Structure mechanical modeling of thin-walled closed- section composite beams, part 2: multi-cell cross section

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1 Loughborough University Institutional epository Structure mechanical modeling of thin-walled closed- section composite beams, part 2: multi-cell cross section This item was submitted to Loughborough University's Institutional epository by the/an author. Citation: WAG, S. and HAG, C., 24. Structure mechanical modeling of thin-walled closed- section composite beams, part 2: multi-cell cross section. Composite Structures, 3, pp Additional Information: This paper was accepted for publication in the journal Composite Structures and the definitive published version is available at Metadata ecord: Version: Accepted for publication Publisher: c Elsevier ights: This work is made available according to the conditions of the Creative Commons Attribution-onCommercial-oDerivatives 4. International (CC BY-C-D 4.) licence. Full details of this licence are available at: Please cite the published version.

2 Structure mechanical modeling of thin-walled closeection composite beams, part 2: multi-cell cross section S Wang*, C hang Department of Aeronutical and Automotive Enigneering Loughborough University Loughborough, Leicestershire, LE 3TU, UK * S.Wang@lboro.ac.uk, *Corresponding author. Tel. : Key wor: Closed-section, Composite beams, Multi-cell, Thin-walled Abstract: The methodology used in part [] of the work for single-cell thin-walled closeection composite beams is extended to multi-cell thin-walled closed-section composite beams. The effect of material anisotropies is fully considered on the mid-surface shear strain of all the cross sectional members including skin walls and internal members. umerical comparisons with ABAQUS finite element simulations are performed for three-cell box and elliptical beams with a variety of laminate layups under various loading conditions and excellent agreements are observed. Significant deficiency of some existing models are shown.. Introduction An accurate structure mechanical model has been developed in part [] of the work for TWCSCBs with single-cell cross sections. In this part 2, the model is extended to TWCSCBs with multi-cell cross sections which are much more popular in several industrial sectors. The extension will involve more complex analytical operations than that for TWCSCBs with single-cell cross sections. All the kinematic developments remain the same and are not repeated here. However, details will be presented here to determine the local shell wall axial warping displacement, the mid-surface shear strain and the global beam stiffness matrix. 2. The TWCSCBs model in [2-4] Fig. shows the multi-cell cross section of a TWCSCB with its shear flow diagram. Although it is in a relatively simple one-direction multi-cell arrangement, the principle for the development of present -D TWCSCB modelling will be thoroughly demonstrated and remains the same for any arbitrary multi-cell arrangements. Eq. (3) in part of the work [] gives the local shell wall mid-surface warping displacement ( s, as w ω

3 with w, C [ Φ ( r( s) s r( s) + + s wω ( s, Φ ( γ wω (, () s s ω ( γ ] and C. ote that the circular integration symbol denotes an integration over the whole cross section perimeter including both the skin and internal shell walls and starting point s is arbitrary. It is seen from Eq. () that the shell wall mid-surface shear strain γ nee to be determined first in order to determine the shell wall mid-surface axial warping displacement w ω ( s,. Applying Eq. () to any one complete cell, e.g. the th cell as shown in Fig., gives where γ 2AeΦ ( (2) 2 A e r with Ae being the enclosed area by the th cell. In order to determineγ, Librescu and Song [2] assumes a constant shear flow q developed within each cell under the St.Venent pure torque. Therefore, the resultant shearing force due to torsion on the four walls of the th cell are T q, T q q, T q, T q q+, respectively. Further, the work [2] makes the same treatment as that for single-cell cross section. That is, S T G( s, t( s, γ ( s, where G ( s, is called the equivalent shear modulus and t( s, is the thickness of the shell wall. However, it is not reported in the work [2] how to determine the equivalent shear modulus G ( s, in the case of generally laminated composite materials. ow, using the above two assumptions Eq. (2) becomes, Φ ( δ where This can be rearranged as: where [ q + ( q q ) δ + q δ + ( q q ] z ) δ ) 4 (3a) 2Ae j ij i ( / Gt) δ (3b) Φ ( q δ + q δ q δ ) (4a) (,, +, + 2Ae ( Gt), /, δ (4b) ( Gt) δ, / (4c),

