formulation-equivalent to that of a classical homogeneous beam-in the following manner: degree of freedom:about x axis:0 ·elastic center0:it is such that∫EydS=∫E,zdS=0 .principal axes:tbey are sucb tbat Eyzds equivalent stiffnesses: 〈E）=∑Bli；〈E)=∑PB n=小c(-9++s torsion center:coordinates in principal axes: 1 ye=-(E 「E,zΦdS Ze= 高505 dM, equilibrium relation:- =0 (M.constant) dx de ·constitutive relation:M.=(Gj)） dx 2-2 d6xa 。sbear stresses t=Gra示 (16.4 T:=( .function (y,z):it is the solution to the problem: 子Φ，子Φ =0 in domain D of tbe section. dΦ dn zny-yn:on the external boundary dD. witb internal continuity: ①，=① along internal dΦ -zm,+ym=G aΦ zny +yn: boundaries ti and the uniqueness condition:E ds=0 。strain energy density dw1M R=动 2003 by CRC Press LLCformulation—equivalent to that of a classical homogeneous beam —in the following manner: • degree of freedom: about x axis: • elastic center 0: it is such that • principal axes: they are such that • equivalent stiffnesses: • torsion center: coordinates in principal axes: • equilibrium relation: • constitutive relation: • shear stresses (16.4) • function F(y, z): it is the solution to the problem: with internal continuity: and the uniqueness condition: • strain energy density qx Ei y dS DÚ Ei z dS DÚ = = 0 Ei yz dS DÚ = 0 EIz · Ò EiIzi ; EIy · Ò i  EiIyi i = =  · Ò GJ Gi y ∂F ∂z ------- z∂F ∂ y ------- y 2 z2 – + + Ë ¯ Ê ˆ dS DÚ = yc 1 EIy · Ò ------------ Eiz F dS DÚ = – zc 1 EIz · Ò ------------- Ei y F dS DÚ = dMx dx ---------- = 0 ( ) Mx = constant Mx · Ò GJ dqx dx = -------- txy Gi dqx dx -------- ∂F ∂y ------- – z Ë ¯ Ê ˆ = txz Gi dqx dx -------- ∂F ∂z ------- + y Ë ¯ Ê ˆ = ∂ 2 F ∂y 2 --------- ∂ 2 F ∂z2 + --------- = 0 in domain D of the section. ∂F ∂n ------- = zny – ynz on the external boundary ∂D. Ó Ô Ô Ì Ô Ô Ï Fi = Fj Gi ∂Fi ∂n-------- – zny + ynz Ë ¯ Ê ˆ Gj ∂Fj ∂n-------- – zny + ynz Ë ¯ Ê ˆ = ˛ Ô ˝ Ô ¸ along internal boundaries
ij EiF dS DÚ = 0 dW dx -------- 1 2 -- Mx 2 · Ò GJ = ------------ TX846_Frame_C16 Page 314 Monday, November 18, 2002 12:32 PM © 2003 by CRC Press LLC