Linear Elastic Fracture Mechanics Applied on Ships

John E. Kokarakis1, Robert K. Taylor2

1Bureau Veritas, 2Design Research Engineering, MI, USA Abstract Applications of linear elastic fracture mechanics in ships are presented to depict the utility and potential to make critical operational decisions, protecting crew, property and environment. Repair alternatives can be chosen by means of optimum cyclic stress distribution. Causal factors of fracture can trace back the as-built fabrication and determine if the problem is design based or an epidemic event. Even a simple model of crack propagation with basic statistics’ principles can provide answers and save a ship from being at an impasse. It is proposed that the design is based on the fracture tolerance. The higher a defect the structure can tolerate the more fracture-resistant it is. 1. INTRODUCTION Linear damage accumulation is popular due to its relative simplicity but it has major shortcomings like neglect of the loading sequence, effect of surrounding structure and lack of accurate physical meaning of the damage ratio. The alternative approach is one based on crack propagation models. Although this type of fatigue assessment is well founded, it finds limited application in ship design. A notable exception is the utilization of Leak Before Break, LBB, criteria in LNG carriers. This study presents examples of crack-propagation-based design and repairs in the marine work. 2. DETERMINISTIC FRACTURE MECHANICS

An access opening in the second deck passageway was opened in a 2600 TEU container ship. Compensating doublers were fitted to compensate for the lost area. After a year, a nasty crack appeared as shown in Fig. 1 spanning from the second to the main deck. The crack was concealed by bundles of wires running longitudinally to feed the refrigerating units in the reefer containers. Target of the study was to determine the likely cause of cracking which appeared to originate at the fillet weld at the toe of the upper doubler. In the beginning it was thought that cracking was due to a weld defect but a similar crack appeared at the same location in a sister vessel.

Fig. 1: Geometry of Fracture

On one ship, the crack penetrated through the 25 mm thick bulkhead, extending vertically up to the main deck, and downward to the second deck as depicted in Fig.1. The distance between main and second deck was 2.40 meters. In the other ship the crack length was lower due to early detection. The task at hand was to also pinpoint the optimum repair method. The study included strain gauge measurements on an “intact” sister vessel and a linear elastic fracture mechanics evaluation of the doubler tip region. Stress and strain data were obtained for voyages in the North Pacific during a two month window in the winter. These data were utilized to develop predictions of stress histograms experienced by the doubler tip over the ship lifetime of service. Sea state data were recorded simultaneously with wave height radar. Significant wave height, average encounter period and wave height were recorded. Theoretical and experimental work provide persuasive arguments for the applicability of the Weibull distribution to describe long term wave-induced stress histories on ships (IACS 2006). Strain gages were recording the far field stress. The actual stress at the toe of the doubler was determined by Finite Element Analysis which provided the transfer functions to go from the far field stress to the hot spot stress at the toe of the doubler. The stress analysis also provides the distribution of the through-thickness plate stress at the toe of the weld, i.e., the stress gradient at the areas of fracture initiation. Crack propagation analysis requires both the stress level and the stress gradient as inputs. During the study of repair alternatives, it became clear that sufficient reduction of the cyclic stress amplitudes could not be achieved with any alternative that included doublers which induce high stress concentration. A design without doublers was proposed. The proposed removal of the doublers resulted in unacceptably high stresses at the corners of the access opening. Therefore, the modification included the addition of a 150 by 25 mm flange-coaming is fitted in the access opening. The structural integrity and fatigue life of the proposed configuration were evaluated to extend far beyond the service life of the vessel The cause of cracking was investigated study of fatigue crack initiation and propagation at the doubler tip. The predicted stress histogram and the spatial distribution of stress around the doubler tip obtained from a 3-D finite element analysis were combined to produce inputs to the fracture mechanics analysis. A linear damage accumulation analysis based on S-N data was used to evaluate the time required to initiate a fatigue crack in a 'good weld' subjected to the predicted/measured fatigue stress spectrum. 'Good weld' is defined as a weld without any pre-existing flaws larger than 0.5 mm. This is an experience-based typical size for weld flaws found in welds made using good welding procedures. A crack growth analysis was performed utilizing spatial distributions of stress normal to the crack plane obtained from the 3-D analysis. The stress at the toe of the weld is on the average 1.33 times greater than the far field stress. A fatigue crack initiation analysis was performed to determine the length of time required to initiate a fatigue crack in the absence of any major weld defects. S-N data for steel grade AH36 from (Matsumoto 1985), were curve-fitted to obtain a relation between stress range and number of cycles to failure for smooth round bar fatigue specimens containing a weld. A linear damage summation was computed utilizing Miner's rule and the

