# The Assembly Process 25 1

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 5 The Assembly Process 5 1

2 Chapter 5: THE ASSEMBLY PROCESS TABLE OF CONTENTS Page 5.1 Introduction Simplifications Simplified Assemblers A Plane Truss Example Structure Simplified Assembler Implementation MET Assemblers A Plane Stress Assembler MET Assembler Implementation MET-VFC Assemblers Node Freedom Arrangement Node Freedom Signature The Node Freedom Allocation Table The Node Freedom Map Table The Element Freedom Signature The Element Freedom Table A Plane Trussed Frame Structure MET-VFC Assembler Implementation *Handling MultiFreedom Constraints Notes and Bibliography Exercises

3 5. SIMPLIFICATIONS 5.1. Introduction Chapters 0, and dealt with element level operations. Sandwiched between element processing and solution there is the assembly process that constructs the master stiffness equations. Assembler examples for special models were given as recipes in the complete programs of Chapters 1 and. In the present chapter assembly will be studied with more generality. The position of the assembler in a DSM-based code for static analysis is sketched in Figure 5.1. In most codes the assembler is implemented as a element library loop driver. This means that instead of forming all elements first and then assembling, the assembler constructs one element at a time in a loop, and immediately merges it into the master equations. This is the merge loop. Model definition data: geometry element connectivity material fabrication freedom activity Assembler K Modify Eqs for BCs ^ K Equation Solver merge loop Element Stiffness Matrices K e Some equation solvers apply BCs and solve simultaneously ELEMENT LIBRARY Figure 5.1. Role of assembler in FEM program. Nodal displacements To postprocessor Assembly is the most complicated stage of a production finite element program in terms of data flow. The complexity is not due to mathematical operations, which merely involve matrix addition, but interfacing with a large element library. Forming an element requires access to geometry, connectivity, material, and fabrication data. (In nonlinear analysis, to the state as well.) Merge requires access to freedom activity data, as well as knowledge of the sparse matrix format in which K is stored. As illustrated in Figure 5.1, the assembler sits at the crossroads of this data exchange. The coverage of the assembly process for a general FEM implementation is beyond the scope of an introductory course. Instead this Chapter takes advantage of assumptions that lead to coding simplifications. This allows the basic aspects to be covered without excessive delay. 5.. Simplifications The assembly process is considerably simplified if the FEM implementation has these properties: All elements are of the same type. For example: all elements are -node plane bars. The number and configuration of degrees of freedom at each node is the same. There are no gaps in the node numbering sequence. There are no multifreedom constraints treated by master-slave or Lagrange multiplier methods. The master stiffness matrix is stored as a full symmetric matrix. 5

4 Chapter 5: THE ASSEMBLY PROCESS (a) E = 000 and A = for all bars 1 (1) () y (b) 1 (1,) (1) (,) () () () (5) (5,6) () () (5) x 1 (1,) (7,8) assembly (,) (5,6) (7,8) Figure 5.. Left: Example plane truss structure. (a): model definition; (b) disconnection assembly process. Numbers in parentheses written after node numbers are global DOF numbers. If the first four conditions are met the implementation is simpler because the element freedom table described below can be constructed on the fly from the element node numbers. The last condition simplifies the indexing to access entries of K. 5.. Simplified Assemblers In this section all simplifying assumptions listed in 5. are assumed to hold. This allows us to focus on local-to-global DOF mapping. The process is illustrated on a plane truss structure A Plane Truss Example Structure The plane truss shown in Figure 5.(a) will be used to illustrate the details of the assembly process. The structure has 5 elements, nodes and 8 degrees of freedom. The disconnection-to-assembly process is pictured in Figure 5.(b). Begin by clearing all entries of the 8 8 master stiffness matrix K to zero, so that we effectively start with the null matrix: 1 K = (5.1) The numbers written after each row of K are the global DOF numbers. Element (1) joins nodes 1 and. The global DOFs of those nodes are 1 1 = 1, 1 = 1, 1 = and =. See Figure 5.(b). Those four numbers are collected into an array called the element freedom table, or EFT for short. The element stiffness matrix K (1) is listed on 5

5 5. SIMPLIFIED ASSEMBLERS the left below with the EFT entries annotated after the matrix rows. On the right is K upon merging the element stiffness: (5.) Element () joins nodes and. The global DOFs of those nodes are 1 =, =, 1 = 5 and = 6; thus the EFT is {,,5,6 }. Matrices K () and K upon merge are Element () joins nodes 1 and. Its EFT is { 1,,7,8 }. Matrices K () and K upon merge are 5 6 (5.) (5.) Element () joins nodes and. Its EFT is {,,7,8 }. Matrices K () and K upon merge are (5.5) 5 5

6 Chapter 5: THE ASSEMBLY PROCESS Finally, element (5) joins nodes and. Its EFT is { 5,6,7,8 }. Matrices K (5) and K upon merge are (5.6) Since all elements have been processed, (5.6) is the master stiffness before application of boundary conditions PlaneTrussMasterStiffness[nodxyz_,elenod_,elemat_,elefab_, eleopt_]:=module[{numele=length[elenod],numnod=length[nodxyz], e,ni,nj,eft,i,j,ii,jj,ncoor,em,a,options,ke,k}, K=Table[0,{*numnod},{*numnod}]; For [e=1, e<=numele, e++, {ni,nj}=elenod[[e]]; eft={*ni-1,*ni,*nj-1,*nj}; ncoor={nodxyz[[ni]],nodxyz[[nj]]}; Em=elemat[[e]]; A=elefab[[e]]; options=eleopt; Ke=PlaneBarStiffness[ncoor,Em,A,options]; For [i=1, i<=, i++, ii=eft[[i]]; For [j=i, j<=, j++, jj=eft[[j]]; K[[jj,ii]]=K[[ii,jj]]+=Ke[[i,j]] ]; ]; ]; Return[K] ]; Figure 5.. Plane truss assembler module. nodxyz={{-,},{0,},{,},{0,0}}; elenod= {{1,},{,},{1,},{,},{,}}; elemat= Table[000,{5}]; elefab= Table[,{5}]; eleopt= {True}; K=PlaneTrussMasterStiffness[nodxyz,elenod,elemat,elefab,eleopt]; Print["Master Stiffness of Plane Truss of Fig 5.:"]; K=Chop[K]; Print[K//MatrixForm]; Print["Eigs of K=",Chop[Eigenvalues[N[K]]]]; Master Stiffness of Plane Truss of Fig 5.: Eigs of K={5007., 7.6, 56.8, 8.78, 6.70, 0, 0, 0} Figure 5.. Plane truss assembler tester and output results. 5 6

