Buckling analysis and design of anisogrid composite lattice conical shells

Composite Structures 93 (2011) 3150–3162

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Composite Structures

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Buckling analysis and design of anisogrid composite lattice conical shells

E.V. Morozov a,⇑, A.V. Lopatin b, V.A. Nesterov b

a School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra, Australiab Department of Aerospace Engineering, Siberian State Aerospace University, Krasnoyarsk, Russia

a r t i c l e i n f o

Article history:Available online 2 July 2011

Keywords:Anisogrid conical shellsComposite lattice structuresBuckling analysisFinite-element modellingSpacecraft adapter

0263-8223/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compstruct.2011.06.015

⇑ Corresponding author. Tel.: +61 2 6268 9542; faxE-mail address: [email protected] (E.V. Moro

a b s t r a c t

Composite lattice anisogrid shells have now become a popular choice in many aerospace applications.Their use in various structural components, such as rocket interstages, payload adapters for spacecraftlaunchers, fuselage components for aerial vehicles, and parts of the deployable space antennas requiresthe development of more advanced finite-element models and analysis techniques capable of predictingbuckling behaviour of these structures under variety of loadings. A specialised finite-element model gen-eration procedure (design modeller) is developed and applied to the buckling analysis of the compositeanisogrid conical shells treated as three-dimensional frames composed of the curvilinear ribs made ofunidirectional composite material. Featuring a dedicated control procedure for positioning the beam ele-ments, the design modeller enables a close approximation of the original twisted geometry of the curvi-linear ribs. The parametric finite-element buckling analyses of the anisogrid conical shells subjected toaxial compression, transverse bending, pure bending, and torsion showed the robustness and potentialof the modelling approach. It was demonstrated that the buckling resistance can be significantlyenhanced by either increasing the stiffness of a few hoop ribs located in the close proximity to the sectionwith the larger diameter, or by introducing the additional hoop ribs in the same part of the conical shell.The effectiveness of the design analyses is demonstrated using particular examples. It has been shownthat the resultant optimised designs can produce up to 22% mass savings in comparison with the non-optimised lattice shells.

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1. Introduction

Anisogrid (anisotropic grid) composite lattice conical shells arecomposed of curvilinear helical and circumferential (hoop) ribsmade of a unidirectional composite material having high specificstrength and stiffness. In these structures, the ribs are the mainload-bearing members that provide both the high membrane andbending stiffnesses of the shell structure. The most advanced andwidely used method of fabrication of the lattice anisogrid shellsis a continuous filament winding. In this process, continuous fibretows impregnated with resin are placed into the grooves formed inthe elastic silicone rubber coating of the conical mandrel.

Anisogrid composite shells are highly efficient and extensivelyused in various structural applications, such as rocket interstages,payload adapters for spacecraft launchers, fuselage componentsfor aerial vehicles, and components of the deployable spaceantennas.

The history of the lattice structures development, review of thefounding studies, and analysis of design approaches and fabrica-tion techniques are given by Vasiliev et al. [1–3].

ll rights reserved.

: +61 2 6268 8276.zov).

One of the important applications of the composite conicalanisogrid shells is in the design of the payload adapters for space-craft launchers (see Fig. 1a and b). These structures should providethe necessary structural mechanical interface between the space-craft and launch vehicle. The high intensity loadings are exertedon them at the orbital injection leg. They normally include axialcompression, transverse and pure bending, and torsion. Bucklingunder these loadings is one of the typical modes of failure of theseshells.

Buckling of the cylindrical anisogrid composite lattice shells hasbeen studied by many researchers. Various approaches have beendeveloped based on the use of continuum models that employ var-ious modifications of rib smearing techniques (see e.g. [1,2]). Inthis case, the lattice structure is modelled as a shell having theaveraged (smeared) stiffnesses. The corresponding stiffness coeffi-cients are calculated using the rib smearing techniques and dependon the number and orientation of the ribs and their stiffnesses.However, the applications of these methods to the buckling analy-sis of the conical lattice shells are rather limited. This is, appar-ently, due to complications in the modelling caused by variablegeometry and stiffness of the shell along the meridian. The stiffnessparameters of the composite lattice conical shell depend on thedistance between the ribs which is decreasing along the shell

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Fig. 1. Anisogrid composite conical payload adapter. (Courtesy of ISS-Reshetnev Company).

E.V. Morozov et al. / Composite Structures 93 (2011) 3150–3162 3151

meridian starting from the section with the large diameter to thesmaller diameter end. This results in the necessity of solving thesystem of differential equations with variable coefficients which,in turn, requires the use of numerical methods. Efficient bucklinganalyses of the composite cylindrical lattice shells can be per-formed using finite-element modelling. A review of the bucklingsolutions obtained for the cylindrical lattice shells using contin-uum models as well as finite-element methods can be found inthe authors’ article [4].

As for the buckling analysis of the anisogrid composite conicalshells, very few publications can be found. The first finite-elementmodelling and buckling analysis of the conical lattice shell sub-jected to axial loading has been undertaken by Hou and Gramoll[5]. The numerical solution was obtained using COSMOS/M. How-ever, the curvatures of the helical and hoop ribs were neglected intheir finite-element model and the actual curvilinear ribs were re-placed by the straight segments located between the rib intersec-tions in the lattice structure.

