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Imperfection sensitivity of flat plates under combined compression and shear
International Journal of Non-Linear Mechanics 36 (2001) 249}259
Imperfection sensitivity of #at plates under combined compression and shear C.A. Featherston Cardiw School of Engineering, Cardiw University, P.O. Box 685, Cardiw CF2 3TB, UK
Abstract Finite element analysis allows fully non-linear analysis of shells containing geometric imperfections. However, such an analysis requires information on the exact size and shape of the imperfection to be modelled, in order to produce accurate results on which designs can be based. In the absence of such data it is generally recommended that the imperfection be modelled on the "rst eigenmode with an amplitude selected according to manufacturing procedure. This paper presents the e!ects of imperfection shape and amplitude on the buckling and postbuckling behaviour of one speci"c case, to test the accuracy of such recommendations. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Buckling; Postbuckling; Imperfection sensitivity; Finite element analysis
1. Introduction It has long been accepted that the existence of imperfections in thin-walled structures can substantially reduce their buckling load and a!ect their postbuckling behaviour. These imperfections can take the form of non-uniformity or local variations in the physical properties of the material, geometric imperfections (deviations in shape, eccentricities, local indentations) and load eccentricities, although the majority of the research carried out to date has been based on geometric imperfections. Attempts to develop a general theory of buckling and postbuckling theory incorporating imperfection sensitivity began with Koiter [1,2]. Further contributions were provided by Budiansky and Hutchinson [3], Stein [4], Arbocz [5], and many others. With the event of "nite element analysis the engineer was presented with a tool which allows him to model the buckling and postbuckling of
shells under complex load and boundary conditions whilst fully incorporating the e!ects of imperfections and other non-linearities. However, di$culties still arise in directly using numerical shell buckling analysis in design due to the need to convert numerical buckling loads based on any of several di!erent types of buckling analysis available into a design load for a particular structure. Several approaches have been considered by di!erent researchers and code writing committees [6}9]. Two main types of approach exist. The "rst involves the use of a linear elastic bifurcation buckling analysis (often known as an eigenvalue analysis) to determine the bifurcation loads of a structure. Reduction factors are then applied to account for both geometric imperfections and plasticity. This method is attractive in that it is similar to existing techniques based on simple load cases, and a linear bifurcation analysis is relatively quick and simple to perform. However, di$culties exist in determining reduction factors for various loading
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C.A. Featherston / International Journal of Non-Linear Mechanics 36 (2001) 249}259
and boundary conditions. Work has been done in this area by Schmidt and Krysik [6] and Samuelson and Eggwertz [8]. The alternative approach involves the performance of a fully non-linear analysis with geometric imperfections, plasticity and also large de#ections being accurately modelled. Although this technique has obvious advantages, the di$culty now lies in deciding upon the amplitude and form of the imperfection to be used. The most accurate method is obviously to base any analysis on real imperfections. Accurate measurement of imperfections in the laboratory and also full-scale shells, have been carried out by Arbocz [10,11], Arbocz and Babcock [12], Arbocz and Hol [13], Elishako! et al. [14], Singer [15], Singer and Abromovich [16], and Weller et al. [17]. These and other results have been incorporated into an International Imperfections Data Bank [10,11]. Unfortunately, detailed information on the amplitude and form of real imperfections in the structure being designed are not always available. In such cases Speicher and Saal [7] have recommended that an equivalent imperfection having the same form at the "rst bifurcation or eigenmode be used. They have also derived the required magnitude of these imperfections to produce a safe design for the particular case of cylinders, based on existing experimental results. Most commercially available "nite element codes recommend a similar approach to designers, who are advised again to base the form of the imperfection on the "rst bifurcation/eigenmode, and set its maximum amplitude equal to that expected in the component
itself, based on a consideration of manufacturing methods used. This paper presents the results of such an analysis performed for the speci"c case of a #at plate under complex load and boundary conditions, for which a model previously validated by comparison with experimental results [18] already exists. This was selected speci"cally to represent part of a component commonly found in aeroengine structures, which are often susceptible to failure by buckling. A simple linear analysis was performed "rst to determine the eigenmodes of the plate. A series of non-linear analyses based on imperfection in the form of these mode shapes was then carried out to investigate the e!ect of varying the modal shape used to represent this imperfection, and its amplitude.
2. Finite element analysis Analysis were carried out using the FE package ABAQUS on a series of #at plates with aspect ratios 1 : 1, 1.5 : 1 and 2 : 1. These plates were built in at one end, simply supported along the two long edges and restrained to prevent out-of-plane displacement and rotation along the fourth edge. A shear load was then applied across this fourth edge, thus creating a combination of bending and shear forces which varied throughout the plate as shown in Fig. 1. This case was selected to provide a highly simpli"ed model of one-half of an aeroengine fan blade, built in at one end (the hub) and loaded in shear across the other end with each
Fig. 1. Load and boundary conditions.
