Finite Element Analysis of Shrinkage in the Interface of Functionally Graded Concrete Segment Us
连续刚构桥运营过程中跨中下挠的分析
持续刚构桥运营过程中跨中下挠旳分析摘要：对于目前持续刚构桥梁状况，对其运营过程中跨中下挠旳变化状况展开了考察，得出了某些因素，也给出了某些相应措施。
核心词：持续刚构桥，跨中下挠In the process of the continuous rigid frame bridge operation across themiddle and lower torsion analysisAbstract：For the case of continuous rigid frame bridge, in the process of its operations across the middle and lower torsion for inspection, the changes of some reasons, and gives some corresponding measures.Keywords: Continuous rigid frame bridge; Across the middle and lower scratching1 前言近来几十年，持续刚构桥梁在我国发展旳速度十分快。
由于使用强度高旳材料制作混凝土构件，可以大幅度减少截面尺寸，减少构造自重，对实现大跨径规定很有协助，但正由于如此桥跨旳变大，持续刚构桥在预定寿命年限中旳问题也暴露出来，一种不可忽视旳状况就是由于服役时间旳增长，如果持续刚构跨中旳大幅度下挠，必然引起桥梁在运作期内引起驾驶人员与乘车人员旳不适感，进一步威胁到车辆安全甚至车辆人员旳生命安全[1]。
持续刚构桥跨中下挠幅度过大，将导致梁体浮现裂缝，对构造整体安全产生威胁，因此大跨径持续刚构桥旳跨中下挠问题也成了这种类型旳桥梁在建设和发展过程中务必完善旳问题。
大跨径持续刚构桥旳明显病害之一便是跨中下挠问题，从目前来看跨径在80 m到100 m如下旳，病害较少；跨径在100 m 到160 m之间旳，病害多；而跨径在160 m以上旳，体现旳比较突出。
英文有限元方法Finite element method讲义 (1)
MSc in Mechanical Engineering Design MSc in Structural Engineering LECTURER: Dr. K. DAVEY(P/C10)Week LectureThursday(11.00am)SB/C53LectureFriday(2.00pm)Mill/B19Tut/Example/Seminar/Lecture ClassFriday(3.00pm)Mill/B192nd Sem. Lab.Wed(9am)Friday(11am)GB/B7DeadlineforReports1 DiscreteSystems DiscreteSystems DiscreteSystems2 Discrete Systems. Discrete Systems. Tutorials/Example I.Meshing I.Deadline 3 Discrete Systems Discrete Systems Tutorials/Example IIStart4 Discrete Systems. Discrete Systems. DiscreteSystems.5 Continuous Systems Continuous Systems Tutorials/Example II. Mini Project6 Continuous Systems Continuous Systems Tutorials/Example7 Continuous Systems Continuous Systems Special elements8 Special elements Special elements Tutorials/Example III.Composite IIDeadline *9 Special elements Special elements Tutorials/Example10 Vibration Analysis Vibration Analysis Vibration Analysis III Deadline11 Vibration Analysis Vibration Analysis Tutorials/Example12 VibrationAnalysis Tutorials/Example Tutorials/Example13 Examination Period Examination Period14 Examination Period Examination Period15 Examination Period Examination Period*Week 9 is after the Easter vacation Assignment I submission (Box in GB by 3pm on the next workingday following the lab.) Assignment II and III submissions (Box in GB by 3pm on Wed.)CONTENTS OF LECTURE COURSEPrinciple of virtual work; minimum potential energy.Discrete spring systems, stiffness matrices, properties.Discretisation of a continuous system.Elements, shape functions; integration (Gauss-Legendre).Assembly of element equations and application of boundary conditions.Beams, rods and shafts.Variational calculus; Hamilton’s principleMass matrices (lumped and consistent)Modal shapes and time-steppingLarge deformation and special elements.ASSESSMENT: May examination (70%); Short Lab – Holed Plate (5%); Long Lab – Compositebeam (10%); Mini Project – Notched component (15%).COURSE BOOKSBuchanan, G R (1995), Schaum’s Outline Series: Finite Element Analysis, McGraw-Hill.Hughes, T J R (2000), The Finite Element Method, Dover.Astley, R. J., (1992), Finite Elements in Solids and Structures: An Introduction, Chapman &HallZienkiewicz, O.C. and Morgan, K., (2000), Finite Elements and Approximation, DoverZienkiewicz, O C and Taylor, R L, (2000), The Finite Element Method: Solid Mechanics,Butterworth-Heinemann.IntroductionThe finite element method (FEM) is a numerical technique that can be applied to solve a range of physical problems. The method involves the discretisation of the body (domain) of interest into subregions, which are known as elements. This enables a continuum problem to be described by a finite system of equations. In the field of solid mechanics the FEM is undoubtedly the solver of choice and its use has revolutionised design and analysis approaches. Many commercial FE codes are available for many types of analyses such as stress analysis, fluid flow, electromagnetism, etc. In fact if a physical phenomena can be described by differential or integral equations, then the FE approach can be used. Many universities, research centres and commercial software houses are involved in writing software. The differences between using and creating code are outlined below:(A) To create FE software1. Confirm nature of physical problem: solid mechanics; fluid dynamics; electromagnetic; heat transfer; 1-D, 2-D, 3-D; Linear; non-linear; etc.2. Describe mathematically: governing equations; loading conditions.3. Derive element equations: convert governing equations into algebraic form; select trial functions; prepare integrals for numerical evaluation.4. Assembly and solve: assemble system of equations; application of loads; solution of equations.5. Compute:6. Process output: select type of data; generate related data; display meaningfully and attractively.(B) To use FE software1. Define a specific problem: geometry; physical properties; loads.2. Input data to program: geometry of domain, mesh generation; physical properties; loads-interior and boundary.3. Compute:4. Process output: select type of data; generate related data; display meaningfully and attractively.DISCRETE SYSTEMSSTATICSThe finite element involves the transformation of a continuous system (infinite degrees of freedom) into a discrete system (finite degrees of freedom). It is instructive therefore to examine the behaviour of simple discrete systems and associated variational methods as this provides real insight and understanding into the more complicated systems arising from the finite element method.Work and Strain energyFLuxConsider a metal bar of uniform cross section, A , fixed at one end (unrestrained laterally) and subjected to an axial force, F , at the other.Small deflection theory is assumed to apply unless otherwise stated.The work done, W , by the applied force F is .a ()∫′′=uau d u F WIt is worth mentioning at this early stage that it is not always possible to express work in this manner for various reasons associated with reversibility and irreversibility. (To be discussed later)The work done, W , by the internal forces, denoted strain energy , is se22200se ku 21u L EA 2121EAL d EAL d AL W ==ε=ε′ε′=ε′σ=∫∫εεwhere ε=u L and stiffness k EA L=.The principle of virtual workThe principle of virtual work states that the variation in strain energy is equal to the variation in the work done by applied forces , i.e.()u F u u d u F du d W u ku u ku 21du d ku 21W u0a 22se δ=δ⎟⎟⎠⎞⎜⎜⎝⎛′′=δ=δ=δ⎟⎠⎞⎜⎝⎛=⎟⎠⎞⎜⎝⎛δ=δ∫()0u F ku =δ−⇒Note that use has been made of the relationship δf dfduu =δ where f is an arbitrary functional of u . In general displacement u is a function of position (x say) and it is understood that ()x u δ means a change in ()u x with xfixed. Appreciate that varies with from zero to ()'u F 'u ()u F F = in the above integral.Bearing in mind that δ is an arbitrary variation; then this equation is satisfied if and only if F , which is as expected. Before going on to apply the principle of virtual work to a continuous system it is worth investigating discrete systems further. This is because the finite element formulation involves the transformation of a continuous system into a discrete one. u ku =Spring systemsConsider a single spring with stiffness independent of deflection. Then, 2F21u1F1u2k()()⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡−−=−=2121212se u u k k k k u u 21u u k 21W()()()⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡−−δδ=δ−δ−=δ21211212se u u k k k k u u u u u u k W()⎟⎟⎠⎞⎜⎜⎝⎛δδ=δ+δ=δ21212211a F F u u u F u F W , where ()111u F F = and ()222u F F =.Note here that use has been made of the relationship δ∂∂δ∂∂δf f u u f u u =+1122, where f is an arbitrary functional of and . Observe that in this case is a functional of 1u u 2W se u u u 2121=−, so()()(121212*********se se u u u u k u u ku 21du d u du dW W δ−δ−=−δ⎟⎠⎞⎜⎝⎛=δ=δ).The principle of virtual work provides,()()()()0F u u k u F u u k u 0W W 21221121a se =−−δ+−−−δ⇒=δ−δand since δ and δ are arbitrary we have. u 1u 2F ku ku 11=−2u 2 F ku k 21=−+represented in matrix form,u F K u u k k k k F F 2121=⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡−−=⎟⎟⎠⎞⎜⎜⎝⎛=where K is known as the stiffness matrix . Note that this matrix is singular (det K k k =−=220) andsymmetric (K K T=). The symmetry is a result of the fact that a unit deflection at node 1 results in a force at node 2 which is the same in magnitude at node 1 if node 2 is moved by the same amount.Could also have arrived at equation above via()⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡−−=⎟⎟⎠⎞⎜⎜⎝⎛⇒=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡−−−⎟⎟⎠⎞⎜⎜⎝⎛δδ=δ−δ2121211121a se u u k k k k F F 0u u k k k k F F u u W WBoundary conditionsWith the finite element method the application of displacement constraint boundary conditions is performed after the equations are assembled. It is an interest to examine the implications of applying and not applying the displacement boundary constraints prior to applying the principle of virtual work. Consider then the single spring element above but fixed at node 1, i.e. 0u 1=. Ignoring the constraint initially gives()212se u u k 21W −=, ()()1212se u u u u k W δ−δ−=δ and 2211a u F u F W δ+δ=δ.The principle of virtual work gives 2211ku ku ku F −=−= and 2212ku ku ku F =+−=, on applicationof the constraint. Note that is the force required at node 1 to prevent the node moving and is the reaction force.21ku F −=21ku F =−Applying the constraint straightaway gives 22se ku 21W =, 22se u ku W δ=δ and 22a u F W δ=δ. The principle of virtual work gives with no information about the reaction force at node 1.22ku F =Exam Standard Question:The spring-mass system depicted in the Figure consists of three massless springs, which are attached to fixed boundaries by means of pin-joints at nodes 1, 3 and 5. The springs are connected to a rigid bar by means of pin-joints at nodes 2 and 4. The rigid bar is free to rotate about pivot A. Nodes 2 and 4 are distances and below pivot A, respectively. Each spring has the same stiffness k. Node 2 is subjected to an external horizontal force F 2/l 4/l 2. All deflections can be assumed to be small.