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1. Once finish input all the members of the space truss structure press on the OK button below the table Subsequently the space truss structure will be generated and displayed on the axes generated before The number of the nodes and elements will be displayed on the axes as well The coordinates of the nodes are also displayed in the left hand side table with heading Data Node and Element The appearance of the interface will become as shown in Figure 4 4 below z El Spacetruss E AAD x Data Node and Element x coordinate y coordinate z coordinate Support Fx kN Fy kN Fz kN Young Modulus E Area 0 kN m 2 m 2 Li 2 3 0 4 0 3 0 0 0 4 0 s e qaqa Analyse Results Node and Element Rx kN Ry kN Rz kN Ux m Uy m Uz m Force in element Stress in element Type of stress nt kN kN m 2 2 L3 la 4 3 4 Save Results Exit Figure 4 4 Model of the space truss structure Now we proceeding to the next stage that is input all the required data for the nodes and elements The support condition of each node is identified and can be selected from the choices available there that is ball and socket constrained from translation roller constrained from movement in the z direction or no support at all For this Example 1 the support condition of node 1 is none and the support
2. METHODOLOGY 3 1 Assumption used in the Research 3 2 Research Design and Procedure 3 3 Modelling of Truss Structure 3 4 Analysis of Truss Structure 3 4 1 Discretize the Problem 3 4 2 Determine the Element Stiffness Matrix 3 4 3 Assemble the Global Stiffness Matrix 3 4 4 Apply the Boundary Conditions 3 4 5 Solve the Equations 3 4 6 Post Processing Stage 35 MATLAB Graphical User Interface Development Environment GUIDE 3 5 1 Figure 3 5 2 Static Text 3 5 3 Edit Text 3 5 4 Table 3 5 5 Push Button 3 5 6 Axes 3 5 7 Panel RESULTS AND ANALYSIS 4 1 Introduction 4 2 Space Truss Modelling User Manual 4 3 Comparison of Results CONCLUSIONS AND RECOMMENDATIONS 5 1 Conclusions 5 2 Recommendations 10 12 12 13 14 15 15 17 22 23 23 24 25 26 26 26 27 28 29 30 32 32 33 38 49 49 50 viii REFERENCES APPENDIX A APPENDIX B 51 52 60 TABLE NO 4 1 4 2 4 3 4 4 4 5 4 6 4 7 4 8 4 9 4 10 4 11 4 12 LIST OF TABLES TITLE Displacements at nodes Example 1 Reactions at nodes Example 1 Force in elements Example 1 Stress in elements Example 1 Displacements at nodes Example 2 Reactions at nodes Example 2 Force in elements Example 2 Stress in elements Example 2 Displacements at nodes Example 3 Reactions at nodes Example 3 Force in elements Example 3 Stress in elements Example 3 PAGE 39 39 40 40 43 43 44 44 47 47 48 48 FIGURE NO 2 1 3 1 3 2
3. for end i 1 t Plot the nodes of the space truss structure plot3 node xyzf i l node xyz i 2 node xyz i 3 Marker o MarkerFaceColor b text node_xyz i 1 xmax xmin 50 node_xyz i 2 ymax ymin 50 node_xyz i 3 zmax zmin 50 num2str i Color r FontWeight bold hold on i 1 n 2 Plot the elements of the space truss structure plot3 node xyz dof 2 i 1 1 node xyz dof 2 i 1 node xyzidof 2 i 1 2 node xyz dof 2 i 2 node xyzidof 2 i 1 3 node xyz dof 2 i 3 hold on text node xyz dof 2 i 1 1 node xyz dof 2 i 1 2 xmax xmin 40 node xyz dof 2 i 1 2 node xyzidof 2 i 2 2 ymax ymin 40 node_xyz dof 2 i 1 3 node_xyz dof 2 i 3 2 zmax zmin 40 num2str i EdgeColor k hold on 30 3 5 7 Panel Panels group GUI components and can make a GUI easier to understand by visually grouping related controls It can contain user interface controls with which the user interacts directly A panel can contain panels and button groups as well as axes and user interface controls such as push buttons sliders pop up menus etc The position of each component within a panel is interpreted relative to the lower left corner of the panel If you move the panel the components within the panel automatically move with it and maintain their positions relative to the panel Figure 3 8 below shows the list of a p
4. 3 end Z find M 2 S size 2 2 for i 1 s result node Z i 1 result node Z i 2 ll rr w N H result node Z i 3 F 3 Z 1i1 end for i 1 t result nodefi 4 U 3 i 2 result nodefi 5 U 3 i 1 result node i 6 U 3 i end global result element result element n 2 3 Il for i l n 2 u i U 3 dof 2 i 1 2 3 dof 2 i 1 U 3 dof 2 1 2 3 dof 2 1 J result element i l ElementForce E i A i L i theta i u i result_element i 2 ElementStress E i L i theta i u i if result_element i 2 gt 0 result_element i 3 Tensile lseif result_element i 2 lt 0 result_element i 3 Compressive else result _element i 3 end end global tused tused toc t handles textl4 string Time taken num2str tused s t handles table node data data node set handles table element data data element t handles table node2 data result node t handles table element2 data result element Executes on button press in pushbutton 7 function pushbutton7 Callback hObject eventdata handles hobj oe oo Data even handles ect handle to pushbutton7 see GCBO tdata reserved to be defined in a future version of MATLAB structure with handles and user data see GUIDATA clear global node node new n t dof node xyz get han
5. oe hObject handle to figure eventdata reserved to be defined in a future version of MATLAB handles structure with handles and user data see GUIDATA Get default command line output from handles structure varargout 1 handles output Executes on button press in pushbuttonl function pushbuttonl Callback hObject eventdata handles hObject handle to pushbuttonl see GCBO 53 eventdata reserved to be defined in a future version of MATLAB 5 handles structure with handles and user data see GUIDATA set handles editl visible off set handles edit2 visible off set handles edit3 visible off set handles edit4 visible off set handles edit5 visible off set handles edit 6 visible off set handles textl visible off set handles text2 visible off set handles text3 visible off set handles text4 visible off set handles text5 visible off set handles text6 visible off set handles text10 visible off set handles panell visible off set handles pushbuttonl visible off set handles pushbutton4 visible on set handles pushbutton5 visible on set handles pushbutton6 visible on set handles table element visible on set handles table node visible on set handles table element2 visible on set handles table
6. 180 w theta 2 pi 180 v theta 3 pi 180 Cx cos x Cy cos w Cz cos v y E A L Cx Cy Cz Cx Cy Cz u end Element Stiffness Matrix to form Structure Stiffness 62 To Calculate Stress in Element function y x theta 1 w theta 2 v theta 3 Cx cos x Cy cos w Cz cos v y E L Cx end ElementStress E L theta u pi 180 pi 180 pi 180 Cy Cz Cx Cy Cz u
7. 3 3 3 4 3 5 3 6 3 7 3 8 3 9 4 1 4 2 4 3 4 4 4 5 4 6 4 7 4 8 4 9 4 10 LIST OF FIGURES TITLE Types of truss Research design and procedure A simple space truss example A truss element in three dimensional space A bar element in local direction Layout Editor with a blank GUI in GUIDE Figure static text and edit text in GUI Table and push button in GUI List of a panel and its child objects in object browser Axes and panel in GUI Space truss for Example 1 Input the range of axes Input the coordinates of nodes of the elements Model of the space truss structure Input the required data for analysis Analysis and display of the results Space truss for Example 2 Analysis and display of the results for Example 2 Space truss for Example 3 Analysis and display of the results for Example 3 PAGE 13 15 16 17 25 2 28 30 31 33 34 35 36 37 38 4l 42 45 46 xi k tg Ul K Qj F x Ni gi xii LIST OF SYMBOLS Young s Modulus Cross sectional area of an element Length of an element Element stiffness matrix in the local coordinate system Element displacement in local coordinate system Element nodal force component in local coordinate system Element stiffness matrix in global coordinate system Element displacement in global coordinate system Element nodal force component in global coordinate system Axial displacement at any position along the length of bar Shape function Displa
8. 4 Analysis of Truss Structure Finite element formulation is used as the method for the analysis of truss in this research There are several approaches that can be used to define and formulate the finite element problem The most widely used method in this research is the direct stiffness method where it can yield the displacements and forces directly The internal force stress and strain in each element can then be determined by back substitution of displacements into individual element equations The detail of space truss finite element formulation is presented in the following section 3 4 1 Discretize the Problem Figure 3 2 A simple space truss example 16 Figure 3 3 A truss element in three dimensional space In general pin jointed structures which do not have loading between joints or moments at joints such as trusses can be subdivided into elastic spring bar elements for analysis A spring bar element is a one dimensional element and has linear interpolation function Figure 3 2 shows a simple space truss example Figure 3 3 shows one of the elements of the space truss The truss is discretized into six finite elements which are connected to each other at joints referred to as nodes Degree of freedom is defined as the possible displacements of the node For a three dimensional space truss problem there are three degrees of freedom for each node of the structure These degrees of freedom are the displacements in x direc