4 ( Gt), + /, + δ (4d) δ,- represents the integration on the wall bounded by the th cell and the (-)th cell; δ, denotes the closed integration on the th cell and δ,+ represents the integration on the wall bounded by the th cell and the (+)th cell. Assembling Eq. (4) for all the cells shown in Fig. gives where [ ]{ q} { I}Φ H (5a) { q} { q } T q2... q... q q { I} { } T (5b) (5c) [ H ] H H, 2, H H,2 2,2 H 2,3 H, H, H, + H, 2 H H,, H H,, (5d) where the elements Hi,j are expressed as: δ (5e) H i, j i j 2A, ei From Eq. (5), the shear flow of each cell, as a function of sectional rate of twist, Φ '(, is given as where: { } { J} Φ' q (6a) { J} [ H ] { I} (6b) Finally, the shear flow distribution in each wall can be calculated as: T T { q q q... q q } { J J J... J J J J J J } Φ (7) q 2, +,, 2 + where the subscript of denotes the walls of the th cell that are not bounded with any other cells, e.g. the wall -2 and 3-4 in Fig.. Whereas the subscript of +, denotes the wall bounded by (+)th and th cells, for example the wall 4- in Fig.. By using the earlier assumption G( s, t( s, γ ( s, the distribution of shell wall mid-surface shear S T

5 strain for a multi-cell section in the work [2] becomes: 2, +,, T { γ γ... γ γ... γ γ } J /( Gt) ( J J ( J 2 J ) /( Gt) ) /( Gt), J 2,... J /( Gt) /( Gt) Eq. (8a) can be written in a more compact form as ( J + J ) /( Gt) +,... T Φ (8a) γ ( s, ψ ( s) Φ'( ψ ( s) Κ ( (8b) where the location of wall segments is represented as s(),(2,), (), (+,), (,- ), (). ψ is called the torsional function of a multi-cell closed-section. Instead of assuming constant shear flow S T G( s, t( s, γ ( s, on each shell wall segment [2], the work [3,4] assumes a constant quantityγ t. The torsional function ψ in the work [3,4] therefore becomes: ψ { J / t ( J J ) / t... J / t ( J J ) / t... ( J J ) / t J / t } T ( s) 2 2, + +,, Theδ integrations corresponding to Eqs. (4b,c,d) change to be ( t), /, XY δ (a) ( t) δ, / (b),, + /, + ( t) δ (c) The mechanical meaning of constant quantity γ t is unclear for composite materials. ow, substituting Eq. (8b) into Eq. () gives the shell wall mid-surface axial warping displacement. where ω wω ( s, ω( s) Φ ( () ( s) η( s) C η is the warping function with ( s) s [ r( s) ψ ( s)]. (9) η It has same form as that for single-cell cross section as expected. Then, the shell wall mid-surface axial strain ε zz is expressed as in terms of the global beam strain and curvatures, which is the same as that in Eq. (8) in part [] and is recorded here. ε zz + Y ( s) K X + X ( s) KY + ω( s) K (2) The -D multi-cell TWCSCBs constitutive equations can then be established in the same way as that for the -D single-cell TWCSCBs by replacing the single-cell torsional function with the multi-cell ones given here. The details can be found in Eqs. (A8-32) in part [] of ω

