{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Maple Input" -1 256 "Courier" 0 0 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 12 0 0 255 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 12 0 0 255 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 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1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 1 12 0 0 255 1 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "N ormal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 4" -1 20 1 {CSTYLE "" -1 -1 "T imes" 1 10 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 24 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Title" -1 258 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 } {PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 24 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 24 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 24 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 264 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 265 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 266 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Heading 3" -1 267 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 268 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2 " -1 269 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 } 3 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 270 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 268 "" 0 "" {TEXT 284 15 "FINITE ELEMENTS" }} {PARA 269 "" 0 "" {TEXT -1 29 "SYMBOLIC PROGRAMMING IN MAPLE" }}{PARA 19 "" 0 "" {TEXT 285 16 "by Artur Portela" }{TEXT 286 0 "" }}{PARA 270 "" 0 "" {TEXT 283 20 "Elastics Example 1: " }{TEXT 287 27 "Tractio n of a Square Plate\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res tart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "interface(verbosep roc=3):printlevel:=3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "li bname := \"C:/mylib/fem\", libname;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%(libnameG6$Q-C:/mylib/fem6\"Q=C:\\Program~Files\\Maple~6/libF'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(Plotter);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#7:%+draw_labelG%,draw_symbolG%*draw_textG%+draw_ titleG%)move_penG%*new_arrowG%)new_axesG%*new_colorG%.new_pen_styleG%1 new_plot_shadingG%/new_plot_styleG%.new_thicknessG%)pen_downG%'pen_upG %-plot_elementG%2show_2D_animationG%-show_2D_plotG%2show_3D_animationG %-show_3D_plotG%3start_2D_animationG%.start_2D_plotG%3start_3D_animati onG%.start_3D_plotG%/upgrade_framesG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "with(Cst_fem); with(G_cst_fem);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$%(cst_femG%/read_save_dataG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*%/animate_3D_defG%/animate_3D_rotG%6animate_displacem entsG%3plot_contour_linesG%3plot_displacementsG%*plot_meshG%2plot_prob lem_dataG%.plot_stressesG" }}}{SECT 0 {PARA 20 "" 0 "" {TEXT 258 17 " \+ Data Preparation" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "The best way t o input data for " }{TEXT 267 7 "Cst_fem" }{TEXT -1 25 " is to use the procedure " }{TEXT 266 14 "read_save_data" }{TEXT -1 50 ", which read s a file with the following structure:" }}{PARA 0 "" 0 "" {TEXT 259 10 "*elements*" }{TEXT -1 127 " [element, node1, node2, node3, mate rial]\n1 1 2 4 1 \n2 4 3 \+ 1 1 \n" }}{PARA 0 "" 0 "" {TEXT 260 7 "*nodes*" }{TEXT -1 123 " [node, x, y]\n1 1 1 \n2 \+ 0 1\n3 1 0 \n4 0 0 \n" }} {PARA 0 "" 0 "" {TEXT 261 11 "*materials*" }{TEXT -1 132 " [materi al, Young modulus, Poisson coef, specific weight, thickness]\n1 \+ 210000000000. .28 -77000. 