东南大学：《建筑力学 Architectural Mechanics》课程教学课件（英文讲稿）A11 Torsion

. Determine the average angulary0.04 in.deformation or shearing strain of the0Eblock.2in.E0.04in.AYxy = 0.020radYxy ~ tan Yxy2in.B: Apply Hooke's law for shearingstress and strain to find thecorresponding shearing stress.Txy, =Gxy = (90×103 psi)(0.020rad)=1800psi. Use the definition of shearing stressto find the force P.P = txyA = (1800psi)(8in.)(2.5in)= 36×1031bP = 36.0kips16

• Determine the average angular deformation or shearing strain of the block. 0.020rad 2in. 0.04in.  xy  tan xy   xy  • Apply Hooke’s law for shearing stress and strain to find the corresponding shearing stress. 90 10 psi0.020rad 1800psi 3  xy  G xy    • Use the definition of shearing stress to find the force P. 1800psi8in.2.5in. 36 10 lb 3 P   xyA    P  36.0kips 16

Static Equivalency: Multiplying the previous equation by theshear modulus, Gy - PGymaxpFrom Hooke's Law, t = Gy, so t =TmaxRThe shearing stress varies linearly with thepradial position in the section.: Recall that the sum of the moments fromthe internal stress distribution is equal tothe torque on the shaft at the section,=max IT=[pt dA="max[p?dA:. The results are known as the elastictorsion formulas, (I, - Polar moment ofTcTpinertia)and tmaxIIp17

max max 2 T dA dA I p c c          • Recall that the sum of the moments from the internal stress distribution is equal to the torque on the shaft at the section, max and p p Tc T I I      • The results are known as the elastic torsion formulas, (Ip – Polar moment of inertia) • Multiplying the previous equation by the shear modulus, max    G c G  max    c From Hooke’s Law,   G , so  The shearing stress varies linearly with the radial position in the section. Static Equivalency 17 c

Torsional Stress & Angle of Twist. Torsional section modulus: [m3]: W, = I p /c.. Torsional stressTTTcTppUTmaxTI,W1WCpppp: Angle of twist per unit lengthT11TL1Tpd2YLy=pp =DL1GIGIpGpGpPL: GI,: Torsional rigidity [N.m?]. The equations are limited to bars of circular cross section (eithersolid or hollow) that behave in a linearly elastic manner.18

. W I c p P  max , p p p p Tc T T T I W I W c          • Torsional section modulus: [m3 ]: 1 1 1 p p p T T TL L L G G I GI GI                   • Angle of twist per unit length • Torsional stress • GIp : Torsional rigidity [N•m2 ] Torsional Stress & Angle of Twist 18 • The equations are limited to bars of circular cross section (either solid or hollow) that behave in a linearly elastic manner