ADVANCED MECHANICS OF COMPOSITE MATERIALS Episode 15 doc

Chapter 8. Optimal composite structures 477 and no further optimization is required. So, following this procedure, we should express h, φ, δ h , and δ c in terms of the safety factors n s ,n b , and n l . Using Eq. (8.101), we get c 2 = Pn s 2πDσhδ h (8.105) Substitution of this result into Eqs. (8.102) and (8.103) yields δ c δ h = 3D 2 n 2 b σ 2 2n 2 s E h E c h 2 (8.106) s 2 = 6n l Dσ 2 h πkP n 2 s E h δ h (8.107) Substituting further Eqs. (8.105) and (8.107) into Eq. (8.104), we obtain δ h = 6n l σ 2 Dh πkP n 2 s E h + Pn s 2πDσh (8.108) Now, Eqs. (8.106) and (8.108) enable us to express the mass of the structure, Eq. (8.90) in terms of only one design variable – the shell thickness h, i.e., M = Lρ c  12D 2 n l σ 2 h 2 Pn 2 s kE h + 3Pn 2 b D 2 σ ρ 4n s E h E c h 2 + 9D 4 σ 4 n l n b ρ Pn 4 s kE 2 h E c + Pn s σ  (8.109) Applying the condition ∂ M/ ∂ h = 0, we have h 4 = P 2 n 2 b n s ρ 16n l E c σ (8.110) Substituting this result into Eq. (8.109), we arrive at M = Lρ c  9D 4 σ 4 n l n 2 b ρ Pn 4 s kE h + 6D 2 σn b n s E h  n l σ ρ kn s E c + Pn s σ  (8.111) It follows from this equation that the mass of the structure M increases with an increase in the buckling safety factors n b and n l , and to minimize the mass, we must take the minimum allowable values of these factors, i.e., n b = 1 and n l = 1. This means that the buckling constraints in Eqs. (8.102) and (8.103) are active. To find the strength safety 478 Advanced mechanics of composite materials factor n s , we need to put ∂ M/ ∂ n s = 0, where M is defined by Eq. (8.111). As a result, we have n s = σ  144D 4 ρ P 2 kE 2 h E c  1/5 (8.112) Taking into account that n s ≥ 1, then equation Eq. (8.112) yields P ≤ P s = 12D 2 σ 2 E h  σ ρ kE c (8.113) So, we have two design cases. For P<P s , we have n s > 1, and the strength constraint, Eq. (8.101) is not active. There exists some safety factor for this mode of failure specified by Eq. (8.112). For P>P s , we have n s = 1, and the strength constraint becomes active, so all three constraints are active in this case. To study these two cases, introduce the following mass and force parameters m = 4M πD 2 L ,p= 4P πD 2 (8.114) Then, Eq. 8.113 gives p s = 4P s πD 2 = 48 σ 2 πE h  σ ρ kE c (8.115) Consider the case p ≤ p s . Substituting n s specified by Eq. (8.112) into Eqs. (8.105), (8.110), (8.111) and using Eq. (8.114), we arrive at the following equations for the parameters of the optimal structure h = h D = 1 4  48π 4 k 2 ρ 3 E h E 3 c p 4  1/10 tan φ = 1 2 ,φ= 26.565 ◦ δ h = 5 4π  108π 2 E c k 4 E 3 h ρ p 2  1/10 δ c = δ c 2ρ m = 25ρ h 8  72 ρp 3 π 2 kE 2 h E c  1/5 (8.116) Chapter 8. Optimal composite structures 479 Consider the case p ≥ p s , repeating the derivation of Eqs. (8.116) and taking n s = 1, we have h = h D =  π 2 kρ E c σ p 2  1/4 tan 2 φ = p s 4p δ h = 2 π sin 2φ  3σ kE h δ c = p s δ h 2ρp m = pρ h σ  1 + p s 4p  2 (8.117) For p = p s Eqs. (8.116) and (8.117) yield the same results. Note that these equations are universal ones, i.e., they do not include the structural dimensions. The Eqs. (8.116) and (8.117) are valid subject to the conditions in Eqs. (8.99). Substi- tuting the parameters following from Eqs. (8.117) in the second of these conditions, we can conclude that the axisymmetric mode of shell