13/09/2009 Analysis Of A Fuselage Crack Anoop Retheesh Fracture Mechanics and Fatigue CONTENTS Title Page Contents Abstract List of Figures List of Tables i ii iii iv iv 1. Analysis of a Fuselage Crack 1. 1 Introduction 1. 2 State of Stress in the absence of the Crack 1. 3 Geometrical Stress Intensity Factor at the Crack Tip 1. 4 Fracture Analysis using Finite Element Methods 1. 4. 1 Finite Element Model of the Fuselage Crack 1. 4. 2 The Solution 1. 4. 3 Grid Independence Study 1. 5 Variation in Stress Intensity Factor with Crack Length 1. 5. 1 Conclusion 1 1 2 3 5 5 8 9 10 12 2. References 13
Analysis of a Fuselage Crack ii Fracture Mechanics and Fatigue ABSTRACT Fracture & Fatigue are the most common engineering concerns that limit the useful life of mechanical components. In fact, it was estimated that the occurrence or prevention of fatigue failures costs the US economy about 3% of its gross national product (R P Reed, J H Smith, B W Christ; 1983). The term ‘fatigue’ represents the permanent structural changes occurring in a material subjected to fluctuating stresses that builds up cracks in it and leads to its complete fracture after a sufficient number of fluctuations (ASTM E-1150; 1987).
Fracture mechanics, meanwhile deals with the microscopic aspects of fracture and the failure of metals due to fracture. During the past 50 years, the subject has evolved a lot and it helped in understanding and predicting not only fracture failure but also crack growth processes such as fatigue. Fracture mechanics, combined with the conventional fatigue design modals now became an integral part of mechanical engineering design. In this report, an analysis is carried out on the cracks developed on an aircraft fuselage skin using Linear Elastic Fracture Mechanics (LEFM) assumptions.
The cracks, emanating from each side of the circular rivet holes in the fuselage is assumed to be in a state of stress equivalent to a wide thin plate subjected to uniform stress (? ). A finite element analysis is a carried out for the same configuration to obtain the stress intensity factor. The analysis is then repeated for different lengths of cracks and it is compared with the values obtained from theoretical calculations. Analysis of a Fuselage Crack iii Fracture Mechanics and Fatigue LIST OF FIGURES Figure 1. 1 Typical Fuselage Structure of an Aircraft Figure 1. 2 Cracks in the Fuselage Figure 1. State of Stress without crack Figure 1. 4 Correction Factor f(a/r) for a double crack at a circular hole Figure 1. 5 Simplified Model of Crack Figure 1. 6 Mesh near the Crack Tip Region Figure 1. 7 Loads and Boundary Conditions for the Fuselage Crack Model Figure 1. 8 Stress Intensity near the Crack Tip Figure 1. 9 Stress Intensity Factor Vs the Crack Length 1 2 2 3 5 7 8 9 11 LIST OF TABLES Table 1. 1 Variation of KI with respect to the size of the Singular Element Table 1. 2 Stress Intensity Vs the Crack Length 10 11 Analysis of a Fuselage Crack iv ANALYSIS OF A FUSELAGE CRACK 1. 1 Introduction
An aircraft fuselage assembly is the main structure or body of the aircraft to which all other components (such as wings and landing gears) are attached. It provides the space for power plant, cargo, controls, accessories, passengers and other requirements depending on the purpose (i. e. , civil, military or transport). It consists of a thin shell stiffened by longitudinal axial elements called stringers and longerons; and supported by traverse frames or rings (bulkheads) along its length. The stringers and longerons are usually riveted to the fuselage skin which is made up of Aluminum alloys.