C.A. Featherston / International Journal of Non-Linear Mechanics 36 (2001) 249}259
half of the blade simply supporting the other half where they are joined together, i.e, along their long edges. The plates modelled were aircraft grade duraluminium (BS1470 Grade 6082 T6) with dimensions 100 mm;100 mm, 100 mm;150 mm and 100 mm;200 mm and thickness 0.5 mm. 2.1. Model The model for each plate was constructed from 5 mm square elements. The elements used were quadrilateral elements (S8R5), which behave in a manner consistent with thin shell theory. These elements have eight nodes and edge behaviour is modelled using quadratic equations. Following calculation of the displacement of each node, stresses and strains are determined at four points and then extrapolated to give the values at each node. The nodes have "ve degrees of freedom, three displacements and two in surface rotations. The boundary conditions modelled can be described by reference to Fig. 2. The "rst three degrees of freedom represent displacements and the last three rotations. Edge 1 of the plate was restricted in all six degrees of freedom to model a clamped condition. Out-of-plane displacement x and rotation about z in edges 2 and 3 were prevented to represent simple support. Edge four was restricted in out-of-plane displacement x and rotations about y and z. Loading was applied through a beam attached to the end of the plate. The beam was modelled using 5 mm long three-noded quadratic beam elements. A shear load was then applied at one end.
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2.2. Linear eigenmode analysis In order to obtain the mode shapes necessary to model geometric imperfections, a linear eigenmode analysis was "rst performed. This is carried out in ABAQUS using a linear perturbation procedure. First the sti!ness matrix corresponding to the base state loading on the structure is stored and then a small perturbation or live load is applied. In this case, the base state is the unloaded condition and the matrix used is the original sti!ness matrix. The program then derives the initial stress matrix due to the live load and an eigenvalue calculation is performed to determine a multiplier to the live load at which the structure becomes unstable. This can be written as follows. If the elastic sti!ness matrix is K , the initial stress and load sti!ness matrix is K , and the load pattern is N de"ned by vector P , then load multipliers (eigenvalues) j, and buckling mode shapes (eigenvectors) u are required which satisfy the equation [K #jK ]u "0. N The critical buckling loads can then be calculated by multiplying j and P . ABAQUS uses subspace iteration (creation of a small set of base vectors de"ning a subspace which is then transformed by iteration into the space containing the lowest few eigenvectors of the overall system) for eigenvalue extraction, which allows extraction of several modes simultaneously. Fig. 3 shows the mode shapes corresponding to the "rst four positive eigenvalues calculated for a plate with aspect ratio 1 : 1. (For each positive
Fig. 2. Boundary conditions for "nite element analysis.
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Fig. 3. First four predicted eigenmodes for a plate with aspect ratio 1 : 1.
eigenvalue a negative eigenvalue of equal magnitude was calculated indicating that the structure would buckle at the same load if loaded in the opposite direction.) 2.2.1. Non-linear imperfection sensitivity and postbuckling analysis Once the mode shapes had been calculated, these were used to model imperfections which were then incorporated into the &perfect geometry' of the original model. A non-linear analysis was then performed to predict the buckling load and postbuckling behaviour of each component. This was done using the Rik's method which uses nonlinear static equilibrium equations to solve unstable problems, where the load}displacement response is such that either the load or the displacement may decrease as the solution evolves. The method is based on the assumptions that the loading is proportional, i.e., all load magnitudes vary with a single scalar parameter, and that bifur-
cations do not occur in the response (this is obviously a serious limitation). Solution of the problem involves following a single equilibrium path in a space de"ned by nobal variables and the loading parameter. At any time there is a "nite radius of convergence. Some of the materials and loadings of interest have path-dependant responses. For these reasons the load increment size has to be limited. This is done by moving a given distance (determined depending on the convergence rate at that point) along the tangent to the current solution point, and then searching in the plane that passes through the point thus obtained and that is orthogonal to the same tangent line for equilibrium as shown in Fig. 4. This step is repeated until the total path length traversed reaches the required value. The results of these analyses are shown in Figs. 5}8. Fig. 5 compares the load versus displacement plots for plates with imperfections in the form of the "rst positive eigenmode, but with varying amplitudes ranging from 0.1t (where t is the thick-
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3. Interpretation of results 3.1. Ewect of varying imperfection amplitude
Fig. 4. Risk analysis.
ness of the plate) to 3t, for plates with each of the three aspect ratios. In each case displacement is measured in-plane at the point of load application and in the direction of load application. Fig. 6 also compares load versus displacement plots, this time for plates with maximum imperfection amplitude 1t but based on varying eigenmodes, both singly and in combination. Again results are presented for plates with each of the three aspect ratios. Note that in the case of imperfections formed from a combination of eigenmodes, the maximum amplitude of each eigenmode-based imperfection introduced is 1t (i.e., the contribution magnitude of each of the modes is equal). The total maximum imperfection therefore, may be greater than 1t by a maximum factor equal to the number of modes introduced. In practice it will be between this value and 1t, however its exact magnitude will not be known by the designer, who is only able to specify the maximum contribution of each mode and not the overall maximum amplitude. This substantially reduces the value of analyses based on these &combined mode' imperfections since they cannot be directly compared with those based on imperfections based on single mode shapes. Finally, Figs. 7 and 8 compare the collapse loads (the maximum load which the plate can support) for plates having imperfections incorporating each possible combination of mode shape and amplitude. These results will be discussed in the next section.
