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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 153104, 2686]*) (*NotebookOutlinePosition[ 153772, 2709]*) (* CellTagsIndexPosition[ 153728, 2705]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Formulation of finite element matrices for a 2d elastic \ isoparametric 4-node quadrilateral element - Motivation for numerical \ integration\ \>", "Title", FontSize->12], Cell[BoxData[ \(Clear["\"]; Off[General::spell, General::spell1];\)], "Input",\ FontSize->12], Cell["Shape functions and derivatives", "Subtitle", FontSize->12], Cell[BoxData[{ \(\(\[Phi]\_1\ = \ \(1\/4\) \((1 + \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\);\)\), \ "\[IndentingNewLine]", \(\(\[Phi]\_2\ = \ \(1\/4\) \((1 - \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\);\)\), \ "\[IndentingNewLine]", \(\(\[Phi]\_3\ = \ \(1\/4\) \((1 - \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\);\)\), \ "\[IndentingNewLine]", \(\(\[Phi]\_4\ = \ \(1\/4\) \((1 + \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\);\)\), \ "\[IndentingNewLine]", \(\(d\[Phi]1\[Xi]1\ = \ D[\[Phi]\_1, \[Xi]\_1];\)\), "\[IndentingNewLine]", \(\(d\[Phi]1\[Xi]2\ = \ D[\[Phi]\_1, \[Xi]\_2];\)\), "\[IndentingNewLine]", \(\(d\[Phi]2\[Xi]1\ = \ D[\[Phi]\_2, \[Xi]\_1];\)\), "\[IndentingNewLine]", \(\(d\[Phi]2\[Xi]2\ = \ D[\[Phi]\_2, \[Xi]\_2];\)\), "\[IndentingNewLine]", \(\(d\[Phi]3\[Xi]1\ = \ D[\[Phi]\_3, \[Xi]\_1];\)\), "\[IndentingNewLine]", \(\(d\[Phi]3\[Xi]2\ = \ D[\[Phi]\_3, \[Xi]\_2];\)\), "\[IndentingNewLine]", \(\(d\[Phi]4\[Xi]1\ = \ D[\[Phi]\_4, \[Xi]\_1];\)\), "\[IndentingNewLine]", \(\(d\[Phi]4\[Xi]2\ = \ D[\[Phi]\_4, \[Xi]\_2];\)\)}], "Input", FontSize->12], Cell["Element displacement interpolation matrix", "Subtitle", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"H", " ", "=", " ", RowBox[{"(", GridBox[{ {\(\[Phi]\_1\), "0", \(\[Phi]\_2\), "0", \(\[Phi]\_3\), "0", \(\[Phi]\_4\), "0"}, {"0", \(\[Phi]\_1\), "0", \(\[Phi]\_2\), "0", \(\[Phi]\_3\), "0", \(\[Phi]\_4\)} }], ")"}]}]], "Input", FontSize->12], Cell[BoxData[ \({{1\/4\ \((1 + \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\), 0, 1\/4\ \((1 - \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\), 0, 1\/4\ \((1 - \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\), 0, 1\/4\ \((1 + \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\), 0}, {0, 1\/4\ \((1 + \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\), 0, 1\/4\ \((1 - \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\), 0, 1\/4\ \((1 - \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\), 0, 1\/4\ \((1 + \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\)}}\)], "Output"] }, Open ]], Cell["\<\ Element nodal displacement vector and dispacement \ interpolation\ \>", "Subtitle", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[{ \(Un\ = \ Table[U\_\(a, i\), {a, 4}, {i, 2}] // Flatten\), "\n", \(ue\ = \ H . 