7 5. MET ASSEMBLERS 5... Simplified Assembler Implementation Figure 5. shows a a Mathematica implementation of the assembly process just described. This assembler calls the element stiffness module PlaneBarStiffness of Figure 0.. The assembler is invoked by K = SpaceTrussMasterStiffness[nodxyz,elenod,elemat,elefab,prcopt] (5.7) The five arguments: nodxyz, elenod, elemat, elefab, and prcopt have the same function as those described in 1.1. for the SpaceTrussMasterStiffness assembler. In fact, comparing the code in Figure 5. to that of Figure 1.1, the similarities are obvious. Running the script listed in the top of Figure 5. produces the output shown in the bottom of that figure. The master stiffness (5.6) is reproduced. The eigenvalue analysis verifies that the assembled stiffness has the correct rank of 8 = MET Assemblers The next complication occurs when elements of different types are to be assembled, while the number and configuration of degrees of freedom at each node remains the same. This scenario is common in special-purpose programs for analysis of specific problems: plane stress, plate bending, solids. This will be called a multiple-element-type assembler, or MET assembler for short. The only incremental change with respect to the simplest assemblers is that an array of element types, denoted here by eletyp, has to be provided. 5 in bar () 1 in 1 bar () y 5 in 5 1in quad () trig (1) in E (plate, bars) = ksi ν (plate)= 0.5 h (plate) = 0. in A (bars) = sq in 6 in x () 1(1,) () (7,8) 1(1,) (7,8) () (,) (,) (1) Figure 5.5. A plane stress structure used to test a MET assembler. 5(9,10) (5,6) assembly 5(9,10) The assembler calls the appropriate element stiffness module according to type, and builds the local to global DOF mapping accordingly A Plane Stress Assembler This kind of assembler will be illustrated using the module PlaneStressMasterStiffness. As its name indicates, this is suitable for use in plane stress analysis programs. We make the following assumptions: 1 All nodes in the FEM mesh have DOFs, namely the {u x, u y } displacements. There are no node gaps or MFCs. The master stiffness is stored as a full matrix. Element types can be: -node bars, -node linear triangles, ot -node bilinear quadrilaterals. These elements can be freely intermixed. Element types are identified with character strings "Bar", "Trig" and "Quad", respectively. 5 7 (5,6)

8 Chapter 5: THE ASSEMBLY PROCESS As noted above, the chief modification over the simplest assembler is that a different stiffness module is invoked as per type. Returned stiffness matrices are generally of different order; in our case, 6 6 and 8 8 for the bar, triangle and quadrilateral, respectively. But the construction of the EFT is immediate, since node n maps to global freedoms n 1 and n. The technique is illustrated with the -element, 5-node plane stress structure shown in Figure 5.5. It consists of one triangle, one quadrilateral, and two bars. The assembly process goes as follows. Element (1) is a -node triangle with nodes, 5 and, whence the EFT is { 5,6,9,10,7,8 }. The element stiffness is K (1) = (5.8) Element () is a -node quad with nodes,, and 1, whence the EFT is {,,5,6,7,8,1, }. The element stiffness computed with a Gauss rule is K () = (5.9) Element () is a -node bar with nodes 1 and, whence the EFT is { 1,,7,8 }. The element stiffness is K () = (5.10) Element () is a -node bar with nodes 1 and, whence the EFT is { 1,,, }. The element stiffness is K () = (5.11) Upon assembly, the master stiffness of the plane stress structure, printed with one digit after the 5 8

9 5. MET ASSEMBLERS PlaneStressMasterStiffness[nodxyz_,eletyp_,elenod_, elemat_,elefab_,prcopt_]:=module[{numele=length[elenod], numnod=length[nodxyz],ncoor,type,e,enl,neldof, i,n,ii,jj,eftab,emat,th,numer,ke,k}, K=Table[0,{*numnod},{*numnod}]; numer=prcopt[[1]]; For [e=1,e<=numele,e++, type=eletyp[[e]]; If [!MemberQ[{"Bar","Trig","Quad"},type], Print["Illegal type", " of element e=",e," Assembly interrupted"]; Return[K]]; enl=elenod[[e]]; n=length[enl]; eftab=flatten[table[{*enl[[i]]-1,*enl[[i]]},{i,1,n}]]; ncoor=table[nodxyz[[enl[[i]]]],{i,n}]; If [type=="bar", Em=elemat[[e]]; A=elefab[[e]]; Ke=PlaneBarStiffness[ncoor,Em,A,{numer}] ]; If [type=="trig", Emat=elemat[[e]]; th=elefab[[e]]; Ke=TrigIsoPMembraneStiffness[ncoor,Emat,th,{numer}] ]; If [type=="quad", Emat=elemat[[e]]; th=elefab[[e]]; Ke=QuadIsoPMembraneStiffness[ncoor,Emat,th,{numer,}] ]; neldof=length[ke]; For [i=1,i<=neldof,i++, ii=eftab[[i]]; For [j=i,j<=neldof,j++, jj=eftab[[j]]; K[[jj,ii]]=K[[ii,jj]]+=Ke[[i,j]] ]; ]; ]; Return[K]; ]; Figure 5.6. Implementation of MET assembler for plane stress. decimal point, is (5.1) The eigenvalues of K are [ ] (5.1) which displays the correct rank MET Assembler Implementation An implementation of the MET plane stress assembler as a Mathematica module called PlaneStressMasterStiffness is shown in Figure 5.6. The assembler is invoked by K = PlaneStressMasterStiffness[nodxyz,eletyp,elenod,elemat,elefab,prcopt] (5.1) The only additional argument is eletyp, which is a list of element types. 5 9

