有限元分析英文文献
工程有限元分析英文课件：Formulation of Isoparametric
0 0
1 j (1 i )
0 0
e
y ati j
v
(3.57)
x
v
1 4
Jij
1
0 0
1 j 1 i
0 0
(1 j ) 1 i
0 0
(1 j ) (1 i )
0 0
1 j (1 i
)
e
y ati j
with
T e
u1
v1
u2
v2
u3
v3
u4
v4
- the vector listing the element nodal point displacements8
1
B 2
B i
B
n
(3.42)
N
i
x
[B]i 0
Ni y
0
Ni y
Ni x
x
J
1
y
x y
J
x y
3
Formulation of Isoparametric Finite Element Matrices
Inverse of Jacobian Operator at a Specific Point
Formulation of Isoparametric Finite Element Matrices 3.4 Formulation of Isoparametric Finite Element Matrices
for Plane Elasticity (平面弹性问题)
The interpolation of the element coordinates and element displacements using the same interpolation functions, which are defined in a natural coordinate system, is the basis of the isoparametric finite element formulation.
MARC有限元分析
毕业设计（论文）中文题目：京沪高铁腕臂结构及其定位装置静力学强度有限元分析英文题目：The Structural Static Strength Analysis Of Cantilever and Positioning Device OfBeijing-Shanghai High-Speed Rail By FEM学院：机械与电子控制工程学院专业：机械工程及自动化学生姓名：陈奕舟学号：08221003指导教师：冯超2012年 6 月 5 日学号：学士论文版权使用授权书本学士论文作者完全了解北京交通大学有关保留、使用学士论文的规定。
特授权北京交通大学可以将学士论文的全部或部分内容编入有关数据库进行检索，提供阅览服务，并采用影印、缩印或扫描等复制手段保存、汇编以供查阅和借阅。
（保密的学士论文在解密后适用本授权说明）学士论文作者签名：指导教师签名：签字日期：年月日签字日期：年月日中文摘要本文以京沪高铁的正定位腕臂为研究对象，研究腕臂结构的静力学强度。
通过对京沪高铁所用正定位腕臂结构分别采用梁单元、壳和实体单元两种网格划分方案进行了腕臂结构的有限元模拟仿真，论文的主要工作以及研究成果表现为：1.通过对京沪高铁某直线段正定位腕臂的受力进行了分析，给出了200km/h和350km/h下的腕臂载荷表。
2.对京沪高铁正定位腕臂结构进行了梁单元、壳和实体单元的有限元模型构建。
3.实体和壳单元分网策略下能够较真实地反映出定位钩和定位环之间的接触状态，能够较清楚地显示结构上的应力分布彩色云图。
4.通过对腕臂结构有限元仿真结果的分析，腕臂结构200km/h工况下的静力学强度分析中应力最大值为74MPa，小于材料许用应力150MPa，结构安全。
5.腕臂结构在350km/h工况下，其应力为162MPa，大于结构材料的许用应力（即铝合金材料的屈服极限），会导致腕臂结构塑性变形。
但是由于铝合金材料的断裂应力为300MPa，所以静力情况下，腕臂结构基本安全。
外文翻译--有限元分析软件的发展
中文3240字Steps in Finite Element AnalysisIntroductionRecently there is a trend towards using it in the early stages of design. A designer may use FEA just to validate the structural integrity of a design or she may use it for structural optimization along with the parametrized design techniques.This paper examines the requirements of a structural analysis agent and proposes an architecture to facilitate FEA in a concurrent design environment. The next section briefly describes how FEA is used in a typical industrial set up.Section 3 presents a survey of existing FE tools. Section 4 discusses some issues related to the development of an FEA agent. Section 5 proposes an architecture for the FEA agent that addresses the issues described in Section 4 and finally Section 6 presents the concluding remarks.Steps in Finite Element AnalysisThe process of FEA starts with identification of the region of interest and the formulation of the physical problem。
有限元分析英文文献
The Basics of FEA Procedure有限元分析程序的基本知识2.1IntroductionThis chapter discusses the spring element,especially for the purpose of introducing various concepts involved in use of the FEA technique.本章讨论了弹簧元件，特别是用于引入使用的有限元分析技术的各种概念的目的A spring element is not very useful in the analysis of real engineering structures;however,it represents a structure in an ideal form for an FEA analysis.Spring element doesn’t require discretization(division into smaller elements)and follows the basic equation F=ku.在分析实际工程结构时弹簧元件不是很有用的;然而,它代表了一个有限元分析结构在一个理想的形式分析。
弹簧元件不需要离散化(分裂成更小的元素)只遵循的基本方程F=ku We will use it solely for the purpose of developing an understanding of FEA concepts and procedure.我们将使用它的目的仅仅是为了对开发有限元分析的概念和过程的理解。
2.2Overview概述Finite Element Analysis(FEA),also known as finite element method(FEM)is based on the concept that a structure can be simulated by the mechanical behavior of a spring in which the applied force is proportional to the displacement of the spring and the relationship F=ku is satisfied.有限元分析(FEA),也称为有限元法(FEM)，是基于一个结构可以由一个弹簧的力学行为模拟的应用力弹簧的位移成正比,F=ku切合的关系。
车架有限元分析外文文献翻译
南京林业大学本科毕业设计（论文）外文资料翻译翻译资料名称(外文)Stress analysis of heavy duty truck chassis as apreliminary data for its fatigue life predictionusing FEM翻译资料名称(中文)利用重型载货汽车的有限元应力分析的初步数据预测其疲劳寿命院（系）：汽车与交通工程学院专业：机械制造及其自动化（汽车设计方向）姓名：学号：指导教师：完成日期： 2012/5/31利用重型载货汽车的有限元应力分析的初步数据预测其疲劳寿命Roslan Abd Rahman, Mohd Nasir Tamin, Ojo Kurdi马来西亚工程大学机械工程系81310 UTM, Skudai,Johor Bahru摘要本文对一重型货车底盘做了应力分析。
应力分析能够确定零件的最大受力点，是分析零部件疲劳研究和寿命预测的重要手段。
前人已有用商用有限元软件ABAQUS软件对底盘模型进行分析的。
本次研究的底盘长12.35米，宽2.45米，材料是ASTM低合金钢710（3级），屈服极限552MPa，抗拉强度620MPa。
分析结果显示，最大应力点出现在底盘与螺栓连接的空缺处，最大应力为386.9MPa，底盘的疲劳破坏将会从最大应力点开始向车架各部位蔓延。
关键字：应力分析，疲劳寿命预测，货车底盘1.0简介在马来西亚，很多货车的车架寿命都有20多年，20多年架就会有使用安全的问题。
因此，为了确保底盘在工作期间的安全性能，就有必要对底盘作疲劳研究和寿命预测。
利用有限元法作应力分析能够确定受最大应力的关键点，这个关键点是导致底盘疲劳损伤的因素之一。
应力的大小能够预测底盘的寿命，所以可以根据应力分析的结果精确地预测底盘的寿命，应力分析越精确，底盘寿命预测的越合理。
本文是用商用有限元软件ABAQUS 软件完成底盘应力分析的。
汽车工业（汽车总成及各部件）在马来西亚的工业中占据非常重要的地位。
有限元分析中英文对照资料知识讲解
有限元分析中英文对照资料The finite element analysisFinite element method, the solving area is regarded as made up of many small in the node connected unit (a domain), the model gives the fundamental equation of sharding (sub-domain) approximation solution, due to the unit (a domain) can be divided into various shapes and sizes of different size, so it can well adapt to the complex geometry, complex material properties and complicated boundary conditions Finite element model: is it real system idealized mathematical abstractions. Is composed of some simple shapes of unit, unit connection through the node, and under a certain load.Finite element analysis: is the use of mathematical approximation method for real physical systems (geometry and loading conditions were simulated. And by using simple and interacting elements, namely unit, can use a limited number of unknown variables to approaching infinite unknown quantity of the real system.Linear elastic finite element method is a ideal elastic body as the research object, considering the deformation based on small deformation assumption of. In this kindof problem, the stress and strain of the material is linear relationship, meet the generalized hooke's law; Stress and strain is linear, linear elastic problem boils down to solving linear equations, so only need less computation time. If the efficient method of solving algebraic equations can also help reduce the duration of finite element analysis.Linear elastic finite element generally includes linear elastic statics analysis and linear elastic dynamics analysis from two aspects. The difference between the nonlinear problem and linear elastic problems:1) nonlinear equation is nonlinear, and iteratively solving of general;2) the nonlinear problem can't use superposition principle;3) nonlinear problem is not there is always solution, sometimes even no solution. Finite element to solve the nonlinear problem can be divided into the following three categories:1) material nonlinear problems of stress and strain is nonlinear, but the stress and strain is very small, a linear relationship between strain and displacement at this time, this kind of problem belongs to the material nonlinear problems. Due to theoretically also cannot provide the constitutive relation can be accepted, so, general nonlinear relations between stress and strain of the material based on the test data, sometimes, to simulate the nonlinear material properties available mathematical model though these models always have their limitations. More important material nonlinear problems in engineering practice are: nonlinear elastic (including piecewise linear elastic, elastic-plastic and viscoplastic, creep, etc.2) geometric nonlinear geometric nonlinear problems are caused due to the nonlinear relationship between displacement. When the object the displacement is larger, the strain and displacement relationship is nonlinear relationship. Research on this kind of problemIs assumes that the material of stress and strain is linear relationship. It consists of a large displacement problem of large strain and large displacement little strain. Such as the structure of the elastic buckling problem belongs to the large displacement little strain, rubber parts forming process for large strain.3) nonlinear boundary problem in the processing, problems such as sealing, the impact of the role of contact and friction can not be ignored, belongs to the highly nonlinear contact boundary. At ordinary times some contact problems, such as gear, stamping forming, rolling, rubber shock absorber, interference fit assembly, etc., when a structure and another structure or external boundary contact usually want to consider nonlinear boundary conditions. The actual nonlinear may appear at the same time these two or three kinds of nonlinear problems.Finite element theoretical basisFinite element method is based on variational principle and the weighted residual method, and the basic solving thought is the computational domain is divided into a finite number of non-overlapping unit, within each cell, select some appropriate nodes as solving the interpolation function, the differential equation of the variables in the rewritten by the variable or its derivative selected interpolation node value and the function of linear expression, with the aid of variational principle or weighted residual method, the discrete solution of differential equation. Using different forms of weight function and interpolation function, constitute different finite element methods. 1. The weighted residual method and the weighted residual method of weighted residual method of weighted residual method: refers to the weighted function is zero using make allowance for approximate solution of the differential equation method is called the weighted residual method. Is a kind of directly from the solution of differential equation and boundary conditions, to seek the approximate solution of boundary value problems of mathematical methods. Weighted residual method is to solve the differential equation of the approximate solution of a kind of effective method. Hybrid method for the trial function selected is the most convenient, but under the condition of the same precision, the workload is the largest. For internal method and the boundary method basis function must be made in advance to meet certain conditions, the analysis of complex structures tend to have certain difficulty, but the trial function is established, the workload is small. No matter what method is used, when set up trial function should be paid attention to are the following:(1) trial function should be composed of a subset of the complete function set. Have been using the trial function has the power series and trigonometric series, spline functions, beisaier, chebyshev, Legendre polynomial, and so on.(2) the trial function should have until than to eliminate surplus weighted integral expression of the highest derivative low first order derivative continuity.