The figure 1. 1 shows the typical riveted fuselage structure of an Airbus A 380 aircraft. Figure 1. 1 Typical Fuselage Structure of an Aircraft (Courtesy: Airbus Corporation) Due to their criticality, the aircraft fuselages are periodically inspected using NonDestructive Testing (NDT) methods for any potential flaws. The discovered flaws or cracks are then analyzed for their criticality. The fracture mechanics plays a major role in these analyses by estimating the residual strength of the fuselage with the crack and the probability of the crack to grow to critical size. Fracture Mechanics and Fatigue
In this problem, the cracks of lengths ranging from 1mm to 4mm were discovered to be emanating from each side of a circular rivet hole in the fuselage lap joint of the aircraft. The fuselage skin is made up of Aluminum alloy 2024-T651 with a plain strain fracture toughness of 40 MPavm. The state of stress existing in the portion of the fuselage between the rivet holes is considered to be similar to that of a wide thin plate subjected to uniform stress containing a circular hole in the middle (Figure 1. 2). Figure 1. 2 Cracks in the Fuselage 1. 2 State of Stress in the absence of the Crack
In the absence of a crack the fuselage skin is considered to be subjected to uniform tensile stress – ?. The stress distribution on the plate is altered by the existence of the stress concentration factor due to the existence of the discontinuity – circular hole. The tangential stress at any point on the plate is given by, ?? = ? /2 [1 + a/r2 – (1 + 3a4/r4)cos2? ] Where, ‘r’ & ‘? ’ is the angular coordinated of the point with respect to the centre of the circle; ? being measured from the vertical axis. ‘a’ is the radius of the circle. At the root of the notch, r = a & ? = ±90o.
Substituting these values at the above equation we get, ?? = 3? Radial stresses at the root of the notch Figure 1. 3 State of Stress without crack (Warren C Young, Richard G Budynas; 2002) Analysis of a Fuselage Crack 2 Fracture Mechanics and Fatigue will be zero as the direction of loading is perpendicular to the radial direction at the root of the notch. Therefore, in Cartesian components ? x = 0; ? y = 3? & ? xy = 0 (Figure 1. 3) 1. 3 Geometrical Stress Intensity Factor at the Crack Tip In the general case of a round hole on a plate of infinite width with symmetrical cracks on it’s both sides (Figure 1. ); the stress intensity factor (if neither plane strain or plane stress is applied) is given by the expression, KI = f(a/r).?. (?. a)1/2 (Arun Shukla; 2005) Where, ‘a’ is the length of each crack & ‘r’ is the radius of the circular hole at the centre of the plate. The correction factor f(a/r) is given by a design curve based on experimental data (Figure 1. 4) Figure 1. 4 Correction Factor f(a/r) for a double crack at a circular hole (Barsom, J. M. ; Rolfe, S. T. 1987) From the figure, it is clear that the correction factor f(a/r) increases with the decrease in the value of the crack length – ‘a’ with respect to the radius of the hole.
For a short length of crack, the value of f(a/r) becomes a maximum and its value can be tends to be, f(a/r) = 3. 36 Analysis of a Fuselage Crack 3 Fracture Mechanics and Fatigue Thus, the expression for stress intensity becomes, KI = 3. 36?. (?. a)1/2 The validity of the above expression can be verified by applying the principle of superposition to the two separate cases of plate with a hole subjected to tensile stress & the plate with an edge crack subjected to tensile stress. Thus we obtain the value of 3 as the stress concentration factor at a round hole (first case), and 1. 2 as the free-surface correction used earlier in stress-concentration formulas dealing with edge cracks in plates. By superposing the two, we obtain f(a/r) = 3 X 1. 12 = 3. 36; which is same as the maximum value obtained from the experimentally tabulated graph in figure 1. 4. For a large crack length, the value of ‘a’ becomes large compared to the hole radius ‘r’. Thus, the crack tip falls out of the zone of influence of the circular hole. From, the figure 1. 4 it is clear that as the value of a/r increases the value of f(a/r) decreases. For large values of a/r, the value of f(a/r) becomes a minimum and its value tends to be f(a/r) = 1. 2. Thus, the expression for stress intensity becomes, KI = 1. 12?. (?. a)1/2 The above expression is similar to the stress intensity expression of an infinite plate with an edge crack subjected to uni-axial tensile stress. This similarity can be explained by the fact that with increase in crack length, the crack grows out of the zone of influence of the hole and behaves like an edge crack. And this explanation provides the necessary theoretical validation to the above equation. Another way of approach follows that, as the crack becomes longer it behaves as if the hole were an extension of the crack.