Buckling analysis of the anisogrid conical spacecraft adaptersubjected to combined (axial compression and bending) loadinghas been performed by Vasiliev et al. [6].

It should be noted, that the finite element models of latticestructures based on the shell or solid elements are usually exces-sively large and require substantial computational resources. Thishampers their efficient use in design analyses and solutions of opti-misation problems.

The use of the methodology and analytical tool based on the fi-nite element discrete modelling of the cylindrical lattice shells pre-sented in the authors’ article [4] is extended in this paper to thebuckling analysis of anisogrid composite lattice conical shells.The dedicated design modeller is developed and applied to the

modelling and buckling analysis of the shells subjected to axialcompression, transverse and pure bending, and torsion.

The conical lattice shells are modelled as three-dimensionalframe structures composed of the curvilinear ribs subjected to ten-sion–compression, bending in two planes, and twisting. Usinginternal coding facilities of COSMOS/M [7], the model generatorprogram has been developed to enable the geometry and finite-element model of the lattice shell to be automatically created.The generator/design modeller is capable of modelling the originalgeometry of the naturally twisted shapes of the helical ribs com-posing the lattice structure with maximum accuracy and authen-ticity. An additional advantage is that the size of the generatedfinite element model does not impede the efficiency of the compu-tational process even for the lattice shells having a large number ofribs. The effects of the lattice structural parameters (such as theribs’ orientation and their cross-sectional dimensions) on the crit-ical loads and buckling modes have been studied for the conicalshells composed of the ribs made of the unidirectional carbon fibrereinforced plastic.

For the shells under consideration, the buckling loads are af-fected by a number of structural parameters (such as a numberof ribs, ribs’ cross-sectional dimensions, ribs’ orientation andspacing) that should be identified at the design stage. The auto-mated model generation procedure presented in this work facil-itates multiple modifications of the finite-element models andsubsequent buckling analyses that allow the influence of thestructural parameters on the value of critical load to be effi-ciently and accurately assessed. The effectiveness of the model-ling approach in the structural design is demonstrated usingparticular examples. The structural parameters providingweight-efficient designs subject to constraints imposed on the

Fig. 3. Shell geometry and dimensions.

Fig. 4. Typical elementary unit (a) of the lattice structure (b).

Fig. 5. Finite element BEAM3D.

3152 E.V. Morozov et al. / Composite Structures 93 (2011) 3150–3162

critical loads under axial loading have been calculated using theproposed methodology.

2. Finite-element modelling

Consider a truncated composite lattice conical shell composedof two symmetric systems of filament wound helical and circum-ferential (hoop) ribs as shown in Fig. 2. The helical ribs are placedalong the geodesic trajectories making angles ±/ with the cone’sgenerator line (see Fig. 3). The hoop ribs are passing through themiddle points of the segments of helical ribs situated betweentheir intersections creating anisogrid lattice structure of the shell.The set of input data for the finite-element modelling includes:the height of the shell, H, diameters of the top and bottom sections,D1 and D2, number of the helical ribs of one direction (either +/, or�/), n and the angle of the rib orientation at the bottom section ofthe shell /2, (see Fig. 3). The finite-element model of the shell isgenerated using a typical elementary unit of the grid structurecomposed of two segments of the helical ribs AB and BC and twosegments of the hoop ribs ED and FG as shown in Fig. 4a and b.The lattice shell is modelled as a three-dimensional frame struc-ture using the BEAM3D element available within COSMOS/M [7].The element has an h � b rectangular cross-section and is referredto the local coordinate frame xyz as shown in Fig. 5.

Each of the element nodes, P1 and P2 is characterised by six de-grees of freedom, i.e. by three linear displacements and three rota-tions. The orientation of the element within the global coordinateframe XYZ is performed using the key point P3 which is situatedin the local plane xy, so a triangle P1P2P3 is generated as shownin Fig. 5. Changing the location of the point P3 within the frameXYZ, the required orientation of the beam element can be achieved.

The typical elementary units of the lattice structure (see Fig. 4a)are meshed as shown in Fig. 6. All the beam elements of the unitABED have the common reference key point P3ED whereas all theelements of the unit BCFG have the common reference key pointP3FG. Both points P3ED and P3FG are located at the intersections ofthe cone axis, Y with the planes of the hoop ribs ED and FG, respec-tively (see Fig. 6). The selected positions of the key points allow thereal geometry of the lattice shell composed of the twisted helicalribs to be reliably reflected in the model. The actual constructionof the finite-element mesh is performed starting from the unit lo-cated at the bottom of the shell (see Fig. 7). The meshing is thencarried on for the units located on top of each other along the conegenerator line A1Bi. As shown in Fig. 7, i is the number of the last(top one) unit in this sequence. As mentioned before, the helicalribs follow the geodesic trajectories which are defined by Clairaut’slaw: r(Y) � sin /(Y) = const = (D2/2) sin /2, where r(Y) is the current

Fig. 2. Conical anisogrid shell.

