AE350_Torsion

AE350_Torsion - TORSION TorsionofUniformBars

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TORSION Torsion of Uniform Bars Aircraft structures subjected to torque; need to determine resulting shear stress and twist angle. “Center of Twist” of a cross section is the line about which cross section rotates during twisting , i.e., u = v = 0 along z –axis (u and v are in plane displacements ). Constant cross section (arbitrary shape) z y x T T AE/ME350 Jha Torsion-1
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St. Venant’s Principle St.Venant’s assumption: During torsional deformation , the plane sections may “warp,” but the projection on the x yplane rotates as a rigid body. Displacement field : is a warping function Twist angle, , where = total angle of rotation (twist) at z relative to the end at z = 0 ( = twist angle per unit length) Based on the above displacement field: / z ,,( , ) uz y vz x w x y   (, ) xy 0 xx yy zz xy   0 xx yy zz xy   xy vu x y     0 yz 0 xz AE/ME350 Jha Torsion-2   0, 0 zy zx 
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Torsion Shear Stresses Due to assumed displacements, only two shear stresses need to be evaluated. They are independent of z, that is, the shear stresses are same everywhere along length of bar. and need to be determined as functions of x and y only. For bars of arbitrary cross section, warping (out of plane displacement) of the section occurs when twisted. Integrate to obtain induced warping. Y X y x dy dx Z T yz xz () yz yz xz xz wv GG yz wu x z   yz xz & ww x y AE/ME350 Jha Torsion-3 Y
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Bars with Circular Cross Section Center of the circular cross section is the center of twist Torque (see derivation in Sun, Ch 3) Polar moment of inertia of the section (a = radius) GJ = torsional rigidity (G = Shear modulus) TG J 24 1 2 A Jr d A a   AE/ME350 Jha Torsion-4
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Torsion–Circular Cross Section ; yz xz Gx G y   (, ) 0 xy rz / Tr J  2 2 a J AE/ME350 Jha Torsion-5
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Torsion Narrow Rectangular Cross Section
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AE350_Torsion - TORSION TorsionofUniformBars

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