(3) the trial function should be special solution with analytical solution of the problem or problems associated with it. If computing problems with symmetry, should make full use of it. Obviously, any independent complete set of functions can be used as weight function. According to the weight function of the different options fordifferent weighted allowance calculation method, mainly include: collocation method, subdomain method, least square method, moment method and galerkin method. The galerkin method has the highest accuracy.Principle of virtual work: balance equations and geometric equations of the equivalent integral form of "weak" virtual work principles include principle of virtual displacement and virtual stress principle, is the floorboard of the principle of virtual displacement and virtual stress theory. They can be considered with some control equation of equivalent integral "weak" form. Principle of virtual work: get form any balanced force system in any state of deformation coordinate condition on the virtual work is equal to zero, namely the system of virtual work force and internal force of the sum of virtual work is equal to zero. The virtual displacement principle is the equilibrium equation and force boundary conditions of the equivalent integral form of "weak"; Virtual stress principle is geometric equation and displacement boundary condition of the equivalent integral form of "weak". Mechanical meaning of the virtual displacement principle: if the force system is balanced, they on the virtual displacement and virtual strain by the sum of the work is zero. On the other hand, if the force system in the virtual displacement (strain) and virtual and is equal to zero for the work, they must balance equation. Virtual displacement principle formulated the system of force balance, therefore, necessary and sufficient conditions. In general, the virtual displacement principle can not only suitable for linear elastic problems, and can be used in the nonlinear elastic and elastic-plastic nonlinear problem.Virtual mechanical meaning of stress principle: if the displacement is coordinated, the virtual stress and virtual boundary constraint counterforce in which they are the sumof the work is zero. On the other hand, if the virtual force system in which they are and is zero for the work, they must be meet the coordination. Virtual stress in principle, therefore, necessary and sufficient condition for the expression of displacement coordination. Virtual stress principle can be applied to different linear elastic and nonlinear elastic mechanics problem. But it must be pointed out that both principle of virtual displacement and virtual stress principle, rely on their geometric equation and equilibrium equation is based on the theory of small deformation, they cannot be directly applied to mechanical problems based on large deformation theory. 3,,,,, the minimum total potential energy method of minimum total potential energy method, the minimum strain energy method of minimum total potential energy method, the potential energy function in the object on the external load will cause deformation, the deformation force during the work done in the form of elastic energy stored in the object, is the strain energy.The convergence of the finite element method, the convergence of the finite element method refers to when the grid gradually encryption, the finite element solution sequence converges to the exact solution; Or when the cell size is fixed, the more freedom degree each unit, the finite element solutions tend to be more precise solution. Convergence condition of the convergence condition of the finite element finite element convergence condition of the convergence condition of the finite element finite element includes the following four aspects: 1) within the unit, the displacement function must be continuous. Polynomial is single-valued continuous function, sochoose polynomial as displacement function, to ensure continuity within the unit. 2) within the unit, the displacement function must include often strain. Total can be broken down into each unit of the state of strain does not depend on different locations within the cell strain and strain is decided by the point location of variables. When the size of the units is enough hours, unit of each point in the strain tend to be equal, unit deformation is uniform, so often strain becomes the main part of the strain. To reflect the state of strain unit, the unit must include the displacement functions often strain. 3) within the unit, the displacement function must include the rigid body displacement. Under normal circumstances, the cell for a bit of deformation displacement and displacement of rigid body displacement including two parts. Deformation displacement is associated with the changes in the object shape and volume, thus producing strain; The rigid body displacement changing the object position, don't change the shape and volume of the object, namely the rigid body displacement is not deformation displacement. Spatial displacement of an object includes three translational and three rotational displacement, a total of six rigid body displacements. Due to a unit involved in the other unit, other units do rigid body displacement deformation occurs will drive unit, thus, to simulate real displacement of a unit, assume that the element displacement function must include the rigid body displacement. 4) the displacement function must be coordinated in public boundary of the adjacent cell. For general unit of coordination is refers to the adjacent cell in public node have the same displacement, but also have the same displacement along the edge of the unit, that is to say, to ensure that the unit does not occur from cracking and invade the overlap each other. To do this requires the function on the common boundary can be determined by the public node function value only. For general unit and coordination to ensure the continuity of the displacement of adjacent cell boundaries. However, between the plate and shell of the adjacent cell, also requires a displacement of the first derivative continuous, only in this way, to guarantee the strain energy of the structure is bounded. On the whole, coordination refers to the public on the border between neighboring units satisfy the continuity conditions. The first three, also called completeness conditions, meet the conditions of complete unit is complete unit; Article 4 is coordination requirements, meet the coordination unit coordination unit; Otherwise known as the coordinating units. Completeness requirement is necessary for convergence, all four meet, constitutes a necessary and sufficient condition for convergence. In practical application, to make the selected displacement functions all meet the requirements of completeness and harmony, it is difficult in some cases can relax the requirement for coordination. It should be pointed out that, sometimes the coordination unit than its corresponding coordination unit, its reason lies in the nature of the approximate solution. Assumed displacement function is equivalent to put the unit under constraint conditions, the unit deformation subject to the constraints, this just some alternative structure compared to the real structure. But the approximate structure due to allow cell separation, overlap, become soft, the stiffness of the unit or formed (such as round degree between continuous plate unit in the unit, and corner is discontinuous, just to pin point) for the coordination unit, the error of these two effects have the possibility of cancellation, so sometimes use thecoordination unit will get very good results. In engineering practice, the coordination of yuan must pass to use "small pieces after test". Average units or nodes average processing method of stress stress average units or nodes average processing method of stress average units or nodes average processing method of stress of the unit average or node average treatment method is the simplest method is to take stress results adjacent cell or surrounding nodes, the average value of stress.1. Take an average of 2 adjacent unit stress. Take around nodes, the average value of stressThe basic steps of finite element method to solve the problemThe structural discretization structure discretization structure discretization structure discretization to discretization of the whole structure, will be divided into several units, through the node connected to each other between the units; 2. The stiffness matrix of each unit and each element stiffness matrix and the element stiffness matrix and the stiffness matrix of each unit (3) integrated global stiffness matrix integrated total stiffness matrix integrated overall stiffness matrix integrated total stiffness matrix and write out the general balance equations and write out the general balance equations and write out the general balance equations and write a general equation 4. Introduction of supporting conditions, the displacement of each node 5. Calculate the stress and strain in the unit to get the stress and strain of each cell and the cell of the stress and strain and the stress and strain of each cell.For the finite element method, the basic ideas and steps can be summarized as: (1) to establish integral equation, according to the principle of variational allowance and the weight function or equation principle of orthogonalization, establishment and integral expression of differential equations is equivalent to the initial-boundary value problem, this is the starting point of the finite element method. Unit (2) the area subdivision, according to the solution of the shape of the area and the physical characteristics of practical problems, cut area is divided into a number of mutual connection, overlap of unit. Regional unit is divided into finite element method of the preparation, this part of the workload is bigger, in addition to the cell and node number and determine the relationship between each other, also said the node coordinates, at the same time also need to list the natural boundary and essential boundary node number and the corresponding boundary value. (3) determine the unit basis function, according to the unit and the approximate solution of node number in precision requirement, choose meet certain interpolation condition basis function interpolation function as a unit. Basis function in the finite element method is selected in the unit, due to the geometry of each unit has a rule in the selection of basis function can follow certain rules. (4) the unit will be analysis: to solve the function of each unit with unit basis functions to approximate the linear combination of expression; Then approximate function generation into the integral equation, and the unit area integral, can be obtained with undetermined coefficient (i.e., cell parameter value) of each node in the algebraic equations, known as the finite element equation.(5) the overall synthesis: after the finite element equation, the area of all elements inthe finite element equation according to certain principles of accumulation, the formation of general finite element equations. (6) boundary condition processing: general boundary conditions there are three kinds of form, divided into the essential boundary conditions (dirichlet boundary condition) and natural boundary conditions (Riemann boundary conditions) and mixed boundary conditions (cauchy boundary conditions). Often in the integral expression for natural boundary conditions, can be automatically satisfied. For essential boundary conditions and mixed boundary conditions, should be in a certain method to modify general finite element equations satisfies. Solving finite element equations (7) : based on the general finite element equations of boundary conditions are fixed, are all closed equations of the unknown quantity, and adopt appropriate numerical calculation method, the function value of each node can be obtained.有限元分析有限元法求解区域是由许多小的节点连接单元（域），该模型给出了切分的基本方程（子域名）的近似解，由于单位（域）可以分为不同的形状和大小不同的尺寸，所以它能很好的适应复杂的几何形状、材料特性和边界条件复杂，复杂有限元模型：它是真实系统的理想化的数学抽象。
