This technical note discusses the theoretical static buckling load of thin-walled plastic cylinders under axial and radial loading. While the buckling of thin-walled cylinders is a well-researched topic, a variety of new design and manufacturing methods, as well as advances in polymer technologies, have brought plastic structures into prominence in recent years, necessitating a re-examination and exploration of their behavior under buckling loads. This work provides a background review of the problem, a discussion of the appropriate buckling equations, an extensive case study to demonstrate the concepts, and a brief review of some previously developed thin-walled structure reinforcement techniques. The presented work and results are intended to provide a helpful perspective, background review, and starting place for future research on the buckling behavior of thin plastic structures. Keywords: Structural design, thin-walled structures, buckling failure, polymer testing 1. Introduction Thin-walled elements such as domes, shells, plates, and membranes are widely used in engineering design, as they allow the construction of strong but relatively light-weight structures in many applications [1, 2]. The most common applications are to the design of load-bearing structures, such as buildings subjected to high winds and earthquakes [3, 4], and aircraft [5, 6], where strength, exibility, and low mass are vital. Other applications include boats and ships, piping, pressure vessels, tanks, and machine components. While there are many advantages to employing thin-walled structures in design, including exibility and mass reduction, their use is not without complications and trade-o s. The mass of the parts must typically be balanced against the strength, resulting in a system that is optimal for both mass and strength, but not for each individually. For load-bearing thin-walled structures, the impact of buckling is of paramount importance, as it is often the primary limiting factor for the safe design load. Typically, the buckling load is signi cantly lower than the compressive yield strength of the material. Buckling failures can happen very suddenly, with catastrophic results, so stable design of the structures is very important to achieve [7]. Figure 1 gives some examples of useful thin-walled structures which may be made from plastic (polymer or polymer-composite) materials, including trusses, machine components, and aircraft parts. Special consid- eration must be taken when designing structural components made from plastic materials, as they typically have much lower compressive strength and toughness when compared to metals [8{10]. In spite of this, plastic materials are sometimes the best choice for a variety of reasons, such as manufacturing cost [11{13], electrical and thermal resistance [14{17], vibration absorption [18{20], and recycling needs [21{23]. This technical note is structured as a hybrid article-review paper, providing a view of the problem combined from a variety of sources and a case study completed by the authors. It will be presented in several sections, beginning with the introduction (Section 1), followed by a discussion of the loading state Rocha Pereira et al. Buckling Load of Thin Plastic Cylinders (a)(d) (c)(b) Figure 1: Examples of useful thin walled cylindrical and semi-cylindrical structures including (a) topologically-optimized trusses, (b) aircraft landing gear components, (c) machine structural components, and (d) aircraft skin panels and important buckling equations (Section 2), a nite element case study using polycarbonate cylinders under the loading discussed in Section 2 for several cases (Section 3), a discussion of some previous work to reinforce thin-walled structures (Section 4), and nally discussion and conclusions about the presented work (Section 5). 2. Thin Cylinder Buckling There are two basic modes of compressive loading for typical thin cylinders under which buckling may occur: Axial (such as column loading) and radial (such as pressure vessel loading) as shown in Figure 2. The axial loads may be uniformly applied (Figure 2a) or non-uniform (Figure 2b); practical examples of each are shown, where the wind turbine tower has an approximately uniform load and the trac sign/light pole has a highly biased loading. Axial loading may be applied along the length of the cylindrical shell for some angle +(Figure 2c), at one point or small area (Figure 2d), or uniformly along the entire surface. The case where load is applied at an angle is well-represented by a culvert under a road (Figure 2c), while