Fig. 6. Meshing the typical elementary unit.

radius of the cone [8]. The helical angle /(Y) of the rib orientationchanges according to this law from the bottom section to the topone. This change is taken into account when constructing the fi-nite-element model from the elementary units as shown inFig. 7. The meshed column group of these units shown in Fig. 7 isthen mirror copied with respect to the cone generator line A1Bi cre-ating the basic finite-element cell of the lattice structure shown in

Fig. 7. Meshed typical elements placed along the cone generator.

Fig. 8. Basic finite-element cell of the lattice structure.


Fig. 8. Using copying and rotation of this cell around the cone axis,the complete finite-element model of the lattice conical shell isautomatically generated. Once this procedure is complete, themeshed top and bottom end rings are added to the model. Inputof the geometry parameters and material properties of the ribscompletes the construction of the finite-element model.

The aforementioned algorithm has been implemented in theform of the finite-element model generator program specificallydeveloped for the problem under consideration. The generatorwas created using the internal COSMOS/M code [7] and has beenapplied to the automated construction of the finite-element mod-els of conical lattice shells having various structural parameters.

3. Numerical analysis

The buckling analyses have been performed for the filamentwound composite conical lattice shells subjected to axial compres-sion, transverse loading, pure bending and torsion. The dimensionsof the shells under consideration are: H = 0.6 m, D1 = 0.8 m,D2 = 1.6 m. The shells are made of unidirectional carbon-fibre rein-forced plastic (CFRP) having the following properties: modulus ofelasticity, Ex = 100 GPa, shear moduli, Gxy = Gxz = 5.5 GPa, -Gyz = 2.5 GPa, and density, q = 1550 kg/m3. The number of helicalribs of one helical direction (either +/, or �/) is n = 60. The analy-ses have been performed for the shells having the following helicalangles of the rib orientation at the bottom section /2 = 5�, 10�, 15�,20� and 25�. The corresponding finite-element models are shownin Fig. 9. As can be seen, the highest density of the lattice structureis achieved for the shell with the angle /2 = 25�. This is expectedsince the increase in helical angle for the shell composed of oneand the same number of helical ribs leads to the increase in thenumber of their intersections. As a result, more hoop ribs are addedto construct the anisogrid lattice structure of the shell.

It is assumed, that all the ribs have the cross-sectional area of 16mm2, whereas the height-to-width ratio h/b (see Fig. 5) can vary.The following three combinations were considered in the analyses:(1) h = 2 mm, b = 8 mm, h/b = 0.25; (2) h = 4 mm, b = 4 mm, h/b = 1;(3) h = 8 mm, b = 2 mm, h/b = 4. The corresponding fragments ofthe lattice structures are shown in Fig. 10.

Thus, the parametric analyses have been performed for the var-ious angles /2 and ribs’cross-sections characterised by the differ-ent values of the ratio h/b. The latter parameter affects not onlythe ability of the shell to resist buckling loads, but also may

influence the cost of the shell fabrication. The ratio h/b would affectthe cost and complexity of the manufacturing of the elastic siliconerubber matrices containing the helical grooves in which the contin-uous fibre tows are placed by winding machine. The lesser value ofh/b reduces the cost of the fabrication of these matrices. Besides,the removal process of the wound lattice shell becomes rathercomplicated when the helical grooves are too deep (h/b > 1).

The weight efficiencies of the above mentioned shell designshave been estimated. The results have been used in the compara-tive analysis. In each case, the mass of a lattice shell was calculatedconcurrently with the finite-model generation. The results are pre-sented in Table 1. Clearly, the mass of the shell structure does notdepend on the ratio h/b for given /2 since the cross-sectional areaof the ribs is kept constant for all the shells.

The lattice shell structures are modelled using the beam finiteelements with the length ranging from 3.5 to 20 mm depending onthe angle /2. For instance, the lattice structure of the shell havingthe angle /2 = 10� has been modelled with 10800 10 to 20-mm longfinite elements. The finite-element model of the lattice shell having/2 = 25� has been built of 25200 3.5 to 15-mm long finite elements.

Buckling analyses have been performed for the conical shellssubjected to loadings that are deemed to be most appropriate forpractice: axial compression, transverse loading, pure bending andtorsion (see Fig. 11).

3.1. Axial compression

Consider buckling of the conical lattice shell subjected to an ax-ial compressive load as shown in Fig. 11a. The bottom end of theshell is fully clamped and the compressive force P is appliedthrough the rigid ring attached to the top section. The shell bucklesunder the critical load P = Pcr. The critical loads Pcr(h/b, /2) obtainedfrom the finite-element buckling analyses of the shells having var-ious ratios h/b and angles of the helical ribs’ orientation at the bot-tom section, /2 are shown in Table 2. Analysis of these resultsyields certain conclusions with regard to the effects of varyingthe parameter h/b and angle, /2 on the magnitude of buckling loadPcr. It follows from Table 2 that the increase in the values of angle/2 and ratio h/b leads to substantial increase in the magnitude ofcritical load Pcr. As expected, the shells with the highest ratio h/b(h = 8 mm and b = 2 mm) are the most resistant to buckling.