(i) Write expressions for the extension of each spring in terms of the displacement of node 2.(ii) In terms of the degrees of freedom at node 2, write expressions for the total strain energy W of the spring-mass system. In addition, specify the variation in work done se a W δ resulting from the application of the force.2F (iii) Use Use the principle of virtual work to find a relationship between the magnitude of and the horizontal components of displacement at node 2.2F (iv) Use the principle of virtual work to show that the net vertical force imposed by the springs on the rigid-bar at node 2 is zero.Solution:(i) Directional vectors for springs are: 2112e 21e 23e +=, 2132e 21e 23e +−= and 145e e =. Extensions for bottom springs are: 221212u 23u e =⋅=δ, 223232u 23u e −=⋅=δ.Note that 2u u 24=, so 2u245−=δ.(ii)()2222222245232212se ku 87u 212323k 21k 21W =⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛−+⎟⎟⎠⎞⎜⎜⎝⎛=δ+δ+δ=, 222u F W δ=δ(iii) 2222a 22se ku 47F u F W u ku 47W =⇒δ=δ=δ=δ(iv) Need additional displacement degree of freedom at node 2. Let 22122e v e u u += and note that2221212v 21u 23u e +=⋅=δ and 2223232v 21u 23u e +−=⋅=δ.()⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛+−+⎟⎟⎠⎞⎜⎜⎝⎛+=δ+δ=222222232212se v 21u 23v 21u 23k 21k 21W ⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛δ+δ−⎟⎟⎠⎞⎜⎜⎝⎛+−+⎟⎟⎠⎞⎜⎜⎝⎛δ+δ⎟⎟⎠⎞⎜⎜⎝⎛+=δ22222222se v 21u 23v 21u 23v 21u 23v 21u 23k W Setting and gives0v 2=0u 2=δ2vert 222222se v F v 0v 21u 23v 21u 23k W δ=δ=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛δ⎟⎟⎠⎞⎜⎜⎝⎛−+⎟⎠⎞⎜⎝⎛δ⎟⎟⎠⎞⎜⎜⎝⎛=δ hence . 0F vert 2=Method of Minimum PotentialConsider the expression,()()F u u u TT 21212121c se K 21F F u u u u k k k k u u 21W W P −=⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡−−=−=where W F and can be considered as a work term with independent of . u F u c =+1122F i u iThe approach of minimising P is known as the method of minimum potential .Note that,()()u F 0F -u u =F u u u +u u K K K K 21W W P T T T T c se =⇒=δδ−δδ=δ−δ=δwhere use has been made of the fact that δδu u =u u T TK K as a result of K 's symmetry.It is useful at this stage to consider the minimisation of an arbitrary functional ()u P where()()3T T O H 21P P u u u u u δ+δδ+∇δ=δand the gradient ∇=P P u i i ∂∂, and the Hessian matrix coefficients H P u u ij i j=∂∂∂2.A stationary point requires that ∇=, i.e.P 0∂∂Pu i=0.Moreover, a minimum point requires that δδu u TH >0 for all δu ≠0 and matrices that possess this property are known as positive definite .Setting P W W K se c T=−=−12u u u F T provides ∇=−=P K u F 0 and H K =.It is a simple matter to check that with u 10= (to prevent rigid body movement) that K is positive definite and this is a property commonly associated with FE stiffness matrices.Exam Standard Question:The spring system depicted in the Figure consists of four massless unstretched springs, which are attached to fixed boundaries by means of pin-joints at nodes 1 to 4. The springs are connected to a slider at node 5. Theslider is constrained to move in a frictionless channel whose axis is to the horizontal. Each spring has the same stiffness k. The slider is subjected to an external force F 0453 whose direction is along the axis of the frictionless channel.(i)The deflection of node 5 can be represented by the vector 25155v u e e u +=, where and areunit orthogonal vectors which are shown in the Figure. Write the components of deflection and in terms of , where is the magnitude of , i.e. e 1e 25u 5v 5U 5U 5u 25U 5u =. Show that the extensions of eachspring, in terms of , are: 5U ()22/31U 515+=δ, ()22/31U 525−=δ, and2/U 54535−=δ−=δ.(ii) In terms of k and write expressions for the total strain energy W of the spring-mass system. Inaddition, specify the variation in work done 5U se a W δ resulting from the application of the force . 5F (iii) Use the principal of virtual work to find a relationship between the magnitude of and thedisplacement at node 5.5F 5U (iv) Use the principal of virtual work to determine an expression for the force imposed by the frictionless channel on the slider.(v)Form a potential energy function for the spring system. Assume here that nodes 1, 3 and 4 are fixed and node 5 is restricted to move in the channel. Use this function to determine the reaction force at node 2.Solution:(i) Directional vectors for springs and channel are: ()2115e e 321e +=, ()2125e e 321e +−=, 135e e −=, 45e e = and (21c 5e e 21e +=). Deflection c 555e U u =, so 2U v u 555==. Extensions springs are: ()3122U u e 551515+=⋅=δ, ()3122U u e 552525−=⋅=δ, 2Uu e 553535−=⋅=δand 2Uu e 554545=⋅=δ(ii)()()()252522245235225215se kU U 83131k 8121k 21W =⎟⎠⎞⎜⎝⎛+−++=δ+δ+δ+δ=, 55a U F W δ=δ(iii)5555a 55se kU 2F U F W U kU 2W =⇒δ=δ=δ=δ(iv) Need additional displacement degree of freedom at node 3. A unit vector perpendicular to the channel is(21p 5e e 21e +−=) and let p 55c 555e V e U u += and note that()()3122V3122U u e 5551515−++=⋅=δ and ()()3122V3122U u e 5552525++−=⋅=δ, 2V 2U u e 5553535+−=⋅=δ and 2V 2U u e 5554545−=⋅=δ()()()()()()()()⎟⎠⎞⎜⎝⎛−+++−+−++=δ+δ+δ+δ=255255255245235225215se V U 831V 31U 31V 31U k 8121k 21W ()()()()()555se V 0V 831313131kU 81W δ=δ−+−+−+=δ, where variation is onlyconsidered and is set to zero. Principle of virtual work .5V δ3V 0F V F V 0W p 55p 55se =⇒δ=δ=δ(v)()3122U u e 551515+=⋅=δ, ()()5552525V 3122Uu u e −−=−⋅=δ, where 2522e V u =. ()()()223333223232233245235225215V F U F U V 3122U 3122U k 21V F U F k 21P −−⎟⎟⎠⎞⎜⎜⎝⎛+⎥⎦⎤⎢⎣⎡−−+⎥⎦⎤⎢⎣⎡+=−−δ+δ+δ+δ=and ()0F V 3122U k V P 2232=−⎥⎦⎤⎢⎣⎡−−−=∂∂, which on setting 0V 2= gives ()⎥⎦⎤⎢⎣⎡−−=3122U k F 32.The reaction is .2F −System AssemblyConsider the following three-spring system 2F 21u 1F 1u 2kF 3F 4u 3u 4k 1k2334()()()234322322121se u u k 21u u k 21u u k 21W −+−+−=,()()()()()()343432323212121se u u u u k u u u u k u u u u k W δ−δ−+δ−δ−+δ−δ−=δ,44332211a u F u F u F u F W δ+δ+δ+δ=δ,and δδ implies that,W W se a −=0u F K u u u u k k 0k k k k 00k k k k 00k k F FF F 43213333222211114321=⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−−+−−+−−=⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛=where again it is apparent that K is symmetric but also it is banded, i.e. the non-zero coefficients are located around the principal diagonal. This is a property commonly associated with assembled FE stiffness matrices and depends on node connectivity. Note also that the summation of coefficients in individual rows or columns gives zero. The matrix is singular and 0K det =.Note that element stiffness matrices are: , and where on examination of K it is apparent how these are assembled to form K .⎥⎦⎤⎢⎣⎡−−1111k k k k ⎥⎦⎤⎢⎣⎡−−2222k k k k ⎥⎦⎤⎢⎣⎡−−3333k k k kIf a boundary constraint is imposed then row one is removed to give:0u 1=u F K u u u k k 0k k k k 0k k k F F F 432333322221432=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−+−−+=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=. If however a boundary constraint (say) is imposed then row one is again removed but a somewhatdifferent answer is obtained: 1u 1=u F K u u u k k 0k k k k 0k k k F F k F 4323333222214312=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−+−−+=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛+=)Direct FormulationIt is possible to formulate the stiffness matrix directly by moving one node and keeping the others fixed and noting the reactions.The above system can be solved for u , once possible rigid body motion is prevented, by setting u (say) to give 10=⇒=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−+−−+=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=u F K u u u k k 0k k k k 0k k k F F F 432333322221432⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−+−−+=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−4321333322221432F F F k k 0k k k k 0k k k u u uThe inverse stiffness matrix, K −1, is known as the flexibility matrix and, for this example at least, can be assembled directly by noting the system response to prescribed forces.In practice K −1is never calculated and the system K u F = is solved using a modern numerical linear system solver.It is a simple matter to confirm thatu u K 21u u u u k k 0k k k k 00k k k k 00k k u u u u 21W T 4321333322221111T4321se =⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−−+−−+−−⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛= with F u T4321T4321a F F F F u u u u W δ=⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛δδδδ=δThus,()u F F u u K 0K W W Ta se =⇒=−δ=δ−δExample:k1F 2u23k2u3F321With use a direct method to find the assembled stiffness and flexibility matrices.0u 1=Solution:The equations of interest are of the form: 3232222u k u k F += and 3332323u k u k F +=.Consider and equilibrium at nodes 2 and 3. At node 2, 0u 3=()2212u k k F += and at node 3,.223u k F −=Consider and equilibrium at nodes 2 and 3. At node 2, 0u 2=322u k F −= and at node 3, . 323u k F =Thus: , , 2122k k k +=223k k −=232k k −= and 233k k =.For flexibility the equations of interest are of the form: 3232222F c F c u += and . 3332323F c F c u +=Consider and equilibrium at nodes 2 and 3. At node 2, 0F 3=122k F u = and at node 3,1223k F u u ==.Consider and equilibrium at nodes 2 and 3. At node 2, 0F 2=122k F u = and at node 3,()2133k 1k 1F u +=.Thus: 122k 1c =, 123k 1c =, 132k 1c = and 2133k 1k 1c +=.Can check that ⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡+⎥⎦⎤⎢⎣⎡−−+1001k 1k 1k 1k 1k 1k k k k k 2111122221 as required,It should be noted that the direct determination requires boundary constraints to be applied to ensure that the flexibility matrix exists, which requires the stiffness to be non-singular. However, the stiffness matrix always exists, so boundary conditions need not be applied prior to constructing the stiffness matrix with the direct approach.Large deformation theory for spring elementsThus far small deflection theory has been applied where the strains are measured using the Cauchy strainxu11∂∂=ε. A conjugate stress can be obtained by differentiating with respect the expression for strain energy density (energy per unit volume) 11ε211E 21ε=ω, i.e. 111111E ε=ε∂ω∂=σ, where E is Young’s Modulusand is the Cauchy stress (sometimes referred to as the Euler stress). 