the single location load case could be seen in a horizontal feed transfer system (Figure 2d) where the middle of the span experiences the most stress. Loads could certainly be applied at an angle or in the reverse direction for any of these cases, but this would tend to reduce the load into components [24] so the shown cases will generally be the most extreme. When designing structural members, it is often the best choice to select the worst expected case for analysis when this will not produce a large design cost in terms of weight, production time, or material cost [25, 26]. Assuming no signi cant material defects, the cylinders are assumed to deform relative to the applied load. In the case of the axial loading, it is assumed that the load will be applied around the entire rim of the cylinder, whether this is uniform or not and regardless of the force direction. Note that a realistic structure may have several of these modes simultaneously and therefore may be under bending or torsion loading as well; however, this is not considered here as this focus of this work is on understanding and describing the basic modes of buckling for these cylinders. For a generic smooth thin-walled cylinder in axial compression, structural collapse will take place at the point of buckling [27, 28], so the buckling load should be considered the ultimate strength of the structure. For the present study, the term \thin cylinder" is de ned to comprise a wall thickness such that, 20tD (1) wheretis the wall thickness and Dis the internal diameter of the cylinder. According to the NASA Space Vehicle Design Manual [27, 29], a reasonable buckling equation for a supported cylinder in axial compression is: Pcr=kx2S l2(2) Page 2 Rocha Pereira et al. Buckling Load of Thin Plastic Cylinders Inside diameter dWall thickness tlength lP σlongitudinal σhoop P (a) Inside diameter dWall thickness tlength lP(𝜽) σlongitudinal σhoopP (b) 𝜽𝑷𝝓 σlongitudinalσhoop P (c)σlongitudinalσhoop P (d)𝑷 𝝓 𝜹 Figure 2: (a) uniform axial load (e.g., wind turbine tower), (b) non-uniform axial load (e.g., trac light and sign pole), (c) angle-speci c uniform radial loading (e.g., drainage culvert), and (d) angle-speci c local radial loading (e.g., feed transfer system). wherePcrdescribes the critical load, Sdescribes the wall exure sti ness per unit width, ldescribes the length of the cylinder, and kxis the buckling coecient. The buckling coecient kxis de ned as, kx=m2(1 + 2)2+12 4 2Z2 m2(1 + 2)2(3) while the wall exure sti ness is, S=Et3 12(12)(4) where describes the buckling aspect ratio, mis the number of buckle half-waves, is the correlation factor, Zis the curvature parameter, Eis the elastic modulus of the material, tis the wall thickness, and is the Poisson ratio. The values of ,m, ,Z, andtare determined by the exact geometry of the structure; likewise, the values of Eandare determined by material choice. In the case of uniform loading, it is assumed that the single load will cause the buckling. This is not true for the non-uniform loading, as the structure will buckling at the point of highest loading; this will likely be at a much lower force than for the uniform loading due to the higher force being applied to a smaller area of the cylinder. For the axial buckling cases, the de ection of the cylinder wall in this con guration is given by [7, 30, 31]: w=w1sin2 v 2! (5) wherewis the de ection of the area under load, w1is the de ection amplitude, vis the variable of angle, andis the angle of the applied load. In the case of uniform pressure over the entire cylinder, will be 2 radians and will be = + (6) for the cases where the force is applied at an angle, both for cylinder-length loads and point loads. There are several methods for solving for the critical load under such conditions; the potential energy method [7, 30] best describes the e ects relative to an angle of load (i.e. a non-uniform loading pattern). For the case described, the potential energy  can be described as [7]:  =EI 2r3Z 0(w+ w)2d+Z 0N2 2EFrdZ 0Pcrwr(w+ w)2d (7) Page 3 Rocha Pereira et al. Buckling Load of Thin Plastic Cylinders whereNdescribes the hoop compressive force on the cylinder, Pcris the external pressure accumulated during the buckling process, ris the radius of the cylinder in the loaded region, EFis the tensile sti ness of the hoop, andEIis the exure sti ness. This equation should be solved relative the given boundary conditions and angles of force application. Note that the integration of force area would be used to distinguish between the cylinder-length and point-load cases. Most simple