10 Chapter 5: THE ASSEMBLY PROCESS Em=10000; ν=1/; h=/10; A=; Emat=Em/(1-ν^)*{{1,ν,0},{ν,1,0},{0,0,(1-ν)/}}; nodxyz= {{0,1},{1,0},{5,0},{0,6},{5,6}}; elenod= {{,5,},{,,,1},{1,},{1,}}; eletyp= {"Trig","Quad","Bar","Bar"}; elemat= {Emat,Emat,Em,Em}; elefab= {h,h,a,a}; prcopt={true}; K=PlaneStressMasterStiffness[nodxyz,eletyp,elenod,elemat,elefab,prcopt]; Print["Master Stiffness Matrix of Plane Stress Structure:"]; Print[SetPrecision[Chop[K],5]//MatrixForm]; Print["Eigs of K:",Chop[Eigenvalues[N[K]]]]; Master Stiffness Matrix of Plane Stress Structure: Eigs of K: {88., , 59., 91.7, 8.99, 5.06, 11.0, 0, 0, 0} Figure 5.7. Script for assembling the plane stress model of Figure 5.5 and output results. The script listed in the top of Figure 5.7 runs module PlaneStressMasterStiffness with the inputs appropriate to the problem defined in Figure 5.5. The output shown in the bottom of the figure reproduces the master stiffness matrix (5.1) and the eigenvalues (5.1) MET-VFC Assemblers The next complication beyond multiple element types is a major one: allowing nodes to have different freedom configurations. An assembler that handles this feature will be called a MET-VFC assembler, where VFC is an acronym for Variable Freedom Configuration. A MET-VFC assembler gets closer to what is actually implemented in production FEM programs. A assembler of this kind can handle orientation nodes as well as node gaps automatically. Only two advanced attributes remain: allowing treatment of MFCs by Lagrange multipliers, and handling a sparse matrix storage scheme. The former can be done by an extension of the concept of element. The latter requires major programming modifications, however, and is not considered in this Chapter. The implementation of a MET-VFC assembler requires the definition of additional data structures pertaining to freedoms, at both node and element levels. These are explained in the following subsections before giving a concrete application example Node Freedom Arrangement A key decision made by the implementor of a FEM program for structural analysis is: what freedoms will be allocated to nodes, and how are they arranged? While many answers are possible, we focus 5 10

11 5.5 MET-VFC ASSEMBLERS PackNodeFreedomSignature[s_]:=FromDigits[s,10]; UnpackNodeFreedomSignature[p_,m_]:=IntegerDigits[p,10,m]; NodeFreedomCount[p_]:=DigitCount[p,10,1]; Figure 5.8. Node freedom manipulation utility functions. here on the most common arrangement for a linear three-dimensional FEM program. 1 At each node n three displacements {u xn, u yn, u zn } and three infinitesimal rotations {θ xn,θ yn,θ zn } are chosen as freedoms and ordered as follows: u xn, u yn, u zn, θ xn, θ yn, θ zn (5.15) This is a Node Freedom Arrangement or NFA. Once the decision is made on a NFA, the position of a freedom never changes. Thus if (5.15) is adopted, position # is forever associated to u yn. The length of the Node Freedom Arrangement is called the NFA length and often denoted (as a variable) by nfalen. For the common choice (5.15) of D structural codes, nfalen is Node Freedom Signature Having picked a NFA such as (5.15) does not mean that all such freedoms need to be present at a node. Take for example a model, such as a space truss, that only has translational degrees of freedom. Then only {u xn, u yn, u zn } will be used, and it becomes unnecessary to carry rotations in the master equations. There are also cases where allocations may vary from node to node. To handle such scenarios in general terms a Node Freedom Signature or NFS, is introduced. The NFS is a sequence of zero-one integers. If the j th entry is one, it means that the j th freedom in the NFA is allocated; whereas if zero that freedom is not in use. For example, suppose that the underlying NFA is (5.15) and that node 5 uses the three freedoms {u x5, u y5,θ z5 }. Then its NFS is { 1,1,0,0,0,1 } (5.16) If a node has no allocated freedoms (e.g, an orientation node), or has never been defined, its NFS contains only zeros. It is convenient to decimally pack signatures into integers to save storage because allocating a full integer to hold just 0 or 1 is wasteful. The packed NFS (5.16) would be This technique requires utilities for packing and unpacking. Mathematica functions that provide those services are listed in Figure 5.8. All of them use built-in functions. A brief description follows. p = PackNodeFreedomSignature[s] Packs the NFS list s into integer p. s = UnpackNodeFreedomSignature[p,m] Unpacks the NFS p into a list s of length m, where m is the NFA length. (The second argument is necessary in case the NFS has leading zeros.) k = NodeFreedomCount[p] Returns count of ones in p. 1 This is the scheme used by most commercial codes. In a geometrically nonlinear FEM code, however, finite rotation measures must be used. Consequently the infinitesimal node rotations {θ x,θ y,θ z } must be replaced by something else. NFA positions #7 and beyond are available for additional freedom types, such as Lagrange multipliers, temperatures, pressures, fluxes, etc. It is also possible to reserve more NFA positions for structural freedoms. 5 11

12 Chapter 5: THE ASSEMBLY PROCESS NodeFreedomMapTable[nodfat_]:=Module[{numnod=Length[nodfat], i,nodfmt}, nodfmt=table[0,{numnod}]; For [i=1,i<=numnod-1,i++, nodfmt[[i+1]]=nodfmt[[i]]+ DigitCount[nodfat[[i]],10,1] ]; Return[nodfmt]]; TotalFreedomCount[nodfat_]:=Sum[DigitCount[nodfat[[i]],10,1], {i,length[nodfat]}]; Figure 5.9. Modules to construct the NFMT from the NFAT, and to compute the total number of freedoms. Example 5.1. PackNodeFreedomSignature[{ 1,1,0,0,0,1 }] returns The inverse is UnpackNodeFreedomSignature[110001,6], which returns { 1,1,0,0,0,1 }. NodeFreedomCount[110001] returns The Node Freedom Allocation Table The Node Freedom Allocation Table, or NFAT, is a node by node list of packed node freedom signatures. In Mathematica the list containing this table is internally identified as nodfat. The configuration of the NFAT for the plane stress example structure of Figure 5.5 is nodfat = { ,110000,110000,110000, } = Table[110000,{ 5 }]. (5.17) This says that only two freedoms: {u x, u y }, are used at each of the five nodes. A zero entry in the NFAT flags a node with no allocated freedoms. This is either an orientation node, or an undefined one. The latter case happens when there is a node numbering gap, which is a common occurence when certain mesh generators are used. In the case of a MET-VFC assembler such gaps are not considered errors The Node Freedom Map Table The Node Freedom Map Table, or NFMT, is an array with one entry per node. Suppose that node n has k 0 allocated freedoms. These have global equation numbers i + j, j = 1,...,k. This i is called the base equation index: it represent the global equation number before the first equation to which node n contributes. Obviously i = 0ifn = 1. Base equation indices for all nodes are recorded in the NFMT. In Mathematica code the table is internally identified as nodfmt. Figure 5.9 lists a module that constructs the NFMT given the NFAT. It is invoked as nodfmt = NodeFreedomMapTable[nodfat] (5.18) The only argument is the NFAT. The module builds the NFMT by simple incrementation, and returns it as function value. The map qualifier comes from the use of this array in computing element-to-global equation mapping. 5 1

