东南大学：《建筑力学 Architectural Mechanics》课程教学课件（英文讲稿）A16 Stress State

Plane Stress Transformation - Alternative Waysingsinecos0cosaOT.TxyxV-sinecosacosO-sina06Tx'yOx+o,Ox-0,x-a,sin 20 + t., cos 20cos20 + tr.sin20222Gra+a,ax-o-0sin20+t..cos20cos20-T..sin2022216

cos sin cos sin sin cos sin cos cos 2 sin 2 sin 2 cos 2 2 2 2 sin 2 cos 2 cos 2 sin 2 2 2 2 T x x y x xy x y y xy y x y x y x y xy xy x y x y x y xy xy                                                                                               16 Plane Stress Transformation – Alternative Way

Mohr's CircleTrt. The previous equations are combined toyield parametric equations for a circle.Do.+0Qx-Oycos20+tsin20min220.-0xysin20+t.cos20T2a.AoBa.= R222E·Principal stresses0.-02O=trysin20+t..cos20=2Ominaw±R=max,minCx=Ove2Omax1Tytan20Oma(g-0 ) /2Note: defines two angles separated by 90°ItishardtopredictwhichofthetwoanglesresultsOmininthemaximum/minimumprincipalstress17

• The previous equations are combined to yield parametric equations for a circle,   2 2 2 2 2 cos 2 sin 2 2 2 sin 2 cos 2 2 , ; 2 2 x y x y x xy x y x y xy x y x y x ave x y ave xy R R                                                           • Principal stresses   2 2 max,min o 0 sin 2 cos 2 2 2 2 tan 2 2 Note: defines two angles separated by 90 x y x y xy x y x y x ave xy xy p x y R                                                   17 Mohr’s Circle • It is hard to predict which of the two angles results in the maximum/minimum principal stress