The situation is similar to the case of an infinite plate with a central crack, subjected to uni-axial tensile stress. Thus, the effective length of the crack can becomes, ae = 2(a + r) The expression for stress intensity in this case is, KI = ?. (?. ae)1/2 Note, that in these case, the correction factor f(a/r) = 1; but, the crack length ‘a’ in the previous equations is replaced with an effective crack length ‘ae’. Both, these equations are used for the analysis of stress intensity factor for a large crack. Analysis of a Fuselage Crack 4 Fracture Mechanics and Fatigue 1. 4 Fracture Analysis using Finite Element Methods
The finite element method (FEM) is a numerical approximation technique for solving the complex partial differential equations. The fundamental principle behind the method is the reduction in number of degrees of freedom of the problem by discretization. There are many commercial tools available for solving the real life problems using FE methods. ANSYS® Mechanical™ is a common FE simulation tool used for mechanical design and optimisation applications. In this section, the fuselage crack is modelled for FE analysis and the results are obtained for different lengths of crack. 1. 4. 1
Finite Element Model of the Fuselage Crack The simplified fuselage crack problem as shown in figure 1. 2 can be further simplified for FE analysis by taking advantage of symmetry. By observation, it is clear that the problem is symmetrical about both the vertical axis as well as horizontal axis. The simplified geometrical representation (1/4th of the original geometry) used for FE analysis is shown in figure 1. 5. The dashed lines at the left and the bottom boundary represent the symmetry conditions. Another assumption which is used for the FE analysis is the plane stress assumption. The condition is mainly Figure 1. Simplified Model of Crack prevalent in relatively thin plates in which there is no considerable variation in displacements along the direction of thickness. The stresses along the direction of the thickness are assumed to be zero. This condition is not valid for thick plates in which there are considerable stresses developed normal to the surface of the plate. Both the situations can be explained by considering the below example. Suppose a given plate with a crack, is loaded in tension. The plastic strains are developed in the region near the crack tip, as the material yield stress is exceeded there.
Analysis of a Fuselage Crack 5 Fracture Mechanics and Fatigue Now, the material within the crack tip stress field, situated close to the free surface of the plate, can deform laterally (normal to the plate surface) because there can be no stresses normal to the free surface. The state of stress in this case is biaxial as there is no stress normal to the free surfaces of the plate. Thus, plane stress assumption is valid in this case. However for a thick plate, material away from the free surfaces is not free to deform laterally as it is constrained by the surrounding material.
The resulting stress state under these conditions tends to be tri-axial and there is zero strain along the thickness of the plate. The condition prevailing in this case is known as plane strain case. A general empirical relation which is used for the validity of plane strain or plane stress assumption, is given below, B ? 2. 5 (KIC / ? y)2 Where, B KIC ? y Thickness of the plate Plane strain fracture toughness of the material & Yield strength of the material (E E Gdoutos; 1993) For the given plate of Aluminum 2024-T651, B = 10mm, KIC = 40MPavm & ? y = 345MPa (www. matweb. om;September 2009) Therefore, it is clear that [B = 10mm] < [2. 5 (KIC / ? y)2 = 33. 61mm] Hence, we can assume that the plate is sufficiently thin for plane stress assumption. In ANSYS®, the element type for the FE model is chosen as ‘Plane-183’, which is a higher order (quadratic) element suitable for handling stress singularities. Since, the stress singularities are bound to be present at the crack tip, the use of ‘Plane-183’ becomes a necessity. Another advantage of the Plane-183 elements is that it accommodates the use of a fair amount of irregular meshes. This makes it suitable to use with automatically generated meshes.
The element type is particularly recommended for modeling 2 dimensional fracture problems (Release 11. 0 Documentation for ANSYS; 2007). The linear elastic fracture mechanics (LEFM) assumption is used to ignore the effects of plasticity and the nonlinearities at the crack tip region. This makes it possible to calculate the stress intensity directly using ‘KCALC’ method in ANSYS®. The dimension of the quarter model of the plate is chosen as 100mm X 100mm. Thus, the plate can be considered as infinite compared to the crack size (1mm to 4mm). The diameter of the hole is taken as 25mm and the crack length ‘a’ is chosen as 1mm.
In Analysis of a Fuselage Crack 6 Fracture Mechanics and Fatigue order to model the crack tip, a key-point is defined at the crack tip and the mesh is concentrated at the crack tip using the ‘concentrate key-point’ option in ANSYS®. This is essential to accurately capture the high gradients of stress variations near the crack tip. A type of special elements called ‘singular elements’ is created near the crack tip by skewing the mid-size nodes of the elements near the crack tip to its quarter position. This makes the elements near the crack tip capable of handling the stress singularity near the crack tip.