11σIn the case of large deformation theory we will restrict our attention to hyperelastic materials which are materials that possess an expression for strain energy density Ω (say) that is analytical in strain.The strain used in large deformation theory is Green’s strain (see Appendix II) which for a uniformly loadeduniaxial bar is 211x u 21x u E ⎟⎠⎞⎜⎝⎛∂∂+∂∂=.An expression for strain energy density (energy per unit volume) 211EE 21=Ω and the derived stress is 111111EE E S =∂Ω∂=, where E is Young’s Modulus and is known as the 211S nd Piola-Kirchoff stress . 2F21u1F1u2kBar subject to longitudinal deformationConsider a bar of length L and cross sectional area A represented by a spring element and subject to nodal forces and . 1F 2FThe strain energy is∫∫∫∫⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛∂∂+∂∂==Ω=Ω=212121x x 22x x 211x x V se dx x u 21x u EA 21dx E EA 21dx A dV WConsider further a linear displacement field of the form ()21u L x u L x L x u ⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛−= and note thatL u u xu 12−=∂∂. ()()221212x x 221212se u u L 21u u L EA 21dx L u u 21L u u EA 21W 21⎥⎦⎤⎢⎣⎡−+−=⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛−+−=∫ ()()()⎥⎦⎤⎢⎣⎡−+−+−=4122312212se u u L 41u u L 1u u k 21W()()()(12312221212se u u u u L 21u u L 23u u k W δ−δ⎥⎦⎤⎢⎣⎡−+−+−=δ) and 2211a u F u F W δ+δ=δ.The principle of virtual work gives()()()⎥⎦⎤⎢⎣⎡−+−+−−=3122212121u u L 21u u L 23u u k F and()()(⎥⎦⎤⎢⎣⎡−+−+−=3122212122u u L 21u u L 23u u k F ), represented in matrix form as()()()()()⎟⎟⎠⎞⎜⎜⎝⎛⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−+−−−−−−−+−+⎥⎦⎤⎢⎣⎡−−=⎟⎟⎠⎞⎜⎜⎝⎛21121212121221u u L 3u u 1L 3u u 1L 3u u 1L 3u u 1L 2u u k 3k k k k F Fwhich is of the form[]u F G L K K += where is called the geometrical stiffness matrix and is the usual linear stiffnessmatrix. G K L KA common approximation used, depending on the magnitude of L /u u 12−, is⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡−−+⎥⎦⎤⎢⎣⎡−−=⎟⎟⎠⎞⎜⎜⎝⎛2121u u 1111L 2P 3k k k k F F where ()12u u k P −=.The fact that is non-linear (even in its approximate form) means that iterative solution procedures are required to be employed to determine the unknown displacements. G KNote that the approximate form is arrived at using the following strain energy expression()()⎥⎦⎤⎢⎣⎡−+−=312212se u u L 1u u k 21WExample:The strain energies for the springs in the above system (fixed at node 1) are k 1 F 2u 23k 2u3F321⎥⎦⎤⎢⎣⎡+=1322211seL u u k 21W and ()()⎥⎦⎤⎢⎣⎡−+−=323222322se u u L 1u u k 21WUse the principle of virtual work to obtain the assembled linear and geometrical stiffness matrices.()()()3322a 2322322322122212se1sese u F u F W u u u u L 23u u k u L 2u 3u k W W W δ+δ=δ=δ−δ⎥⎦⎤⎢⎣⎡−+−+δ⎥⎦⎤⎢⎣⎡+=δ+δ=δThus ()(⎥⎦⎤⎢⎣⎡−+−−⎥⎦⎤⎢⎣⎡+=2232232122212u u L 23u u k L 2u 3u k F ) and ()()⎥⎦⎤⎢⎣⎡−+−=22322323u u L 23u u k F⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡αα−α−α+α+⎥⎦⎤⎢⎣⎡−−+=⎟⎟⎠⎞⎜⎜⎝⎛32222212222132u u k k k k k F F where 1211L 2u k 3=α and ()23222u u L 2k 3−=α.Note that the element stiffness matrices are[][]111k K α+= and ⎥⎦⎤⎢⎣⎡αα−α−α+⎥⎦⎤⎢⎣⎡−−=222222222k k k k Kand it is evident how these should be assembled to form the assembled linear and geometrical stiffness matrices.2v21u 1v1u 2kxBar subject to longitudinal and lateral deflectionConsider a bar of length L and cross sectional area A represented by a spring element and subject to longitudinal and lateral displacements u and v, respectively.The normal strain is 2211x v 21x u 21x u E ⎟⎠⎞⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂+∂∂= and the associated strain energy∫∫∫∫⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂+∂∂==Ω=Ω=212121x x 22x x 211x x V se dx x v 21x u 21x u EA 21dx E EA 21dx A dV W ∫⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛∂∂⎟⎠⎞⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂≈21x x 232se dx x v x u x u x u EA 21WConsider further a linear displacement field of the form ()21u L x u L x L x u ⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛−= and()21v L x v L x L x v ⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛−=, and note thatL u u x u 12−=∂∂ and L v v x v 12−=∂∂. ()()()()⎥⎦⎤⎢⎣⎡−−+−+−=L v v u u L u u u u L EA 21W 21212312212se()()()()()()()1212121221221212se v v L v v u u k u u L 2v v L 2u u 3u u k W δ−δ⎦⎤⎢⎣⎡−−+δ−δ⎥⎦⎤⎢⎣⎡−+−+−=δ2v 22h 21v 11h 1a v F u F v F u F W δ+δ+δ+δ=δ and the principle of virtual work gives()()()⎥⎦⎤⎢⎣⎡−+−+−−=L 2v v L 2u u 3u u k F 21221212h1and ()()⎥⎦⎤⎢⎣⎡−−−=L v v u u k F 1212v1 ()()()⎥⎦⎤⎢⎣⎡−+−+−=L 2v v L 2u u 3u u k F 21221212h2and ()()⎥⎦⎤⎢⎣⎡−−=L v v u u k F 1212v2()()⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−−−−−+⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−−−+⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−−=⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛22111212v 2h 2v 1h 1v u v u 101005.105.1101005.105.1Lu u k 1010000010100000L2v v k 0000010100000101k F F F FExam Standard Question:The spring system depicted in the Figure consists of two massless springs of equal length , which are attached to fixed boundaries by means of pin-joints at nodes 1 and 2. The springs are connected to a slider atnode 3. The slider is constrained to move in a frictionless channel whose axis is 45 to the horizontal. Each spring has the same stiffness . The slider is subjected to an external force F 1L =0L /EA k =3 whose direction is along the axis of the frictionless channel.FigureAssume the springs have strain density ⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛∂∂⎟⎠⎞⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂=Ω232x v x u x u x u E 21.(i) Write expressions for the longitudinal and lateral displacements for each spring at node 3 in terms of thedisplacement along the channel at node 3.(ii) In terms of displacement along the channel at node 3, write expressions for the total strain energy W of thespring-mass system. In addition, specify the variation in work done se a W δ resulting from the application of the force .3F (iii) Use the principle of virtual work to find a relationship between the magnitude of and the displacementalong the channel at node 3. 3FSolution:(i) Directional vectors for springs and channel are: ()2113e e 321e +=and ()2123e e 321e +−= and (21c 3e e 21e +=). Perpendicular vectors are: ()2113e 3e 21e +−=⊥and ()2123e 3e 21e +=⊥Deflection c 333e U u =, so 2U v u 333==.Longitudinal displacement: ()3122U u e 331313+=⋅=δ, ()3122U u e 332323−=⋅=δ.Lateral displacement: ()3122U u e 331313+−=⋅=δ⊥⊥, ()3122U u e 332323+=⋅=δ⊥⊥(ii) The strain energy density for element 1 is ⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛δ⎟⎠⎞⎜⎝⎛δ+⎟⎠⎞⎜⎝⎛δ+⎟⎠⎞⎜⎝⎛δ=Ω⊥21313313213L L L L E 21 The strain energy density for element 2 is ⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛δ⎟⎠⎞⎜⎝⎛δ+⎟⎠⎞⎜⎝⎛δ+⎟⎠⎞⎜⎝⎛δ=Ω⊥22323323223L L L L E 21 The total strain energy with substitution of 1L = gives()()()()[]()()()()[][]3322312232332322321313313213se U Uk 21k 21k 21W α+α=δδ+δ+δ+δδ+δ+δ=⊥⊥where and are constants determined on collecting up terms on substitution of and .1α2α231313,,δδδ⊥⊥δ2333a U F W δ=δ.(iii) The principle of virtual work gives⎥⎦⎤⎢⎣⎡α+α=⇒δ=δ=δ⎥⎦⎤⎢⎣⎡α+α=δ32133332323231se U 23kU F U F W U U 23U k WPin-jointed structuresThe example above is a pin-jointed structure. A reasonable good approximation reported in the literature for strain energy density, commonly used with pin-jointed structures, is⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛∂∂⎟⎠⎞⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂=Ω22x v x u x u E 21This arises from strain-energy approximation 211x v 21x u E ⎟⎠⎞⎜⎝⎛∂∂+∂∂=. Can be used when 22x v x u ⎟⎠⎞⎜⎝⎛∂∂<<⎟⎠⎞⎜⎝⎛∂∂.。
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。
选择性激光烧结翘曲变形模拟
文章编号：1001-4934(2021)02-0044-07选择性激光烧结翘曲变形模拟蔡令令1,丁浩亮2,孟娟1,温馨1,严波1*(1.上海交通大学材料科学与工程学院，上海200030；2.航天材料及工艺研究所，北京100076)摘要：建立了烧结过程的温度场、弹性应力应变场的有限元计算模型，开发了C++模拟软件,模拟了选择性激光烧结(SLS)过程的温度、残余热应力的演化，以及最终的翘曲变形。
提出的改进移动热源加载算法提高了计算速度。
研究结果表明：激光烧结过程中温度场不均匀，导致了热应力的产生，进而使制件变形，制件的上下部分收缩不一致，从而向上翘曲。
随着烧结制件增厚，其弯曲所需的应力增大，翘曲变形减小；选择合理的扫描方式可以使温度场更加均匀，从而减少制件的翘曲变形；随着激光功率增大，温度场更不均匀，热应力增大，翘曲变形增大。
关键词：选择性激光烧结(SLS)；弹性应力应变关系；有限元法;数值模拟;翘曲中图分类号：TQ320.6文献标识码：AWarping simulation of selective laser sinteringCAI Lingling1,DING Haoliang2,MENG Juan1,WEN Xin1,YAN Bo1*Abstract:The finite element model of selective laser sintering(SLS)process was built The corresponding C++software was developed which involving temperature，elastic stress and strainfields.Thenthetemperature，thethermalresidualstress，andthewarpingintheSLS processweresimulated.Animprovedloadingalgorithmofmovingheatsourcewaspresented，which reduced the time consumption and accelerates the computation greatly.It is found that in the process of laser sintering,the temperature field is mot uniform,which leads to thermal sressandwarpingdeformaion.Theupwardwarpingofaworkpieceiscausedbyinconsisen< shrinkageof<heupperandlowerparsof<heworkpiece.As<he<hicknessof<hesinered parsincreases，<herequired bending sress wi l increase，leading<o reduced warping.Selecingareasonablescanningmodecanmake he emperaurefieldmoreuniformandreduce the warping of the parts.As the laser power increases,the unevenness of the temperature fieldincreases#leading<ogreaer<hermalsressandwarping.Key words:selective laser sintering(SLS)；elastic constitutive model；finite element method；num rical simulation$warping收稿日期：2020-11-20基金项目：国家重点研发计划(2016YFB1106700)；国家自然科学基金项目(51005151)作者简介：蔡令令(1996—)，男，安徽省宿州市人，硕士生,从事选区激光烧结仿真模拟研究)通讯作者：严波(1979—)，男，副教授)0引言与传统注塑成型制品相比,塑料粉末选择性激光烧结(SLS)工艺产品质量较差受关注最多的是产品的翘曲问题。
3.The finite element method-its____ basis and fundamentals
3.1 Introduction The finite element process, being one of approximation, will seek the solution in the approximate form †
where Na are shape functions prescribed in terms of independent variables (such as the coordinates x, y, etc.) and all or most of the parameters ũa are unknown.
