110 5-5 Plot showing the relationship between the reduction in buckling. reduction of the buckling load of a cylinder subjected to lateral external pressure. 121 5- 18 Plot showing the relationship between the reduction in buckling. load , tap er and imperfection for the case of combined loading for.
N umerical analysis of buckling
The number of bending loads that can be achieved for the model in question is equal to the number of degrees of freedom of the model. The basis of the theory will be outlined and the equations that determine the bending loads will be produced.
The Energy Method
Th in flat plates
Therefore, the four corners of the element will each deflect a different amount in the z direction. The strain energy stored in a plate element is the sum of the work done by the bending moments Mx, Mll> and by the torsional moments M2:JJ and M JJx.
Thin cylindri cal shells
Differential equations
Axial loading of the cylinder
Li terature surv ey
However, once the buckling begins, the load of the middle face in the circumferential direction must also be taken into account. This form of the equation was given in [37], but several other references ([55J ,[65)) cited this equation as.
Numerical analys is
Instead of using the energy method as above, it can be shown that the differential equations for symmetric deformation can also be used in the calculation of the critical load. Quick findings show that the ultimate condition may not be met even when the ratio of length h to radius is much less than one, and his study includes:s cy linders up to a length to radius ratio of only 0.769. .
Lateral external pressure
Literature survey
If, for example, the ends of the cylinder are simply supported, the boundary conditions require that the rod f)2wj8x2 becomes zero at the ends. The small terms, which have very little effect on the magnitude of the critical pressure, are omitted.
Numerical analysis
External pressure - Both lateral and axial
Literature survey
As long as it does not meet any of the c curves, the shell is in stable equilibrium. Each axial pressure reduces the critical value of the lateral pressure, and each lateral pressure reduces the critical value of the axial pressure. 2.116). Figure 2-27 shows that the number of circumferential slots into which the cylinder clamps folds as the length and thickness of the shell decreases.
Numerical analysis
Combi ned hydrostatic pressure and axial l oading
Tennyson, Booton et al [36] determined an interactive equation for the buckling of cylinders subjected to both axial load and hydrostatic pressure. Tennyson, Booton et al comparison was found not applicable to steel cylinders, while accurate for appropriate epoxy plastics. Galletly and Pemsing showed that this relationship is true, but only for steel cylinders with large values of Z (for Z > 200), as seen in figure 2-30.
Numerical analysis
When these buckling results are compared to pure hydrostatic pressure buckling, it can be seen that the axial load reduces the buckling load by a very small margin, as seen in Figure 2-32. If this axial load is then varied, as the axial load increases, the critical buckling pressure decreases as expected. The imperfection, such as out-of-roundness, grows as the external pressure is introduced and increases, whereas it is 'nullified' in the case of internal pressure.
Axial loading
Literatu re su rvey
The variables lx and la are the half-wavelengths of the deflections in the axial and c circumferential directions of the cylinder wall, respectively. In 1999, Yamada and Croll [66] theoretically demonstrated the physical theory of the reduced stiffness method. The authors concluded that the modified form of the reduced stiffness method provides a simple but safe basis r either de:sigu or axially compressed cylinders, especially for the design of cylinder liners where weight is not an issue.
Nu merical an alysis. ,
The results of the numerical analysis under axial load for different degrees of imperfection for the investigated cylinders are shown in Figure 3-10. It is also important to note that the effect of the imperfection becomes more pronounced as the thickness decreases. T l w dt'pendencf' of the buckling load on t lw f'ccf'ntr icity can also be seen in Figure 3-11 where the buckling load decreases sharply as the eccentricity increases.':iE'::i.
Lateral external pressure
Literature survey
Numerical an alysis
The effect of imperfection and constriction on the reduction of bending load for the case of hydrostatic pressure is shown in Figures 5-14 and 5-15. They also introduced a stiffness variation parameter to access the effect of thickness variation on bending load reduction. 1969) "Effect of general imperfections on buckling of cylindrical shells". 7hzsactions of ASME.
Hydrostatic press ure
Literature survey
The initial imperfection in the cylinders was believed to be similar to one of the forms of buckling. But this research did not provide a definitive answer as to whether there was a reduction formula. The 1989 study concluded that the complexity of the buckling behavior exhibited by this shape of the shell was demonstrated through the use of numerical experiments using different levels of geometric imperfection.
Numerical analysis
The effect of the thickness variation on the buckling load reduction is shown using a plot of the thickness variation parameter versus imperfection for ·various values of taper. In the buckling of perfect cylinders subjected to axial loading, buckling equations were definitive and it was found that as the radius of the cylinder increased or wall thickness decreased, the load carrying capacity decreased. Numerical analysis also showed that as the Batdorf parameter increased, the effect of the taper on the buckling load increased, and vice versa.