stress histogram. The individual damage fractions were summed over the entire stress spectrum to obtain a total predicted accumulation per year of 0.199. This analysis was performed to determine the initial size of a fatigue crack at the weld toe that would propagate to critical size within one year. Fatigue threshold in steel is about 6.5 MPa m1/2 and fracture toughness Kic equals typically55 MPa m1/2(Fortner and Preager 1980). The doubler tip crack was modeled as a semi-elliptical surface crack in a plate, (Harris 1988). The crack was centered at the toe of the doubler weld having three degrees of freedom, one at each surface crack tip and one at the deepest point of the semi-ellipse, as shown in Fig. 2. Each degree of freedom can grow at a different rate depending on the magnitude of the local range of stress intensity factor. Fatigue crack propagation analysis showed that a semi-elliptical surface crack 2.2 mm deep by 5.2 mm surface length, located at the toe of the doubler tip weld would propagate to a critical size of 8.1 deep by 24.5 mm surface length in one year. It was thus concluded that the initial defect size was unacceptably high with a surface length of 5.2 mm. Also the initial flaw in the doubler tip weld must have been 2.2 mm deep or larger. A crack propagation analysis also showed that a semi-circular surface crack 0.5 mm deep, expected in a “good” weld, would grow to critical size in a little more than 18 years. Then, if a fatigue crack 0.5 mm deep was to initiate in a good quality weld after 5 years, as resulting from the crack initiation analysis, the predicted life of a good quality weld at the doubler tip is more than 20 years. Proposed repair alternatives were: 1) Removal of doublers, no reinforcement of access door corners. 2) Removal of doublers, 150 mm by 25 mm ring reinforcement at door. 3) Extension of doublers aft, up to the nearest frame. 4) Extension of doublers aft, 100 mm past the nearest frame. 5) Same as (4) but with lower thickness doubler. 6) Removal of doublers with smaller access door.

Fig.2: Crack Geometry Modeled.

The proposed alternatives were evaluated from the stress standpoint using a 2-D FEA model. The optimum modification was alternative (2). This configuration resulted in significantly reduced bulkhead stresses without any local hot spots. A fatigue crack propagation analysis was performed on the door corner for the proposed modification (2). The same semi-elliptical crack model was used as was used for the doubler analysis. Stress distributions were obtained from the 2-D finite element analysis for light load hogging and sagging conditions. Results of the fatigue crack propagation analysis for the door corner with the proposed modification showed the predicted life of a 0.5 mm deep crack (“good” weld) oriented

transverse to the coaming weld to be greater than 100 years. The tolerable crack size for 20 years of remaining life was approximately 5 mm by 10 mm surface length. The later is another way to judge a fracture tolerant design i.e. by the size of tolerable initial defect. 3. A POOR MAN’S RISK ANALYSIS OF SEA WORTHINESS A vessel had a transverse fracture at one of his watertight bulkheads separating the double bottom of No 1 and No 2 ballast tanks. The fracture was not caused by fatigue but it was the result of tearing due to grounding. The presence of the transverse fracture necessitates special attention due to the likelihood of crack propagation. It was inquired if the vessel could cross the Atlantic from Belem in Brazil to Turkey for final repair. Vessel was loaded and could not be dry-docked since there was no dry-dock of the required capacity. It was thus given that the risk of performing the crossing must be evaluated in a thorough but rather conservative way. Arrest holes were drilled at the ends of the fracture but they are not a panacea to stop a crack. Crack propagation is governed by Paris’ law which in its simplest form can be written as:

