桥梁设计外文翻译.docx

外文文献翻译7.2 equilibrium equations7.2.1 equilibrium equation and virtual work equationfor 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 pointsat 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；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 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 asandwherex,y,andz 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;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 setequilibrium setorwhich states that the external virtual work(wext) equals the internal virtual work(wint).here the integration is over the whole area a,or volune v,of the body.the stress field ij, body forces fi,and external surface forces ti are a statically admissible set that satisfies eqs.(7.1ac). similarly, the strain field ij and the displacement uiare a compatible kinematicsset 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 ij,fi,and ti(figure 7.3a)nor the compatible setij 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 elementsfor 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 fem or the displacement method, it will result in the following element equilibrium equations:figure 7.4 plane truss memberend forces and displacements.(source: meyers,v.j.,matrix analysis of structures,new york: harper & row,1983. with permission.)where fe and de are the element nodal force vector and displacement vector,respectively,whileke 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 obtained by enforcing overall equilibrium on one member, which giveswhere l= the element force transformation matrix and pe =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 hasfigure 7.5 coordinate transformation.andwhere ea/l=axial stiffness of the truss member and p=axial force of the truss member.7.2.3 coordinate transformationthe 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 necessary to reorient the reference axes and evaluate new values for the components of v in the new coordinate system.assuming that v has components vi and viin two sets of right-handed cartesian coordinate systems xi (old)and xi(new)having the same origin (see figure 7.5), and ei,eiare the unit vectors of xi and xi, respectively. thenwhere ,that is,the cosines of the angles between xiand xj axes for i and j ranging from 1 to 3;and =(lij)33 is called coordinate transformation matrix from the old system to the new system.it should be noted that the elements of lij or matrix are not symmetrical,lijlji.for example,l12 is the cosine of angle from x1to x2 and l21is that from x2to x1(see figure 7.5). the angle is assumed to be measured from the primed system to the unprimed system.for a plane truss member(see figure 7.4),the transformation matrix from local coordina tesystem to global coordinate system may be expressed aswhere is the inclined angle of the truss member which is assumed to be measured from the global to the local coordinate system.7.2.4 equilibrium equation for structuresfor discretized structure,the equilibrium of the whole structure is essentially the equilibrium of each joint. after assemblage,for ordinary fem or displacement methodfor force method where f=nodal loading vector;k=total stiffness matrix;d=nodal displacement vector;a=total forces transformation matrix;p=total primary internal forces vector.it should be noted that the coordinate transformation for each element from local coordinates to the global coordinate system must be done before assembly.in the force method, eq.