IMPROVEMENT IN PLATE MODELING AND PLATE DESIGN BY THE FINITE ELEMENT METHOD Thesis for the Degree of Ph. D. MICHEGAN STATE UNIVERSITY CHENG-KORG CHOU 1975 This is to certify that the thesis entitled Improvement in Plate Modeling and Plate Design by the Finite Element Method presented by Cheng-Kong Chou has been accepted towards fulfillment of the requirements for __Ph_.2._degree in mm 73)» _ (7 , 4 Lame.“ 5/ 914151.]: Major professor Dfie Niki \L \QWS 0-7639 5', T l- H' .c O Q VON-‘9 be.“ 'S S C‘- c- “ u h l . v o ‘- . ‘ T“ .: “5-; l'l ABSTRACT IMPROVEMENT IN PLATE MODELING AND PLATE DESIGN BY THE FINITE ELEMENT METHOD By Cheng—kong Chou The objectives of the studies contained in this thesis are : (1). to develOp refined finite element computer programs for plane stress and plate bending with the plan that these can be combined in the future for use in folded plate analysis, especially for the complex nonprismatic multiple-folded plate structures. (2). to compare analysis results using the refined elements with results of the standard ELAS program. and (3). to apply the standard program to study some possibilities in Optimization in design of a uniformly loaded clamped or simply-supported square plate. A The refined plane stress finite element program assumed a complete 2 nd. order polynomial as the displacement function with mid-side nodes used to form a conforming element. This function gives a linear stress distribution within the element, compared with the constant Q .4 h 0"? Q R _.. or: ‘.$..‘. "‘t~ . ...v ‘- - 3' . "“2... v —""" ~ 3-"; "1 r - .u "' .g u ...._ .....v I- ' ' .~ , . .0 o ' .o'r-fl FV‘ ' - “-1 P‘ >- "u . ,.-o.vd . ~~ . v . ' p.~~-r.: ; ' D — y w v "Q , ,-.. .4... - o . ash _ - m..- . a "p- 1.“v :A;w;r0 n: J-:~bfi—Vta.a.. . . ... .w.-.-"~ .9.. < ‘ P‘V t: “‘:‘0‘0---' :4 .14. ‘ o . ‘ - . 'dll‘. A. A I‘ .1 7‘ o “H on: .u-‘_, .v . u _ ‘ .-.'-- "‘NP ‘na~,‘ ' .P H >— .. 7' “-4 5‘du l.._._. \ ' “ ha. . u ‘ ,1 V‘ ...~~.--“ ‘v1.. 4 n. I ' . . ‘ Lv,’ h: Pa..r~a”.' ..... “n. ~, A,“ ' U .._. ‘ = A“; Few .‘r .. ~ V‘-\-. k. 'u‘ 6 ll . . ‘.. "an '5‘ ..........”;.. . p -‘ or- . .,. ‘ “:1. . nag - ‘-. v -. . \ “p -A “'“1.9.~"‘ 2rd . twa|~ .‘ ‘ l '- \ I . ‘* . " ‘V‘ o. .‘h‘ u- 4.“' h" I ‘ V ‘5' _ ‘0 , Tut-:1}. . . ~ .-~4.'\ 2* r...“ b H‘- h: . ., ;,--“o:_‘: "d A Q“:.".“ ‘v.” ¢.. -v' Q In s :.‘:" § . v.‘-_ 611-..."“ "4 . “1‘ \ u l. I\ .‘.~. f‘ .. JY§‘ :v N r A “‘4“L’ .a i 9. .‘ V‘ .1 ‘ A! a... "ne S‘Avd' v t". P M' : Hr... “0‘: "v A ‘ V«' F H” ‘ “#1.. Cheng-kong Chou stress of the standard program. In calculating the center diSplacement of a simple beam, the standard ELAS program with 256 elements gives a result with an error of 31.26% ; the T—6 refined program gives only a 3.41% error when only 16 elements are used. The refined plate bending finite element program assumed a complete 5 th. order polynomial as the displacement function with the additional compatibility equation being the normal lepe of the mid-side node to form a conforming element. To eliminate the mid-side nodes, which increase the band width by a substantial amount, a cubic function variation is assumed along the boundary line in order to replace the mid-side normal SlOpe by the corner parameters. Thus, the 21 degrees of freedom are reduced to 18 but the element is still conforming. The major advantage of the T-18 refined element program is that the nodal parameters include not only the displacement and lepes but also the 3 curvature terms; this is the only way to truly represent the boundary line of a non-prismatic multiple-folded plate structure. A comparison of results shows that in the thin plate problem, a subdivision into A elements using the T-18 program can give a result equivalent to about 100 elements using standard ELAS. Optimization of the internal moment is studied by using the standard ELAS program. Both isotrOpic and orthotrOpic clamped square plates combined with homogeneous ‘_-~‘n h ' .—-V"', \ V' ..,. ' :r.‘_-J-r v- - -A - ' 5’:n\ o -"' r~'-l .u.o- "’ T» n ' ‘." ’-—-V\. K ..- .t‘.. .__‘... . ._..- I-O" . - a F " .--u"‘ ‘ A . _, p- .m-"' o ‘ on- .vv‘I-r"‘ = V a 5-0. "5.4- uAu-vv ..-.. . a. q Au-HO-Otw-Rc r. . v-\ II ‘ - - ~v.....v~~ .. “-Q. V -\.."’A-1 . . '- Aer-pp..- .0; y-.- ... _ a . “0 past-'— ~ AI J- >.¢a.--r- 1-. .. . q “°-r-y\~ A. w ' 1 ' D -I-vtn-a.> .4- 0‘ ,_ __ . .-- ~ C . v'u-o. . p c' V P: ‘N “~'C': A" 'ttv v L. d v ‘ v - .— ‘lvo..". Cheng—kong Chou or nonhomogeneous conditions, have been considered. A reduction of the maximum moment has been achieved for the different cases. For example, a 33.2% reduction of the maximum moment has been achieved for the isotrOpic nonhomogeneous plate with prOper modifications from the homogeneous plate. This occurs when the boundary region has a stiffness of 0.16 times that of the center region. In conventional design we usually strengthen the regions of maximum moment. This study shows that a stiffening of a region will attract even more moment; so, to reach an Optimum design on internal moment a prOper rearrangement of stiffness should be made. A guideline for the clamped or simply—supported square plate has been presented. . 2"“ a. o; l5 cs1¢ IMPROVEMENT IN PLATE MODELING AND PLATE DESIGN BY THE FINITE ELEMENT METHOD By Cheng—kong Chou A THESIS Submitted to Michigan State University in partial fulfillment of the requirement for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics and Materials Science 1975 TO MY LOVELY WIFE MEI-SHENG ii . ' ‘ * I! ~-..-- .I. arc" f, ' 3.-.; “I UQ-A-‘O -. - ‘ A .1 .. - a C- n-. _‘ ~~ - u n - a F.‘ ' . u... '7' - “I 4| .s...._,' . . 4 ‘-J-¢~A¢v .4-~—... . , ~ . a ‘ .P-v-v ‘1. 0,. V a a x “ ~. H...-.v~' -5 - v‘ : "' I v .. , n 7 ~ .- ACKNOWLEDGMENTS Sincere appreciation is extended to Professor William A. Bradley for his continuous guidance and advice. The author also wishes to take this Opportunity to thank Dr. G. Mase, Dr. G. Cloud and Dr. N. Hills for serving as guidance committee. Particular thanks to my lovely wife, Mei-Sheng, is for her understanding, encouragement and unfailing assistance during the past six years. iii ...--|-v-\- o .1.» 4.. .~.. ICU. .o‘l' .—.~‘ — -- a‘fi-‘o- an ' ' - II \ -u "uaun—-'-U .— -’ ;— .‘H c\o v- y...- -...g A .5' site a,” 1 H, P. . 7-. ‘0‘. V" “.‘.- n | c -J N k.) n: I\) ‘o'——~ c a ~“ 7““... ‘ I- '-.u a- kn C,“ 7"~'~ -g-n—V ‘4‘... DEDICATION TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS II. INTRODUCTION 1.1. General Remarks and Previous DevelOpments 1.2. Objectives of this Study FINITE ELEMENT METHOD 2.1. Introduction and General Remarks 2.2. Constant Strain Element in General 2.2.1. Element Stiffness Matrix 2.2.2. Overall Stiffness Matrix 2.2.3. Constant Strain Plane Stress Triangular Element 2.3. High Order Plane Stress Element (T-6) 2.4. Plate Bending Element in General 2.5. 5 th. Order Polynomial Triangular Plate Element (T-18) 2.6. The DevelOpment Of T-6 and T-18 Finite Element Computer Programs 2.6.1. T-6 Triangular Plane Stress Element Programming 2.6.2. T-18 Triangular Plate Bending Element Programming iv Page ii iii vii viii xiv H o \J'fl 4? -P m he 18 24 27 34 36 38 I‘ .,‘ v ~ .. "a: In pug" ‘ l.‘ h..-“ u 0““ “-uvn; l 401. (3:? v!- ’ Iao' ' \ rt ,. v- “ ' Lafltv‘..- ”I A ‘ ”Tr-r ': . V A p to .I. .‘W" l a~ '- . fi 0 r I ' 9 :- :— .~. c-.." 1 7 A 4 I j... - 1 " ' 10-05.. V C .‘2 ~;~' " 1". "-'~"- " A 1 . ‘ JO} . ‘1 I C J 3. d 30 3. . f‘flsun,“~' Ve.‘ \.f\‘1,. III. APPLICATION AND COMPARISON OF RESULTS 3.1. Plane Stress Problem 3.2. Plate Bending Problem 3.2.1. Introduction 3.2.2. Structural Modeling 3.3. Results and Optimum Design 3.3.1. IsotrOpic Nonhomogeneous Plate (A). Effect of Varying the Boundary Stiffness Uniformly (B). Effect Of Varying the Properties Within the Boundary Region 3.3.2. IsotrOpic Homogeneous Plate --- Effect of Varying the Thickness of the Plate Page 41 41 49 49 49 53 53 53 55 96 3.3.3. OrthotrOpic Nonhomogeneous Plate —— Effect of Varying the PrOperties in Different Directions 3.3.4. Simply-supported Square Plate 3.3.5. Results Comparison of T-18 and Simpler Elements IV. CONCLUSIONS AND RECOMMENDATIONS 4.1. Conclusions 4.2. Recommendations BIBLIOGRAPHY APPENDICES A-1. Simple Example To Demonstrate the Formulation of the Overall Stiffness Matrix K A-2. Integration Formulas for a Plane Triangle 97 98 125 142 142 144 146 151 152 156 on I a- I’d. . a. "-'.. . ,..uvo Ar“ R. H 4». p. .A >- I-*.. "-f‘cfi“ .VIDh—u - A- r v A .J ~ . p‘ \v Ar-A“‘ “ 0 -.-§.v-- F. 0-. p A a a. C. ‘0 ‘v‘:“ ‘ Vita... ~ 0 A \-l. U-‘. ‘lP‘P l,— . .e‘ _ ‘0 .. .r"‘y- -‘-...~: A u \" o ~--¢ ‘v ‘0‘- . A ‘—A PF~$ UV..“- ~‘ I. "I. ."v ‘- v‘r- F. ... F», V‘- .0 r5 I _\:..Y\ N... A-5. A-6. A-7. A-8. A-9. A-10. A-11. A-120 Nonzero Terms of the Integration “I; QT D Q dA for the IsotrOpic Material Nonzero Terms of the Integration .1; QT D Q dA for the OrthotrOpic Material Nonzero Terms of the Multiplication Of P: D Pg for the IsotrOpic Material Nonzero Terms of the Multiplication Of PT D P for the OrthotrOpic Material g g Nonzero Terms Of the Generalized Element Stiffness Kg for the IsotrOpic 21 Material Nonzero Terms of the Generalized Element Stiffness Kg for the OrthotrOpic 21 Material Complete Listing of Square Matrix GT The Transformation Matrix A Relating the Mid-side Normal SlOpe to the Corner Parameters User's Manual for T-6 & T—18 Programs Listing of T-6 & T-18 Programs vi Page 157 158 159 164 169 174 179 184 185 192 q R v , u ‘- A p ‘ I a "'... v....-. y’ “ rig-~- _V d u- H b--.-.._,.,v . A .. P'p" . .4 I ,. \ a--. — ‘Py-~ V‘ A , ‘h-A.. "‘ o O U!- l _V‘ a . \ r ""‘vI-.\_ ." in. '~ l4 " _ ‘I ~-- .. ‘- ...~"V‘.~ I‘ ~~.‘:.. .3 n- F‘- \-.~ / - .. Table 3.1. 3.2. 3.3a. 3.3b. 3.30. LIST OF TABLES Page Simple Beam X-Displacements ..................... 47 Simple Beam Y-Displacements ..................... 48 Displacements Along y=a/2 ford/3:1.0 ( 9 Element Rectangular an 18 Element Triangular Models ) ......................... 57 Moments Along y=a/2 for =1.0 ( 9 Element Rectangular and 18 Element Triangular Models coo-00000000000000...coo- 59 Moments Along y=O for"[3=1.0 ( 9 Element Rectangular and 18 Element Triangular Models ) ......................... 61 vii - ‘Y‘RT-Y‘:7‘-' “ h i v $-85 ~~o v. nr" ’ . ”I. ..-‘-" I. LIST OF FIGURES Figure Page 2.1, Elastic Body Subjected to Static Loading ........ 8 2.2, Triangular Plane Stress Element ................. 13 2.3. High Order Plane Stress Triangular Element ...... 18 2.4. Improved Triangular Element Types for Plane Stress AnalySiS 0....O...C.........C..O...OOOOO 19 2.5. Nodal Point System and Nodal Parameters for T-21 Element 0......I.000......OIIIOOOIOIOOOOOO 32 2.6. Variation Of the Normal SlOpe Along an ArbitraryEdge O00.0.00...0.000.000.0000.0.0... 33 2.7. Local and Global Coordinate Systems ............. 37 3.1. Simple Beam ELAS 16 Elements Configuration ...... 42 3.2. Simple Beam ELAS 32 Elements Configuration ...... 43 3.3. Simple Beam ELAS 64 Elements Configuration ...... 44 3.4. Simple Beam ELAS 256 Elements Configuration ..... 45 3.5. Simple Beam T-6 16 Elements Configuration ....... 46 3.6. oz ,/3 and 7’ Ratios 52 3.7a. ELAS Rectangular 9 Elements Configuration ....... 56 3.7b. ELAS Triangular 18 Elements Configuration ....... 56 3.8a. Displacements Along y=a/2 for B=1.0 ( 9 Elements Rectangular and 18 Elements Triangular Models ) ......................... 58 3.8b. Moments Along y=a/2 /3=l.O ( 9 Elements Rectangular and 18 Elements Triangular MOdelS ) 00.00.000.00...0.00000... 6O viii r.- o . r.» c . .. .. _ I I I - . V“ a g o“ f. .1. V. ... ... w .‘a. w r.. . .. I I I . . L. “A w. 5.. c. L. w . .. .2 . .. . ... L“ ._ . . 1 C. ar. . l .. .r. . , . I Ann —.. 7. n —.. .. w“ L. ... I“ .u. .3. “a . .n. ”1 F: Pv a: Pa .: FL I. r . T. . L. r . A: A . av. . . . . . . x A: n; a . hex. . .2 . . S c. . . 5. .1 T. 2. .L 2. T. . . r. .w.... Md. . J a: ”W .. s a s . at. ad .. a a .3 n1. . ad a . rJ a a . r 4 a: . .. . n S .. .. . . . . . . . .4. . .9 . 4 .r.a-u r” .rA—pu : .p.../ ruiz rnz.’ vuz/ ....:. v..4:. v..4np. .v....n_. ~x..r-/ M.»Jrl r..qr/ 5.. odio o Aw. .4 -\~ 9 . az~ .wv fixu .u. n/h AH. ai~ R.\V “val. A 0 Ana ‘5‘ O AN.‘ IL 6 n~.