Fig. 5 shows the e!ect of varying the amplitude of the imperfection on the load}displacement pro"les. The results presented are all calculated for plates having imperfections in the form of the "rst positive eigenmode, although those for other eigenmodes and combinations of eigenmodes are similar. Data showing the e!ect of varying imperfection amplitude on the collapse load only, is presented in Fig. 7, this data covers all eigenmodes. Reference to Fig. 5 shows that each load versus displacement pro"le follows a similar path. In each case the graph is initially linear, however, as the load increases the increase in displacement becomes disproportionately larger, until the &collapse load' is reached (i.e., the maximum load which the plate can support). After this point, the load begins to decrease whilst the displacement continues to increase. In each case, however there remains some residual load carrying capacity in the postbuckling region. Increasing the size of the imperfection introduced does however, reduce the prebuckling sti!ness and the collapse load as would be expected. In terms of prebuckling sti!ness, Fig. 5 shows that this reduction is minimal for imperfections less than 0.5 times the thickness of the plate for plates with aspect ratios 1 : 1 and 2 : 1 and 1.0 times the thickness of the plate for plates with aspect ratio 1.5 : 1. This is not the case for imperfections in the shape of other higher eigenmodes where a reduction in sti!ness is seen with each increase in imperfection amplitude. In all cases the load versus displacement pro"les converge for higher displacements so the e!ect of imperfection is signi"cantly less in the postbuckling region. In this region, the non-linearity introduced by the large out-of-plane displacements occurring is much greater than that caused by initial imperfections which therefore become relatively insigni"cant. In terms of collapse load Fig. 7 shows the reduction with increase in imperfection size to be almost linear for imperfections based on a single eigenmode although nonlinearity increases for more complex shapes formed from combined eigenmodes. It is possible therefore
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Fig. 5. Comparison of load versus displacement pro"les for plates having imperfections of varying amplitudes based on the "rst positive eigenmode and aspect ratios (a) 1 : 1, (b) 1.5 : 1 and (c) 2 : 1.
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Fig. 6. Comparison of load versus displacement pro"les for plates having imperfections based on varying eigenmodes with maximum amplitude 1t and aspect ratios (a) 1 : 1, (b) 1.5 : 1 and (c) 2 : 1.
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Fig. 7. Comparison of collapse loads for plates having imperfections based on varying each combination of mode shape and amplitude with aspect ratios (a) 1 : 1, (b) 1.5 : 1 and (c) 2 : 1.
C.A. Featherston / International Journal of Non-Linear Mechanics 36 (2001) 249}259
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Fig. 8. Comparison of collapse loads for plates having imperfections based on varying each combination of mode shape and amplitude with aspect ratios (a) 1 : 1, (b) 1.5 : 1 and (c) 2 : 1.
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that the overall relationship between imperfection size and collapse load could be derived by performing only two non-linear analyses, for example on plates with maximum imperfection amplitudes of 1t and 2t. It can also be seen that the rate of reduction in collapse load with increasing imperfection amplitude is generally greater for higher eigenmodes. This reduction is even larger for combined eigenmodes, although this is probably a direct result of the fact that with combined eigenmode imperfections, the maximum imperfection amplitude may be larger than that for single eigenmode imperfections as discussed earlier. Finally, in relation to the e!ect of aspect ratio on the sensitivity to imperfection amplitude of the prebuckling sti!ness and collapse load, reference to Fig. 7 shows that the rate of reduction in both of these with increasing imperfection amplitude is much greater for plates with aspect ratio 1 : 1 than for those with aspect ratio 2 : 1. This indicates that imperfection sensitivity of these plates decreases with increased aspect ratio. This is as anticipated, since it can be shown that for plates of equal thickness and end shortening, as aspect ratio increases, the ratio (imperfection introduced (t)/maximum eigenmode amplitude for a perfect structure) becomes smaller, which in turn can be shown to result in a smaller reduction in critical load due to the same size of imperfection.
3.2. Ewect of varying eigenmode Fig. 6 shows the e!ect of varying the shape of the imperfection on the load versus displacement pro"les. The results are all for plates with an imperfection of maximum amplitude equal to the thickness of the plate. Data is presented for each of the three aspect ratios analysed. Fig. 8 compares the calculated collapse loads generated by varying the shape of the imperfection for each imperfection amplitude. It can again be seen that whilst the load versus displacement plots for plates with imperfections in the form of each of the "rst four eigenmodes singly and combined are similar, they have di!erent prebuckling sti!nesses and collapse loads whilst converging in the postbuckling region.
