Research Journal of Recent Sciences _________________________________________________ ISSN 2277-2502 Vol. 2(2), 33-39, February (2013) Res.J.Recent Sci. International Science Congress Association 33 Buckling of Cracked Conical Frusta under Axial Compression Dadrasi Ali Department of Mechanics, Shahrood Branch, Islamic Azad University, Shahrood, IRANAvailable online at: www.isca.in Received 28th September 2012, revised 3rd October 2012, accepted 6th November 2012Abstract Presence of cracks or similar imperfections can considerably reduce the buckling load of a shell structure. In this paper, the buckling of thin conical frusta with cracks under axial loads has been studied. At first, a frustum without any imperfection has been analyzed. In continuation, sensitivity of the buckling load to the crack presence with different length and orientation has also been investigated. This procedure has been investigated on three types of frusta with different heights and constant semi-apical angles. Some effective parameters on buckling have been studied separately and the required data for analysis have been gained through experimental tests. The finite element ABAQUS software has been used for the numerical analyses. Key words: Buckling, frusta, crack, FEM. Introduction Shell structures have been widely used in many fields like pipelines, aerospace and marine structures, large dams, shell roofs, liquid-retaining structures and cooling towers. Shell’s buckling is one of the important considerations within designing this structures2-3. Presence of defects, such as cracks, may seriously compromise their buckling behavior and endanger the structural integrity4-6. The post-buckling analysis of cracked plates and shells showed that the buckling deformation could cause a considerable amplification of the stress intensity around the crack tip. On the other hand, increasing the load can cause propagation of the local buckling leading to the catastrophic failure of the structure. The nonlinear buckling of thin cylindrical shells with longitudinal cracks subject to above-mentioned loading was studied by Starnes and Rose9-10. It was concluded that the non-linear interaction between in-plane stress resultants and the out-of-plane displacements near a crack significantly affects the buckling behavior of the shells. These studies indeed revealed that the sensitivity of the buckling behavior of both plates and shells to the presence of defects highly depends on the loading condition. As an example, Estekanchi et al.11-12 showed the buckling behavior of the cylindrical shells under torsional loading is less sensitive to the presence of a crack than that of a similar axially compressed cylindrical shell. Postlethwaite and Mills13 performed the axial crushing of conical shells of semi-apical angles ranging from 5\r to 20\r and studied their energy absorption capacity. Mamalis and Johnson14 have studied the axial compression of aluminium conical frusta of semi-apical angles from 5\r to 10\r under quasi-static loading. Mamalis et al.15 have also performed the axial compression on steel thin-walled frusta of semi-apical angles from 5\r to 10\r at elevated strain rates. The load–deformation behavior and the collapse in this case were similar to those in14. With respect to Presented papers, it's observed that the category of crack presence in frusta despite of its importance hasn't been probed up to now. In this paper, the effect of crack presence with different angles on the buckling behavior of various types of conical thin frusta has been investigated. Material and MethodsFigure 1 shows the total geometry of specimens. In all of cones, the large diameter is D=50mm and the semi-apical angle is =4.5\r and both of these parameters are constant. Also, the center of crack is laid in the middle of cone's height. Crack lengths are a=10, 20 and 30 mm that have been analyzed with angles =0\r, 30\r, 60\r and 90\r and the shell's thickness is considered 1mm. The mechanical properties of the investigated steel shells have been obtained through the standard tensile test ASTM E8 and an INSTRON 8802 servohydraulic machine. It’s stress-strain curve has been displayed in figure 2. The obtained elasticity modulus from linear elastic region is equal to E=201 GPa. Moreover the Poisson's ratio is set equal to =0.3. Figure-1 The total geometry of specimens q Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(2), 33-39, February (2013) Res. J. Recent Sci. International Science Congress Association 34 10020030040050060000.050.10.150.20.25Strain(mm/mm) Stress (Mpa) Figure-2 The stress-strain curve resulted by the experimental test The data of the plastic region of the stress-strain curve has been used for the analysis