— 4“ l v~.dl.§ I A.. 1 I pui- Al I ~.. -.. u." .I .. ~,. ... -.. 3. w” w. w. nC ». w- . u.” .. ... . ... - _.. ...il .. ...-.\ ..II\ ..T\ ..II\ 5.1... ..f\ .5le — i . ‘l. p-v - 1 OD. "I. ll. Flu R J “In "In. "Tn. “M “It. \l'a. ,. i - 2“ .H . .. a - - - - . .. .. . u i n a . . I ..,I, At A» A. I 3‘ I . . I. I l“ t I. 1 VI. 0 V. .a V. lo§ L1 />.- I!” .4» [M44 . . s a.l) ‘4'! 1J l- [I I Q . 4 .1 a a- J ~.H. .1. ...- Figure Page 3.80. Moments Along y=O for B=1.0 ( 9 Elements Rectangular and 18 Elements Triangular Models ) ......................... 62 3.9. Moments and Deflection for Variation in.CX ( 9 Elements Rectangular Model ) .............. 63 3.10. ELAS Rectangular 25 Elements Configuration ..... 64 3.11a. Displacements Along y=a/2 for B=1.0 ( 25 Elements Rectangular Model ) ............. 65 3.11b. Moments Along y=a/2 for B=1.0 ( 25 Elements Rectangular Model ) ............. 66 3.11c. Moments Along y=O for [3=1.0 ( 25 Elements Rectangular Model ) ............. 67 3.12. Moment and Deflection for Variation in CM ( 25 Elements Rectangular Model ) ............. 68 3.13. ELAS Rectangular 100 Elements Configuration .... 69 3.14a. Displacements Along y=a/2 for ‘fl3=1.0 ( 100 Elements Rectangular Model with .8a by .88. Center)'0'....0...........COOCCCUCCCCCC 7O 3.14b. Moments Along y=a/2 for [[3:1.0 ( 100 Elements Rectangular Model with .8a by 083. Center)OOOCCOOCIOOOCIOOCCOOCC.0.......0 71 3.14c. Moments Along y=O for* [3=1.0 ( 100 Elements Rectangular Model with .8a by .8a Center).0...I.......U............I..... 72 3.15. Moments and Deflection for Variation in C1 ( 100 Elements Rectangular Model with .8a by 08a Center)00000000.ooooooooooooooooooooooo 73 3.16a. Displacements Along y=a/2 for ‘[3=1.0 ( 100 Elements Rectangular Model with .9a by 098. Center)OCUOCCOOOCOOCCCOOCCO'CC'COCCO0.0 7n 3.16b. Moments Along y=O for ‘/3=1.0 ( 100 Elements Rectangular Model with .9a by .9a Center ) ococoon-00000000000.000.00.00.00 75 3.16c. Moments Along y=a/2 for* 13:1.0 ( 100 Elements Rectangular Model with .9a by 093 Center ) coo-00000000000000.0000.0.00.000 76 ix '.9' .cl' ‘7' . . rh- ' .uAy-OH — \ I ........ .5 Figure Page 3.17. Moment and Deflection for Variation in C1 ( 100 Elements Rectangular Model with .9a by .93 Center)OOOOOOCOIICOOOOCICIOOCCOIOUOIOI 77 3.18. Displacements Along y=a/2 for CX==O.5 ( 25 Elements Rectangular Model with .8a by .88. Center)0.....O....C...’O..'......IO... 81 3.19a. Moments Along y=O for (1:0.5 ( 25 Elements Rectangular Model with .8a by .8a Center).0....'....0...‘..............O 82 3.19b. Moments Along y=a/2 for O(=O.5 ( 25 Elements Rectangular Model with .8a by 08a Center)O.I.UCIOUICCCIOOCIOOOOCOOOII.0. 83 3.20a. Displacements Along y=a/2 for CX==O.1 ( 25 Elements Rectangular Model with .8a by .8a Center)00.0.0...0......00.........I..O 8n 3.20b. Moments Along y=O for (X =O.1 ( 25 Elements Rectangular Model with .8a by .83- Center)0"....O...‘..............OC... 85 3.20c. Moments Along y=a/2 for CX =0.1 ( 25 Elements Rectangular Model with .8a by .8a Center) OICOOOOCCICCO'CCCOUOOCCCCCCCCCO 86 3.21. Curves to Approximate Optimum Value of CX &4/3 . 87 3.22a. Displacement for O(=o.16 & /3=7.o ( 25 Elements Rectangular Model with .8a by .8a Center)OICOCCCOIOCCCOCOOOOO0......U... 88 3.22b. Moments for CX=O.16 & [3=7.0 ( 25 Elements Rectangular Model with .8a by '88: Center).ICOCCOOCCOOCCOCCCCI.CCOOOOOOOO 89 3.23a. Displacements Along y=a/2 for a=0.1 ( 100 Elements Rectangular Model with .9a by .9a Center ) ............................... 9O 3.23b. Moments Along wo for O(=o.1 ( 100 Elements Rectangular Model with .9a by .98. Center ) 00....O...OOOOOOIOIOOOOOOOOOOIO 91 3.23c . Moments Along y=a/2 for (1 =0.1 ( 100 Elements Rectangular Model with .9a by 093 Center)OCCCIOOOCQOOCCO.....00......I.. 92 .o' In' — I f. 5.. ‘ r O n“.§ Ar u "";r’ : "v~.v-.' A H, h A‘s." :V“ 'Ivot ' he“. .'. .’ ~-\.. Figure Page 3.24a. Displacements Along y=a/2 for (X =0.5 ( 100 Elements Rectangular Model with .9a by .98. Center).0.......................I...II 93 3.24b. Moments Along y=0 for CK: -0. ( 100 Elements Rectangular Model with .9a by .9a Center ) ............................ . 94 3.24c. Moments Along y=a/2 for CX=0.5 ( 100 Elements Rectangular Model with .9a by .98. Center )OCCICC00......00000.0CCCCUOOOOOO 95 3.25a. Idealization of Plate with Homogeneous Central Portion coconut-cocoouoocoo-00000000000000.o.o 99 3.25b. Moment Results ................................. 100 3.26a. Idealization of Plate with Homogeneous Central Portion Except for 3 rd. Set Elements From center0.....00000000000O'COOCOOICUOU.COOOIOOO 101 3.26b. Moment Results ................................. 102 3.27a. Idealization of Plate with Homogeneous Central Portion Except 4 th. Set of Elements From center 0000000000000000000.0000000000000000... 103 3027b. Moment ReSUlts 0....OOOOOIOOOOOOOOOOIOOOII..0... 1024' 3.28a. Idealization of Plate with Homogeneous Central Portion Except 2 nd. Set of Elements From Center O...00.0.0.0...OOOOOIICOOOOOIOOOOIOOOOO 1.05 3.28b. Moment Results ................................. 106 3.29a. Idealization of Plate with OrthotrOpic Homogeneous Central Portion Except 3 rd. Set of Elements From Center ...................... 107 3.29b. Moment Results ................................. 108 3.30a. Idealization of Plate with Homogeneous Central Portion and OrthotrOpic Nonhomogeneous Boundary Portion 0.0.0.0....IOOOOOIOIOOIOOOOI. 109 3.30b. Moment Results ................................. 110 3.31a. Idealization of Plate with Homogeneous Central Portion and OrthotrOpic Nonhomogeneous Boundary Portion ............................. 111 xi . . J a: CV e... O U r” .v. ... a—- v 4 :- b p . q-t “ASV ‘ 9 fiv-‘CIJ‘- .- ‘A a; .- on ._ .-.,.o- .. C 1 - nls..-sh 0 ..~ 1' 40’ .c I u b w. .1 4‘ r” .4 AV 2. Re. ”'V’fi o A ”r .""UDQU Q! I . 0- CYQ .’ a A.‘ «.4-.. w n A: "v. ‘v n ‘tha An 1!. 1'». .1 5’54» -1;. 4.. Figure Page 3.31b. Moment Results ................................. 112 3.32a. Idealization of Plate with Homogeneous Central Portion and OrthotrOpic Boundary Portion ..... 113 3.32b. Moment Results ................................. 114 3.33a. Idealization of Plate with Homogeneous Central Portion Except 3 rd. Set of Elements From Center and OrthotrOpic Nonhomogeneous Boundary Portion ............................. 115 j3.33b. Moment Results ................................. 116 j3.34a. Idealization of Plate with Homogeneous Central Portion Except 3 rd. Set of Elements From Center and OrthotrOpic Nonhomogeneous Boundary Portion ............................. 117 :3-3htu Mement Results ................................. 118 3.:35a. Idealization of Plate with Nonhomogeneous central Portion 0......0................00.... 119 Bollsb. Moment Results ................................. 120 3- 36a. Idealization of Plate with Homogeneous Central Portion Except 3 rd. Set of Elements From Center and OrthotrOpic Nonhomogeneous Boundary Portion ............................. 121 3.361). Moment Results 0.0.0.0000...IOOOOOOOOOOIOOOCOOOO 122 :3-l37au Idealization of Plate with Homogeneous Central Portion Except 3 rd. Set of Elements From Center and OrthotrOpic Nonhomogeneous Boundary Portion ............................. 123 3.37b0 Moment ReSUlts 0.00.00...0.00.00.00.00...0...... 124 3 ‘ 38a. Idealization of Nonhomo eneous Simply-supported Square Plate with =1.0 ................... 126 3.381). Moment ReSUltS 00......OIOOOOOOOOOOOOOOOI00.00.. 127 :3'i3598” Idealization of Nonhomo eneous Simply-supported Square Plate with =O.5 ................... 128 3.39b0 Moment Results 00.00.00...OOOOOOOOOOOOOOOO.00...129 xii r. rm”. . . - . . _ . A . . . g . D .H . a c o . V F; o. . o A - . .. . Nb. 7. .. . a}. a; .. m a A .H . . L. .J P u . wv . . .u. o s. Auv ..h. nay AAA 0 a r . .. a; 5. —~ . r3 . a. AP. :. n3 5. ... w. .7 a; a; . r. .. . V . . w . a a . . “u. o . a. . . . . . a .. : . . .. o A; o . .. . PA» 0 . .. . .J . n . .1 F” - Q q . J r“ . A r . ~ ~ . r . A Q 4. .p. a—,.. .4 FA‘ .5» . 1 S. 7» a .. C. 2 a u S. a. \J . Te 2. .a. n\..~ ... .—. qu -.. .v. A\.J -. “—h N Am. 0.. «a. uRA . ~ . n ..a. ".1". o.“ uflvm on“ v.4“ . . nun“ - . gnu . . ...... ... ..u .H z“ .. - 2“ ..u - - .... . v. . a u .n a no . ... n ..v a A... 1 W unwl U . . y, . 1 I n . . .. , 1 Figure Page 3.40a. Idealization of Nonhomogeneous Simply-supported Square Plate with 7:20 130 30Ll’ObO Moment Results OOOOOIOOOOOOIOOOOOIOIOOIOOOOOOOO' 131 3.41a. Idealization of Nonhomo eneous Simply-supported Square Plate with :5.0 ................... 132 3.41b.M0mentReSUltS Oil...0.0.00IOOOIOIOOOIOOOOOIOOOO 133 3.42a. Idealization of Nonhomo eneous Simply—supported Square Plate with ‘=7.5 ................... 134 3.42b. Moment Results ................................. 135 3.43a. Idealization of Nonhomogeneous Simply-supported Square Plate with y=10.0.................. 136 3.43b. Moment Results ................................. 137 3.44. T-18 Results for Clamped Square Plate .......... 140 3.45. Results Comparison ............................. 141 A—1. Idealization for the Pin-jointed Bar ........... 145 xiii a O *4-.- W'i-u'. .n-L-_I ( ’1 . . L 1. On U' gnu o l r ...- . "9 a o . . ~.. ~ a. ’ C ,, .. . a . O I V .. .1 . 5 ‘ . . . A . . t. . . '1 . . . . v ' | a Q . ‘ ~ 1 I .. . fi . -. ' 1 . I 1 J . J' Y -. , .‘ v .. ! v q n .l H ‘l l LIST OF SYMBOLS Side Dimension of Square Plate Side Dimension of Square Element of the Plate Uniform Load Per Unit Area Applied Normal to Plate Thickness of Plate Et3 12(1-U2) Stiffness of Plate Elements in Central Region of the Plate Plate Stiffness — Stiffness of Plate Element in the Boundary Region at the Corner of the Plate Stiffness of the Plate Element in Boundary Region and Next to Center of the Plate Modulus of Elasticity Modulus of Elasticity in X-Direction Modulus of Elasticity in Y-Direction Poisson's Ratio Poisson's Ratio in X-Direction Poisson's Ratio in Y-Direction Shear Modulus E __ X - 1-2/va xiv — '— .- n | I a . r " ”‘ -..—é— " .. _ . 1 .. d .' A v v n“ V- - ‘ u 3 1 v ‘1 - V J " , .J ‘ A “ .. f . . D“ u - ' ‘ J n O V v b I . v 6 I I . 'J . 5.! b . . I " r: J .n, E" LTDI " 1- UX Dy 2 ”x9. z 25%. 1- UK Uy 14?ny Measure of Ratio of Boundary Region Stiffness to Stiffness of Center Region for Plate with Clamped Edge Boundary Condition Measure of Variation of Element Stiffness within the Boundary Region of Elements for Clamped Edge Plate Measure of Ratio of Boundary Region Stiffness to Stiffness of Center Region for Simply- Supported Plate Bending Moment Per Unit Width on Face Normal to Y—Direction Twisting Moment Per Unit Width Applied Concentrated Force Rectangular Local Cartesian Coordinates Rectangular Global Cartesian Coordinates Cartesian Displacements Cartesian Normal Stress Components Cartesian Shear Stress Component Cartesian Normal Strain Components Cartesian Shear Strain Generalized Element Stiffness Matrix XV Ky .21 1.8 Generalized Element Stiffness Matrix Related to the Normal Stress Generalized Element Stiffness Matrix Related to the Shear Stress Element Stiffness Matrix . 3 EX t 1 c+ UN El ‘ 1 N E"t3 12 Gt3 12(1- 21x Uy) Element Stiffness Matrix for T-21 Element Element Stiffness Matrix for T-18 Element Area of the Triangular Element xvi I ....v- - I. a. - ‘- A u A u o-.A-v .... pl-Ayfifl' . \ v.v-- ‘ - v ..4.- v 0 . . . .‘ ...v—a p“. N -t....-.-~' a...‘ ' M-Op- n A A FH*‘ .....--.-.. ..’ ‘ Q ... ...: :r- .. h... ~.'.J 4‘ u‘ ' ~ .- h c H: I. ~ av-‘.."‘i : U ;’ ,AP I -.Z . ~~“‘v.~ .“"...C . ... . y... I ‘40. n" I. ...4'.‘ r V ‘ 7p. ' ..‘t- “he. . . 