有限元分析报告报告材料法英文简介
The finite element analysisFinite element method, the solving area is regarded as made up of many small in the node connected unit (a domain), the model gives the fundamental equation of sharding (sub-domain) approximation solution, due to the unit (a domain) can be divided into various shapes and sizes of different size, so it can well adapt to the complex geometry, complex material properties and complicated boundary conditionsFinite element model: is it real system idealized mathematical abstractions. Is composed of some simple shapes of unit, unit connection through the node, and under a certain load.Finite element analysis: is the use of mathematical approximation method for real physical systems (geometry and loading conditions were simulated. And by using simple and interacting elements, namely unit, can use a limited number of unknown variables to approaching infinite unknown quantity of the real system. Linear elastic finite element method is a ideal elastic body as the research object, considering the deformation based on small deformation assumption of. In this kind of problem, the stress and strain of the material is linear relationship, meet the generalized hooke's law; Stress and strain is linear, linear elastic problem boils down to solving linear equations, so only need less computation time. If the efficient method of solving algebraic equations can also help reduce the duration of finiteelement analysis.Linear elastic finite element generally includes linear elastic statics analysis and linear elastic dynamics analysis from two aspects. The difference between the nonlinear problem and linear elastic problems:1) nonlinear equation is nonlinear, and iteratively solving of general;2) the nonlinear problem can't use superposition principle;3) nonlinear problem is not there is always solution, sometimes even no solution. Finite element to solve the nonlinear problem can be divided into the following three categories:1) material nonlinear problems of stress and strain is nonlinear, but the stress and strain is very small, a linear relationship between strain and displacement at this time, this kind of problem belongs to the material nonlinear problems. Due to theoretically also cannot provide the constitutive relation can be accepted, so, general nonlinear relations between stress and strain of the material based on the test data, sometimes, to simulate the nonlinear material properties available mathematical model though these models always have their limitations. More important material nonlinear problems in engineering practice are: nonlinear elastic (including piecewise linear elastic, elastic-plastic and viscoplastic, creep, etc.2) geometric nonlinear geometric nonlinear problems are caused due to the nonlinear relationship between displacement. When the object the displacement is larger, the strain and displacement relationship is nonlinear relationship. Research on this kind of problemIs assumes that the material of stress and strain is linear relationship. It consists of a large displacement problem of large strain and large displacement little strain. Such as the structure of the elastic buckling problem belongs to the large displacement little strain, rubber parts forming process for large strain.3) nonlinear boundary problem in the processing, problems such as sealing, the impact of the role of contact and friction can not be ignored, belongs to the highly nonlinear contact boundary. At ordinary times some contact problems, such as gear, stamping forming, rolling, rubber shock absorber, interference fit assembly, etc., when a structure and another structure or external boundary contact usually want to consider nonlinear boundary conditions. The actual nonlinear may appear at the same time these two or three kinds of nonlinear problems.Finite element theoretical basisFinite element method is based on variational principle and the weighted residual method, and the basic solving thought is the computational domain is divided into a finite number of non-overlapping unit, within each cell, select some appropriate nodes as solving the interpolation function, the differential equation of the variables in the rewritten by the variable or its derivative selected interpolation node value and the function of linear expression, with the aid of variational principle or weighted residual method, the discrete solution of differential equation. Using different forms of weight function and interpolation function, constitutedifferent finite element methods. 1. The weighted residual method and the weighted residual method of weighted residual method of weighted residual method: refers to the weighted function is zero using make allowance for approximate solution of the differential equation method is called the weighted residual method. Is a kind of directly from the solution of differential equation and boundary conditions, to seek the approximate solution of boundary value problems of mathematical methods. Weighted residual method is to solve the differential equation of the approximate solution of a kind of effective method. Hybrid method for the trial function selected is the most convenient, but under the condition of the same precision, the workload is the largest. For internal method and the boundary method basis function must be made in advance to meet certain conditions, the analysis of complex structures tend to have certain difficulty, but the trial function is established, the workload is small. No matter what method is used, when set up trial function should be paid attention to are the following:(1) trial function should be composed of a subset of the complete function set. Have been using the trial function has the power series and trigonometric series, spline functions, beisaier, chebyshev, Legendre polynomial, and so on.(2) the trial function should have until than to eliminate surplus weighted integral expression of the highest derivative low first order derivative continuity.(3) the trial function should be special solution with analytical solution of the problem or problems associated with it. If computing problems with symmetry, should make full use of it. Obviously, any independent complete set of functionscan be used as weight function. According to the weight function of the different options for different weighted allowance calculation method, mainly include: collocation method, subdomain method, least square method, moment method and galerkin method. The galerkin method has the highest accuracy.Principle of virtual work: balance equations and geometric equations of the equivalent integral form of "weak" virtual work principles include principle of virtual displacement and virtual stress principle, is the floorboard of the principle of virtual displacement and virtual stress theory. They can be considered with some control equation of equivalent integral "weak" form. Principle of virtual work: get form any balanced force system in any state of deformation coordinate condition on the virtual work is equal to zero, namely the system of virtual work force and internal force of the sum of virtual work is equal to zero. The virtual displacement principle is the equilibrium equation and force boundary conditions of the equivalent integral form of "weak"; Virtual stress principle is geometric equation and displacement boundary condition of the equivalent integral form of "weak". Mechanical meaning of the virtual displacement principle: if the force system is balanced, they on the virtual displacement and virtual strain by the sum of the work is zero. On the other hand, if the force system in the virtual displacement (strain) and virtual and is equal to zero for the work, they must balance equation. Virtual displacement principle formulated the system of force balance, therefore, necessary and sufficient conditions. In general, the virtual displacement principle can not only suitable for linear elastic problems, and can be used in the nonlinearelastic and elastic-plastic nonlinear problem.Virtual mechanical meaning of stress principle: if the displacement is coordinated, the virtual stress and virtual boundary constraint counterforce in which they are the sum of the work is zero. On the other hand, if the virtual force system in which they are and is zero for the work, they must be meet the coordination. Virtual stress in principle, therefore, necessary and sufficient condition for the expression of displacement coordination. Virtual stress principle can be applied to different linear elastic and nonlinear elastic mechanics problem. But it must be pointed out that both principle of virtual displacement and virtual stress principle, rely on their geometric equation and equilibrium equation is based on the theory of small deformation, they cannot be directly applied to mechanical problems based on large deformation theory. 3,,,,, the minimum total potential energy method of minimum total potential energy method, the minimum strain energy method of minimum total potential energy method, the potential energy function in the object on the external load will cause deformation, the deformation force during the work done in the form of elastic energy stored in the object, is the strain energy. The convergence of the finite element method, the convergence of the finite element method refers to when the grid gradually encryption, the finite element solution sequence converges to the exact solution; Or when the cell size is fixed, the more freedom degree each unit, the finite element solutions tend to be more precise solution. Convergence condition of the convergence condition of the finite element finite element convergence condition of the convergence condition of thefinite element finite element includes the following four aspects: 1) within the unit, the displacement function must be continuous. Polynomial is single-valued continuous function, so choose polynomial as displacement function, to ensure continuity within the unit. 2) within the unit, the displacement function must include often strain. Total can be broken down into each unit of the state of strain does not depend on different locations within the cell strain and strain is decided by the point location of variables. When the size of the units is enough hours, unit of each point in the strain tend to be equal, unit deformation is uniform, so often strain becomes the main part of the strain. To reflect the state of strain unit, the unit must include the displacement functions often strain. 