textbook solutions are formulated assuming a uniformly- applied pressure over the entire surface [0 ;2] of the cylinder and true analytical solutions are dicult to nd using these equations. Typically, nite element models [32{34] are used to nd the true buckling loads; the background and equations are reproduced in this section for completeness and to give the reader a good basis for understanding the physical mechanisms at play. 3. Nonlinear FEA Case Study To further explore the theoretical buckling behavior of polymeric thin-walled cylinders, a case study was completed using nite element analysis (FEA). This study was done in ve parts, corresponding to each of the loading cases shown in Figure 2, with Case (d) completed with two di erent boundary conditions. For all cases, a nonlinear buckling analysis was done using AutodeskR NastranR 2019 for the geometry shown in Figure 3a. A 2 mmquadrilateral shell mesh (Figure 3b) was used to model the 25 mmdiameter cylinders, with two wall thicknesses of 0 :5mmand 1:0mm. The material used for all cases was acrylonitrile butadiene styrene (ABS) and the analysis used the built-in AutodeskR material model (Table 1) at STP conditions. The material used was assumed to be isotropic (such as what would ideally be observed in molded ABS). 100 mm25 mm 25 mm0.5 mm shell 1.0 mm shell (a) (b) Figure 3: FEA cases: example contours Table 1: AutodeskR non-linear ABS material model properties Property ABS Units Elastic modulus 2.24 GPa Yield stress 20.0 MPa Yield strain 0.0089 mm=mm Ultimate strength 29.6 MPa Poisson's ratio 0.38 Material density 1050 kg=m3 First, the cylinder (de ned as previously described in Figure 3) was pinned at one end (no translation but rotation was allowed around all axes) and a load of 1 Napplied symmetrically to the other. Upon running the nonlinear buckling analysis, the eigenvalues for the rst three modes were collected and used to determine the critical load for the structure; these modes are shown in Figure 4. The rst mode for each is theoretically the most likely case, as it will fail with a signi cantly smaller load than the other modes; Page 4 Rocha Pereira et al. Buckling Load of Thin Plastic Cylinders however, small material defects and small variations in force angle for a real case or physical experiment may drive other modes, so it is important to consider several. The full analysis required approximately 30 seconds of time using a DellR InspironR desktop computer with an Intel i5-7400 processor (3.0 GHz) and 12 GB of RAM. In order to determine whether the predicted failure was due to elastic buckling or yielding and then plastic buckling, the yielding load was calculated using the cross sectional area of the cylinders (Figure 3a) and the given yield stress of 20 MPa (Table 1). The thinner cylinder was predicted to yield at a force of 801 N, while the thicker was calculated to require 1634 N. It was noted that the thin cylinder (wall-to-diameter ratio of 50, well within the limit set by Equation 1) failed under elastic buckling for all three modes, but the thicker walled cylinder (wall-to-diameter ratio of 25) yielded before buckling plastically. Table 2 gives the critical load and failure mode for each of the shown buckling modes. (a) (b)(c) (d) (e) (f)t= 0.5 mm Mode 1 Critical load = 681.1 Nt= 0.5 mm Mode 2 Critical load = 732.7 Nt= 0.5 mm Mode 3 Critical load = 749.6 N t= 1.0 mm Mode 1 Critical load = 2470 Nt= 1.0 mm Mode 2 Critical load = 3014 Nt= 1.0 mm Mode 3 Critical load = 3070 NABS, symmetric loading ABS, symmetric loading ABS, symmetric loading ABS, symmetric loading ABS, symmetric loading ABS, symmetric loading Figure 4: FEA models for symmetric axial loading cases. Shown are 0 :5mm wall (a) mode 1, (b) mode 2, (c) mode 3 and 1:0mm wall (d) mode 1, (e) mode 2, and (f) mode 3 Table 2: Buckling modes, observed critical loads, and failure modes for the symmetric axial loading case ModeCritical Load (N) and Failure Mode 0:5mm Wall Failure 1 :0mm Wall Failure 1 681.1 Elastic buckling 2470 Yield + plastic buckling 2 732.7 Elastic buckling 3014 Yield + plastic buckling 3 749.6 Elastic buckling 3070 Yield + plastic buckling This analysis was then repeated for the case where the load was asymmetric; all detailed were identical except all the of the load was applied to only 180of the cylinder edge. Figure 5 and Table 2 show the results for the rst three modes for each of the two wall thicknesses. Analysis time was slower for this case, but still required under one minute to complete. As in the symmetric case, the thin cylinder buckled elastically before the