9. analysis of space truss analysis only The mathematical model for a space truss analysis consists of a set of joints nodes which are connected by straight member in three dimensional spaces All loads acting on the structure must be consists of concentrated loads act on the joints of the structure only Intermediate loads acting on the members or moments acting on the joints of the structure are not permitted There is no shear force or bending moment exists in the truss members The truss members only subjected to compression forces or tension forces There are only reactions forces exist at the support joints of the structure 13 vi The self weight of truss members is small and can be neglected in the analysis vii Forces acting on the structure are only permitted for directions in x axis y axis Or z axis viii The truss members are assumed having linear elastic behaviour ix The supports of the structure are assumed lie on the horizontal plane X The inclined support of the structure is not permitted in the analysis 3 2 Research Design and Procedure The research design is shown in figure 3 1 below Identify the Discretize the problem problem ja Determine the member stiffness matrix Model the structure Establish the structure stiffness matrix K in MATLAB GUIs um Apply boundary conditions Analyze the Vw F 4 Display of results Post processing stage Figure 3 1 Research design and proce
10. are analysed and solved by using finite element formulation programmed in MATLAB The efficiency of the program developed and the accuracy of the results obtained have been validated and verified by comparing the results with those generated by existing commercial software which is STAAD Pro in this research The comparison of the two results shows that the results obtained from the program developed are accurate and acceptable The displacements at the nodes the reactions at the nodes which have support the internal force and stress inside the elements can be determined effectively by using the space truss analysis program The research has been carried out successfully The objective of the research is achieved that is developing a valid space truss analysis program by applying finite element method using MATLAB R2009b 50 5 2 Recommendations Although in this research the space truss analysis program has been successfully developed by using MATLAB the program can be upgraded and improved in various aspects especially regarding the assumptions and limitations stated in this research As mention earlier in the Chapter 3 the space truss analysis program is limited to the analysis of space truss structure subjected to external forces which acting in directions along x axis y axis or z axis only In the future research the space truss analysis program can be improved by making it able to analysis space truss structure subjected to
11. eguation only gives the element stiffness matrix in the element coordinate system which is in one dimensional In order to analyze a three dimensional space truss the element stiffness matrix for each member of the truss must be transformed and expressed in global coordinate system first For a space truss each node of the member has three degrees of freedom or nodal displacements in global coordinates The relationship between the element displacements in the element coordinate system and the global coordinate system can be defined as q Qix x Qiy Ay 0141 3 15 92 Q x x Q5 Ay t 0221 3 16 where Qix Qiy Q z displacements at node I in global x axis y axis and Z axis directions respectively Q2 Q2 Q2 displacements at node 2 in global x axis y axis and z axis directions respectively cos 0 cos 0 cos x Ny Eguation 3 15 and 3 16 can be written in matrix form as qu pL Bis 0 ds 3 pe D e d Qix Qiy Qi Qn Q0 QnLy qj ITI Q 3 17 where 0 0 0 mel s JE 3 18 21 T is referred to as the displacement transformation matrix because it transform the global displacements into the element local displacement The relationship between the global force components and the local forces can be defined as Ehh Fy fRA Fiz 5d Po fod Fay h Faz fear which can be written in matrix form as A A 0 Pi Big ug iE ey ENE Fix Fiy Fiz Fox Fay Foz ua a
12. node2 visible on set handles panel2 visible on set handles panel3 visible on set handles panel4 visible on ele set handles axesl visible on axes handles axesl cla reset global xLimit yLimit zLimit xInterval yInterval zInterval xLimit 1 str2double get handles editl string yLimit 1 str2double get handles edit2 string zLimit 1 str2double get handles edit3 string xLimit 2 str2double get handles edit4 string yLimit 2 str2double get handles edit5 string zLimit 2 str2double get handles edit6 string axis xLimit 1 xLimit 2 yLimit 1 yLimit 2 zLimit 1 zLimit 2 xlabel x ylabel y zlabel z grid on hold on Executes on button press in pushbutton2 function pushbutton2 Callback hObject eventdata handles hObject handle to pushbutton2 see GCBO eventdata reserved to be defined in a future version of MATLAB handles structure with handles and user data see GUIDATA uestion ans questdlg Are you sure you want to exit s EXIT Yes s No yi if strcmp question ans Yes close SpaceTruss end 54 o gt Executes on butt function pushbutton4 C hObject handle to oe on press in pushbutton4 allback hObject eventdata pushbutton4 see GCBO oo oo tut TS D ventdata reserved an
13. oe oo 59 NewData EditData or its converted form set on the Data property Empty if Data was not changed Error error string when failed to convert EditData to appropriate value for Data handles structure with handles and user data see GUIDATA data node get handles table node data support data_node 4 global t for i 1 t b strcmp support i Ball and socket c stremp support i Roller if b data nodefi 5 data nodefi 6 data_node i 7 elseif c data_node i 7 else end end set handles table node data data_node Executes on button press in pushbuttonll function pushbuttonll Callback hObject eventdata handles hObject handle to pushbuttonll see GCBO eventdata reserved to be defined in a future version of MATLAB handles structure with handles and user data see GUIDATA global t U n result element tused Resultfile path uiputfile doc Save file name if Resultfile 0 Result fopen path Resultfile w fprintf Result s n Node Displacement fprintf Result s n Node Ux Uy Ug Yi for i 1 t fprintf Result 5 0f 7 4e 7 4e 7 4e Nn i U 3 i 2 U 3 i 1 U 3 i end fprintf Result n n s n Element Internal Force And Stress Ya fprintf Result s n Element Internal Force I
14. pushbutton8 visible off o o oe SpaceTruss fig File Edit View Layout Tools Help cQOH m50c s amp B md sv I x i ieee pers ra pa Coordinates af ra E E First Node x First Node y First Node z End Node x End Node y End Node z ma Rand fel Ba Rang kl Type of stress Tag figurel CuNpt Point 1034 565 Position 520 205 1162 595 Table Push Button Figure 3 7 Table and push button in GUI 3 5 6 Axes images While the basic purpose of an axes object is to provide a coordinate system Axes components enable your GUI to display graphics such as graphs and for plotted data axes properties provide considerable control over the way MATLAB displays data The current axes is the target for functions that draw image line patch rectangle surface and text graphics objects Axis manipulates commonly used axes properties Code axis xmin xmax ymin ymax zmin zmax sets the x y and z axis limits of the current axes The axes component in this research is shown in Figure 3 9 Examples of code used in this research to set the axis limits of axes and plot the space truss structure in axes are as following 5 t the x y and z axis limits of the axes axis xLimit 1 xLimit 2 yLimit 1 yLimit 2 zLimit 1 zLimit 2 Plot the space truss structure in the axes fot end
15. space truss structure always involved with tedious steps and lengthy calculations Most of the existing commercial engineering software is very expensive Conseguently a cheaper local developed computer software which can analyse a space truss structure efficiently in a shorter time should be developed to assist the user in analysing a space truss structure This paper presents the development of space truss analysis software by using MATLAB The method of analysis used in this research is finite element method FEM The derivation and formulation of finite element method involved in analysing a space truss will be discussed in detail in this paper Graphical user friendly interfaces are developed using Graphical User Interface Development Environment GUIDE function in MATLAB to ease the users in using the software An user manual of guidelines in using the space truss analysis software with example is included in this paper as well The results generated from the space truss analysis software are compared with those obtained from the existing engineering software STAAD Pro for validation vi ABSTRAK Analisis struktur kekuda ruang selalu melibatkan langkah dan pengiraan yang panjang Kebanyakan perisian komersil yang sedia ada sangat mahal Oleh sebab itu satu perisian komputer tempatan murah yang dapat menganalisis suatu struktur kekuda ruang dengan cekap dalam masa yang pendek harus dibangunkan untuk membantu pengguna dalam menganalisis s