mKCdN

da)(∆=

where α is the crack length, N is the number of cycles acting on the crack, C, m are constants and ∆K is the cyclic stress intensity range, which can be written further as:

aYK πσ∆=∆ Where Y is a factor accounting for geometry and can be taken in this case as equal to 1.12 (closer to one but lets consider the higher value), σ∆ is the cyclic stress range and a in this case is half the crack length. It was estimated that the crossing of the Atlantic will last about 14 days (Belem to Gibraltar is 3293 nautical miles and a speed of 10 miles per hour was assumed). Every day the vessel will be subject to about 10000 wave induced cyclic stresses. Overall the maximum number of cyclic stresses is about N=14x10000=140000 cycles. Also the cyclic stress amplitude will be due to the wave putting the bottom in tension. It is expected/advised that the vessel will be placed in hogging for the still/static bending moment. For conservative reasons the reduction due to hogging loading will not be considered (this stress reduction is of the order of 3.3 MPa). The IACS rule sagging wave bending moment is equal to 700000 KNm from our MARS analysis file. On the other hand this value is based on 108 cycles or 20 years of life for the ship. It can be proven, on the basis of Weibull distribution for the sea loads that the maximum sagging bending moment during the 14 day trip will be:

KNmMMsag 450000420.18

85.11700000

)10ln(

)140000ln(8108 ===

which given that the section modulus at that area is 18 m3 will result to a cyclic stress of 25 KN/mm2. Consequently: =∆σ 25 KN/mm2. Furthermore, constants can be taken as C=2x10-13 and m=3. Integration of Paris’ law equation above yields:

)11

()(

23

fi aaYCN −

∆=

σπ

Where i stands for initial (half length) and f stands for final. It is noted that this relatively simple analysis does not account for load sequence effects and crack blunting due to the generation of compressive residual stresses at the crack tip (Willenborg model). More importantly, the computed cyclic stress level is the maximum expected over the period of 14 days. It is thus very conservative to consider it acting for all 140000 cycles. Moreover, not all stresses will cause crack propagation. Then the integrated form of Paris’ law above with =∆σ 25 KN/mm2 and initial crack length varying between 2000 and 3000 mm yields final crack length depicted on Fig. 3 here below:

Fig. 3: Crack growth versus initial crack length The predicted increase of the size of the fracture is relatively benign and definitely can be tolerated. On the other hand these calculations are typically order of the magnitude. If the more realistic daily maximum instead of the overall maximum is utilized, then the crack growth is slashed. The calculations are conservative but everything depends on the encountered wave spectrum. It is highly recommended that a weather routing service is consulted to plan the best route for the trip. A last but not least check has to do with the critical crack length. This is the length beyond which the crack will propagate with the speed of sound in steel i.e. about 4 km/sec. This will occur when the stress intensity factor K at the crack tip equals the fracture toughness i.e.:

aYKK IC πσ==

this then yields for the critical crack length:

2)(2

σπ Y

Ka IC

critical =

Fracture toughness is a highly varying property. It typically lies between 50 and 100

MPa m for mild steel. On the other hand, the minimum required Charpy impact

energy for such steels is equal to 27 J at 0 degrees C. Utilization of the well known