buckling exists if p ≤ p 0 = p s     1 2   2E h ρ E c +  2E h ρ E c −1  (8.118) Analysis of this result confirms that the calculated value of p 0 corresponds to an axial force that is much higher than the typical loads for existing aerospace structures. So, the nonsymmetric mode of buckling does not occur for typical lattice structures. As an example, consider an interstage section of a space launcher with D = 4m designed to withstand an axial force P = 15 MN. The ribs are made from carbon– epoxy composite with the following properties: E h = E c = 90 GPa, σ = 450MPa, ρ h = ρ c = 1450 kg/m 3 . Taking k = 4 and calculating p, p s , and p 0 using Eqs. (8.114), (8.115), and (8.118), we get p = 1.2 MPa,p s = 1.45 MPa,p 0 = 1.6 MPa. As can be seen, p<p s and the optimal parameters of the structure are specified by Eqs. (8.116) which give the following results h = 0.009,φ= 26.565 ◦ , δ h = 0.05 δ c = 0.025,m= 6.52 kg/m 3 Consider a design in which there are 120 helical ribs in the shell cross section and that the lattice structure corresponds to that in Fig. 8.20b. In this case, the calculation yields 480 Advanced mechanics of composite materials a h = 188 mm and a c = 210 mm. For a structure with D = 4m, we have h = 36 mm, δ h = 9.4 mm, δ c = 2.35 mm. The mass of the unit surface is 6.52 kg/m 2 . To confirm the high weight efficiency of this lattice structure, note that the composite section with this mass corresponds to a smooth or stringer stiffened aluminum shell with the efficient thickness h = 2.4 mm. The axial stress induced in this shell by an axial force P = 15 MN is about 500 MPa, which is higher than the yield stress of typical aluminum alloys. 8.4. References Bakhvalov, Yu.O., Molochev, V.P., Petrovskii, S.A., Barynin, V.A., Vasiliev, V.V. and Razin, A.F. (2005). Proton-M composite interstage structures: design, manufacturing and performance. In Proc. European Conf. Aerospace Sci., July 4–7, 2005, Moscow, CD-ROM. Kyser, A.C. (1965). Uniform-stress spinning filamentary disk. AIAA Journal. July, 1313–1316. Obraztsov, I.F. and Vasiliev, V.V. (1989). Optimal design of composite structures. In Handbook of Composites: Vol. 2, Structure and Design (C.T. Herakovich and Yu.M. Tarnopol’skii eds.). Elsevier, Amsterdam, pp. 3–84. Rehfield, L.W., Deo, R.B. and Renieri, G.D. (1980). Continuous filament advanced composite isogrid: a promis- ing design concept. In Fibrous Composites in Structural Design (E.M. Lenoe, D.W. Oplinger and J.L. Burke, eds.). Plenum Publishing Corp., New York, pp. 215–239. Vasiliev, V.V. (1993). Mechanics of Composite Structures. Taylor & Francis, Washington. Vasiliev, V.V., Barynin, V.A. and Razin, A.F. (2001). Anisogrid lattice structures – survey of development and application. Composite Struct. 54, 361–370. Vasiliev, V.V. and Razin, A.F. (2001). Optimal design of filament-wound anisogrid composite lattice structures. In Proc. 16th Annual Tech. Conf. American Society for Composites, September 9–12, 2001, Blacksburg, VA, USA. (CD-ROM). Vasiliev, V.V. and Razin, A.F. (2006). Anisogrid composite lattice structures for spacecraft and aircraft applications. Composite Struct. 