3) within the unit, the displacement function must include the rigid body displacement. Under normal circumstances, the cell for a bit of deformation displacement and displacement of rigid body displacement including two parts. Deformation displacement is associated with the changes in the object shape and volume, thus producing strain; The rigid body displacement changing the object position, don't change the shape and volume of the object, namely the rigid body displacement is not deformation displacement. Spatial displacement of an object includes three translational and three rotational displacement, a total of six rigid body displacements. Due to a unit involved in the other unit, other units do rigid body displacement deformation occurs will drive unit, thus, to simulate real displacement of a unit, assume that the element displacement function must include the rigid body displacement. 4) the displacement function must be coordinated in public boundary of the adjacent cell.For general unit of coordination is refers to the adjacent cell in public node have the same displacement, but also have the same displacement along the edge of the unit, that is to say, to ensure that the unit does not occur from cracking and invade the overlap each other. To do this requires the function on the common boundary can be determined by the public node function value only. For general unit and coordination to ensure the continuity of the displacement of adjacent cell boundaries. However, between the plate and shell of the adjacent cell, also requires a displacement of the first derivative continuous, only in this way, to guarantee the strain energy of the structure is bounded. On the whole, coordination refers to the public on the border between neighboring units satisfy the continuity conditions. The first three, also called completeness conditions, meet the conditions of complete unit is complete unit; Article 4 is coordination requirements, meet the coordination unit coordination unit; Otherwise known as the coordinating units. Completeness requirement is necessary for convergence, all four meet, constitutes a necessary and sufficient condition for convergence. In practical application, to make the selected displacement functions all meet the requirements of completeness and harmony, it is difficult in some cases can relax the requirement for coordination. It should be pointed out that, sometimes the coordination unit than its corresponding coordination unit, its reason lies in the nature of the approximate solution. Assumed displacement function is equivalent to put the unit under constraint conditions, the unit deformation subject to the constraints, this just some alternative structure compared to the real structure. But the approximatestructure due to allow cell separation, overlap, become soft, the stiffness of the unit or formed (such as round degree between continuous plate unit in the unit, and corner is discontinuous, just to pin point) for the coordination unit, the error of these two effects have the possibility of cancellation, so sometimes use the coordination unit will get very good results. In engineering practice, the coordination of yuan must pass to use "small pieces after test". Average units or nodes average processing method of stress stress average units or nodes average processing method of stress average units or nodes average processing method of stress of the unit average or node average treatment method is the simplest method is to take stress results adjacent cell or surrounding nodes, the average value of stress.1. T ake an average of 2 adjacent unit stress. Take around nodes, the average value of stressThe basic steps of finite element method to solve the problemThe structural discretization structure discretization structure discretization structure discretization to discretization of the whole structure, will be divided into several units, through the node connected to each other between the units; 2. The stiffness matrix of each unit and each element stiffness matrix and the element stiffness matrix and the stiffness matrix of each unit (3) integrated global stiffness matrix integrated total stiffness matrix integrated overall stiffness matrix integratedtotal stiffness matrix and write out the general balance equations and write out the general balance equations and write out the general balance equations and write a general equation 4. Introduction of supporting conditions, the displacement of each node 5. Calculate the stress and strain in the unit to get the stress and strain of each cell and the cell of the stress and strain and the stress and strain of each cell. For the finite element method, the basic ideas and steps can be summarized as: (1) to establish integral equation, according to the principle of variational allowance and the weight function or equation principle of orthogonalization, establishment and integral expression of differential equations is equivalent to the initial-boundary value problem, this is the starting point of the finite element method. Unit (2) the area subdivision, according to the solution of the shape of the area and the physical characteristics of practical problems, cut area is divided into a number of mutual connection, overlap of unit. Regional unit is divided into finite element method of the preparation, this part of the workload is bigger, in addition to the cell and node number and determine the relationship between each other, also said the node coordinates, at the same time also need to list the natural boundary and essential boundary node number and the corresponding boundary value. (3) determine the unit basis function, according to the unit and the approximate solution of node number in precision requirement, choose meet certain interpolation condition basis function interpolation function as a unit. Basis function in the finite element method is selected in the unit, due to the geometry of each unit has a rule in the selection of basis function can follow certain rules. (4) the实用标准文案unit will be analysis: to solve the function of each unit with unit basis functions to approximate the linear combination of expression; Then approximate function generation into the integral equation, and the unit area integral, can be obtained with undetermined coefficient (i.e., cell parameter value) of each node in the algebraic equations, known as the finite element equation. (5) the overall synthesis: after the finite element equation, the area of all elements in the finite element equation according to certain principles of accumulation, the formation of general finite element equations. (6) boundary condition processing: general boundary conditions there are three kinds of form, divided into the essential boundary conditions (dirichlet boundary condition) and natural boundary conditions (Riemann boundary conditions) and mixed boundary conditions (cauchy boundary conditions). Often in the integral expression for natural boundary conditions, can be automatically satisfied. For essential boundary conditions and mixed boundary conditions, should be in a certain method to modify general finite element equations satisfies. Solving finite element equations (7) : based on the general finite element equations of boundary conditions are fixed, are all closed equations of the unknown quantity, and adopt appropriate numerical calculation method, the function value of each node can be obtained.精彩文档。
有限元分析英文文献教程文件
The Basics of FEA Procedure有限元分析程序的基本知识2.1 IntroductionThis chapter discusses the spring element, especially for the purpose of introducing various concepts involved in use of the FEA technique.本章讨论了弹簧元件，特别是用于引入使用的有限元分析技术的各种概念的目的A spring element is not very useful in the analysis of real engineering structures; however, it represents a structure in an ideal form for an FEA analysis. Spring element doesn’t require discretization (division into smaller elements) and follows the basic equation F = ku.在分析实际工程结构时弹簧元件不是很有用的;然而,它代表了一个有限元分析结构在一个理想的形式分析。
弹簧元件不需要离散化(分裂成更小的元素)只遵循的基本方程F = ku We will use it solely for the purpose of developing an understanding of FEA concepts and procedure.我们将使用它的目的仅仅是为了对开发有限元分析的概念和过程的理解。
2.2 Overview概述Finite Element Analysis (FEA), also known as finite element method (FEM) is based on the concept that a structure can be simulated by the mechanical behavior of a spring in which the applied force is proportional to the displacement of the spring and the relationship F = ku is satisfied.有限元分析(FEA),也称为有限元法(FEM)，是基于一个结构可以由一个弹簧的力学行为模拟的应用力弹簧的位移成正比,F = ku切合的关系。
有限元分析文献
