Substitution of Equations(9.10)and (9.11)into (9.17)yields (9.18) Note that when the length-to-width ratio,,Lc/w→o，∞(qy)a→Si6/Si,as given by Equation (9.9c). Pindera and Herakovich [3]examined the influence of end constraint on the evaluation of shear modulus,G2,from the off-axis tension specimen using the elasticity solution of Halpin and Pagano[2]and found that the procedure outlined in Section 9.1 leads to error in G2.The main source of error is the neglect of the shear stress txy in Equation (9.7).The proper transformation is [3] t2=-mnox +(m2-n2)txy (9.19) This equation,combined with the definition of G2(Equation(9.6)),yields an expression for the correct value of the shear modulus in terms of the apparent modulus(G2),evaluated using the procedure in Section 9.1 [3], 3m2-n2） 1+ Ga=(G2.1-B56/5 2mn (9.20) where m cos0,n sine,and B is defined in Equation (9.16).As (w/Lc) e∞，B→0，and the apparent shear modulus approaches Gi2: The above expressions for apparent off-axis properties(E)a,(vxy)a,and (nx)a may be used to correct measured values,or to design the off-axis specimen for minimum error resulting from end constraint.An obvious way to reduce the error is to use specimens with large aspect ratios,Lc/w.As mentioned early in this chapter,specimens are typically 230 mm long and between 12.7 and 25.4 mm wide.For 38-mm-long tabs at the ends,this corresponds to aspect ratios between 6 and 12.For a carbon/polyimide composite specimen with an aspect ratio of 10 and 10 off-axis angle,Pindera and Herakovich [3]found an error in E,of about 2 to 4%.The error in shear coupling ratio is larger,as will be discussed later. The error in shear modulus G2 for a 10 off-axis carbon/polyimide speci- men at LG/w=10 is approximately 12 to 15%[3].For proper determination of G12,Pindera and Herakovich [3]recommend use of coupons with an aspect ratio of 10 or more and an off-axis angle of 45. 2003 by CRC Press LLCSubstitution of Equations (9.10) and (9.11) into (9.17) yields (9.18) Note that when the length-to-width ratio, LG/w → ∞, (ηxy)a → , as given by Equation (9.9c). Pindera and Herakovich [3] examined the influence of end constraint on the evaluation of shear modulus, G12, from the off-axis tension specimen using the elasticity solution of Halpin and Pagano [2] and found that the procedure outlined in Section 9.1 leads to error in G12. The main source of error is the neglect of the shear stress τxy in Equation (9.7). The proper transformation is [3] τ12 = –mnσx + (m2 – n2)τxy (9.19) This equation, combined with the definition of G12 (Equation (9.6)), yields an expression for the correct value of the shear modulus in terms of the apparent modulus (G12)a, evaluated using the procedure in Section 9.1 [3], (9.20) where m = cosθ, n = sinθ, and β is defined in Equation (9.16). As (w/LG) → ∞, β → 0, and the apparent shear modulus approaches G12. The above expressions for apparent off-axis properties (Ex)a, (νxy)a, and (ηxy)a may be used to correct measured values, or to design the off-axis specimen for minimum error resulting from end constraint. An obvious way to reduce the error is to use specimens with large aspect ratios, LG/w. As mentioned early in this chapter, specimens are typically 230 mm long and between 12.7 and 25.4 mm wide. For 38-mm-long tabs at the ends, this corresponds to aspect ratios between 6 and 12. For a carbon/polyimide composite specimen with an aspect ratio of 10 and 10° off-axis angle, Pindera and Herakovich [3] found an error in Ex of about 2 to 4%. The error in shear coupling ratio is larger, as will be discussed later. The error in shear modulus G12 for a 10° off-axis carbon/polyimide speci￾men at LG/w = 10 is approximately 12 to 15% [3]. For proper determination of G12, Pindera and Herakovich [3] recommend use of coupons with an aspect ratio of 10 or more and an off-axis angle of 45°. ηxy a 1 2 G 2 S S 1 3 2 w L S S S S ( ) = +       −                       − 16 11 66 11 16 11 S S 16 11 G G m n mn S S a 12 12 2 2 16 11 1 3 2 1 = ( ) + ( ) − − ( ) β β / TX001_ch09_Frame Page 136 Saturday, September 21, 2002 5:01 AM © 2003 by CRC Press LLC