4's. ‘ .....J‘)‘ (l .... . y“ I. Introduction 1.1. General Remarks and Previous DevelOpments Plate structures play a very important role in almost every structural system built by modern man. Buildings, bridges, hydraulic structures, pavements, containers, airplanes, missiles, ships, instruments, machine parts, etc., all may contain different types of plates. Closed form solutions to particular plate problems appear in the literature and have been a great asset to designers. These solutions are. however, generally for plates with relatively simple support and loading conditions;and they are generally restricted to elastic, homogeneous, isotrOpic plates of a constant thickness. In the early days of engineering construction, plates were generally designed with much greater strength than was normally required. In recent years, the develOpment of Operational research has contributed to the understanding of structural behavior. The Optimum design for the plate panel has been introduced by Lowe and Melchers [1] and Kirsch [:2] . With the advent of the electronic digital computer, many investigators have turned their efforts toward the study of numerical techniques such as finite difference methods and the finite element . ' pnnA fl ... .'.“. :4. -..:- o ‘ a v. V. p- .. S u ".- L ' u.‘ .A " ' 4 .- n-- ...... AAV.“‘ ’A '1, -....oooV ' “L... A Awav“. .’ . a u- u- . ..‘.0.>~ 4-"".. ‘ ‘v V r» 50$ch .-a-..~u44w - . ..0. ~00. O r. ' . . -y\°w Hr» ‘-.-i.v‘ll v-A . an... A. v'p A. .-. ~c ... on-cn d ‘.I_-_. ' r RI“ ‘ I\ ‘ ~v . we. ..J ' v.‘. ‘ b 4 5 . o. . " m ...fi 7‘ u- " "" ~-.. ...1 l4 -. "‘ 'y- .... "'00..- vv . .' a ‘IA ..‘ 1. “‘ C‘Y‘"A‘ .v-‘.‘ H.. .1, . ‘ " :F'q ‘ I- #.h ~" um.“ “...: --. ... ...,p~~ ‘ ..,:.;.c :“O Q ‘0 .4- . I‘-.‘ ‘ nae . v-\ 1 p ‘ “‘5 G T -..... "El-AE.. . . ... ‘rn A: n .u ‘u :*‘»~ '0' A- A x.» . .-‘A. . ., ‘ is.“ 5“", ‘0 .. ~ ~ .. "31‘ r‘ ’, J-~ ‘A‘ -:.,~\C-“‘ 6".“ .. ”‘01 . H In. I u :“'v- 1 ..‘J N ..ul.. -3 ch 1.. ‘ In ' ‘F . approach to obtain solutions to plate structures. Especially within the last decade, a significant number of papers in the field of finite element plate theory have been introduced. Zienkiewicz [3], Melosh [4], Clough [5,6], Utku [7,8], Bell [9], Butlin [10] and Whang [if] have made a great contribution with the develOpment of the plate bending element. Some finite element textbooks [12,13,14] which introduce the basic theorems and applications in the different engineering fields have also stimulated the speed of develOpment. General computer programs, such as ELAS [15], SAP [16], and STARDYNE [17] are regularly used in the United States in the design of hydraulic dams, cooling towers, ships and reactor containment, missile and aircraft structures. Because of the limitation of computer size and the time consumed in their develOpment, these programs are all in the same category in that they use the simplest element type and the lowest order of polynomial for the displacement functions. For some structures, a lower order displacement function cannot adequately represent the true structural behavior; therefore, great caution is needed in using these programs for solutions. In particular, for complex structural systems such as multiple- folded plate structures, the reliability of solutions using these programs is really in doubt. 1.2. Objectives of this Study The objectives of this work are as follows (1). (2). (3). To develOp finite element plate programs using refined element types. Compare the results Of the refined element programs and the ELAS program for plate structures. Investigate the Optimum design Of clamped or simply-supported, isotrOpic or orthotrOpic, homogeneous or nonhomogeneous square plates using different types of finite element modeling. The Optimization goal is to make the absolute maximum moment as small as possible. v- 0‘. , “v“ .-',;...\o' . . L ' uu‘v'.‘ ...... I ~. ‘ ...: V" “7‘. v"" Av . o.-. A.v‘ 5 ‘ . n _ ' - An .' ., ,. any! fl_ ’ ‘>V‘ -U ‘ -.. ‘ b"" o ' ‘ .' lion A‘ ..--'7 a fi,. ‘5: ..t ..t...-‘ a‘o . . I ~ r' . ..~'~v~ tr up; > ‘ lb .: :.,......,.£: ~. ...Jy-J. «u P a . , “A ...... a ”- .. .. “\ / ,. ""-‘-~~ us .1. - ~ . . A: °|'" P A .. "‘ H b-J " u.u --. - ‘ 1 .. .’- u. new ~1.‘\' d“ - I ~. .' ‘V‘ .A .. v .v.‘_-.‘... U n,.‘_‘ ‘ 5‘: i'v ..~ "'~-..., .' ‘ . _ v. A ~ \- n .. . v... . . \ "“hnp ....i. .. o "‘ ‘9‘ v .. x » m -- :.7‘".‘\"‘y\’ , II. Finite Element Method 2.1. Introduction and General Remarks The concept Of the finite element method was originally introduced by Argyris [18], Turner, Clough, Martin and TOpp [19] in a series Of papers published in 1954 to 1956. The method was develOped originally for aircraft structures. It strongly relies on the matrix transformation theory of structures which was first brought out by Langefors [20] and Denke [21]. Since 1956 the rapid develOp— ment of the electronic computer has changed the nature of the structural problem from that of searching for a closed solution to one of applying numerical approximation methods. Within the last decade, the solution Of structural problems by using finite element methods has received considerable attention. Progress has been made through several important conferences [22,23,24]. The theory of the method is well established at this point, and an understanding of the requirements necessary to achieve adequate solutions and guarantee convergence has been gained. Two Of the approaches to the finite element method are : (a) Mathematical and (b) Physical. The mathematical concept is based on the energy theory and variational princi— ples; this approach will not be discussed in detail in this study. In the physical approach the basic concept is to assume that each structural system may be considered as an assemblage of a finite number of individual structural ,- I o -.rfl 7‘ 'f a -oh' .w‘: .--""' . o .—...v ‘ VIF'I' :"‘: -“ I _. .,',r A “A ”‘I- '.- .Av' ' u .n":o‘o :V'4 .._-0"‘ " w.‘ I -Lo“ . .-..- . . c 0 Y u . 'F -pva“ C ‘ ‘ _. ' t d‘J‘.‘ 'au-~. . q “7" ..r. o c... — -. ‘. .A'oa --" _‘.~ 0 .. ~W' ,. A“ Orv-5:-" p i" ,. ...-vv ‘- .‘ ‘0 ‘ ..rA ”.1 — a .fi- ‘fi‘ . 1. ‘ - . an EA ’:'; A, H .. ...-4-- ¢*‘ 5' ' 4 . a .y ""'-‘W" A no . — —- 'Afi .a -OO.UIV.5‘-..“. .1. .. .1 . ... .. ,. .. . : or " a~¢a~ov da§“‘ a .. - -‘v--.. n,. ““ ... , ' I \ ' K -«u-‘-...y v. n A ‘ \ “7"! N d . ...... h.’ " .-. .-.... . .\ u: 7‘ ~Arpp¢ --..~4 " 1*“ t— _ "“ “44..., , ‘Ino‘ :h‘u h~;'"' A “W‘v d-"“ LI. .r‘, A . ~~ ... . ~ v .. ‘ .,;.1” c M .. , :‘r-J ,-~. "“--:.n '5 r! -.Bv ... .. s q A. ."- J. ““A-k‘ “VP . r“~«':.. I. I \ -..’_ . ~, ‘ s4.-v‘." l: a-v,. 9 ; .- "9.: v . ~ N "-~-‘C‘“"~ “ "-.‘;‘ r: - '0. 3'” " "an" F ”A ‘ ‘ Iv ‘. 1’10 ..."u ~ V- C I F: :. 3:»1: ‘ A "'0“: "HM; " ... T‘A" u - 334‘ ‘F‘n "-'\J :r ‘I. 9-. I , ... .v 3 ‘§ ~.:"‘ns., "‘33 aw: *“ in -.‘ :V‘ elements, interconnected at a finite number of nodal points. The structural system should satisfy equilibrium, compatibility and force-displacement requirements simultaneously for each element and for each of the nodal points. The finite element method can be applied to a one- two- or three-dimensional continuum. A two-dimensional elastic body such as a dam, plate or shell can be treated as an indeterminate structure having an indeterminacy of infinite degree. Closed solutions are available only for certain special boundary or loading conditions of such structural systems. The majority of practical design problems-owing to their complexity and irregular geometric form or non-linearity and inhomogeneity in material prOperties-fall outside the reach of closed solutions. In such cases, only approximate solutions can be obtained. The conventional analytical approach for this type of problem in general is to establish a simplified and idealized model of the real structure for which a closed solution is available and then to interpret the solution considering the degree to which the ideal model may fit the real situation. For the finite element approach we still have to model the real structure as a simplified system, but the idealization is merely the subdivision of the original system into an assemblage of discrete segments. Because the analysis actually is performed on this substitute structure and the results can be valid only to the extent that the behaviour of the substitute structure . 'A A .... "a a“ - :un- ..v 1“... V A .' o . ...; '8 ...- .6 I .0- ' "1..-."- . Q ‘ Q .0 ;_ pup-0g rp“ .* v P as .a-" -'-1 av -' y I p.- p. .... aI-t 'r.; C . ' “ hnov ..- as, -' v Q '- A'v‘ n‘V‘ ”E I ....-- AU- V . 1 v~ ... ~v'-'-V‘" A > ~ ;.;~ou".‘:'-v . ‘0 O - ‘ . .0 ... o,- o y- a c ... ...a .....1' ~‘ A ... . on“ c- .... ‘ .4 t a "O- C" h‘ ‘ «I V ..- . . h - "'z ' :' an. .. .. ...- a-_~-.av..-- o ..... 3.. ‘ l _‘ ‘ .. , r v- r A... .v- It‘ss.."‘ . > . o;op»fl '“CCS -_‘ I I "" *‘vvv--.: . .‘V- I a ."~I F A «...- a‘l c. 4 “\-0.' ~J v - "VII . . . .. v- “.-A“ " -o‘ H A “we“-‘. '9 6 . ‘ 7‘ 1 ...? V3. “ _ A- u ‘flhaoc ‘ a..'. ‘ .‘vn. In "A... ~- .. ‘ ..‘l‘..‘.‘ : A..-\‘_“‘\I y V ‘n 0‘ . ' 9 . ‘V ._ - "'b ‘ y‘ “+HV‘ Md‘ "v 1‘ - "w-A ‘5‘- E .. ”F. I ‘ ~ \ H " C A ‘ ~' :h ‘-' "A ‘ :~1’\“ “v I“ simulates the actual structure, good judgement is necessary when making the modeling, However, an improved result can usually be obtained by the use of a finer mesh idealization as long as the computer capacity is not exceeded. For complex structural systems such as inhomogeneous and anisotrOpic plates, the closed solution is not available and the finite element method becomes one of few available tools for study of the phenomenon. In general,either the force or the displacement method can be used. It has been found that for highly complex structural systems, the displacement method presents the simpler formulation and computer program- .ming task, so only the displacement method will be described here in detail. Structural members which are already made up from discrete elements. such as trusses or rigid frames, present no difficulty in the formulation of their discrete models. If the structural elements are made up with fictitious boundaries,such as for plates or shells,exact discrete element representations are not possible; and so we must assume force or displacement distributions within the elements. The assumed distributions must be such that when the size of elements is decreased, i.e..the number of discrete elements increased. the matrix solutions for the stresses and displacements must tend to the exact values for the continuous system. To ensure convergence, the assumed force or displacement function must satisfy certain criteria. Which we will discuss in a later section. In general, assuming the force distribution O 'AF‘V‘ . _ fl ' orfi ,. w‘_‘_.. "' ...: ..-- "u, . o .n . -50“ _ - .o- -.. .-‘I‘ ‘ *'. .t».--“'b n .. ... v -. "‘ r- v _:,.’..: w‘ 'l. “‘8 b‘l“‘ ::,.....vu ‘0' ... ...-:r. WV ‘ ’ ' _.. any on ...-.--:n- ' ..v.op'o ‘0‘ ‘7‘ I. v. .I.| 11..., ...: “ n I.‘ ..‘~.'. ~ . I‘ h’ .‘-l.| d--..--.. . ." -.. . O ... ...... . A- ., \ \ Po \ “...".-g' .... 5 ‘ R "an ‘v~ - ‘ . . '_Y\ P0 o y . ”...... ‘. .-. ' on n ." P~~ '- F .+ . " A ‘5‘," .uv “‘l“"" ., ~ g . A-v I . .- ...... _" C a . -.‘ u- v- '3." V“! ‘V" _ .‘1“_..‘ ~"’ «A...‘ I. I I..:". R ‘5 n...__‘-.. '.. Q. i- o.‘:‘ ...,A ' .- .-. {A ,H yi A." V .‘ .yp. -¥ ' ..._‘ ‘c A. “*7 "N \A ‘ '; -~~'-_. . ... ‘s A .. -'\ a; ‘. y “v‘2- n “ l‘ . - :‘;VQ . ‘ A“ ‘ n-.o. “A“. A ‘ “4‘ ‘— ‘ ‘ ~_. - h :‘-.,: QV§~ a .-.‘v\ fit w r- u, ‘ ... .," a .‘ 9' A. ‘ 'N.Vf"Y‘. C. . v -, \‘Vfin: . "1 w ‘ “‘ *Q n 5 ~ .. q‘p “\ -. "~c“:v.'. 'I -..v S .1 . . 'n.. no ‘ v“ 1,. ¢ y 4.“! “I . .“— ‘.,‘:‘ ‘1‘" ‘- ‘vL-‘\(‘~ ‘ ~A within the element will lead to the formulation of the flexibility matrix of the structure, and the assumption of the displacement distribution will lead to develOpment of the stiffness matrix. In this research only the stiffness- displacement method will be discussed. 2.2. Constant Strain Element In General 2.2.1. Element Stiffness Matrix In the finite element matrix method of structural analysis, the structure is separated by imaginary lines into a number of "finite elements". These elements are attached to the adjacent elements at a discrete number of nodal points which may be either the actual joints or fictitious points obtained by intersecting boundary lines. The displacements of these nodal points will be the basic unknown parameters. To determine the stiffness characteristics of each finite element we first have to select a diSplacement function (or functions). From the displacement function the diSplacement field within each finite element will be uniquely defined in terms of its nodal displacements. The state of strain within each element can also be uniquely defined in terms of nodal dis- placements according to the strain-displacement relation. For a known elastic material,the relation between stress and strain is also known. Hence the state of stress throughout the element and its boundaries can be represented in terms of unknown nodal displacement parameters. By applying the unit-displacement energy theorem, the minimum potential A; ku . . . . . .. p 9/ I V . \ L . .l. .v v W. O” u a 'yw A.» WV“ an; ufl‘ “a“ um I |_ 5. a: r . .. . .r” . . 