được xây dựng cuối cùng cong vênh w = 0 và kể từ khi w = -2ARd8/dz sau đó làm / dz = 0 vào cuối được xây dựng trong. (2) Tại gr kết thúc = 0, như không có hạn chế và không tải được áp dụng trực tiếp bên ngoài. Vì vậy, từ Eq. (1 1,54), d'O / d2 = 0 vào cuối miễn phí. Từ (1) -T/GJ = Từ (2) B = (T / GJ) tanh PL | 474 Structural constraint rR follows from the moment of inertia of the wire about an axis through its centre of gravity. Hence ihd - . . íhdỸ hd h d V which simplifies to Equation that is d0 d30 T GJ _ET may now be solved for dớ dz. Rearranging and writing g2 GJ ETR we have d30 2dớ 2 T . The solution of Eq. iii is of standard form . dớ T t v A cosh uz Bsinh fiz az GJ The constants A and B are found from the boundary conditions. 1 At the built-in end the warping w 0 and since w 2ARd0 dz then d0 dz 0 at the built-in end. 2 At the free end rr 0 as there is no constraint and no externally applied dhect load. Therefore from Eq. d2ổ dz2 0 at the free end. From 1 A T GJ From 2 B T GJ tanh zL so that dớ T . r ._v .X 1 cosh zz tanh fiL sinh fiz dz GJ or dớ T r cosh fi L z 1 . . 1----------y------- IV dz GJ L cosh zZ. J The first term in Eq. iv is seen to be the rate of twist derived from the St. Venant torsion theory. The hyperbolic second term is therefore the modification introduced by the axial constraint. Equation iv may be integrated to find the distribution of angle of twist Ớ the appropriate boundary condition being 0 at the built-in end. Thus _ T sinh fi L z sinh fiL GJ fl cosh fiL fl cosh fiL Constraint of open section beams 475 8 Fig. Stiffening effect of axial constraint. and the angle of twist ớp E at the free end of the beam is TL tanh zZA 0F E 757 1------77 vi GJ fiL J Plotting 0 against z Fig. illustrates the stiffening effect of axial constraint on the beam. The decrease in the effect of axial constraint towards the free end of the beam is shown by an examination of the variation of the St. Venant 77 and Wagner rr torques along the beam. From Eq. iv dớ r coshu L-z . dz cosh ịiL J v and m d3ớ cosh z Z z . Tỵ - TR T viii dz3 cosh Ị1L Tj and Tr are now plotted against z as fractions of the total torque T Fig. . At the built-in end the entire torque is carried by the Wagner stresses but although the constraint effect .