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m .03716 .25514 L s .68874 0 m .96935 .42924 L s .96935 .42924 m 1 .6535 L s 1 .6535 m .70298 .24544 L s .70298 .24544 m .68874 0 L s .03716 .25514 m 0 .48963 L s 0 .48963 m .70298 .24544 L s .70298 .24544 m .68874 0 L s .68874 0 m .03716 .25514 L s .40529 .59895 m .96935 .42924 L s .96935 .42924 m 1 .6535 L s 1 .6535 m .39787 .81114 L s .39787 .81114 m .40529 .59895 L s 0 0 m 1 0 L 1 .81114 L 0 .81114 L closepath clip newpath .5 Mabswid .687 .578 .728 r .37637 .74564 .39806 .80578 .43733 .78007 .41565 .71971 Metetra .68 .574 .73 r .41565 .71971 .43733 .78007 .47681 .75658 .45512 .69595 Metetra .672 .57 .732 r .45512 .69595 .47681 .75658 .51656 .73527 .49486 .67435 Metetra .664 .565 .734 r .49486 .67435 .51656 .73527 .55668 .71614 .53495 .65488 Metetra .655 .561 .736 r .53495 .65488 .55668 .71614 .59724 .69918 .57547 .63754 Metetra .646 .556 .739 r .57547 .63754 .59724 .69918 .63835 .6844 .61651 .62233 Metetra .637 .551 .741 r .61651 .62233 .63835 .6844 .68009 .67179 .65818 .60926 Metetra .627 .545 .742 r .65818 .60926 .68009 .67179 .72256 .66139 .70056 .59835 Metetra .618 .54 .744 r .70056 .59835 .72256 .66139 .76587 .65323 .74376 .58962 Metetra .608 .534 .746 r .74376 .58962 .76587 .65323 .81012 .64735 .7879 .58313 Metetra .597 .528 .748 r .7879 .58313 .81012 .64735 .85545 .64379 .8331 .57891 Metetra .587 .522 .749 r .8331 .57891 .85545 .64379 .90198 .64262 .87948 .57703 Metetra .576 .516 .75 r .87948 .57703 .90198 .64262 .94985 .64393 .92718 .57756 Metetra .565 .509 .752 r .92718 .57756 .94985 .64393 .99922 .6478 .97636 .58059 Metetra .696 .596 .741 r .35429 .69145 .37637 .74564 .41565 .71971 .39366 .66526 Metetra .688 .592 .743 r .39366 .66526 .41565 .71971 .45512 .69595 .43319 .64122 Metetra .679 .588 .746 r .43319 .64122 .45512 .69595 .49486 .67435 .47298 .6193 Metetra .67 .583 .749 r .47298 .6193 .49486 .67435 .53495 .65488 .51311 .59949 Metetra .661 .578 .751 r .51311 .59949 .53495 .65488 .57547 .63754 .55366 .58176 Metetra .652 .573 .754 r .55366 .58176 .57547 .63754 .61651 .62233 .59472 .56614 Metetra .642 .568 .757 r .59472 .56614 .61651 .62233 .65818 .60926 .63638 .55262 Metetra .631 .562 .759 r .63638 .55262 .65818 .60926 .70056 .59835 .67875 .54122 Metetra .62 .557 .761 r .67875 .54122 .70056 .59835 .74376 .58962 .72194 .53197 Metetra .609 .55 .763 r .72194 .53197 .74376 .58962 .7879 .58313 .76604 .5249 Metetra .598 .544 .765 r .76604 .5249 .7879 .58313 .8331 .57891 .81119 .52007 Metetra .586 .537 .767 r .81119 .52007 .8331 .57891 .87948 .57703 .85751 .51753 Metetra .574 .53 .769 r .85751 .51753 .87948 .57703 .92718 .57756 .90515 .51735 Metetra .562 .523 .77 r .90515 .51735 .92718 .57756 .97636 .58059 .95425 .51963 