As for the buckling modes, it was found that the shells underconsideration buckle in one of the two configurations. The firstmode is characterised by formation of a few buckle waves in thearea adjacent to the bottom section which is typical for the axiallyloaded conventional homogeneous conical shells. The shells withh/b = 0.25 and 1 buckle in this mode (see Fig. 12). The number ofbuckling waves depends on the angle /2. For example, the shellsbuckle in the mode having four waves along the circumferencefor /2 = 5�. If the angle /2 = 25�, then the shell with h/b = 0.25 buck-les in the mode with eight waves and the buckling mode shape forthe shell with h/b = 1 has seven waves evenly distributed along thecircumference.

For certain combinations of the helical angles and dimensions hand b, shells buckle in the second mode characterised by the localbuckling of ribs (see Fig. 13). This is typical for the lattice shellshaving h/b = 4 and the angles /2 = 5�, 10�, 15� and 20�. As shownin Fig. 13, the segments of the helical ribs located in the vicinityof the bottom section buckle as the beams in the plane of the rib’sminimal bending stiffness (see Fig. 10c). However, the shell with/2 = 25� and h/b = 4 buckles in the first (shell-like) mode with fivewaves distributed in hoop direction in the area adjacent to the bot-tom section. Note that irrespective of the mode, all the shellsbuckle in the vicinity of the bottom end. This is caused by the re-duced stiffness of the shell due to relatively sparse lattice structurein this part of the cone.

Fig. 9. Finite-element models of the lattice shells with different orientation of helical ribs.

Fig. 10. Fragments of the lattice structures.


It is clear that the resistance to buckling can be improved byincreasing the stiffness of the structure in the area close to the bot-

tom end of the shell. One of the possible ways of doing this is dem-onstrated for the shell with /2 = 15�, h = 8 mm and b = 2 mm. It

Table 1Mass of the lattice shell as a function of angle /

�

2.

/�

25 10 15 20 25

m(kg) 2.908 2.263 3.320 3.792 4.567

Table 2Buckling loads Pcr (kN) (axial compression).

/�

2h = 2 mm, b = 8 mm h = 4 mm, b = 4 mm h = 8 mm, b = 2 mm

5 13.863 34.416 44.33410 32.969 70.541 110.00215 60.601 109.475 179.71420 96.801 150.397 252.12325 139.542 192.213 316.574


follows from Tables 1 and 2 that the critical load for this shell isPcr = 179.714 kN and the mass of the structure is m = 3.32 kg. Asmentioned above, this shell buckles in the ribs’ local bucklingmode. If the widths of the two closest to the bottom section hoopribs are increased by factor of two, i.e. b = 4 mm (see Fig. 14), thenthe corresponding critical load obtained from finite-element anal-ysis is equal to Pcr = 230.507 kN, which is by 28% higher than theoriginal design. The mass of the new structure is increased by only7%. Note, that the new shell having the two hoop ribs reinforced asabove buckles in different, shell-like mode with five waves in thehoop direction in the area adjacent to the bottom section.

Another way of the improvement of the shell design involvesintroduction of the additional hoop ribs of the same cross-sectionas the original ones. This will lead to the increase in the stiffnessof the shell and, subsequently to the increase in critical load.

It is important to note that the both above mentioned tech-niques can be implemented in practice with relative ease as faras the respective fabrication methods are concerned.

One of the common design requirements is the mass or weightefficiency of the lattice structure. The suitable criterion, reflectingthis requirement, can be adopted in the form of the following ratio:Kp = Pcr/mg, in which m is the mass of the shell and g = 9.81 m/sec2.Using data shown in Tables 1 and 2, the values of Kp have been cal-culated and are shown in Table 3. It was found that the ratio Kp in-creases with the increase in both, angle /2 and ratio h/b. Note thatthe values of the above criterion can be used when selecting the

Fig. 11. Lattice conical shells (meshes) under: (a) axial compressi

design parameters of the lattice shells subject to the constraintsimposed on the critical load and on the cost of the structure.

3.2. Transverse bending

Consider buckling of the conical lattice shell subjected to trans-verse bending. The transverse load Q is applied to the rigid ring at-tached to the top section as shown in Fig. 11b. The bottom end ofthe shell is fully clamped.

The buckling analyses of the shells having the angles /2 = 5�,10�, 15�, 20� and 25� and dimensions of the ribs’ cross-sectionschanging within the same ranges as in the previous case of axialcompression yield the values of the critical load Qcr presented inTable 4. Analysis of these results shows that the increase in the an-gle /2 leads to a substantial increase in the critical buckling load. Itfollows from Table 4 that the value of the critical force for the shellcomposed of the ribs with h = 2 mm and b = 8 mm with the angle/2 = 25� is 18.9 times higher in comparison with the shell having/2 = 5� and the ribs of the same cross-section. If the shells are madeof the ribs with the square cross-sections (h = b = 4 mm), then thecritical load for the shell having /2 = 25� is 7.8 times higher thanthat for the shell wound with /2 = 5�. The highest increase, i.e. bythe factor of 33.4, is observed for the shells composed of the ribs

on, (b) transverse bending, (c) pure bending, and (d) torsion.

Fig. 12. Shell-like buckling mode under compression (/2 = 15�, h = b = 4 mm).