16. 009a script Space truss modelling function varargout SpaceTruss varargin gui Singleton 1 gui State struct gui Name mfilename gui Singleton gui Singleton gui OpeningFcn SpaceTruss OpeningFcn gui OutputFcn SpaceTruss OutputFcn gui LayoutFcn mer gui Callback Il if nargin amp amp ischar varargin 1 gui State gui Callback str2func varargin 1 end if nargout varargout l nargout gui mainfcn gui State varargin else gui mainfcn gui State varargin end End initialization code DO NOT EDIT Q Executes just before SpaceTruss is made visible function SpaceTruss OpeningFcn hObject eventdata handles varargin This function has no output args see OutputFen oo hObject handle to figure eventdata reserved to be defined in a future version of MATLAB handles structure with handles and user data see GUIDATA varargin command line arguments to SpaceTruss see VARARGIN oo Choose default command line output for SpaceTruss handles output hObject Update handles structure guidata hObject handles grid on UIWAIT makes SpaceTruss wait for user response s UIRESUME uiwait handles figurel Outputs from this function are returned to the command line function varargout SpaceTruss OutputFcn hObject eventdata handles varargout cell array for returning output args see VARARGOUT
17. 1 2 K 3 i 2 3 i K 3 i 2 3 i k 1 3 K 3 1 2 3 2 K 3 i 2 3 J 2 k 1 4 K 3 i 2 3 j 1 K 3 i 2 3 j 1 k 1 5 K 3 i 2 3 j K 3 i 2 3 3 k 1 6 K 3 i 1 3 i 2 K 3 i 1 3 i 2 k 2 1 K 3 i 1 3 i 1 K 3 i 1 3 i 1 k 2 2 K 3 i 1 3 i K 3 i 1 3 i k 2 3 K 3 i 1 3 j 2 K 3 i 1 3 j 2 k 2 4 K 3 i 1 3 j 1 K 3 i 1 3 j 1 k 2 5 K 3 i 1 3 3 K 3 i 1 3 j k 2 6 K 3 i 3 i 2 K 3 i 3 i 2 k 3 1 K 3 i 3 i 1 K 3 i 3 i 1 k 3 2 K 3 i 3 i K 3 i 3 i k 3 3 K 3 i 3 j 2 K 3 i 3 j3 2 k 3 4 K 3 i 3 j 1 K 3 i 3 j 1 k 3 5 K 3 i 3 3 K 3 1 3 j k 3 6 K 3 j 2 3 i 2 K 3 j 2 3 i 2 k 4 1 K 3 j 2 3 i 1 K 3 j 2 3 i 1 k 4 2 K 3 j 2 3 i K 3 j 2 3 i k 4 3 K 3 j 2 3 3 2 K 3 j 2 3 j 2 k 4 4 K 3 j 2 3 j 1 K 3 j 2 3 j 1 k 4 5 K 3 j 2 3 3 K 3 5 2 3 3 k 4 6 K 3 j 1 3 i 2 K 3 j 1 3 i 2 k 5 1 K 3 j 1 3 i 1 K 3 j 1 3 i 1 k 5 2 K 3 j 1 3 i K 3 j 1 3 i k 5 3 K 3 j 1 3 3 2 K 3 j 1 3 j 2 k 5 4 K 3 7 1 3 1 K 3 7J 1 3 7J 1 k 5 5 7 K 3 j 1 3 3 K 3 3 1 3 3 k 5 6 K 3 J 3 2 2 K 3 9 2 2 k 6 1 K 3 j 3 i 1 K 3 jJ 3 i 1 k 6 2 K 3 3 i K 3 j 3 i k 6 3 K 3 j 3 j 2 K 3 j 3 j 2 k 6 4 K 3 j 3 3 1 K 3 3 3 j 1 k 6 5 K 347 347 K 3 5 3 j k 6 6 y K end To Calculate Force in Element function y ElementForce E A L theta u x theta 1 pi
18. 1563e 03 Tensile 4 1 1915e 15 10 0117 16 6862 0 0 0 3 19 4593 6 4864e 03 Tensile 5 8 5849 8 4148e 16 10 7311 0 0 0 4 13 7425 4 5808e 03 Compressive Time taken 0 016718 s Save Results Exit Figure 4 8 Analysis and display of the results for Example 2 The results obtained from the space truss analysis program are compared with those from STAAD Pro as shown in Table 4 5 Table 4 6 Table 4 7 and Table 4 8 below 43 Table 4 5 Displacements at nodes Example 2 Node Displacements Number Space Truss Analysis Program m STAAD Pro m of node Ux U U U U U 1 2 3509e 3 6713e 2 5875e 0 235e 0 367e 0 000 2 0 0 0 0 000 0 000 0 000 3 0 0 0 0 000 0 000 0 000 4 0 0 0 0 000 0 000 0 000 5 0 0 0 0 000 0 000 0 000 Table 4 6 Reactions at supports Example 2 Reactions Number Space Truss Analysis Program kN STAAD Pro kN of node R R R Rx Ry R 2 1 1887e P 9 9883 16 6471 0 000 9 988 16 647 3 6 4151 7 6349e 10 6919 6 415 0 000 10 692 4 1 1915e 10 0117 16 6862 0 000 10 012 16 686 5 8 5849 8 4148e 10 7311 8 585 0 000 10 731 Table 4 7 Force in elements Example 2 Force In Element Number of Bienes Space Truss Analysis Program STAAD Pro kN kN 1 19 4137 19 414 2 12 4688 12 469 3 19 4593 1
19. 9 459 4 13 7425 13 742 Table 4 8 Stress in elements Example 2 ee eae Stress In Element Element Space Truss se Program STAAD Pro kN n kN m 1 6 4712e Compressive 6 471e Compressive 2 4 1563e Tensile 4 156 Tensile 3 6 4864e Tensile 6 486 Tensile 4 4 5808e Compressive 4 581e Compressive From the comparison as shown in Table 4 5 Table 4 6 Table 4 7 and Table 4 8 above we can see that the results obtained from the space truss analysis program and STAAD Pro are approximately the same except that the results values obtained from STAAD Pro show less decimal places than the results values obtained from space truss analysis program developed Example 3 Figure 4 9 Space truss for Example 3 Data available Young s modulus E 200 GPa 200e kN m Cross sectional area A15 0 001 m A13 0 002 m A14 0 005 m Determine i The x direction y direction and z direction displacements at nodes ii The reactions at nodes 1 2 3 and 4 iii The force in each element iv The stress in each element 45 46 The appearance of the interface of space truss analysis program after the space truss problem as in Example 3 is modelled and analysed using the program is shown in Figure 4 10 below 5 Bl SpaceTruss AAD Data Node and Element x coordinate y coordinate 2 coordinate Support Fx KN Fy kN
20. Fz kN Young Modulus E Area 1 6 0 ORoler 0 EE kN m 2 m 2 eal 0 0 4 Bal and s v 2 1 200000000 1 0000e 03 0 3 0 Bal and s y E z 2 200000000 0 0020 0 3 6 Bal and s v 5 E 3 200000000 BETTE Results Node and Element Rx kN Ry kN Rz kN Ux m Uy m Uz m Force in element Stress in element Type of stress aul B 35 2596 0 0016 0 0037 0 kN kN m 2 2 30 0000 22078e 15 20 0000 0 0 o a 36 0555 3 6056e 04 Compressive 3 14 7404 7 3702 1 0091e 15 0 0 0 2 16 4803 8 2402e 03 Tensile 4 15 2596 7 6298 15 2596 0 0 0 3 22 8893 4 5779e 03 Tensile 4 13 7425 4 5808e 03 Compressive Insert Clear Time taken 0 0031908 s Save Results Exit Figure 4 10 Analysis and display of the results for Example 3 Similarly the results obtained from the space truss analysis program are compared with those from STAAD Pro as shown in Table 4 9 Table 4 10 Table 4 11 and Table 4 12 below Table 4 9 Displacements at nodes Example 3 47 Node Displacements Number Space Truss Analysis Program m STAAD Pro m of node U U U Ux Uy U 1 0 0016 0 0037 0 1 562e 3 743e 0 000 2 0 0 0 0 000 0 000 0 000 3 0 0 0 0 000 0 000 0 000 4 0 0 0 0 000 0 000 0 000 Table 4 10 Reactions at suppots Example 3 Reactions Number Space Truss Analysis Program kN S
21. In order to validate the space truss analysis program and verify its results the results from space truss analysis program developed is compared with results from existing engineering software STAAD Pro Three examples of space truss problems are modelled and analysed using both the space truss analysis program developed and STAAD Pro in this chapter The results obtained from the program and the software should be the same or approximately same 33 4 2 Space Truss Modelling User Manual A simple example of space truss problem is taken from MATLAB Guide to Finite Elements Peter I Kattan 2006 as following Example 1 Figure 4 1 Space truss for Example 1 Data available Young s modulus E 200 GPa 200e kN m Cross sectional area A15 0 001 m A13 0 002 m A14 0 001 m Determine i The x direction y direction and z direction displacements at nodes ii The reactions at nodes 2 3 and 4 iii The force in each element iv The stress in each element 34 First of all the range of the axes is determined from the space truss structure diagram in order to display the space truss structure within the axes range In the case of this Example 1 the range of x axis is from 4 m till 4m the range of y axis is from 3 m till 0 m and the range of z axis is from 0 m till 5 m Example of inputs for this Example 1 is shown as Figure 4 2 below Bl SpaceTruss z ALD z Coordinates Range of x axis f
22. PSZ 19 16 Pind 1 07 UNIVERSITI TEKNOLOGI MALAYSIA DECLARATION OF THESIS UNDERGRADUATE PROJECT PAPER AND COPYRIGHT Author s full name CHANG CHEE BOON Date of birth 4 APRIL 1987 Title COMPUTER PROGRAM DEVELOPMENT FOR SPACE TRUSS ANALYSIS Academic Session 2010 2011 2 declare that this thesis is classified as iai CONFIDENTIAL Contains confidential information under the Official Secret Act 1972 RESTRICTED Contains restricted information as specified by the organization where research was done OPEN ACCESS agree that my thesis to be published as online open access full text I acknowledged that Universiti Teknologi Malaysia reserves the right as follows The thesis is the property of Universiti Teknologi Malaysia The Library of Universiti Teknologi Malaysia has the right to make copies for the purpose of research only The Library has the right to make copies of the thesis for academic exchange Certified by SIGNATURE SIGNATURE OF SUPERVISOR 870404 08 5215 DR AIRIL YASREEN MOHD YASSIN NEW IC NO PASSPORT NO NAME OF SUPERVISOR Date 5 MAY 2011 Date 5 MAY 2011 NOTES If the thesis is CONFIDENTAL or RESTRICTED please attach with the letter from the organization with period and reasons for confidentiality or restriction hereby declare that I have read this report and in my opinion this thesis is sufficient in terms of scope and qua
23. TAAD Pro KN of node Rx Ry R Rx Ry R 1 0 0 35 2596 0 000 0 000 35 260 2 30 0000 2 2078e 20 0000 30 000 0 000 20 000 3 14 7404 7 3702 1 00916 14 740 7 370 0 000 4 15 2596 7 6298 15 2596 15 260 7 630 15 260 48 Table 4 11 Force in elements Example 3 Force In Element Number of Space Truss Analysis Program Element j STAAD Pro kN kN 1 36 0555 36 056 2 16 4803 16 480 3 22 8893 22 889 Table 4 12 Stress in elements Example 3 Stress In Element Number of Space Truss Analysis Program Element STAAD Pro kN m kN m 1 3 6056e Compressive 36 056e Compressive 2 8 2402e Tensile 8 240 Tensile 3 4 5779e Tensile 4 578 Tensile From the comparison as shown in Table 4 9 Table 4 10 Table 4 11 and Table 4 12 above we can see that the results obtained from the space truss analysis program and STAAD Pro are again approximately the same The analysis of the three examples of space truss problems by using space truss analysis program and STAAD Pro give the results with approximately same values Consequently this validates the space truss analysis program developed and verifies its results CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS 5 1 Conclusions Through the research carried out the space truss analysis program has been developed using MATLAB Space truss problems