The main point of note, however is that in contrast with the e!ect of increasing the size of imperfection introduced into the plate, Fig. 8 illustrates the fact that there is no obvious correlation between the shape of the imperfection and the prebuckling sti!ness and collapse loads. The mode shape that results in the lowest collapse load for example for a speci"c plate, di!ers for each size of imperfection and aspect ratio. Following the recommended procedure of performing an analysis on a model with an imperfection in the shape of the "rst eigenmode with amplitude representative of the maximum imperfection anticipated to give the lowest collapse load therefore, will not necessarily give the correct result. This suggests that analysis for any particular case should be performed on a series of models having imperfections in the form of the "rst few eigenmodes and compared to ensure that the lowest value of collapse load is detected. Finally, as noted when examining the e!ect of modifying imperfection amplitude, it can be seen that the imperfection sensitivity of the plate decreases with an increase in aspect ratio. 4. Conclusions In examining the e!ect of imperfection shape and amplitude on a #at plate under combined compression and shear, the following points have been noted: E Increasing the amplitude of the imperfection reduces both the prebuckling sti!ness and the collapse load of the plate as would be expected. E The reduction in collapse load with the increasing amplitude is approximately linear for imperfections in the form of a single eigenmode. E The e!ect of initial imperfection amplitude on the postbuckling behaviour is minimal. E Modifying the shape of the imperfection again changes the prebuckling sti!ness and the collapse load. E Analyses based on imperfections formed from a combination of di!erent mode shapes are of limited use, since only the maximum amplitude of each contributing eigenmode can be de"ned, and not the overall maximum amplitude.
C.A. Featherston / International Journal of Non-Linear Mechanics 36 (2001) 249}259
E There is no obvious relationship between the eigenmode used to represent the imperfection and the prebuckling sti!ness and collapse load. E However, in no case does the "rst eigenmode give the lowest collapse load. E The imperfection sensitivity of the plate decreases with an increase in aspect ratio. In this particular case therefore, the introduction of an imperfection in the form of the "rst eigenmode, with a maximum amplitude equal to that of the maximum imperfection anticipated (based on manufacturing technique, tolerances, etc.) will not necessarily provide a lower limit to the prebuckling sti!ness and collapse load. Analyses would need to be carried out based on imperfections in the form of each of the "rst few eigenmodes to ensure that the results were conservative and could safely be used in design calculations.
[7]
[8] [9]
[10]
[11]
[12]
[13]
References [14] [1] W.T. Koiter, On the stability of elastic equilibrium, Ph.D. Thesis, University of Delft, 1945. [2] W.T. Koiter, Elastic stability and postbuckling behaviour, Proceedings, Symposium on Nonlinear Problems, University of Wisconsin Press, Madison, 1963, pp. 257}275. [3] B. Budiansky, J.W. Hutchinson, Dynamic buckling of imperfections sensitive structures, Proceedings of the 11th International Congress of Applied Mechanics, Munich, 1964, pp. 636}651. [4] M. Stein, Some recent advances in the investigation of shell buckling, AIAA J. 6 (1968) 2239}2245. [5] J. Arbocz, The e!ect of initial imperfections on shell stability, in: Y.C. Fung, E.E. Sachier (Eds.) Thin-Shell Structures, Prentice-Hall, Inc., Englewood Cli!s, NJ, 1974, pp. 205}245. [6] H. Schmidt, R. Krysik, Toward recommendations for shell stability design by means of numerically determined buckling loads, in: J.F. Jullian (Ed.) Buckling of Shell Struc-
[15]
[16]
[17]
[18]
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tures, on Land, in the Sea, and in the Air, Elsevier Applied Science, London, 1991, pp. 508}519. G. Speicher, H. Saal, Numerical calculation of limit loads for shells of revolution with particular regard to the applying equivalent initial imperfection, in: J.F. Jullian (Ed.) Buckling of Shell Structures, on Land, in the Sea, and in the Air, Elsevier Applied Science, London, 1991, pp. 466}475. L. Samuelson, S. Eggwertz, Shell Stability Handbook, Elsevier Applied Science, London, 1992. J.G. Teng, J.M. Rotter, Stability assessment of complex shell structures by numerical analysis, Austr. Civil Engng. Trans. CE37 (1) (1995) 61}69. J. Arbocz, The imperfection data bank, a mean to obtain realistic buckling loads, in: E. Ramm (Ed.) Buckling of Shells, Springer, Berlin, 1982, pp. 535}567. J. Arbocz, Towards an improved design procedure for buckling critical structures, in: J.F. Jullian (Ed.) Buckling of Shell Structures, on Land, in the Sea, and in the Air, Elsevier Applied Science, London, 1991, pp. 270}276. J. Arbocz, C.D. Babcock Jr., Computerised stability analysis using measured initial imperfections, Proceedings of 12th Congress of the International Council of the Aeronautical Sciences, October 12}17, Munchen, 1981, pp. 688}701. J. Arbocz, J.M.A.M. Hol, Collapse of axially compressed cylindrical shells with random imperfections, AIAA J. 29 (12) (1991) 2247}2256. I. Elishako!, S. van Manen, P.G. Vermeulen, J. Arbocz, First order second-moment analysis of the buckling of shells with random imperfections, AIAA J. 25 (8) (1987) 1113}1117. J. Singer, Vibration correlation techniques for improved buckling predictions of imperfect sti!ened shells, in: J.E. Harding, P.J. Dowling, W. Agoliolis (Eds.) Buckling of Shells in O!shore Structures, Granada, London, 1982, pp. 285}329. J. Singer, H. Abromovich, The development of shell imperfection measurement techniques, Thin-Walled Struct. 23 (14) (1995) 379}398. T. Weller, H. Abromovich, J. Singer, Application of nondestructive vibration correlation techniques for buckling of spot-welded and riveted stringer sti!ened shells, Seitschrift Flugwessenshafter Weltraumforschung 10 (3) (1986) 183189. C.A. Featherston, C. Ruiz, Buckling of #at plates under bending and shear, J. Mech. Engng. Sci. Proc. Part C 212 (1998) 249}261.