6 the work. The shell wall mid-surface shear strain γ in Eq. (8b) and the axial strain ε zz in Eq. (2) serve the basis for the mechanical modelling in the work [2-4]. It is worth to repeat the following point made in part [] of the present work. Eq. (8b) is the key feature in works [2-4] in which the material anisotropies in TWCSCBs are not considered. It gives the solution of the local shell wall mid-surface warping displacement ( s, in Eq. () which produces the local shell wall mid-surface warping axial strain ω( s) K in Eq. (2) and. The part [] of the present study has proved that ω( s) Kω has small effect on the modelling accuracy. Hence, the axial strain in Eq. (2) is kept in use in the present work although a slightly more accurate approach can be achieved with much more complications to use the γ determined in next section. Part [] of the present study has shown that the adoption of Eq. (8b) can introduce gross error on γ and consequently on the accuracy of modelling. In next section theγ will be determined by using the material constitutive laws of local shell wall. 3. The present TWCSCBs model From the constitutive equations of local shell wall given in Eq. (22) or (23) in part [], the shell wall mid-surface shear strain γ rea S4 zz + S42kzz + S43k + S44 w ω γ ε (3) where S ij are given in Appendix Eq. (A2) of part []. Substituting Eq. (3) into Eq. () gives, ( S4 zz + S42κ zz + S43κ + S44 ) AeΚ XY ε 2 (4) The resultant shear force can be considered in two parts, namely the constant St.Venant torsional shear flow T and the variable bending shear flow B, i.e. T + B. Then, Eq. (4) becomes, Since ( ) ( S4 zz + S42κ zz + S43κ + S44T + S44B ) AeΚ XY ε 2 (5) S for symmetrical cross-section and negligibly small for most of the 44 B closed-cross sections in common applications, the bending shear term ( ) S 44 B ω T can be solved from Eq (5) by neglecting ( S ) A K ( S ) ( S κ ) ( S κ ) 44 T 2 e XY 4ε zz 42 zz 43 (6a) eplacing T by shear flow q for simplicity, the left-hand side of Eq. (6) can be considered for each wall segment as:

7 where: S ( q + q δ q δ ) δ (6b) 44 T,, +, + δ, S44 (6c), δ (6d), S 44, + S44, + δ (6e) Using Eq. (2) for ε zz and k zz, k in Eq. (8) in part [], the left hand side of Eq. (6a) becomes, 2A e z Κ + Κ XY ( S ε ) ( S κ ) ( S κ ) ( S4) + Κ X ( ( S4Y S24 cosα ) ) + ΚY ( ( S4X + S24 sinα ) ) XY ( 2 S34 + 2Ae ) + Κω ( ( S4ω S24q) ) 4 zz 42 zz 43 (6f) ote that the quantity q in Eq. (6f) is the normal distance from the global axis O to an arbitrary point (s, on the shell wall mid-surface as shown in Fig. in part []. ow, assembling Eq. (6) for all the cells shown in Fig. gives where [ P] [ δ ] δ, δ 2, S 4 S 4 S 4 δ δ,2 2,2 δ 2,3 δ [ δ ]{ } [ P]{ ε}, q (7a) δ, δ, + δ, 2 δ, δ, δ δ, ( S Y S cosα ) ( S X + S sinα ) 2 S + 2A ( S ω S q) 4, (7b) ( S Y S cosα ) ( S X + S sinα ) 2 S + 2A ( S ω S q) 4 ( ) ( ) ( ) S Y S cosα S X + S sinα 2 S + 2A S ω S q e e e (7c) ε Κ Κ Κ Κ (7d) { } { } T z X Y XY ω 4 24

8 The shear flow developed in each cell { } curvatures{ ε }. That is where q is then solved in terms of the global strain and { } [ ζ ]{ ε} q (8a) [ ] [ δ ] [ P] ζ (8b) Finally, the shear flow distribution in each wall can be calculated as: where { } ζ (, 2,..., 5, j q q q q+ q, q 2,, { ζ, j} { ζ } { ζ } 2, j { ζ, j} { ζ } { ζ } +, j { ζ, j} { ζ, j} { ζ }, j, j, j { ε} j ) is the th row of the matrix [ ] (9) ζ. eplacing the shear flow symbol q by the symbol used in local shell wall constitutive equations, the constant shear flow at each wall segment is given as: ( ξ ) + ( ξ ) K + ( ξ ) K + ( ξ ) K ( ξ ) K ω ( s) + (2) T s, z s,2 X s,3 Y s,4 XY s, 5 The location of the wall segment is s(),(2,), (), (+,), (,-), () corresponding to Eq. (9), and ξ ( j, 2,..., 5) can be easily calculated from Eq. (9) s, j using the matrix [ ζ ] in Eq. (8b). ow, substituting Eq. (2) in to Eq. (3) and neglecting bending shear flow give the shell wall mid-surface shear strainγ. Finally, the -D multi-cell TWCSCBs constitutive equations can then be established in the same way as that for the -D single-cell TWCSCBs in part [] of the work. That is, T { F M M M M } [ E ]{ K K K K } T X Y The details of the stiffness matrix [ E ij ] are given in the Appendix. 4. umerical validations ω (2) ij The -D multi-cell TWCSCB model in Eq. (2) has been implemented in a -D beam finite element similar to that in the work [3,4]. For the purpose of convenient comparison, the model in the work [2,3,4] and the present model are designated as Model and 2 in the X Y XY ω

