1、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
2、 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； ji,j represents the first derivative of ij with respect to xj； and Fi is the body force intensity.Any set of stresses ij,body forcesFi,and external surface forces
3、 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 Where x, y,and z are the normal stress in(x,y,z) direction respectively； xy, yz,and so on,are the corresponding shear stresses in(x,y,z) notation；andFx,Fy,andFzard the body f
4、orces 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
5、and AT represent displacement and stress boundary),as follows: compatible set equilibrium set or which states that the external virtual work( Wext) equals the internal virtual work( Wint). Here the integration is over the whole area A,orvoluneV,of the body. The stress field ij,body forces Fi,and ext
6、ernal surface forces Tiare a statically admissible set that satisfies Eqs.(7.1a c).Similarly, the strain field ij 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
7、only to small strain or small deformation. The important point to keep in mind is that, neither the admissible equilibrium set ij,Fi,andTi(Figure 7.3a)nor the compatible set ij and ui ( Figure 7.3b)need be the actual state,nor need the equilibrium and compatible sets be related to each other in any
8、way.In the other words, these two sets are completely independent of each other. 7.2.2 Equilibrium Equation for Elements For an infinitesimal material element,equilibrium equations have been summarized in Section 7.2.1,which will transfer into specific expressions in different methods.As in ordinary
9、 FEM or the displacement method, it will result in the following element equilibrium equations: FIGURE 7.4 Plane truss member end forces and displacements.(Source: Meyers, V.J.,Matrix Analysis of Structures,New York: Harper & Row,1983. With permission.) Where F e and d e are the element nodal force
10、vector and displacement vector,respectively,whilek e is element stiffness matrix；theoverbar here means in local coordinate system. In the force method of structural analysis, which also adopts the idea of discretization,it is proved possible to identify a basic set of independent forces associated w
11、ith each member, in that not only are these forces independent of one another, but also all other forces in that member are directly dependent on this set.Thus,this set of forces constitutes the minimum set that is capable of completely defining the stressed state of the member.The relationship between basic and local forces may be