4 .. . L. a t v L. o . a: A: r” 0 r!. v u. .a 4 o . ..J a: ..V¢ "In A: n. o y a a . 3» fl . .5 AAA. . H u. 3. DIIIL a. r . [IL I v.1 I o . A v w l. V . tr . . 1‘ § w h t In H . .7 2 h: 71 ..rv... 5: .5». A- .- . r .- «V. .n‘ . . I o . no a a u. n- ray he . uv; . n o ... pp. A . p a ;~. at. .. o v" ..¢ .» v a . ... .. - .o . . a . ..... —_ . . . .. n... . . ...., u: .u . . energy theorem, or the virtual displacement theorem we may relate the external forces to the nodal displacements in terms of stress and strain. This relationship between.force and displacement in each element is given by the element stiffness matrix. The derivation, using the unit- displacement theorem, will be shown below. We consider an elastic element subjected to a group of n external forces P:= [p1 p2 ....... pj pk ..... pn] -------- (2-1) The displacements corresponding to the force P will be denoted by U=[u1 L12 ....... uj uk.....un] ....... (2-2) Figure 2.1. Elastic Body Subjected to Static Loading By apply%Pg the unit—displacement theorem, I. P=I€ 0’ dv ----------------------------- (2-3) 51. . nrn ....A ' - -~V' ""JI' '- '“fi~h ' -.. -.—v .4. u 'vqgn-A'OAfl F-v . _.— — ' .v....¢-.o-d t. III I -F "n— .‘ ‘ q "'v Q] \ M 5-ifi, v, ... v- I — ‘v H . a. . >~ .. , II UN where I is the unit diagonal matrix, or identity matrix. €.=[€i €2 ------ €j €}{°°' '€}J represents the compatible strains corresponding to the unit displacements in the direc- tions of P1 P2.... Pj Pk ...Pn . (7 IS the exact stress matrix corresponding to the applied forces P. Then,the strain and displacement relation can be simply represented by E=B U -------------------------------------- (2-4) where B is a differential Operator to the displacement function. By relating the stress and strain according to the elastic prOperty we have 6=DE=DBU ----------------------------- (2-5) where D is the stress-strain relation matrix. Substituting equation (2-5) into (2-3) we have T sz EDBUdv=K U __________________ (2-6) Vol. g T where K = _€_ D B dv is the generalized element Vol stiffness matrix. But we have to note here that .E: is the compatible strain correSponding to the unit displacement so §:= B. Then the generalized element stiffness matrix becomes .q 1'-.- . ' ": 2 3. a.- - gt..a0 u" 'r a» S PF ’oOsF y...“ L... uvh "’ .. ' “In.“ ,4 . fl...- Orp cu...— " _. unov _' ... H""';~ - ‘ "‘..“ :1‘.a-' ‘__,.J.a- ~ .. _ ...»wv- c '“ -ld v i] ‘ - unamb- ' 1 ‘:.u'~ 9‘“ . f°-v-..i 4‘ e .. 'V‘;Ao v.— 1..."‘ P‘ 'K 5.-., . q i” a», .‘J . '1‘.e ‘I Q I -U . . . ,“\. "“5 .3" 4: . ._ .‘ H"; . ._ ‘ “ F. ‘ “‘14'3‘ .‘ ... Dr. I ‘F. I u" .‘Ne- ._ U “Y‘h V‘s~°"f"_ -"f’ n .":. ~u .. ‘h' ‘ ‘Q‘fi v“;Y‘c~ H ‘ '~ 10 2.2.2. Overall Stiffness Matrix After each element stiffness matrix is evaluated, we may form the overall stiffness matrix for the complete structural system by using the transformation matrix A. A simple example to demonstrate the formulation of the overall stiffness matrix K will be given in Appendix (A—1). K=AKA ------------------------------- 2-8 g ( ) where the matrix A is the transformation matrix relating the element displacement to the nodal displacement for the complete structure. In practice, the multiplication of AT Kg A is never carried out. since this Operation is equivalent to the placing of elements from each element stiffness K: in their correct positions into the overall complete structural stiff- ness matrix K and then summing all the overlapping terms. This Operation is much simpler than finding the matrix prod— uct ATKgA, and it can be programmed directly into a digital computer without actually setting up the transformation matrix A . Whenever the stiffness matrix for the complete structure is being formulated, the relationship between the force and displacement for this particular structural system is known. So, for any given set of externally applied forces, the corresponding displacement can be evaluated by inverting the stiffness matrix. Usually a few different numerical Q P ‘ A ' .rv' P” IJ: ' ... ’ no ' “ -.avt ' c O ‘ R w"? w .o"' .0 . "'4 . n.- v- ...-t." h'd . I -.II' '...--u 'f‘ ‘ ,'¢ "‘ ‘ - I. .4- »;.‘_-o , ..' ,9 *9: V". I u...- ' ‘ ,. was 0-.-. u I". r""‘1“ ”a ‘ — ..as- — ... ”4.....- ‘w a : 9‘ '9‘ a ‘ ~< OOOU Nay..- . D I 1. ‘P- I“ . 2"1 ... "r O ., -..---¢ ...- 4 . ny’y‘n~».~po I ‘ ’- "- 45L-~-'.-..._. . , ' n, 0'“ ,‘5‘7-'y.n .- "~ ---- \.—..‘-..J. ." "P"o- a, AA ' .”“~~' a "" 'UI‘ '-A'.." .'-...:,,, . ’_ ‘ ‘ A . -L. . \f' . N “"H ~..-.A-_‘ \ FA ._~ ell :— V‘ .“" IuAA ..- ‘ .. F_:;r¢.‘~ ..r ... . .~ ~--..-“5 V.- A '5». ‘z: ””2 A: cha- - "~ v. .‘MN I 3‘ -. I ‘l' ”D A}- ...- ‘An- -."II§.‘ I. . ‘I‘51 F‘ ‘-~o '- ~ 'QV‘-“' ,4 .- ,A‘ ‘\ J}. -r.e a} r V.- :.\1-: ~- “ y-~ ~-A-.,‘..c‘ C“ 9‘ \l ‘ “- I ,v. n.- H:‘x~ ' “and 7... ‘1 ~‘ lA "v- . ~-_"“~‘L‘ . “"V.'_ th.‘ ‘ ‘ . ... ‘ $ v F q s v.,E:e ‘th u 4 y ‘ . I "l a s \.-...;.‘a. ,3" H ,r ”I .‘. I Q‘. a ‘ A r332“-.. ...}. ”q :m" ~uQ 13hr." P} “H. c. ‘v ‘A... ‘J ZZZ-4:“ V i \ -V' I: .,\ m I. .M ‘.A~ b 11 methods can be used to perform the inversion. The Gauss Elimination method is most commonly used in finite element analysis to solve a system of linear equations instead of inversion of a matrix. The method itself can be seen in most numerical or finite element analysis textbooks. The basic requirement for the finite element method as applied here is to assume a displacement function which will approximate the displacement distribution for the continuum system by a discrete system. To ensure the convergence to the correct result the assumed displacement function should satisfy the following criterion. (1). The number of coefficients in the displacement function representing the unknown function must at least equal the degrees of freedom associated with the element. (2)- The chosen displacement function should provide compatibility across element interfaces. (3). The chosen displacement function should not permit straining of an element to occur when the nodal displacements are caused by a rigid-body displacement,and it should also be compatible with a constant strain condition. The net effect of these two requirements is to make it mandatory that the polynomial starts with the lowest-ordered terms; that is. With a constant and then the linear terms. These lower-order terms provide for the constant-strain state and the rigid— body modes. (4). The assumed displacement function must be continuous p ,.-v . P... . ||| I l .. .I .. v . 5. 5. ur. , _ .. . w 5* r. a; my on. ht . ..r.. r|\ (K . 0‘ n “We De 5“ ‘. CO‘ s n d . I . av u .n.. .r“ w. ... S a a . . ... n. + T h. o .. , . .... S. .. . . b . a: a: “L n av 1 I.“ Cw no . a: vu . I r“ A. ... \m. «Iv v .. D. ... L . . .. an. r. a a .fi .1 .r“ w”. 3 . Ff. v” be .p .. o . 2. S .1 .f. I I . m... . . ..n .. - .. X - .. a". u .. .rm .p. ..F.. "—u ail; ....\m In“ NH]! WNW 5" f. ....“ v“ v. .,n. . . .. 7.. 7.. . o . . g . . - IHQ .r- J v 12 within the element and be differentiable to an order consistent with the variational principle eXpressing the problem. In general this criterion will be satisfied automatically if a complete polynomial is used. Of the various shapes of finite elements the triangle is perhaps the most attractive shape for plate problems. With the triangular shape, it is possible to treat plates with irregular boundaries and it is easy to vary the element size in the vicinity of stress concentrations. In this research only triangular elements will be discussed in detail. 2.2.3- Constant Strain Plane Stress Triangular Element For the simplest triangular plane stress element, as shown in Figure 2.2, assume the diSplacement distribution within or on the boundary of the element as u(x,y)==CX1 + szxz+ CXBy \ngy0=:CXM'+-C15x + CKéy where the six coefficients CX1--—-C16 can be uniquely defined from the displacements of the three vertices of the triangle. Applying the six boundary conditions, at (xi'yi) u: ui & v: vi . . = . & = . ___________________ 2-10 at (xJ,yJ) u uJ v vJ ( ) at (xk’yk) u= uk & v: vk to equations (2-9) to evaluate the unknown coefficients (11' ' ' (16 in terms of nodal displacements. "IV n'u - v. I I. .l .].v “In 1., 1 9 Val- . x X VI ..|§ .. n... N\U ]&\A/~ 1‘ AC 1 1. l I. I b : : ~‘ ”It WIN Alli]! ‘1 0. a , .94 K ? OI. I . a o p/~ I »/~ ‘ 4 m." ..u. X Figure 2.2. Triangular Plane Stress Element “Tu” {-1 x. y. RX, -y - I.1 X- y IFCX - i 1 1 1 i 1 i 4 - = 1 X- - 3 V. = 1 x. . ---’2-11 uJ J yJ CXZ J J yJ C15 ‘ ) u“. . 1 Xk yki Loc3 J ka -1 xk Jki .a 6J and then {X . .1 x y ._1 'u , 1 1 i 1 = 1 x. . u ----------------------- 2—12 CXZ J yJ J ( ) [03. . 1 Xk ykJ ukl Substituting equation (2-12) into (2-9) we have 1 -- o c o o o ' O " U(X.Y) 29l<31+y fi'i’X] y>u |(a +y| X+X 1y)u.+(a]+y. .XIXle)U]| 1 = + O o o 0 O o o + O + O v(x,y) -§A{(ai kax+kay)ul+(aJ+yklx+xlky)uJ+(ak yin ley)u£] _______________ (2-13) ,.n ‘ ‘ q . I I ‘ ‘ ‘. a ' ’ '1 ‘r: ‘F‘ .""'° 1 r. Vb‘ u u , . -.-~.L4-:‘ ' ‘ v _. ‘_ (D H m m m H — - 11.. '1 14 1 . 1 1 where A: area of the triangle =-%-1 xj yj 1 Xk yk 1 ‘ 2 (inyki-inyji) and in= xj- Xi ; yji: yj- yi etc. .==X. "X . a1 Jyk kyJ a5: xkyi- Xiyk ak= xiyj- ijk Equations (2-13) shows that the displacement within the triangular element varies linearly and depends upon displacements of the vertices. Since the two adjacent triangular elements share the same boundary grid line the displacement compatibility is automatically satisfied. The relation between strains and displacements can be represented by r , 5“ 6*; .5x _. _ 75V 8 “ ‘Ey " igy u v ficy _b_+b___ . . Lby bx . r . rka 0 yki 0 yij 0 “i =.l_ vi 2A 0 xkj O xik O xji u. ——-— (2-14) J v. L k3 ka ik yk1 31 le u: .ka OI‘ .. p C 6 ‘ ‘ ' of I U . Q . ...-..— 1’3“: f-l .1 / _n ..up. 1 ’ r‘. ... th-F ~_.-‘..-~av-6 a a lop. ~vpp-p-- P' v .A f - man-A.- —~ . -. 'I“" A“ A ~—\ r-J f ...‘.a-~ «unv ' ' -‘ - A-nv. ,. o .. >‘ ...”..u.‘ .‘.’ . . . II a... 5' ...“ A 'vod-gv‘o. A p . s I...- -~ :1 a ” '. A 4- C “In: - I 1 ‘ “:- 0-. ..‘QI. .v‘c U< I ! ! A ‘5'». --=re D . u . ‘ F“ .35. L. 15 yjk O yki O yij O __1_ where '2A 0 xkj O xik O xji kaj yjk Xik yki in yij d From equation (2-14) we can see that the assumed displacement function will not only lead to linear displace— ment variation within the triangular element, but it also provides the constant strain distribution over the triangular element. The stress field after applying the stress-strain relation will satisfy the stress-equilibrium equations since it is a constant independent Of x and y. For a linear isotrOpic elastic material the stress— strain relationship for plane stress is (7:: D e E z/ o where D=‘1j_l7W 1 O L . or 17x i 1 b’ O -P€}{ ] 6y :—E'2U 1 O €y ----------- (2—15) 147 LTxy 0 O Ail/“J 7’ny where E is the Young's modulus, I} is the Poisson's ratio. Then the stress will relate to the nodal displace— ment by substituting equation (2-14) into equation (2-15). ‘ 5.....- h. ......“ 16 We have 6x 6 E v Y ‘ 2A(1- 5) ' 7. ll L XV ’ ' n i u. ij ng yki Xik yiJ’ xji 1 Vi . ka ka yki Xik yij in uJ V. _ _ _ _ J (14mm <1 wyjk (WM-1k <1 Wm <1 ”)in <1 Wm u, . 2 2 2 2 2 2 j v L kl ............... (2-16) Hence the generalized element stiffness matrix can be formulated by _ T ______________________ _ Kg—fiolB Dde -- (217) The elements of B and D are independent of x and y so can be removed from integration. For a uniform thickness plate, equation (2-17) becomes Kg = A t BT D B ------------------------ (2-18) If we decompose the stiffness matrix into two parts- one related to the normal stress and one to the shearing stress- it becomes KzK +K g gn 55 where ... AC .- U C I h [.h J lIIIIIL L AV 1! \JV . . .. an t. o. , |-“ a... L .