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  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 \documentclass[a4paper]{article} \usepackage{graphicx} \title{Ocframe Manual \\ {\large Structural Analysis functions of the Mechanics package}} \author{Johan Beke} \begin{document} \maketitle \tableofcontents % for F in *.m; do octave -q --eval "get_help_text_from_file(\"$F\")"; done > output.texi % for F in *.m; do octave -q --eval "disp(get_help_text_from_file(\"$F\"))"; done > ref.texi ; makeinfo --plaintext ref.texi > ref.txt \section{Introduction} \begin{sloppypar} The structural analysis functions of the \emph{Mechanics} packages where written during a FEM course. The following posibilities are in the package: \begin{itemize} \item The analysis of 2D frames with rigid connections with the function {\emph{SolveFrame}}. The solutions for the reaction forces, displacements and member end forces are given. \item Solution of multiple load cases at once with the function \emph{SolveFrameCases} \item A plot of the frame, with nodal displacements if needed, with the function \emph{PlotFrame}. The nodes and members are numbered. \item Calculation the member internal forces for each member with the function \emph{MSNForces} \item Plot of the internal member forces diagram with the function \emph{PlotDiagrams} \end{itemize} \end{sloppypar} \section{Conventions} \subsection{Units} \begin{sloppypar} The user can use any units as long as they are consistent. So if the forces are provided in kN, kN/m and kNm, the geometry and member properties must be in similar units (m, m$^2$, m$^4$ and kN/m$^2$). \end{sloppypar} \subsection{Global and local axis} \begin{sloppypar} For the nodes and the members, care must be taken for the axes. The following images show the used coordinate systems (figure \ref{fig:axis.png}) and the conventions for the member forces (figure \ref{fig:dist.png} and \ref{fig:point.png}). The local axes are always from the near node to the far node. \end{sloppypar} \begin{figure}[h] \includegraphics[width=0.50\linewidth]{axes.png} \caption{Local and global axis convention} \label{fig:axis.png} \end{figure} \begin{figure}[tb] \includegraphics[width=0.50\linewidth]{dist.png} \caption{Conventions for a distributed load on a member} \label{fig:dist.png} \end{figure} \begin{figure}[tb] \includegraphics[width=0.50\linewidth]{point.png} \caption{Conventions for a point load on a member} \label{fig:point.png} \end{figure} \begin{figure}[tb] \includegraphics[width=0.50\linewidth]{sign_conv.png} \caption{Sign conventions for internal forces} \label{fig:sign_conv.png} \end{figure} \newpage \section{Example} \begin{sloppypar} An example will clarify the usage of the different functions. \end{sloppypar} \subsection{Forces and geometry} \begin{figure}[h] \includegraphics[width=0.75\linewidth]{example5_7.png} \caption{Example frame} \label{fig:example_frame} \end{figure} \begin{sloppypar} An example frame, which was taken from the book Matrix Structural Analysis, is shown in figure \ref{fig:example_frame}. The following code snippet is used to enter the geometry: \end{sloppypar} \begin{verbatim} joints=[0,0,1,1,1; 7.416,3,0,0,0; 8+7.416,3,1,1,1]; # first cells of each row are the x and y coordinates # next cells are the x, y and z constraints. # node 1 and 3 are fully fixed, node 2 is free # member data E = 210.0e3; # N/mm^2 = MPa A = 6000;# mm^2 I = 200.0e6;# mm^4 # convert units to kN and m E = E*10^3; A = A*(10^-3)^2; I = I*(10^-3)^4, #connectivity data members=[1,2,E,I,A; 2,3,E,I,A]; \end{verbatim} \subsection{Loads} \begin{sloppypar} The following code snippet is used to enter the loads: \end{sloppypar} \begin{verbatim} # point load on node 2 # Fx = 18.75 kN # Fy = -46.35 kN # Mz = 0 kNm nodeloads=[2, 18.75,-46.35, 0.0]; loc = 1; glob = 0; # distributed load on member 2 # Fx = 0 kN/m # Fy = -4 kN/m # same for the end of the load # a = b = 0 m load on full span # local load dist=[2,0,-4.0,0,-4.0,0.0,0.0,loc]; #no point loads on members point=[]; \end{verbatim} \subsection{Solutions} \begin{sloppypar} The following code snippet is used to find the basic solution: \end{sloppypar} \begin{verbatim} [P,D,MemF]=SolveFrame(joints,members,nodeloads,dist,point); \end{verbatim} \begin{sloppypar} The basic solution are the reactions, the displacements and the member end forces: \end{sloppypar} \begin{verbatim} P = 130.497 55.677 13.374 NaN NaN NaN -149.247 22.673 -45.356 D = 0.0000000 0.0000000 0.0000000 0.0009476 -0.0047441 -0.0005088 0.0000000 0.0000000 0.0000000 MemF = 141.8530 2.6758 13.3742 -141.8530 -2.6758 8.0315 149.2473 9.3266 -8.0315 -149.2473 22.6734 -45.3557 \end{verbatim} \begin{sloppypar} Each row of the reaction matrix (matrix P in this case) corresponds to the node. (First row to first node, etc.). The columns are R$_x$, R$_y$ and M$_z$. For node 1 the reactions are thus: R$_x$ = 130.497 kN, R$_y$ = 55.677 kN and M$_z$ = 13.374 kNm. In case of a free component without reactions, the value is represented by NaN.