CHAPTER 3Truss Element3.1 IntroductionThe single most important concept in understanding FEA, is the basic understanding of various finite elements that we employ in an analysis. Elements are used for representing a real engineering structure, and therefore, their selection must be a true representation of geometry and mechanical properties of the structure. Any deviation from either the geometry or the mechanical properties would yield erroneous results.The elements used in commercial codes can be classified in two basic categories:1.Discrete elements: These elements have a well defined deflection equation that canbe found in an engineering handbook, such as, Truss and Beam/Frame elements. The geometry of these elements is simple, and in general, mesh refinement does not affect the results. Discrete elements have a very limited application; bulk of the FEAapplication relies on the Continuous-structure elements.2.Continuous-structure Elements: Continuous-structure elements do not have a welldefine deflection or interpolation function, it is developed and approximated by using the theory of elasticity. In general, a continuous-structure element can have anygeometric shape, unlike a truss or beam element. The geometry is represented by 1-D, 2-D, or a 3-D solid element. Since elements in this category can have any shape, it is very effective in calculation of stresses at a sharp curve or geometry, i.e., evaluation of stress concentrations. Since discrete elements cannot be used for this purpose,continuous structural elements are extremely useful for finding stress concentration points in structures.NodesAs explained earlier, for analyzing an engineering structure, we divide the structure into small sections and represent them by appropriate elements. Nodes define geometry of the structure and elements are generated when the applicable nodes are connected. Resultsare always obtained for node points – and not for elements - which are then interpolated to provide values for the corresponding elements.For a static structure, all nodes must satisfy the equilibrium conditions and the continuity of displacement, translation and rotation. However, the equilibrium conditions may not be satisfied in the elements.In the following sections, we will get familiar with characteristics of the basic finite elements.3.2 Structures & ElementsMost 3-D structures can be analyzed using 2-D elements (idealization), which require relatively less computing time than the 3-D solid elements. Therefore, in FEA, 2-D elements are the most widely used elements. However, there are cases where we must use 3-D solid elements. In general, elements used in FEA can be classified as: - Trusses-Beams-Plates-Shells-Plane solids-Axisymmetric solids-3-D solidsSince Truss element is a very simple and discrete element, let us look at its properties and application first.3.3Truss ElementsThe characteristics of a truss element can be summarized as follows:Truss is a slender member (length is much larger than the cross-section).It is a two-force member i.e. it can only support an axial load and cannot support a bending load. Members are joined by pins (no translation at the constrained node, but free to rotate in any direction).The cross-sectional dimensions and elastic properties of each member are constant along its length.The element may interconnect in a 2-D or 3-D configuration in space.The element is mechanically equivalent to a spring, since it has no stiffness against applied loads except those acting along the axis of the member.However, unlike a spring element, discussed in previous chapters, a truss element can be oriented in any direction in a plane, and the same element is capable of tension as well as compression.jiFigure 3.2 A Truss Element3.3.1 Stress – Strain relation :As stated earlier, all deflections in FEA are evaluated at the nodes. The stress and strainin an element is calculated by interpolation of deflection values shared by nodes of the element. Since the deflection equation of the element is clearly defined, calculation of stress and strain is rather simple matter. When a load F is applied on a truss member, the strain at a point is found by the following relationship. xor, ε = δL/LL + δLFigure 3.3 Truss member in Tensionwhere, ε = strain at a pointu = axial displacement of any point along the length LBy hook’s law,Where, E = young’s modulus or modulus of elasticity.From the above relationship, and the relation,F = A σdxdu =εεσE =the deflection, δL, can be found asδL = FL/AE (3.1) Where, F = Applied loadA = Cross-section areaL = Length of the element3.3.2 Treatment of Loads in FEAFor a truss element, loads can be applied on a node only. If loads are distributed on a structure, they must be converted to the equivalent loads that can be applied at nodes. Loads can be applied in any direction at the node, however, the element can resist only the axial component, and the component perpendicular to the axis merely causes free rotation at the joint.3.3.3 Finite Element Equation of a Truss StructureIn this section, we will derive the finite element equation of a truss structure. The procedure presented here is the basis for all FEA analyses formulations, wherever h-element are used.Analogues to the previous chapter, we will use the direct or equilibrium method for generating the finite element equations. Assembly procedure for obtaining the global matrix will remain the same.In FEA, when we find deflections at nodes, the deflections are measured with respect to a global coordinate system, which is a fixed frame of reference. Displacements of individual nodes with respect to a fixed coordinate system are desirable in order to see the overall deformed structural shape. However, these deflection values are not convenient in the calculation of stress and strain in an element. Global coordinate system is good for predicting the overall deflections in the structure, but not for finding deflection, strain, and stress in an element. For this, it’s much easier to use a local coordinate system. We will derive a general equation, which relates local and global coordinates.In Figure 3.4, the global coordinates x-y can give us the overall deflections measured with respect to the fixed coordinate system. These deflections are useful for finding the final shape or clearance with the surroundings of the structure. However, if we wish to find the strain in some element, say, member 2-7 in figure 3.4, it will be easier if we know the deflections of node 2 and 3, in the y’ direction. Thus, calculation of strain value is much easier when the local deflection values are known, and will be time- consuming if we have to work with the x and y values of deflection at these nodes.Therefore, we need to establish a trigonometric relationship between the local and global coordinate systems. In Figure 3.4, xy coordinates are global, where as, x’y’ are local coordinates for element 4-7Nodes2 3 4 5 xFigure 3.4. Local and Global Coordinates3.3.4 Relationship Between Local and Global DeflectionsLet us consider the truss member, shown in Figure 3.5. The element is inclined at an angle θ, in a counter clockwise direction. The local deflections are δ1 and δ2. The global deflections are: u1, u2, u3, and u4. We wish to establish a relationship between these deflections in terms of the given trigonometric relations.δ2, R2R1, δ1Figure 3.5 Local and Global DeflectionsBy trigonometric relations, we have,δ1 = u1x cosθ + u2 sinθ = c u1x + s u1yδ2 = u2x cosθ + u2y sinθ = c u2x + s u2ywhere, cosθ = c, and sinθ = sWriting the above equations in a matrix form, we get,u1xδ1 c s 0 0 u1y= u2x (3.2) δ10 0 c s u2yOr, in short form, δ = T uWhere T is called Transformation matrix.Along with equation (3.2), we also need an equation that relates the local and global forces.3.3.5 Relationship Between Local and Global ForcesBy using trigonometric relations similar to the previous section, we can derive the desired relationship between local and global forces. However, it will be easier to use the work-energy concept for this purpose. The forces in local coordinates are: R1 and R2, and in global coordinates: f1, f2, f3, and f4, see Figure 3.6 for their directions.Since work done is independent of a coordinate system, it will be the same whether we use a local coordinate system or a global one. Thus, work done in the two systems is equal and given as,W = δT R = u T f, or in an expanded form,R1 f1W = δ1 δ2 = u1x u1y u2x u2y f2R2 f3f4= {δ}T {R} = {u}T{f}Substituting δ = T u in the above equation, we get,[[T] {u}]T {R} = {u}T {f}, or{u}T [T]T {R} = {u}T {f}, dividing by {u}T on both sides, we get,[T]T {R} = {f}(3.3)Equation (3.3) can be used to convert local forces into global forces and vice versa.2, δ2 R 1, δ1 11x Figure 3.6 Local and Global Forces3.3.6 Finite Element Equation in Local Coordinate SystemNow we will derive the finite element equation in local coordinate system. This equation will be converted to global coordinate system, which can be used to generate a global structural equation for the given structure. Note that, we can not use the elementequations in their local coordinate form, they must be converted to a common coordinate system, the global coordinate system.Consider the element shown below, with nodes 1 and 2, spring constant k, deflections δ1, and δ2, and forces R 1 and R 2. As established earlier, the finite element equation in local coordinates is given as,R 1 k -k δ1 = δ1, R 1 R 2 -k k δ1 22Figure 3.7 A Truss Element Recall that, for a truss element, k = AE/LLet k e = stiffness matrix in local coordinates, then,AE/L -AE/Lk e = Stiffness matrix in local coordinates-AE/L AE/L3.3.7 Finite Element Equation in Global CoordinatesUsing the relationships between local and global deflections and forces, we can convert an element equation from a local coordinate system to a global system.Let k g = Stiffness matrix in global coordinates.In local system, the equation is: R = [k e]{ δ} (A)We want a similar equation, but in global coordinates. We can replace the local force R with the global force f derived earlier and given by the relation:{f} = [T T]{R}Replacing R by using equation (A), we get,{f} = [T T] [[k e]{ δ}],and δ can be replaced by u, using the relation δ = [T]{u}, therefore,{f} = [T T] [k e] [T]{u}{f} = [k g] { u}Where, [k g] = [T T] [k e] [T]Substituting the values of [T]T, [T], and [k e], we get,c 0[k g] = s 0 AE/L -AE/L c s 0 00 c -AE/L AE/L 0 0 c s0 sSimplifying the above equation, we get,c2cs -c2-cscs s2-cs -s2[k g] = -c2-cs c2cs (AE/L)-cs -s2cs s2This is the global stiffness matrix of a truss element. This matrix has several noteworthy characteristics:The matrix is symmetricSince there are 4 unknown deflections (DOF), the matrix size is a 4 x 4.The matrix represents the stiffness of a single element.The terms c and s represent the sine and cosine values of the orientation of element with the horizontal plane, rotated in a counter clockwise direction(positive direction).The following example will illustrate its application.ExamplesFind displacements of joints 2 and 3Find stress, strain, & internal forcesin each member.A AL = 200 mm2 , A ST = 100 mm2All other dimensions are in mm.SolutionLet the following node pairs form the elements:Element Node Pair(1) 1-3(2) 2-1(3) 2-3Using Shigley’s Machine Design book for yield strength values, we have, S y(AL) = 0.0375kN/mm2 (375 Mpa)S y (ST) = 0.0586kN/mm2 (586 Mpa)E (AL) = 69kN/mm2 , E (ST) = 207kN/mm2A(1) = A(2) = 200mm2 , A(3) =100mm2Find the stiffness matrix for each elementu1y u3y Element (1)L(1 ) = 260 mm, u1x u3x E(1) = 69kN/mm2A(1) = 200mm2θ = 0c = cosθ = 1, c2 = 1s = sinθ = 0, s2 = 0cs = 0EA/L = 69 kN/mm2 x 200 mm2 x 1/(260mm) = 53. 