Rolfe-Barsom correlation yields a lower bound limit of 70 MPa m . The maximum expected total tensile stress will be used in this case i.e. (25-3.3) N/mm2 (or 22.7 MPa). The critical crack length will be between 5 and 10 meters practically unreachable on the basis of the calculations above. 4. DAMAGE TOLERANT SHIP DESIGN The essence of the damage-tolerant ship design is that the structure must contain successfully the growth of an assumed initial damage for a specified period of service. The safe growth period is coupled to either the design life of the ship or to the scheduled in-service inspection intervals. The structure is designed such that an initial defect will grow at a stable, slow rate under service environment and it will not achieve a size large enough to cause rapid unstable/brittle propagation. Residual strength capacity is necessary as well. For example it is also required that a main deck crack will not reach such a size to cause the section modulus to be reduced below the required minimum by IACS. It is assumed that sub-critical growth with sufficient residual strength capacity will be either detected during the surveys or the crack will be below critical for several ship design life times. The product of the analysis is the determination of inspection-rejection criteria, which when applied will ensure sub-critical growth preferably for the ship lifetime or between inspection intervals. The most critical location for the initial flaw should be determined by reviewing all elements of the ship structure. The basic premise in arriving at the initial damage size is the assumption that the as-fabricated structure contains flaws of a size just smaller than the maximum undetectable flaw size found with the non-destructive inspection procedures used in the ship production line. The assumptions relative to the shape, size and location are based on review of existing NDT methods, NDT data, based on 90% probability of detection. Application of these ideas on a new-building containership commenced with the determination of the critical areas in the vessel, not only from the stress standpoint, but also with respect to accessibility and criticality. Examples of such areas are thickness transitions on main deck and bottom, misaligned plates, top plates of hatch coamings with or without thickness transition, drain and vent holes on stiffeners, deck penetrations and fixtures welded on deck and coamings. Depending on location semi-elliptical surface cracks or corner cracks grew through a design life of 35 years and the initial size to avoid critical crack size was determined. The higher the acceptable initial flaw size the more damage tolerant the structure is. Utilization of butt welds at the most highly stressed region of the vent holes rendered these details as the most critical with respect to weld quality and tolerable defect. The “acceptable” defect sizes are then specified as the inspection-rejection criteria to be utilized by the shipyard in the NDT assessment of the welds. The effect of accelerated crack growth due to the presence of sodium chloride is also accounted for by considering different parameters in the growth equation. Sub-surface defects can also be modeled by buried elliptical cracks subject to bi-modal crack growth rates. The slower rate applies during the sub-surface growth, whereas as the crack reaches the surface the catalyzed growth due to sodium chloride the rate increases. The spatial variation of the stresses was necessary since the gradients in the

crack growth directions (through thickness and on plate surface) are necessary for the analysis. The stress gradients were determined by finite element analysis or by utilization of published solutions for the stress concentration factors, (HSE 1997). The study produced results similar to the one shown on Fig. 4. In this plot the remaining life for subcritical crack growth is depicted as a function of the initial defect size. It can be thus concluded from this plot, that the initial defect should have length and

Fig. 4: Life versus Defect Size (e = 3 mm misaligned butt weld)

depth greater than about 4.3 mm and 1.7 mm respectively to ensure subcritical crack growth during the 35 years design life of the ship. These results can be utilized as inspection-rejection criteria for the NDT inspections. Similar results can be developed for all critical ship details following the method described above. 5. CONCLUSIONS Crack propagation-fracture mechanics based approach might be more realistic for assessment of the cracking resistance of a ship. Fracture mechanics based evaluations have been routinely applied to study a variety of cracking problems. Fracture mechanics based evaluation of the cause and repair of cracking at the toe of a doubler in a containership is presented in this study as an example of how to choose the most crack-resistant repair method. Cracking on the deck of a container-ship is studied with probabilistic fracture mechanics. Monte Carlo simulation is utilized to evaluate the optimum inspection schedule and the corresponding inspection criteria. Finally, a method for fracture mechanics based ship design is proposed. Its application on a container-ship is depicted. Experience gained from the application of such method, will provide advance knowledge of areas of potential problems.

References Fortner E, Preager M., 1980, “Fracture toughness of wrought and cast steels”, Report MPC-13, American Society of Mechanical Engineers, New York. Forman R.G., 1986, “Derivation of crack growth properties of materials for NASA/FLAGRO”, NASA, Materials Branch Report 86-ES5-1, Houston. Harris D. et al, 1988, “NASCRAC A fracture mechanics analysis code”, Proceedings of the Advanced Earth-to-Orbit Propulsion Technology Conference, Huntsville, Alabama. Health and Safety Executive , HSE, 1997, “Geometric stress concentration factors for classified details”, Report, OTO- 97-024. International Association of Classification Societies, IACS, 2006 “Common Structural Rules for Oil Tankers and Bulk Carriers”. Matsumoto S., 1985, “Fatigue properties of high strength hull structural steels manufactured by thermo mechanical control process”, Kawasaki Steel Technical report No. 13, Kobe.