Metetra .705 .617 .756 r .33171 .64299 .35429 .69145 .39366 .66526 .37122 .61651 Metetra .697 .613 .759 r .37122 .61651 .39366 .66526 .43319 .64122 .4109 .59216 Metetra .688 .609 .762 r .4109 .59216 .43319 .64122 .47298 .6193 .45081 .56991 Metetra .678 .604 .766 r .45081 .56991 .47298 .6193 .51311 .59949 .49105 .54973 Metetra .668 .599 .769 r .49105 .54973 .51311 .59949 .55366 .58176 .53169 .53163 Metetra .658 .594 .772 r .53169 .53163 .55366 .58176 .59472 .56614 .57284 .51559 Metetra .647 .589 .775 r .57284 .51559 .59472 .56614 .63638 .55262 .61459 .50163 Metetra .635 .583 .778 r .61459 .50163 .63638 .55262 .67875 .54122 .65703 .48976 Metetra .624 .577 .781 r .65703 .48976 .67875 .54122 .72194 .53197 .70027 .48001 Metetra .611 .57 .783 r .70027 .48001 .72194 .53197 .76604 .5249 .74442 .47241 Metetra .598 .563 .786 r .74442 .47241 .76604 .5249 .81119 .52007 .78961 .46701 Metetra .585 .556 .788 r .78961 .46701 .81119 .52007 .85751 .51753 .83597 .46387 Metetra .572 .548 .79 r .83597 .46387 .85751 .51753 .90515 .51735 .88363 .46304 Metetra .558 .54 .791 r .88363 .46304 .90515 .51735 .95425 .51963 .93275 .46463 Metetra .716 .641 .773 r .3085 .60008 .33171 .64299 .37122 .61651 .34823 .57329 Metetra .707 .638 .777 r .34823 .57329 .37122 .61651 .4109 .59216 .38811 .5486 Metetra .698 .634 .781 r .38811 .5486 .4109 .59216 .45081 .56991 .42822 .52599 Metetra .687 .629 .785 r .42822 .52599 .45081 .56991 .49105 .54973 .46864 .50544 Metetra .677 .624 .789 r .46864 .50544 .49105 .54973 .53169 .53163 .50946 .48695 Metetra .665 .619 .793 r .50946 .48695 .53169 .53163 .57284 .51559 .55077 .4705 Metetra .653 .613 .797 r .55077 .4705 .57284 .51559 .61459 .50163 .59267 .45611 Metetra .64 .607 .801 r .59267 .45611 .61459 .50163 .65703 .48976 .63526 .44379 Metetra .627 .601 .804 r .63526 .44379 .65703 .48976 .70027 .48001 .67864 .43356 Metetra .613 .594 .807 r .67864 .43356 .70027 .48001 .74442 .47241 .72293 .42546 Metetra .599 .586 .81 r .72293 .42546 .74442 .47241 .78961 .46701 .76825 .41953 Metetra .584 .578 .813 r .76825 .41953 .78961 .46701 .83597 .46387 .81473 .41582 Metetra .568 .57 .815 r .81473 .41582 .83597 .46387 .88363 .46304 .86252 .41441 Metetra .552 .561 .817 r .86252 .41441 .88363 .46304 .93275 .46463 .91176 .41538 Metetra .729 .67 .792 r .28452 .56262 .3085 .60008 .34823 .57329 .32454 .53547 Metetra .719 .666 .797 r .32454 .53547 .34823 .57329 .38811 .5486 .3647 .51042 Metetra .709 .663 .803 r .3647 .51042 .38811 .5486 .42822 .52599 .40507 .48743 Metetra .697 .659 .808 r .40507 .48743 .42822 .52599 .46864 .50544 .44575 .4665 Metetra .686 .654 .813 r .44575 .4665 .46864 .50544 .50946 .48695 .48682 .44761 Metetra .673 .649 .818 r .48682 .44761 .50946 .48695 .55077 .4705 .52837 .43075 Metetra .659 .643 .823 r .52837 .43075 .55077 .4705 .59267 .45611 .5705 .41593 Metetra .645 .637 .827 r .5705 .41593 .59267 .45611 .63526 .44379 .61332 .40316 Metetra .63 .63 .831 r .61332 .40316 .63526 .44379 .67864 .43356 .65692 .39248 Metetra .614 .622 .835 r .65692 .39248 .67864 .43356 .72293 .42546 .70144 .3839 Metetra .598 .614 .839 r .70144 .3839 .72293 .42546 .76825 .41953 .74698 .37747 Metetra .581 .605 .842 r .74698 .37747 .76825 .41953 .81473 .41582 .79368 .37325 Metetra .563 .596 .844 r .79368 .37325 .81473 .41582 .86252 .41441 .84169 .37131 Metetra .545 .586 .846 r .84169 .37131 .86252 .41441 .91176 .41538 .89116 .37172 Metetra .742 .703 .814 r .25961 .53057 .28452 .56262 .32454 .53547 .3 .50302 Metetra .732 .7 .821 r .3 .50302 .32454 .53547 .3647 .51042 .34051 .47756 Metetra .721 .697 .827 r .34051 .47756 .3647 .51042 .40507 .48743 .38123 .45417 Metetra .708 .694 .834 r .38123 .45417 .40507 .48743 .44575 .4665 .42224 .43283 Metetra .695 .689 .84 r .42224 .43283 .44575 .4665 .48682 .44761 .46364 .41353 Metetra .681 .684 .846 r .46364 .41353 .48682 .44761 .52837 .43075 .50551 .39625 Metetra .666 .679 .852 r .50551 .39625 .52837 .43075 .5705 .41593 .54796 .38101 Metetra .65 .672 .858 r .54796 .38101 .5705 .41593 .61332 .40316 .59108 .36781 Metetra .633 .665 .863 r .59108 .36781 .61332 .40316 .65692 .39248 .635 .35668 Metetra .614 .657 .868 r .635 .35668 .65692 .39248 .70144 .3839 .67982 .34765 Metetra .595 .648 .872 r .67982 .34765 .70144 .3839 .74698 .37747 .72567 .34077 Metetra .575 .638 .876 r .72567 .34077 .74698 .37747 .79368 .37325 .77269 .33608 Metetra .555 .627 .879 r .77269 .33608 .79368 .37325 .84169 .37131 .82103 .33365 Metetra .533 .616 .881 r .82103 .33365 .84169 .37131 .89116 .37172 .87083 .33356 Metetra .757 .742 .839 r .23361 .50391 .25961 .53057 .3 .50302 .27445 .47592 Metetra .746 .74 .847 r .27445 .47592 .3 .50302 .34051 .47756 .3154 .45003 Metetra .733 .738 .855 r .3154 .45003 .34051 .47756 .38123 .45417 .35654 .42621 Metetra .72 .735 .863 r .35654 .42621 .38123 .45417 .42224 .43283 .39797 .40444 Metetra .705 .731 .871 r .39797 .40444 .42224 .43283 .46364 .41353 .43977 .3847 Metetra .689 .726 .879 r .43977 .3847 .46364 .41353 .50551 .39625 .48204 .367 Metetra .672 .721 .886 r .48204 .367 .50551 .39625 .54796 .38101 .52489 .35133 Metetra .653 .714 .893 r .52489 .35133 .54796 .38101 .59108 .36781 .56841 .33771 Metetra .633 .706 .899 r .56841 .33771 .59108 .36781 .635 .35668 .61272 .32615 Metetra .612 .698 .905 r .61272 .32615 .635 .35668 .67982 .34765 .65795 .31669 Metetra .589 .688 .91 r .65795 .31669 .67982 .34765 .72567 .34077 .70421 .30938 Metetra .565 .677 .914 r .70421 .30938 .72567 .34077 .77269 .33608 .75165 .30426 Metetra .54 .664 .917 r .75165 .30426 .77269 .33608 .82103 .33365 .80042 .3014 Metetra .515 .651 .919 r .80042 .3014 .82103 .33365 .87083 .33356 .85067 .30087 Metetra .771 .787 .866 r .20633 .48272 .23361 .50391 .27445 .47592 .2477 .45424 Metetra .759 .786 .875 r .2477 .45424 .27445 .47592 .3154 .45003 .28917 .42787 Metetra .745 .785 .885 r .28917 .42787 .3154 .45003 .35654 .42621 .33082 .40358 Metetra .73 .783 .895 r .33082 .40358 .35654 .42621 .39797 .40444 .37275 .38136 Metetra .713 .78 .904 r .37275 .38136 .39797 .40444 .43977 .3847 .41504 .36118 Metetra .694 .776 .914 r .41504 .36118 .43977 .3847 .48204 .367 .4578 .34304 Metetra .674 .77 .923 r .4578 .34304 .48204 .367 .52489 .35133 .50114 .32694 Metetra .652 .763 .931 r .50114 .32694 .52489 .35133 .56841 .33771 .54515 .3129 Metetra .628 .755 .938 r .54515 .3129 .56841 .33771 .61272 .32615 .58996 .30093 Metetra .602 .745 .945 r .58996 .30093 .61272 .32615 .65795 .31669 .63568 .29107 Metetra .575 .734 .95 r .63568 .29107 .65795 .31669 .70421 .30938 .68246 .28335 Metetra .546 .72 .954 r .68246 .28335 .70421 .30938 .75165 .30426 .73042 .27784 Metetra .516 .706 .957 r .73042 .27784 .75165 .30426 .80042 .3014 .77973 .27459 Metetra .484 .689 .958 r .77973 .27459 .80042 .3014 .85067 .30087 .83055 .27369 Metetra .784 .837 .892 r .17755 .46713 .20633 .48272 .2477 .45424 .21955 .4381 Metetra .77 .838 .904 r .21955 .4381 .2477 .45424 .28917 .42787 .26163 .41121 Metetra .754 .838 .915 r .26163 .41121 .28917 .42787 .33082 .40358 .30388 .38642 Metetra .736 .837 .927 r .30388 .38642 .33082 .40358 .37275 .38136 .3464 .36372 Metetra .716 .834 .938 r .3464 .36372 .37275 .38136 .41504 .36118 .38928 .34307 Metetra .694 .83 .948 r .38928 .34307 .41504 .36118 .4578 .34304 .43262 .32448 Metetra .669 .824 .958 r .43262 .32448 .4578 .34304 .50114 .32694 .47654 .30795 Metetra .642 .817 .967 r .47654 .30795 .50114 .32694 .54515 .3129 .52114 .29349 Metetra .612 .807 .975 r .52114 .29349 .54515 .3129 .58996 .30093 .56654 .28112 Metetra .581 .795 .981 r .56654 .28112 .58996 .30093 .63568 .29107 .61286 .27088 Metetra .546 .781 .985 r .61286 .27088 .63568 .29107 .68246 .28335 .66026 .26279 Metetra .511 .765 .988 r .66026 .26279 .68246 .28335 .73042 .27784 .70886 .25693 Metetra .473 .746 .989 r .70886 .25693 .73042 .27784 .77973 .27459 .75883 .25335 Metetra .434 .725 .987 r .75883 .25335 .77973 .27459 .83055 .27369 .81035 .25213 Metetra .791 .89 .915 r .14702 .45735 .17755 .46713 .21955 .4381 .18975 .42772 Metetra .774 .892 .928 r .18975 .42772 .21955 .4381 .26163 .41121 .23254 .40025 Metetra .755 .893 .94 r .23254 .40025 .26163 .41121 .30388 .38642 .27549 .37492 Metetra .732 .892 .952 r .27549 .37492 .30388 .38642 .3464 .36372 .31869 .3517 Metetra .707 .889 .964 r .31869 .3517 .3464 .36372 .38928 .34307 .36226 .33056 Metetra .679 .884 .974 r .36226 .33056 .38928 .34307 .43262 .32448 .40629 .31151 Metetra .648 .877 .983 r .40629 .31151 .43262 .32448 .47654 .30795 .45089 .29454 Metetra .614 .867 .99 r .45089 .29454 .47654 .30795 .52114 .29349 .49618 .27967 Metetra .576 .854 .996 r .49618 .27967 .52114 .29349 .56654 .28112 .54228 .26691 Metetra .536 .838 .999 r .54228 .26691 .56654 .28112 .61286 .27088 .58932 .2563 Metetra .493 .819 1 r .58932 .2563 .61286 .27088 .66026 .26279 .63745 .24788 Metetra .448 .796 .998 r .63745 .24788 .66026 .26279 .70886 .25693 .68681 .24171 Metetra .401 .771 .993 r .68681 .24171 .70886 .25693 .75883 .25335 .73758 .23785 Metetra .354 .744 .986 r .73758 .23785 .75883 .25335 .81035 .25213 .78992 .23639 Metetra .786 .94 .928 r .11447 .45365 .14702 .45735 .18975 .42772 .15803 .42335 Metetra .764 .942 .939 r .15803 .42335 .18975 .42772 .23254 .40025 .20164 .39527 Metetra .739 .941 .95 r .20164 .39527 .23254 .40025 .27549 .37492 .24539 .36935 Metetra .71 .939 .96 r .24539 .36935 .27549 .37492 .31869 .3517 .28939 .34557 Metetra .677 .933 .968 r .28939 .34557 .31869 .3517 .36226 .33056 .33375 .32393 Metetra .64 .925 .975 r .33375 .32393 .36226 .33056 .40629 .31151 .37857 .30439 Metetra .599 .912 .979 r .37857 .30439 .40629 .31151 .45089 .29454 .42397 .28698 Metetra .555 .896 .981 r .42397 .28698 .45089 .29454 .49618 .27967 .47006 .2717 Metetra .507 .876 .979 r .47006 .2717 .49618 .27967 .54228 .26691 .51698 .25856 Metetra .456 .852 .975 r .51698 .25856 .54228 .26691 .58932 .2563 .56486 .24761 Metetra .403 .825 .967 r .56486 .24761 .58932 .2563 .63745 .24788 .61384 .23889 Metetra .348 .794 .956 r .61384 .23889 .63745 .24788 .68681 .24171 .6641 .23245 Metetra .293 .76 .943 r .6641 .23245 .68681 .24171 .73758 .23785 .7158 .22837 Metetra .237 .724 .927 r .7158 .22837 .73758 .23785 .78992 .23639 .76912 .22674 Metetra .76 .976 .919 r .07955 .45644 .11447 .45365 .15803 .42335 .12406 .42541 Metetra .73 .974 .926 r .12406 .42541 .15803 .42335 .20164 .39527 .16861 .39663 Metetra .695 .969 .931 r .16861 .39663 .20164 .39527 .24539 .36935 .21329 .37008 Metetra .656 .96 .933 r .21329 .37008 .24539 .36935 .28939 .34557 .2582 .34571 Metetra .612 .947 .933 r .2582 .34571 .28939 .34557 .33375 .32393 .30347 .32352 Metetra .564 .93 .93 r .30347 .32352 .33375 .32393 .37857 .30439 .3492 .30349 Metetra .512 .908 .924 r .3492 .30349 .37857 .30439 .42397 .28698 .39551 .28562 Metetra .456 .881 .914 r .39551 .28562 .42397 .28698 .47006 .2717 .44253 .26993 Metetra .398 .85 .901 r .44253 .26993 .47006 .2717 .51698 .25856 .4904 .25644 Metetra .338 .815 .885 r .4904 .25644 .51698 .25856 .56486 .24761 .53924 .24518 Metetra .276 .777 .866 r .53924 .24518 .56486 .24761 .61384 .23889 .58923 .23619 Metetra .216 .737 .845 r .58923 .23619 .61384 .23889 .6641 .23245 .64053 .22955 Metetra .156 .695 .822 r .64053 .22955 .6641 .23245 .7158 .22837 .69331 .22531 Metetra .099 .652 .799 r .69331 .22531 .7158 .22837 .76912 .22674 .74778 .22359 Metetra .704 .984 .876 r .04187 .46621 .07955 .45644 .12406 .42541 .08747 .43437 Metetra .662 .973 .873 r .08747 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