equations
能源与建筑工程学院
Contents
3 Generalization of the finite element concepts. Galerkin-weighted residual and variational approaches 3.9 Establishment of natural variational principles for linear, self-adjoint, differential equations 3.10 Maximum, minimum, or a saddle point? 3.11 Constrained variational principles. Lagrange multipliers 3.12 Constrained variational principles. Penalty function and perturbed lagrangian
能源与建筑工程学院
3.Generalization of the finite element concepts. Galerkin-weighted residual and variational approaches
有限元分析法英文简介
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 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 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 ofthe 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 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 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, 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 willdrive 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 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. 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 establishintegral 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 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.。
高英力教授 - 长沙理工大学欢迎您!
高英力教授作者： 2010-04-01 15:12:48个人基本情况：姓名：高英力性别：男出生年月：1977年5月民族：汉政治面貌：党员职称职务：教授，硕士生导师，系支部书记最后学历、学位：博士研究生学历、博士、博士后工作单位：长沙理工大学交通运输工程学院办公电话：E－mail ： yingligao@从事研究的学科专业领域及主要研究方向：从事研究的学科专业领域：土木工程材料主要研究方向：道路结构新材料、新型建筑材料研发、工业固体废弃物综合利用主要工作经历及业绩：1999年建筑工程专业本科毕业，获学士学位，2000年考入中南大学土木建筑学院攻读硕士学位，2002年4月获得提前攻读博士学位资格，2005年12月获道路与铁道工程专业工学博士学位，2006年1月进入武汉理工大学材料学院进行博士后研究工作，2008年3月博士后出站进入长沙理工大学交通运输工程学院。
2013.5～2014.5，作为国家公派访问学者，在加拿大Ryerson University 留学访问。
主要从事建筑材料的科研与教学工作，承担了本科生《土木工程材料》、《建筑材料》、《工程复合材料》等课程的教学任务。
在科研方面，多年来一直从事土木工程结构高性能水泥基材料、道路工程新材料方面的研究工作，包括：①道路结构新材料的研发及应用；②公路路面高性能水泥混凝土及沥青混凝土研究及应用；③工业固体废弃物综合利用；④地下工程隧道衬砌结构设计、开发、研究与应用。
⑤混凝土用高性能化学外加剂研究及应用等。
先后主持或参加了国家及省部级科研项目10余项。
发表科研学术论文70余篇，其中SCI、EI、ISTP检索达32篇次，申请国家发明专利10项，其中已授权专利8项。
作为项目负责人和主要参加人获省部级奖励7项。
目前主持的主要科研项目:①湖南省自然科学基金项目“碳化、氯盐及荷载耦合作用下钢筋轻骨料混凝土劣化过程及机理研究（编号13JJ3070）”②湖南省交通科技计划项目“湘西地区低温潮湿沥青路面防冻抗滑关键材料设计及施工技术研究”③教育部博士点基金项目“非荷载多因素作用下高强混凝土微裂缝演化机制的工业CT 研究”（编号：20104316120001）④湖南省教育厅重点项目“相变控温梯度复合水泥混凝土路面结构及性能研究” （编号：11A005）⑤中国博士后科学基金特别资助项目“梯度复合混凝土界面区微裂缝动态演化机制的工业CT研究”（编号：201003507）⑥长沙理工大学特殊环境道路工程湖南省重点实验室开放基金资助项目“高强轻骨料混凝土微裂缝动态演化机制的CT研究”（编号：kfj110407）已完成的主要科研项目:作为主持或主要完成人，完成主要科研项目及成果包括：①“水泥混凝土结构材料功能优化及其在公路工程中的应用技术研究”，通过湖南省科技厅组织的技术成果鉴定，成果整体国际先进，部分国际领先，项目主持②湖南省自然科学基金项目“功能梯度水泥基材料界面特征及体积变形机理研究（编号09JJ4023）”，项目主持③中国博士后科学基金面上项目“梯度复合混凝土界面区体积变形与整体性能劣化的相关性(编号20090450986)”，项目主持④湖南省交通厅科技项目子项“冰雪气候下桥梁的桥面抗滑技术探索研究”，项目主持⑤中国博士后科学基金面上项目“复杂环境下超细粉煤灰功能梯度混凝土体积变形及机理研究(编号20070410969)”，项目主持⑥国家973项目“高性能水泥制备和应用”子题：“化学外加剂对高性能水泥水化及结构的作用机理”（编号：2001CB610704），技术负责⑦国家863计划：“高抗渗长寿命大管径隧道管片材料结构设计与工程应用”（编号：2005AA332010），技术负责⑧国家十五科技攻关项目：“新型高性能混凝土及其耐久性的研究”课题“不同强度等级水泥配制高性能混凝土的研究（编号：2001BA307B0302-03）”，技术负责⑨重庆交通大学（桥梁）结构工程重点实验室开放基金项目“盾构隧道衬砌管片功能梯度设计及性能优化研究”（编号CQSLBF-Y07-1），项目主持⑩湖北省博士后科研活动专项资助计划项目“隧道管片功能梯度设计方法与优化技术”（编号：408-32305），项目主持11长沙理工大学博士科研启动基金项目“梯度复合混凝土收缩变形及抗裂性研究”（编号1004212），项目主持12武汉理工大学科学研究基金自由探索类项目: “功能梯度水泥基材料界面特征及体积变形研究”(编号：xjj2007033)，项目主持13长沙理工大学道路结构与材料交通行业重点实验室开放基金资助项目“功能梯度混凝土的界面特征与变形机理”（编号：kfj080208），项目主持完成的主要论文:1. Gao Ying-li, MA Baoguo, ZHOU Shiqiong. Production and Engineering Application of C60 High-Performance Pump Pebble Concrete Containing Ultra-Fine Fly Ash [J]. Canadian Journal of Civil Engineering, 2008, 35(8): 757-763. (SCI, EI收录)2 Gao Ying-li, Cheng Ling. Effects of different mineral admixtures on carbonation resistance of lightweight aggregate concrete, Construction and Building Materials, 2013, 43: 506-510. (SCI,EI收录)3 Gao Ying-li, Zhang Hai-lun. Study on early autogenous shrinkage and crack resistance of fly ash high-strength lightweight aggregate concrete. Magazine of Concrete Research, 2013, 65(15): 906-913. (SCI, EI收录) 4. Gao Ying-li, Zhou Shi-qiong. Influence of Ultra-Fine Fly Ash on Hydration Shrinkage of Cement Paste. Journal of Central South University of Technology (English edition), 2005, 12(5): 596-600. (SCI,EI收录) 5. Gao Ying-li, Zhou Shi-qiong, Yin Jian. Integrated utilization of ultra-fine fly ash - Fluoro gypsum for highway repair. Key Engineering Materials, 2006, v302-303: 255-262. (SCI,EI收录)6. Gao Ying-li, Ma Bao-guo, Cai Mao-tao. Finite Element Analysis of shrinkage in the Interface of Functionally Graded Concrete Segment Used in Shield Tunneling. Journal of Wuhan University of Technology-Materials Science, 2006, 21(Suppl.): 94-98. (SCI、EI收录)7. Gao Ying-li, Ma Bao-guo. Production of high-performance composite concrete segment and experimental investigation of materials. Advanced Materials Research, 2011, (168-170): 1042-1045. (EI收录)sequestration property of cement 8. Gao Ying-li, Chen Ling. Study on CO2based composite cementitious materials containing steel slag used in road. Advanced Materials Research, 2011, 311-313(8): 1949-1952. (EI收录)9. 高英力, 胡柏学, 贺敬, 杨文剑. 相变控温水泥混凝土路面板设计及模型试验. 中国公路学报, 2011, 24(2): 13-16. (EI收录)10. 高英力, 马保国, 王信刚, 等. 盾构隧道功能梯度混凝土管片的研发及性能研究. 岩石力学与工程学报, 2007, 26(11): 2341-2347. (EI收录)11. 高英力, 周士琼, 马保国. 铁路桥梁用超细粉煤灰高性能混凝土的试验研究. 铁道学报, 2006, 28(3):121-127. (EI收录)12. 高英力, 马保国, 王信刚. 钢筋保护层混凝土细观界面过渡区优化及耐久性. 土木建筑与环境工程, 2009, 31(4): 19-24. (EI收录)13. 高英力, 陈瑜. 粉煤灰-脱硫石膏水泥基材料水化活性及微结构. 土木建筑与环境工程, 2011, 33(5): 137-142. (EI收录)14. 高英力, 马保国, 周士琼. UFA道路混凝土早期自收缩及开裂敏感性研究. 沈阳建筑大学学报, 2007, 23(2): 271-275. (EI收录)15. 高英力, 马保国, 周士琼. UFA水泥基材料早期自干燥及自收缩研究. 武汉理工大学学报, 2007, 29(3): 45-49. (EI收录)16. 高英力, 周士琼, 马保国. UFA水泥基材料早期自收缩与干燥收缩研究. 深圳大学学报(理工版), 2008, 25(1): 97-102. (EI收录)17. 高英力, 陈瑜, 马保国, 等. 脱硫石膏－粉煤灰活性掺合料设计及水化特性. 四川大学学报（工程科学版）, 2010, 42(2):225-231. (EI收录)18. 马保国, 高英力, 王信刚, 等. 无细观界面过渡区水泥基材料的设计及性能. 硅酸盐学报, 2007, 35(7): 886-892. (EI收录)19. 周士琼, 高英力, 尹健. 道路用复合超细粉煤灰的应用研究. 建筑材料学报, 2004, (7):311-316.（EI收录）已授权主要专利:1. 高英力. 一种含相变储能材料的防冻抗滑混凝土路面板, 专利号: ZL 2010 2 0142424.9.2. 高英力, 贺敬, 杨文剑, 杜月凡, 汪文渊. 一种具有梯度功能的高耐久混凝土路面板. 专利号: ZL 2010 2 0142449.9.功能的混凝土路面材料、路面3. 高英力, 程领. 具有吸附固化汽车尾气中CO2及其制备和应用方法. 专利号: ZL 2011 1 0029970.0.4. 高英力等. 水泥基复合材料硬化后体积变形实时测试装置及方法. 专利号: ZL 2013 2 0063365.X.5. 马保国, 高英力, 王信刚, 王凯, 金宇, 罗忠涛. 盾构隧道混凝土管片的功能梯度材料界面强化工艺. 专利号: ZL 2006 1 0019657.8.6. 马保国, 高英力, 王信刚, 王凯, 金宇, 罗忠涛. 一种利用FRP筋强化盾构管片的功能梯度材料界面的工艺. 专利号: ZL 2006 1 0124482.7.7. 马保国, 高英力, 王信刚, 王凯, 金宇, 罗忠涛. 盾构隧道管片的保护材料及其制备方法. 专利号: ZL 2006 1 0124481.2.8. 高英力, 陈瑜. 一种粉煤灰-脱硫石膏-水泥自膨胀抗裂型干粉砂浆及其制备方法. 专利号: ZL 2009 1 0042971.1.科研获奖:中国公路学会科技进步二等奖, “水泥混凝土结构材料功能优化及其在公路工程中的应用技术研究”，2014年，排名1；中国公路学会科技进步二等奖, “水泥基复合材料多尺度理论及应用技术”，2011 年，排名2；湖南省科学技术进步三等奖, "土木工程结构材料多尺度理论、耐久性优化及核心应用技术"，2013年，排名2；教育部高等学校科学研究优秀成果科学技术进步二等奖, “高抗渗长寿命大管径隧道管片材料关键制备技术与应用”；2010年，排名6；教育部高等学校科学研究优秀成科学技术进步奖二等奖, “高掺量硅铝质固体废弃物制备承重墙体材料的关键技术与成套装备”；2007年，排名9；中国公路学会科技进步三等奖, “燃煤电厂工业固体废料脱硫石膏－粉煤灰综合开发及利用”，2010 年，排名2；中国公路学会科技进步二等奖, “高速公路隧道路面结构与材料应用技术研究”，2010 年，排名9 。
Finite Element Analysis(MSC基础理论)
