Buckling of Shells—pitfall for Designers

PANDA2, a computer program for the minimum-weight design of elastic perfect and imperfect stiffened cylindrical panels and shells under multiple sets of combined loads, is used to obtain optimum designs of uniformly axially compressed elastic internal T-ring and external T-stringer stiffened cylindrical shells with initial imperfections in the form of the general buckling mode. The optimum designs generated by PANDA2 are verified by STAGS elastic and elastic-plastic finite element models produced automatically by a PANDA2 processor called STAGSUNIT. Predictions from STAGS agree well with those from PANDA2. Improvements to PANDA2 during the past year are summarized. Seven different optimum designs are obtained by PANDA2 under various conditions. The most significant condition is whether or not PANDA2 is permitted automatically to make the initial user-specified amplitude of the general buckling modal imperfection directly proportional to the axial half-wavelength of the critical general buckling mode. A survey is conducted over (m,n) space to determine whether or not the critical general buckling modal imperfection shape computed by PANDA2 with (m,n)critical (m=axial, n=circumferential) half-waves is the most harmful imperfection shape. It is found that indeed (m,n)critical is, for all practical purposes, the most harmful imperfection mode shape if PANDA2 is permitted automatically to make the general buckling modal imperfection amplitude directly proportional to the axial half-wavelength of the critical general buckling mode (inversely proportional to m). It is concluded that for axially compressed, stiffened, globally imperfect cylindrical shells the optimum designs obtained with the condition that PANDA2 is NOT allowed to change the initial user-specified imperfection amplitude are probably too heavy. One of the cases investigated demonstrates that the optimum design of a perfect shell obtained via the commonly used condition that a likely initial imperfection be replaced by an increase in the applied load by a factor equal to the inverse of a typical knockdown factor is too heavy. A new input index, ICONSV, is introduced into PANDA2 by means of which optimum designs of various degrees of conservativeness can be generated. Optimum designs are obtained with ICONSV = -1, 0, and +1, which represent increasing degrees of conservativeness in the PANDA2 model. It is concluded that, when obtaining optimum designs with PANDA2, it is best to allow PANDA2 to enter its branch in which local post-buckling behavior is determined, thereby avoiding the generation of designs that may be unsafe because of excessive local bending stresses in the panel skin and stiffener parts. In most cases both nonlinear static and nonlinear dynamic analyses are required in order to obtain collapse loads with STAGS. A table is included that demonstrates how to use STAGS to evaluate an optimum design obtained by PANDA2. In most cases the elastic STAGS models predict collapse in one of the ring bays nearest an end of the cylindrical shell. With the effect of elastic-plastic material behavior included in the STAGS models, collapse most often occurs in an interior ring bay where the finite element mesh is the most dense. From Section 7 of the same AIAA Paper 2007-2216: 7.0 TWO MAJOR EFFECTS OF A GENERAL BUCKLING MODAL IMPERFECTION (For some of the tables referenced, see 2007.axialcomp.pdf) Much of the following appears in Section 11.1 on p. 19 of [1K] (Reference [1K] and other PANDA2 and STAGS references are given below). It is repeated here because this section is especially important. It briefly describes the behavior of a stiffened cylindrical shell with a general buckling modal imperfection shape. This behavior plays a major role in the evolution of the design during optimization cycles in PANDA2. Here it is assumed that the shortest wavelength of the general buckling modal imperfection is greater than the greatest stiffener spacing, as holds in Figs. 1 and 2, for example (disregarding the component of stringer bendingtorsional deformation displayed in the expanded insert in Fig. 1a). A general buckling modal imperfection in a stiffened shell has two major effects: 1. The imperfect stiffened panel or shell bends as soon as any loading is applied. This pre-buckling bending causes significant redistribution of stresses between the panel skin and the various stiffener parts, thus affecting significantly many local and inter-ring buckling and stress constraints (margins). 2. The "effective" circumferential curvature of an imperfect cylindrical panel or shell depends on the amplitude of the initial imperfection, on the circumferential wavelength of the critical buckling mode of the perfect and of the imperfect shell, and on the amount that the initial imperfection grows as the loading increases from zero to the design load. The "effective" circumferential radius of curvature of the imperfect and loaded cylindrical shell is larger than its nominal radius of curvature because the larger "effective" radius corresponds to the maximum local radius of the cylindrical shell with a typical inward circumferential lobe of the initial and subsequently load-amplified buckling modal imperfection. In PANDA2 this larger local "effective" radius of curvature is assumed to be the governing