— If. I . .... a... .n a : .h. v . «s. ... .2. ..h AR. 2: in :- ‘5 i 1? '2 l ka ijyjk yjkyki Xikyjk yjkyij yjkxji 2 xkj ijyki ijxik xkjyij ijxji 2 _Et yki xikyki ykiyij xjiyki K = en 4A(1- ) 2 L? Xik Xikyij Xikxji 2 X y symmetric yij ji ij 2 yjij and k2. x .y. x .x. x .y . x .x.. x .y .~ kj kj 3k k3 1k kJ k1 kj 31 kJ ij 2 yjk yjkxik yjkyki yjkxji yjkyij 2 Et Xik Xikyki Xikxji Xikyij Kgs=8At1ivD 2 yki ykixji ykiyij 2 Symmetric in inyij 2 L. yij . As mentioned above, the displacements vary linearly along the element boundaries. When the nodal displacements are the same, the displacements of two adjacent elements coincide along the common boundary. Thus. complete continui- ty in displacements is preserved. The derivatives of the displacements, and consequently the stress, are constant in each element but discontinuous across the boundaries. The relation of these computed constant stresses for each individual element to the actual continuum system is difficult to determine. Although the constant strain triangular elements do I“: «V-F ‘ 4 ‘ :- Aan 1". Y ‘:':'~' ~\ IVCOO'OOV'U - .... 0...; .A IA»: — .11.. .-.“.v id at -‘ ..' I°‘~7\A R . 0-0 h. \ "O‘v0‘-v 5. (h . '7' 9‘ HUG E“- m ( V‘ .... 4.. . - . . “ '3 3r "-‘~- - F ‘9" A n In 7‘ ' ...E 3 “w“ n (I) 0 3,. U (6 1 m J '- A :- O 0 18 give good results when stress gradients are moderate, eXpe- rience with these elements has demonstrated the need for more refined element. The improvement is Obtained by increasing either the number of nodes for each element, or the number of kinematic degrees of freedom at each node. Figure ( 2.4 ) lists a group of improved element types and references. In most cases the improved element type needs more time to compute the stiffness matrix, but for the structure system with step stress gradients the improved element gives better results. Comparison of results between the constant strain triangular element and the improved T-6 triangular element will be discussed in Chapter III. 2.3. High Order Plane Stress Element ( T-6 ) The improved triangular element introduced here is the one shown in row one of Figure ( 2.4 ) . This element has twelve degrees of freedom in total and has three additional mid-side nodes. Figure 2L3. High Order Plane Stress Triangular Element \c ...i..lI.-... 122.4. I N .91 . VI 19 NO. OF DEGREES ELEMENT NODAL OF FREEDOM DEGREE OF TYPE PARAMETERS PER ELEMENT POLYNOMIAL REFERENCE. ux uy 12 2 [12] ' 18 ux uy +2 INTERNAL 3 [26] u oux oux x: bx: ‘by Q 18 [ ] +2 INTERNAL 3 29 buy' buy Jr OX: ‘Oy Figure 2.4. Improved Triangular Element Types for Plane Stress Analysis U" ‘9‘ -n.’ . r4 A..~‘ / Aor- / \ 4' a C I- .-v.‘ o J . . A "on? '7.» .4 " ; to.“ “.“ ‘. FA l". ..r‘ " ‘- -‘ . ...v n' '.. ,2-.- ‘ , . : - r- . - --.-l1-.. "- g n a ' ..-..-‘ -‘ -.. ’ -- ' b .-.—‘4‘“- ~ _ a .”..'..--v' If" '_ .-..J- "' "‘ .. 1 ' {a V’s-F" I‘ V. --- 'A-" “‘ , - ...,p.-F--r- Q .— ... -~..~ "~.dl.. ‘ . --’-"~P Ora. ‘--..vd;\ou a..- n w - ‘ . . a .v -.. I- '7' —-< .-.---«--~ u. a J...- O ‘ ‘ "' ‘v -pA ,- ...... -.. ...: . . ..- r A: o.” 4.4 ' - .. v- ...- C n.. .- r III! -oov C a ~.: 1' ‘2’ F: ‘f‘ a.” . ‘.__'- ‘I v - .x. I H Dc, 5' a 20 The displacement functions are described by two d order polynomials; each has 6 coefficients which complete 2n match the 6 nodal displacements for each displacement compo— nent. The displacements at the corner and mid-side nodes can uniquely define a second degree displacement curve, the compatibility criterion is satisfied and the complete continuity of displacement is preserved. Stresses and strains will vary linearly within each element, which is the major improvement in comparison with the constant strain element. Although the stresses and strains are discontinuous along the element boundaries, the gaps become substantially less severe than in the case of the constant strain element. The deriva- tion of the stiffness matrix is straightforward. The assumed displacement functions are, in complete Quadratic form, u(x, y): a 11+a2x+a3y+a4x2+a5xy+a6y2 V(X 1 Y): (X 7+Cx8X+a9y+a1 Ox2+allxy+a1 23,2 01? if). u 1 x y x2 xy y2 0 o o o o o f:[ ] 2 2 = O O O O O O 1 x y x xy y = chx "“ (2-19) ”§?-__--______- 12. (I) L. . V pp, "" .0.“ U .. onf‘. ”7“, "" .. : CL- - .“,-.',::“~v rs” .. ..&.a-o-" o .Aun-A'O: ' ‘ 4 .---':—--u J‘--d- O ' I .... .. or Q,’ a a At- .- .uv-‘~- v..- -‘I ' Q o.~,: 0" A: n. ”-..- av ~..-- 4 v """~0wo -n 6' ' '\~A l ""-*‘.J -\J - “..-. V .. .4..,:" a I A . . “I Av- ' Q ‘0 "flA’ e D~A ‘1‘-.. ‘vn..: o O . .‘3 37“ ”‘w 0. - v “>-‘.. - V. :‘Mn‘. ~ -... - a f‘ '. I". 1-“, h- - . w ‘P§ I Cfrr‘. A ."'--. f‘ F H _‘-‘¢"‘ 1 ‘ 4 . H I ‘v- . . - A ¢. ‘ “Ma-.13 .‘- ~4" 21 where u & v are the displacement components of the element in the x & y directions respectively.- C113.....CX12 are 12 coefficients to be determined. The boundary conditions used to determine the constants are nodal displacements u & v at the six nodes. 12 boundary conditions are sufficient to define the 12 constants uniquely, so the element is a conforming one. Because the inverse of a 12 by 12 matrix is not an easy thing to calculate, a numerical integration technique is necessary to form the element stiffness matrix. For this purpose, a local coordinate system should be established for each element. If we choose the centroid of the triangle as the origin of the local coordinate system, the numerical integration becomes much simpler. Formulas for the area integration are given in Appendix ( A-2 ) . After each element stiffness matrix has been formulated a transformation multiplication is necessary to form the overall stiffness matrix. Applying the 12 boundary conditions in equation (2-1) we have F 1 PwN C)» u C>JC>1<:h‘Ch C>JC>1 O\U\ F/L .1.“ . _ V . J Y.“ n O V}. r. . . A _ w." ac A v ‘1 Auv .{u H“ “A In. , n, . "1 ru .3 O a: S ab .— g n‘ A d a aflu C. ..1 an .Hu Alv Y“ AV ”It. Alv . k I‘ « FV A; Mi 3.. ‘ a d . . 11‘. . . t . ll . . 2 . . v. F 1. h u. NM 3 f b .. 22 = c O( ---------------------------------------- (2-20) From equation (2-20) we can solve CX in terms of nodal displacement by inversion of the square matrix C provided ICI % 0. So we have (X = c ‘1 69 —————————————————————— (2—21) Substitute equation (2—21) into equation (2—19); then -1 e f = P C q 6 For the linear elastic material the strain-displace— ment relation is 0. fiu‘: ’u'w‘ *n; ‘ Szrava - ... 23 r 1 an . .p - €x bx 0102xyoooooooO(1 v 5 e = 6y. = -%:7- = o o o o o o o o 1 o x 2y ; u v ' [,ny g—y- a: _0 0-1 0 x2y0102xy OJICXIZJ - J ._.Q a =Q 0-169 ----------------------- (2-22) And the stress-displacement relation is simply 6=dy =De-—-DQG=DQC_1(5e T ________________________ (2—23) where D is the stress-strain relation matrix and has the value of '11! o1 I) ==3€3722l 1 O L 24 for isotrOpic materials with E: Young's modulus, bi: Poissons ratio. "n nL’ O E 2 D '-'-' (1-1‘1 2) 112/2 1 02 _O O m(1-nU2)d for orthotrOpic materials, where E1, bi are associated with the behaviour in plane of the strata in the x-direction and E2, U2, G2 with a E1 G1 direction normal to these and n: E , m: G . 2 2 oh - .r G ...e V‘ .q, r o “ bony : a.» D II ‘ . ,‘ , .. y r p . -- ' ache :e ‘0 T‘ .htv r u.,. a H A c ~-_~r a ¢ ,_ .v as v...: ..IA ‘ H. II.‘ I .. F‘d-e 17" ‘ A.._ ."3 . A'“*:n‘. ..~.."LV‘AS EA ‘- ' ;.Y“ RI§‘X f-Q +‘r 'l 7' C. I W 24 Then the element stiffness matrix can be eXpressed K =va].([C]-1)T Q T D Q [C]‘1 dv ---- (2—24) The elements of matrix C are independent of x and as y, so it can be placed outside of the integration, and equation (2-24) becomes K =([c]'1)TL01 QT D Q dv. [c2]‘1 By using the integration formulas as given in Appendix ( A—2 ). the results for the integration of _£;ea9 T D Q dx dy are as given in Appendices (A-3) & (A-u) for isotrOpic & orthotrOpic cases. The application of this type of element will be given in Chapter III. 2.h. Plate Bending Element in General The plate bending problem can be treated in the same manner as the plane stress problem; namely, by subdividing the plate into a finite number of elements interconnected by fictitious boundary lines, and then forming the stiffness matrix for the idealized structural system. In the case of bending of a plate, the state of deformation can be described by one quantity. That is the lateral displacement, w, of the "middle plane of the plate". Hence, W and its derivatives up to a chosen order are taken as nodal parameters. The assumed displacement function in general must satisfy the following requirements: (1). Completeness: All rigid body modes and uniform strain states (constant curvature) must be included in the . . pu- -- 2"...5n -' . ".... , 0-. n .' on. . an‘ on.-‘ “ "-t’ :J ‘..~Av . ,— . I -. “A'v-"“ avab' . ;‘ 'mr,. ,..-. : :.n~-~*~‘~ ... : zpz O ...: ,--......‘-| I" 'v; I q 9 'I' v --~q l' 4;. - u- A...'dv-.‘ «--. any. “ I ..4 ...v ‘ "Q - ‘ a .... out: ,' 'v‘- I -V v fin h o. n QA'.F I Mur q "..V"-.QAAE 9“ II... a C ... “ “S‘Igrt I .‘ V“‘irc. t .' a». ... ‘H L‘- W 4. ' yd...“ Far ‘ '1‘ \ ’Grg ~ are ah . =E‘“c . . ... will d! v.‘ "a 9 ' J' J? ‘ , an." Us. .-- hi 25 displacement function. In other words, the lower terms such as 1. x, y, x2, xy, y2 (or their equivalent in other coordinate systems) must be included in the polynomial. (2). Compatibility: (a). The assumed displacement function w(x,y) must be continuous and have continuous first derivatives inside the element. (b). w(x,y) and its normal derivatives must be uniquely defined along the boundary lines. The assumed functions which satisfy both of the above requirements lead to conforming elements. By using Sobolev's inclusion theorem (See page 401 , ref. 6 ) it can be shown that the displacements will converge uniformly and the SlOpeS and curvatures, as well as bending moments, will converge in the mean for the conforming element. Some non- conforming elements which violate the SlOpe continuity requirement have been used and have given even better results in some particular cases; they may also be simpler to derive. There are always certain disadvantages such as: (1) conver- gence will depend on the pattern of subdivision (2) curva- tures and bending moments may not converge at all even though the strain energy does. If a conforming element satisfies not only the above two requirements but also has continuous second derivatives (curvatures) inside the element a quick convergence toward much better results can be always exPected. For such an element a minimum of 6 degrees of freedom (w 5‘" bw £3! bb__2___w Aiv!) is required at each corner 5X, 5y! X2, by' by2 ‘ .' ’A'V'. - “: Pva" ’ .o" . 4 A p r' 'v" ov‘ 5“.'U .... v ,. 1 O .. ..,. A v- ,. ; .-.“ ... . .. :A’\ -"a c v- 0'. t‘ l-..”- -\‘ .l: --....4.. .. ... - “a. W .\ .. . .... ,. § ' ~‘ ‘. A U . . . "" 5' q‘ L s. ‘,F " ,IC H'u‘u: “' mu,- ' f‘ *v s.‘ “2‘ ‘ Q” a” R v '~“~‘ I. o“ ‘A ... s .p ‘ (~ "‘1’.sz ‘._ T. --II - Au s,“ VI,’ ““§: +0 '- ‘ ' in,“ . o' 7 . l N J 'T A. . f‘ ‘v'c.’ 27 flexural rigidity of the plate. (h), The element stiffness matrix is T K: I P D P dV ----------------------- (2-28) Vol.g g (5), The formulation of the overall stiffness matrix for the complete structural system follows the same pattern as described in the plane stress section. 2.5, 5th Order Polynomial Triangular Plate Element (T-18) As discussed in the previous section, the assumed displacement function must satisfy certain criteria as minimum requirements. Practically speaking, to satisfy those requirements does not guarantee a quick convergence to a good result for any kind of a structure system. Adini [25] suggested the following form of displace— ment function w(x,y)=(X1+(X2xi-(13y+ CX “162+ a5y2+a6x3+a7x2y+a 8xy2+C19y3 It will be noted that only 9 terms are included, corresponding with the 9 degrees of freedom of the triangular plate. The complete polynomial series includes 10 terms up to the cubic elements. Adini chose to omit the uniform twist term from his eXpression in order to maintain symmetry, thus making it impossible to represent constant twist, and leading to an element which is far too stiff. (See page 520, ref. 5 ) Tocher [27] tried the following assumption: w ( x , y )=a14a2x+%y+q,x2+(x5xy+%y2+a7x3+% (x2y+xy2 )+%y3 ~ ”I. D “ . -..v “s ' - ...P‘“-r I. ‘__~.