Fig. 13. Local buckling of ribs for the shell under compression (/2 = 15�, h = 8 mm, b = 2 mm).


having h = 8 mm and b = 2 mm. Also, it can be seen (see Table 4)that the critical load depends on the ratio h/b. The lowest bucklingresistance is found to be for the shells with h/b = 0.25. The increaseof h/b from 0.25 to 1.0 leads to the increase in Qcr for all the angles/2. However, for the grid structures with h/b = 4 and /2 = 5� and10� some reduction of the critical load has been observed in com-parison with the shells having h/b = 1. It follows from the Table 4that the critical load for the shell with h/b = 4 becomes greater thanthat for the shell having h/b = 1 if the angle /2 = 15�.

The corresponding buckling modes are determined by the pres-ence of the areas of compression and shear. It has been shown thatthe lattice shells under consideration can buckle in either of thetwo modes: the shell-like mode or the ribs’ local buckling modedepending on the angle /2 and ratio h/b. The typical shell-likebuckling mode is shown in Fig. 15 for the shell with /2 = 15�and

h = b = 4 mm. As can be seen, the buckling pattern is characterisedby a few lobes oriented at certain angles to the direction of theshear load applied to the shell. This buckling mode is realised forthe lattice shells with h/b = 0.25 and 1 for all the angles /2. Thestructures having h/b = 4 and /2 = 5�, 10�, 15�, 20� buckle in theribs’ local buckling mode. This is illustrated in Fig. 16 for the shellhaving /2 = 15�, h = 8 mm and b = 2 mm. As can be seen, the seg-ments of the helical ribs at the bottom section buckle as the beamsin the planes of their minimal bending stiffness.

Note, that the lattice shell with /2 = 25� and h/b = 4 buckles inthe shell-like mode similar to that shown in Fig. 15.

Clearly, the buckling resistance of the lattice shells can be in-creased to certain extent by the increase in the stiffness of somehoop ribs. This has been demonstrated for the shell having/2 = 15�, h = 8 mm and b = 2 mm. For this shell, the corresponding

Fig. 14. Lattice shell with the two reinforced hoop ribs.

Table 3Parameter Kp for the shell under axial compression.

/�


5 485.96 1206.44 1554.1110 1029.96 2203.72 3436.4915 1860.67 3361.33 5517.9520 2602.25 4043.04 6777.6825 3114.64 4290.28 7066.07

Table 4Buckling loads Qcr (kN) (transverse bending).

/�


5 5.066 15.908 6.88310 21.732 41.429 36.14415 42.139 65.003 87.76220 66.842 92.868 166.26525 95.805 123.492 229.752

Fig. 15. Shell-like buckling mode under tran


buckling critical load Qcr = 87.762 kN (see Table 4). However, if thewidths of the two, closest to the bottom section, hoop ribs are in-creased from 2 mm to 4 mm, then the corresponding calculatedcritical load becomes Qcr = 98.256 kN, which is higher by the factorof 1.12.

The mass efficiency of the shells has been estimated using thefollowing criterion: KQ = Qcr/mg using the data presented in Tables1 and 4. The results of the calculations are shown in Table 5. It fol-lows from this analysis that for the angles /2 = 5�, 10� the shellshaving h = b = 4 mm represent the most efficient structures. Forthe angles /2 = 15�, 20� and 25� the maximum values of the coeffi-cient KQ are achieved for the shells with h = 8 mm and b = 2 mm.

3.3. Pure bending

The shell subjected to pure bending is shown in Fig. 11c. Thebending moment, M acting in the diametral plane is applied tothe rigid ring attached to the top end. The bottom end of the shellis fully clamped. The buckling analyses have been performed forthe same sets of angles /2 and cross-sectional dimensions of theribs as in the cases of axial and transverse loadings. The resultsof calculations are presented in Table 6. It follows from these datathat the critical moment Mcr increases for all the combinations of hand b with the increase in the value of angle /2 from 5� to 25�. Themost intensive growth has been observed for the shell composed ofthe ribs with h = 8 mm and b = 2 mm. The slowest rate of increasein critical load was registered for the shells made of the ribs havingthe square cross-sections.

The value of critical moment also depends on the ratio h/b. Itcan be found (see Table 6) that for the angles /2 = 15�, 20�, and25� the critical moment increases with the increase in h/b from0.25 to 4. For the angles /2 = 5� 10�, a maximum value of Mcr couldbe achieved if the lattice structure is composed of the ribs with thesquare cross-section (h = b = 4 mm).

Buckling modes depend on the angle /2 and ratio h/b. The shell-like buckling mode is shown in Fig. 17. This mode is characterisedby the presence of a few buckling lobes located symmetrically withregard to the plane of the moment application in the compressedzone of the shell. The shell-like buckling mode takes place if h/b = 0.25 and /2 = 10�, 15�, 20� and 25�. The lattice structures,

sverse bending (/2 = 15�, h = b = 4 mm).

Fig. 16. Local buckling of ribs for the shell under transverse bending (/2 = 15�, h = 8 mm, b = 2 mm).

Table 5Parameter KQ for the shell under transverse bending.