Combi ned hydrostatic pressure and axial loading
Literature survey
The load-deflection curve decreases after bifurcation, and th is bifurcation value is therefore the failure load. The eigenvalue equation obtained in [23], which was shown to be applicable in the case of external pressure is. Hutchin son pointed out that the assumed imperfection pattern has essentially no effect on the buckling load, so this equation is still valid for other assumed harmonic initial imperfection distribution, as is the case in this study.
Numerical analys is
In this study, cylinders with a taper, et, ranging from 0.002° to 0.04° are considered, where the wall thickness at the mean length, tm, is considered to have similar values equal to those considered in the study of cylinders with constant wall thickness. . Hardly any research has been carried out on the bending of cylinders with variable wall thickness and even less on the thickness variations of the symmetric axis in the part.r. The influence of thickness variation on bending load has received little attention so far from researchers in this field.
Axial loading
Literature s urvey
Koiter et al [55] finally determined the reduction factor due to thickness variation using a hybrid perturbation-weighted res id uals method, and it was found. The theoretical relationships hip between the reduction in buckling load and the size parameter can be seen graphically in figure 4-2, for both cases A and B.
Numerical analysis
Literature survey
Numerical analysis
As in the case of axial loading, the effect of the impact of the individual and the analyzed narrow red values are bent in the same way. A general indication of the buckling of a perfect cylinder under axial compress ion can be executed using Equations 2.66 and 2.67. Shell bending: Procedure from a modern bending of cylinders under combined external pressure and axial compression.
Hydrostatic pressure
Literature survey
Pioneers in investigating the bending of cylinders with variable wall thickness, Biezeno and Koch [91 investigated this for cylinders subjected to external hydrostatic pressure as early as 1938. However, they considered the general case of varying thickness too complicated and decided to replace the tapered wall cylinder with a cylinder constructed of a finite number of rings, each of constant thickness. They then concluded that the bending of a cylinder of this shape is similar to the bending of a rod composed of a finite number of segments, as long as ~ «1 and the radius and length of the cylinder were of the same order of magnitude.
Numerical analysis
It is clear that as tupf's increase, the behavior of the cylinder ('viatCs and marf' from t h that of tlw constant wall thickness cylinder. The effect of the individual imperfection and taper values we investigated and is shown in Figure 5 -8.In other words, as the degree of tapering increased, the buckling waves moved to where the cylinder is thinner.
Combined hydrostatic pressure and axial loading
Numerical analysis
Much literature over the past century has been devoted to the buckling of cylinders with constant wall thickness, and the buckling of cylinders with geometric imperfections has been investigated for the past half century. Even the buckling of cylinders with thickness variations was briefly investigated. However, no similar research appears to be available for the combined effect of thickness variations and geometric imperfections on the buckling of cylinders subjected to external pressure.
Axial loading
Literature survey
Numerical analysis
Numerical analysis
In the study of buckling under external pressure, it was found that as the length to radius r atio increases, the buckling load decreases. In the case of axial loading, it was seen that, as expected, the buckling load decreases for. 1999) "Influence of localized imp erfections on the buckling of cylindrical shells under axial compression." International Journal of Solids and Structures.
Literature s urvey
Numerical analysis
It was also found that as the radius increases, the wall thickness decreases or the length decreases. However, when numerical analysis was performed for lateral and hydrostatic pressure, it was found that as the imperfection was introduced, the reduction of the buckling load increased. When a numerical analysis was performed, it was found that as the taper increased, the breaking lobes moved higher on the cylinder.
Overview
Numerical analyzes were performed using finite element analysis and MSC Nastran f or Windows ftware. The numerical analyzes performed in this study turn out to be correct in terms of trends, but over-predicted in terms of absolute values. This was found to be due to the fact that the MSC Nastran thus ftw are implicated did not have the nonlinear buckling solution set needed to predict buckling of such rear walls.
Recommendat ion for furth er work
1962) "Theoretical interaction equation for tll.e bending of circular shells under axial compression and external pressure". AIAA Journal v Elastic Stability of Thin-Walled Cylindrical and Conical Shells under Combined External Pressure and Axial Compression.". Buckling and Postb uckling Behavior of Cylindrical Shells und er Combined External Pressure and Axial Compression." Thin-wall constructions.
A Comparison of Shell Theories in the Analysis of Cylindrical Shells with Thickness Discontinuity." Journal of Computers and Structures. Chapman and Ha ll, London Effect of Primary Boundary Conditions on Buckling of Shallow Cylindrical Shells." Journal of Constructional Steel Research.