9 following text, respectively. The numerical results calculated from both models are compared with results from an ABAQUS shell model. The four-node linear quad-4 S45 shell element from ABAQUS element library is employed [6]. The first validation example concerns a three-cell cantilever box beam as shown in Fig. 2. The box beam consists of three equal cells. The material properties are summarized as E 48GPa, E GPa, G GPa, ν 2. 3 Similar to that in part [] of the work, four loading cases are considered. They are the axial load F, two transverse loa F X, FY and torsional moment M. The loa are applied individually at the free end of the box. Five different layups are studied. They are layups in [5] [45 /9 ], [6 /7 /8 ], symmetric layups [-75 /75 ]s, anti-symmetric layups [7 /-2 ]2 and quasi isotropic layups [75 /9 /-75 ]. ote that all the layups are counted in the LSWCS n from n-t/2 and the fibre angle θ in Fig. of part [] of the work is measured relative to s and not the usual axial axis z. The direction of s axis is anti-clockwise along the skin shell wall and downwar and upwar along the left and right webs, respectively. Table shows the free end displacements and rotations from both models and the four-node linear quad-4 S45 shell element from ABAQUS. For symmetric and quasi-isotropic laminate layups, the results predicted by both models are identical and compares well with ABAQUS simulations. For layups [45/9], the difference between the two models becomes clearly visible but not significant. Model 2 achieves better accuracy than Model comparing with the ABAQUS benchmark. For arbitrary layups [6/7/8] and antisymmetric layup [7/-2]2, Model 2 achieves consistently close agreement with ABAQUS whereas gross errors are found in Model. The failure of Model can be seen as a consequence of its inappropriate application of the shell wall mid-surface shear strain in Eq. (8) to a composite beam. The shear strainγ for a composite beam should be explicitly defined by Eq. (3) indicating that the shear strainγ of a composite shell is not only related with shear flow but also depen on the material coupling with axial strainε zz, and curvatures k zz and k. The causes of significant errors in calculating the bending stiffness are similar to those observed in the analysis of the single-cell TWCSCBs in part [] of the work. The A6 and A26 decide the magnitude of k2 and k23, which ultimately affect the magnitude of E22 and E33. Unlike the single-cell TWCSCBs of which axial, axial-twist coupling and torsional stiffnesses were unchanged between the Model and 2 regardless of laminated layups, the axial, axial-twist coupling and torsional stiffnesses of the multi-cell box beam varies between the two models for different laminate layups, as shown in Table. Assessments of the stiffness difference, e.g. E and E44, can be carried out in the similar approach as E22 and E33 from part [] of the work for single-cell TWCSCB model. The E is studied as a demonstration example below. The constitutive relationship between the global axial force F and the global axial strain in Model is defined by the stiffness term E given as

10 ( E ) M k (22) where the subscript M denotes Model. Its counterpart in the present model is given as 2 ( E ) k + k k + S s ( E ) + M / 22 4ξ 2 2, M (23) where the underlined part is the difference between the two models and is represented by as + 2 (24a) with 2 k2 / k 22 (24b) 2 S4ξ s, (24c) Table 2 lists the stiffness parameters and variations of the axial stiffness between the two models for the five layups considered. Upon comparisons, the following observations were obtained: Generally, E due to terms and 2 are less significant compared to the E22 and E33 given in Table 2 of part [] of the work. The material coupling between γ and ε zz are mainly contributed to the warping which is negligible for closed cross section beams. For symmetrical layups and quasi-isotropic layup, A6A26. Effectively, k2s4. Therefore, vanishes so that the axial displacements calculated by the two models are identical. For layup [45/9], the k2 is much smaller than k22. As a result, the magnitude of z become insignificant compared to E from Model. Therefore, the difference of the axial stiffness of the two models is small. For arbitrary layups [8/7/6] and [7/-2] 2 where the k2 is much greater than k22, the is greater than 4% of E from Model and need to be considered. To validate present model on a commonly used multi-cell TWCSCB with curved walls, a cantilever multi-cell elliptical beam with same material properties and layups as those used for multi-cell box beam is assessed. Fig. 3 shows the geometrical specifications of the beam. The results from both models are tabulated in Table 3 along with ABAQUS simulations. The observations are the same as those concluded from the analysis of the three-cell box beam. 5. Conclusions The accurate structure mechanical model for single-cell TWCSCBs in part [] of the work is extended to multi-cell TWCSCBs. umerical comparisons with ABAQUS simulations are performed for box and cylindrical beams with a variety of laminate layups under various loading conditions and excellent agreements are observed. It concludes that the present model is applicable with truly arbitrary layups of laminates. eglecting the effect of material anisotropies can lead to gross overestimate of global beam bending stiffness while neglecting the axial warping effect lea to slight overestimate of the global beam extension