24. The use of MATLAB is available for many computer systems including MS windows Linux and Macintosh The MATLAB system consists of five main parts which are i The MATLAB Language ii Desktop Tools and Development Environment ii Graphics iv The MATLAB Mathematical Function Library v The MATLAB External Interfaces API The MATLAB Language is a high level matrix array language with control flow statements functions data structures input output and object oriented programming features It allows the user to program in the small creating quickly throw away programs and program in the large creating large and complicated application specific programs Desktop Tools and Development Environment is a set of tools and facilities that help user in using MATLAB functions and files Many of these tools are graphical user interfaces GUIs Examples of the tools and facilities are the MATLAB desktop and Command Window a command history an editor and debugger and browsers for viewing help the workspace files and the search path Graphics is the MATLAB graphic system MATLAB has extensive facilities for displaying annotating and printing various graphs It can be used for two dimensional and three dimensional data visualization image processing animation 10 and presentation by using high level functions It also allows the user to fully customize the appearance of graphics and build complete graphical user interfaces GUIs
25. and internal forces of members 1 5 Contents of Thesis There are five chapters in this report Chapter 1 is the introduction chapter which contains brief introductory about the research of computer program development for space truss analysis Chapter 2 provides an overview of truss structure including types of truss and its advantages It also covers the overview of software MATLAB and its advantages Alternative methods to analyse a truss are included in this chapter as well In chapter 3 the assumptions used in the research are established The detail finite element formulation for space truss problem is shown Chapter 4 demonstrate the analysis procedure and the application of the space truss analysis program The results obtained are compared with those from software STAAD Pro For chapter 5 it is the conclusion of the research and recommendations for future research CHAPTER 2 LITERATURE REVIEW 2 1 Definition The finite element analysis or finite element method is a numerical method for solving problems of engineering and mathematical physics including structural analysis heat transfer fluid flow etc A First Course in the Finite Element Method Fourth Edition Daryl L Logan 2007 MATLAB is the premier software packages for technical computation data analysis and visualization in education and industry Learning MATLAB The MathWorks Inc 2001 In fact MATLAB is a popular computer software used as a matrix calc
26. anel named Results Node and Element and its child objects in object browser The panel in GUI of the space truss analysis program is shown in Figure 3 9 G Object Browser sj uipanel panel Results Node and Element 9t uicontrol pushbutton2 Exit E uitable table node2 ES uitable table element2 T uicontrol text14 uicontrol pushbuttonll Save Results Figure 3 8 List of a panel and its child objects in object browser 31 P SpaceTruss fig le EG File Edit View Layout Tools Help Dead saeot ITE Nj Data Node and Element De x coordinate y coordinate z coordinate Support Fx kN Fy kN Fz kN Young Modulus Ares eg me kN m 2 m 2 Analyse Results Node and Element Ry kN Rz kN Ux m Uy m Uz m Force in element Stress in element Type of stress KN kN m 2 la 2 L3 4 Save Results lt Tag figurel Position 520 205 1162 595 Axes Panel Figure 3 9 Axes and panel in GUI CHAPTER 4 RESULTS AND ANALYSIS 41 Introduction In this chapter an example of space truss problem is used to demonstrate the steps to model and analysis a space truss structure in the space truss analysis program developed The results obtained from the space truss analysis program are then compared with the results obtained from STAAD Pro in which analysis the same space truss problem
27. cement at nodes in local coordinate system Displacement at nodes in global coordinate system The deflection of an elastic bar element Axial load Strain component Displacement transformation matrix Element stiffness terms in global stiffness matrix xiii LIST OF APPENDICES APPENDIX TITLE PAGE A MATLAB R2009a script Space truss modelling 52 B MATLAB R2009a script Space truss analysis 60 CHAPTER 1 INTRODUCTION 1 1 General A truss is a simple skeletal structure where its individual members are only subjected to tension or compression forces Bending force and moment are explicitly excluded in a truss Triangle is the most important shape and design in a truss due to its structural stability A triangle is the simplest geometric figure that will not change shape when the lengths of the sides are fixed A truss that composed entirely of triangles is known as a simple truss A truss that lies in a single plane is called planar truss Planar trusses are commonly seen in most of the structures such as bridges and roofs On the other hand a space truss is a three dimensional framework of members pinned at the joints The simplest shape of a space frame is a tetrahedron which consists of three triangles meet at six edges A tetrahedron consists of six individual members One of the examples of application of space truss is electricity pylon 1 2 Problem Statement Analysis of a space truss structure always involving many p
28. computation by incorporating the LAPACK and BLAS libraries MATLAB has evolved over a period of years It is now the standard instructional tool for introductory and advanced courses in mathematics engineering and science in university environments On the other hand it is the tool of choice for high productivity research development and analysis in industry 2 4 2 Applications in MATLAB MATLAB is a high performance programming language developed by The MathWorks for technical computing It is a great tool for simulation and data analysis It allows us integrate computation visualization and programming in a convenient way and express the solution in familiar mathematical notation It provides an excellent computational language built in state of the art algorithms for mathematics and excellent visualization using ready made functions Furthermore MATLAB is a modern programming language environment it has sophisticated data structures contains built in editing and debugging tools and supports object oriented programming These factors make MATLAB an excellent tool for teaching and research The most common uses of MATLAB including i Math and computation ii Matrix manipulation iii Algorithm development iv Data acquisition v Plotting of functions and data vi Modelling simulation and prototyping vii Data analysis exploration and visualization viii Application development including graphical user interface building
29. condition of node 2 3 and 4 are all ball and socket If the node is supported by roller or no support at all the external forces acting at the nodes along x direction y direction and z direction are identified and input into the node data table Therefore the external forces at node 1 are Fx Fy and F which are equal to 12 kN 0 kN and 0 KN respectively Other than that the Young s Modulus and area of each element are identified and input into the element data table For this Example 1 the Young s Modulus value is the same for the three elements which is 200e kN m On the other hand the area of element 1 and 3 are both 0 001 m while the area of element 2 is 0 002 m The figure 4 5 below shows the appearance of the interface after input all the important data E SpaceTruss AXD Data Node and Element x coordinate y coordinate z coordinate Support Fx kN Fy kN Fz kN Young Modulus E Area 0 0 SNone 12 0 kN m 2 m 2 0 4 0 Bal and s 1 200000000 1 0000e 03 3 0 0 Bal and s y 2 200000000 0 0020 0 4 0 Bal and s v 3 200000000 Analyse Results Node and Element T Rx KN Ry KN Rz kN Ux m Uy m Uz m Force in element Stress in element Type of stress kN kN m 2 amu gt Insert Clear Exit Save Results Figure 4 5 Input the reguired data for a
30. dles structure set handles table inse handles pushbutton handles pushbutton Executes on butt function pushbutton5 C hObject handle to oo to be defined in a future with handles and user data rt vasible on 7 sisible on 8 visible on on press in pushbutton5 allback hObject eventdata pushbutton5 see GCBO oe eventdata reserved handles structure cla set handles table node set oo o to be defined in a future with handles and user data data handles table element data 1 Executes on button press in pushbuttone function pushbutton6 Callback hObject eventdata pushbutton6 see GCBO to be defined in a future hObject handle to eventdata reserved handles structure data node get handle with handles and user data S table node data data element get handles table element data support data node tic global node n t dof U 4 for i 1 t a stremp support i None b strcmp support i Ball and socket c stremp support i Roller if a M i 0 elseif b 1 M i 1 elseif c 1 M i 2 else end end F zeros 3 t 1 for i 1 t F 3 i 2 data node i 5 F 3 i 1 data node i 6 F 3 i data node i 7 end E zeros n 2 1 A zeros n 2 1 K zeros 3 t 3 t tor i 1 n 2 E i da
