Then 111 = In + 2zl,2 " ^Y^is ^~^%2 " m^s + y%3 



"^13 =Il3 +^l23 -yl33 

 22 = ^22' I23 " 23' I33 " 33 



Since 1^2 I33 - I23 ^iH i" general be nonzero, it is possible to solve for y and z such that 

 I = I = (i.e., select y, z coordinate system such that 1^2 = I13 "" ^)- I*" ''^'^ barred 

 coordinate system is used, it is customary to callT^^ = l/EA,l22 = Iyy/E(lyylzz - ly^^), 

 I23 = Iyz/E(IyyI.z - lyz')' and T33 = KJEil^^l^^ - Iy,2); Equation [10] of Appendix A.2 

 validates the expressions for I221 1231 133- (These terms are defined conventionally either 

 as geometric integrals, or by the geometric summations given below.) Thus* 



(9U V 



X X ^ -, 



= [la] 



ax EA 



dd I M +l,JA^ 

 -1 = '' ' '^ ^ [lb] 



'' Edyyizz -I^z) 



— = [Ic] 



"^^ E(I I -l2 ) 



^yyzz yz' 



Thus the choice of this coordinate system (elastic axis coordinates y , "z) uncouples 

 the tension and pure bending elastic equations. For the other three equations, a coordinate 

 system (generally not the barred system for bending) y, z can be found to uncouple the tor- 

 sion from the shear. The center of this system is called the shear center. 



V [2a] 



dx 



1 = 1 • 



y 1 



KAyyG y KA^^G 





1 _ 1 . 



V + 

 KA G y KA^ G 



[2b] 



dd 



1 = 



M [2c] 



*( GJ^ '^ 



The foregoing equations follow from the discussion in the footnote on page 48, that the shear 

 and torsion deformations can be expressed in the form: 



e^ = N,,V,. + N,^V, + N,,M, 



ax 



'1 1 V "^ ' 12 'z ^ 'M3'"x 



*An alternative method of derivation is given in Appendix A.2. 