11 and torsional stiffness. It will be valuable work to extend the present model to broader areas such as including vibration analysis [7], transverse shear effect [8], geometrical nonlinearity [9], and etc. Appendix T Elements of the global cylinder beam stiffness matrix [ E ] [ ] in Eq. (2) in the present model ij E ij T { F M M M M } [ E ]{ K K K K } T X Y + ( SY S2 ) + S4 s, ( SX + S2 ) + S4 s, ( S3) + S4 s, ( S S2q) + S4 s, 2 ( SY S2Y cosα ) + S4Yξ s,2 2 ( ) + S2Y cosα S22 cos α S24 cosαξs,2 ω : E S S4ξ s, (A) 2 E2 cosα ξ (A2) E3 sinα ξ 3 (A3) 4 E4 2 ξ (A4) E E E 5 ξ ij ω (A5) ( S XY + S Y sinα ) 2 + ( S X cosα + S cosα sinα ) S cosαξ S Yξ 4 ( 2S Y ) + S Y ( 2S cosα ) S αξ s,3 24 s,3 X Y XY ω (A6) (A7) E ξ s, cos s, 4 (A8) E E ( S Yω S Yq) 2 + ( S cosαω S cosαq) S cosαξ S Yξ 4 s,5 24 s,5 2 ( SX + S2xsinα ) + S4Xξs,3 + 2 ( + ) + S2X sinα S22 sin α S24 sinαξs,3 ( 2S X ) + S X + ( 2S sinα ) + S αξ (A9) (A) E ξ s, sin s, 4 (A)

12 E E E E 35 ( S Xω S Xq) 2 + ( S sinαω S sinαq) S sinαξ S 4 Xξ s,5 24 ( 4S ) S ξ s,5 (A2) rξ + + s, s, 4 (A3) 44 2 ( 2S ω 2S q) S ξ rξ + + s, s, 5 (A4) s ( ) 2 + ( ) + S S2ωq S4ωξs,5 S2ωq S22q S24q s, 5 ω (A5) 55 ξ eferences [] hang C, Wang S. Structure mechanical modeling of thin-walled closed-section composite beams, part : single-cell cross section. Composite Structures 23 (submitted). [2] Librescu L, Song O. Thin-Walled Composite Beams: Theory and Application. Berlin: Springer; 26. [3] Lee J, Lee S. Flexural-torsional behavior of thin-walled composite beams. Thin- Walled Structures 24;42(9): [4] Vo TP, Lee J. Flexural-torsional behavior of thin-walled closed-section composite box beams. Eng.Struct. 27;29(8): [5] Kollar LP, Pluzsik A. Analysis of Thin-Walled Composite Beams with Arbitrary Layup. Journal of einforced Plastics and Composites 22 ovember ;2(6): [6] Simulia. ABAQUS/Standard User's Manual. 29;version 6.9. [7] Vo TP, Lee J. Free vibration of thin-walled composite box beams. Composite Structures 28;84():-2. [8] Vo TP, Lee J. Flexural-torsional behavior of thin-walled composite box beams using shear-deformable beam theory. Eng.Struct. 28;3(7): [9] Vo TP, Lee J. Geometrically nonlinear theory of thin-walled composite box beams using shear-deformable beam theory. Int.J.Mech.Sci. 2;52():65-74.