31. dles table insert data global node 57 for i 1 15 j Data i 1 if isempty j break else node 2 i 1 str2double Data i 1 3 node 2 i str2double Data i 4 6 end end global n dof n size node 2 N zeros 1 n zeros 1 n 1 for i 1 n k i while dof i 0 dof i p p ptl end while k lt n k k 1 if node i node k N k 1 dof k else gontinue end end end dof i global node_new Z find N 1 s size Z 2 node new node p 0 for i 1 s node new Z i p Il p ptl end global t node_xyz t size node_new 2 for i 1 t node num node new i ode xyz i 1 node num 1 ode xyz i 2 node num 2 ode xyz i 3 node num 3 a Ey 5 end T t 4 Il for i 1 t T il End element n 2 2 set handles table node data node xyz T set handles table element data element 58 set handles table insert visible off set handles pushbutton7 visible off set handles pushbutton8 visible off set handles axesl visible on axes handles axesl een xLimit yLimit zLimit xmin xLimit 1 xmax xLimit 2 ymin yLimit 1 ymax yLimit 2 zmin zLimit 1 zmax zLimit 2 cla hold on for i 1 t plot3 node xyzf i l node xyz i 2 node xyz i 3 Marker o MarkerFac
32. dure 14 a Identify the model of the space truss structure to be analysed including all the coordinates of the joints of the structure b Model the space truss structure in graphical user interfaces GUIs of software MATLAB Inputs all the important values such as Young s Modulus and area for material properties type of support for support condition and any external forces acting on the structure c Run the analysis after all the inputs needed have been input in the software MATLAB The software MATLAB will analysis the problem through the steps in finite element formulation as following i Discretize the problem into elements and nodes ii Determine the stiffness matrix for each element iii Assemble the element member stiffness matrix to form the structure stiffness matrix global stiffness matrix iv Apply the boundary conditions of the problem v Define the loadings vi Solve the equation K Q F to obtain nodal displacements and consequently reaction forces at the supports vii Proceed with post processing stage the computation of stresses and strains d Display all the important results in software MATLAB 3 3 Modelling of Truss Structure In order for the software MATLAB can analysis the truss problem user has to model it and input all the needed input in the software first Graphical user interfaces GUIs in MATLAB is used as the medium for user input values and model the structure 15 3
33. e De F TT where 4 0 0 or Z 0 0 0 4 8 E A 3 19 3 20 The results are combined together to determine the stiffness matrix for a member which relates the member s global force components F to its global displacement Q Substitute equation 3 17 and 3 19 into 3 13 yields k tq3 f x IT to3 f T TNO F K Q F where Ale M Ay Ag 0 0 OP ae pd I ay A 0 Hes 0 0 0 Ax y a E 1 ls 0 0 A 3 21 3 22 3 23 22 2 de A A Se XX Ax AO xh xx dy Ay ha hA Ah Xt OG Wh SKI i AS Lh Ak w XX XX dee h Dyke X 32 dyke a dO d 0 XL x a 3 24 The member stiffness matrix expressed in global coordinates for each member of the space truss can be determined by using equation 3 24 above 3 4 3 Assemble the Global Stiffness Matrix Once the member stiffness matrix for each of the member of the truss is determined the structure stiffness matrix can then be determined by assembling all the member stiffness matrices to represent the entire truss The rows and columns of the member stiffness matrix are designated by the code numbers which are used to identify the global degrees of freedom that occur at each end of the member The structure stiffness matrix will have an order that equal to the highest code number assigned to the truss For example a tetrahedron will have structure stiffness matrix with order 12 x 12 corresponding to total of twelve degrees of freedom wh
34. e E Young modulus of elasticity of the material From equation 3 4 the equivalent spring constant of an elastic bar can be obtained as k 2 4 3 5 When a bar element is subjected to a uniaxial loading the normal strain component is defined as _ du EE 3 6 Substitute equation 3 2 into equation 3 6 will give e 11 a 3 7 19 The strain is constant in an element that has constant cross sectional area and subjected to constant forces at the points By Hooke s law the axial stress in the bar element is defined as a Ee 11 fa 3 8 This axial stress is then related to the axial force gives P 04 4 11 3 9 Equation 3 9 can be used to relate nodal forces with the nodal displacements Be noted that equation 3 9 has a positive value when the element is subjected to tension force and has a negative value when the element is subjected to compression force Nodal forces of an element at node 1 fi and node 2 f2 must be of same magnitude but opposite direction for equilibrium Hence AE q fim p P3 3 10 pn 3 11 Equation 3 10 and 3 11 can be combined to yield 6 20 0 en or k q3 f 3 13 where the element stiffness matrix for the bar element is given as MU 7 3 14 20 Eguation 3 14 shows that the element stiffness matrix for a bar element is symmetric and in order of 2 x 2 corresponding to two degrees of freedom or two nodal displacements in a bar element This
35. eColor b text node_xyz i 1 xmax xmin 50 node_xyz i 2 ymax ymin 50 node_xyz i 3 zmax zmin 50 num2str i Color r FontWeight bold hold on end for i 1 n 2 plot3 node xyz dof 2 i 1 1 node xyz dof 2 i 1 node xyz dof 2 i 1 2 node xyz dof 2 i 2 node xyz dof 2 i 1 3 node xyz dof 2 i 3 hold on text node xyz dof 2 i 1 1 node xyz dof 2 i 1 2 xmax xmin 40 node_xyz dof 2 i 1 2 node_xyz dof 2 i 2 2 ymax ymin 40 node_xyz dof 2 i 1 3 node_xyz dof 2 i 3 2 zmax zmin 40 num2str i EdgeColor k hold on end o Executes on button press in pushbutton8 function pushbutton8 Callback hObject eventdata handles hObject handle to pushbutton8 see GCBO ventdata reserved to be defined in a future version of MATLAB andles structure with handles and user data see GUIDATA handles table insert visible off handles pushbutton7 visible off handles pushbutton8 visible off Executes when entered data in editable cell s in table node function table node CellEditCallback hObject eventdata handles hObject handle to table node see GCBO eventdata structure with the following fields see UITABLE Indices row and column indices of the cell s edited PreviousData previous data for the cell s edited EditData string s entered by the user oe oe oo
36. ere three degrees of freedom for each of the four nodes When the member stiffness matrices are assembled each element in the matrix will then placed in its same row and column designation in the structure stiffness matrix 23 3 4 4 Apply the Boundary Conditions Once the structure stiffness matrix is obtained the boundary conditions will be applied where there are supports or external forces at the nodes The external forces at the nodes can be directly assigned to the elements in the external force matrix with corresponding degrees of freedom and rows code numbers On the other hand the supports at the nodes will restrain the nodes from moving and therefore do not have displacement Direction of movement restraint is governed by the type and orientation of the support These displacements which are zero value are assigned to the elements in the displacement matrix with corresponding degrees of freedom and rows code numbers An equation relating unknown displacements to known external forces can then be formed Kreduce Keu Fi 3 25 where Kreduce reduced structure stiffness matrix Qu unknown displacements matrix Fx known external forces matrix 3 4 5 Solve the Equations The values of unknown displacements can be determined by solving the equation 3 25 Subsequently the components of reaction forces can be determined by substituting the values of the unknown displacements obtained above into the structure stif
37. external forces acting in any directions Besides that the space truss analysis program is also not able to analysis space truss structure with incline support There are only two types of horizontal support conditions available in this space truss analysis program which are ball and socket constrained from movement in the x direction y direction and z direction and roller constrained from movement in the z direction In order to improve the usability and flexibility of the program the incline support can be included in the program in the future to make the program more applicable to various types of space truss problems 51 REFERENCES Airil Yasreen Mohd Yassin and Ahmad Kueh Beng Hong 2008 Differential Variational and Finite Element Formulations for Structural Beams Daryl L Logan 2007 A First Course in the Finite Element Method 6th edition India Rahul Print O Pack Delhi 20 Getting Started with MATLAB Version 7 The MathWorks Inc J N Reddy 1993 An Introduction to the Finite Element Method 2nd edition McGraw Hill Inc Patrick Marchand and O Thomas Holland 2003 Graphics and GUIs with MATLAB 3rd edition Chapman amp Hall CRC Peter I Kattan 2006 MATLAB Guide to Finite Elements An Interactive Approach 2nd edition New York Springer Verlag Berlin Heidelberg R C Hibbeler 2005 Mechanics of Materials 6th edition Prentice Hall Inc 52 APPENDIX A MATLAB R2