13 5.5 MET-VFC ASSEMBLERS ElementFreedomTable[enl_,efs_,nodfat_,nodfmt_,m_]:=Module[ {nelnod=length[enl],eft,i,j,k=0,ix,n,s,es,sx}, eft=table[0,{sum[digitcount[efs[[i]],10,1],{i,1,nelnod}]}]; For [i=1,i<=nelnod,i++, n=enl[[i]]; s=integerdigits[nodfat[[n]],10,m]; ix=0; sx=table[0,{m}]; Do [If[s[[j]]>0,sx[[j]]=++ix],{j,1,m}]; es=integerdigits[efs[[i]],10,m]; For [j=1,j<=m,j++, If [es[[j]]>0, k++; If [s[[j]]>0, eft[[k]]=nodfmt[[n]]+sx[[j]] ]]]; ]; Return[eft]]; Figure Module to construct the Element Freedom Table for a MET-VFC assembler. Example 5.. The NFAT for the plane truss-frame structure of Figure 5.11 is nodfat = { ,110000, ,000000, }. The node freedom counts are {,,,0, }. The call nodfmt = NodeFreedomMapTable[nodfat] (5.19) returns { 0,,5,8,8 } in nodfmt. This can be easily verified visually: 0+=, 0++=5, etc. Figure 5.9 also lists a function TotalFreedomCount that returns the total number of freedoms in the master equations, given the NFAT as argument The Element Freedom Signature The Element Freedom Signature or EFS is a data structure that shares similarities with the NFAT, but provides freedom data at the element rather than global level. More specifically, given an element, it tells to which degrees of freedom it constributes. The EFS is best described through an example. Suppose that in a general FEM program all defined structural nodes have the signature , meaning the arrangement (5.15). The program includes -node space bar elements, which contribute only to translational degrees of freedom {u x, u y, u z }. The EFS of any such element will be efs = { , } (5.0) For a -node space beam element, which contributes to all six nodal freedoms, the EFS will be efs = { , } (5.1) For a -node space beam element in which the third node is an orientation node: efs = { ,111111, } (5.) This information, along with the NFAT and NFMT, is used to build Element Freedom Tables. 5 1

14 Chapter 5: THE ASSEMBLY PROCESS The Element Freedom Table The Element Freedom Table or EFT has been encountered in all previous assembler examples. It is a one dimensional list of length equal to the number of element DOF. If the i th element DOF maps to the k th global DOF, then the i th entry of EFT contains k. This allows to write the merge loop compactly. In the simpler assemblers discussed in previous sections, the EFT can be built on the fly simply from element node numbers. For a MET-VFC assembler that is no longer the case. To take care of the VFC feature, the construction of the EFT is best done by a separate module. A Mathematica implementation is shown in Figure Discussion of the logic, which is not trivial, is relegated to an Exercise, and we simply describe here the interface. The module is invoked as eft = ElementFreedomTable[enl,efs,nodfat,nodfmt,m] (5.) The arguments are enl The element node list. efs The Element Freedom Signature (EFS) list. nodfat The Node Freedom Allocation Table (NFAT) described in 5.5. nodfmt The Node Freedom Map Table (NFMT) described in 5.5. m The NFA length, often 6. The module returns the Element Freedom Table as function value. Examples of use of this module are provided in the plane trussed frame example below. Remark 5.1. The EFT constructed by ElementFreedomTable is guaranteed to contain only positive entries if every freedom declared in the EFS matches a globally allocated freedom in the NFAT. But it is possible for the returned EFT to contain zero entries. This can only happen if an element freedom does not match a globally allocated one. For example suppose that a -node space beam element with the EFS (5.1) is placed into a program that does not accept rotational freedoms and thus has a NFS of at all structural nodes. Then six of the EFT entries, namely those pertaining to the rotational freedoms, would be zero. The occurrence of a zero entry in a EFT normally flags a logic or input data error. If detected, the assembler should print an appropriate error message and abort. The module of Figure 5.10 does not make that decision because it lacks certain information, such as the element number, that should be placed into the message A Plane Trussed Frame Structure To illustrate the workings of a MET-VFC assembler we will follow the assembly process of the structure pictured in Figure 5.11(a). The finite element discretization, element disconnection and assembly process are illustrated in Figure 5.11(b,c,d). Although the structure is chosen to be plane to facilitate visualization, it will be considered within the context of a general purpose D program with the freedom arrangement (5.15) at each node. The structure is a trussed frame. It uses two element types: plane (Bernoulli-Euler) beam-columns and plane bars. Geometric, material and fabrication data are given in Figure 5.11(a). The FEM idealization has nodes and 5 elements. Nodes numbers are 1,, and 5 as shown in Figure 5.11(b). Node is purposedly left out to illustrate handling of numbering gaps. There are 11 DOFs, three 5 1

15 5.5 MET-VFC ASSEMBLERS (a) m E=00000 MPa A=0.001 m E=0000 MPa, A=0.0 m, I zz =0.000 m y x m m E=00000 MPa A=0.00 m E=00000 MPa A=0.001 m (b) y 1 Beam-column Beam-column 5 FEM idealization Bar Bar Bar disassembly x (node : undefined) (c) 1(1,,) (1) () (6,7,8) () () (5) 5(9,10,11) assembly (d) 1(1,,) (6,7,8) 5(9,10,11) (,5) (,5) Figure Trussed frame structure to illustrate a MET-VFC assembler: (a) original structure showing dimensions, material and fabrication properties; (b) finite element idealization with bars and beam column elements; (c) conceptual disassembly; (d) assembly. Numbers written in parentheses after a node number in (c,d) are the global freedom (equation) numbers allocated to that node. at nodes 1, and 5, and two at node. They are ordered as Global DOF #: DOF: u x1 u y1 θ z1 u x u y u x u y θ z u x5 u y5 θ z5 Node #: (5.) The NFAT and NFMT can be constructed by inspection of (5.) to be nodfat = { ,110000,110001,000000, } nodfmt = { 0,,5,8,8 } (5.5) The element freedom data structures can be also constructed by inspection: Elem Type Nodes EFS EFT (1) Beam-column { 1, } { , } { 1,,,6,7,8 } () Beam-column {,5 } { , } { 6,7,8,9,10,11 } () Bar { 1, } { , } { 1,,,5 } () Bar {, } { , } {,5,6,7 } (5) Bar {,5 } { , } {,5,9,10 } (5.6) Next we list the element stiffness matrices, computed with the modules discussed in Chapter 0. The EFT entries are annotated as usual. 5 15