Imperfection sensitivity of #at plates under combined compression and shear C.A. Featherston Cardiw School of Engineering, Cardiw University, P.O. Box 685, Cardiw CF2 3TB, UK
Abstract Finite element analysis allows fully non-linear analysis of shells containing geometric imperfections. However, such an analysis requires information on the exact size and shape of the imperfection to be modelled, in order to produce accurate results on which designs can be based. In the absence of such data it is generally recommended that the imperfection be modelled on the "rst eigenmode with an amplitude selected according to manufacturing procedure. This paper presents the e!ects of imperfection shape and amplitude on the buckling and postbuckling behaviour of one speci"c case, to test the accuracy of such recommendations. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Buckling; Postbuckling; Imperfection sensitivity; Finite element analysis
1. Introduction It has long been accepted that the existence of imperfections in thin-walled structures can substantially reduce their buckling load and a!ect their postbuckling behaviour. These imperfections can take the form of non-uniformity or local variations in the physical properties of the material, geometric imperfections (deviations in shape, eccentricities, local indentations) and load eccentricities, although the majority of the research carried out to date has been based on geometric imperfections. Attempts to develop a general theory of buckling and postbuckling theory incorporating imperfection sensitivity began with Koiter [1,2]. Further contributions were provided by Budiansky and Hutchinson [3], Stein [4], Arbocz [5], and many others. With the event of "nite element analysis the engineer was presented with a tool which allows him to model the buckling and postbuckling of
shells under complex load and boundary conditions whilst fully incorporating the e!ects of imperfections and other non-linearities. However, di$culties still arise in directly using numerical shell buckling analysis in design due to the need to convert numerical buckling loads based on any of several di!erent types of buckling analysis available into a design load for a particular structure. Several approaches have been considered by di!erent researchers and code writing committees [6}9]. Two main types of approach exist. The "rst involves the use of a linear elastic bifurcation buckling analysis (often known as an eigenvalue analysis) to determine the bifurcation loads of a structure. Reduction factors are then applied to account for both geometric imperfections and plasticity. This method is attractive in that it is similar to existing techniques based on simple load cases, and a linear bifurcation analysis is relatively quick and simple to perform. However, di$culties exist in determining reduction factors for various loading
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and boundary conditions. Work has been done in this area by Schmidt and Krysik [6] and Samuelson and Eggwertz [8]. The alternative approach involves the performance of a fully non-linear analysis with geometric imperfections, plasticity and also large de#ections being accurately modelled. Although this technique has obvious advantages, the di$culty now lies in deciding upon the amplitude and form of the imperfection to be used. The most accurate method is obviously to base any analysis on real imperfections. Accurate measurement of imperfections in the laboratory and also full-scale shells, have been carried out by Arbocz [10,11], Arbocz and Babcock [12], Arbocz and Hol [13], Elishako! et al. [14], Singer [15], Singer and Abromovich [16], and Weller et al. [17]. These and other results have been incorporated into an International Imperfections Data Bank [10,11]. Unfortunately, detailed information on the amplitude and form of real imperfections in the structure being designed are not always available. In such cases Speicher and Saal [7] have recommended that an equivalent imperfection having the same form at the "rst bifurcation or eigenmode be used. They have also derived the required magnitude of these imperfections to produce a safe design for the particular case of cylinders, based on existing experimental results. Most commercially available "nite element codes recommend a similar approach to designers, who are advised again to base the form of the imperfection on the "rst bifurcation/eigenmode, and set its maximum amplitude equal to that expected in the component
itself, based on a consideration of manufacturing methods used. This paper presents the results of such an analysis performed for the speci"c case of a #at plate under complex load and boundary conditions, for which a model previously validated by comparison with experimental results [18] already exists. This was selected speci"cally to represent part of a component commonly found in aeroengine structures, which are often susceptible to failure by buckling. A simple linear analysis was performed "rst to determine the eigenmodes of the plate. A series of non-linear analyses based on imperfection in the form of these mode shapes was then carried out to investigate the e!ect of varying the modal shape used to represent this imperfection, and its amplitude.
