520 Mechanics of Materials 2 §12.4 with a corresponding hoop stress: E6, OH= r This stress sets up a force in the circumferential direction of E8, FH=OH XA=- ×t×1. This force has an outward radial component from both sides of the element of: de de 2E8,tde Fk=2FHsin 2-2F82-2 Eδ，td8 r and since the strip is of unit width,rde =1 E8,t FR二2 This force can be considered as a distributed load along the strip (since equal values will apply to all other unit lengths)and will act in opposition to the mis-match displacements caused by the membrane stresses. It the strip were considered to be a simple beam then,the differential equation of bending would be: Eldy E8,t dx= but,as for the case of the deformation of circular plates in 7.2,the restraint on distortion produced by adjacent strips needs to be allowed for by replacing EI by the plate stiffness constant or flexural rigidity Et3 D=121-2): E8,t ie. r2 E32 =-4D848,=-4DB4y (1) where 6产=31-) r212 and y=6,. The solution to egn.(1)is of the form: y=8,ef (AI Cos Bx +A2 sin Bx)+e (A3 cos Bx +A4 sin Bx) (2) Now as x→o,8,→o and A1=A2=0. Atx=0.M=Ma and Ddy =-MA.520 Mechanics of Materials 2 $12.4 with a corresponding hoop stress: Ear UH = - r This stress sets up a force in the circumferential direction of This force has an outward radial component from both sides of the element of: ES, t de r -- and since the strip is of unit width, rde = 1 ES,t .. FR = - r2 This force can be considered as a distributed load along the strip (since equal values will apply to all other unit lengths) and will act in opposition to the mismatch displacements caused by the membrane stresses. It the strip were considered to be a simple beam then, the differential equation of bending would be: EId4y E6,t dx4 r2 __. - -- but, as for the case of the deformation of circular plates in 7.2, the restraint on distortion produced by adjacent strips needs to be allowed for by replacing EI by the plate stiffness constant or flexural rigidity Et3 D= 12(1 - LJ2) : i.e. where 3(1 - w4) and y = 6,. r2t2 s”= The solution to eqn. (1) is of the form: Y = 6, = eS*(Al COS Bx + A2 sin Bn) + e-@*(A, cos #?x + A4 sin #?x) (2) Now as x -+ 00, S, -+ 00 and AI = A2 = 0. d2Y At x = 0, M = MA and D- = -MA. dx2