16 Chapter 5: THE ASSEMBLY PROCESS Element (1): This is a plane beam-column element with stiffness K (1) = (5.7) Element (): This is a plane beam-column with element stiffness identical to the previous one K () = (5.8) Element (): This is a plane bar element with stiffness K () = Element (): This is a plane bar element with stiffness K () = Element (5): This is a plane bar element with stiffness K (5) = (5.9) (5.0) (5.1) Upon merging the 5 elements the master stiffness matrix becomes K = (5.) 5 16

### The Basics of FEA Procedure

CHAPTER 2 The Basics of FEA Procedure 2.1 Introduction This chapter discusses the spring element, especially for the purpose of introducing various concepts involved in use of the FEA technique. A spring

### COMPUTATIONAL ENGINEERING OF FINITE ELEMENT MODELLING FOR AUTOMOTIVE APPLICATION USING ABAQUS

International Journal of Advanced Research in Engineering and Technology (IJARET) Volume 7, Issue 2, March-April 2016, pp. 30 52, Article ID: IJARET_07_02_004 Available online at http://www.iaeme.com/ijaret/issues.asp?jtype=ijaret&vtype=7&itype=2

### Stress Recovery 28 1

. 8 Stress Recovery 8 Chapter 8: STRESS RECOVERY 8 TABLE OF CONTENTS Page 8.. Introduction 8 8.. Calculation of Element Strains and Stresses 8 8.. Direct Stress Evaluation at Nodes 8 8.. Extrapolation

### The elements used in commercial codes can be classified in two basic categories:

CHAPTER 3 Truss Element 3.1 Introduction The single most important concept in understanding FEA, is the basic understanding of various finite elements that we employ in an analysis. Elements are used for

### Finite Element Formulation for Plates - Handout 3 -

Finite Element Formulation for Plates - Handout 3 - Dr Fehmi Cirak (fc286@) Completed Version Definitions A plate is a three dimensional solid body with one of the plate dimensions much smaller than the

### Stiffness Methods for Systematic Analysis of Structures (Ref: Chapters 14, 15, 16)

Stiffness Methods for Systematic Analysis of Structures (Ref: Chapters 14, 15, 16) The Stiffness method provides a very systematic way of analyzing determinate and indeterminate structures. Recall Force

### Types of Elements

chapter : Modeling and Simulation 439 142 20 600 Then from the first equation, P 1 = 140(0.0714) = 9.996 kn. 280 = MPa =, psi The structure pushes on the wall with a force of 9.996 kn. (Note: we could

### CAD and Finite Element Analysis

CAD and Finite Element Analysis Most ME CAD applications require a FEA in one or more areas: Stress Analysis Thermal Analysis Structural Dynamics Computational Fluid Dynamics (CFD) Electromagnetics Analysis...

### Benchmark Problems 24 1

24 Benchmark Problems 24 1 Chapter 24: BENCHMARK PROBLEMS TABLE OF CONTENTS Page 24.1 Introduction..................... 24 3 24.1.1 Problem Information.............. 24 3 24.2 Benchmark Problem Modules...............

### Tutorial for Assignment #2 Gantry Crane Analysis By ANSYS (Mechanical APDL) V.13.0

Tutorial for Assignment #2 Gantry Crane Analysis By ANSYS (Mechanical APDL) V.13.0 1 Problem Description Design a gantry crane meeting the geometry presented in Figure 1 on page #325 of the course textbook

### Introduction to the Finite Element Method (FEM)

Introduction to the Finite Element Method (FEM) ecture First and Second Order One Dimensional Shape Functions Dr. J. Dean Discretisation Consider the temperature distribution along the one-dimensional

### Introduction to the Finite Element Method (FEM)

Introduction to the Finite Element Method (FEM) Lecture 1 The Direct Stiffness Method and the Global StiffnessMatrix Dr. J. Dean 1 Introduction The finite element method (FEM) is a numerical technique

### 1. a) Discuss how finite element is evolved in engineering field. (8) b) Explain the finite element idealization of structures with examples.

M.TECH. DEGREE EXAMINATION Branch: Civil Engineering Specialization Geomechanics and structures Model Question Paper - I MCEGS 106-2 FINITE ELEMENT ANALYSIS Time: 3 hours Maximum: 100 Marks Answer ALL

### Applied Finite Element Analysis. M. E. Barkey. Aerospace Engineering and Mechanics. The University of Alabama

Applied Finite Element Analysis M. E. Barkey Aerospace Engineering and Mechanics The University of Alabama M. E. Barkey Applied Finite Element Analysis 1 Course Objectives To introduce the graduate students

### FUNDAMENTAL FINITE ELEMENT ANALYSIS AND APPLICATIONS

FUNDAMENTAL FINITE ELEMENT ANALYSIS AND APPLICATIONS With Mathematica and MATLAB Computations M. ASGHAR BHATTI WILEY JOHN WILEY & SONS, INC. CONTENTS OF THE BOOK WEB SITE PREFACE xi xiii 1 FINITE ELEMENT

### Finite Element Method (ENGC 6321) Syllabus. Second Semester 2013-2014

Finite Element Method Finite Element Method (ENGC 6321) Syllabus Second Semester 2013-2014 Objectives Understand the basic theory of the FEM Know the behaviour and usage of each type of elements covered

### New approaches in Eurocode 3 efficient global structural design

New approaches in Eurocode 3 efficient global structural design Part 1: 3D model based analysis using general beam-column FEM Ferenc Papp* and József Szalai ** * Associate Professor, Department of Structural

### Exercise 1: Three Point Bending Using ANSYS Workbench

Exercise 1: Three Point Bending Using ANSYS Workbench Contents Goals... 1 Beam under 3-Pt Bending... 2 Taking advantage of symmetries... 3 Starting and Configuring ANSYS Workbench... 4 A. Pre-Processing:

### Module 4: Buckling of 2D Simply Supported Beam

Module 4: Buckling of D Simply Supported Beam Table of Contents Page Number Problem Description Theory Geometry 4 Preprocessor 7 Element Type 7 Real Constants and Material Properties 8 Meshing 9 Solution

### 2. A tutorial: Creating and analyzing a simple model

2. A tutorial: Creating and analyzing a simple model The following section leads you through the ABAQUS/CAE modeling process by visiting each of the modules and showing you the basic steps to create and

### Finite Elements for 2 D Problems

Finite Elements for 2 D Problems General Formula for the Stiffness Matrix Displacements (u, v) in a plane element are interpolated from nodal displacements (ui, vi) using shape functions Ni as follows,

### List of Problems Solved Introduction p. 1 Concept p. 1 Nodes p. 3 Elements p. 4 Direct Approach p. 5 Linear Spring p. 5 Heat Flow p.