1 kN/mmc2cs - c2-cs[K g](1) = (AE/L) x cs s2-cs - s2-c2-cs c2cs-cs -s2cs s21 0 -1 0[K g](1) = (53.1) x 0 0 0 0-1 0 1 00 0 0 0u1y Element 2θ = 900c = cos 900 = 0, c2 = 0 u1x s = sin 900 = cos 00 = 1, s2 = 1cs = 0EA/L = 69 x 200 x (1/150) = 92 kN/mmu2xu2x u2y u1x u1y0 0 0 0 u2x0 1 0 -1 u2y[k g](2) = (92) 0 0 0 0 u1x0 -1 0 1 u1yElement 3θ = 300c = cos 300 = 0.866, c2 = 0.753x s = cos 600 = .5, s2 = 0.25cs = 0.433 uEA/L = 207 x 100 x (1/300) = 69 kN/mmu2x u2y u3x u3yu2x .75 .433 -.75 -.433u2y -.433 .25 -.433 -.25 (69)[k g](3) = u3x-.75 -.433 .75 .433u3y-.433 -.25 .433 .25Assembling the stiffness matricesSince there are 6 deflections (or DOF), u1 through u6, the matrix is 6 x 6. Now, we will place the individual matrix element from the element stiffness matrices into the global matrix according to their position of row and column members.Element [1]u1x u1y u2x u2y u3x u3yu1x 53.1 -53.1u1yu2xu2yu3x -53.1 53.1u3yThe blank spaces in the matrix have a zero value.Element [2]u1x u1y u2x u2y u3x u3yu1x92 -92u1yu2xu2y -92 92u3xu3yElement [3]u1x u1y u2x u2y u3x u3yu1xu1yu2x51.7 29.9 -51.7 -29.9u2y29.9 17.2 -29.9 -17.2u3x-51.7 -29.9 51.7 29.9u3y-29.9 -17.2 29.9 17.2Assembling all the terms for elements [1] , [2] and [3], we get the complete matrix equation of the structure.u1x u 1yu 2x u 2y u 3x u 3y 53.1 0 0 0 -53.1 0 u 1x F 1 0 92 0 -92 0 0 u 1y F 1 0 0 51.7 29.9 -51.7 -29.9 u 2x = F 1 0 -92 29.9 109.2 -29.9 -17.2 u 2y F 1 -53.1 0 -51.7 -29.9 104.8 29.9 u 3x F 1 0 0 -29.9 -17.2 29.9 17.2u 3y F 1Boundary conditionsNode 1 is fixed in both x and y directions, where as, node 2 is fixed in x-direction only and free to move in the y-direction. Thus,u 1x = u 1y = u 2x = 0.Therefore, all the columns and rows containing these elements should be set to zero. The reduced matrix is:Writing the matrix equation into algebraic linear equations, we get,29.9u 2y - 29.9 u 3x - 17.2u 3y =0 -29.9u 2y + 104 u 3x + 29.9u 3y = 0 -17.2u 2y + 29.9u 3x + 17.2u 3y = -0.4solving, we get u 2y = -0.0043 u 3x = 0.0131 u 3y = -0.0502Sress, Strain and deflectionsElement (1)Note that u 1x , u 1y , u 2x , etc. are not coordinates, they are actual displacements.−= −−−−4.0002.179.292.179.298.1049.292.179.292.109332y x y u u uL = u3x = 0.0131= L/L = 0.0131/260 = 5.02 x 10-5 mm/mm= E = 69 x 5.02 x 10-5 = 0.00347 kN/mm2Reaction R = A = 0.00347 kNElement (2)L = u2y = 0.0043= L/L = 0.0043/150 = 2.87 x 10-5 mm/mm= E = 69 x 2.87 x 10-5 = 1.9803 kN/mm2Reaction R = A = (1.9803 x 10-3) (200) = 0.396 kNElement (3)Since element (3) is at an angle 300, the change in the length is found by adding the displacement components of nodes 2 and 3 along the element (at 300). Thus,L = u3x cos 300 + u3y sin 300 – u2y cos600= 0.0131 cos300 -0.0502 sin300 + 0.0043 cos600= -0.0116= L/L = -0.0116/300 = -3.87 x 10-5 = 3.87 x 10-5 mm/mm= E = 207 x -3.87.87 x 10-5 = -.0080 kN/mm2Reaction R = A = (-0.0087) (100) = 0.-0.800 kNFactor of SafetyFactor of safety ‘n’ is the ratio of yield strength to the actual stress found in the part.The lowest factor of safety is found in element (3), and therefore, the steel bar is the most likely to fail before the aluminum bar does.Final Notes- The example presented gives an insight into how the element analysis works. The example problem is too simple to need a computer based solution; however, it gives the insight into the actual FEA procedure. In a commercial FEA package, solution of a typical problem generates a very large stiffness matrix, which will require a computer assisted solution.- In an FEA software, the node and element numbers will have variable subscriptsso that they will be compatible with a computer-solution- Direct or equilibrium method is the earliest FEA method.0.0375(1)10.80.003470.0375(2)18.90.001980.0586(3)7.3250.0080yyyS Element n S Element n S Element n σσσ=========Example 2Given:Elements 1 and 2: Aluminum Element 3: steel A (1) = 1.5in 2 A (2) = 1.0in 2 A (3) = 1.0in 2Required:Find stresses and displacements using hand calculationsy.SolutionCalculate the stiffness constants:Calculate the Element matrix equations.inlb L AE K in lb L AE K in lb L AE K 563562561100.650101030105.24010101105.72010105.1×=××==×=××==×=××==Element (1)u2x Denoting the Spring constant for element (1) by k1, and the stiffness matrix by K(1), the stiffness matrix in global coordinates is given as,u1x u1y u2x u2yc2cs -c2-cs u1xcs s2-cs -s2 u1y[K g](1) = K1c2-cs c2cs u2x-cs -s2cs s2 u2yFor element (1), θ = 00, thereforec =1, c2 = 1s = 0, s2 =0, and cs = 0u1x u1y u2x u2y1 0 -1 0 u1x0 0 0 0u1y[k(1)] = k1 1 0 1 0 u2x0 0 0 0u2yElement (2) 2xFor this element, θ = 900, Therefore,c= cosθ = 0, c2 = 0 0s = sinθ = 1, s2 = 1 u3xcs= 0The stiffness matrix is, u3yu3x u3y u2x u2yc2cs -c2-cs u3xcs s2-cs -s2 u3y[k g](2) = k2c2-cs c2cs u2x-cs -s2cs s2 u2yu3x u3y u2x u2y0 0 0 0 u3x0 1 0 -1u3y[k g](2) = k20 0 0 0 u2x0 -1 0 1u2yElement 3 uFor element (3), θ = 126.90.c= cos(126.90) = -0.6, c20 s = sin(126.90, s2 = .64cs = -0.484yu4x u4y u2x u2y.36 -.48 -.36 .48 u4x-.48 .64 .48 -.64u4y[k g](3) = k3-.36 .48 .36 -.48 u2x.48 -.64 -.48 .64u2yAssembling the global MatrixFollowing the procedure for assembly described earlier, the assembled matrix is,[K g] =111133333232332233333333 00000010000000020.36.4800.36.48300.48.640.48.644000000005000000600.36.4800.36.48700.48.6400.48.648K KK K K K K KK K K K K KK KK K K KK K K K−−+−−+−−−−−−−The boundary conditions are:u1x = u1y = u3x = u3y = u4x = u4y = 0We will suppress the corresponding rows and columns. The reduced matrix is a 2 x2, given below,u2x u2yK1-.48K3 u2x[K g] =-.48K3 K2 + .64 K3 u2yThe final equation is,K1 + .36 K3 -.48K3 u2x -4000=-.48K3 K2 + .64 K3 u2x 8000Substituting values for k1, k2, and k3, we getChapter 3 Truss ElementFEA Lecture Notes © by R. B. Agarwal 3-21 9.66 -2.88 u 2x -4000 105 = -2.88 6.34 u 2y 8000 u 2x = - 0.0000438 in. u 2y = - 0.01241 in.1 = P/A = K 1 u/A 1 = [(7.5 x 105)(- 0.0000438)] / (1.5) = 214 psi2= P/A = K 2 u/A 2 = [(2.5 x 105)(- 0.012414)] / (1.0) = 3015 psi3 = P/A = K 3 u/A 3 = [(6 x 105)(- 0.0000438 cos 53.10 + 0.012414 sin 53.10)] / (1.0) = 6119 psi。
利用有限元模型对钻机进行动态分析的研究外文文献翻译、中英文翻译
Studies in dynamic design of drilling machine using updated finite element modelsAbstractThe aim of the present work is to develop updated FE models of a drilling machine using analytical and experimental results. These updated FE models have been used to predict the effect of structural dynamic modifications on vibration characteristics of the drilling machine. Two studies have been carried out on the machine. In the first study, modal tests have been carried out on a drilling machine using instrumented impact hammer. Modal identification has been done using global method of modal identification. For analytical FE modeling of the machine, a computer program has been developed. The results obtained using FEM, have been correlated with the experimental ones using mode shape comparison and MAC values. Analytical FE model has been updated, with the help of a program, which has been developed using direct methods of model updating. In the second study, modal testing has been carried out using random noise generator and modal exciter. Global method has been used for modal identification. Analytical FE modeling has been done using I-DEAS software. Correlation of FE results with the experimental ones has been carried out using FEMtools software. Updating of the analytical FE model has also been done using the above software, based on an indirect technique viz. sensitivity based parameter estimation technique. The updated FE models, obtained from both the studies have been used for structural dynamic modifications (SDM), for the purpose of dynamic design and the results of SDM predictions are seen to be reasonably satisfactory.Article Outline1. Introduction2. Modal testing and identification3. Finite element formulation of drilling machine4. Comparison of analytical FE and experimental results (model correlation)5. Finite element model updating6. SDM studies using updated models for dynamic design7. ConclusionsReferences1. IntroductionDynamic design aims at obtaining desired dynamic characteristics in machines and structures, which may include shifting of natural frequencies, desired mode shapes and vibratory response. The ultimate objectives are to have a quieter and more comfortable environment, higher reliability and better quality of product. The conventional dynamic design is basically hit and trial method in which we try to achieve desired dynamic characteristics by making several prototypes. The disadvantage of this technique is that actual design cycle takes a lot of time and therefore it is not cost effective. However, model updating based dynamic design saves design cycle time as well as reduces the cost involved. Various tools used for updating based dynamic design are: experimental modal analysis (EMA) including modal testing and modal identification, model updating and structural dynamic modification.Ewins [1]and Maia and Silva [2]have explained the basic concepts of modal testing, which is an experimental approach to obtain mathematical model of a structure. In a modal test, the structure under test is excited either by an impact hammer or by a modal exciter, and the response of the structure is recorded at several experimental points, in the form of frequency response functions (FRFs), using a dual channel FFT analyzer. The experimental modal model gives information about the natural frequencies, corresponding mode shapes and modal damping factor and is useful for model updating. The model updating techniques helps us to bring analytical finite element models closer to real systems. In model updating an initial analytical FE model constructed for analyzing the dynamics of a structure is refined or updated using test data measured on actual structure such that the updated model describes the dynamic properties of the structure more correctly. The inaccuracies in FEM, when applied to dynamic problems are due to uncertainties in boundary conditions and structural damping etc.Friswell and Mottershead [3] have discussed the finite element model updating in structural dynamics. Baruch and Bar-Itzhack and Baruch [4] and [5] considered analytical mass matrix to be exact and developed a direct method for updating using test data. Berman and Nagy [6]developed a method of model updating, which uses measured modes and natural frequencies to improve analytical mass and stiffness matrices. Structural dynamic modification (SDM) techniques [7]and [8]are the methods by whichdynamic behaviour of the structure is improved by predicting the modified behaviour brought about by adding modifications like those of lumped masses, rigid links, dampers etc. Thus the dynamic design using updated model is expected to be helpful in order to predict accurately and quickly, the effect of possible modifications on the dynamic characteristics of the structure at computer level itself, thus saving time and cost.Sestieri [7]has discussed SDM application to machine tools and engines. Kundra [8]gave the method of structural dynamic modification via models. Modak [9] has discussed SDM predictions using updated FE model for an F-structure. He used constrained nonlinear optimization method for updating of a machine tool using stiffness parameters at the boundary [10]. The present paper deals with the FE model updating using direct as well as indirect method, and to use this updated FE model for dynamic design based on SDM predictions of a machine tool viz. a drilling machine. Two different studies are reported using different techniques for analytical and experimental analysis and for updating. Various objectives with which the present research work has been carried out are• To develop updated FE models of a complex structure like that of a drilling machine and to use these updated models to predict the effect of various modifications on modal properties of the machine.