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青岛理工大学毕业设计（论文）
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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；σ
ji,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
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青岛理工大学毕业设计（论文）
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 ε
﹡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 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 compatible setε
﹡
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 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
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青岛理工大学毕业设计（论文）
been summarized in Section 7.2.1,which will transfer into specific expressions in different methods.As in ordinary 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 {}e and {}e are the element nodal force vector and displacement vector,respectively,while{}e is element stiffness matrix;the overbar 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 with 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
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青岛理工大学毕业设计（论文）
obtained by enforcing overall equilibrium on one member, which gives
Where [L]= the element force transformation matrix and {P}e =the element primary forces vector.It is important to emphasize that the physical basis of Eq.(7.5)is member overall equilibrium.
Take a conventional plane truss member for exemplification(see Figure 7.4),one has
FIGURE 7.5 Coordinate transformation.
and
where EA/l=axial stiffness of the truss member and P=axial force of the truss member.
7.2.3 Coordinate Transformation
The values of the components of vector V,designated by v1,v2,and v3 or simply,are associated with the chosen set coordinate axes.Often it is
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