Preface p. v List of Problems Solved p. xiii Introduction p. 1 Concept p. 1 Nodes p. 3 Elements p. 4 Direct Approach p. 5 Linear Spring p. 5 Heat Flow p. 6 Assembly of the Global System of Equations p.

### Finite Element Formulation for Beams - Handout 2 -

Finite Element Formulation for Beams - Handout 2 - Dr Fehmi Cirak (fc286@) Completed Version Review of Euler-Bernoulli Beam Physical beam model midline Beam domain in three-dimensions Midline, also called

### EML 5526 FEA Project 1 Alexander, Dylan. Project 1 Finite Element Analysis and Design of a Plane Truss

Problem Statement: Project 1 Finite Element Analysis and Design of a Plane Truss The plane truss in Figure 1 is analyzed using finite element analysis (FEA) for three load cases: A) Axial load: 10,000

### Bedford, Fowler: Statics. Chapter 4: System of Forces and Moments, Examples via TK Solver

System of Forces and Moments Introduction The moment vector of a force vector,, with respect to a point has a magnitude equal to the product of the force magnitude, F, and the perpendicular distance from

### Indeterminate Analysis Force Method 1

Indeterminate Analysis Force Method 1 The force (flexibility) method expresses the relationships between displacements and forces that exist in a structure. Primary objective of the force method is to

### Advanced Structural Analysis. Prof. Devdas Menon. Department of Civil Engineering. Indian Institute of Technology, Madras. Module - 5.3.

Advanced Structural Analysis Prof. Devdas Menon Department of Civil Engineering Indian Institute of Technology, Madras Module - 5.3 Lecture - 29 Matrix Analysis of Beams and Grids Good morning. This is

### Introduction to Solid Modeling Using SolidWorks 2012 SolidWorks Simulation Tutorial Page 1

Introduction to Solid Modeling Using SolidWorks 2012 SolidWorks Simulation Tutorial Page 1 In this tutorial, we will use the SolidWorks Simulation finite element analysis (FEA) program to analyze the response

### Variable Base Interface

Chapter 6 Variable Base Interface 6.1 Introduction Finite element codes has been changed a lot during the evolution of the Finite Element Method, In its early times, finite element applications were developed

### FEA Good Modeling Practices Issues and examples. Finite element model of a dual pinion gear.

FEA Good Modeling Practices Issues and examples Finite element model of a dual pinion gear. Finite Element Analysis (FEA) Good modeling and analysis procedures FEA is a powerful analysis tool, but use

### Back to Elements - Tetrahedra vs. Hexahedra

Back to Elements - Tetrahedra vs. Hexahedra Erke Wang, Thomas Nelson, Rainer Rauch CAD-FEM GmbH, Munich, Germany Abstract This paper presents some analytical results and some test results for different

### Introduction. 1.1 Motivation. Chapter 1

Chapter 1 Introduction The automotive, aerospace and building sectors have traditionally used simulation programs to improve their products or services, focusing their computations in a few major physical

### Programming the Finite Element Method

Programming the Finite Element Method FOURTH EDITION I. M. Smith University of Manchester, UK D. V. Griffiths Colorado School of Mines, USA John Wiley & Sons, Ltd Contents Preface Acknowledgement xv xvii

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### AN INTRODUCTION TO THE FINITE ELEMENT METHOD FOR YOUNG ENGINEERS

AN INTRODUCTION TO THE FINITE ELEMENT METHOD FOR YOUNG ENGINEERS By: Eduardo DeSantiago, PhD, PE, SE Table of Contents SECTION I INTRODUCTION... 2 SECTION II 1-D EXAMPLE... 2 SECTION III DISCUSSION...

### Solving FEM Equations

. 26 Solving FEM Equations 26 1 Chapter 26: SOLVING FEM EQUATIONS 26 2 TABLE OF CONTENTS Page 26.1. Motivation for Sparse Solvers 26 3 26.1.1. The Curse of Fullness.............. 26 3 26.1.2. The Advantages

### 1. Start the NX CAD software. From the start menu, select Start All Programs UGS NX 8.0 NX 8.0 or double click the icon on the desktop.

Lab Objectives Become familiar with Siemens NX finite element analysis using the NX Nastran solver. Perform deflection and stress analyses of planar truss structures. Use modeling and FEA tools to input

### CHAPTER 9 MULTI-DEGREE-OF-FREEDOM SYSTEMS Equations of Motion, Problem Statement, and Solution Methods

CHAPTER 9 MULTI-DEGREE-OF-FREEDOM SYSTEMS Equations of Motion, Problem Statement, and Solution Methods Two-story shear building A shear building is the building whose floor systems are rigid in flexure

### TUTORIAL FOR RISA EDUCATIONAL

1. INTRODUCTION TUTORIAL FOR RISA EDUCATIONAL C.M. Uang and K.M. Leet The educational version of the software RISA-2D, developed by RISA Technologies for the textbook Fundamentals of Structural Analysis,

### Module 4. Analysis of Statically Indeterminate Structures by the Direct Stiffness Method. Version 2 CE IIT, Kharagpur

Module Analysis of Statically Indeterminate Structures by the Direct Stiffness Method Version CE IIT, haragpur esson 7 The Direct Stiffness Method: Beams Version CE IIT, haragpur Instructional Objectives

### Stability Of Structures: Basic Concepts

23 Stability Of Structures: Basic Concepts ASEN 3112 Lecture 23 Slide 1 Objective This Lecture (1) presents basic concepts & terminology on structural stability (2) describes conceptual procedures for

### Feature Commercial codes In-house codes

A simple finite element solver for thermo-mechanical problems Keywords: Scilab, Open source software, thermo-elasticity Introduction In this paper we would like to show how it is possible to develop a

### NonLinear Materials. Introduction. Preprocessing: Defining the Problem

NonLinear Materials Introduction This tutorial was completed using ANSYS 7.0 The purpose of the tutorial is to describe how to include material nonlinearities in an ANSYS model. For instance, the case

### FINITE ELEMENT METHOD (FEM): AN OVERVIEW. Dr A Chawla

FINITE ELEMENT METHOD (FEM): AN OVERVIEW Dr A Chawla ANALYTICAL / MATHEMATICAL SOLUTIONS RESULTS AT INFINITE LOCATIONS CONTINUOUS SOLUTIONS FOR SIMPLIFIED SITUATIONS ONLY EXACT SOLUTION NUMERICAL (FEM)

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### MATH 2030: SYSTEMS OF LINEAR EQUATIONS. ax + by + cz = d. )z = e. while these equations are not linear: xy z = 2, x x = 0,