Finite Element Analysis（MSC. Software）Basic Theory IntroductionOverview1. What is a Finite Element?2. Common terminology3. Elements4. Constraints5. Loads6. Analysis types7. Units1. What is a Finite Element?It is a mathematical model that interpolates displacements of points in space to get displacements anywhere in the domain of the element.Because of the nature of the interpolation functions, a the gradient of the displacement field is continuous, meaning that a continuous strain field is implied.In other words, the strain field in an element is continuous. This indirectly means that the stress field is continuous in an element.Elements next to each other do not share any information about their interpolation methods, so the strain, and therefore stress field at element boundaries are almost always discontinuous.In a typical first-order quad-element, strain can vary linearly across an element.This capability is used to model more complex stress fields by using a series of linear approximations to model a nonlinear stress-field. In general, the more elements, the better the approximation.It’s already said that the stresses across element boundaries are not continuous. What now? We simply average the values at the nodes.Isn’t this dangerous? Yes it is: If the mesh is not fine enough to accurately represent the stress state.The error at the nodes can however also be used to check for convergence of the mesh: A small error (ie. (max predicted) –(min predicted) ) gives a good indication that the mesh has converged.This brings us to one of the most important concepts in Finite Elements: Mesh ConvergenceA mesh is converged if it is able to represent the stress-state of a component with the required level of accuracy.Mesh Convergence has absolutely nothing to do with convergence settings in Non-linear analysis! Both are needed for Non-linear analysis, while only Mesh convergence are required for linear analysis.A linear analysis has a linear relationship between forces, displacements and stresses.A nonlinear analysis has a nonlinear relationship between forces and displacements due to nonlinear material properties, large rotations, loads changing as the structure deforms or if boundary conditions are dependent on displacements. Convergence for a non-linear analysis has to do with the difference between the applied load and the reaction-forces generated by the elements.2. Common TerminologyNode: A point in space that is used to connect the elements to. Loads / Boundary conditions can only be applied to nodes. Even pressure values or gravitational loads are reduced to loads at nodes by the finite element solver.DOF (Degree of Freedom): This is the displacement of a NodeElement stiffness matrix: The matrix that maps forces at nodes to displacements at the nodes of an elementGlobal stiffness matrix: The assembled matrix that contains the stiffness of each element in the model, thereby having the stiffness of each nodeConstraint: Applying an enforced motion (zero or otherwise) to a node. This is the same as connecting an infinite stiffness at this node.Plane stress element: A thin element that can only move in a plane. The stress normal to the element is zero. Used to model thin structures loaded in 2DPlane strain element: An element that is constrained in it’s thickness direction. The strain normal to the element is zero. Used to approximate thick structures loaded in 2DRigid body mode: The ability of a body to move in space without generating stresses. A model without any constraints can move in 6 directions without generating strain. NO rigid body modes are allowed in a static analysis. Why?3. ElementsSolid elements have only translational DOF at each node. A solid element is able to translate or rotate by only translating the nodes.Because the element has the ability to generate internal moments by applying a couple, no rotational DOF are needed.Constraining the rotational DOF of a solid element will have EXACTLY ZERO EFFECTShell elements usually have 5 DOF at each node: 3 Translations and 2 Rotations. Some Finite Element implementations actually has a third rotation as well, but it is not a necessity to have it to be able to fully load such an element.Shell elements do not have nodes spaced apart in the thickness direction to be able to apply a couple. Therefore, rotational DOF are needed to be able to bend a shell element out-of-plane.In-plane bending is possible using only forces because of the possibility of applying a couple. It is this rotational DOF about the normal to the shell that is commonly neglected in element formulationsBeam elements has 6 DOF per node: Three translations and three rotations.Without the 3 rotational DOF, the element would not be able to load or constrain the two bending directions or to apply a torque.Special cases exist like a truss element (Only axial- and torque stiffness) and Cable elements (only tension stiffness, no compression stiffness)Second order elements gives more accurate answers than first order elements in most cases. They are a lot more expensive in computer resources, so they end up being selectively usedOne important thing to remember about second order elements is that they are a lot more sensitive to distortion than their first-order equivalentsFirst order Tri and Tet elements have a constant stress state, while First order Quad and Hex elements has a Linear stress distribution.This means that first order Tri and Tet elements do a really bad job of representing a linear stress gradient as found in bending problems, while Quads and Hexes can (in theory at least) represent it with a single element.Second order Tri and Tet elements has an accuracy very similar to that of first order Quad and Hex elements.Their drawback is the large increase in number of nodes for the same accuracy.A factor that impacts element-choice for an analysis is how easy it is to create a mesh using that type of element.Meshing of Curves into beam elements is trivialIt is also easy to mesh surfaces to get Tri or Quad elements. Getting a good mesh may take a bit longer when you need to specify where to refine a model.Meshing solid elements into Tet elements is simple. To generate a hex-mesh takes a lot longer because there is no consistent way to mesh complex solids with good-quality Hex elements.This brings us to the most important question in meshing: What element should I use?The answer is very simple: The simplest element that will answer the question that I have!Using too complex an element for the required job leads to a severe penalty in runtime and resource-usageImagine modeling a Crane using solid elements. Of course it does not make sense. Beams do a much better job by being able to use a lot less elements to get the forces and moments. Catalogues exist that tells you what the force- and moment-handling capabilities of beams are, so in a lot of cases you may not even care what the stress is in the model!What happens to the connections between beams? Can beam elements model this accurately? No, of course not. The beam mesh models a “perfect”load-transfer path from one beam to the next, but no local stress-distribution are taken into account.For this, a shell model may be more appropriate. The loads found by a beam model is applied to a shell-element sub-model to find the local stresses and stiffness of the connection.Solid elements are used when there are significant through-thickness stresses at some location. This happens at each location where the thickness changesTake note: Even though a structure is made up of plates, it does not mean that shell elements are the correct choice! If the ratio of the plate thickness to the other dimensions of the plateis too small, the through-thickness stress becomes very significant.A ratio of 1:10 is about the limit for any kind of accuracy for Thin-Shell elements and 1:5 for thick shells. More sensible would be 1:20 and 1:10. The difference between thin- andthick-shell elements is that thin shells ignores through-thickness shear stiffness, whilethick-shells include them. Recommendation: Always use thick shell elements as they will give more accurate answers. They take slightly longer to run, which is their only “drawback”Further recommendations:Use Quad elements rather than Tri elementsUse first order elements unless it is Tri or Tet elementsUse Hex elements ifComputer resources are a problem (Not enough disk / Memory)The meshing-time will be less than the run-time (Nonlinear analysis having to perform thousands of iterations)Use second order Tet elements for most cases where an answer is needed quickly for linear analysis. They are about as accurate a Hex-8 element but takes a lot less time to mesh a complex structure. It’s also easy to check convergence: Simply re-mesh and re-run.4. ConstraintsConstraints are there to generate reaction forces that balances the applied loadsIn a static analysis, the minimum number of DOF that needs to be constrained (even for an unloaded model) is 6: The model may not be able to rotate or displace without reaction forces being generated.There are two types of constraints:An SPC (Single-point constraint) is a constraint equation that prescribes the motion of a single nodeAn MPC (Multi-point constraint) is a constraint equation that links DOF with a linear equation. This makes possible modeling a lever action between two nodes (Displacement 1 = 10* Displacement 2) or rigid connection between nodes (Displacement 1 = Displacement 2).A series of “rigid”elements are implemented in most FEA codes that defines the coefficients of the MPC constraint equations implicitly. This makes it a lot easier to use them.In Nastran, the “R”type elements are implemented through MPC equations. The RBE2 element is a Rigid connection, while the RBE3 is an averaging connection.SPCs are easy to use and to understand the implications of applying them.