- a '. ,. -’_:V‘_ " ~ :,-:.t.-..~v ,...-. O '.. '..-r ‘V‘ V‘ ' i H“ ...v. ;. ..-... a— . harp .V'v..- Q '!.I"AH poo ,- —l ' / 1.4.-nov‘ \- V a v . "“ . “w- - ~ r“:— ..J' M->-‘a~.r.v..- . ""“- cuva‘Q. \ a ......_.__ ~ .~ A v A,Q v " .' “C22“; .‘.. “'"-'\§‘ ‘ 2 a. ‘ .. ,_ ..‘IU" I‘vfiei :, .. -.. r: vhf‘fic . U l. ‘ n" ‘4?" s I I. *2 “h "a... .§.‘: .J\~ I UV . . I. .: r.“ . ‘AJJ (‘v- s. \ fa- . h- e A '0“, __~ “'h.€‘r- ..- K I y.‘.‘v-~.n . .. ..‘.."VJQ -~. .1. . ‘uh‘v,n‘. .. . .' ‘1 D Q‘-IGC '\ , . s‘ -'€- {34: ... "l T‘H1 c 9-: ‘r. 28 The symmetry of this expression has been maintained by combining two of the cubic terms into a single coordinate. Reasonably good results have been obtained in some cases with this element, but unfortunately a serious lack of invariance has been introduced. For certain orientations of the element sides with respect to the coordinate axes, the transformation matrix becomes singular. (See page 215 , ref. 26 ) Tocher's second element stiffness approximation was obtained by assuming the complete 10—term polynomial displacement expression and then reducing it to a 9-degree of freedom system by the Ritz method. This procedure has the advantage of producing a stiffness matrix which is invariant with regard to the orientation of the element in the x—y plane; however, the element flexibility is greatly increased in the process. (See page 520 , ref. 5 ) Zienkiewicz [12] suggested a stiffness matrix for the triangular element with 9 degrees of freedom by making use of area coordinates. Displacement compatibility is not obtained. The deformation is continuous across the element boundaries but the edge normal lepe is not continuous. Zienkiewicz also indicated that if continuity of derivatives higher than the first is accepted at nodes (thus imposing a certain constraint on non-homogeneous situations) the generation of SlOpe and deflection compatible elements ° - . ow ow 32w presents less difficulty. If one con81ders “D's; 163;- g—Z— D I X i Y: byZ as nodal degrees of freedom, a triangular element 29 will involve at least eighteen degrees of freedom. But a complete fifth-order polynomial contains twenty-one terms; therefore , if we add three normal SlOpeS at the mid-side as additional degrees of freedom, a sufficient number of equations appears to exist for which the displacement function can be found. In general, the existence of the mid-side nodes with their single degree of freedom along the sides is a complication. They have a severe disadvantage in that they cause a significant increase of the width of the band in the banded complete structure stiffness matrix. It is possible, however, to constrain these by allowing only a cubic variation of the normal lepe along each triangle side and reducing the total degrees of freedom to 18 and maintaining complete displacement & lepe compatibility. This type of element is named the T-18 element. It is, in fact, the more useful element in practice. However, it should always be borne in mind that it involves an inconsistency when non-homogeneous, stepwise variation occurs. The T-18 triangular plate bending element was develOp- ed by Kolbein Bell [9] and was first published in the International Journal for Numerical Method in Engineering, Vol. 1 No. 1 Jan-March 1969, under the title of "A refined triangular plate bending finite element". Later the subject was modified and republished in the book of "Finite Element Methods in stress Analysis" edited by I. Holand and K- Bell. This study mainly follows his theoretical approach but extends m". J ..o a." . I O ".":. .: (\FV‘ C L. .- Vfil.‘ . .- .. 100' ‘. J I '."A.q . a. A"? ‘ ’ . \ P l “...avon ‘U V"" ‘ . ~ 9‘ fu-n . . r on ... .t ’ 3"» arm-...: «A ..:. Tue u ...i r c ‘..e ‘v'e ,_ a v .rfi.: u“. .f‘ wTS wt‘ c": Atop .a 4‘-‘9Y‘cv. 30 it by writing a computer program with both isotrOpic and orthotrOpic cases. The practical application for the computer program is contained in the next chapter. A brief summary of the method will be introduced here. The theoretical approach is explained in more detail in the papers referenced above. Outline of T-18 element develOpment procedure is as following: (1). Select the complete fifth-order polynomial as the displacement function which contains 21 terms w ( x , y ) =03+%X+%y+%x2+%xyfl6y2a7x340gx2y+ogxy2+0i 03/ 3 +05 1 {nos £3Y+q 3X2 yzqul'oi 5yu+oi 6X5+Oi 7X“y ‘lq 8x3y2+q 9x2y3+oéoxyu+cx21 y5 = WT O( ...................................... (2-29) where the vector W contains the assumed displacement th. functions which form a complete 5 order polynomial. (2), The curvature vector C is obtained by taking the differential of equation (2-29) (9.23.4. 1 M2 2 C : 3y; = Pga ----------------------- (2‘30) b2 2 W L 825V; (3). The moment vector is defined by H A 0 V : .'. 00 ‘I I .. ‘7 I ‘- u .' r unw‘ n ‘C ‘ ,.'v- M -.. . A-O' -: I h] ' In (.1) 0 4 ) . a,- ": -u - . _» ,— ." “a.--“ ..A': . In 5 Q A- .. 7‘ " » ”Aw f':r; n, = K D . X',’ \r w} H. 3:» 31 M nyl where D is the material prOperty matrix. In the case of isotrOpic plates, r1 2] 3 Et 2} D = 12(1-i?) 3 and E 18 Young s modulus, 2/ IS P01sson s ratio. 12(1"L;) is called the flexural rigidity of the plate. In the case of orthotrOpic material Where E t3 E t3 U E t3 D = X2711) ‘ D = y = D = x ‘V 7 x 12(1- X y y 12H-quy) 1 12““wa Gt3 . D :12(1'VxVy) With UE =UE xy y X X y (4) The generalized element stiffness matrix is obtained by K =f PgDPgdv=t.f PEDPgdAu-(Z-BZ) g21 Vol ’ ' Area ’ The results of the multiplication of FED Pg for isotrOpic and orthotrOpic materials will be listed in I? $175,“ a ~.__ I -' Fo~h C "3:.3- ' V“ w"- . p- R ‘7 ' O .. w-v ... .4 " .a-.4b-v ..-- ' . b --n ‘\ ..- VI 0 x c u u“ . .a c“ r- ‘L V; 1“? 1 2 "TAT-ha : V. 1 F‘ L V. mp «.‘C. ‘r: :u..u‘ n.‘ L, ‘IA 'O-— (L U F.A~ 32 Appendices ( A-5 ) and ( A-6 ) respectively. If we locate the origin of the local coordinate system at the centroid of the triangle, then area integration can be much simpler. The results of the area integration for both the isotrOpic and orthotrOpic cases will be completely listed in Appendices (A-7) and (A—8) (5). The nodal parameters are listed in the vector Figure 2.5. Nodal Point System and Nodal Parameters for T-21 Element T V =[ v1 v2 v3 v4 v5 V6 ] whe re 2) w b 52 T “11:“ "fit? 3'; W gt]: 2 S 0’ m 2 9’s 2 T 2) [vi] = to: 7:} j=u.5.6 The indices i and j denote the local nodal point number as shown in Figure (2.5). (6). Applying boundary conditions by inserting all nodal coordinate data into equation (2-29) the relation ,.,,-; .. ‘ A '. 0:-.- .C' . a . run- u- ‘ v'b " ,A‘ .. V V' .. r‘ 4... .."~. «1.5.. ' ‘ . ..--.— w 1?;vu- .- .4 ..v. . o : :‘zr‘ ~ 2 I-u-J~.o4 .';, .20! N 1“ 33 between nodal parameter V and generalized displacement (X is obtained as V = GT 0( --------------------------------- (2-34) The elements of the 21 by 21 square matrix GT are listed in Appendix (A-9) . If the square matrix is not singular, then C! = ( GT )'1 v = BT v ------------------- (2-35) where BT is the inverse of GT. (7). The element stiffness matrix for the T—21 element is obtained by K = B K BT ---------------------------- (2-36) 21 g21 (8). To modify the T—21 element into T—18 element we eliminate the mid-side degrees of freedom by introducing a cubic variation on the normal SlOpe égg- along the element edges. This is obtained by expressing égg— at the mid-side nodes in terms of the corner parameters. bW %.¥.(T)=cubic function of T _ 2 3 —a0+a1T+a2T +a3T Figure 2.6. Variation of the Normal SlOpe Along an Arbitrary Edge *‘ rl -.. a A. ....- 2 ¢ 4 ...-d‘ -.. l"‘ . .'. -- ' IA .4 c p Q - fl ... v__ - ... ..-v -r a ... - V '.' ~-vv- I l— ~ .- .-- Ib‘ .- . 2 ‘ Q 'v'v-v-v n ‘ ' a.“ ..-...— J,- - «.‘W3nc Ay- o'-.-|-.~ V.- ., , W'n-.. .a.v:_ures t! I ”A ... ....”c A .v ‘ . ‘r. .__ A; "Aro-w. -‘ VVL‘MA o ‘A ... * l I A Q¥ sgu' _ a I b ’V 31:? o v Q‘vlj “i I “ a . ' n0 3h The transformation matrix, A, relating the mid-side normal SlOpe to the corner parameters is given in Appendix ( A-10). (9). By applying the transformation matrix, A, onto element stiffness of T-21, the element stiffness matrix for the T-18 element is obtained by _ T K18” A K21 A ---------------------------- (2-37) where K18 is the stiffness matrix for a fully compatible triangular plate bending element with nodal points at the corners only and with 3 by 6 = 18 degrees of freedom. (10). The boundary condition for a plate can be specified more precisely in the T-18 element because the curvatures at the nodal points are taken as parameters. For more detail about how to convert the boundary conditions into computer language the original paper [9] is referred to. 2.6. The DevelOpment of T-6 and T-18 Finite Element Computer Programs The initial goal of this study was to develOp a suit- able computer program which could apply to folded plate structures with isotrOpic and orthotrOpic prOperties. The difficulty in handling the multiple-folded plate structure is the stress concentration at the joint of multiple—folded intersection lines. Each plate comes from a different angle and plane in all the three dimensions. The discontin- uity of stress and strain across the boundary line causes a ’:AAAv‘ .-.. 3 ‘vv-- ' . J "' ‘.'-' - o v .A‘V‘- :- "3 --.-' _. unv ; O . "e o :--v' u. .. .... “'7': a .~— ,_ .o ...v , s A - .. c ...I " Hood‘ .n ‘- q a c . .. .PP ‘~‘ ~02 , "’ '-.a- .. ...v 0" o ..--.,. . -5, .— ...-v. -.a‘u- .< u- ' -,. lit-ways. a-I . '- --‘nv. Ua-yv - a o . a”- -h .» - n A A o u..- v-v..-..-~ 'F- Fan 1,, “hp-\a :. : --vc~n~u ‘5‘ . .. I‘a-a -..c m: 1 ....»v.;,.. ‘ ... “ I '3 cy-O‘1 ‘“ Io.» ‘- $4. ‘Q a “Wed -g-‘~‘ 35 sharp discontinuity when a constant strain element is used. At the joint of intersection lines it is even harder to predict the stress distribution. In the practical design case when a simple folded structure is handled, we assume it as a shell section with a much more refined element size at the folded line. This arrangement does help to give a better interpretation on stress distribution at the folded line. Experience has shown that a good number of small size elements are required to make this procedure effective. Also the elements which are adjacent to those small ones should increase gradually in size as the distances from the joint in— crease.The ratio between the largest and the smallest element in the entire structure idealization should be less than about 10. Because of all those limitations to the analysis of a folded-plate structure, a large number of nodal points and elements is needed. However, there is always a limit to the capacity of the computer facility. In the design of the containment internal shield wall of nuclear power plants the commonly used computer programs, such as SAP, STARDYNE, STRUDL, (Approved by Atomic Energy Commission) are used for the analysis. A safety factor of up to 2.5 is used to cover the dynamic impact. Above that, AEC recently requested that another AO% margin should be used on the reinforcement design to cover the unpredictable items such as the difference between the real and the idealized finite element solution. So the commonly used computer program is not suitable for the analyses of complex folded-plate structures. The ‘ au- 4- _'_.‘r’ J- "-L‘..p . . a- F .r - '. f ‘ -...— I‘ _. v ‘ -.. ‘- .. CI'IA “v.4 r g - o" . “ ..- d - ...- _,..¢ on -v- .4" , Q u ,_ _ can I' . - r‘ _. w-. —¢.‘. ..na-v“ ~"" -\- ‘ ..--1..¢ w4~ . ~ ..--Aq ca- v‘ .-~-- v n o . ;;';v‘ 9.... --.~..ad ..y - T 8’-.. - :4“ VA»- "“"w- 5.- ‘ ‘it‘;fi-‘-va .. ..s..- ..-. .. . I .q - fl ~O.... ‘-\J ”N . ... -.“ ‘I “ .:.“~~ o.“~. 7‘: V —-.-.. 7-. V a 5‘ . ..-: fa flay 1... Va 8 . .1- IA ‘flii “3‘ .~ 36 original objective was to develOp a general computer program by using a refined element. HOpefully, the develOped program would serve as a tool for the analyses of the entire line of folded plate structures and eliminate the possible errors. Due to both the theoretical and technical difficulty this goal has not been achieved. As an alternative, the present subject uses the develOped program to study the clamped edge square plate instead. For the commonly used computer programs such as ELAS, SAP or STARDYNE the element types were introduced in Sections 2.2 and 2.4. The refined element types T-6 and T-18 which we used to develOp our computer programming were discussed in Sections 2.3 and 2.5 respectively. 