12.1.3.1 Principle of Calculation Apparent moduli of the laminate:The matrix A]evaluated using Equation 12.8 can be inverted,and one obtains Equation 12.9 as: Eox _P四 n E Ey Ey We have already determined the apparent moduli and the coupling coefficients of the laminate. Nonrupture of the laminate:Let o,o,and te be the stresses in the orthotropic axes t,t of one of the plies making up the laminate that is subjected to the loadings NN.T Let b be the thickness of the laminate (unknown at the moment)so that the rupture limit of the ply using the Hill-Tsai failure criterion is just reached. One then has for this ply: _+ -1 rupture rupture rupture rupture Multiplying the two parts of this equation with the square of thickness b: (Gb)b)(b)(b)(b)=b o o Tit (12.10) rupture rupture rupture rupture To obtain the values (ob),(ob),(tb),one has to multiply with b the global stresses t that are applied on the laminate,to become (ab),(b), (Tab)which are the known stress resultants: N =(Goxb);Ny =(Gob);T=(Tob) Then,for a ply,the calculation of the Hill-Tsai criterion can be done by substituting for the unknown global stresses the known stress resultants NN.T This leads to the calculation of the thickness b so that the ply under consideration does not fracture. In this way,each ply number k leads to a laminate thickness value denoted as b.The final thickness to be retained will the one with the highest value. For the Hill-Tsai failure criterion,see Section 5.2.3 and detailed explanation in Chapter 14. 2003 by CRC Press LLC
12.1.3.1 Principle of Calculation Apparent moduli of the laminate: The matrix [A] evaluated using Equation 12.8 can be inverted, and one obtains Equation 12.9 as: We have already determined the apparent moduli and the coupling coefficients of the laminate. Nonrupture of the laminate: Let s, st , and tt be the stresses in the orthotropic axes , t of one of the plies making up the laminate that is subjected to the loadings Nx, Ny, Txy. Let h be the thickness of the laminate (unknown at the moment) so that the rupture limit of the ply using the Hill–Tsai failure criterion is just reached. One then has for this ply8 : Multiplying the two parts of this equation with the square of thickness h: (12.10) To obtain the values (sh), (st h), (tth), one has to multiply with h the global stresses sox, soy, toxy that are applied on the laminate, to become (soxh), (soyh), (toxyh) which are the known stress resultants: Then, for a ply, the calculation of the Hill–Tsai criterion can be done by substituting for the unknown global stresses the known stress resultants Nx, Ny, Txy. This leads to the calculation of the thickness h so that the ply under consideration does not fracture. In this way, each ply number k leads to a laminate thickness value denoted as hk. The final thickness to be retained will the one with the highest value. 8 For the Hill–Tsai failure criterion, see Section 5.2.3 and detailed explanation in Chapter 14. 1 h -- e ox e oy Óg oxy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ 1 E x ----- nyx E y –------- h xy G xy -------- n xy E x –------- 1 E y ----- m xy G xy -------- h x E x ----- m y E y ----- 1 G xy -------- sox soy Ót oxy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ = s 2 s 2 ----- st 2 st 2 ----- sst s 2 – ---------- t t 2 t t 2 + + ----- = 1 rupture rupture rupture rupture ( ) sh 2 s 2 --------------- ( ) sth 2 st 2 --------------- ( ) sh ( ) sth s 2 – --------------------------- ( ) t th 2 t t 2 + + ---------------- h2 = rupture rupture rupture rupture Nx == = ( ) soxh ; Ny ( ) soyh ; Txy ( ) t oxyh TX846_Frame_C12 Page 240 Monday, November 18, 2002 12:27 PM © 2003 by CRC Press LLC
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 LLC
12.1.3.2 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
(2)Simplified calculation:One can write more rapidly the Equation 12.10 if one knows at the beginning for each orientation the stresses due to a global uniaxial state of unit stress applied on the laminate:first oox 1 (for example,1 MPa),then ooy =1 MPa,then too=1 MPa. Assume first that the state of stress is given as: ox=1(MPa) Cosx =0 Toxy=0 Inverting the Equation 12.9 leads to (1 MPa Ey 0 E 是 0 which can be considered as "unitary strains"of the laminate.These allow the calculation of the stresses in each ply by means of Equations 11.8 and then 11.4, successively,as: 「E E12 E13 E2 E22 E的 t E31 E32 plyn plyn Laminate and in the orthotropic coordinates of the ply (Equation 11.4): =c0s0 2cs s=sin0 SC -Sc (c2-52 ply n plynk plynk Consider then the state of stresses: o“=0 =1 (MPa) t=0 Following the same procedure,one can calculate o,o,,and e in the orthotropic axes of each ply for a global stress on the laminate that is reduced to oy=1 MPa. 2003 by CRC Press LLC
(2) Simplified calculation: One can write more rapidly the Equation 12.10 if one knows at the beginning for each orientation the stresses due to a global uniaxial state of unit stress applied on the laminate: first (for example, 1 MPa), then MPa, then MPa. Assume first that the state of stress is given as: Inverting the Equation 12.9 leads to which can be considered as “unitary strains” of the laminate. These allow the calculation of the stresses in each ply by means of Equations 11.8 and then 11.4, successively, as: and in the orthotropic coordinates of the ply (Equation 11.4): Consider then the state of stresses: Following the same procedure, one can calculate , , and in the orthotropic axes of each ply for a global stress on the laminate that is reduced to MPa. sox ¢ = 1 soy ¢¢ = 1 t oxy ¢¢¢ = 1 s¢ ox = 1 MPa ( ) s¢ ox = 0 t ¢ oxy = 0 e ¢ ox e ¢ oy g ¢ Ó oxy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ 1 E x ----- nyx E y –------- hxy G xy -------- n xy E x –------- 1 E y ----- m xy G xy -------- h x E x ----- m y E y ----- 1 G xy -------- 1 MPa 0 Ó 0 ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ = s ¢ x s ¢ y t ¢ Ó xy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ E11 E12 E13 E21 E22 E23 E31 E32 E33 e ¢ ox e ¢ oy g ¢ Ó oxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = ply n∞k ply n∞k laminate s ¢ s ¢ t t ¢ Ó t ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ c 2 s 2 –2cs s 2 c 2 2cs sc sc – c 2 s 2 ( ) – s ¢ x s ¢ y t ¢ Ó xy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ c = cosq s = sinq = ply n∞k ply n∞k ply n∞k sox≤ = 0 soy≤ = 1 MPa ( ) t oxy ≤ = 0 s ¢¢ st ¢¢ t t ¢¢ soy ¢¢ = 1 TX846_Frame_C12 Page 242 Monday, November 18, 2002 12:27 PM © 2003 by CRC Press LLC