“R” type elements are more difficult to use, but mostly still straightforward.Explicit MPCs can be very difficult to use, and it’s very easy to make a severe modeling mistake.In practice, SPCs are used to represent the connection to the rest of the universe. Think very carefully before applying a constraint:Is the part of the structure you’re replacing with SPCs really infinitely stiffer then the part you are modeling? If not, at least the answers at the SPCs will be wrong. Further away it may be correct assuming the constraints are equivalentDoes the structure want to deform in a way in real life that the constraint does not allow?Stress concentrations occur where parts of different stiffness are connected together. Because an SPC is infinitely stiff, the stress-concentration effect is extreme. Therefore, except where the constraints exactly model a physical phenomenon, the results at a constraint will be wrong.The same goes for MPCs: Remember that they are infinitely rigid, thereby causing severe stress-concentrations. They can also severely affect the load-path because of the severe stiffness.Symmetry is used where both the loads and the structure have mirror symmetry.Anti-symmetry can also be used to advantageAdvantages:Less computer resources are needed for a symmetry model than a full modelBoundary conditions are greatly simplified. The symmetry constraints removes certain rigid body modes without introducing any artificial stiffnessDisadvantages:Symmetry is slightly more difficult to interpret than a full modelVery difficult to present results to non-technical people (especially anti-symmetry!).5. LoadsLoads applied can be point-, edge, face or body loads.To apply a load that is not impossible to re-create in real life, a Force per unit area (pressure loading) or Force per unit volume (inertial loading) must be applied.To generate a “pressure”type loading an Area is needed for the load to act on:For solid elements, this can only be a pressure load to have an area over which to actFor the shells, it needs at least a load per unit length because the shell has thickness defined already. (Thickness * Length = Area for load to act)Beam elements has cross-sectional Area. Therefore a point load can be applied to a beam without a stress-concentration: The load is effectively distributed evenly over the whole cross-sectional areaThe meaning of a point load applied to a shell- or solid element is actually a pressure load on the element that varies across the element. Therefore, the finer the mesh, the higher the stress: The load is distributed over a smaller areaExactly the same is true for a constraint: It is effectively smeared across an element.6. Analysis typesStatic analysis:A static analysis is useful to model a structure where the loading is applied slow enough so that dynamic effects are not excited. If a structure is loaded at a frequency of about 1/3 of it’s first mode, a static analysis can be performed with good results.The model must be fully constrained (No rigid body modes)Mass is only needed if inertial (gravity) loading is appliedIt’s the most common type of analysis performed. The key requirements of a good static analysis are:A converged mesh (Relatively easy)Sensible constraints (Requires careful thought)Correct loading (Sometimes difficult to obtain)Good material properties (Usually easy)Modal analysis: Used to find the un-damped modes of a structureThere are no loads in a modal analysis, only boundary conditions. For the analysis to run, boundary conditions are not a requirement, but unless it is an aero-plane or a satellite, constraints needs to be used.The model must have mass and stiffness!Stresses can be requested. Yes they have meaning, but NOT to determine if the structure will fail or not. The eigen-vectors (mode-shapes) reported by the FEM code are arbitrarily scaled. The stress corresponds to this scaled deformation shape. Nothing more, nothing less. If you make a change to a model and the modal stress comes down, it does NOT mean it is a better design!Buckling analysis: Used to find the scale factors for all the applied loads for the structure to become unstableBuckling loads for some structures are extremely sensitive to the initial shape. No structure is perfect, so introduce small imperfections into the model to determine how sensitive your model is to initial shape.One way to do this is to run a buckling analysis on the perfect shape, then to offset the mesh with the buckling shape within manufacturing tolerances, and then to re-run the analysisFrequency response: This is a transfer-function between input loads and every node in the modelTransient dynamic:Similar to a static analysis, but with dynamics starting to play a role.Model should have mass, otherwise no dynamics wil be present.No constraints are needed for the analysis to run because the inertia of the structure can balance the applied loads7. UnitsFEA codes uses a set of consistent units. This means that basic equations such as F=m*a should hold for the units chosen.SI is a consistent set of units and it is strongly advised that it be used.A seemingly attractive set of Engineering units is based on mm-N-s, which has pressures (stresses) in MPa, velocity in mm/s, acceleration in mm/s^2 etc.The catch is that mass is in Mg (tons) and density in Mg/mm^3!Units for thermal analysis are even more involved. It is very easy to make a mistake during the process of calculating the correct units and the required scale factors.Having consistent units for any dynamic-type of analysis is especially important. Be very careful when not using SI!。
地暖管地面混凝土收缩应力有限元
2012年11月下第41卷第377期施工技术CONSTRUCTION TECHNOLOGY55地暖管地面混凝土收缩应力有限元分析丁璐瑶1，于德湖1，张同波2，王胜2（1．青岛理工大学土木工程学院，山东青岛266000；2．青建集团股份公司，山东青岛266000）［摘要］针对地暖管地面裂缝产生的主要原因，结合某实际工程，采用ABAQUS 有限元软件，建立有限元模型，将模型划分为4种工况进行分析，对比4种工况下地暖管地面的收缩应力，从而得出设置界格缝能显著减小混凝土地面收缩应力，有利于地暖管地面混凝土裂缝控制。
［关键词］混凝土；裂缝；收缩应力；有限元分析［中图分类号］TU755［文献标识码］A［文章编号］1002-8498（2012）22-0055-02Finite Element Analysis of Shrinkage Stress forConcrete Floor with Heating PipesDing Luyao 1，Yu Dehu 1，Zhang Tongbo 2，Wang Sheng 2（1．School of Civil Engineering ，Qingdao Technological University ，Qingdao ，Shandong266000，China ；2．Qingjian Group Co．，Ltd．，Qingdao ，Shandong266000，China ）Abstract ：With reference to a practical project ，finite element model was established using ABAQUS software to analyze the main reasons that cause cracks of cement floor with heating pipes．Four analysis scenarios were proposed and shrinkage stress of the cement floor with heating pipes were analyzed and compared for these four scenarios．It is concluded that the construction joints can significantly reduce the shrinkage stress and control the floor cracking．Key words ：concrete ；crack ；shrinkage stress ；finite element analysis ［收稿日期］2012-04-12［基金项目］2010年青建集团股份公司自主科研项目“地暖管地面抗裂关键技术研究”［作者简介］丁璐瑶，硕士研究生，E-mail ：luyaoa1987@163．com 目前，低温地板辐射采暖系统（简称地暖管地面）在国内外已被广泛使用，然而，在具体施工过程中发现，地暖管地面由于混凝土收缩极易产生裂缝，不仅会造成对精装修层的破坏，而且会由于局部应力集中破坏地暖管系统，造成楼面渗漏，直接影响使用功能。