76, 182–189. AUTHOR INDEX [Plain numbers refer to text pages on which the author (or his/her work) is cited. Boldface numbers refer to the pages where bibliographic references are cited.] Abdel-Jawad, Y.A. 83 131 Abu-Farsakh, G.A. 83 131 Abu-Laila, Kh.M. 83 131 Adams, R.D. 402 434 Adkins, J.E. 137 253 Aleksandrov, A.Ya. 280 320 Alfutov, N.A. 227 253 Anderson, Ya.A. 406 433–434 Andreevskaya, G.D. 127 131 Annin, B.D. 325 357 Aoki, T. 94 131 Apinis, R.P. 404 434 Artemchuk, V.Ya. 374 434 Ashkenazi, E.K. 335 357 Ashton, J.E. 303 320 Azzi, V.D. 201 253 Baev, L.V. 325 357 Bakhvalov, Yu.O. 472, 475 480 Barbero, E.J. 334 357 Barnes, J.A. 369 434 Barynin, V.A. 470, 472, 475 480 Belyankin, F.P. 326 357 Birger, I.A. 148 253 Bogdanovich, A.E. 16 30 98 131 Brukker, L.E. 280 320 Bulavs, F.Ya. 101, 127 132 239 253 322 357 385, 399 434 Bulmanis, V.N. 383 434 Chamis, C.C. 172 253 Chen, H J. 281 320 Cherevatsky, A.S. 222 253 Chiao, T.T. 88 131 202 253 Chou, T.W. 16 30 407 434 Crasto, A.S. 121–122 131 Curtis, A.R. 121 132 Deo, R.B. 472 480 Doxsee, L. 410 435 Dudchenko, A.A. 201 254 Egorov, N.G. 127 132 Elpatievskii, A.N. 196, 201 253–254 Ermakov, Yu.N. 401–402 435 Farrow, G.J. 369 434 Fukuda, H. 16 30 82 131 Fukui, S. 233 253 Gere, J.M. 116 132 Gilman, J.J. 62 131 Gol’denblat, I.I. 321, 326, 338, 343 357 Golovkin, G.S. 128 131 Gong, X.J. 167, 176 254 Goodey, W.J. 17 30 70 131 Grakova, T.S. 383 434 Green, A.E. 137 253 Griffith, A.A. 64, 66 131 Gudmundson, P. 201 253 Gunyaev, G.M. 126 131 Gurdal, Z. 43 56 Gurvich, M.R. 101, 127 132 239 253 322 357 385, 399 434 Gutans, Yu.A. 66 132 Ha, S.K. 375 434 Hahn, H.T. 159, 201 253 321 357 Hamilton, J.G. 369 434 Haresceugh, R.I. 121 132 Hashimoto, S. 233 253 Hashin, Z. 101 131 201 253 Herakovich, C.T. 157, 162, 182 253 Hondo, A. 233 253 Hyer, M.W. 429 434 481 482 Author index Ilyushin, A.A. 147, 153 253 Ishida, T. 408 434 Ivanovskii, V.S. 422 434 Jackson, D. 369 434 Jeong, T.H. 94 131 Jones, R.M. 98, 104, 115 131 158 253 328 357 Kanagawa, Y. 408 434 Kanovich, M.Z. 129 131 Karmishin, A.V. 303 320 Karpinos, D.M. 21 30 Katarzhnov, Yu.I. 330 357 Kawata, K. 233 253 Kharchenko, E.F. 128–129 131 Khonichev, V.I. 404 434 Kim, H.G. 94 131 Kim, R.Y. 121–122 131 Kincis, T.Ya. 105, 122 132 Kingston-Lee, D.M. 366 434 Ko, F.K. 16 30 Kobayashi, R. 233 253 Koltunov, M.A. 129 131 Kondo, K. 94 131 Kopnov, V.A. 321, 326, 338, 343 357 Kruklinsh, A.A. 101, 127 132 239 253 322 357 385, 399 434 Kurshin, L.M. 280 320 Kyser, A.C. 465 480 Lagace, P.A. 104 131 212, 222 253 Lapotkin, V.A. 374 434 Lee, D.J. 94 131 Li, L. 410 435 Limonov, V.A. 406 434 Lungren, J E. 201 253 Margolin, G.G. 326 357 Mikelsons, M.Ya. 66 132 406 433–434 Mileiko, S.T. 83 132 Milyutin, G.I. 383 434 Miyazawa, T. 82 131 Molochev, V.P. 472, 475 480 Morozov, E.V. 177, 252 253 296, 298 320 328 357 431, 433 434 Murakami, S. 408 434 Nanyaro, A.P. 335 357 Natrusov, V.I. 129 131 Ni, R.G. 402 434 Obraztsov, I.F. 451, 465 480 Otani, N. 233 253 Pagano, N.J. 251 253 Pastore, C.M. 16 30 98 131 Patterson, J.M. 369 434 Peters, S.T. 10, 16 30 102 132 Petrovskii, S.A. 472, 475 480 Phillips, L.M. 366 434 Pleshkov, L.V. 129 131 Polyakov, V.A. 16 30 247 253 Popkova, L.K. 431, 433 434 Popov, N.S. 383 434 Prevo, K.M. 369 434 Protasov V.D. 402, 407–408 435 Prusakov, A.P. 280 320 Rabotnov, Yu.N. 397 434 Rach, V.A. 422 434 Razin, A.F. 470, 472, 475–476 480 Reese, E. 407 434 Rehfield, L.W. 472 480 Reifsnaider, K.L. 201 253 Renieri, G.D. 472 480 Rogers, E.F. 366 434 Roginskii, S.L. 127, 129 131, 132 Roze, A.V. 94 132 424 435 