the yield point. With an asymmetric load, the rst two modes of the thicker cylinder was also predicted to fail by elastic buckling before the yield point of the material. Page 5 Rocha Pereira et al. Buckling Load of Thin Plastic Cylinders (a) (b)(c) (d) (e) (f)t= 0.5 mm Mode 1 Critical load = 387.3 Nt= 0.5 mm Mode 2 Critical load = 394.5 Nt= 0.5 mm Mode 3 Critical load = 465.4 N t= 1.0 mm Mode 1 Critical load = 1525 Nt= 1.0 mm Mode 2 Critical load = 1566 Nt= 1.0 mm Mode 3 Critical load = 2026 NABS, asymmetric loading ABS, asymmetric loading ABS, asymmetric loading ABS, asymmetric loading ABS, asymmetric loading ABS, asymmetric loading Figure 5: FEA models for symmetric axial loading cases. Shown are 0 :5mm wall (a) mode 1, (b) mode 2, (c) mode 3 and 1:0mm wall (d) mode 1, (e) mode 2, and (f) mode 3 Table 3: Buckling modes, observed critical loads, and failure modes for the asymmetric axial loading case ModeCritical Load (N) and Failure Mode 0:5mm Wall Failure 1 :0mm Wall Failure 1 387.3 Elastic buckling 1525 Elastic buckling 2 394.5 Elastic buckling 1566 Elastic buckling 3 465.4 Elastic buckling 2026 Yield + plastic buckling After completion of the axial loading cases, three radial loading cases were analyzed. These included (R1) the loading shown in Figure 2c with = = 60and applied the full length of the cylinder and (R2 and R3) the loading shown in Figure 2d with the load applied at the center of the cylinder over a 60angle and= 1mm. In the case of R1, the section de ned by was pinned, while R2 and R3 used pinned and xed constraints, respectively, applied to the ends of the cylinder. The results are shown in Table 4 for all three cases; analysis was completed identically to that of the axial cases, except that only one mode was calculated successfully for these loading cases. The buckling modes for each case are shown in Figure 6a and b (for R1), Figure 7a and b (for R2), and Figure 8a and b (for R3). Each of the analyses required about three minutes to complete using the same settings and assumptions as the axial loading cases. After completion of the buckling studies and calculation of the critical loads, a simple linear static analysis was done on each case to determine if the failure was due to elastic buckling or yielding and then plastic buckling; it was found that the stress in the material at the critical load was signi cantly higher than the yielding point for all cases, so it was found that all the radial loading cases experienced material yielding before buckling. These results of these analyses, including maximum stress values for each case, can be seen in Figure 6c and d (R1), Figure 7c and d (R2), and Figure 8c and d (R3). There was some di erence observed between the results of the pinned and xed cases for the local radial loading, with the xed ends having a slightly higher critical load. Page 6 Rocha Pereira et al. Buckling Load of Thin Plastic Cylinders t= 0.5 mm Mode 1 Critical load = 98.6 N(a) (b) (c) (d)t= 1.0 mm Mode 1 Critical load = 774.4 N ABS, 100 mm/60 °loading, pinned bottom BC ABS, 100 mm/60 °loading, pinned bottom BC Figure 6: Buckling modes for the (a) thinner and (b) thicker-walled cylinders under the R1 radial loading case, with simple linear static analysis completed at the buckling load of the (c) thin and (d) thick cases to determine if the failure was due to elastic buckling or yielding and then plastic buckling t= 0.5 mm Mode 1 Critical load = 229.6 N(a) (b) (c) (d)t= 1.0 mm Mode 1 Critical load = 1747 N ABS, 1 mm/60 °loading, pinned ends BC ABS, 1 mm/60 °loading, pinned ends BC Figure 7: Buckling modes for the (a) thinner and (b) thicker-walled cylinders under the R2 radial loading case, with simple linear static analysis completed at the buckling load of the (c) thin and (d) thick cases to determine if the failure was due to elastic buckling or yielding and then plastic buckling Page 7 Rocha Pereira et al. Buckling Load of Thin Plastic Cylinders t= 0.5 mm Mode 1 Critical load = 242.6 N(a) (b) (c) (d)t= 1.0 mm Mode 1 Critical load = 1832 N ABS, 1 mm/60 °loading, fixed ends BC ABS, 1 mm/60 °loading, fixed ends BC Figure 8: Buckling modes for the (a) thinner and (b) thicker-walled cylinders under the R3 radial loading case, with simple linear static analysis completed at the buckling load of the (c) thin and (d) thick cases to determine if the failure was due to elastic buckling or yielding and then plastic buckling Table 4: Buckling modes, observed critical loads, and failure modes for the three radial loading cases Radial Loading CaseCritical Load (N) and Failure Mode 0:5mm Wall Failure 