土木工程桥毕业设计-外文文献翻译.docVIP

外文文献翻译 7.2 Equilibrium Equations 7.2.1 Equilibrium Equation and Virtual Work Equation For any volume V of a material body having A as surface area, as shown in Figure 7.2, it has the following conditions of equilibrium: FIGURE 7.2 Derivation of equations of equilibrium. At surface points At internal points Where ni represents the components of unit normal vector n of the surface;Ti is the stress vector at the point associated with n；σji,j represents the first derivative of σij with respect to xj；and Fi is the body force intensity.Any set of stresses σij, body forces Fi,and external surface forces Ti that satisfies Eqs.(7.1a-c) is a statically admissible set. Equations(7.1b and c)may be written in(x,y,z) notation as and Whereσx,σy,andσz are the normal stress in(x,y,z) direction respectively;τxy,τyz,and so on,are the corresponding shear stresses in(x,y,z) notation;and Fx,Fy,and Fz ard the body forces in(x,y,z,)direction,respe- ctively. The principle of virtual work has proved a very powerful technique of solving problems and providing proofs for general theorems in solid mechanics. The equation of virtual work uses two independent sets of equilibrium and compatible(see Figure 7.3, where Au and AT represent displacement and stress boundary),as follows: compatible set equilibrium set or which states that the external virtual work(δWext) equals the internal virtual work(δWint). Here the integration is over the whole area A,or volune V,of the body. The stress field δij, body forces Fi,and external surface forces Ti are a statically admissible set that satisfies Eqs.(7.1a–c). Similarly, the strain field εij﹡ and the displacement ui﹡are a compatible kinematics set that satisfies displacement boundary conditions and Eq.(7.16) (see Section 7.3.1). This means the principle of virtual work applies only to small strain or small deformation. The important point to keep in mind is that, neither the admissible equilibrium set δij,Fi,and Ti(Figure 7.3a)nor the c