This is Fig. 2 from the paper, "Optimization of an axially compressed ring and stringer stiffened cylindrical shell with a general buckling modal imperfection", by David Bushnell, AIAA 48th Structures, Structural Dynamics, and Materials Conference, Paper no. AIAA-2007-2216, 2007.

This slide shows the outer fiber effective stress (psi) at axial load, Nx= -3000 x 1.00516 lb/in. These results are from Case 4 in Table 4: no Koiter, yes change imperfection, ICONSV=1 . (See Slides 16-18 in the "Buckling Comments" slide show for an explantion of this jargon.)

Shown here is the nonlinear equilibrium state from STAGS at the load factor, PA= 1.00516. The imperfect shell has two initial buckling modal imperfection shapes:

1. a general buckling modal imperfection shown in Fig. 1a of the paper with amplitude, Wimp1=+0.0625 inch, and

2. a local buckling modal imperfection shown in Fig. 61 of the paper with amplitude, Wimp2= -0.0005 inch.

Prebuckling bending of the imperfect shell causes redistribution of stresses among the shell skin and the stiffener segments.

Also, prebuckling bending gives rise to “flattened” regions with an “effective” circumferential radius of curvature that is greater than the nominal curvature of the perfect shell and that therefore causes early general buckling. (See the right-most expanded insert).

This is a STAGS "compound" model. In a "compound" model the stringers are modeled as little flexible shell units over part of the circumference of the cylindrical shell and smeared out over the remainder of the circumference.

ABSTRACT from the paper from which this slide is taken: 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 halfwavelength 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) halfwaves 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 halfwavelength 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 postbuckling 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.

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