(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