366 Mechanics of Materials s14.8 14.8.Effect of lateral restraint (a)Restraint in one direction only Consider a body subjected to a two-dimensional stress system with a rigid lateral restraint provided in the y direction as shown in Fig.14.4.Whilst the material is free to contract laterally in the x direction the "Poisson's ratio"extension along the y axis is totally prevented. 一可x m Rigid restraint Fig.14.4.Material subjected to lateral restraint in the y direction. Therefore strain in the y direction with a,and a,both compressive,i.e.negative, 1 =6,=-Ea,-G)=0 Gy=VGx Thus strain in the x direction =ex=- (ox-voy) E 】 一 E(0x-v2a,) =-1-2) (14.9) E Thus the introduction of a lateral restraint affects the stiffness and hence the load-carrying capacity of the material by producing an effective change of Young's modulus from E to E/(1-v2) (b)Restraint in two directions Consider now a material subjected to a three-dimensional stress system o,and o:with restraint provided in both the y and z directions.In this case, 1 8,=-Eo,-0x-o）=0 (1) and e:=-E(o.-ox-o)-0 (2)366 Mechanics of Materials $14.8 14.8. Effect of lateral restraint (a) Restraint in one direction only Consider a body subjected to a two-dimensional stress system with a rigid lateral restraint provided in the y direction as shown in Fig. 14.4. Whilst the material is free to contract laterally in the x direction the “Poisson’s ratio” extension along they axis is totally prevented. UY I I Rigid restraint Fig. 14.4. Material subjected to lateral restraint in the y direction. Therefore strain in the y direction with a, and oy both compressive, i.e. negative, 1 ’E =E = -- (ay-va,) = 0 .. a, = va, Thus strain in the x direction 1 E = E, = --(a, - va,) 1 E = - - (a, - v%,) (14.9) Thus the introduction of a lateral restraint affects the stiffness and hence the load-carrying capacity of the material by producing an effective change of Young’s modulus from E to E/(1 -v*) (b) Restraint in two directions Consider now a material subjected to a three-dimensional stress system a,, a, and a, with restraint provided in both the y and z directions. In this case, (1) (2) 1 ’E 1 ‘E E = --(ay-va,-va,) =o and E = --(a,-vo,-vay)=0