2.6.1. T-6 Triangular Plane Stress Element Programming. The T-6 triangular plane stress element was develOped in the second half of 1971. The entire programming and debugging work took about 800 man hours. The program now has the capability of analysing any plane stress problem in both the isotrOpic and orthotrOpic linear elastic materials. With the help of mid-side nodal points as parameters, the program can handle the uniform load more precisely. The prOgram was develOped with a very simple form of input and an easily understood users' manual. The text of the program is listed in Appendix ( A-12 ) and the users' manual is given in Appendix ( A-11 ). To save the users' input Preparation time, a subroutine to calculate the mid-side . ‘nfi“‘ ..h u- ’ P .v"- ..O" m‘ ’0‘ ‘9 .- .v 0",“o. . -a '_ 7‘0- ‘0'“ ‘ OfiF -p" f - 1'” ...: .-- 'u . u - n- 'F‘1‘, 'D b - v..a V —' ...- a VG . ‘___‘... co- — ;: ‘ .'-....— v - ‘ ‘ --<- g,- a. u“ A ~ ‘ ..--Ab '- . O A " "s f‘ r: ... noov v-I-.. 9.. ‘Cu- ‘ \’ 37 node coordinates is included, so the user does not have to calculate the coordinates for the mid-side nodes. The only difficulty encountered during the develOpment was the transformation matrix which transforms the generalized element stiffness matrix from the local coordinate system into the global coordinate system. The formulation of the overall stiffness matrix is not just a summation of all the overlap terms from each element stiffness matrix as discussed in the constant strain plane stress element in Section 2.2. We used the global coordinate system to form the element stiffness matrix instead. The relation between the two systems and its effect on the numerical integration is shown in Figure ( 2.7 ) X = a+x Yzb+y etc. 0 P”X: where A is the area of the triangle. Figure 2.7. Local and Global Coordinate Systems f- .4 "5v 9 .-v- -9” tr" ‘—.J,~- - ’n . ,. “ l >- I. I - . ..-l~. . -v III-u A P... ...V . a ,,-' nn""- ...“: givi“ " . .. '3; nus—v- 1‘: l -v‘.‘ ..., .-.”..p' . " ' ‘.l 0.. ~ P. ‘ 0‘ A” -4-.-~--v.«c. s . 0‘ ‘ c n..- o f‘ d — —r I s--—.-..4 a- l I ‘:'°.vv‘v-‘ A n- ..v.-'-b‘v 'u. a ’l.‘ . - ,“S a 1 'vcdcv don‘t I q A Pi v ' ,f 1‘. t-r .A. ‘U' . r‘ J vu ‘ C O '9’-A~ - ~ b-3‘a ‘ A, .U-K. “‘4‘. IT 2 “o A.'. v '5. .. l' — ... ‘ P 7-, x‘. {-- -4 150:: 3.4" ,‘ 4 d *r! M 'r‘ f‘ -5" ‘ l ', ‘5':1\ / (91": (T. ~. ‘wn 11 --, :1 y‘fi,‘ m 38 The solution of problems using the T—6 element computer program will be shown in Chapter III. 2.6.2. T-18 Triangular Plate Bending Element Programming The T-18 triangular plate bending element program took about 2400 manhours for develOpment and debugging. The complexity of each matrix and its transformation multiplication made the work difficult. The 19th by 19th element of the 21 by 21 element stiffness matrix for the isotrOpic case can be as long as 4y6 ~48L/x2yu + 36x4y 2 before the numerical integration, and becomes after the numerical integration. where _1_-_5. 6 P06 14A P041302 BT( y1 +y2 +y3) 2 P42 TEX (6P40P02+24P20P22+30P30P12+“8P31 P11+75P21) 11A 4 2 4 2 4 2 -'§35'(x1 y1 +x2 y2 +x3 y3 ) qu— —-4- (6P O4P20+24P02P 22+3OPOZP21+48P13P11+75PEZ ) _%fi%. (XE y? +x§ y; +x2y g) and P04: 3%'( y? +yg +yg ) _ A 4 4 4 P4O__ 30 ( x1 +x2 +x3 ) P =1-A-( 3’1 WE +y§) u... \v ‘ ‘fiwfih O". ‘I 39 _._A. 2 2 2 P20” 12 (x1 +x2 +x3 ) ...iL 2 2 2 P12‘ 30 ( x13’1 +X2Y2 +X3y} ) P ='-A-( x y +x y +x y ) 11 12 1 1 2 2 3 3 ___£L 2 2 2 2 2 P22‘ 30 ( X1 y1 +X2 y2 +X3 y3 ) ...fiL 2 2 2 P21‘ 30 ( x1 y1 +X2 y2 +X3 y3 ) _ i 3 3 3 _ .5; 3 3 3 P30— 30 ( x1 +x2 +x3 ) 31 3o 1 1 2 2 3 3 ...lL 3 3 3 P13 30 ( X1Y1 +x2Y2 +x3y3 ) where xi , yi are the x and y coordinates for the ith corner of the triangle and A is the area of the triangle. All of these computations for both the isotrOpic and orthotrOpic cases have to be eXplicitly calculated before the programming starts. To convert each computation into the computer language is another challenging job,Changes in the University computer system periodically during the past few years has also created some additional complexity to the work. The physical interpretation of some of the terms in the T-18 element theory is obscure. For example, the energy is the product of force and displacement or moment and lepe, but what is the physical quantity which combines with the curvatures to produce energy ? Without ... 1. v ... v” ... ...“ .1 a 7. ~.. 0. . , .3 ’5. S. 2. p2 . . a. r” ~ ad a, C. ..i ... ... v“ I; . . .. 2. ... r” . . ~.. ... .C r” L. C. o . .1 v». .. Y... .. . w” I . 2. 5.. 2* a ... .. a... . . . . n: . . ... .mu ... ...; Y. n. r” C.» .... .. . r” ... L. .'H r. ... J. .4 Au. .... C. t. . . .p» . .. v. ...M . . . . .4 n w_ ..A o . A . A. r». a: up. '4. ... .va .. . up. . u. ”aw w»_ H u “\h ...“ c. .. uh... I‘muu. .F ... no knowing this, a transformation to the curvature terms becomes impossible. In his book on Finite Element in Stress Analysis, Bell indicated that the physical interpretation is not possible. Since Bell was not combining bending with in-plane action, such an interpretation wasn't necessary. For a general folded—plate program a transformation must take place for the combined in—plane and bending parameters. Without knowing its energy derivation a transformation matrix can not be formulated. After considerable research and discussion we decided to adjust our previous goal from combining T-6 and T-18 elements to the present objective of studying the square plate panel. Hopefully, in the near future a theoretical interpretation can be found and T—6 & T-18 elements can be combined to serve as a general computer program for the folded-plate structures. The users' manual for the program is given in Appendix ( A—11 ) . A listing of the T-18 element general plate bending computer program can be found in Appendix ( A—12 ) . The program can be used for either isotrOpic or orthotrOpic cases. The loading can be a uniformly distributed load or a concentrated force. The boundary condition can be either simply-supported or clamped edges. Use of the T—18 element computer program to study the square plate with clamped edges will be shown in Chapter III. o ‘- .. A or .1 r w v -. - -s . -. J...» _--- ,- - ... .-ro:_v~ - .- J, --.. a- , . v- .‘d‘ a -r' — ~ ..--» 9"- a. o a .. .AnA ‘v a .— .... .«.-..v.a.-- 1 - 's. Aha-pay . .1 .- p.- ‘-v~~c-- -v~ A ’- ‘V ‘ a .- \' ...: up...’ .- a . a ‘ D t ';Y‘ (‘9 *4"~-~ J- - ‘ :l _ A 1' ~35 . ‘7‘ i ‘ - a- P/ f” ..- "“ v'. ' ’b. ‘ . ' — our ”fr! ....,__" r“‘ J- ‘ - ..,‘ A” .u‘i“: '1 ‘ u CA PV- ~ --...C"‘,.::rn» "uv..v Ln 4 I . ‘~ a. ‘: rc"V‘I-,.: vQ‘.“:‘. . 1 t 7. 'O we I. V ~:~“:=A o, ... o w- III. Application and Comparison of Results 3.1. Plane Stress Problem The ELAS [15] constant strain element program and the T-6 refined plane stress program were used to study a simply-supported beam with a concentrated load applied at the center of the beam. The finite-element idealization models for ELAS program are given in Figures 3.1 through 3.4. The modeling for the T-6 program can be found in Figure 3.5. Displacements in both x and y directions for all the models are shown in Table 3.1 and 3.2 respectively. For the same number of 16 elements, the T-6 program has an error of 3.414% for the y-direction center displacement compared with 86.7 % error of ELAS output. For the ELAS 32 and 64 element models the error improved from 86.5 % to 64.0 %. The ELAS 256 element model has a much better result with a error of 31.26 % in comparison with the exact solution. Obviously the ELAS constant strain element has a very slow convergence toward the exact solution as the size of elements is refined. The main reason is that the displacement function of the constant strain element program is not suitable for representing a beam deflection problem. The comparison demonstrates that a refined element type such as the T—6 can provide much better results with a much smaller number of elements. 41 42 l:\\j)2‘\ \2‘\FV;‘\‘\.‘\‘V.§J COHpmhsmHMCOU mpooaoam ma mass swam oaasam .H.m marmaa o.osnm Ho.ammuo o.oooHum mpCmEon ma mpcflom nmmz ma _Jufl: 43 l;\\ ‘RTYL‘\ \x‘\.‘\‘\.‘\ \a‘fl SoapmssmmeOU mpcosoam mm msqm smom oagEam .m.m marmaa o.osum Ho.smmuo o.oooHum mpzoSon mm mpcaom zoos mm "u‘f >< 44 L)\ \C‘\g\.‘\‘V.‘\‘\.\c‘\‘\J CowpwssmHMcoo wpcososm so m<;:1>2] . M 95/ Di 1-1) iDe (n-l)h ALL EDGES FIXED a Y a. (X and [3 Ratios ! x ? De 7 =15— c __.- __..L- ._._ _J__.-_____.-._% m Di =Dc[1+(7—1)(%1€11.-)2] DD DJ jD D.— 10: pl Di : ID: 1D 33 .20 D ’ 1 é i-l (n-1)h 1 ALL EDGES SIMPLY-SUPPORTED a Y . 7 Ratio Figure 3.6. (X , /3 and 7 Ratios ..rflon .v..~ vv‘ n u a nu” V"--v-‘- "V‘F :- n§, U.‘ PU a: H 53 3.3. Results and Optimum Design 3.3.1. IsotrOpic Nonhomogeneous Plate (A). Effect of Varying the Boundary Stiffness Uniformly. A square plate with all edges fixed is used. PrOperties at the central portion of the plate were assumed constant. The stiffness of the elements in the boundary region is also a constant but prOportional to the center stiffness by the ratio C! as defined before. A uniform load with intensity p is applied on the plate. By taking advantage of symmetry about each 2, only one-quarter of the plate is needed for the analysis. Boundary conditions of the lepes normal to the centerlines of the cutting lines are imposed to represent the symmetrical characteristics. Finite-element structural idealization for the ELAS solutions-9 elements rectangular, 18 elements triangular, 25 elements rectangular & 100 elements rectangular are given in Figures 3.7a, 3.7b, 3.10, 3,13 For each idealization case, the following quantities are obtained through the ELAS computer outputs. (a). Deflections along a centerline. (b). Variations Of the moment per unit width, My, along the boundary, y=0. (0). Variations of the moment per unit width, My, along the centerline. y=a/2. The moments of (b) and (o) are calculated by using the nodal moments from the finite-element solution and ;.-'- 8" .1 .‘. ‘ fA'W‘Jn SIJ‘.‘a bu 4:151 ', . n‘; uremc)‘. $ Ty: 54 calculating the corresponding distributed moment intensity. For the 9 element and 18 element mOdels, the center portion is a square which is-g-a by-g-a. The 25 element model has a %-a by'%'a region as the center portion. The 100 elements idealizations have both .8a by .8a and .9a by .9a center region as shown in Figures 3.13a and 3.13b. The purpose of the variation is to show the effect on the moment distri- bution of the plate of the size of the boundary region. To study the effect of varying the boundary prOperty uniformly in an isotrOpic non-homogeneous plates, only the CX is to vary./3 is kept equal to 1.0. The results are given in Table 3.3a and Figures 3.8a, 3.11a, 3.14a and 3.16a for the displacements of each of the different models. The moments 3L 2 3.14b &3.16b. The moments along y=0 can be found in Table along y= are shown in Table 3.3b and Figures 3.8b, 3.11b, 3.30 and Figures 3.80, 3.110, 3.140 and 3.160 respectively. The deflections along the centerline increase as the boundary stiffness decreases. This has been found in each different size of modeling and also for different size of boundary region. As C! and h both approach zero, the plate would approach to the condition of a simply-supported square plate. The displacement at the center of the plate for the case of 01:1 and {3:1, as given by the 9 elements model is 11.713 x 10-u-2%-—-. There is a 7.04 % error compared 14. with the 12.6 x 10-4-2%-— exact solution. The 18 element 4 model has 12.134 X 10-4-2%—- with 3.7 % error, the 25 4 element one has 12.319 x 1o”“-E%- with 2.23 % error and the 2-995'n'r s" x ‘ ' \ ‘:.uv.n~ ' ...o~ u . ‘r .I-fl‘v‘: UA‘ ... ~- v4"' v . 0 .‘,.¢ A‘V“ A |,. “in._;...1 fa I1: I ah '_.. ..-... ‘\ x .3... ~"“' . IA-F“ .‘A F ... v- C -v>nI-'V v.4 . 1] .R” I. .1. Vttv I ......Ar “AH 1m. 1. J” LC.» . V w. . :- ‘ . A '14“! ...