The figure 1. 6 shows the resultant mesh created near the crack tip. Figure 1. 6 Mesh near the Crack Tip Region In the next step, a uniform stress of 100 N/mm2 is applied at the top edge of the meshed model. The symmetry boundary condition is applied on the lines on left of the plate and the line on the bottom of the plate and ahead of the crack tip. The resulting loads and the boundary conditions are shown in the figure 1. 7. The red arrows on the left edge and the bottom edge of the plate represent the symmetry boundary conditions applied on them.
The applied stress of 100N/mm2 is shown as uniform pressure acting on the top edge of the plate. Analysis of a Fuselage Crack 7 Fracture Mechanics and Fatigue Figure 1. 7 Loads and Boundary Conditions for the Fuselage Crack Model 1. 4. 2 The Solution In the final step, the FE model is solved and the stress intensity factor is obtained using KCALC method in ANSYS®. The direction of the crack is defined by selecting the three adjacent left nodes from the crack tip. The condition for stress intensity calculation is given as plane stress. The stress intensity near the crack tip is plotted and is shown in the figure 1. . Note that the crack is expanded in the figure. This is because the figure is plotted on the final deformed shape of the geometry. The stress intensity factor KI is obtained as, K1 = 525. 34 MPavmm. From the theoretical calculations in section 1. 3, for a crack length a = 1mm & the radius of the hole r = 12. 5 mm. a/r = 1/12. 5 = 0. 08; therefore from the chart on figure 1. 4 f(a/r) = 2. 8 Thus, KI = f(a/r).?. v(?. a) = 2. 8 X 100MPa X v(?. 1mm) = 496. 29 MPavmm Analysis of a Fuselage Crack 8 Fracture Mechanics and Fatigue The percentage deviation from the theoretical value is obtained as around -5. 5%. Thus, it can be said that the FE solution is valid. This error can be attributed to the plane stress assumption used for FE analysis and the errors due to the automated meshing procedure. 1 NODAL SOLUTION STEP=1 SUB =1 TIME=1 SINT (AVG) DMX =. 149919 SMN =17. 657 SMX =2348 SEP 7 2009 09:41:14 MX 17. 657 276. 558 535. 458 794. 358 1053 1312 1571 1830 2089 2348 Figure 1. 8 Stress Intensity near the Crack Tip 1. 4. 3 Grid Independence Study The size of the singular elements which were used to model the region near the crack tip is an important factor in determining the accuracy of the results.
Generally, the size of the singular elements can be varied from a/10 to a/100; where ‘a’ is the length of the crack. The ANSYS® user manual recommends a value of a/8 for reliable results (Release 11. 0 Documentation for ANSYS; 2007). For the initial analysis in the section 1. 4. 3, the length of the singular elements was fixed at a/10 = 0. 1mm. For the verification of its results, the analysis is repeated by varying the length of singular elements. The aim is to verify that the obtained solution is independent of the size of the grid selected for solving the problem. The results are tabulated in table 1. . From, the table it is clear that the results obtained were, in fact independent of the size of the singular elements within a reasonable Analysis of a Fuselage Crack 9 Fracture Mechanics and Fatigue accuracy. The percentage deviations from the previous values in all the cases were less than 2%. Thus, it can be concluded that within ±2% accuracy, the solution obtained can be considered as independent of the grid size. Length of the Crack Size of Singular Element a/5 = 0. 2mm a/10 = 0. 1mm a/20 = 0. 05mm a/40 = 0. 025mm a/80 = 0. 0125mm KI MPavmm 518. 26 525. 34 530. 36 531. 90 529. 56 % Deviation -1. 7% -0. 96% -0. 29% 0. 44% a = 1mm Table 1. 1 Variation of KI with respect to the size of the Singular Element 1. 5 Variation in Stress Intensity Factor with Crack Length As per the theoretical expressions in section 1. 3, the stress intensity factor K I is dependent on the length of the crack. The two important parameters: f(a/r) and the term v(?. a) in the theoretical equation for the stress intensity factor is determined by the length of the crack. The FE analysis can be repeated for different lengths of cracks in order to find out the relationship between the stress intensity factor and the crack length.