2. Finite element analysis Analysis were carried out using the FE package ABAQUS on a series of #at plates with aspect ratios 1 : 1, 1.5 : 1 and 2 : 1. These plates were built in at one end, simply supported along the two long edges and restrained to prevent out-of-plane displacement and rotation along the fourth edge. A shear load was then applied across this fourth edge, thus creating a combination of bending and shear forces which varied throughout the plate as shown in Fig. 1. This case was selected to provide a highly simpli"ed model of one-half of an aeroengine fan blade, built in at one end (the hub) and loaded in shear across the other end with each
Fig. 1. Load and boundary conditions.
C.A. Featherston / International Journal of Non-Linear Mechanics 36 (2001) 249}259
half of the blade simply supporting the other half where they are joined together, i.e, along their long edges. The plates modelled were aircraft grade duraluminium (BS1470 Grade 6082 T6) with dimensions 100 mm;100 mm, 100 mm;150 mm and 100 mm;200 mm and thickness 0.5 mm. 2.1. Model The model for each plate was constructed from 5 mm square elements. The elements used were quadrilateral elements (S8R5), which behave in a manner consistent with thin shell theory. These elements have eight nodes and edge behaviour is modelled using quadratic equations. Following calculation of the displacement of each node, stresses and strains are determined at four points and then extrapolated to give the values at each node. The nodes have "ve degrees of freedom, three displacements and two in surface rotations. The boundary conditions modelled can be described by reference to Fig. 2. The "rst three degrees of freedom represent displacements and the last three rotations. Edge 1 of the plate was restricted in all six degrees of freedom to model a clamped condition. Out-of-plane displacement x and rotation about z in edges 2 and 3 were prevented to represent simple support. Edge four was restricted in out-of-plane displacement x and rotations about y and z. Loading was applied through a beam attached to the end of the plate. The beam was modelled using 5 mm long three-noded quadratic beam elements. A shear load was then applied at one end.
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2.2. Linear eigenmode analysis In order to obtain the mode shapes necessary to model geometric imperfections, a linear eigenmode analysis was "rst performed. This is carried out in ABAQUS using a linear perturbation procedure. First the sti!ness matrix corresponding to the base state loading on the structure is stored and then a small perturbation or live load is applied. In this case, the base state is the unloaded condition and the matrix used is the original sti!ness matrix. The program then derives the initial stress matrix due to the live load and an eigenvalue calculation is performed to determine a multiplier to the live load at which the structure becomes unstable. This can be written as follows. If the elastic sti!ness matrix is K , the initial stress and load sti!ness matrix is K , and the load pattern is N de"ned by vector P , then load multipliers (eigenvalues) j, and buckling mode shapes (eigenvectors) u are required which satisfy the equation [K #jK ]u "0. N The critical buckling loads can then be calculated by multiplying j and P . ABAQUS uses subspace iteration (creation of a small set of base vectors de"ning a subspace which is then transformed by iteration into the space containing the lowest few eigenvectors of the overall system) for eigenvalue extraction, which allows extraction of several modes simultaneously. Fig. 3 shows the mode shapes corresponding to the "rst four positive eigenvalues calculated for a plate with aspect ratio 1 : 1. (For each positive
Fig. 2. Boundary conditions for "nite element analysis.
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Fig. 3. First four predicted eigenmodes for a plate with aspect ratio 1 : 1.
eigenvalue a negative eigenvalue of equal magnitude was calculated indicating that the structure would buckle at the same load if loaded in the opposite direction.) 2.2.1. Non-linear imperfection sensitivity and postbuckling analysis Once the mode shapes had been calculated, these were used to model imperfections which were then incorporated into the &perfect geometry' of the original model. A non-linear analysis was then performed to predict the buckling load and postbuckling behaviour of each component. This was done using the Rik's method which uses nonlinear static equilibrium equations to solve unstable problems, where the load}displacement response is such that either the load or the displacement may decrease as the solution evolves. The method is based on the assumptions that the loading is proportional, i.e., all load magnitudes vary with a single scalar parameter, and that bifur-
cations do not occur in the response (this is obviously a serious limitation). Solution of the problem involves following a single equilibrium path in a space de"ned by nobal variables and the loading parameter. At any time there is a "nite radius of convergence. Some of the materials and loadings of interest have path-dependant responses. For these reasons the load increment size has to be limited. This is done by moving a given distance (determined depending on the convergence rate at that point) along the tangent to the current solution point, and then searching in the plane that passes through the point thus obtained and that is orthogonal to the same tangent line for equilibrium as shown in Fig. 4. This step is repeated until the total path length traversed reaches the required value. The results of these analyses are shown in Figs. 5}8. Fig. 5 compares the load versus displacement plots for plates with imperfections in the form of the "rst positive eigenmode, but with varying amplitudes ranging from 0.1t (where t is the thick-
C.A. Featherston / International Journal of Non-Linear Mechanics 36 (2001) 249}259
253
3. Interpretation of results 3.1. Ewect of varying imperfection amplitude
Fig. 4. Risk analysis.