\\ The same convention holds for the displacement matrix (matrix D in this case). For node 2 the displacements are thus: x = 0.0009476 m, y = -0.0047441 m and rotation = -0.0005088 rad.\\ A similar principle holds for the member-end-forces. Each row corresponds to the element. The columns are: F$_x$, F$_y$, M$_z$, F$_x$, F$_y$ and M$_z$ where the first three components are for the first node and the last three components are for the last node. \end{sloppypar} \newpage \section{Function reference} \begin{verbatim} -- Function File: [X, M, S, N] = MSNForces (JOINTS, MEMBERS, DIST, POINT, MEMF, MEMBERNUM, DIVISIONS) This function returns the internal forces of a member for each position x. The member is divided in 20 subelements if the argument is not given. The used sign convention is displayed in the help file. Input parameters are similar as with SolveFrame and PlotFrame with extra arguments: membernum = Number of the member to calculate divisions = Number of divisions for the member -- Function File: ocframe_ex1 () Example of a planar frame. -- Function File: ocframe_ex2 () Example of a beam. -- Function File: ocframe_ex3 () Example of a planar frame. -- Function File: ocframe_exLC () Example of a beam with generation of eurocode ULS load cases -- Function File: ocframe_railwaybridge () Example taken from a real railwaybridge. -- Function File: ocframe_tests () Various tests for the entire package. Test 1, 2 & 3 are simple beams (tested for reactions and internal forces) Test 4 & 5 are frames (tested for reactions) -- Function File: PlotDiagrams (JOINTS, MEMBERS, DIST, POINT, MEMF, DIAGRAM, DIVISIONS, SCALE) This function plots the internal forces for all members. The force to be plotted can be selected with DIAGRAM which will be "M", "S" or "N" for the moment, shear or normal forces. Input parameters are similar as with SolveFrame and PlotFrame. -- Function File: PlotFrame (JOINTS, MEMBERS, D, FACTOR) Plots a 2D frame (with displacements if needed) using the following input parameters: joints = [x , y, constraints ; ...] constraints=[x , y, rotation] free=0, supported=1 members = [nodeN, nodeF, E, I, A; ...] Optional arguments: D = [x,y,rotation;...] Displacements as returned by SolveFrame factor= Scaling factor for the discplacements (default: 10) -- Function File: [RESULTS] = SolveFrameCases (JOINTS, MEMBERS, LOADCASES) Solves a 2D frame with the matrix displacement method for the following input parameters: joints = [x , y, constraints ; ...] constraints=[x , y, rotation] free=0, supported=1 members = [nodeN, nodeF, E, I, A; ...] loadcases is a struct array with for each loadcase the fields - nodeloads = [node, Fx, Fy, Mz; ...] - dist = [membernum,FxN,FyN,FxF,FyF,a,b,local ; ...] - point = [membernum,Fx,Fy,a,local; ...] input is as for the function SolveFrame. Output is a struct array with the fields: Displacements, Reactions and MemF (output formated as for the function SolveFrame.) -- Function File: [REACTIONS, DISPLACEMENTS, MEMF] = SolveFrame (JOINTS, MEMBERS, NODELOADS, DIST, POINT) Solves a 2D frame with the matrix displacement method for the following input parameters: joints = [x , y, constraints ; ...] constraints=[x , y, rotation] free=0, supported=1 members = [nodeN, nodeF, E, I, A; ...] nodeloads = [node, Fx, Fy, Mz; ...] loads on members: dist = [membernum,FxN,FyN,FxF,FyF,a,b,local ; ...] for distributed loads where FxN and FyN are the loads on distance a from the near node (same with far node and distance b) local=1 if loads are on local axis, 0 if global point = [membernum,Fx,Fy,a,local; ...] where Fx and Fy are the loads on distance a from the node near local=1 if loads are on local axis, 0 if global Output is formated as follows (rownumber corresponds to node or member number): Reactions = [Fx,Fy,Mz;...] where NaN if it was a non supported dof Displacements = [x,y,rotation;...] MemF = [FxN, FyN, MzN, FxF, FyF, MzF; ...] \end{verbatim} \end{document}