12.1.3.2 Calculation Procedure 1.Complete calculation:The ply proportions are given,the matrix I[A]of the Equation 12.7 is known,and then-after inversion-we obtain the elastic moduli of the laminate (Equation 12.9).Multiplying 12.9 with the thickness b (unknown)of the laminate: bEox _ ix (N Ey bEoy 1 E G bYoxy 是 Txy Then introducing a multiplication factor of b for the stresses in the ply-or the group of plies-corresponding to the orientation k(see Equation 11.8): bOx E1 Ev bOy bEoy bixy E31 E32 bYoxy ply ply nk laminate and in the orthotropic coordinates of the ply (see Equation 11.4): 、2 -2cs bO: 2cs c=cose;s sin0 bter SC -Sc (c2-s2) bTxy】 ply nok ply nok plynk Saturation of the Hill-Tsai criterion leads then to Equation 12.10 where the above known stress resultants values appear in the numerator as: (ba (bo)(bo)(ba=x1 -2 O rupture rupture rupture rupture After having written an analogous expression for each orientation k of the plies, one retains for the final value of the laminate thickness,the maximum value found for b. One can read directly these moduli in Tables 5.1to5.15 of Section 5.4.2 for balanced laminates of carbon,Kevlar,and glass/epoxy with V=60%fiber volume fraction. 2003 by CRC Press LLC12.1.3.2 Calculation Procedure 1. Complete calculation: The ply proportions are given, the matrix [A] of the Equation 12.7 is known, and then—after inversion—we obtain the elastic moduli of the laminate (Equation 12.9). 9 Multiplying 12.9 with the thickness h (unknown) of the laminate: Then introducing a multiplication factor of h for the stresses in the ply—or the group of plies—corresponding to the orientation k (see Equation 11.8): and in the orthotropic coordinates of the ply (see Equation 11.4): Saturation of the Hill–Tsai criterion leads then to Equation 12.10 where the above known stress resultants values appear in the numerator as: After having written an analogous expression for each orientation k of the plies, one retains for the final value of the laminate thickness, the maximum value found for h. 9 One can read directly these moduli in Tables 5.1 to 5.15 of Section 5.4.2 for balanced laminates of carbon, Kevlar, and glass/epoxy with Vf = 60% fiber volume fraction. 1 h -- he ox he oy Óhg oxy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ 1 E x ----- nyx E y –------- h xy G xy -------- nxy E x –------- 1 E y ----- m xy G xy -------- h x E x ----- m y E y --- 1 G xy -------- Nx Ny ÓTxy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ = hsx hsy Óhtxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ E11 E12 E13 E21 E22 E23 E31 E32 E33 he ox he oy Óhg oxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = ply n∞k ply n∞k laminate hs
hst Óht
t˛ Ô Ô Ì ˝ Ô Ô Ï ¸ c 2 s 2 –2cs s 2 c 2 2cs sc sc – c 2 s 2 ( ) – hsx hsy Óhtxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = == c cosq; s sinq ply n∞k ply n∞k ply n∞k hs
( )2 s
2 --------------- hst ( )2 st 2 --------------- hs
( ) hst ( ) s
2 – --------------------------- ht
t 2 ( ) t
t 2 + + -------------- h2 = ¥ 1 rupture rupture rupture rupture TX846_Frame_C12 Page 241 Monday, November 18, 2002 12:27 PM © 2003 by CRC Press LLC