Finite Element Analysis（FEA）的应用与优化指南说明书
▪▪▪▪Agenda▪Session Approach▪Finite Element Analysis (FEA) Overview ▪FEA Parameters▪FEA Best Practices▪FEA Software Introduction▪Analysis WalkthroughThis session is not just theoretical information… there’s just not enough time to teach everyone matrix algebraWe are going to apply and leverage modern technology to gain insights into the use cases and capabilitiesFEA DescriptionFinite Element Analysis (FEA) is a computerized method for predicting how a real-world object will react to forces, vibration, heat, etc. to determine whether or not it will function as plannedFEA Benefits▪Predict Product Performance▪Reduce Raw Materials▪Ensure Optimal Design▪Verification▪Reduce Manual Testing and Prototypes ▪Test What If Scenarios▪Shorten Design CycleWho Uses Simulation?The Engineer The AnalystFEA Process Overview1.CAD Model Creation2.Simulation Setup3.Solve Simulation4.Review Results5.OptimizeMeshed 3D Model ExampleElement OverviewAn element is a mathematical relation that defines how the DOFs of one node relate to the next.Node OverviewA node is a coordinate location in space where the Degrees of Freedom (DOFs) are defined.Types of Elements▪1D elements▪ A line connecting 2 nodes only for items like beams and springs▪2D elements▪Planar or axisymmetric elements with either three or four edges enclosing an area▪Plates or Shell Elements: Planar elements that are triangular or quadrilateral with a specifiedthickness▪3D (solid) elements▪Enclosed 3D volumes with 4, 5, 6 or 8 corner nodesStandard Element BondingElements within a single part body can only communicate to one another via common nodes for transferring data information No Connection Connected NodesMaterial Assignment▪Material properties define the structure characteristics of the part ▪Material property information can be located at LoadsStructural loads are forces applied to a part or assembly during operation and cause the model to displace, deflect, and induce stresses and strainsConstraintsStructural constraints restrict or limit the displacement of the model mesh nodesContact ConditionsContact conditions are used to establish relationships between the nodes of contacting parts within an assemblySimulation SolvingRunning or solving the simulation processes and calculates the results based on the parameters establishedResultsThe simulation results can be reviewed and exported as a report to make intelligent decisionsReviewing Results▪Simulation does not always replace the need for physical testing ▪The engineer / analyst still needs to interpret the results to make final decisionsAnalysis Types▪LinearFocus for this presentation ▪Nonlinear▪Thermal / Electrostatic▪Natural Frequency / Modal Analysis▪Vibration▪Fatigue AnalysisLinear vs. Nonlinear▪LinearFocus for this presentation ▪Structure returns to original form▪Small changes in shape stiffness▪No changes in loading direction or magnitude▪Material properties do not change▪Small deformation and strain▪Nonlinear▪Geometry changes resulting in stiffness change▪Material deformation that may not return to original form▪Supports changes in load direction and constraint locations ▪Support of nonlinear load curvesMild Steel Material Properties▪Density = 0.284 lbmass/in^3▪Young’s Modulus = 3.193E+4 ksi▪Poisson’s Ratio = 0.275▪Yield Strength = 3.004E+4 psi▪Ultimate Tensile Strength = 5.007E+4 psi▪Thermal Conductivity = 1.259E+3 btu in/(ft^2 hr*F) ▪Linear Expansion = 21.600E-6 in/(in*F)▪Specific Heat = 0.356 btu/(lbmass*F)Strain Hardening Mild Steel Stress Strain Curve S t r e s s =Strain = Yield Strength (Elastic Limit) Ultimate StrengthFailure NeckingChange in LengthOriginal LengthA r e a F o r c eDisplacementThe displacement results show the magnitude of the model deformation from the original shapeVon Mises StressFormula for combining three principal stresses into an equivalent stress to compare to the material stress propertiesSafety FactorProvides a ratio of how much stronger the object is than it usually needs to be for an intended loadMaterial Yield Strength Safety Factor =Maximum Von Mises Stress40,000 psi2 =20,000 psiConvergenceConvergence is the process of altering element sizes in high stress areas to ensure the specified result criteria has convergedStress SingularitiesA localized high stress area where the stress becomes infinite resulting in distorted resultsBest Practices▪Setup simulation to match real world▪Verify material properties▪Use engineering knowledge judgment▪Avoid putting loads on nodes or small edges ▪Ensure solution type (Linear / Nonlinear)▪Identify stress singularities▪Ensure your results convergeFEA Software▪Conceptual Simulation (Apps)▪FEA Features Built into Design Tools▪Purpose Built Simulation Software Focus for thispresentationAutodesk Simulation Mechanical▪Importing geometry▪Assigning loads and constraints▪Evaluating results▪Results based optimizationRecap▪Finite Element Analysis (FEA) Overview ▪FEA Parameters▪FEA Best Practices▪FEA Software Introduction▪Analysis WalkthroughThank You!Questions?▪Contact: ***********************▪Products: /simulation ▪Community: Session Feedback▪Via the Survey Stations, email or mobile device ▪AU 2015 passes given out each day!▪Best to do it right after the session▪Instructors see results in real-time。
finite and discrete element analysis -回复
finite and discrete element analysis -回复基于有限元和离散元分析的主题。
有限元分析（FEA）和离散元分析（DEA）是两种广泛应用于工程领域的数值模拟方法。
它们能够对结构和材料的力学行为进行准确的预测和分析。
在本文中，我们将详细介绍有限元分析和离散元分析的原理、步骤和应用。
有限元分析（FEA）是一种基于数学方法的工程模拟技术，它将复杂的物体或结构分割成小的有限元素，利用该元素的单元特性和材料力学行为进行模拟分析。
有限元方法基于数学微分方程和力学原理，将问题建模为一组代数方程，然后通过求解这些方程得到所需的物理结果。
有限元分析的步骤主要包括前处理、求解和后处理。
前处理阶段是有限元分析的第一步，它包括几个重要的步骤。
首先，需要将实际结构转化为几何模型。
对于复杂结构，可以使用计算机辅助设计软件进行建模。
接下来，选择适当的网格划分方法将结构划分为小的有限元素。
常用的划分方法包括四边形划分、三角划分和六面体划分。
然后，需要定义边界条件和加载条件，这些条件对于模拟结构的约束和外部加载非常重要。
求解阶段是有限元分析的核心，它涉及到数值计算和求解代数方程。
有限元分析通过建立线性或非线性代数方程组，使用数值方法求解该方程组来确定结构的响应。
线性求解方法常见的有直接法（如高斯消元法）和迭代法（如雅可比法和高斯-赛德尔迭代法）。
非线性求解通常需要使用迭代方法，例如牛顿-拉普森法。
后处理阶段是有限元分析的最后一步，它用于分析和解释求解结果。
后处理涉及到计算和显示各种物理量，例如应力、应变、位移等。
通过后处理可以对结构的性能进行评估和优化。
离散元分析（DEA）是一种基于颗粒或刚体的数值计算方法，它主要用于模拟离散体系在力学行为中的相互运动和相互作用。
离散元分析通过将连续的物体划分成一系列离散的粒子或刚体，使用牛顿第二定律和力学原理来计算这些粒子或刚体之间的相互作用。
离散元分析的步骤主要包括网格生成、求解和后处理。
A geometrically nonlinear finite element
finite element analysis chap_01
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What is Finite Element Analysis (FEA)?
To answer this question, we need to solve the following equations:
∂u j ∂u 1⎛ eij = ⎜ i + 2⎜ ⎝ ∂X j ∂X i ⎞ ⎟ ⎟ ⎠
What is Finite Element Analysis (FEA)?
FEA is originally developed for solving solid mechanics problem. Solid Mechanics:
Object: A Solid with known mechanical properties. (Mr. Potato; a shaft; human tissue……) Concepts:
y
y = x2 + 6
y
y = x2 + 6
x
N=1, F=12, Error= -5.26% N=2, F=12.5, Error= -1.32%
x
How does FEA work?
Integration using numerical methods:
Numerical integration
y
y = x2 + 6
y
y = x2 + 6
x Different functions can be chosen to approximate f (x).
x
How does FEA work?
Integration using numerical methods: Two key steps: 1. 2. Divide the interval of integration. In each sub-interval, choose proper simple functions to approximate the true function.
使用有限元分析的注射模具设计
使用有限元分析的注射模具设计摘要归功于电脑模拟分析运用，通过Moldflow模流分析仿真系统几个步骤就可以对塑件及模具进行开发。
基于注塑模具的仿真结果是在Pro/E 三维辅助设计的帮助下完成的。
计算机辅助工程的运用产品的开发阶段尤为突出，这也使得工具及模具的成本减少。
关键词：注射模具，CAE仿真，Moldflow1.介绍塑料的最常见的处理方法之一是注塑。
这一进程的主要优点包括其在塑造的各种产品的多功能性，它的自动化可以引进，高的生产率和较小的物件公差与制造的可能性。
注射成型件的生产与注塑机和注塑模具一起进展。
相对于形状复杂的零件等的生产方法，这个过程运行在相对短的周期。
注塑标准的要求是：高精确度和型腔表面光洁度最大刚度，模具组件的强度，整个注射模具可抵抗夹紧力，注塑压力适当的浇注系统，注射系统，排气系统和冷却系统最佳生命周期。
注射模具设计师的任务是：分析成型零件的设计，考虑形状、尺寸、浇口位置和零件回弹确定分型面，考虑功能和成型零件的外观，定义型腔草图确定模腔数，模腔尺寸及布局和考虑收缩，几何公差，及注塑机选择适当的浇注系统，冷却和通风系统•对模架选择材料和其他研究过程中加载的条件和模具组件•注塑周期。
•选择合适的注塑机。
现在注射成型技术的制图不适合用二维绘图来描述复杂的形状。