MATH 23: SYSTEMS OF LINEAR EQUATIONS Systems of Linear Equations In the plane R 2 the general form of the equation of a line is ax + by = c and that the general equation of a plane in R 3 will be we call

### Start the program from the Desktop Icon or from Start, Programs Menu.

5. SHELL MODEL Start Start the program from the Desktop Icon or from Start, Programs Menu. New Create a new model with the New Icon. In the dialogue window that pops up, replace the Model Filename with

### Lab Project: Bridge Construction

MATH 5485 Introduction to Numerical Methods I Fall 2008 Marcel Arndt (modified by Duane Nykamp) Lab Project: Bridge Construction In this lab project, we investigate the static behavior of bridges built

### 820446 - ACMSM - Computer Applications in Solids Mechanics

Coordinating unit: 820 - EUETIB - Barcelona College of Industrial Engineering Teaching unit: 737 - RMEE - Department of Strength of Materials and Structural Engineering Academic year: Degree: 2015 BACHELOR'S

### Study of Analysis System for Bridge Test

Study of Analysis System for Bridge Test Chen Ke, Lu Jian-Ming, Research Institute of Highway, 100088, Beijing, China (chenkezi@163.com, lujianming@263.net) Summary Analysis System for Bridge Test (Chinese

### An Overview of the Finite Element Analysis

CHAPTER 1 An Overview of the Finite Element Analysis 1.1 Introduction Finite element analysis (FEA) involves solution of engineering problems using computers. Engineering structures that have complex geometry

### Building Better Boundary Conditions elearning

Building Better Boundary Conditions elearning Peter Barrett December 2012 2012 CAE Associates Inc. and ANSYS Inc. All rights reserved. Assembly Modeling Webinar Do you effectively manage your boundary

### Nonlinear analysis and form-finding in GSA Training Course

Nonlinear analysis and form-finding in GSA Training Course Non-linear analysis and form-finding in GSA 1 of 47 Oasys Ltd Non-linear analysis and form-finding in GSA 2 of 47 Using the GSA GsRelax Solver

### Introduction to Beam. Area Moments of Inertia, Deflection, and Volumes of Beams

Introduction to Beam Theory Area Moments of Inertia, Deflection, and Volumes of Beams Horizontal structural member used to support horizontal loads such as floors, roofs, and decks. Types of beam loads

### Input Output. 9.1 Why an IO Module is Needed? Chapter 9

Chapter 9 Input Output In general most of the finite element applications have to communicate with pre and post processors,except in some special cases in which the application generates its own input.

### Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli

Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli Prompt: A mentor teacher and student teacher are discussing

### AN AD OPTIMIZED METHOD FOR THE STRUCTURAL ANALYSIS OF PRESSURE VESSELS

Proceedings of ICAD2014 ICAD-2014-24 AN AD OPTIMIZED METHOD FOR THE STRUCTURAL ANALYSIS OF PRESSURE VESSELS ABSTRACT Andrea Girgenti a.girgenti@unimarconi.it Alessandro Giorgetti a.giorgetti@unimarconi.it

### Plates and Shells: Theory and Computation - 4D9 - Dr Fehmi Cirak (fc286@) Office: Inglis building mezzanine level (INO 31)

Plates and Shells: Theory and Computation - 4D9 - Dr Fehmi Cirak (fc286@) Office: Inglis building mezzanine level (INO 31) Outline -1-! This part of the module consists of seven lectures and will focus

### Unit 21 Influence Coefficients

Unit 21 Influence Coefficients Readings: Rivello 6.6, 6.13 (again), 10.5 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Have considered the vibrational behavior of

### Modeling solids: Finite Element Methods

Chapter 9 Modeling solids: Finite Element Methods NOTE: I found errors in some of the equations for the shear stress. I will correct them later. The static and time-dependent modeling of solids is different

### ASEN 3112 - Structures. MDOF Dynamic Systems. ASEN 3112 Lecture 1 Slide 1

19 MDOF Dynamic Systems ASEN 3112 Lecture 1 Slide 1 A Two-DOF Mass-Spring-Dashpot Dynamic System Consider the lumped-parameter, mass-spring-dashpot dynamic system shown in the Figure. It has two point

### Learning Module 6 Linear Dynamic Analysis

Learning Module 6 Linear Dynamic Analysis What is a Learning Module? Title Page Guide A Learning Module (LM) is a structured, concise, and self-sufficient learning resource. An LM provides the learner

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### 3 Concepts of Stress Analysis

3 Concepts of Stress Analysis 3.1 Introduction Here the concepts of stress analysis will be stated in a finite element context. That means that the primary unknown will be the (generalized) displacements.

### Partitioned Matrices and the Schur Complement

P Partitioned Matrices and the Schur Complement P 1 Appendix P: PARTITIONED MATRICES AND THE SCHUR COMPLEMENT TABLE OF CONTENTS Page P1 Partitioned Matrix P 3 P2 Schur Complements P 3 P3 Block Diagonalization

### Programming a truss finite element solver in C

CHAPTER 4 Programming a truss finite element solver in C In this section, some C functions are developed that allow to compute displacements and internal efforts of a two-dimensional truss structure. 4.1

### Solution of Linear Systems

Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We start

### Linear Static Analysis of a Cantilever Beam Using Beam Library (SI Units)

APPENDIX A Linear Static Analysis of a Cantilever Beam Using Beam Library (SI Units) Objectives: Create a geometric representation of a cantilever beam. Use the geometry model to define an MSC.Nastran

### ETABS. Integrated Building Design Software. Composite Floor Frame Design Manual. Computers and Structures, Inc. Berkeley, California, USA

ETABS Integrated Building Design Software Composite Floor Frame Design Manual Computers and Structures, Inc. Berkeley, California, USA Version 8 January 2002 Copyright The computer program ETABS and all

### THE CONCEPT OF HIERARCHICAL LEVELS; AN OVERALL CONCEPT FOR A FULL AUTOMATIC CONCRETE DESIGN INCLUDING THE EDUCATION OF CONCRETE.