• To see whether hammer excitation yields good results for fairly complex structures like drilling machine or not, and to compare these results with those obtained from modal exciter.• To analyze the results of SDM predictions obtained using the updated models derived in the studies.2. Modal testing and identificationIn the two studies mentioned earlier, different techniques have been used, for modal testing and identification. In the first study, impact hammer is used to excite the drilling machine structure, at various points as shown in Fig. 1 and Fig. 2. Response is taken at a fixed point with the help of an accelerometer.(12K)Fig. 1. Experimental setup (Study 1).(4K)Fig. 2. Hammer excitation locations.In the present study, the drilling machine is excited at 30 locations and therefore, 30 FRFs are obtained. These FRFs are recorded in the form of inertance. The experimental FRFs, thus obtained are transferred to computer. Modal identification or modal parameter extraction consists of curve fitting a theoretical expression for an individual FRF to the actual measured data obtained. The experimental FRFs are analyzed by GRF-M method using modal analysis software ICATS [11] to obtain modal parameters of the drilling machine. In the second study, the machine tool structure has been excited at the base at point 28, referring to Fig. 2, using modal exciter and response has been measured at various points using piezoelectric accelerometers. The modal identification of the FRFs, thus measured has been carried out using global method GRF-M method in ICATS software.Table 1compares the experimental natural frequencies obtained from both the methods, which shows minor differences in the two modal frequenciesTable 1.Mode110.29 Hz8.67 HZ0.95311.20 Hz8.79 Hz0.946Mode 265.14 Hz47.34 HZ0.90163.37 Hz44.40 Hz0.9063. Finite element formulation of drilling machineSeveral books have given the basic concepts of finite element analysis, some of them are: Zienkiewicz [12] and Bathe [13].The drilling machine structure is very complicated with different mountings and accessories. Therefore exact modeling and analysis of the actual structure is difficult and it takes more computational effort. However for analytical FE analysis, simplified model of drilling machines has been considered. In study 1, the finite element modelling has been done using a program developed in MATLAB. Beam elements have been used for the analysis. The joints and boundary conditions are considered to be rigid and influence of structural damping on modal model parameters, is ignored. The relevant data used for the drilling machine is given below:25 mm pillar type, height = 1.655 m, mass density = 7800 kg/m3, Young’s modulus= 200 Gpa, number of nodes = 30, number of elements = 29, number of nodes per element = 2, degrees of freedom per node = 3.Fig. 3shows the structure of the drilling machine with the node numbers given for study 1.(7K)Fig. 3. Drilling machine structure for FE analysis.The eigenvalues and eigenvectors have been calculated. The analytical FE model of the structure consists of 90 ×90-size mass and stiffness matrices (30 ×3, 30 nodes and 3 d.o.f. per node). But by experiment only 30 coordinates can be measured. Therefore FE model has been reduced using Guyan [14] reduction method with the help of a program developed in MATLAB.In study 2, the finite element modelling has been done using I-DEAS software. The model has been made using beam mesh. Although the FE model has been simplified but the beam elements has rotational degree of freedom, which cannot be measured experimentally. Therefore the FE model needs to be reduced. The FE model has been reduced using model reduction utilityin FEMtools software. Fig. 4 and Fig. 5 shows the mode shape animation for the first and second mode respectively, using I-DEAS software.(31K)Fig. 4. Mode shape animation (first mode).(23K)Fig. 5. Mode shape animation (second mode).4. Comparison of analytical FE and experimental results (model correlation)The first stage of any reconciliation exercise is to determine how closely the experimental and analytical models correspond. If we are unable to obtain a satisfactory degree of correlation between the initial analytical FE model and the test data, then it is extremely unlikely that any form of model updating will succeed. Thus, a successful correlation is crucial for the success of model updating. Table 1gives the comparison between experimental and analytical natural frequencies. There are differences between analytical FE model predictions and experimental results. Thus the FE models need to be updated. However, the differences between the corresponding results of both studies are minor.Apart from natural frequency comparison (as given in Table 1), another method of model correlation is mode shape comparison. To compare the mode shapes, we plot the deformed shapes of the structure for a particular mode, using experimental as well as analytical model. These mode shapes are plotted side-by-side for quick comparison. Mode shape corresponding to second mode is shown in Fig. 6, using ICATS software. It shows a fairlygood level of correlation between the experimental and analytical FE model.(87K)Fig. 6. Mode shape comparison.Several researchers have developed techniques for quantifying the comparison between measured and predicted mode shapes. As an alternative to the graphical approach, Model Assurance Criterion i.e. MAC, [15]) is a widely used technique to estimate the degree of correlation between mode shape vectors. This provides a measure of the least squares deviation or ‘scatter’ of the points from the straight-line correlation. The MAC between a measured and analytical mode is:(1)wherem andarepresents measured or experimental and analytical modeshapes respectively. MAC is a scalar quantity whose value is between 0 and 1. A value of MAC close to 1 shows a good degree of correlation between experimental and analytical FE model. We can see in Table 1 that the MAC numbers are close to 1, though somewhat lower for the second mode. Table 1 also shows that the results obtained from both the studies are quite close to each other.5. Finite element model updatingModel updating can be defined as “the process of correcting the numerical values of individual parameters in a analytical FE model using data obtained from an associated experimental model such that the updated model correctly describes the dynamic properties of the subject structure”.Various model updating methods can be classified into two major groups: • Direct matrix methods• Indirect or iterative methodsDirect methods are capable of reproducing measured data exactly, but they provide no opportunity for the user to select parameters for updating. Here parameter means any physically realizable quantity like Young’s modulus, Poisson’s ratio, mass density etc. When using the direct methods, the entire stiffness and mass matrices are updated in a single(non-iterative) solution step. Consequently any physical meaning, which the initial finite element model might have possessed, is lost in the updating process.Techniques like the indirect methods, allow the updating parameters to be selected. So, considerable physical insight is required if the model is to be improved, not only in its ability to reproduce test results, but also in interpreting the parameters physically. Methods in this second group are iterative and, as such, considerably more expensive of computer effort.Two studies have been carried out for model updating and computer programs for the same were developed in the present work using MATLAB. In the first study, two direct methods are applied to update the analytical FE model of the drilling machine structure.Baruch and Bar-ltzhack [4] and Baruch [5] considered the mass matrix of the analytical model to be exact. The measured eigenvectors are corrected by using the relation:(2)The stiffness matrices of the analytical FE model after updating is given as:K=K a-K a T M a+M a T K a+M a T K a T M a+M aΛT M a(3)uBerman and Nagy [6] used a method similar to that of Baruch. They update mass and stiffness matrices while the mass matrix is updated to ensure the orthogonality of the exact FE model modes. The mass matrix is updated as:(4)(5) The stiffness matrix is updated using following equation:(6) The updated mass matrix obtained will be symmetric and stiffness matrix will be close to that of exact stiffness matrix. Results obtained fromBaruch and Berman–Nagy model updating methods are tabulatedin Table 2. It is clear from Table 2, that the updated model reproduces the measured frequencies. After updating, MAC values have been calculated again using Eq. It can be observed that updated MAC values show some improvement over the initial MAC values.Table 2.Comparison of experimental and updated FE frequencies and MAC valuesModenumberStudy 1Study 2Measured frequency Baruch method Berman method MeasuredfrequencySensitivitymethodUpdatedfrequencyMACUpdatedfrequencyMACUpdatedfrequencyMACMod e 18.67 Hz8.67 Hz0.9258.67 Hz0.9348.79 Hz8.79 Hz0.947Mod e 247.34 Hz47.34 Hz0.96647.34 Hz0.94744.40 Hz44.43 Hz0.913The results after updating have been tabulated in Table 2. It clearly shows that after updating, the updated FE model closely represents the actual machine tool structure. It also shows that MAC values have also been improved after updating.6. SDM studies using updated models for dynamic designStructural dynamic modification (SDM) techniques are methods by which dynamic characteristics of the structure can be improved by adding the modifications like changing mass, spring, damping etc. The mass modification has been considered here for predicting dynamic characteristics using updated FE model.A mass modification on drilling machine is introduced in the form of a lumped mass of 14.3 kg at the top of the vertical pillar, i.e. node 20 as in Fig. 3. The modal test for the mass modified machine is carried out by Modak [10], using impact hammer for excitation. The FRFs are analyzed in ICATS in order to obtain an experimental estimate of the altered dynamic characteristics of the drilling machine, as given in Table 3.Table 3.Comparison of measured and predicted frequencies, after mass modification, using updated FE modelMode no.MeasuredfrequenciesSDM prediction using updated FE modelBaruch’s method Berman and Nagy’s method Sensitivity method18.37 Hz8.34 Hz8.17 Hz8.40 Hz246.05 Hz47.15 Hz46.96 Hz43.20 HzThe effect of the same mass modification on the dynamic characteristics of the drilling machine has also been predicted by updated FE model. Table 3 gives a comparison of the predictions based on the updated FE models obtained by direct methods of Baruch, Berman and Nagy and by indirectmethod based on sensitivity analysis, with that of the measured modified characteristics.It is seen from the Table 3 that the updated FE model predictions of the natural frequencies are quite close to the measured value of natural frequencies. This shows the capability of the updated FE model to accurately predict the effect of structural modifications on the dynamic properties of the structure. Now, this updated FE model has been further used to predict, at computer level, the effect of various mass modifications on the structural dynamics of the drilling machine, for the purpose of dynamic design.The mass modification can bring about significant changes in the natural frequencies of a structure. For predicting the effect of modifications using updated FE model, modification at node 20 and node 25, on the drilling machine has been considered. The effects on FRFs due to these modifications have been shown in Fig. 8and Fig. 9respectively. The values of the first two natural frequencies have been predicted using updated FE models and MODIFY module of ICATS software and are shown in Table 4and Table 5.