/�


5 177.59 557.65 241.2810 678.91 1294.25 1129.1515 1293.84 1995.85 2694.6520 1796.88 2496.52 4469.6125 2138.41 2756.39 5128.16

Table 6Buckling loads Mcr (kN m) (pure bending).

/�


5 2.618 8.089 3.55210 11.077 20.455 19.19315 21.236 33.310 50.52120 34.141 47.662 85.76025 49.619 63.411 111.069


consisting of the ribs with the square cross-sections (h/b = 1),buckle in the same mode as conventional homogeneous shells forthe angles /2 = 5�, 10�, 15�, 20� and 25�. This also holds for theshells with h/b = 4 if the angles /2 = 20�and 25�.

The shells having h/b = 0.25 and /2 = 5 buckle in the secondmode, i.e. the local buckling of ribs, and so do the shells with h/b = 4 and /2 = 5�, 10�, 15� (see Fig. 18). In these cases, the segmentsof the ribs located in the two symmetrically placed (with regard tothe plane of action of bending moment) areas of the shell sectionbuckle as the beams in the planes of their minimal bendingstiffness.

As in the cases of axial compression and transverse bending, thebuckling resistance of the shell subjected to pure bending can beimproved if some hoop ribs are replaced with those having higherbending stiffness. Consider the shell with /2 = 15� and h = 8 mmand b = 2 mm. For this shell, the corresponding buckling criticalmoment Mcr = 50.521 kN m (see Table 6). Increasing the widthsof the two, closest to the bottom section, hoop ribs from 2 mm to4 mm, the critical value of bending moment can be improved by12% reaching the value of 56.650 kN m.

The mass efficiency of the lattice structure has been estimatedusing the following criterion: KM = Mcr/mg. The values of KM areshown in Table 7. It can be seen, that the shells having /2 = 5�,10� and composed of the ribs with the square cross-section arethe most efficient. The coefficients KM have the maximum valuesfor the shells made of the ribs having h = 8 mm and b = 2 mmand wound with the angles /2 = 15�, 20� and 25�.

3.4. Torsion

Buckling under torsion has been analysed for the lattice shellsloaded by the torque T as shown in Fig. 11 (d). The results of calcu-lations of the critical torques Tcr obtained from the finite-elementbuckling analyses performed for the shells having various angles/2 and dimensions of the ribs’ cross-sections h and b are presentedin Table 8.

As can be seen, the values of critical torques increase with theincrease in the values of angles /2. For example, the ratio Tcr

(/2 = 25�)/Tcr (/2 = 5�) takes the following values: 23.13 forh = 2 mm and b = 8 mm; 8.32 for h = 4 mm and b = 4 mm, and36.19 for h = 8 mm and b = 2 mm. Clearly, the critical torque alsodepends on the size of the ribs’ cross-sections (h and b). It followsfrom the data shown in Table 8 that the lattice shell made of theribs having h = 2 mm and b = 8 mm has the least resistance tobuckling under torsion. The magnitude of the critical load increasesfor the shells composed of the ribs with the square cross-sections(4 � 4 mm). The structure made of the ribs with h = 8 mm andb = 2 mm has the maximum critical buckling load for the angles/2 = 15�, 20� and 25�. However, if the angles /2 = 5�, 10�, thenthe critical load values are reduced below those found for the shellcomposed of the 4 � 4 mm ribs.

The shells can buckle in either shell-like or ribs’ local buck-ling modes. The typical example of the former is shown inFig. 19. As can be seen, the buckling pattern is characterisedby the presence of the helical buckling waves stretching fromthe top to the bottom sections of the shell. This type of bucklingmode is typical for the shells having the ribs with h = 2 mm andb = 8 mm, and h = 4 mm and b = 4 mm for all the angles /2 con-sidered in this work. The same holds for the lattice shells withh = 8 mm and b = 2 mm and /2 = 20� and 25�. The number of

Fig. 17. Shell-like buckling mode under pure bending (/2 = 15�, h = b = 4 mm).

Fig. 18. Local buckling of ribs for the shell under pure bending (/2 = 15�, h = 8 mm, b = 2 mm).

Table 7Parameter KM (m) for the shell under pure bending.

/�


5 91.77 283.56 124.5110 346.05 639.02 599.5915 652.03 1022.75 1551.9920 917.79 1281.27 2305.4425 1107.52 1415.36 2479.11

Table 8Buckling loads Tcr (kN m) (torsion).

/�


5 4.076 12.439 5.45810 20.289 34.668 30.25715 39.574 52.612 76.08320 64.071 76.816 146.64025 94.260 103.475 197.499


the helical buckling waves depends on the angle /2 and the ratioh/b. For example, for the lattice structure with h/b = 0.25 thenumber of waves changes from five, for the shell wound with/2 = 5�, to ten for the shell having /2 = 25�. If the shell is com-

posed of the ribs with square cross-sections (h/b = 1), the shellbuckles in the mode in which six waves are generated for/2 = 5�. The same shell having the helical ribs wound with/2 = 25� buckles with nine helical waves evenly distributed in

Fig. 19. Shell-like buckling mode under torsion (/2 = 15�, h = b = 4 mm).