UNIFORM radius in the buckling equations pertaining to the imperfect shell. For the purpose of computing the general buckling load, the imperfect shell is replaced by a new perfect cylindrical shell with the larger “effective” circumferential radius. By means of this device a complicated nonlinear collapse analysis is converted into a simple approximate bifurcation buckling problem a linear eigenvalue problem. For each type of buckling modal imperfection (general, inter-ring, local [1E]) PANDA2 computes a "knockdown" factor based on the ratio: (buckling load factor: panel with its “effective” circumferential radius)/ (buckling load factor: panel with its nominal circumferential radius) (7.1) Figures 1a,b,c show a STAGS model of a typical general buckling modal imperfection shape (amplitude exaggerated) for an optimized “compound” model [1K] of an axially compressed cylindrical shell with external stringers and internal rings (Case 4 in Table 4 in this paper). In this compound model a 45-degree sector has both external stringers and internal rings modeled as flexible branched shell units. A 315-degree sector, the remainder of the cylindrical shell, has smeared stringers and internal rings modeled as flexible branched shell units. Figure 2 shows the deformed state of the imperfect compound model as loaded by the design load, NX = -3000 lb/in axial compression (STAGS load factor PA is close to 1.0). One observes three characteristics: 1. The stresses in the imperfect axially compressed shell have been redistributed as the globally imperfect shell bends under the applied axial compression. The maximum effective (von Mises) stress in this case, sbar(max) = 66.87 ksi, occurs in the outstanding stringer flanges where the pre-buckling deformation pattern of the imperfect shell has a maximum inward lobe. 2. The typical maximum “effective” circumferential radius also occurs where the deformation pattern has a maximum inward lobe. This larger-than-nominal circumferential radius is highlighted most clearly by the in-plane circumferential deformation of the interior ring located one ring spacing in from the right-hand curved edge of the STAGS model shown in Fig. 2. See the right-most expanded insert in Fig. 2. 3. There is an important phenomenon that occurs when imperfect cylindrical shells are optimized. This phenomenon has been described in previous papers [1K]. It occurs in the case of a stiffened cylindrical shell with an imperfection in the form of the critical general buckling mode of the perfect shell. The optimum design of an imperfect stiffened cylindrical shell has a general buckling load factor that is usually considerably higher than load factors that correspond to various kinds of local and “semi-local” buckling, such as local buckling of the panel skin and stiffener segments, rolling of the stiffeners, and inter-ring buckling. The general buckling margin of such a shell is usually not critical (near zero). In contrast, when a perfect stiffened cylindrical shell is optimized the general buckling load factor is usually very close to at least one local buckling load factor and is usually lower than many other local and “semi-local” buckling load factors. The general buckling margin of an optimized perfect shell is usually critical (near zero). The cases explored in this paper exhibit this characteristic. Take, for example, the optimum designs called Case 1 and Case 2 in Table 4. In Case 1 a perfect shell is optimized. The margins for the Case 1 optimum design are listed in Table 10. (See Table 10 in panda2.papers/2007.axialcomp.pdf) Several of the margins for local and “semi-local” buckling are essentially equal to or greater than that for general buckling, and the general buckling margin is near zero (critical). In Case 2 a shell with a general buckling modal imperfection is optimized. The margins for the imperfect optimized shell are listed in Table 6, and those for the same optimum configuration but with the amplitude of the general buckling modal imperfection set equal to zero are listed in Table 7. In both Tables 6 and 7 of the paper, panda2.papers/2007.axialcomp.pdf, the margin for general buckling of the optimized imperfect shell is considerably higher than many of the margins corresponding to local and “semi-local” buckling. The general buckling margin of the optimized imperfect shell is well above zero (not critical). Why does this happen? The general buckling margin of optimized IMPERFECT stiffened shells is forced higher during optimization cycles because PREBUCKLING BENDING OF THE IMPERFECT SHELL increases with applied load approximately hyperbolically as the applied load approaches the general buckling load of the imperfect shell [1E]. If the general buckling load of the optimized imperfect shell were close to the design load, that is, if the general buckling margin were near zero (almost critical), there would be so much prebuckling bending near the design load that LOCAL STRESS AND BUCKLING MARGINS FOR THE STIFFENER PARTS AND FOR THE PANEL SKIN WOULD BECOME NEGATIVE BECAUSE THESE PARTS OF THE STRUCTURE WOULD BECOME HIGHLY STRESSED. A table and several figures from the 2007 AIAA Paper 2007-2216, April, 2007 follow. (See panda2.papers/2007.axialcomp.pdf for the complete paper.) Table 4 Optimum designs from PANDA2 suitable for analysis by STAGS (dimensions in inches)