THE CONCEPT OF HIERARCHICAL LEVELS; AN OVERALL CONCEPT FOR A FULL AUTOMATIC CONCRETE DESIGN INCLUDING THE EDUCATION OF CONCRETE. THE CASE MatrixFrame VERSUS EuroCadCrete. ABSTRACT: Ir. Ron Weener 1. The

### A Direct Numerical Method for Observability Analysis

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL 15, NO 2, MAY 2000 625 A Direct Numerical Method for Observability Analysis Bei Gou and Ali Abur, Senior Member, IEEE Abstract This paper presents an algebraic method

### (Seattle is home of Boeing Jets)

Dr. Faeq M. Shaikh Seattle, Washington, USA (Seattle is home of Boeing Jets) 1 Pre Requisites for Today s Seminar Basic understanding of Finite Element Analysis Working Knowledge of Laminate Plate Theory

### Piston Ring. Problem:

Problem: A cast-iron piston ring has a mean diameter of 81 mm, a radial height of h 6 mm, and a thickness b 4 mm. The ring is assembled using an expansion tool which separates the split ends a distance

### CHAPTER 3. INTRODUCTION TO MATRIX METHODS FOR STRUCTURAL ANALYSIS

1 CHAPTER 3. INTRODUCTION TO MATRIX METHODS FOR STRUCTURAL ANALYSIS Written by: Sophia Hassiotis, January, 2003 Last revision: February, 2015 Modern methods of structural analysis overcome some of the

### 0 Introduction to Data Analysis Using an Excel Spreadsheet

Experiment 0 Introduction to Data Analysis Using an Excel Spreadsheet I. Purpose The purpose of this introductory lab is to teach you a few basic things about how to use an EXCEL 2010 spreadsheet to do

### MATERIAL NONLINEAR ANALYSIS. using SolidWorks 2010 Simulation

MATERIAL NONLINEAR ANALYSIS using SolidWorks 2010 Simulation LM-ST-1 Learning Module Non-Linear Analysis What is a Learning Module? Title Page Guide A Learning Module (LM) is a structured, concise, and

### METHODS FOR ACHIEVEMENT UNIFORM STRESSES DISTRIBUTION UNDER THE FOUNDATION

International Journal of Civil Engineering and Technology (IJCIET) Volume 7, Issue 2, March-April 2016, pp. 45-66, Article ID: IJCIET_07_02_004 Available online at http://www.iaeme.com/ijciet/issues.asp?jtype=ijciet&vtype=7&itype=2

### Vectors 2. The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996.

Vectors 2 The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996. Launch Mathematica. Type

### Nonlinear Finite Element Method 01/11/2004

Nonlinear Finite Element Method 01/11/2004 Nonlinear Finite Element Method Lectures include discussion of the nonlinear finite element method. It is preferable to have completed Introduction to Nonlinear

### STRUCTURAL ANALYSIS SKILLS

STRUCTURAL ANALYSIS SKILLS ***This document is held up to a basic level to represent a sample for our both theoretical background & software capabilities/skills. (Click on each link to see the detailed

### Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 01

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture - 01 Welcome to the series of lectures, on finite element analysis. Before I start,

### Tower Cross Arm Numerical Analysis

Chapter 7 Tower Cross Arm Numerical Analysis In this section the structural analysis of the test tower cross arm is done in Prokon and compared to a full finite element analysis using Ansys. This is done

Frequently Asked Questions about the Finite Element Method 1. What is the finite element method (FEM)? The FEM is a novel numerical method used to solve ordinary and partial differential equations. The

### Version default Titre : SSNP161 Essais biaxiaux de Kupfer Date : 10/10/2012 Page : 1/8 Responsable : François HAMON Clé : V6.03.161 Révision : 9783

Titre : SSNP161 Essais biaxiaux de Kupfer Date : 10/10/2012 Page : 1/8 SSNP161 Biaxial tests of Summarized Kupfer: Kupfer [1] was interested to characterize the performances of the concrete under biaxial

### CAE -Finite Element Method

16.810 Engineering Design and Rapid Prototyping CAE -Finite Element Method Instructor(s) Prof. Olivier de Weck January 11, 2005 Plan for Today Hand Calculations Aero Æ Structures FEM Lecture (ca. 45 min)

### When a user chooses to model the surface component using plate elements, he/she is taking on the responsibility of meshing.

Concrete slab Design: Yes, you can design the concrete slab using STAAD, with plate elements and meshing it appropriately. But it is best practice to take the analysis results from the STAAD and do the

### EFFECTS ON NUMBER OF CABLES FOR MODAL ANALYSIS OF CABLE-STAYED BRIDGES

EFFECTS ON NUMBER OF CABLES FOR MODAL ANALYSIS OF CABLE-STAYED BRIDGES Yang-Cheng Wang Associate Professor & Chairman Department of Civil Engineering Chinese Military Academy Feng-Shan 83000,Taiwan Republic

### NAPA/MAESTRO Interface. Reducing the Level of Effort for Ship Structural Design

NAPA/MAESTRO Interface Reducing the Level of Effort for Ship Structural Design 12/3/2010 Table of Contents Introduction... 1 Why Create a NAPA/MAESTRO Interface... 1 Level of Effort Comparison for Two

### Memory Systems. Static Random Access Memory (SRAM) Cell

Memory Systems This chapter begins the discussion of memory systems from the implementation of a single bit. The architecture of memory chips is then constructed using arrays of bit implementations coupled

### Finite Element Analysis (FEA) Tutorial. Project 2: 2D Plate with a Hole Problem

Finite Element Analysis (FEA) Tutorial Project 2: 2D Plate with a Hole Problem Problem Analyze the following plate with hole using FEA tool ABAQUS P w d P Dimensions: t = 3 mm w = 50 mm d = mm L = 0 mm

### arxiv: v1 [physics.gen-ph] 10 Jul 2015

Infinite circuits are easy. How about long ones? Mikhail Kagan, Xinzhe Wang Penn State Abington arxiv:507.08v [physics.gen-ph] 0 Jul 05 Abstract We consider a long but finite ladder) circuit with alternating

### 4. MATRICES Matrices

4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:

### CAE -Finite Element Method

16.810 Engineering Design and Rapid Prototyping Lecture 3b CAE -Finite Element Method Instructor(s) Prof. Olivier de Weck January 16, 2007 Numerical Methods Finite Element Method Boundary Element Method

### Instructors Manual Finite Element Method Laboratory Sessions

Instructors Manual Finite Element Method Laboratory Sessions Dr. Waluyo Adi Siswanto 6 July 2010 Universiti Tun Hussein Onn Malaysia (UTHM) This document is written in LYX 1.6.7 a frontend of LATEX Contents

### Universal Format of Shape Function for Numerical Analysis using Multiple Element Forms

, pp.63-67 http://dx.doi.org/0.4257/astl.206. Universal Format of Shape Function for Numerical Analysis using Multiple Element Forms Yuting Zhang, Yujie Li, Xiuli Ding Yangtze River Scientific Research

### LECTURE NOTES: FINITE ELEMENT METHOD

LECTURE NOTES: FINITE ELEMENT METHOD AXEL MÅLQVIST. Motivation The finite element method has two main strengths... Geometry. Very complex geometries can be used. This is probably the main reason why finite