Rosen, B.W. 101 131 Rowlands, R.E. 321 357 Salov, O.V. 204 254 Salov, V.A. 204 254 Schapery, R.A. 395 434 Schulte, K. 407 434 Shen, S.H. 378 434 Sibiryakov, A.V. 413 435 Simms, I.J. 369 434 Skudra, A.M. 101, 127 132 239 253 322 357 385, 399 434 Sobol’, L.A. 374 434 Soutis, C. 375 434 Springer, G.S. 375, 378, 384 434 Strife, J.R. 369 434 Sukhanov, A.V. 374 434 Author index 483 Takana, N. 233 253 Tamuzh, V.P. 402, 407–408 433, 435 Tarashuch I.V. 406 433 Tarnopol’skii, Yu.M. 16, 21 30 68, 94, 105, 122 132 244, 246–247 253–254 424 435 Tatarnikov, O.V. 298 320 328 357 Tennyson, R.C. 335 357 Tikhomirov, P.V. 83 132 Timoshenko, S.P. 116 132 Toland, R.H. 321 357 Tomatsu, H. 82 131 Tsai, S.W. 159, 166, 201 253 281 320 321 357 380–381, 383–384, 404 435 Tsushima, E. 408 434 Turkmen, D., 375 434 Van Fo Fy (Vanin), G.A. 93–94, 97 132 Varshavskii, V.Ya. 124 132 Vasiliev, V.V. 21 30 43 56 68 132 177, 190, 196, 201, 204, 206, 244, 246–247, 252 253–254 280, 286, 304, 311 320 344, 347 357 413 435 451–452, 465, 470, 474–476 480 Verchery, G. 166–167, 176 254 303 320 Verpoest, I. 410 435 Vicario, A.A. Jr. 321 357 Vorobey, V.V. 298 320 328 357 Wada, A. 16 30 Wharram, G.E. 335 357 Whitford, L.E. 251 253 Whitney, J.M. 303 320 Woolstencroft, D.H. 121 132 Wostenholm, G. 369 434 Wu, E.M. 321 357 Yakushiji, M. 16 30 Yates, B. 369 434 Yatsenko, V.F. 326 357 Yushanov, S.P. 83 132 Zabolotskii, A.A. 124 132 Zakrzhevskii, A.M. 383 434 Zhigun, I.G. 16 30 244, 247 253 Zinoviev, P.A. 227 253 401–402 435 SUBJECT INDEX actual axial stiffness 276 adhesion failure 106 advanced composites 10 carbon/graphite fiber 11 glass fiber 10 mineral fiber 10 quartz fiber 10 aging 377, 384 aging theory 397 angle variation 227 angle-ply laminate 443 orthotropic layer 208, 211, 224, 226, 320 angular velocity 465–467 anisogrid, See anisotropic grid lattice 451 anisotropic grid 470 layer 13, 165, 255, 257, 368 antisymmetric laminates 293 approximation criterion 327, 331 aramid fibers 13, 82, 109, 120 aramid epoxy composite 105, 128, 157, 175–176 aromatic polyamide fibers, See aramid fibers Arrhenius relationship 383 axial compression 307–308 axial displacement 177 axial force/strain 179, 232 axisymmetric buckling 475 ballistic limit 417–418 basic deformations 257 beam torsional stiffness 287 bending 257–259, 274, 426 bending moment 179, 276, 280, 289, 304 bending–shear coupling effect 296 bending–stretching coupling effects 275, 290 biaxial tension 441 body forces 44, 54 boron fibers 13, 66–67 boron–aluminium composite material 85, 157, 182–183 unidirectional composite 162–163 boron–epoxy composite material 105 borsic 13 boundary conditions 207, 231, 466 braiding 23, 25 two-dimensional 25 three-dimensional 28 brittle carbon matrix 120 buckling constraint 475, 477 safety factors 477 bulk materials 243 burst pressure 200, 297, 351–352 carbon–carbon technology 244 carbon–carbon unidirectional composites 22, 28, 120, 122 carbon–epoxy composite material 25–26, 60–61, 104, 157, 175–176, 208 fibrous composite 354 layer 170 ply 79 strip, deflection of 180 carbon–glass epoxy unidirectional composite 125 carbonic HM-85 fibers 13 carbonization 12, 22 carbon–phenolic composites 22 Cartesian coordinate 31–32, 35, 37, 41, 468 Castigliano’s formula 138, 140 ceramic fibers 14 circumferential deformation 330 circumferential ribs 241 circumferential