Fig. 20. Local buckling of ribs for the shell under torsion (/2 = 15�, h = 8 mm, b = 2 mm).


the hoop direction. The buckling mode for the shells with h/b = 4and /2 = 20�, 25� contains seven buckling waves.

The local buckling of ribs occurs for the shells having h/b = 4 and/2 = 5�, 10�, 15� (see Fig. 20). As can be seen, the segments of theribs located in the bottom part of the shell buckle like beams inthe planes of their minimal bending stiffness.

The resistance of the shells to buckling under torsion can be en-hanced in the same way which was shown in previous sections.Namely, this can be achieved by increasing the stiffness of a fewhoop ribs located in the close proximity to the bottom section ofthe shell. For example, the critical torque can be increased by1.16 times (from 76.083 kN m to 88.089 kN m) for the shell madeof the ribs with h = 8 mm and b = 2 mm and having the angle

/2 = 15� if the widths of the two hoop ribs, closest to the bottomend of the shell, are doubled (see Fig. 14).

The weight efficiency of the shells subjected to torsion is as-sessed using the following criterion: KT = Tcr/mg. Using the data pre-sented in Tables 1 and 8, the values of KT are calculated for variousangles /2 and ribs’ cross-sectional dimensions h and b. The resultsof the calculations are shown in Table 9. It follows from the analysisof the data presented in this Table that the value of the coefficient KT

grows with the increase in /2, h and b. Changing the ratio h/b from0.25 to 4 leads to the increase in the value of KT for /2 = 15�, 20�, and25�. If the shells are wound with the angles /2 = 5�, 10�, the maxi-mum buckling resistance is observed for the structures composedof the ribs having the square cross-sections (4 � 4 mm).

Table 9Parameter KT (m) for the shell under torsion.

/�

2h = 2 mm, b = 8 mm h = 4 mm, b = 4mm h = 8 mm, b = 2 mm

5 142.88 436.04 191.3310 633.83 1083.04 945.2415 1215.08 1615.40 2336.0620 1722.39 2065.00 3942.0425 2103.92 2309.61 4408.26

Fig. 21. Functions Pcr(/2).


4. Design for buckling resistance

A specialised finite-element model generation procedure (de-sign modeller) developed in this work can be efficiently employedfor designing the composite anisogrid lattice conical shells underconstraints imposed on the level of critical load. As shown above,the buckling performance of the shells under consideration de-pends, for given dimensions H, D1 and D2, on the geometrical andstiffness parameters of the shell’s lattice structure. These parame-ters can vary within certain ranges normally determined by overallstructural design limitations and/or manufacturing constraints. Forgiven (desired) critical load the design procedure normally impliesfinding the required combination of the lattice structure parame-ters that would also deliver the minimal mass of the shell. Irrespec-tive of the selection of one or another design algorithm, this processinvolves the generation of the suitable finite-element model andsubsequent numerical solution of the corresponding buckling prob-lem. It has been demonstrated that the design modeller developedin this work can efficiently generate the finite-element formula-tions closely reflecting main geometrical and structural featuresof the conical anisogrid lattice shells. The use of such an automatedmodel generation procedure enhances substantially the design pro-cess and makes the search for optimum design solutions feasible.

As indicated in previous Sections, the lattice structure of theconical shells is characterised by the mechanical properties ofthe rib materials, number of helical ribs, trajectories of the filamentwinding (represented by the helical angle of the rib orientation atthe bottom section of the shell: /2), and the rib’s cross-sectionaldimensions. The change of any of these parameters causes thechange in the value of critical load. The solution of the design opti-misation problem including the comprehensive search of all thesevariables is computationally expensive. The examples of the struc-tural designs presented in this article and demonstrating the capa-bilities of the proposed approach are based on the limited numberof the design parameters.

It is assumed, that the dimensions of the shells (H, D1 and D2),composite material properties, and number of helical ribs are thesame as in the previously considered numerical examples. It is alsosuggested that all the ribs have the square cross-section with thefollowing dimensions: h = b = 4 mm, 4.5 mm and 5 mm. The angle/2 changes from 5� to 25�. The corresponding buckling problemshave been solved for the shells subjected to axial loading. Theresulting functions Pcr (/2) are shown in Fig. 21. The design solu-tions have been obtained for the three levels of axial critical load:Pcr = 100 kN, 150 kN, and 185 kN. It follows from Fig. 21 that foreach level of the buckling load, the angles /2 corresponding tothe three different sizes of the ribs’ cross-sections ((1) 4 � 4 mm,(2) 4.5 � 4.5 mm, and (3) 5 � 5 mm) are found to be:

Fig. 22. Functions m(/ ).

/ð1Þ2;100 ¼ 6�;/ð2Þ2;100 ¼ 9� and /ð3Þ2;100 ¼ 14� for Pcr ¼ 100 kN; ð1Þ



The graphs representing the mass of the shell as the functions ofthe angle /2 have been plotted in Fig. 22 for the three shell designswith the ribs having the three different aforementioneddimensions.