38. fness equation as shown below K Q F 3 26 where K structure stiffness matrix Q global displacements matrix F global force components matrix 24 3 4 6 Post Processing Stage The final computational step in finite element analysis of a truss is to determine the internal force strain and stress in each element of the truss by making use of the global displacements obtained in the steps previously The internal force or member force can be determined using eguation 3 21 k ITKO 63 anten SS 1 TA 0 0 0 E ae 0 0 A Ay A Qix Qiy Q z Q2 Q y Q22 amp 3 27 The strain in the element of the truss can be determined by substituting equation 3 17 into 3 7 e 1 1 fq 1 1 ITKQJ Ab y a 0 0 0 0 0 0 Ae A Qu Qiy Qi Qu Qa oz 3 28 11 The stress in the element of the truss can be determined by substituting equation 3 28 into 3 8 o Es E A A 0 0 0 11 7 7 zl lle 0 0 A Ay Qix Qiy Qi Qox Q5 Qny6 3 29 25 3 5 MATLAB Graphical User Interface Development Environment GUIDE The graphical user interface development environment GUIDE function in MATLAB provides a set of tools for creating graphical user interfaces GUIs These tools simplify the process of laying out and programming GUIs When a GUI is opened in GUIDE it is displayed in the Layout Editor which is the control panel for all of the GUIDE tools The
39. following figure shows the Layout Editor with a blank GUI template E SpaceTruss fig bd File Edit View Layout T ools Help moc sBs hd Pes P gt Current Point Ng 434 Position 520 205 1162 595 Component Palette Figure Layout area Figure 3 5 Layout Editor with a blank GUI in GUIDE You can lay out your GUI by dragging components such as panels push buttons pop up menus or axes from the component palette at the left side of the Layout Editor into the layout area Some of the important components used in developing the graphical user interfaces in this research will be discussed in this section 26 3 5 1 Figure Figure objects are the individual windows on the screen in which the MATLAB software displays graphical output In this research the figure is as shown in Figure 3 6 3 5 2 Static Text Static text boxes display lines of text Static text 1s typically used to label other controls provide directions to the user or indicate values associated with a slider Users cannot change static text interactively Static text controls do not activate callback routines when clicked Example of static text in GUI of the space truss analysis program is shown in Figure 3 6 3 5 3 Edit Text Editable text fields enable users to enter or modify text values Editable text is often used when text is wanted as input If Max Min gt 1 then multiple lines are allowed For multi line edit boxes a vertical sc
40. he property values of a table can be set and query using the set and get functions Examples of code used in this research to obtain data from tables and set the data in tables are as following Obtain data from table node and table element ata node get handles table node data ata element get handles table element data QQ Ne Set the data in table node and table element set handles table node data data node set handles table element data data element 28 3 5 5 Push Button The push button is perhaps the most prevalent MATLAB user interface control uicontrol style It is used primarily to indicate that a desired action should immediately take place When the user clicks on a push button it invokes an event immediately The event is dictated by the code that is programmed in the push button s callback function The Cancel button in Figure 3 7 when clicked by the user will close the table and go back to the screen before Example of code for the Cancel button is shown below o Executes on button press in pushbutton8 function pushbutton8 Callback hObject eventdata handles hObject handle to pushbutton8 see GCBO eventdata reserved to be defined in a future version of MATLAB handles structure with handles and user data see GUIDATA set handles table insert visible off set handles pushbutton7 visible off set handles
41. lity for the award of the degree of Bachelor Degree of Civil Engineering Signature H Mm DR AIRIL YASREEN MOHD YASSIN Name of Superior se EN BEG Date PTUS COMPUTER PROGRAM DEVELOPMENT FOR SPACE TRUSS ANALYSIS CHANG CHEE BOON A report submitted in partial fulfillment of the reguirements for the award of the degree of Bachelor of Civil Engineering Faculty of Civil Engineering Universiti Teknologi Malaysia MAY 2011 I declare that this thesis entitled Computer Program Development for Space Truss Analysis is the result of my own research except as cited in the references The thesis has not been accepted for any degree and is not concurrently submitted in candidature of any other degree Signature Wunde SN Name CHANG CHEE BOON Date 5 MAY 2011 Dedicated to my beloved family and friends ACKNOWLEDGEMENT First and foremost I would like to express profound gratitude to my thesis supervisor Dr Airil Yasreen Mohd Yassin for his support guidance and advice throughout my final year project thesis I also would like to take this opportunity to acknowledge the technical supports and valuable assistance from the seniors in Steel Technology Centre STC Thanks to all my friends colleagues and who have involved directly or indirectly during this study Finally thanks to my beloved family for their care love and support during my study Thank you ABSTRACT Analysis of a
42. lized technology Toolboxes are comprehensive collections of MATLAB functions M files that extend the MATLAB environment to solve particular classes of problem Examples of areas in which toolboxes are available including control systems neural networks simulation etc ii Matrix optimized MATLAB is optimized for matrices Thus if a problem can be formulated with a matrix solution MATLAB executes substantially faster than a similar program in a high level language iv Credit MATLAB is well developed and well tested with its long history It makes use of highly respected algorithms and hence you can be confident about your results In general MATLAB is a useful tool for vector and matrix manipulations Since the majority of the engineering systems are represented by matrix and vector equations we can relieve our workload to a significant extent by using MATLAB The finite element method which is used in this research is a well defined candidate for which MATLAB can be very useful as a solution tool Matrix and vector manipulations are essential parts in the method CHAPTER 3 METHODOLOGY 3 1 Assumption used in the Research There are various types of trusses It is ideal to develop a software that can be used for analysis of any type of truss However this research is only focus in space truss analysis The following are the assumptions and limitations in this research il iii iv The program is limited to the
43. nalysis The space truss analysis program is now ready to analysis the space truss problem as in Example 1 Press on the Analyse button below the element data table the space truss problem will be analysed based on finite element method and the results will be displayed in the node result table and element result table as shown in Figure 4 6 below E SpaceTruss AXD Data Node and Element x coordinate y coordinate z coordinate Support Fx kN Fy kN Fz kN YoungModulus E Area 0 0 SNone 12 0 kN m 2 m 2 0 4 0 Bal and s 1 200000000 1 0000e 03 3 0 0 Bal and s y z z i2 200000000 0 4 0 Bal and s v 3 200000000 Results Node and Element Rx kN Ry kN Rz kN Ux m Uy m Uz m Force in element Stress in element Type of stress il L E 0 0015 1 3616e 19 5 2506e 04 kN kN m 2 2 7 8416e 16 8 0000 10 0000 0 0 0 1 12 8062 1 2806e 04 Compressive 3 12 0000 1 4282e 15 20 0000 0 0 0 2 23 3238 1 1662e 04 Tensile 4 7 8416e 16 8 0000 10 0000 0 0 0 3 12 8062 1 2806e 04 Compressive Insert Clear Time taken 0 45007 s Save Results Exit Figure 4 6 Analysis and display of the results 4 3 Comparison of Results The same space truss problem as in Example 1 is modelled and analysed using engineering software STAAD Pro The results obtained are compared with the results obtained previ
44. nternal Stress Type of Stress for i 1 n 2 fprintf Result 5 0f 7 4e 7 4e S n i result_element i 1 result_element i 2 result_element i 3 end fprintf Result n n Time taken for analysis 8 6f s n tused fclose Result end 60 APPENDIX B MATLAB R2009a script Space truss analysis To Calculate function y xl nodel 1 yl nodel 2 zl nodel 3 x2 node2 1 y2 node2 2 z2 node2 3 y sqrt x2 end Element Length ElementLength nodel node2 y2 yl y2 yl z2 z1 z2 z1 To Calculate Element Angle ElementAngle nodel node2 L x1 nodel 1 yl nodel 2 zl nodel 3 x2 node2 1 y2 node2 2 z2 node2 3 thetax acos x2 x1 L 180 pi thetay acos y2 y1 L 180 pi thetaz acos z2 z1 L 180 pi y thetax thetay thetaz end To Determine Element Stiffness Matrix function y ElementStiffness E A L theta x theta 1 pi 180 y theta 2 pi 180 z theta 3 pi 180 Cx cos x Cy cos y Cz cos z w Cx Cx Cx Cy Cx Cz Cy Cx Cy Cy Cy Cz Cz Cx Cz Cy Cz Cz y E A L w w w w end STo Assemble Matrix function y Assemble K k 1i j K 3 i 2 3 i 2 K 3 i 2 3 i 2 k 1 1 K 3 i 2 3 i 1 K 3 1i 2 3 1i 1 k
