This study received support from the Grant Agency of the Czech Technical University in 
Prague, under grant No. SGS21/151/OHK2/3T/12

Malá, A.

Mareš, T.

Padovec, Z.

DSpace at University of West Bohemia

 Efficient methods for calculating equivalent stiffness of circular composite tubes: Insights for structural design optimisation  A. Maláa, Z. Padoveca, T. Mareša   a Department of Mechanics, Biomechanics and Mechatronics, Faculty of Mechanical Engineering, Czech Technical University in Prague, Prague, Czech Republic  This paper presents the comparative analysis of various analytical methods used to estimate the equivalent stiffness of circular cross-section composite tubes manufactured using filament wound technology with long fibre laminates. Four computation methods, namely ABD matrices theory (ABD), minimum total potential energy theory (E), minimum complementary energy theory (C), and an averaged approach (avg), are evaluated. The comparison involves analysing a cantilever beam subjected to bending, torsion, and combined loading with varying force-to-moment ratios (as illustrated in Fig. 1).     Fig. 1. Cantilever beam loaded with bending (bending force F) and torsion (torque moment T)  These theories are derived from the extended Hooke´s law for orthotropic materials, defining the stiffness matrix of a ply in the laminate coordinates 𝐸𝑡𝑖 akin to its inversion (compliance matrix 𝐶𝑡𝑖), as found in references like [1]. Both matrices are determined by material mechanical parameters (Young´s module in the fibre direction 𝐸𝐿 and perpendicular to the fibre direction 𝐸𝑇, shear module 𝐺𝐿𝑇, and Poisson’s ratio 𝜈𝐿𝑇), as well as the ply orientation angle (𝛼𝑖).  Important geometric parameters used in calculations include tube outer diameter (𝑑𝑜𝑢𝑡), inner diameter (𝑑𝑖𝑛), outer radius (𝑟𝑜𝑢𝑡), mean radius (𝑟), inner radius (𝑟𝑖𝑛), the number of plies in the laminate (𝑛), ply thickness (𝑡), and the thickness of the tube (𝑇𝑇). The index '𝑖' refers to '𝑖 − 𝑡ℎ' ply of the laminate.  The equations for determining equivalent stiffnesses, such as tensile (𝐸𝐴)𝑒𝑞, bending (𝐸𝐽)𝑒𝑞, torsional (𝐺𝐽𝑝)𝑒𝑞 and shear (𝐺𝐴)𝑒𝑞), are provided in Table 1. The cross-sectional area ?̃? = 𝜋(𝑟𝑜𝑢𝑡2 − 𝑟𝑖𝑛2 ), the moment of inertia 𝐽 =𝜋(𝑑𝑜𝑢𝑡4 −𝑑𝑖𝑛4 )64, and the polar moment of inertia 𝐽𝑝 =𝜋(𝑑𝑜𝑢𝑡4 −𝑑𝑖𝑛4 )32. ABD matrices theory employs matrix 𝐴 for calculation, representing laminate extensional stiffness. It is defined by the equation (1) as the sum of the product of individual ply stiffnesses and the ply thicknesses.  The laminate tension module 𝐸𝑒𝑞𝐴𝐵𝐷 and shear module 112𝐺𝑒𝑞𝐴𝐵𝐷 are determined by members of the 𝐴 matrix in equations (2) and (3), incorporating the laminate thickness (𝑇𝑇) and auxiliary variable M (4).   A= ∑ Etitini=1  ,  (1)  EeqABD=MTT(A(3, 3)A(2, 2)-A(2, 3)A(2, 3)) ,  (2)  GeqABD=MTT(A(1, 1)A(2, 2)-A(1, 2)A(1, 2)) ,  (3) M=A(1,1)A(2,2)A(3,3)+2A(1,2)A(1,3)A(2,3)-A(1,1)A(2,3)2-A(2,2)A(1,3)2-A(3,3)A(1,2)2 ,  (4)  Table 1. Equations for calculating equivalent stiffnesses according to individual calculation methods   Stiffness Theory (EA)eq (EJ)eq (GJp)eq (GA)eq ABD EeqABDÃ EeqABDJ GeqABDJp GeqABDÃ E ∑ π(routi2 -rini2 )Eti(1,1)ni=1 ∑ πri3tiEti(1,1)ni=1 ∑4Aci2 tiEti(3,3)cini=1 ∑ π(routi2 -rini2 )Eti(3,3)ni=1 C ∑π(routi2 -rini2 )Cti(1,1)ni=1 ∑πri3tiCti(1,1)ni=1 ∑4Aci2 ticiCti(3,3)ni=1 ∑π(routi2 -rini2 )Cti(3,3)ni=1 avg the average of the stiffness values obtained from the previous three methods  The set of tubes were loaded by bending force 𝐹 = 1000 N, torsion moment 𝑇 = 1000 Nm, or by its combination, where the ratios 𝑇/𝐹 = 1, 10, 100 or 1000 with constant value of bending force 𝐹 = 1000 N. The value of displacement for pure bending, rotation of pure torsion, and minimum equivalent energy of combined loading was shown in a dependence graph of the given value on the tube winding angle. These dependencies were plotted for tubes of different thicknesses (𝑡 = 0.2, 0.5, 1, 2, 5, 10 and 20 mm), which winding is composed of four layers with thicknesses 𝑡/4 and winding angles ±𝛼.  Inner diameters of analysed tubes were 𝑑 = 5, 10, 20, 40, 80 mm, and the tube length-to-inner diameter ratios 𝐿/𝑑 = 1, 5, 10, 20 and 50. Material parameters of analysed tubes were set as T700/epoxy laminate properties: 𝐸𝐿 = 128000 MPa, 𝐸𝑇 = 5060 MPa, 𝐺𝐿𝑇 =3400 MPa and 𝜈𝐿𝑇 = 0.345.  A typical angle-to-displacement dependence for the bended beam, with a length-to-diameter ratio greater than 5, is displayed in Fig. 2. Similarly, the angle-to-rotation dependence is presented in Fig. 3. The angle-to-equivalent energy dependence for beams subjected to combined loading highly depends on the torsion moment-to-bending force ratio, which affects the position of the extremes of the given dependencies, as shown in Fig. 4. The top graph shows an example of a load with a bending force of 1000 N and a torque of 1000 Nmm, while the bottom graph shows a load with a bending force of 1000 N and a torque of 1000 Nm.  113  Fig. 2. A typical angle-to-displacement dependence for bended beam    Fig. 3. A typical angle-to-rotation dependence for torsion  114  Fig. 5. Angle-to-total potential energy dependence  As evident from the aforementioned dependencies, the minimum complementary energy theory exhibits the highest absolute values, while the trends remain similar across the selected theories. This underscores the preference for utilizing the minimum complementary energy approach in optimizing the winding parameters for composite tubes. The theory's higher values correspond to lower stiffness, which enhances the safety margin in the design process. It is worth noting that extreme cases involving beams with a length-to-diameter ratio of one have been excluded from consideration for the given structures, as such geometries are no longer deemed as beams in this context. Acknowledgement This study received support from the Grant Agency of the Czech Technical University in Prague, under grant No. SGS21/151/OHK2/3T/12.  References [1] Jones, R. M, Mechanics of Composite Materials, second ed., Taylor & Francis Ltd., London, 1999. 115

Efficient methods for calculating equivalent stiffness of circular  composite tubes: Insights for structural design optimisation

https://dspace5.zcu.cz/bitstream/11025/54939/2/CM2023_Conference_Proceedings-121-124.pdf

Efficient methods for calculating equivalent stiffness of circular composite tubes: Insights for structural design optimisation

Abstract

Similar works

Full text

Available Versions

DSpace at University of West Bohemia