O.N. Grigoriev et al./Composites: Part B 37(2006)530-541 Preliminary results of calculation on the base the experimental dependence F- F(w) We(MKM (MIla) MIla) Enax(M/M) D(HM) EA (MIla) EB(MIla) EA(MM) EB(M/M) GA(MIla) B(MIla 3.55X1 5.50×105500×1040 0.5 3.12×105503 1.61×10-54808 523×105475×104163×10-51.59×10-58.53×10°0.76×10 1.10 295×105 527×10-54546 3.41×10 494×105449×104345×10-53.37×10-517.1×10°151×10 4.80X10 436×104533×10-55.20×10-5256×10°227×10 50505050 m 744×10-54.117 453×105412×104753×10-57.35×10-5341×10°3.03×10 977 431×105392×1049.88×10-5965×10-5426×10° 78×10 243×105 124×1 408×1053.71×10412.1×10-5123×10-5512×10° .54X10 232×1053519 3.88×105353×104154×10-515.0×10-5597×10°5.30×10° 3.78×105344×104180×10-5176×10-5682×10°605×10° 5×105452411×10-533093.60×103327×10521.3×10-520.8×10-5767×1°6.81×10° 7.8 207×10550.27 243×10-53185346×1053.5×10246×10-5240×10-5853×10°1756×10° 2. Longitudinal strains E of the specimen are linearly Let us rewrite the second formula in Eq. ( 3)using the elastic distributed across its thickness, i.e. the hypothesis of moduli Ea and EB and allowing for the third assumption: plane cross-sections is fulfilled(this is determined by the ortion between support points ) and the total thickness of D=E∑3(-a)+E∑3(a-动) the specimen and the layer thicknesses do not change 3. We assume (not finally) that in each loading point, the The terms under the summation signs are the inertia elastic moduli of all layers do not depend on longitudinal moments for the A and B layers relative to the center of strains E, i.e. the elastic moduli of materials A and b do not stiffness C of the specimen cross-section [ 19] change across the thickness of the cross-section 4. The elastic moduli of the layers in compression and tension b b IB <. Let us come to the real heterogeneous laminated system; for With account of the first assumption, let us write down the his system, the following equations for the coordinate of the final equation for stiffness in any loading point as: center of stiffness C and bending stiffness [11](Fig. 10)are D=EACA mlB) The largest deflections of the specimen can be calculated by ∑E(=2-dm) D=3∑E(-)(3 2∑Ek(a06-xom,) Wmax128D where k is the number of a layer calculated from the bottom If experimental we values are taken as the largest deflections, stiffnesses for each loading point can be determined from Eq (7): D= o=1.14*+102*e3-7,34*108*g2+3.18°10°e 20n Strain s,。10m/m Fig.9. Effectiveσ re for the case of a quasi-homogeneous bear Fig. 10. Calculation scheme for the specimen cross-section.2. Longitudinal strains 3 of the specimen are linearly distributed across its thickness, i.e. the hypothesis of plane cross-sections is fulfilled (this is determined by the fact that the beam is in the state of pure flexure over the portion between support points), and the total thickness of the specimen and the layer thicknesses do not change. 3. We assume (not finally) that in each loading point, the elastic moduli of all layers do not depend on longitudinal strains 3, i.e. the elastic moduli of materials A and B do not change across the thickness of the cross-section. 4. The elastic moduli of the layers in compression and tension are equal. Let us come to the real heterogeneous laminated system; for this system, the following equations for the coordinate of the center of stiffness C and bending stiffness [11] (Fig. 10) are used zC Z P 12 kZ1 Ek z 2 06kKz 2 0Hk
2 P 12 kZ1 Ekðz06kKz0Hk Þ ; D Z b 3 X 12 kZ1 Ek z 3 6k Kz 3 Hk
(3) where k is the number of a layer calculated from the bottom upwards. Let us rewrite the second formula in Eq. (3) using the elastic moduli EA and EB and allowing for the third assumption: D Z EA X 6 kZ1 b 3 z 3 62kKz 3 H2k
CEB X 6 kZ1 b 3 z 3 62kK1Kz 3 H2kK1
(4) The terms under the summation signs are the inertia moments for the A and B layers relative to the center of stiffness C of the specimen cross-section [19]: IA Z b 3 X 6 kZ1 z 3 62k Kz 3 H2k
; IB Z b 3 X 6 kZ1 z 3 62kK1Kz 3 H2kK1
: (5) With account of the first assumption, let us write down the final equation for stiffness in any loading point as: D Z EAðIA CmIBÞ: (6) The largest deflections of the specimen can be calculated by the equation: wmax Z Fl3 128D (7) If experimental we values are taken as the largest deflections, stiffnesses for each loading point can be determined from Eq. (7): D Z Fl3 128we (8) Fig. 10. Calculation scheme for the specimen cross-section. Table 5 Preliminary results of calculation on the base the experimental dependence F–w F (w) We (MKM) Ey3 (MPa) smax (MPa) 3max (M/M) D (H M2 ) EA (MPa) EB (MPa) 3A (M/M) 3B (M/M) sA (MPa) sB (MPa) 0 0 3.55!105 0 0 5.060 5.50!105 5.00!104 0000 5 0.52 3.12!105 5.03 1.61!10K5 4.808 5.23!105 4.75!104 1.63!10K5 1.59!10K5 8.53!106 0.76!106 10 1.10 2.95!105 10.05 3.41!10K5 4.546 4.94!105 4.49!104 3.45!10K5 3.37!10K5 17.1!106 1.51!106 15 1.70 2.86!105 15.08 5.27!10K5 4.412 4.80!105 4.36!104 5.33!10K5 5.20!10K5 25.6!106 2.27!106 20 2.40 2.70!105 20.11 7.44!10K5 4.117 4.53!105 4.12!104 7.53!10K5 7.35!10K5 34.1!106 3.03!106 25 3.15 2.57!105 25.13 9.77!10K5 3.968 4.31!105 3.92!104 9.88!10K5 9.65!10K5 42.6!106 3.78!106 30 4.00 2.43!105 30.16 12.4!10K5 3.750 4.08!105 3.71!104 12.1!10K5 12.3!10K5 51.2!106 4.54!106 35 4.90 2.32!105 35.19 15.2!10K5 3.571 3.88!105 3.53!104 15.4!10K5 15.0!10K5 59.7!106 5.30!106 40 5.75 2.26!105 40.22 17.8!10K5 3.478 3.78!105 3.44!104 18.0!10K5 17.6!10K5 68.2!106 6.05!106 45 6.80 2.15!105 45.24 21.1!10K5 3.309 3.60!105 3.27!104 21.3!10K5 20.8!10K5 76.7!106 6.81!106 50 7.85 2.07!105 50.27 24.3!10K5 3.185 3.46!105 3.15!104 24.6!10K5 24.0!10K5 85.3!106 7.56!106 Fig. 9. Effective s–3 curve for the case of a quasi-homogeneous beam. 538 O.N. Grigoriev et al. / Composites: Part B 37 (2006) 530–541