winding 27, 323, 420 Clapeyron’s theorem 53 coefficient of thermal expansion (CTE) 365–367, 370, 374 cohesion failure 109 485 486 Subject index compliance coefficient 167, 260 matrix 250 composite beam theory 177 bundles 70 flywheels 451 laminates of uniform strength 445, 447 layer, mechanics of 133 composite material 9 filled 9 reinforced 10 unidirectional 61, 236 compression 101, 159, 204 constant of integration 311, 315, 468 convolution theorem 393 cooling 426 coupling coefficients 259 stiffness coefficient 296–297, 303 stiffnesses 289 crack 197, 351 macrocracks 66 microcracks 66, 97,187 surface 192 vicinity 189, 192, 194 creep compliance/kernel 386–388, 392, 396 strain 9 cross-over circles 295 cross-ply couples 287 layer 183, 184, 186, 197 nonlinearity 187 nonlinear models 187 transverse shear 186 curing reaction 19 deformable thermosetting resin 206 deformation 40, 228, 430 creep 7 elastic 7 in-plane/out-of-plane 372 plastic 7 symmetric plies 229 theory 141, 146, 152 delamination 345 densification 22 density 102, 128, 204 diffusivity coefficient 377, 381 direct impregnation 23 displacement 38–39, 77–78, 117, 119, 371 decomposition 257 formulation 51 dissipation factor 401–403 dry bundles 70 dry/prepreg process 23–24 durability 399 evaluation 399 elastic constants 199 potential 46, 64 potential energy 401 solid 44 strain 9, 141 waves 413 elasticity modulus 233, 240, 243, 450 theory 147 elastic–plastic material 8, 182 energy dissipation 401 energy loss, ratio of 401 environmental factors 359 temperature 359 epoxy composites aramid–epoxy 114 boron–epoxy 114 carbon–epoxy 114 glass–epoxy 114 equilibrium condition 94 equation 33–34, 44, 51, 54, 71–72, 91, 98, 118, 190 state 33 Euclidean space 43 Euler formula 474 Euler integral 456 extension–shear coupling coefficient 48 fabric layers 233 strength 419 fabric composites 237 density 237 fiber volume fraction 237 in-plane shear strength 237 [...]... absorption 383 concentration 380–381 monotropic model 438 MSC NASTRAN 296 multi-dimensionally reinforced materials 245 natural fibers 15 Newton flow law 389 Newton’s method 148–149 nonlinearity 161 constitutive equations 215 deformation 182 489 hereditary theory 397 models 137, 182, 215 nonsymmetric buckling 475 off-axis tension 177, 183, 240, 328 optimal laminate 442, 446 optimality conditions 439, 441, 448... velocity 417 Fourier’s law 360, 377 fracture 330, 350 mechanics 64–65 toughness 83–85 work 85 free-edge effect 227, 233 free shear deformation 181 487 generalized layer 256 geodesic filament-wound pressure vessels 451 geodesic trajectories 458 geodesic winding 27 geometric parameters 244 glass–epoxy 157 composite 81, 191, 227 glass–epoxy unidirectional composites 205 density 205 fiber volume fraction 205... layer 209 Hooke’s law 4, 123, 133, 142, 148, 150 , 165, 215, 232, 260, 304, 365, 389, 393–394, 453 hoop layer 320 hybrid composites 123, 125 hydrothermal effects 377 impact loading 408 resistance 418–419 inflection point 461 in-plane contraction 260 deformation/twisting 259, 429 displacement 256, 427 extension 260 loading 86 shear 61, 96, 100–101, 110, 122, 159 , 163, 204, 224, 257–258, 260, 323, 389 modulus... 