of the plastic behavior in ABAQUS software. The shells have been laid under axial compression by means of supports that prepare clamp boundary condition. Finite element analysis: FE methods are considered to be the most appropriate tool in cases dealing with structural mechanics problems involving complications in behavior that are analytically intractable. Buckling of cracked shells is such a problem. FE methods have been successfully applied in problems involving shell buckling and cracked shells12. In shell buckling problems, adequacy of the FE mesh for incorporating the shell buckling modes, which is sometimes unexpectedly complicated, is of utmost importance. Failing to observe this condition can result in omission of the first few buckling modes and grossly overestimated buckling loads. Shell structures usually buckle in complex modes relative to their initial geometry. Therefore, the analyst should resort to past experience and empirical rules for selecting an appropriate FE mesh in the buckling analyses. Numerous shell element formulations are available for buckling and large displacement analysis. Among them, popular isoparametric formulations seem to be most convenient and reliable in general applications. Quadrilateral elements are usually recommended for problems involving buckling behavior. In cracked plate and shell problems, appropriate modeling of the singular stress field at the crack tip area is of prime importance. The numerical simulations were carried out using the general finite element program ABAQUS 6.6.3. At first we must do the meshing in order that we can analyze the specimen. The nonlinear element S8R5, which is suitable for analysis of thin shells, was used16. This element is an eight-node element with five-degree of freedom including three displacements in three directions of the coordinate axes and two rotations for each node. Figure 3 shows a meshed specimen with horizontal crack. We should change meshing so much that we reach an optimized meshing. That is, we should continue meshing to the extent that there isn't any considerable difference in buckling load coefficient that is achieved by linear analysis. Linear analysis must be done for getting eigenvalue by using the Buckle solver in ABAQUS software and getting the buckling modes. These modes have smaller eigenvalues and buckling usually occurs in these mode shapes. Of course we should notice that eigenvalue analysis overestimates the value of buckling load, because in this analysis, we don't consider the plastic properties of material. In continuation, considering the plastic properties of material, a non-linear analysis is done by using Static Risks solver. Then, in this analysis, we take into consideration the effect of mode shapes of buckling gotten from the linear analysis. Figure 4 shows crack tip meshing. As a result a load-displacement curve is obtained and the maximum value of this curve displays the buckling load. Figure-3 Meshed specimen with H=100 mm and a=20 mm Figure-4 Meshing the tip of the crack The point that should be considered in this analysis is the imperfection in the shells. Shells mainly due to their construction don't follow the ideal shape of a cone. This deviation from the ideal shape must be considered in the analysis. For example, about conical shells which are produced through the process of drawing because of fluidizing material during the process of forming, some small waves are created on the shell that make it out of ideal shape. Usually, in the shells, the imperfection is considered about 0.02th of its thickness16. In the cracked conical shell, the occurrence of buckling phenomenon is more likely in these two positions. The first position is in which shell has the minimum diameter, and the second position is in which shell has the lowest strength because of the presence of crack in the shell. Now considering how stress is distributed in the shell and which area will be yielded soon, the buckling is occurred. Even the geometry of shell and crack and also stress distribution may be in such a way that both areas are yielded. Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(2), 33-39, February (2013) Res. J. Recent Sci. International Science Congress Association 35 Results and Discussion Analysis of shells 75mm in height: At first, conical shells with D: 50mm, H: 75mm, and semi-apical angle of 4.5\r are analyzed. One uncracked sample of this cone has buckling load equal to 43.168kN. Now if we create a horizontal crack with the length of a=10mm, the amount of 0.65kN of shell buckling load is reduced. Figure 5 demonstrates the buckling of a sample with mentioned characteristics. As you notice in this figure, buckling has been occurred in the necking part of shell and a plastic ring on the top of shell has been created symmetrically. The point that is clear in distributing stress of these shells is the fact that by increasing loading, severe stress concentration is created in both crack tips which in continuation, more severe stress concentration is created in necking part of shell that finally can lead to yielding of shell in that part. Figure-5 Distributing stress in a shell with a=10 mm and =0\r In continuation, we rotate the angle of crack a=10mm in length and gradually increase it to 90 degrees. It can be seen that by increasing the angle of crack, the amount of buckling load of shell is also increased. In figure 6, the diagram shows buckling load variations with angle of crack for a crack with length of a=10mm. For better understanding, these changes are also given in table 1. 1015202530354045500123456Displacement(mm)Load(kN) Perfect =0 =30 =60 =90Figure-6 The diagram of buckling load variations vs. crack angle (a=10 mm) Table-1 The buckling load variations vs. crack angle (a=10 mm) Buckling Load(kN) Crack angle(\r) 43.168 No crack 42.497 0\r 42.690 30\r 42.977 60\r 43.001 90\r Analyses show that for a crack with the length of 10mm, angle variations don't have any considerable effect on the place of shell buckling. And the shell, by forming the plastic ring in the necking part, will always have symmetrical buckling. In the next step, we double the length of crack and analyze the question with a=20 mm in different angles. Load-displacement diagram of these series of shells are shown in figure 7 and their exact values are given in table 2. 1015202530354045500123456Displacement(mm)Load(kN) Perfect =0 =30 =60 =90Figure-7 The diagram of buckling load variations vs. crack angle (a=20 mm) Table-2 The buckling load changes vs. crack angle (a=20 mm)Buckling Load(kN) Crack angle(\r) 43.168 No crack 39.777 0\r 40.535 30\r 42.443 60\r 43.033 90\r The process of changing shows that by increasing the angle of crack, buckling load is also increased. Because the crack length is raised, these changes are more compared to the shells with crack length of 10 mm. Increasing crack length and parallel to it, reducing shell strength cause that buckling mode of the shell changes and also buckling occurs in an asymmetric mode. Figure 8 shows buckling of a sample with a=20 mm and =0\r. The reason of choosing a shell with horizontal crack is to show asymmetric buckling of the shell in the situation that geometry has symmetry. In these series of shells, buckling is asymmetric in all crack angles except the one with vertical crack and it is Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(2), 33-39, February (2013) Res. J. Recent Sci. International Science Congress Association 36 because of the presence of crack in buckling mode. In continuation, the length of crack is increased to a=30 mm. Considering load-displacement diagram in figure 9 and related data table 3, the effect of increasing crack length in parallel with striking reducing of buckling load can be revealed. Figure-8 Distributing stress in a shell with a=2 mm and =0\r 1015202530354045500123456Displacement(mm)Load(kN) Perfect =0 =30 =60 =90Figure-9 The diagram of buckling load variations vs. crack angle (a=30 mm) Table-3 Buckling load changes vs. crack angle (a=30 mm) Buckling Load(kN) Crack angle(\r) 43.168 No crack 36.024 0\r 37.564 30\r 41.214 60\r 42.970 90\r Figure 10 shows that because of crack geometry, buckling occurs in several areas and also it is completely asymmetric. In figure 11 the diagram of buckling load variations versus crack length in constant crack angle is depicted. Diagram reveals that shell buckling load is reduced while crack propagates. Also, horizontal crack has the most significant effect on reducing buckling load of shell while vertical crack has little effect on it. Analysis of shells 100 mm in height: If the previous process of analysis for shells 100 mm in height is repeated, the graphs similar to previous graphs are obtained. Therefore, only buckling load variations for different crack length with mentioned angles are shown in table 4. Figure-10 Stress distribution in a shell with a=30 mm and =30\r 35363738394041424344450510152025303540Crack Length(mm)Buckling Load(kN) =0 =30 =60 =90Figure-11 The diagram of buckling load variations vs. crack length in constant crack angles (H=75 mm) Table-4 Buckling load changes vs. crack angle change in different lengths Buckling Load(kN) a=30 mm a=20 mm a=10 mm Crack angle(\r) 38.694 38.694 38.694 No crack 33.990 37.304 38.567 0\r 35.643 37.759 38.634 30\r 38.005 38.495 38.672 60\r 38.685 38.686 38.690 90\r Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(2), 33-39, February (2013) Res. J. Recent