使用三维CAD模型的确切原因是确切描述部件形状，很容易看到剖面图及细节图。
更重要的原因，使用三维CAD模型可以利用CAM软件对注塑和模具制造的过程仿真。
（2）为CAM使用三维CAD模型已成为模具制造的一个组成部分。
（4）辅助工程的注射成型过程模拟，逐步成为了模具制造过程的组成部分。
随着CAE模拟模具设计人员的帮助可以测试未经注塑模具的使用前需要解决的各种费用及耗时模具的更正。
2.注射模具过程仿真这部分是描述在Mlodflow模流分析6.0的帮助下的顶部注射情况。
这次仿真的材料是Cycoloy C2100HF。
分析准备的是有限元网格模型。
光机系统胶接结构固化时间对面形精度的分析
光机系统胶接结构固化时间对面形精度的分析刘玖;杨洪波;巩岩;袁文全【摘要】在光机系统胶接结构中,为研究胶接剂固化收缩对镜片的影响,对胶接剂采用粘弹性模型进行了仿真计算,通过本构方程得到理论解,并采用当量温差法利用有限元分析软件ANSYS模拟收缩,分析了固化时间对镜片面形误差的影响.通过和线弹性模型对比,分析了粘弹性模型的优势.通过改变固化时间,分析了不同固化时间对镜片的不同影响.结果表明,仿真结果和理论解高度一致,且采用粘弹性模型可以很好地反映最终收缩应力随固化时间变化的关系,更好地模拟了收缩应力的生成过程；同时发现,对同一种胶接剂,通过改变环境适当延长固化时间,减小了对镜片面形误差的影响,因此可以适当延长固化时间以减小收缩应力.%In the bonding structure of the optical-mechanical system, for researching the influence on lens of the adhesive's curing shrinkage, the viscoelastic model is used for adhesive's simulation, and the theory solution can be gotten from the constitutive equation. The method of equivalent temperature and the Finite Element Analysis (FEA) soft ANSYS is used for simulating the shrinkage to analyze the influence on the surface figure error. Compared with the elastic model, the viscoelastic model has its advantages. Change the curing time for analyzing the influence on the lens. The analysis result shows that the simulation result fit the theory solution well, and the viscoelastic model can better describe the relationship between the time and the final shrinkage stress. Moreover, it can better simulate the generation of shrinkage stress. In regard to the same adhesive, through prolonging the curing time properly, there is the smaller influence on thesurface figure error of the lens, so it can prolong the time for reducing the shrinkage stress.【期刊名称】《光电工程》【年(卷),期】2011(038)010【总页数】6页(P140-145)【关键词】光机系统;胶接结构;胶接剂;固化时间;粘弹性【作者】刘玖;杨洪波;巩岩;袁文全【作者单位】中国科学院长春光学精密机械与物理研究所,长春130033;中国科学院研究生院,北京100049;中国科学院长春光学精密机械与物理研究所,长春130033;中国科学院长春光学精密机械与物理研究所,长春130033;中国科学院长春光学精密机械与物理研究所,长春130033【正文语种】中文【中图分类】TH761在光学仪器制造业中，胶接技术起着越来越重要的作用。
几种老化系数计算方法的比较
几种老化系数计算方法的比较饶瑞1，刘爱荣1，王荣辉2【摘要】摘要:以计算结果与Bastgen试验结果的吻合程度作为判别计算精度的标准，比较继效法、老化法、B-K法、王勋文法和中值法5种老化系数计算方法对于长期老化系数和短期老化系数的计算精度，讨论了TROSTH、周建民老化系数建议值的合理性.研究发现:王勋文法无论在长期老化系数还是短期老化系数的计算上均有较高精度.在采用按龄期调整的有效模量法分析具有多次加载过程的结构徐变分析时，宜用该方法计算老化系数.【期刊名称】广州大学学报（自然科学版）【年(卷),期】2011(010)004【总页数】5【关键词】关键词:徐变;按龄期调整的有效模量法;老化系数按龄期调整的有效模量法(AEMM)是分析结构徐变效应的一种有效方法［1－2］.它具有概念明确，便于电算的优点，故最为研究者所乐用.在运用该方法的过程中，除了混凝土的徐变系数外，还涉及到另一个系数——老化系数.老化系数是在计算随时间变化的力所产生的徐变变形时，用于对徐变系数进行修正的系数，也可理解为反映徐变量随着混凝土老化而逐渐衰减的一种系数，是运用按龄期调整的有效模量法进行结构徐变分析必不可少的系数之一.其取值问题，是用按龄期调整有效模量法求解的关键问题.其取值的合适与否，决定着徐变分析质量的高低.TROST H［3］认为老化系数的平均值在0.82左右，并建议一般情况下取0.8.周建民［4］则认为老化系数应该取为0.67.除此之外，学者们还提出了一系列老化系数的计算方法，如基于老化理论和继效理论的计算方法［5］、基于松弛系数经验公式的计算方法［6］、基于考虑龄期影响的徐变本构理论与继效流动理论相结合的计算方法［7］以及基于积分中值定理的计算方法［8－9］.上述方法均有各自的理论依据，孰优孰劣，尚无定论.因此，在运用按龄期调整的有效模量法进行结构徐变分析的过程中，在老化系数计算方法的选取上，一般都是出于经验或习惯.各种方法都不乏应用案例.本文将从结构徐变效应分析的按龄期调整的有效模量法入手，寻找比较的着眼点，进而对上述几种老化系数计算方法进行比较.以期获得有益的结论，为老化系数计算方法的选择提供参考.1 老化系数计算方法为了文章的完整性起见，先扼要介绍几种老化系数计算方法.为了描述方便，记t0为加载龄期，t为计算龄期，x(t，t0)为老化系数，φ(t，t0)为徐变系数，J(t，t0)为徐变柔量，R(t，t0)为松弛系数，E(28)为混凝土28 d弹性模量.其中，龄期以天为单位，老化系数徐变系数和徐变柔量为无量纲量，弹性模量以MPa为单位.在老化系数计算方法中，最为常用的方法是基于继效理论和老化理论的计算方法(分别简称为继效法和老化法)［5］，这两种方法具有相同的计算式，其区别在于式中常数α和β的取值.这两种表达式可分别写为BAZANT等［6］提出了松弛系数的经验公式，并由此推出老化系数的计算公式(B-K法):王勋文［7］运用考虑龄期影响的徐变本构理论与继效流动理论相结合的方法(王勋文法)，推导得出老化系数计算公式为而在徐变电算中较为常用的老化系数计算公式则是由积分中值定理推导得出的［8－9］，文中称为中值法，其表达式可写成2 按龄期调整的有效模量法设t0时刻在混凝土结构上作用恒应力σ(t0)，假定弹性模量不随时间变化，在t 时刻，由Trost-Bazant理论［10］，混凝土的徐变应变(应变增量)和应力存在如下关系:式中，φ(t，t0)为加载龄期t0至t时刻混凝土的徐变系数;Δεcr(t)和Δσcr(t)分别为t时刻的徐变应变和应力增量;Eφ(t，t0)为按龄期调整的有效模量;χ (t，t0)为老化系数.式(6)为按龄期调整的有效模量法(AEMM)［10］的徐变应变—应力基本方程.对于具有多次加载的过程结构，其应力和时间的关系可用图1近似表示.图1中，σ(ti)为ti时刻的瞬时弹性应力增量，σcr(i)为ti－1至ti时段(i时段)的徐变应力.根据式(6)，tn时刻的徐变应变和tn－1时刻的徐变应变可用式(7)和式(8)表示.根据式(7)和式(8)可得n时段的徐变应变增量如式(9).3 比较结果分析从式(9)可知，在用AEMM法计算结构徐变时，必须先求出和Eφ(tn，tn－1)3种按龄期调整的有效模量，也即必须获得和3 种老化系数.这些老化系数的时间区间(以χ(tn，ti－1)为例，时间区间即为tn－ti－1)可能包括一个阶段或者多个阶段，即从时间跨度上看，这些老化系数有长期和短期之分.因此，有效的计算方法必须兼顾到长期老化系数和短期老化系数的计算精度.为此，本文将分别从长期老化系数计算精度和短期老化系数计算精度两个角度对上述几种老化系数计算公式进行对比分析.Bastgen试验［11］是一个研究混凝土老化的试验.该试验以加载龄期28 d，徐变系数终值1.5的混凝土块为试验试件，获得了持载龄期和老化系数的关系曲线.由于相关试验较少，Bastgen试验结果一直以来都用于验证老化系数计算公式计算结果的有效性.本文也将其用于验证各种方法的计算结果，将各种老化系数计算公式的计算结果与Bastgen试验值比较，以计算结果与试验结果的吻合程度作为判别计算精度的标准.从式(1)至(5)可以发现，老化系数与徐变系数有很大关系.不同的徐变系数推算表达式和不同的徐变计算理论都将得到不同的老化系数值.为了使各公式的计算结果具有可比性，统一采用现行公路桥梁规范中的徐变计算模式——CEBFIP1990模式计算得出的徐变系数进行老化系数的计算.此外，为了便于与Bastgen试验值比较，计算过程中，加载龄期取28 d，徐变系数的终值取1.5.计算结果如图2所示.为了表示得更加清晰，图2仅给出按继效法和B-K法计算所得曲线的一部分.从图2中可见:各种老化系数的计算效果显然是不同的，以按王勋文法计算的结果与试验值最为接近.继效法和B-K法的计算结果前期(100 d以前)与试验值相差较大，它们与试验值的偏差随持载时间的增加而减小，到100 d以后，与试验值基本吻合.继效法、B-K法和王勋文法的计算结果与试验结果随持载时间的增加趋向0.81，与TROST H的建议值基本相同.可见，TROST H建议值实际上是老化系数随时间增长的收敛值.该建议值应用于徐变效应计算中可起到简化计算的作用，但也将造成一定的精度损失.由于该建议值是老化系数的下限，因此，用此建议值得出的结果将低估徐变效应.中值法的计算结果一开始与TROST H的建议值比较接近，而且与试验值的偏差也不大，但是随着持载时间的增加，偏差越来越大，无收敛的趋势.至于其他两种方法，老化法计算值和周建民建议值，由始至终均与试验值偏差较大.上述分析表明，继效法和B-K法适合用于计算时间区间较长的老化系数;中值法较适合用于计算时间区间较短的老化系数;老化法不适合用于老化系数的计算;而王勋文法适用面较广，无论时间区间长短，均能得到比较满意的老化系数计算结果.至于TROST H建议值，则适用于计算区间较长的场合.在不同加载龄期的情况下，进一步研究继效法、王勋文法和中值法对计算短期老化系数的适用性.本文以持载两周为例，分别用3种公式计算7种不同的加载龄期的混凝土的短期老化系数，结果列于表1.表1中可见，王勋文法的计算结果大于其它两种方法的计算结果.虽然有部分计算结果与继效法的计算结果比较接近，但从总体上，王勋文法和中值法的计算结果更加接近.究竟哪一组计算结果更合理，下面将结合文献［11］的研究成果进行判断.研究发现，老化系数与混凝土加载龄期成正比［11］.因此，在相同的持载时间下，加载龄期越大的混凝土，其老化系数越大.从不同加载龄期的老化系数随持载龄期变化曲线(图3)可以看出，王勋文法和中值法的计算结果与这个规律是吻合的，而继效法的计算结果则违背了这个规律.另外，加载龄期越大，其老化系数该越接近1［11］，这与王勋文法和中值法计算结果也是吻合的.可见，王勋文法和中值法的计算结果较继效法的合理，更适合用于短期老化系数的计算.4 结论本文从长期和短期两种角度，比较继效法、老化法、B-K法、王勋文法和中值法5种老化系数计算方法的计算精度，同时讨论了TROST H、周建民建议值的合理性.分析发现:(1)在上述5种计算方法中，老化法的计算误差最大;继效法和B-K法适合于计算长期老化系数;中值法适合于计算短期老化系数;而王勋文法的适用范围较广，无论用于计算长期老化系数还是短期老化系数，均能得到比较满意的结果. (2)TROST H建议值实际上是老化系数随时间增长的收敛值，该建议值应用于徐变效应计算中可起简化计算的作用，但将造成一定的精度损失.该建议值是老化系数的下限，因此，用此建议值得出的结果将低估徐变效应.与TROST H建议值相比，周建民建议值更是低估了老化系数.综上，推荐用王勋文法计算老化系数，该公式较好地反映混凝土老化的基本规律，对于长期和短期老化系数的计算结果均有较高精度.References:［1］HU Di，WENG Yun-xin，YU Zhi-wu.Analysis of creep effecton reinforced concrete thin plate［J］.Engineering Mechanics，2011，28(3):185 －190.(in Chinese)［2］WANGWen-wei，HEChu-sheng，FENG Zhu-lin，et al.Incrementl differentialmodeling of creep and shrinkage of steel-concrete composite beams［J］.Journal of Southeast University:Natural Science Edition，2010，40(6):1252-1256.(in Chinese)［3］ TROST H.Auswirkungen des superpositionsprinzips auf kriech-und relaxations probleme bei beton and spannbeton［J］.Beton-und Stahlbetonbau，1967，61:203-238，261-269.［4］ZHOU Jian-min.Analysismethod of RC structures under the consideration of time factor［D］.Shanghai:Tongji University，2006.(inChinese)［5］ XIAO Ru-cheng.Analysis of bridge structure and program system ［M］.Beijing:China Communications Press，2002.(in Chinese)［6］BAZANT Z P，KIM SS.Approximate relaxation function for concrete［J］.Journal of the Structural Divison，1983，105:2695-2705.［7］WANG Xun-wen，PAN Jia-ying.Evaluating aging coefficient in age-adjusted effective modulus method ［J］.China Railway Science，1996，17(3):12-23.(in Chinese)［8］YAN Dong-huang，TIAN Zhong-chu，LIXue-wen.Finite elementmethod and application for the shrinkage and creep of concrete bridges［J］.China Journal of Highway and Transport，2004，17(2):55-58.(in Chinese)［9］WANG Yin-hui，CHEN Shan-lin.Application of compulsive strain method to creep analysis of concrete bridges［J］.Bridge Construction，2007(1):74-77.(in Chinese)［10］BAZANT Z P.Prediction of concrete creep effects using age-adjusted effective modulus method［J］.ACIJournal，1972，69:212-217［11］NEVILLE A M，DILGERW H，BROOK JJ.Creep of plain and structural concrete［M］.London and New York:Construction Press，1983.参考文献:［1］胡狄，翁运新，余志武.钢筋混凝土矩形薄板徐变效应分析［J］.工程力学，2011，28(3):185-190.［2］王文炜，何初生，冯竹林，等.钢－混凝土组合梁混凝土收缩徐变的增量微分方法［J］.东南大学学报:自然科学版，2010，40(6):1252-1256.［4］周建民.考虑时间因素的混凝土结构分析方法［D］.上海:同济大学，2006.［5］肖汝诚.桥梁结构分析及程序系统［M］.北京:人民交通出版社，2002.［7］王勋文，潘家英.按龄期调整有效模量法中老化系数γ的取值问题［J］.中国铁道科学，1996，17(3):12-23.［8］颜东煌，田仲初，李学文，等.混凝土桥梁收缩徐变计算的有限元方法与应用［J］.中国公路学报，2004，17(2):55-58.［9］王银辉，陈山林.强迫应变法在混凝土桥梁徐变分析中的应用［J］.桥梁建设，2007(1):74-77.【责任编辑:刘少华】基金项目:国家自然科学基金项目(50978062);广东省自然科学基金项目(9151065004000002);广东省科技计划项目(2010A030200014);广州市建委专项科技项目(2004047)资助。