Using the angles given by Eqs. (1)–(3) for Pcr = 100 kN, 150 kN,and 185 kN, the corresponding curves showing dependence ofthe mass of the shell on the ribs’ dimensions and angle /2 are plot-ted in Fig. 22. It follows from these diagrams that the lattice shelldesigned for buckling load Pcr = 100 kN has a minimum mass of3.175 kg if h = b = 4 mm and /2 = 14�. This provides a 22 per centsaving in mass in comparison with the shell composed from thesame ribs but having the angle /2 = 6� (see Fig. 22).

Similarly, the shell designed for the buckling load Pcr = 150 kNhas a minimum mass of 3.8 kg for h = b = 4 mm and /2 = 20� withthe mass saving of 12 per cent in comparison with the shell wound

2

Fig. 23. Functions Kp(/2).


at /2 = 9�. It can be found (see Fig. 22) that the minimum mass ofthe third shell design (Pcr = 185 kN) is 4.2 kg which corresponds to/2 = 18� and h = b � 4.3 mm with the mass saving of 8.7 per centcompared to the shell wound at /2 = 11�.

Depending on the design goals, the minimum mass criterioncan be replaced with the requirement to have a maximum valueof the coefficient of weight efficiency Kp = Pcr/mg. The correspond-ing graphs Kp(/2) are shown in Fig. 23 for h = b = 4 mm, 4.5 mm and5 mm. Once again, using the angles given by Eqs. (1)–(3) forPcr = 100 kN, 150 kN, and 185 kN, the corresponding curves show-ing dependence of the coefficient of weight efficiency, Kp on theribs’ dimensions (b and h) and angle /2 are plotted in Fig. 23. Ana-lysing these diagrams, the dimensions of the rib’s cross-sectionsand corresponding angle /2 delivering the maximum weight effi-ciency can be identified.

Note that the aforementioned analysis can be efficiently used atthe early design stages and provide reasonable estimates with re-gard to the selection of main design parameters of the lattice shellsfor given buckling loads. In the cases where more detailed designanalysis is required, the design modeller developed in this workcan be embedded into a finite-element software package as partof the appropriate advanced optimisation procedure.

5. Conclusions

The buckling behaviour of conical composite lattice shells underaxial compression, transverse bending, pure bending, and twisting

has been investigated using finite element modelling and analysis.The specialised algorithm (model generator/design modeller) hasbeen developed, coded and implemented in COSMOS/M. The pro-gram enabled fast and efficient generation of the finite elementmodels for the buckling analysis of the anisogrid conical shells.Using finite element analyses, the effects of the angles of orienta-tion of the ribs and their cross-sectional dimensions on the buck-ling loads and modes have been investigated.

It has been demonstrated that the modelling technique pre-sented in this article is capable of capturing and investigating thevariety of buckling modes of conical lattice shells including thosetypical for 3D frames composed from curvilinear ribs (i.e. themodes characterised by the local buckling of ribs) and, also thebuckling modes reflecting the shell-like buckling behaviour. Theweight efficiencies of the lattice shells under consideration havebeen assessed using calculations of the ratio of the critical loadto the weight of the shell.

The parametric finite-element buckling analyses of the aniso-grid conical shells subjected to various loadings showed therobustness and the potential of the modelling approach. Based onthe results of parametric analyses, it has been shown that thebuckling resistance and weight efficiency of the conical latticeshells considered in this work increase substantially with the in-crease in the angle of orientation of helical ribs in the section withthe larger diameter of the cone (angle /2). Also, it was found thatthe buckling resistance can be significantly enhanced by eitherincreasing the stiffness of a few hoop ribs located in the close prox-imity to the bottom section of the cones, or by introducing theadditional hoop ribs of the same original cross-section in the samearea of the shell.

The effectiveness of the design analyses is demonstrated usingparticular examples. It has been shown that the resultant opti-mised designs can deliver up to 22 per cent mass savings in com-parison with the non-optimised conical lattice shells.

References

[1] Vasiliev VV, Barynin VA, Rasin AF. Anisogrid lattice structures–survey ofdevelopment and application. Compos Struct 2001;54:361–70.

[2] Vasiliev VV, Razin AF. Anisogrid composite lattice structures for spacecraft andaircraft applications. Compos Struct 2006;76:182–9.

[3] Vasiliev VV, Barynin VA, Rasin AF, Petrokovskii SA, Khalimanovich VI. Anisogridcomposite lattice structures–development and space applications. ComposNanostruct 2009;3:38–50.

[4] Morozov EV, Lopatin AV, Nesterov VA. Finite-element modelling and bucklinganalysis of anisogrid composite lattice cylindrical shells. Compos Struct2011;93:308–23.

[5] Hou A, Gramoll K. Compressive strength of composite latticed structures. J ReinfPlast Compos 1998;17(5):462–83.

[6] Vasiliev VV, Razin AF, Totaro G, De Nicola F. Anisogrid conical adapters forcommercial space application. In: AIAA/CIRA 13th international space planesand hypersonics systems and technologies. 16–20 May, 2005 Centro ItalianoRicerche Aerospaziali (CIRA) Capua.

[7] COSMOS/M User Guide. Srtuctural research & analysis corporation; 2003.[8] Struik DJ. Lectures on classical differential geometry. 2nd ed. Reading,

Massachusetts: Addison-Wesley; 1961.

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Buckling analysis and design of anisogrid composite lattice conical shells