Sci. International Science Congress Association 37 It is observed that increasing frusta height strikingly reduces the effect of crack on the buckling load. For example, the most buckling load variations with crack angle, for crack a=10 mm in height, has been 0.127 N and for a=20mm, 1.39N and in a=30 mm has been 4.704 N. figure 12 demonstrates buckling load variations with crack length in constant crack angles. In figure 12, it's observed that horizontal crack still has the most significant effect on reducing buckling load and vertical crack doesn't have any considerable effect on buckling load. An Important point in these series of shells is that buckling in all shells is symmetric and leads to make a plastic ring in the necking part of shell with the exception of the shell with a=30 mm and =30\r which has asymmetric buckling. For example, figure 13 shows stress distribution in a shell with a=10 mm and =60\r. Analysis of shells 150 mm in height: In these series of shells, the effect of height on buckling load is so much that fades greatly the effect of crack and its different angles. For example, from these series of shells, the cracked ones a=20 mm in length were studied and its results are mentioned in table 5. 333435363738390510152025303540Crack Length(mm)Buckling Load(kN) =0 =30 =60 =90Figure-12 The diagram of buckling load variations vs. crack length in constant crack angles (H=100 mm) Figure-13 Stress distribution in a shell with a=10 mm and =60\r Table-5 Buckling load change vs. crack angle (a=30 mm) Buckling Load(kN) Crack angle(\r) 29.995 No crack 29.738 0\r 29.866 30\r 29.915 60\r 29.976 90\r Table 5 shows that the effect of increasing shell height is so much that buckling load variations as a result of presence of a crack 20 mm in length and different angles is equivalent to 0.257 kN. Because of relatively high height of these series of shells, buckling occurs completely at necking part of shell and creates a symmetrical ring. For instance, buckling of a shell with a=20 mm and =90\r is demonstrated in figure 14. Table 6 indicates the effect of increasing height on buckling load for conical shells with a crack 20 mm in length in different angles. It is observed that in equal loading and support conditions, buckling load is reduced with increasing shell height and also buckling load is raised with increasing crack angle. Figure-14 Stress distribution in a shell with a=20 mm and =90\r Table-6 Buckling load variations vs. height with presence of a crack with a=20 mm and different angles Buckling Load(kN) H=150 mmH=100 mmH=75 mmCrack angle(\r) 29.738 37.304 39.777 0\r 29.866 37.759 40.535 30\r 29.915 38.495 42.443 60\r 29.976 38.686 43.033 90\r Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(2), 33-39, February (2013) Res. J. Recent Sci. International Science Congress Association 38 Table 6 is demonstrated as a diagram in figure 15. Figure-15 The diagram of buckling load variations vs. crack angles Comparison of FEM and experimental test results: Different experimental tests were conducted to confirm of the authenticity of the results obtained from the numerical method17-20. For example Gupta et al17 investigated on conical frusta of aluminum of thicknesses between 0.7 and 1.62mm and semi-apical angles range of 16–29 were axially compressed in a universal testing machine. They recorded load–deformation curves and deformed shapes of specimens. All deformation mode of buckling are similar to my investigation and also the behavior of load-displacement was the same. ConclusionAfter buckling phenomenon occurs, load tolerance capacity of shell is severely reduced. The presence of a crack with different angles and obtained stress current has a considerable effect on buckling load of shells. In a constant crack length, with increasing crack angle, buckling load increases and in a constant crack angle, with increasing length, buckling load is reduced. Since vertical crack has little effect on stress current, it isn't so effective on buckling load shell. Where shell at first is buckled depends on several factors. Considering shell geometry, this point may be either in a place that shell has minimum diameter and leads to creating a plastic ring or in a place that as a result of the presence of a crack, plate strength is extremely reduced. In equal loading and support conditions, with increasing shell length, buckling load is considerably reduced. The effect of increasing shell height on buckling load may be to such an extent that the effect of the presence of a crack with definite length is really eliminated; therefore, the effect of the crack presence on buckling load in shells with low height is more noticeable. 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