45. on MATLAB application by using low level functions The MATLAB Mathematical Function Library is an enormous collection of computationally efficient and robust algorithms and functions ranging from elementary functions sine cosine tangent cotangent etc to specialized functions matrix inverse matrix eigenvalues Bessel functions fast Fourier transforms etc commonly used in scientific and engineering practice The MATLAB External Interfaces API is a library that allows the user to write C and FORTRAN programs that interact with MATLAB It includes facilities for calling routines from MATLAB dynamic linking calling MATLAB for computing and processing reading and writing M files etc 2 4 3 Advantages using MATLAB Nowadays there are many other programming softwares that having the ability to analysis structure by using finite element method such as Visual Basic C FORTRAN etc However MATLAB is chosen for this study because of some advantages i Dimensionless One of the most important features of MATLAB compared to other high performance programming languages is that MATLAB does not require dimensioning which allows the user to perform matrix computations efficiently This is due to MATLAB is an interactive system whose basic data element is an array that does not require dimensioning 11 ii Toolboxes MATLAB is extensible with toolboxes in various application areas which allow the user to learn and apply specia
46. ously by using space truss analysis program developed The comparisons of the two results are shown in Table 4 1 Table 4 2 Table 4 3 and Table 4 4 below Table 4 1 Displacements at nodes Example 1 39 Node Displacements Number Space Truss Analysis Program m STAAD Pro m of node Ux U U U U U 1 0 0015 1 3616e 5 2506e 1 536e 0 000 0 525e 2 0 0 0 0 000 0 000 0 000 3 0 0 0 0 000 0 000 0 000 4 0 0 0 0 000 0 000 0 000 Table 4 1 above shows the comparison of the nodes displacements obtained from the two analysis program and software We can see that the results from both the program and the software are almost the same The displacement at node 1 in y direction Uy is 1 3616e m in space truss analysis program but it is 0 m in engineering software STAAD Pro This may be due to the value is too small that STAAD Pro treat it as zero Table 4 2 Reactions at supports Example 1 Reactions Number Space Truss Analysis Program kN STAAD Pro kN of node Rx Ry R Rx Ry R 2 7 8416e 6 8 0000 10 0000 0 000 8 000 10 000 3 12 0000 1 4282e 20 0000 12 000 0 000 20 000 4 7 8416e 6 8 0000 10 0000 0 000 8 000 10 000 40 Comparison of reactions as obtained from space truss analysis program and STAAD Pro is shown in Table 4 2 above Most of the results obtained from space truss analysis p
47. proximated by a model that consists of piecewise continuous simple solutions Finite Element Method For Structural Analysis Redzuan Abdullah 2010 The modern development of the finite element method began in the 1940s in the field of structural engineering with the work by Hrennikoff in 1941 and MCHenry in 1943 who used a lattice of line one dimensional elements bars and beams for the solution of stresses in continuous solids Levy developed the flexibility or force method in 1947 The first treatment of two dimensional elements was by Turner et al in 1956 They derived stiffness matrices for truss elements beam elements and two dimensional triangular and rectangular elements in plane stress They also outlined the procedure commonly known as the direct stiffness method for obtaining the total structure stiffness matrix The work of Turner et al prompted further development of finite element stiffness equations expressed in matrix notation in pace with the development of the high speed digital computer in the early 1950s Clough introduced the phrase finite element in 1960 Extension of the finite element method to three dimensional problems with the development of a tetrahedral stiffness matrix was done by Martin in 1961 by Gallagher et al in 1962 and by Melosh in 1963 From the early 1950s to the present enormous advances have been made in the application of the finite element method to solve complicated engineering problems A Fi
48. rocedures and calculations which are lengthy and troublesome especially when involving a structure that has many members it may take time to be completed The solution for this problem is to develop a computer program which can do all the calculations faster consistently and accurately Although there are guite a number of existing commercial engineering software nowadays that can perform the same task but unfortunately most of them are expensive such as STAAD Pro LUSAS and etc It is not affordable for small companies or individual to get the license of these softwares 1 3 Objectives of Research The main objective in this study is to develop a cheaper local developed space truss analysis program by using MATLAB The following objectives are to be achieved in this study i To create a space truss analysis program by applying finite element method using MATLAB R2009b software ii To validate the result from space truss analysis program developed by comparing it with result from existing engineering software STAAD Pro 1 4 Scope of Research This study is mainly focus on the analysis of space truss structure MATLAB R2009b is used to develop the computer program for space truss analysis The analysis will consider space truss comprised of compression and tension members only and no shear and bending in the structure The analysis will be including calculation of all important values such as resultant displacements reactions
49. rogram are same as the results obtained from STAAD Pro except that some reactions which are very small value in space truss analysis program are shown as zero in STAAD Pro Table 4 3 Force in elements Example 1 Force In Element Number of Space Truss Analysis Program Element P j STAAD Pro kN KN 1 12 8062 12 806 2 23 3238 23 324 3 12 8062 12 806 From Table 4 3 we can see the comparison of forces in the elements of the space truss structure The results of forces in elements are the same when analyse the Example 1 by using space truss analysis program and STAAD Pro Table 4 4 Stress in elements Example 1 Stress In Element Number of Element Space Truss un Program STAAD Pro kN n kN m 1 1 2806e Compressive 12 806e Compressive 2 1 1662e Tensile 11 662 Tensile 3 1 2806e Compressive 12 806 Compressive 41 Table 4 4 above shows the comparison of the elements internal stresses obtained from the two analysis program and software The comparison shows that the analysis results from the space truss analysis program are identical with the results from the engineering software STAAD Pro The positive value of stress means the stress is tensile stress while negative value means the stress is compressive stress In order to check the functionality and performance of the space truss analysis program two more example
50. rollbar enables scrolling as do the arrow keys To obtain the string a user types in an edit box get the string property Examples of code used in this research to obtain text or string from edit boxes are as following Obtain data from editl edit2 edit3 edit4 edit5 and edite xLimit 1 str2double get handles editl string yLimit 1 str2double get handles edit2 string zLimit 1 str2double get handles edit3 string xLimit 2 str2double get handles edit4 string yLimit 2 str2double get handles edit5 string zLimit 2 str2double get handles edit string 27 EY SpaceTruss fig 5 File Edit View Layout Tools Help GET IEL TL BASEN 1 0 Mk E kAL Coordinates Range of x axis from Range of y axis from Range of z axis from Tag figurel Current Point 338535 Position 520 205 1162 595 Figure Static Text Edit Text Figure 3 6 Figure static text and edit text in GUI 3 5 4 Table Tables contain rows of numbers text strings and choices grouped by columns Rows and columns can be named or numbered Entire tables or selected columns can be made user editable Data in each column must of the same type of data The number of rows and columns automatically adjust to reflect the size of the data matrix the table displays Tables are useful in presenting data which is in array structure or matrix T
51. rom Range of y axis from Range of z axis from Figure 4 2 Input the range of axes After input the axes range press on the Next button The axes will then be generated and displayed on the next interface To model the space truss structure press on the Insert button on the bottom left of the interface A table will pop out at the centre of the interface for inputting coordinates of the start node and end node for each element of the space truss structure For this Example 1 the x coordinate y coordinate z coordinate of start node and end node for element 1 are 0 0 5 and 0 4 0 The x coordinate y coordinate z coordinate of start node and end node for element 2 are 0 0 5 and 3 0 0 Lastly the x coordinate y coordinate z coordinate of start node and end node for element 3 are 0 0 5 and 0 4 0 Example of the inputs is shown in Figure 4 3 below 35 E Spacetruse Sex AAD x Data Node and Element x coordinate y coordinate z coordinate Support Fx kN Fy kN Fz kN Young Modulus E Area v kN m 2 m 2 v 1 2 First Node y First Node z End Node x End Node y End Node z 3 5 0 0 4 5 3 0 0 5 0 4 o Analyse element Stress in element Type of stress N kN m 2 OK Cancel Insert Clear Save Results Exit Figure 4 3 Input the coordinates of nodes of the elements
