaluation example. (1) During an inspection, a 9-cm (3.5-in.) crack was found on the downstream flange of a horizontal girder on a tainter gate. The crack is an edge crack similar to that shown in Figure 6-10. Prior to the inspection, no indication of damage had been reported. Since the cracked girder is a main framing element of the tainter gate, an immediate assessment of its critical nature is required. The crack is near the midlength of the girder. The girder flange is 35.6 cm (14 in.) wide and 3.8 cm (1.5 in.) thick. (2) To evaluate this crack, a fracture analysis must be conducted. For this example, a linear-elastic fracture mechanics (LEFM) analysis will be used. The first step in performing the analysis is to obtain data on the three key parameters necessary for any fracture analysis: the crack size and geometry, the nominal stress in the member or component σ, and the critical stress intensity factor, KIc or Kc. (3) The crack size and the geometry have already been determined from the inspection. For an LEFM analysis, the nominal member stress is required. For this case, the nominal girder flange stress can be determined from a plane frame analysis similar to that used in the design of tainter gate girders. An analysis showed that the nominal girder flange stress in the vicinity of the crack was 117.2 MPa (17 ksi) in tension. (4) The next step in the analysis is to determine the fracture toughness. A review of the hypothetical design documents indicated that the gate had been fabricated from A36 steel. Since KIc testing (ASTM E399) of mild steels at reasonable service temperatures is impractical if not impossible, the fracture toughness will be determined from correlations with CVN data. As a first estimate, published CVN data for A36 steel will be used. This can be only an estimate, since KIc values can vary significantly for the same type of steel. KIc is also very dependent on temperature, so a minimum operating temperature for the structure must be established. Based on A36 steel CVN data (Barsom and Rolfe 1987), Figure 7-4 shows the approximation of KIc as a function of temperature. The curve on the left is calculated from the two-stage CVN-KId-KIc correlation (valid for the lower shelf and the lower end of the transition region; see paragraph 7-1b), and the curve on the right is from the upper shelf CVN-KIc correlation (Equation 7-3). The heavy line of each curve indicates the range in which the correlations are valid, as discussed in paragraph 7-1. The minimum service temperature for this example is -31.6 °C (-25 °F). Since neither curve is valid at this temperature, an estimate for KIc is determined by linear interpolation between the two correlations as indicated by the dashed line in Figure 7-4. This (-25 °F). interpolation indicates that KIc is approximately 62.6 Mpa- m (57 ksi- in . ) at -31.6 °C Conservatively, an estimate of KIc of 55 MPa- m (50 ksi- in . ) is selected for use in the analysis. 7-7 EM 1110-2-6054 1 Dec 01 Figure 7-4. CVN-KIc correlations (°C = 5/9 (°F – 32); 1 ksi- in . = 1.099 MPa- m ) (5) Since the crack size and geometry of detail are known and the stress level and material fracture toughness have been estimated, the crack can be evaluated for fracture by calculating the stress intensity factor and comparing to the fracture toughness. For a single-edge crack perpendicular to the stress field in a finitewidth plate, the stress intensity factor incorporating a factor of safety (FS), KIf , is given by  a FS   K If = 1.12σ π a FS • k   b  (7-7) where a = crack size k = function of a and b b = half-width of the plate (Tabulated values for k and stress intensity factor formulas for other crack geometries are given in Chapter 6.) For a factored crack length-to-plate half-width ratio of (a × FS)/b = (3.5 × 2)/7 = 1.0, k = 2.55, then K If = 1.12 (117.2) π (0.09) (2) • 2.55 = 250 MPa - m = 288 ksi - in. (7-8) Since KIf is greater than KIc = 54.95 MPa- m (50 ksi- in. ), an unsafe condition exists for plane-strain conditions. Checking the plane strain assumption with Irwin's β factor from Equation 2-2: 2 1  55  β Ic =   = 1.3 > 0.4 0.038  248  7-8 (7-9) EM 1110-2-6054 1 Dec 01 Since βIc > 0.4, the plane-strain condition assumption is not valid and the fracture toughness is represented by the critical stress intensity factor Kc. Using Equation 7-6 to estimate Kc (even though there is considerable deviation from plane strain condition) gives ( ( ) K c2 = K lc2 1 + 1.4 β Ic = 55 2 (1 + 1.4 ⋅ 1.292 ) = 10,072 MPa- m 2 ( 91 ksi- in. ) = 250 MPa- m ( 228 ksi- 2 ( 8,324 ksi- in.  )  2 (7-10) K c = 100 MPa- m K c < K If ) in. ) (6) Since Kc is less than KIf , an unsafe condition exists. This indicates that an immediate repair plan should be developed and implemented. If the repair will be costly and/or substantially affect the function of the project, a more accurate analysis should be made. The analysis was based on an estimation of KIc that may not accurately reflect the plane-strain fracture toughness of the material, and the approximation of Kc from KIc introduces more uncertainty in the estimation of the fracture toughness of the girder flange. A more exact analysis would require having tests conducted on the girder material so that a more accurate value of Kc may be obtained. A CTOD test, which can be used to estimate Kc (Equation 7-5), would likely be most appropriate because of the uncertainty in correlating CVN data at the service temperature. Alternatively, an elastic-plastic fracture assessment can be performed as outlined in Chapter 6. d. Lock gate fracture example. Cracks of various shapes were revealed on two tension members on a lock gate by nondestructive testing inspection. One member has the cross-sectional dimensions of 10 cm (4 in.) thick by 30.5 cm (12 in.) wide. The other member is 2.5 cm (1 in.) thick by 30.5 cm (12 in.) wide. The crack types and shapes include single-edge crack; through-thickness center crack; surface crack along the 0.3-m (12-in.) side (a/2c = 0.1 and 0.2), and embedded circular cracks. The material properties at the minimum service temperature of –1.1 °C (30 °F) were determined by material testing and are summarized as follows: σys = offset yield strength of 345 MPa (50 ksi) !ult = 552 MPa (80 ksi) E = 206,840 MPa (30,000 ksi) KIc = 66 MPa- m (60 ksi- in. ) KId = 44 MPa- m (40 ksi- in. ) "crit = critical CTOD value of 0.0052 cm (0.002 in.) (static) "crit = 0.0025 cm (0.001 in.) (dynamic) From structural analysis, the maximum applied tensile stress is 207 MPa (30 ksi). For each cracked member, the critical crack size will be determined for each cracking condition under static loading and dynamic loading, respectively: (1) Example for 10-cm (4-in.) by 30-cm (12-in.) plate: 7-9 EM 1110-2-6054 1 Dec 01 1 β Ic = t 2 2  K Ic    = 1  66  = 0.36  σ ys  10  345    βIc < 0.4; therefore, LEFM is applicable. (a) Single-edge crack (see Figure 6-10): a K I = 1.12 σ π a k   b where σ is the nominal stress. C = 1.12 a  π k   in Equation 6-1 b  Assume k (a / b) = 1.0 . The critical discontinuity size is calculated as 2 1  K Ic   = 2.59 cm (1.02 in .) (Equation 6-2 with no factor of safety) a cr =  π  1.12 σ  (a/b) = 0.17 and k(a/b) = 1.06; therefore, iteration is needed for acr and k(a/b). After iteration, acr = 2.34 cm (0.92 in.) (k(a/b) = 1.05). With FS = 2.0, acr = 0.5 (2.34) = 1.17 cm (0.46 in.) for dynamic loading: 2 a cr = 0.5  K Id    = 0.58 cm (0.23 in .) π  1.12 σ  (b) Through-thickness center crack (Figure 6-8). Calculate the stress intensity factor: KI =σ πa 2b  πa  tan   πa  2b  Assume 2b πa  tan   = 1.0 πa  2b  2 a cr = 1  K Ic    = 3.23 cm (1.27 in .) π  σ  2b πa  tan   = 1.02 πa  2b  After iteration, acr = 3.1 cm (1.22 in.). With FS = 2.0, acr = 3.1/2 = 1.55 cm (0.61 in.) and for dynamic loading, 2 a cr = 7-10 0.5  K Id    = 0.71 cm (0.28 in .) π  σ  EM 1110-2-6054 1 Dec 01 (c) Surface crack along the 30.5-cm (12-in.) side (2c is the length of the surface crack along the slope of the component; see Figure 6-15): • a / 2c = 0.1 K I = 1.12 σ π a MK Q σ 207 = = 0.6 σ ys 345 where Q is the flow shape parameter defined by Figure 6-14 and Mk is a variable that describes the effect of a/t on KI. From Figure 6-14, Q = 1.02, assume Mk = 1.0 2 a cr = Q  K Ic    = 2.64 cm (1.04 in .) (a/t = 0.26; Mk = 1.0) π  1.12 σ  With FS = 2.0, acr = 2.64/2 = 1.32 cm (0.52 in.), and for dynamic loading, 2 0.5 Q  K Ic  = 0.58 cm (0.23 in .) a cr = π  1.12 σ  • a/2c = 0.2 From Figure 6-14, Q = 1.24, assume Mk = 1.0 2 Q  K Ic  = 3.2 cm (1.23 in.) (a/t = 0.32; Mk = 1.0) a cr =  π  1.12 σ  With FS = 2.0, acr = 3.2/2 = 1.6 cm (0.63 in.), and for dynamic loading, 2 0.5 Q  K Ic  = 0.71 cm (0.28 in .) a cr = π  1.12 σ  (d) Embedded circular crack (see Figure 6-14). K I =σ π a Q a/2c = 0.5; from Figure 6-14, Q = 2.4 with FS = 2.0, 2 0.5 Q  K Ic  = 3.89 cm (1.53 in.) a cr = π  σ  7-11 EM 1110-2-6054 1 Dec 01 and for dynamic loading: 2 a cr = 0.5 Q  K Ic  = 1.73 cm (0.68 in.). π  σ  (2) Example for 2.5-cm (1-in.) by 30-cm (12-in.) plate: 2 1  K Ic  1  66   = β Ic =    = 1.46. t  σ ys  0.025  345  2 #Ic > 0.4; therefore, elastic-plastic fracture mechanics is applicable. Determine the allowable discontinuity parameter a m (paragraph 6-5b).  δ crit   am = C   ε y  (Equation 6-4) where εy is the yield strain of the material σ ys = 345 = 0.0017 E 206 ,843 σ 207 = = 0.6 σ ys 345 ε y= From Figure 6-20, C = 0.44 For static loading  0.0052   =1.32 cm (0.52 in.) a m = 0.44   0.0017  For dynamic loading  0.0025   = 0.65 cm (0.26 in.) a m = 0.44   0.0017  Critical crack lengths can be determined for various crack shapes from the allowable discontinuity parameter a m (paragraph 6-5b). 7-3. Example Fatigue Analysis This example shows how to apply fatigue analysis to determine expected life given an initial flaw size ai. For this case, consider an initial surface flaw of the type shown in Figure 6-15 with a/2c = 0.25. The member is a 10-cm- (4-in.-) thick plate of ASTM A572/572M Grade 345 (50) steel. The critical stress intensity factor (fracture toughness) KIc of this steel is 66 MPa- m (60 ksi- in . ) at the minimum service temperature. a. The maximum stress level is 207 MPa (30 ksi) and the minimum stress is zero. A curve relating the initial surface flaw size ai to number of cycles to failure Np will be developed. From Figure 6-15 7-12