(7.11)will be adopted to solve for internal forces of a statically determinate structure.the number of basic unknown forces is equal to the number of equilibrium equations available to solve for them and the equations are linearly independent.for statically unstable structures, analysis must consider their dynamic behavior. when the number of basic unknown forces exceeds the number of equilibrium equations,the structure is said to be statically indeterminate.in this case,some of the basic unknown forces are not required to maintain structural equilibrium.these are“extra”or“redundant”forces.to obtain a solution for the full set of basic unknown forces,it is necessary to augment the set of independent equilibrium equations with elasticbehavior of the structure,namely,the forcedisplacement relations of the structure.having solved for the full set of basic forces,we can determine the displacements by back substitution.7.2.5 influence lines and surfacesin the design and analysis of bridge structures,it is necessary to study the effects intrigued by loads placed in various positions.this can be done conveniently by means of diagrams showing the effect of moving a unit load across the structures.such diagrams are commonly called influence lines(for framed structures) or influence surfaces (for plates). observe that whereas a moment or shear diagram shows the variation in moment or shear along the structure due to some particular position of load,an influence line or surface for moment or shear shows the variation of moment or shear at a particular section due to a unit load placed anywhere along the structure.exact influence lines for statically determinate structures can be obtained analytically by statics alone.from eq.(7.11),the total primary internal forces vector p can be expressed asby which given a unit load at one node,the excited internal forces of all members will be obtained,and thus eq.(7.12) gives the analytical expression of influence lines of all member internal forces for discretized structures subjected to moving nodal loads.for statically indeterminate structures,influence values can be determined directly from a consideration of the geometry of the deflected load line resulting from imposing a unit deformation corresponding to the function under study,based on the principle of virtual work.this may better be demonstrated by a two-span continuous beam shown in figure 7.6, where the influence line of internal bending moment at section mb is required.figure 7.6 influence line of a two-span continuous beam.figure 7.7 deformation of a line element for lagrangian and eluerian variables.cutting section b to mb expose and give it a unit relative rotation=1(see figure 7.6) and employing the principle of virtual work givestherefore,which means the influence value of mb equals to the deflection v(x)of the beam subjected to a unit rotation at joint b(represented by dashed line in figure7.6b).solving for v(x)can be carried out easily referring to material mechanics.7.2 平衡方程7.2.1平衡方程和虚功方程对于任何有一定体积的材料都有一个表面积，如图7.2所示，它具有以下平衡条件：在表面的点：图7.2 平衡方程的推导在内部的点其中，ni表示n表面的单位法向量；ti表示与n相关的向量点应力；ji,j表示ij关于xj的一阶导数；而fi表示体积力。任何一系列满足方程（7.1a）-（7.1c）的应力ij、体积力fi、表面力ti都是一个静态的容许集。方程（7.1b和7.1c）可以写成如下所示（x,y,z）的形式，和其中，x,y,和z 分别是（x,y,z）方向的正应力，xy和y等表示（x,y,z）中的剪应力；fx,fy和fz分别表示（x,y,z）方向的体积力虚功原理被证明是一个解决问题的非常有效的方法，它在固体力学领域为一般性定理提供了证明。虚功方程采用两套独立的平衡集和兼容集（见图7.3，其中au 和a t分别表示位移边界和应力边界），如下所示：图7.3 虚功方程的两独立集相容集平衡集或是它表明外力虚功(wext)等于内力虚功(wint)。这个集成包括了物体的整个面积或体积。应力场ij，体积力fi和外部表面力ti是一个满足方程（7.1a-7.1c）的静态容许集。相似的，应变场ij和位移ui是一个满足位移边界条件和方程（7.16）（见7.3.1节）的兼容的运动学集。这意味着虚功原理仅适用于小应变或变形小的情况。需要注意的重要一点是，无论容许均衡集ij,fi,和 ti（图7.3a），还是兼容集ij 和 ui都不需要明确的状态，也不需要平衡集和兼容集以任何方式彼此相关。换句话说，这两个集合是完全相互独立的。7.2.2单元的平衡方程对于一个无穷小单元，平衡方程已经在7.2.1节中总结，这可以转化成不同方法中的具体表达式。正如在普通有限元法、位移法中，它可以导出以下单元平衡方程：图7.4 平面桁架端承力和位移（来源：meyers,v.j.，结构矩阵分析，1983年纽约harper &row出版授权社出版）其中，fe 和 de 分别表示单元节点力向量和位移向量，而ke 表示单元刚度矩阵；这里的上划线表示局部坐标系。在力法的结构分析中采用了离散化的方法，这被证明可以用来确定一套与各构件相关联的基本独立的力，在其中不仅这些力彼此之间相互独立，而且构件中的所有其他的力直接依赖于本集。因此，这些力构成的最小集能够完全定义构件的受力状态。基本力与局部力的关系可以通过乘以整体平衡的一个构件来获得，如下所示：其中，l表示单元力的变换矩阵，pe表示单元基本的向量力。需要强调的是，物理基本方程（7.5）是所有平衡的组成部分。以一个传统的平面桁架构件为例（见图7.4），有图7.5 坐标变换和其中，ea/l表示桁架构件的轴向刚度，p表示桁架构件的轴向力。7.2.3 坐标变换 向量v的分量的数值，是与所选择的坐标系有关，常常被定义为v1,v2,v3或者是些更简单地定义。通常在新的坐标系中必须调整参考轴并且为v的分量评估新值。假设向量v在两个右手笛卡尔坐标系xi（旧）和xi（新）中有两个具有相同起点的分量vi和 vi（见图7.5），而ei,ei分别是坐标系xi和 xi的单位向量。于是有其中，即在轴xi和xj之间角的余弦中的i和j的变化范围是从1到3；=(lij)33 被称为从旧坐标系向新坐标系的坐标变换矩阵。应该注意的是，lij中的元素或是矩阵是非对称的，即lijlji。例如，l12 是从x1到x2的角的余弦，而l21是从x2到x1的角的余弦（见图7.5）。假定角度是从原坐标系到坐标