ness of the plate) to 3t, for plates with each of the three aspect ratios. In each case displacement is measured in-plane at the point of load application and in the direction of load application. Fig. 6 also compares load versus displacement plots, this time for plates with maximum imperfection amplitude 1t but based on varying eigenmodes, both singly and in combination. Again results are presented for plates with each of the three aspect ratios. Note that in the case of imperfections formed from a combination of eigenmodes, the maximum amplitude of each eigenmode-based imperfection introduced is 1t (i.e., the contribution magnitude of each of the modes is equal). The total maximum imperfection therefore, may be greater than 1t by a maximum factor equal to the number of modes introduced. In practice it will be between this value and 1t, however its exact magnitude will not be known by the designer, who is only able to specify the maximum contribution of each mode and not the overall maximum amplitude. This substantially reduces the value of analyses based on these &combined mode' imperfections since they cannot be directly compared with those based on imperfections based on single mode shapes. Finally, Figs. 7 and 8 compare the collapse loads (the maximum load which the plate can support) for plates having imperfections incorporating each possible combination of mode shape and amplitude. These results will be discussed in the next section.
Fig. 5 shows the e!ect of varying the amplitude of the imperfection on the load}displacement pro"les. The results presented are all calculated for plates having imperfections in the form of the "rst positive eigenmode, although those for other eigenmodes and combinations of eigenmodes are similar. Data showing the e!ect of varying imperfection amplitude on the collapse load only, is presented in Fig. 7, this data covers all eigenmodes. Reference to Fig. 5 shows that each load versus displacement pro"le follows a similar path. In each case the graph is initially linear, however, as the load increases the increase in displacement becomes disproportionately larger, until the &collapse load' is reached (i.e., the maximum load which the plate can support). After this point, the load begins to decrease whilst the displacement continues to increase. In each case, however there remains some residual load carrying capacity in the postbuckling region. Increasing the size of the imperfection introduced does however, reduce the prebuckling sti!ness and the collapse load as would be expected. In terms of prebuckling sti!ness, Fig. 5 shows that this reduction is minimal for imperfections less than 0.5 times the thickness of the plate for plates with aspect ratios 1 : 1 and 2 : 1 and 1.0 times the thickness of the plate for plates with aspect ratio 1.5 : 1. This is not the case for imperfections in the shape of other higher eigenmodes where a reduction in sti!ness is seen with each increase in imperfection amplitude. In all cases the load versus displacement pro"les converge for higher displacements so the e!ect of imperfection is signi"cantly less in the postbuckling region. In this region, the non-linearity introduced by the large out-of-plane displacements occurring is much greater than that caused by initial imperfections which therefore become relatively insigni"cant. In terms of collapse load Fig. 7 shows the reduction with increase in imperfection size to be almost linear for imperfections based on a single eigenmode although nonlinearity increases for more complex shapes formed from combined eigenmodes. It is possible therefore
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Fig. 5. Comparison of load versus displacement pro"les for plates having imperfections of varying amplitudes based on the "rst positive eigenmode and aspect ratios (a) 1 : 1, (b) 1.5 : 1 and (c) 2 : 1.
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Fig. 6. Comparison of load versus displacement pro"les for plates having imperfections based on varying eigenmodes with maximum amplitude 1t and aspect ratios (a) 1 : 1, (b) 1.5 : 1 and (c) 2 : 1.
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Fig. 7. Comparison of collapse loads for plates having imperfections based on varying each combination of mode shape and amplitude with aspect ratios (a) 1 : 1, (b) 1.5 : 1 and (c) 2 : 1.
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Fig. 8. Comparison of collapse loads for plates having imperfections based on varying each combination of mode shape and amplitude with aspect ratios (a) 1 : 1, (b) 1.5 : 1 and (c) 2 : 1.