0.01\n" }}{PARA 0 "" 0 " " {TEXT 262 8 "*forces*" }{TEXT -1 62 " [node, fx, fy]\n1 \+ 0 10\n2 0 10\n" }}{PARA 0 "" 0 "" {TEXT 263 13 "*c onstraints*" }{TEXT -1 131 " [node, direction(x, y or angle measured from x), displacement]\n4 x 0\n4 y 0\n3 90 \+ 0\n2 0 0\n" }}{PARA 0 "" 0 "" {TEXT 264 10 "*control* " }{TEXT -1 127 " [title, plane stress/strain, point forces, self weight]\ntitle Plate Under Uniaxial Traction \nplane stress\npoint fo rces\n" }}{PARA 0 "" 0 "" {TEXT 265 5 "*end*" }{TEXT -1 1 "\n" }} {PARA 0 "" 0 "" {TEXT -1 83 "The data blocks, with the respective keyw ord on the top, can be given in any order." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Alternatively, data can be give n manually through the definition of the variables: " }{TEXT 268 5 "ti tle" }{TEXT -1 2 ", " }{TEXT 269 10 "plane_case" }{TEXT -1 2 ", " } {TEXT 270 7 "control" }{TEXT -1 2 ", " }{TEXT 271 4 "nods" }{TEXT -1 2 ", " }{TEXT 272 9 "mat_props" }{TEXT -1 2 ", " }{TEXT 273 5 "elems" }{TEXT -1 2 ", " }{TEXT 274 6 "forces" }{TEXT -1 5 " and " }{TEXT 275 9 "bdr_conds" }{TEXT -1 46 ". See bellow the structure of these variab les." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 20 "" 0 "" {TEXT 257 27 " Traction of a Square Plate" }{TEXT -1 0 "" } }{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "As the first problem, consider th e case of a square plate, with 2" }{TEXT 276 1 "m" }{TEXT -1 19 " of l ength and 0.01" }{TEXT 278 1 "m" }{TEXT -1 41 " thick, under the actio n of a load of 20" }{TEXT 277 3 "kN/" }{XPPEDIT 18 0 "m^2;" "6#*$%\"m G\"\"#" }{TEXT -1 58 ", uniformly distributed on the sides perpendicul ar to the " }{TEXT 279 1 "y" }{TEXT -1 174 " direction. Since the prob lem is symmetric, a quarter of the plate, the first quadrant, will be \+ analysed, with the symmetry boundary conditions. The material properti es are " }{TEXT 280 2 "E=" }{TEXT -1 4 "210*" }{XPPEDIT 18 0 "10^6;" "6#*$\"#5\"\"'" }{TEXT -1 0 "" }{TEXT 281 3 "kN/" }{XPPEDIT 18 0 "m^2; " "6#*$%\"mG\"\"#" }{TEXT -1 5 " and" }{TEXT 282 3 " " }{XPPEDIT 18 0 "nu = .28;" "6#/%#nuG$\"#G!\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 398 "Since the state of str ess of the plate is constant, as is the stress field of the triangular finite element, the finite element analysis will lead to the exact so lution. Hence, the discretization of the plate needs to consider only \+ 2 finite elements, which correctly describe the plate geometry. Note t hat, in the finite element mesh, the distributed load is replaced by t he equivalent nodal loads." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read_save_data();" }}}{EXCHG {PARA 0 "read data from a file (y/n) \+ ? " 0 "" {MPLTEXT 1 0 2 "y;" }}}{EXCHG {PARA 0 "file name: " 0 "" {MPLTEXT 1 0 16 "\"dat_test4.txt\";" }}}{EXCHG {PARA 0 "save data into a file (y/n) ? " 0 "" {MPLTEXT 1 0 2 "y;" }}}{EXCHG {PARA 0 "file nam e: " 0 "" {MPLTEXT 1 0 12 "\"check.txt\";" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "This is the finite element data just red :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "tcase;plane_case;control;nods;mat_p rops;elems;forces;bdr_conds;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q@~Pla te~Under~Uniaxial~Traction~6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q-pl ane~stress6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#Q-point~forces6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&7$\"\"\"F%7$\"\"!F%7$F%F'7$F'F'" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7&$\"-+++++@\"\"!$\"#G!\"#$!&+q(F'$ \"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7&\"\"\"\"\"#\"\"%F%7&F' \"\"$F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%\"\"\"\"\"!\"#57%\"\" #F&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&7%\"\"%%\"xG\"\"!7%F%%\"yGF '7%\"\"$\"#!