1 :0mm Wall Failure R1 98.6 Yield + plastic buckling 774.4 Yield + plastic buckling R2 229.6 Yield + plastic buckling 1747 Yield + plastic buckling R3 242.6 Yield + plastic buckling 1832 Yield + plastic buckling 4. Thin Cylinder Reinforcement One of the many solutions o ered to improve the strength and buckling resistance of thin-walled parts is to ll them with expanding foam or composite material; if done properly, this can result in a part that is signi cantly stronger than an un lled part, but that does not involve a signi cant increase in mass. Often, the structure can be made even lighter, as the addition of foam can allow the reduction of the wall thickness without sacri cing performance [35]. Foam- lled components are also more vibration-resistant and therefore often have longer fatigue lives due to improved energy absorbency by the foam [36]. Several studies [37{43] have explored the production and use of polymer and metal foams for structural applications. In particular, expanding polyurethane foam has been widely used to strengthen static structural components to address a variety of interesting problems. Heim et al [44] combining plastic components and foams to reduce the mass of vehicles, while increasing the safety, sti ness, and crushing resistance. Ashra et al [45], Tuwair et al [46], and Boccaccio et al [6] explored the use of foam to sti en and strengthen honeycomb panels, particularly for aerospace applications. Several studies were also performed using polyurethane foam to strengthen various types of conical and cylindrical tubes, including those made from aluminum [47, 48], brass [49], and steel [45, 50]. Surprisingly, no previous studies were found that examined the foam lling and strength testing of structures made from plastics, even though many thin-walled plastic structures are manufactured and used. Page 8 Rocha Pereira et al. Buckling Load of Thin Plastic Cylinders 5. Summary, Conclusions, and Future Work This work explored the buckling behavior of thin-walled polymeric cylinders, rst reviewing some liter- ature to provide motivation for the importance of the problem, then examining several loading cases (with practical examples) and their basic equations. A case study was performed to further illuminate the behavior of these structures, demonstrating some interesting behavior. Finally, a brief review was completed to ex- plore some of the previous work on strengthening thin-walled structures against crushing and buckling. This paper is meant to provide background and motivation for future work in this area, especially on providing a good perspective for experimental validation of the behavior. The buckling of thin polymeric material structures, especially structural members (not lms), is a topic that has not been explored extensively in the literature this far and needs more attention. The background review and case study from this work provided several interesting conclusions, all of which will be explored in future work. These include: Very little previous work has been done to study the buckling behavior of thin-walled cylindrical structures made from polymeric and polymer-composite materials, even though these are becoming more widely used in engineering design. The FEA results implied that the de nition of thin-walled as being a wall thickness under 1 =20 of the diameter may be too liberal for polymeric materials. This conclusion was driven by the fact that most of the 1 =25 wall thickness produced yielding in the material before buckling, while the thinner one reliability produced elastic buckling. Perhaps it is appropriate to de ne thin-walled structures by their ability to elastically buckle, not by a speci c wall thickness. This would be material-dependent and requires further study. The use of a shell model with quadrilateral mesh elements for the FEA was a good choice for these kinds of thin structures, as all the nonlinear buckling solutions converged reliably and quickly without a single failure. When examining these structures, it should be noted that the axial loading cases had several modes (at least three) each but the radial loading cases only had a single buckling mode. The impact of small material defects and microstructure discontinuities in the polymer material needs to be explored experimentally, as these may cause elastic buckling even of the yielding cases found here. The impact of anisotropy (especially from 3-D printing) needs to be examined with respect to buckling; this is a topic very little discussed in the literature, whether for molded or additively built materials