52. rst Course in the Finite Element Method Fourth Edition Daryl L Logan 2007 The general procedures of the finite element method are consisting of three stages which are pre processing solution and post processing 1 Pre processing e Identify the geometric domain of the problem e Discretize the problem into elements e Identify the type of element to be used e Apply the boundary conditions physical constraints e Define the loadings 2 Solution e Assemble the stiffness matrix K e Solve the governing algebraic equations KQ F to obtain the unknown values of the primary field variables displacement Q and reaction forces F for stress analysis problem 3 Post processing e The computed values are then used to compute derived variables elements stresses and strains by back substitution 24 MATLAB 2 4 1 History By referring to Getting Started with MATLAB The MathWorks Inc 2005 MATLAB was originally written to provide easy access to matrix software developed by the LINPACK a software library for performing numerical linear algebra on digital computers and EISPACK a software library for numerical computation of eigenvalues and eigenvectors of matrices projects In 20th century a newer set of latest set of libraries were rewrote for matrix manipulation known as LAPACK a software library for numerical linear algebra to supersede LINPACK and EISPACK Today MATLAB engines embed the state of the art in software for matrix
53. s of space truss problems are modelled and analysed by using the space truss analysis program and then the answers are compared with those obtained from engineering software STAAD Pro Example 2 Figure 4 7 Space truss for Example 2 Data available Young s modulus E 200 GPa 200e kN m Cross sectional area A 0 003 m 42 Determine i The x direction y direction and z direction displacements at nodes li The reactions at nodes 2 3 4 and 5 iii The force in each element lv The stress in each element Figure 4 8 below shows the appearance of the interface of space truss analysis program after the space truss problem as in Example 2 is modelled and analysed using the program E B SpaceTruss AND Data Node and Element x coordinate y coordinate 2 coordinate Support Fx kN Fy kN Fz kN Young Modulus E Ares 1 0 0 SNone 18 gt kN m 2 m 2 En 0 E 0 Balands y 1 200000000 0 0030 ER 3 0 0 Bal and s y 2 200000000 0 0030 4 0 3 0 Bal and s v 3 200000000 0 0030 Ls 4 0 0 Bal and s v a 200000000 BIENES Results Node and Element Rx kN Ry kN Rz kN Ux m Uy m Uz m Force in element Stress in element Type of stress am E 2 3509e 04 3 6713e 04 2 5875e 07 kN kN m 2 2 1 1887e 15 9 9883 16 6471 0 0 1 19 4137 6 4712e 03 Compressive 3 6 4151 7 6349e 16 10 6919 0 0 0 2 12 4688 4
54. ta element i 1 A i data element i 2 i ElementLength node 2 i 1 node 2 i het handles version of MATLAB see GUIDATA handles version of MATLAB see GUIDATA handles version of MATLAB see GUIDATA theta i ElementAngle node 2 i l node 2 i L i k i ElementStiffness E i A i L i theta i K Assemble K k i dof 2 i 1 dof 2 i end Z find M 1 S size Z 2 kl K fl F p 0 for i 1 s kl 3 Z i 2 p kl 3 2 i 2 p Il kl 3 Z i 2 p Il k1 3 Z i 2 p Il kl 3 Z i 2 p Il k1 3 2 i 2 p TJ E1 3 Z i 2 p E1 3 Z i 2 p 1 F1 3 2 1 2 p Il p p 3 end Z find M 2 S size Z 2 p 0 for i 1 s kl 3 Z i p Il kti 3 4 1 p 3 17 1 3 Z i p p ptl end ul k1 f1 U zeros 3 t 1 Z find M 0 S size Z 2 for i 1 s U 3 A2 x 2 ul 9 r 2 U 3 Z i 1 ul 3 i 1 U 3 Z i ul 3 i End Z find M 2 S size 2 2 for i 1 s U 3 Z i 2 ul 3 i 2 U 3 Zz 2 1 vUl 3 1 1 End F K U result_node t 6 Z find M 0 S size Z 2 for i 1 s result node Z i 1 result node Z i 2 result node Z i 3 end 56 Z find M 1 s size Z 2 for i 138 result node Z i 1 result node Z i 2 result node Z i
55. thod of sections finite element method and etc The method of joints uses the concept of equilibrium of joints and has become the basis for all trusses analysis directed towards finding the unknown forces in the truss structure A truss is considered to be composed of a series of members and joints Member forces are found by considering all the joints are in a state of equilibrium For a plane truss the two independent equations of statics are used simultaneously to find the member forces 3 F 0 and F 0 However a joint will move if there was a net force acting to the joint The method of sections is a method in where firstly cut the truss structure into sections then replace the removed section with unknown member forces acting in the direction of the cut member The forces in the members are then computed by summing the unknown forces by using equilibrium equations X F 20 FE 0 and XY M 0 Since there are only three equilibrium equations the truss section cut should be chosen properly and located at where there are only three unknown member forces 2 3 Finite Element Method The finite element method FEM or sometimes referred to as finite element analysis FEA is a general numerical technique for approximating the behaviour if continua by assembly of small parts elements It is much easier to analyze each element separately than the whole structure due to its simple geometry In essence a complicated solution is ap
56. tion y direction and z direction of the node as shown in Figure 3 3 17 3 4 4 Determine the Element Stiffness Matrix After discretize the truss into bar elements a shape function is needed to be assumed to represent the physical behaviour of an element A shape function is an approximate continuous function expressed in terms of two nodal variables u and up Figure 3 4 below shows an elastic bar element with length L which is affixed an uniaxial coordinate system x with its origin placed at the left hand Figure 3 4 A bar element in local direction The axial displacement at any position along the length of the element is denoted as u x The nodal displacement at x 0 node 1 and x L node 2 are denoted as u 0 q and u L q respectively The axial displacement function can be expressed in term of x by assuming linear interpolation for displacement u x ag a4x 3 1 From the boundary conditions u 0 g and u L g the shape functions can be determined as follow u 0 ag ta 0 q gives Aj q u L ag a L q gt q a L q2 gives a dt 18 Then une nt E gt u x 1 q 5 q2 3 2 IN No P where Nj 21 5 M N q 3 3 By recalling back from the elementary strength of materials the deflection 5 of an elastic bar of length L with uniform cross sectional area A when subjected to an axial load P is given by c DE TF 3 4 wher
57. truktur kekuda ruang Laporan ini menerangkan tentang pembangunan perisian untuk menganalisis kekuda ruang dengan menggunakan MATLAB Kaedah analisis yang digunakan dalam kajian ini adalah kaedah unsur terhingga Langkah langkah dan formulasi tentang kaedah unsur terhingga yang terlibat dalam menganalisis suatu kekuda ruang akan dibincangkan secara terperinci dalam laporan ini Kaedah permodelan grafik dibangunkan dengan menggunakan fungsi Graphical User Interface Development Environment dalam MATLAB untuk memudahkan pengguna dalam menggunakan perisian tersebut Sebuah manual penggunaan perisian analisis kekuda ruang dengan contoh adalah termasuk dalam laporan ini juga Keputusan yang dihasilkan daripada perisian analisis kekuda ruang tersebut dibandingkan dengan keputusan yang diperolehi daripada perisian yang sedia ada STAAD Pro untuk membuktikan ketepatannya CHAPTER TABLE OF CONTENTS TITLE DECLARATION DEDICATION ACKNOWLEDGEMENT ABSTRACT ABSTRAK TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS LIST OF APPENDICES INTRODUCTION 1 1 General 1 2 Problem Statement 1 3 Objectives of Research 1 4 Scope of Research 1 5 Contents of Thesis LITERATURE REVIEW 2 1 Definition 2 2 Analysis of Truss 2 3 Finite Element Method 2 4 MATLAB PAGE ii iii iv vi vii xi xii xiii wW Q N N oO t RR vii 2 4 1 History 2 4 2 Applications in MATLAB 2 4 3 Advantages using MATLAB
58. ulator It is used extensively in doing finite element analysis The name MATLAB actually comes from the combination of the first three letters of matrix and laboratory MATLAB Guide to Finite Elements An Interactive Approach Second Edition Peter I Kattan 2007 In conclusion the main idea of this research is to create a space truss analysis program by applying finite element method and using programming software MATLAB 2 2 Analysis of Truss Trusses are made up of short thin straight members interconnected at joints to form triangulated patterns The joints in the trusses can only transmit forces from one member to another member but not the moment For analysis purpose external forces and reactions from supports are considered to act only at the nodes This result in there is only axial force in the member of the truss This axial force remains constant along the length of the member and must be of the same magnitude but opposite in directions at the two ends of the member The figure 2 1 below shows some common type of truss Pratt truss King post truss Town s lattice truss Figure 2 1 Types of truss Planar truss or plane truss is a truss where all the members and nodes are lying within a two dimensional plane Space truss is a truss that having members and nodes extending into three dimensions In structural analysis there are several methods that can be used to analysis truss which are method of joints me

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