(33K)Fig. 8. Regenerated FRF due to mass modification at node 20.(29K)Fig. 9. Regenerated FRF due to mass modification at node 25.Table 4.Predicted natural frequencies after mass modification using 20 kg at node 20 (PanelA), predicted natural frequencies after mass modification using 40 kg at node 20 (PanelB)Comparisons of SDM predictions using updated FE modelBaruch method Berman method Sensitivity method ICATS MODIFY Measured Frequencies Panel AMass modification of 20 kg8.22 Hz7.99 Hz8.3 Hz8.2 Hz8.3 Hz47.08 Hz46.83 Hz42.87 Hz47.1 Hz42.8 HzPanel BMass modification of 40 kg7.83 Hz7.45 Hz7.76 Hz7.75 Hz7.8 Hz46.86 Hz46.44 Hz41.27 Hz46.6 Hz41.3 HzTable 5.Predicted natural frequencies after mass modification using 20 kg at node 25 (PanelA), predicted natural frequencies after mass modification using 40 kg at node 25 (PanelB)Comparisons of SDM predictions using updated FE modelBaruch method Berman method Sensitivity method ICATS MODIFY Measured frequenciesPanel AMass modification of 20kg8.34 Hz8.44 Hz8.41 Hz8.35 Hz8.4 Hz 42.24 Hz43.74 Hz41.67 Hz42.2 Hz41.6 HzComparisons of SDM predictions using updated FE modelBaruch method Berman method Sensitivity method ICATS MODIFY Measured frequenciesPanel BMass modification of 40kg8.03 Hz8.21 Hz8.04 Hz8.1 Hz8.1 Hz38.89 Hz41.02 Hz39.67 Hz39.15 Hz39.5 Hz7. ConclusionsComparison of results obtained from experimental modal analysis and FE models of a drilling machine, indicate that its finite element models need to be updated. This is necessary in order to predict dynamic behavior of the complex structure with an acceptable accuracy.An experimentation involving modal testing has been carried out on the drilling machine using impact hammer as well as modal exciter. It has been observed that both impact hammer and modal exciter yield good results for fairly complex structures like drilling machine.Analytical FE model has been updated in the light of experimental data using direct as well as indirect methods. Both the methods give results, which are fairly close when used for predicting results of SDM attaching additional mass on the machine at different locations. The predicted results have been validated by comparison with measured results and are seen to be fairly accurate in particular for the indirect method using sensitivity analysis.References[1]D.J. Ewins, Modal Testing: Theory and Practice, John Willey and Sons, New York (2000).[2] N.M.M. Maia and J.M.M. Silva, Theoretical and Experimental Modal Analysis, John Willey and Sons, New York (1997).[3]M.I. Friswell and J.E. Mottershead, Finite Element Model Updating in Structural Dynamics, Kluwer Academic Publishers, Dordrecht (1995).[4] M. Baruch and I.Y. Bar-ltzhack, Optimal weighted orthogonalization of measured modes, AIAA Journal16 (1978), pp. 346–351.[5] M. Baruch, Optimisation procedure to correct stiffness and flexibility matrices using vibration test data, AIAA Journal16 (1978), pp. 1208–1210.[6]A. Berman and E.J. Nagy, Improvement of a large analytical model using test data, AIAA Journal21 (1983), pp. 1168–1173.[7] A. Sestieri, SDM application to machine tools and engines, Sadhana 25 (2000), pp. 305–317.[8] T.K. Kundra, Structural dynamic modification via model, Sadhana25 (2000), pp. 261–276.[9]S.V. Modak, Studies in Finite Element Model Updating and Application to Dynamic Design, Ph.D. Thesis, Department of Mechanical Engineering, IIT Delhi, 2001.[10]S.V. Modak, T.K. Kundra, B.C. Nakra, Dynamic design of machine tool structures using an updated model Pro. IMAC- XX, 2002, pp. 1489–1494.[11] ICATS Reference Manual, Imperial College, London, 1996.[12]O.C. Zienkiewicz, The Finite Element Method in Engineering Science, McGraw-Hill Publishing Company, London (1977).[13] K.J. Bathe and E.J. Wilson, Numerical Methods in Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ (1982).[14] R.J. Guyan, Reduction of stiffness and mass matrices, AIAA Journal 3 (1965), p. 380.[15]R.L. Allemang, BD.L. Rown. A correlation coefficient for modal vector analysis, Proc. 1st IMAC, 1982, pp. 110–116.[16] FEMtools Manual Version 2.0, Dynamic design solutions, 2000.Corresponding author. Steam Turbine Engineering, TCGT Division, Bharat Heavy Electricals Limited, Hyderabad-502032, India译文利用有限元模型对钻机进行动态分析的研究摘要现在的工作的目的是使用分析和实验的结果发展和改进钻机的有限元分析模型，这种有限元分析已经用在监测将钻机的工作特性修改后对钻机动态的结构的影响；对钻机将有两个研究就进行。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
The Basics of FEA Procedure有限元分析程序的基本知识2.1 IntroductionThis chapter discusses the spring element, especially for the purpose of introducing various concepts involved in use of the FEA technique.本章讨论了弹簧元件，特别是用于引入使用的有限元分析技术的各种概念的目的A spring element is not very useful in the analysis of real engineering structures; however, it represents a structure in an ideal form for an FEA analysis. Spring element doesn’t require discretization (division into smaller elements) and follows the basic equation F = ku.在分析实际工程结构时弹簧元件不是很有用的;然而,它代表了一个有限元分析结构在一个理想的形式分析。
弹簧元件不需要离散化(分裂成更小的元素)只遵循的基本方程F = ku We will use it solely for the purpose of developing an understanding of FEA concepts and procedure.我们将使用它的目的仅仅是为了对开发有限元分析的概念和过程的理解。
2.2 Overview概述Finite Element Analysis (FEA), also known as finite element method (FEM) is based on the concept that a structure can be simulated by the mechanical behavior of a spring in which the applied force is proportional to the displacement of the spring and the relationship F = ku is satisfied.有限元分析(FEA),也称为有限元法(FEM)，是基于一个结构可以由一个弹簧的力学行为模拟的应用力弹簧的位移成正比,F = ku切合的关系。
In FEA, structures are modeled by a CAD program and represented by nodes and elements. The mechanical behavior of each of these elements is similar to a mechanical spring, obeying the equation, F = ku. Generally, a structure is divided into several hundred elements, generating a very large number of equations that can only be solved with the help of a computer.在有限元分析中,结构是由CAD建模程序通过节点和元素建立。
每一个元素的力学行为类似于机械弹簧,遵守方程,F =ku。
一般来说,一个结构分为几百元素,生成大量的方程,只能在电脑的帮助下得到解决。
The term ‘finite element’ stems from the procedure in which a structure is divided into small but finite size elements (as opposed to an infinite size, generally used in mathematical integration).“有限元”一词源于一个结构分为小而有限大小元素的过程(而不是无限大小,通常用于数学集成)The endpoints or corner points of the element are called nodes.元素的端点或角点称为节点。
Each element possesses its own geometric and elastic properties.每个元素拥有自己的几何和弹性。
Spring, Truss, and Beams elements, called line elements, are usually divided into small sections with nodes at each end. The cross-section shape doesn’t affect the behavior of a line element; only the cross-sectional constants are relevant and used in calculations. Thus, a square or a circular cross-section of a truss member will yield exactly the same results as long as the cross-sectional area is the same. Plane and solid elements require more than two nodes and can have over 8 nodes for a 3 dimensional element.弹簧，桁架和梁元素，称为线元素，通常分为小节，每端有节点。
截面形状并不影响线元素的特性;只有横截面常数是相关的并用于计算。
因此,一个正方形或圆形截面桁架成员将产生完全相同的结果,只要横截面积是一样的。
平面和立体元素需要超过两个节点,可以有超过8节点的三维元素。
A line element has an exact theoretical solution, e.g., truss and beam elements are governed by their respective theories of deflection and the equations of deflection can be found in an engineering text or handbook. However, engineering structures that have stress concentration points e.g., structures with holes and other discontinuities do not have a theoretical solution, and the exact stress distribution can only be found by an experimental method. However, the finite element method can provide an acceptable solution more efficiently.线元件具有精确的理论解，例如桁架和梁元件由它们各自的偏转理论控制，并且偏转方程可以在工程文本或手册中找到。
然而，具有应力集中点的工程结构，例如具有孔和其他不连续的结构不具有理论解，并且精确的应力分布只能通过实验方法找到。
然而，有限元方法可以更有效地提供可接受的解决方案。
Problems of this type call for use of elements other than the line elements mentioned earlier, and the real power of the finite element is manifested.这种类型的问题要求使用前面提到的行元素以外的元素。
有限元法能真正的来体现证明。
In order to develop an understanding of the FEA procedure, we will first deal with the spring element.为了能深刻理解有限元分析过程,我们将首先处理弹簧元件。
In this chapter, spring structures will be used as building blocks for developing an understanding of the finite element analysis procedure.在这一章,弹簧结构将被用作构建块来使用有利于有限元分析过程的理解。
Both spring and truss elements give an easier modeling overview of the finite element analysis procedure, due to the fact that each spring and truss element, regardless of length, is an ideally sized element and does not need any further division.弹簧和桁架元件给出一个简单的建模概述了有限元分析过程,由于每个弹簧和桁架元件,不计长度,是一种理想的元素不需要任何进一步的细化。
2.3 Understanding Computer and FEA software interaction -Using the Spring Element as an example2.3理解计算机和有限元分析软件交互，使用弹性元件作为一个例子In the following example, a two-element structure is analyzed by finite element method.在接下来的例子中,对一个双元素结构有限元方法进行了分析。
The analysis procedure presented here will be exactly the same as that used for a complex structural problem, except, in the following example, all calculations will be carried out by hand so that each step of the analysis can be clearly understood. All derivations and equations are written in a form, which can be handled by a computer, since all finite element analyses are done on a computer. The finite element equations are derived using Direct Equilibrium method.本文提供的分析过程将一模一样,用于复杂的结构性问题,除了在以下示例中,所有的计算将手算进行,这样可以清楚地理解每一步的分析。