353, 450 loading conditions 215, 441, 460 cyclic 400, 404 frequency 406 direction 402 proportional loading, theorem of 153 longitudinal compression 61, 104–106, 113–114, 122, 323 longitudinal compressive strength 128 longitudinal conductivity 362 longitudinal elasticity modulus 222 longitudinal elongation 204, 207 longitudinal plies 184 failure 188 longitudinal strain 215, 221 longitudinal strength... macroheterogeneity 23 macrostructure 23 man-made fibers 15 mass diffusion coefficient 378 mass moisture concentration 377 material behavior 157 , 197 creep 5 damage, accumulation of 399 deformation 170 delamination 412–413 internal 410 interlaminar 410 density 4 diffusivity coefficient 383 integrity 17 microstructure 66, 80 modulus 124 nonlinearity 215, 223 porosity 60, 127 relaxation time 402 stiffness... high-cycle 407 low-cycle, 407–408 strength 85, 201, 405 fiber buckling 115 116 elasticity modulus 71 failure 109, 351 length 67 modulus 88 orientation angle 177, 221–222, 226, 437 placement 25 strength 66, 88 deviation 68 fiber volume fraction 61, 89, 96, 102, 107, 115, 124, 127, 204, 362 fiberglass–epoxy composite 175–176 fiberglass-knitted composites 239 fiber–matrix deformation 119 interaction 61 interface... PAN/pitch-based fibers 11 parallel fibers 15 plain fabric 236 plastic strains 141 plasticity, theory of 141–142 flow theory 141 152 ply architecture 57 degradation 224 elongation 90 interface 195 microstructure 84, 87 orientation angle 370 waviness 423–425 Poisson’s effect 118, 131, 188, 330, 394 Poisson’s ratio 48, 87, 92, 102, 105, 109, 111, 114, 136, 143, 145–146, 157 , 240, 247, 292–293, 301, 428, 450... restricted shear deformation 181 revolution, surface of 458 Riemannian space 43 rotation angles 42, 177, 180, 217, 220, 256, 282, 371 rotation components 257 safety factor 354 minimization 476 sandwich structure 280, 298 shear deformation 172 failure 114 modulus 87, 102, 128, 136 shear stiffness 276, 347 coefficient 280 shear strain 158 , 215 shear stress 36, 81, 158 shear–extension coupling 240, 290 coefficient... 104–106, 112, 159 , 188, 202, 204, 224, 323, 389 triaxial woven fabric 236 twill fabric 236 twisting 257–259 deformation 284 two-matrix composites 201 two-matrix fiberglass composite 204 ultimate angular velocity 467 ultimate tensile load 70 ultimate tensile strain 125 uniaxial tension 144, 172, 202, 213, 216, 221, 356, 440 unidirectional anisotropic layer 162 unidirectional orthotropic layer 154 unidirectional... elongation 115 failure 353, 355, 357 materials 16 modulus 87, 92 nonlinearity 97 shear modulus 120 stiffness 80, 202, 450 strain 100 Subject index strain energy 118 volume 98 volume fraction 124, 362 maximum moisture content 381 maximum strain criterion 323 failure criterion 329 maximum stress criterion 323–325, 327, 345 maximum tensile stress 413 mean longitudinal strain 194 mechanical properties 6, 101 mechanics . calculation yields 480 Advanced mechanics of composite materials a h = 188 mm and a c = 210 mm. For a structure with D = 4m, we have h = 36 mm, δ h = 9.4 mm, δ c = 2.35 mm. The mass of the unit surface. uniform strength 445, 447 layer, mechanics of 133 composite material 9 filled 9 reinforced 10 unidirectional 61, 236 compression 101, 159 , 204 constant of integration 311, 315, 468 convolution theorem. constraints in Eqs. (8.102) and (8.103) are active. To find the strength safety 478 Advanced mechanics of composite materials factor n s , we need to put ∂ M/ ∂ n s = 0, where M is defined by Eq. (8.111).