*F'7%\"\"#F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "The first thing to do is to check data" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot_problem_data();" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6<-%'CURVESG6&7$7$$\"\"!F)F(7$$\"\"\"F)F(-%&C OLORG6&%$RGBGF)F)F)-%*THICKNESSG6#F)-%*LINESTYLEGF3-F$6&7$F*7$F+F+F-F1 F4-F$6&7$7$F(F+F'F-F1F4-%%TEXTG6'7$$\"+nmmmm!#5$\"+LLLLLFDQ\"16\"%+ALI GNBELOWG-F.6&F0F)F,F)-%%FONTG6%%&TIMESG%%BOLDG\"#5-F$6&7$F9F'F-F1F4-%) POLYGONSG6'7&7$$\"+++++:!#6F(7$$!+++++:FfnF(FgnFY7%Fgn7$F($\"$v$!\"%FY -F.6&F0F,F)F,F1F4-F$6#7*FYFgnFgnFgnF[oFYFYFY-F$6&7$F9F=F-F1F4-FV6'7&7$ $\"++]P45!\"*F+7$$\"+++D1**FDF+7$F_p$\"+LLL$3\"F]p7$F[pFbp7%7$$\"+++]( o*FDFbp7$F+$\"++++D6F]p7$$\"+++DJ5F]pFbp-F.6&F0F,F)F)F1F4-F$6#7*FjoF^p FapFfpFipF\\qFdpFjo-FV6'7&7$$\"++++v$*!#7F+7$$!++++v$*FjqF+7$F\\rFbp7$ FhqFbp7%7$$!++++DJFfnFbp7$F(Fjp7$$\"++++DJFfnFbpF_qF1F4-F$6#7*FgqF[rF^ rFarFdrFerF_rFgq-FV6'7&7$F(Fhn7$F(FZF_sF^s7%F_s7$F\\oF(F^sF_oF1F4-F$6# 7*F^sF_sF_sF_sFasF^sF^sF^s-F?6&7$$\"+nmmm;FD$!\"\"F[tQ%Load6\"F_q-FM6% FOFP\"#7-F?6&7$$\"+++++]FDFjsQ+ConstraintF]tF_oF^t-FV6'7&7$$\"++++:5F] pF(7$$\"++++])*FDF(F]uFjt7%F]u7$F+F\\oFjtF_oF1F4-F$6#7*FjtF]uF]uF]uFau FjtFjtFjt-FV6'7&7$F(F^u7$F(F[uFiuFhu7%Fiu7$F\\oF+FhuF_oF1F4-F$6#7*FhuF iuFiuFiuF[vFhuFhuFhu-F?6&7$$\"+LLLL$)FDFjsQ)MaterialF]tFJF^t-F?6'7$FEF BFGFIFJFL-F$6&7$F'F9F-F1F4-%&TITLEG6$%-Problem~DataG-FM6%FOFP\"#9-%*AX ESSTYLEG6#%%NONEG-%%VIEWG6$%(DEFAULTGFiw" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17 " "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" }} }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "plot_mesh();" }}{PARA 13 " " 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "63-%%TEXTG6&7$$\"\"\"\"\"!F' Q\"16\"-%&COLORG6&%$RGBGF)F)F(-%%FONTG6%%&TIMESG%%BOLDG\"#5-F$6&7$$F)F )F'Q\"2F+F,F0-F$6&7$F'F9Q\"3F+F,F0-F$6&7$F9F9Q\"4F+F,F0-F$6&7$$\"+++++ D!#5$!\"\"FJQ,Node~Number6\"F,-F16%F3F4\"#7-F$6&7$$\"+++++vFHFIQ/Eleme nt~NumberFL-F-6&F/F(F)F)FM-%'CURVESG6&7$7$$\"+++++5FH$\"+++++?FH7$$\"+ ++++!)FH$\"+++++!*FH-F-6&F/F)F)F)-%*THICKNESSG6#F)-%*LINESTYLEGFdo-FY6 &7$F[o7$FgnF^oF`oFboFeo-FY6&7$FjoFfnF`oFboFeo-F$6&7$$\"+nmmmmFH$\"+LLL LLFHF:FVF0-FY6&7$7$F^oF\\o7$FinFgnF`oFboFeo-FY6&7$Fip7$F^oFgnF`oFboFeo -FY6&7$F]qFhpF`oFboFeo-F$6&7$FcpFapF*FVF0-%&TITLEG6$%4Finite~Element~M eshG-F16%F3F4\"#9-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$%(DEFAULTGFbr" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "After checking data, run the finite eleme nt procedure" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "cst_fem(\"a nything here will print intermediate results\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%&PlaneG\"\"\"%'stressGF%%)analysisGF%" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%>Initializing~global~matrices:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'matrixG6#7*7*\"\"!F(F(F(F(F(F(F(F'F'F'F'F'F'F'- %'vectorG6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%3Assembling~elemen tG\"\"\"7&F%\"\"#\"\"%F%F%%\":GF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %'matrixG6#7*7*$\"+;HKR6\"\"!F*$!+;HKR6F*$\"+lT5!>$!\"\"F*F*F*$!+lT5!> $F/7*F*$\"+)\\i:5%F/F3$!+)\\i:5%F/F*F*F5F*7*F+F3$\"+m\"z%\\:F*$!+lmm\" H(F/F*F*F5F-7*F-F5F:F8F*F*F3F+7*F*F*F*F*F*F*F*F*F=7*F*F5F5F3F*F*F3F*7* F0F*F-F+F*F*F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%3Assembling~ele mentG\"\"\"7&\"\"%\"\"$F%F%F%%\":GF%" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%'matrixG6#7*7*$\"+m\"z%\\:\"\"!F*$!+;HKR6F*$\"+lT5!>$!\"\"$!+)\\i: 5%F/$\"+)\\i:5%F/F*$!+jmm\"H(F/7*F*F(F2F0F-F+F4F*7*F+F2F($!+lmm\"H(F/F *F*F0F-7*F-F0F8F(F*F*F2F+7*F0F-F*F*F(F8F+F27*F2F+F*F*F8F(F-F07*F*F4F0F 2F+F-F(F*7*F4F*F-F+F2F0F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9Assem bling~point~forces:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7* \"\"!\"#5F'F(F'F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%CForcing~exa 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