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that the overall relationship between imperfection size and collapse load could be derived by performing only two non-linear analyses, for example on plates with maximum imperfection amplitudes of 1t and 2t. It can also be seen that the rate of reduction in collapse load with increasing imperfection amplitude is generally greater for higher eigenmodes. This reduction is even larger for combined eigenmodes, although this is probably a direct result of the fact that with combined eigenmode imperfections, the maximum imperfection amplitude may be larger than that for single eigenmode imperfections as discussed earlier. Finally, in relation to the e!ect of aspect ratio on the sensitivity to imperfection amplitude of the prebuckling sti!ness and collapse load, reference to Fig. 7 shows that the rate of reduction in both of these with increasing imperfection amplitude is much greater for plates with aspect ratio 1 : 1 than for those with aspect ratio 2 : 1. This indicates that imperfection sensitivity of these plates decreases with increased aspect ratio. This is as anticipated, since it can be shown that for plates of equal thickness and end shortening, as aspect ratio increases, the ratio (imperfection introduced (t)/maximum eigenmode amplitude for a perfect structure) becomes smaller, which in turn can be shown to result in a smaller reduction in critical load due to the same size of imperfection.
3.2. Ewect of varying eigenmode Fig. 6 shows the e!ect of varying the shape of the imperfection on the load versus displacement pro"les. The results are all for plates with an imperfection of maximum amplitude equal to the thickness of the plate. Data is presented for each of the three aspect ratios analysed. Fig. 8 compares the calculated collapse loads generated by varying the shape of the imperfection for each imperfection amplitude. It can again be seen that whilst the load versus displacement plots for plates with imperfections in the form of each of the "rst four eigenmodes singly and combined are similar, they have di!erent prebuckling sti!nesses and collapse loads whilst converging in the postbuckling region.
The main point of note, however is that in contrast with the e!ect of increasing the size of imperfection introduced into the plate, Fig. 8 illustrates the fact that there is no obvious correlation between the shape of the imperfection and the prebuckling sti!ness and collapse loads. The mode shape that results in the lowest collapse load for example for a speci"c plate, di!ers for each size of imperfection and aspect ratio. Following the recommended procedure of performing an analysis on a model with an imperfection in the shape of the "rst eigenmode with amplitude representative of the maximum imperfection anticipated to give the lowest collapse load therefore, will not necessarily give the correct result. This suggests that analysis for any particular case should be performed on a series of models having imperfections in the form of the "rst few eigenmodes and compared to ensure that the lowest value of collapse load is detected. Finally, as noted when examining the e!ect of modifying imperfection amplitude, it can be seen that the imperfection sensitivity of the plate decreases with an increase in aspect ratio. 4. Conclusions In examining the e!ect of imperfection shape and amplitude on a #at plate under combined compression and shear, the following points have been noted: E Increasing the amplitude of the imperfection reduces both the prebuckling sti!ness and the collapse load of the plate as would be expected. E The reduction in collapse load with the increasing amplitude is approximately linear for imperfections in the form of a single eigenmode. E The e!ect of initial imperfection amplitude on the postbuckling behaviour is minimal. E Modifying the shape of the imperfection again changes the prebuckling sti!ness and the collapse load. E Analyses based on imperfections formed from a combination of di!erent mode shapes are of limited use, since only the maximum amplitude of each contributing eigenmode can be de"ned, and not the overall maximum amplitude.
C.A. Featherston / International Journal of Non-Linear Mechanics 36 (2001) 249}259
E There is no obvious relationship between the eigenmode used to represent the imperfection and the prebuckling sti!ness and collapse load. E However, in no case does the "rst eigenmode give the lowest collapse load. E The imperfection sensitivity of the plate decreases with an increase in aspect ratio. In this particular case therefore, the introduction of an imperfection in the form of the "rst eigenmode, with a maximum amplitude equal to that of the maximum imperfection anticipated (based on manufacturing technique, tolerances, etc.) will not necessarily provide a lower limit to the prebuckling sti!ness and collapse load. Analyses would need to be carried out based on imperfections in the form of each of the "rst few eigenmodes to ensure that the results were conservative and could safely be used in design calculations.
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References [14] [1] W.T. Koiter, On the stability of elastic equilibrium, Ph.D. Thesis, University of Delft, 1945. [2] W.T. Koiter, Elastic stability and postbuckling behaviour, Proceedings, Symposium on Nonlinear Problems, University of Wisconsin Press, Madison, 1963, pp. 257}275. [3] B. Budiansky, J.W. Hutchinson, Dynamic buckling of imperfections sensitive structures, Proceedings of the 11th International Congress of Applied Mechanics, Munich, 1964, pp. 636}651. [4] M. Stein, Some recent advances in the investigation of shell buckling, AIAA J. 6 (1968) 2239}2245. [5] J. Arbocz, The e!ect of initial imperfections on shell stability, in: Y.C. Fung, E.E. Sachier (Eds.) Thin-Shell Structures, Prentice-Hall, Inc., Englewood Cli!s, NJ, 1974, pp. 205}245. [6] H. Schmidt, R. Krysik, Toward recommendations for shell stability design by means of numerically determined buckling loads, in: J.F. Jullian (Ed.) Buckling of Shell Struc-
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