13 Figure captions: Fig.. A multi-cell cross section with its shear flow under pure torque M. Fig. 2. Dimensions of a cantilever thin-walled composite three-cell box beam. Fig. 3. Dimensions of a cantilever thin-walled composite three-cell elliptical beam.

14 Table captions: Table Comparions of free end displacements of cantilever thin-walled composite three-cell box beams. Table 2 Stiffness parameters of the shell wall for the composite three-cell box beams. Table 2 Comparions of free end displacements of cantilever thin-walled composite three-cell elliptical beams.

15 Fig.. A multi-cell cross section with its shear flow under pure torque M.

16 Fig. 2. Dimensions of a cantilever thin-walled composite three-cell box beam.

17 Fig. 3. Dimensions of a cantilever thin-walled composite three-cell elliptical beam.

18 Table Comparions of free end displacements of cantilever thin-walled composite three-cell box beams. Anti Quasi Layups in [5] symmetric symmetric isotropic Load Case Disp Models [45/9] [6/7/8] [-75/75]s [7/-2]2 [75/9/-75] F 2.4k ABAQUS 3.89E-4 6.3E E E E-4 W,m Model 3.89E-4 5.7E E E E-4 Model 2 3.9E-4 6.5E E E E-4 ABAQUS -6.2E E-2.E E E-4 Ф, ra Model -6.6E E-2.E+ -4.7E-2 8.4E-4 Model 2-6.9E E-2.E E-2 8.4E-4 ABAQUS 2.E E-3.39E E-3.3E-3 F Y5 V,m Model.9E-3.7E-3.36E E-3.25E-3 Model 2.96E E-3.36E E-3.25E-3 ABAQUS 9.3E-4.6E-3 6.4E-4.79E-3 6.6E-4 F X5 U,m Model 8.56E E-4 6.6E-4.9E E-4 Model 2 8.7E-4.7E-3 6.6E-4.7E E-4 ABAQUS.68E-2 2.4E-2.29E-2 2.6E-2.64E-2 M.m Ф, ra Model.7E-2.99E-2.3E-2 2.2E-2.66E-2 Model 2.7E-2 2.5E-2.3E-2 2.9E-2.66E-2

19 Table 2 Stiffness parameters of the shell wall for the composite three-cell box beams. Anti Quasi Stiffness Layups in [5] symmetric symmetric isotropic parameters [45/9] [6/7/8] [75/-75] s [7/-2] 2 [75/9/-75] A E+7.497E+7.E E+7.E+ A E E+7.E+ 3.33E+7.E+ k E E+7.E+ 4.64E+7.E+ k E E E E+7 2.3E E E+7.E+ -2.3E+7.E+ 2.66E+6 3.9E+7.E+ 2.2E+7.E+ -.53E E+6.E+ -.95E+6.E+ (E ) M 6.32E+7 7.8E+7 9.5E E E+7 E -.24% -4.9%.% -4.2%.%

20 Table 3 Comparions of free end displacements of cantilever thin-walled composite three-cell elliptical beams. Anti Quasi Layups in [5] symmetric symmetric isotropic Load Case Disp Models [45/9] [6/7/8] [-75/75] s [7/-2] 2 [75/9/-75] F 2.4k ABAQUS 5.2E-4 8.3E E-4.23E E-4 W,m Model 5.9E E E-4.4E E-4 Model 2 5.2E-4 8.7E E-4.23E E-4 ABAQUS -9.37E E-2.E E-2.2E-3 Ф, ra Model -9.8E E-2.E+ -6.5E-2.44E-3 Model E-3-5.4E-2.E E-2.44E-3 ABAQUS 3.74E-3 6.9E-3 2.6E-3.E E-3 F Y5 V,m Model 3.59E-3 3.2E E-3 5.9E E-3 Model E-3 6.E E-3.E E-3 ABAQUS 2.58E-3 3.5E-3.8E E-3.68E-3 F X5 U,m Model 2.45E-3 2.9E-3.76E E-3.6E-3 Model 2 2.5E E-3.76E E-3.6E-3 ABAQUS 3.3E E E E E-2 M.m Ф, ra Model 3.7E E E E E-2 Model 2 3.8E-2 3.6E E E E-2