The resulting values can then be compared with the values obtained from the theoretical equation. Table 1. 2 shows the results in tabulated form. Different values of K I are obtained using FE analysis and three different theoretical relations. The first set of theoretical values are obtained using the empirical values of f(a/r) plotted in figure 1. 4. It can be considered as the most accurate set of values as these are based on experiments. The second set is calculated using the approximated equation for a short length crack from section 1. 3. The value of f(a/r) is constant and equal to 3. 6 in this case. The last set of theoretical values are calculated using the approximated equation for long cracks; and the value of f(a/r) is 1. 12 in this case. Analysis of a Fuselage Crack 10 Fracture Mechanics and Fatigue Table 1. 2 Stress Intensity Vs the Crack Length Figure 1. 9 Stress Intensity Factor Vs the Crack Length Analysis of a Fuselage Crack 11 Fracture Mechanics and Fatigue 1. 5. 1 Conclusion The variation of stress intensity factor with respect to crack length according to each method is plotted in a graphical form in figure 1. 9.
From the figure, it is clear that the FEA results are most closely matching the empirical relation based on experimental results. The long and short length approximations are matching with the experimental results on their respective domains. As the crack length becomes shorter the short length approximated equation comes closer to the experimental results. But, the long length approximated equation fail to achieve satisfactory results in this case. Similarly, as the crack length becomes larger the long length approximated equation comes closer to the experimental results.
But, the short length equation fails in this case. Only the FEA results match the experimental results in both the domains. Another important aspect which can be concluded from the graph is that the FEA results are always more than the empirical equation based values. Thus FEA provides a conservative estimation of the stress intensity. This makes the FEA results very much reliable for the practical design applications. The long length approximated values were always less than the values predicted by the empirical relation. Thus, it is dangerous to use this equation for design purposes.
In the case of short length approximation, its values are not matching with the empirical relations for the long cracks; but the values predicted by it is always higher than the values based on the empirical relation. Thus, it is a fairly conservative equation which can be used for a rough estimate of stress intensity for design purposes. In the case of fuselage crack, the stress intensity levels predicted by the empirical relation as well as the FEA falls below the plane strain fracture toughness (40MPavm) of the material of the fuselage.
The maximum value of stress intensity in the given range of crack length’s (1mm to 4mm) is predicted by the short length approximation for 4mm crack length; it is given as 1191. 09MPavmm from the table 1. 2. If we convert it into MPavm for comparison purposes, the value is approximately 37. 67MPavm which is less than the plane strain fracture toughness of the material of the fuselage skin. Similarly, from the FE analysis, the maximum value of stress intensity comes to around 25. 1MPavm. Thus, it can be concluded that for 4mm length of crack, the fuselage is stable from a design point of view. Analysis of a Fuselage Crack 2 REFERENCES 1. 2. 3. ANSYS®, Inc. , Release 11. 0 Documentation for ANSYS, 2007 Arun Shukla; Practical Fracture Mechanics in Design, 2’nd Ed. , Marcel Dekker, 2005. ASTM E1150 – 1987, Standard Definitions of Fatigue, 1995 Annual Book of Standards, ASTM, 1995. 4. E E Gdoutos; Fracture Mechanics an Introduction, 2’nd Ed. , Kluwer Academic Publishers, 2005. 5. John M Barsom, Stanley Theodore Rolfe; Fracture and Fatigue Control in Structures, 2’nd Ed. , Prentice-Hall, 1987. 6. Nitin S Gokhale, Sanjay S Deshpande, Sanjeev V Bedekar, Anand N Thite; Practical Finite Element Analysis, Finite To Infinite, 2008. . R P Reed, J H Smith, B W Christ; The Economic Effects of Fracture in the United States, National Bureau of Standards (Washington, D. C), 1983. 8. Warren C Young, Richard G Budynas; Roark’s Formulas For Stress And Strain, 7’th Ed. , McGraw-Hill Education, 2002. 9. http://www. matweb. com/search/PropertySearch. aspx; September 2009 10. http://www. ndt_ed. org/EducationResources/CommunityCollege/Materials/Mechanical /FractureToughness. htm; September 2009. 11. http://www. efunda. com/formulae/solid_mechanics/fracture_mechanics/fm_intro. cfm; September 2009.