- . {ruo $‘ ‘ F F9 9 “‘3 U... .. ’9 1.1 ‘ Pr». A. e P 1 vu.. .1 *1 6 n MI. E. ‘ “ m U. _, “7.” RAM" . v; n ‘ A.Nar_ v 55 u 100 elements model has 12.585><10_u-E§- with the error of .12%. D This demonstrates that by refining the element size and increasing the number of elements,a better result is achieved when conforming elements are used. As would be eXpected, as the boundary elements become less stiff, the negative moments along the edge decrease and the center moments increase. As shown in Figures 3.12 and 3.15, the maximum positive center moment equals the maximum negative edge moment for a value of aapproximately equal to .2 for the .8a by .8a center. For the 2/3a by 2/3a center portion case the CI value is .16 when the two moments become equal. This data can be found in Figure 3.9. As the center region becomes .9a by .9a square the Figure 3.17 indicates that the two moments are equal in magnitude whenCX= .16. If we consider the case Of the .8a by .8a square center portion, the two moments equal each other at the value of .0343 pa2 and the corresponding center deflection is 23.428"10'u-E%—. The deflection is 1.859 times that which would occur if the plate were homogeneous and of the same stiffness as the center portion of the plate. The maximum moment, however, has been reduced from an absolute value Of .0513 pa2 to the .0343 pa2, or 66.8% of the value for the homogeneous clamped uniformly-loaded, square plate. (B). Effect Of Varying the PrOperties Mfithin the Boundary Region. The effect of varying the stiffness within the boundary region is next to be considered in order to study I p. , . \. r‘ 9v ..4 V. M~Av . u I‘ .~ ~ I uphv “It .. . IIA 56 CENTER ELEMENT \\ 7 § BOUNDARY ELEMENT \ \\ __§L. m h’ 6 (X __.Dboundary element center element // crampet L %a Figure 3.7a- ELAS Rectangular 9 Elements Configuration J—‘(Z W CENTER ELEMENT '5 Q BOUNDARY ELEMENT __ a “—6— (0 '3 do” 0(=]_D;Doundary element C1. g Dcenter element H D V I Clamped L éa Figure 3.7b. ELAS Triangular 18 Elements Configuration ‘ .a k'.“0 I do"“"‘ i"; _ , I l a»..— .- rvc A.“ .~.. ~\H in R\U ...1 an .. h MI ..I», .. q . r . I 4 . . Cy . x . . , 1 .. . \ . : . I . no .2 I» m... A. .5 nm mt .... any ..J E n a: A.“ i. w a J .u a v v. a J, . . . u C/ mun QJ «flu 9mb DJ E V.“ l I . .. r ..-...1Tw r ”w 6H” .le , - .- x (\x .I\ “5 4h at ’51} 0 A, U . . u. . u - u 41. 7 I ...I‘. l:' ’l \ux. ( \ 57 Table 3.3a. Displacements Along y=a/2 for /3 =1.0 ( 9 Elements Rectangular and 18 Elements Triangular Models ) DEFLECTION ALONG Yxa/Z oz ELglggT VAR. X=a/6 X=a/3 X=a/5 9 Eles. E,G 7.6787 19.2760 23.3470 0 1 (ELASEJ ) T 7.5654 19.1640 23.2340 ' 18 Eles. E,G 8.1345 20.2730 24.4320 (ELASZX ) T 8.0227 20.1640 24.3200 9 Eles. E,G 6.7246 17.0780 20.7830 0 2 (ELASIZI) T 6.6786 17.0340 20.7380 ' 18 Eles. E,G 7.1015 17.9080 21.6900 (ELAS A ) T 7.0558 17.8640 21.6450 9 Eles. E,G 4.9614 12.8890 15.8850 0 5 (ELAs[].) T 4.9517 12.8800 15.8750 ' 18 Eles. E,G 5.2026 13.4370 16.4990 (ELAS A ) T 5.1929 13.4280 16.4890 9 Eles. E,G 3.4692 9.3198 11.7130 1 0 (ELAs[j ) T 3.4692 9.3198 11.7130 ° 18 Eles. E,G 3.6148 9.6753 12.1340 (ELAS A) T 3.6148 9.6753 12.1340 1.0 Exact 12.6 2 0 (ELAS [:l) E,G 2.1731 6.2208 8.0992 ' (ELAS A) E,G 2.2513 6.4466 8.3952 5 0 (ELAS Cl) E,G 1.0281 3.4921 4.9263 ' (ELAS A) E,G 1.0594 3.6325 5.1450 2}: 5AA r N 10 \ FIJ 58 01.1 0‘2 OIB 0‘4 0‘5 1. \\ \\ .. \\ \\ 26 \\ \\]Q=O 25- —— [:1 ELEMENT -—-- A ELEMENT w _\4—— Figure 3.8a. Displacements Along y=a/2 for /3 =1.0 ( 9 Elements Rectangular and 18 Elements Triangular Models ) S u -o . "u. :u G. C. . It r. fufinre - .R\U. .N\U ~ .3 . .N..~ . .ruw. a». I .11 D. . -. .1. . PF. 1.. .9. A. a: A. .2 1.. r2 .... a: 1A nu]. .A. A: 1M . . . . o v... u.“ -4 - VJ .m E 7“ .u v. .m 7 0C .~.L .1.) e: ”5 04/ aka .u.. «no #41 .iv an.“ nu... Pub nun... Hum 04/ nun nHu Fun wt... Hula hug rye. ”I” ,v I 1: {It I I\ [111 1‘ I..\\ (u\ 1 ( (\ 1 ( E ({\ ....n , AV$ J I. .../M (I) n1J. AIJ 5.1%- A u .5U l U Q H 0’“ 59 Table 3.3b. Moments Along y=a/2 for [3:460 ( 9 Elements Rectangular and 18 Elements Triangular Models ) MOMENTS ALONG Y=a/2 CX mil??? VAR. X=0 X=a/6 X=a/3 X=a/2 9 Eles. E,G —0.00066 0.00590 0.03441 0.0400 0 1 (ELASEZI) T —0.00066 0.00583 0.03435 0.0400 ’ 18 Eles. E,G -0.00086 0.00784 0.03668 0.04109 (ELAS A ) T -0.00086 0.00776 0.03663 0.04105 9 Eles. E,G -0.00115 0.00499 0.03109 0.03665 0 2 (ELASIZJ) T -0.00115 0.00496 0.03107 0.03664 ’ 18 Eles. E,G -0.00152 0.00681 0.03313 0.03765 (ELAS A ) T -0.00152 0.00677 0.03311 0.03764 9 Eles. E,G -0.00212 0.00344 0.02457 0.03018 0 5 (ELASIZI) T -0.00212 0.00343 0.02457 0.03018 ' 18 Eles. E,G -0.00279 0.00501 0.02625 0.03104 (ELAS A ) T -0.00279 0.00500 0.02625 0.03104 9 Eles. E,G -0.00293 0.00236 0.01882 0.02466 1.() (ELAS E]) T -0.00293 0.00236 0.01882 0.02466 18 Eles. E,G -0.00389 0.00369 0.02034 0.02544 (ELAS A ) T -0.00389 0.00369 0.02034 0.02544 1-0 Exact 0.0231 2.c> (ELAS CJ) E,G —0.00363 0.00171 0.01362 0.01990 (ELAS A ) E,G -0.00485 0.00277 0.01512 0.02061 5.() (ELAS E3) E,G -0.00422 0.00154 0.00787 0.01427 (ELAS £1) E,G -0.00571 0.00224 0.01042 0.01638 60 M __I;_ Paz .04- O<=0.1 (1:0.2 .031 O(=0.5 O(=1.0 .02- O(=2.0 O(=5.0 .01- ‘1 #4659' x/a 0] 1 l I l I 0.1 0.2 0.3 0.4 0.5 Figure 3.8b. Moments Along y=a/2 for /3=fl”() ( 9 Elements Rectangular and 18 Elements Triangular Models ) ”—1” ...“ a... afiv .. nxy . ~ 1 1 . I: II II. I h .. ~ 4 1 . . . 6 I . I .a D II . W was I-m «h. Phw Flu fluw wuL PM r“ ”J .- v Au. ... .1. v . ~n~ n5 .lulz ~|~ . c . «a. A a.) wa~ A...» mi u Ald/ mug FHV ML at an AHJ “L Ali/G find aura Hm IFW In!“ (any E II. {IL (\ (\ 1.. {\ (\ 1 I .\ III. (|\ I . n/b (J NU NJ A|.J AU ~v$ . - . - .ru 7“ - 7“ K... 61 Table 3.30. Moments Along y=0 for ‘/3=1.0 ( 9 Elements Rectangular and 18 Elements Triangular Models ) MOMENTS ALONG Y: O (X ELEMENT TYPE VAR.~ X=0 X=a/6 X=a/3 X=a/2 9 Eles. E,G —0.00029 -0.00523 -0.02864 -0.03634 (ELAleJ) T -0.00028 —0.00513 -0.02871 -0.03639 0'1 18 Eles. E,G -0.00058 -0.00543 -0.02868 -0.03719 (ELAS,A;) T -0.00058 -0.00543 -0.02868 -0.03719 9 Eles. E,G -0.00050 -0.00738 -0.03142 -0.03959 0 2 (ELAs[:]) T -0.00050 -0.00733 -0.03144 -0.03961 18 Eles. E,G -0.00101 -0.00723 -0.03152 -0.04104 (ELASZ§,) T -0.00100 -0.00718 -0.03153 -0.04107 9 Eles. E,G -0.00090 -0.01153 -0.03687 -0.04581 0 5 (ELASEZI) T -0.00090 -0.01151 -0.03688 -0.04582 ' 18 Eles. E,G -0.00180 -.01070 -0.03697 -0.04840 (ELAS A ) T -0.00180 -0 . 01068 -0.03697 -0.04841 9 Eles. E,G -0.00123 -0.01493 -0.04168 -0.05101 1 0 (ELASEZJ) T -0.00123 -0.01493 -0.04168 -0.05101 ' 18 Eles. E,G -0.00244 -0.01357 -0.04157 -o.05461 (ELAS A ) T —0.00244 —0.01357 -0.04157 -0.o5461 1.0 Exact -0.0513 2 0 (ELASIZI) E,G -0.00149 —0.01768 -0.04602 -0.05536 ' (ELAS A ) E,G -0.00294 -0.01594 -0.04553 -0.05992 5 0 (ELASIZI) E,G -0.00169 -0.01984 -0.05007 -0.05898 ' (ELAS A) E,G -0.00333 —0.01790 -0.04898 -0.06451 . 1.. I '44) d 62 0 0.11 0.12 0.13 0.14 O-é x/a -0.01 . -0.02 1 -0.03 . \\;\\‘ (1:0.1 -0.04 .. \ CX=O.2 \ \ \ \\O<=o.5 -0.05 . \ 4 —— C] ELEMENTS \ 1 \ O(=1.0 ....-- A ELEMENTS \ OC=5-O -0.06 1 « _M_ 2 pa JFigure 3.80. Moments Along y=0 for ‘/3=1.0 ( 9 Elements Rectangular and 18 Elements Triangular Models ) 6.014 p AN: J .v n U 63 30-1 k\ -? 20-4 O H 3 21,3010— 0 3 130 230 330 430 a. Center Deflection 0.06~ _______ /”’-—-—— -MaX M 0.0 z”’ 5‘ x’ // /3 M = . . . . & N . 0.041 for 1 0 ax1mum Positive egative Cl=u16 Moments Are Equal 0,031 scfiu p. 0.02- M y 0.0156 0 .5 1.'0 2T0 3.'0 410 5.0 b. Moments Figure 3.9. Moments and Deflection for Variation in CX ( 9 Elements Rectangular Model ) 64 / /// 31 Q, '9 c: "C3 u \ g; .\\ » I ClMped Edge J a/2 v. D Center Element \ Boundary Element __8_ h‘10 CK _ Bbguggary element D center element Figure 3.10. ELAS Rectangular 25 Elements Configuration 65 DEFLECTION ALONG Y = a/2 : Unit 2&4 10-4 Do a VAR. X=0 . 1a X30 . 2a X=0 . 3a X=0 . 4a X=0 . 5a 0 1 E,G 4.7565 13.3950 20.1950 24.4800 25.9370 ' T 4.6840 13.3190 20.1140 24.3970 25.8530 0 2 E,G 3.9178 11.2390 17.1890 20.9940 22.2960 ° T 3.8880 11.2080 17.1570 20.9610 22.2630 0 5 E,G 2.5967 7.7724 12.3440 15.3710 16.4200 ' T 2.5902 7.7660 12.3370 15.3640 16.4130 1 0 E,G 1.6778 5.3509 8.9622 11.4460 12.3190 ' T 1.6778 5.3509 8.9622 11.4460 12.3190 1.0 Exact 12.6 2.0 3.9 0.9916 3.5457 6.4490 8.5311 9.2731 5-0 E'G 0-4504 2.1297 4.4884 6.2601 6.8997 0.1 0.2 o .=.O H 7&9 O(=1.0 CX=O.5 CK=0-2 CXE=0.1 Figure 3.11a. Displacements Along y=a/2 for ‘/3=1.0 ( 25 Elements Rectangular Model ) r. a A... - . a: 1— . qld :Li!. .2. .di F? . . . «J a; any «uh fl 1.4. .H J 7. . .. 4). 0.9 D u 0 v 0 Ply . | . ...n nth. F: ”911“: 66 MOMENTS ALONG Y = a/2 ; Unit —M%— pa CX ‘VAR. x=0 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a 0 1 E,G -0.00132 0.00237 0.02089 0.03037 0.03668 0.03859 ' T -0.00132 0.00204 0.02105 0.03024 0.03668 0.03855 0 2 E,G -0.00218 0.00060 0.01753 0.02666 0.03273 0.03458 ' T -0.00218 0.00045 0.01760 0.02661 0.03273 0.03457 0 5 E,G -0.00361 -0.00210 0.01183 0.02065 0.02627 0.02807 ' T -0.00361 -0.00213 0.01184 0.02064 0.02627 0.02807 1 0 E,G -0.00465 -0.00398 0.00766 0.01636 0.02172 0.02351 ' T -0.00465 -0.00398 0.00766 0.01636 0.02172 0.02351 1.0 Exact 0.0231 2.0 E,G —0.00545 -0.00542 0.00438 0.01306 0.01831 0.02011 5.0 E,G -0.00613 -0.00664 0.00162 0.01031 0.01562 0.01745 J .04 l' =0.1 o(=0.2 .033 0.5 1.0 .02~ .0 NM :2. 5.0 .01- .ggggiEEEL3 032 0.3 0.4 035 X/a -.01 'Figure 3.11b. Moments Along y=a/2 for [3 =1.0 ( 25 Elements Rectangular Model ) P32 VAR. T E,G 6? MOMENTS ALONG Y = O X=0 -0.00027 -0.00027 -0.00043 -0.00043 -0.00066 '0 0 00066 -0.00078 "O o 000 78 Exact O .0 E,G -0.00082 0 E,G -0.00079 -.01« ‘002— ”003-1 -00LL- -.05_ -.06. ,/"._——.._""\ 0.1 x=001a 0.00079 0.00087 -0.00141 -0 o 001 36 -0.00472 -0.00470 -0.00661 -0.00661 -0.00755 -0.00776 0. X=0.2a .01328 .01314 .01612 .01605 .02463 .02061 .02349 .02349 .02523 .02605 2 X=0.3a '0 o 0214'36 -0.02473 -00 02837 .02854 ~03453 .03456 .03856 .03856 013 pa Unit _MIE X=0.4a '00 -00 -00 0.4 1 02649 02612 03192 .03177 .04118 .04117 .04796 .04796 .04130 -0.05323 .0431? -0.05758 X: 0.5a .02857 .02891 .03444 .03459 .04407 .04409 .05119 .05119 .05130 -05705 .06237 Figure 3.110, Moments Along y=0 for [3'=1.0 ( 25 Elements Rectangular Model ) O(=0.1 (1:0.2 O(=0.5 (1:170 O(=2.0 O(=5-0 25... Z J . . 1% «4 rL. n- v filo n 0., 1.! a my LIIIIII 68 40- 304 :f' \ 3 20- 3L? o ‘3'“ 10..- I I I I I 0"015 1.0 2.0 3.0 4.0 5.00< a. Center Deflection 006'“ I’d—’d” ———————— - max. M .05- .04.. Maximum Positive & Negative m Era. Moments Are Equal ~03-1 + I .02._ Max My .01- 1I I I I I T 7r 0 0.5 1.0 2.0 3.0 4.0 5.001 b. Moments Figure 3.12. Moment and Deflection for Variation in CM ( 25 Elements Rectangular Model ) ’ F — F l\ .I.‘I: 'z A A p F NW .0 m6. _ .| . TIM - -- _ __ 1r 1. .1 A A. A M - A | 69 _ CENTER ELEMENT QN—jr’ _JL § BOUNDARY ELEMENT a &\ h = 20 \x \. {i D \ m CK=’ boundary element ‘\\\_ center element ‘~‘\\\:\ \ \ \ \ \ ‘fi‘\ ‘\‘\ L L a/2 J a. 0.8a by 0,8a Center Portion " 1 5 t CENTER ELEMENT §\\_ BOUNDARY ELEMENT or _ a } h‘20 (1:: Dboundary element Dcenter element kaVkaVka L5 a/2 #1 b. 0.2a by 0.9a Center Portion Figure 3.13. ELAS Rectangular 100 Elements Configuration 70 w DEFLECTION ALONG Y = a/2 ; Unlt a4 10—4 DC (X VAR. X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a 0 1 E,G 7.0223 15.6990 22.5530 26.8890 28.3670 ' T 6.9524 15.6260 22.4780 26.8110 28.2880 0 2 E,G 5.0019 12.3400 18.3020 22.1200 23.4280 ' T 4.9728 12.3100 18.2710 22 0880 23.3950 0 5 E,G 2.9959 8.2080 12.7840 15.8100 16.8590 ' T 2.9894 8.2015 12.7780 15.8030 16.8520 1 0 E,G 1.8653 5.6037 9.2295 11.7130 12.5850 ' T 1 8653 5.6037 9.2295 11.7130 12.5850 1.0 Exact 12.6 2.0 E,G 1.0804 3.7260 6.6528 8.7363 9.4769 5.0 E,G 0.4839 2.2781 4.6666 6.4404 7.0786 q_ 011 012 013 014 015 X/a £125 0 10— cx=2.0 0 =1.0 =0. _? C1 5 o 20— H CX=0.2 :- I: cx=0.1 Figure 3.14a. Displacements Along y=a/2 for [3=4.0 ( 100 Elements Rectangular Model with .8a by .8a Center ) 71 Figure 3.14b. Moments Along y=a/2 for B =1.0 ( 100 Elements Rectangular Model with .8a by .8a Center ) .JAY. MOMENTS ALONG Y = a/2 ; Unit paz CX VAR x=0 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a 0 1 E,G -0.00274 0.00502 0.02194 0.03130 0.03690 0.03877 ° T -0.00274 0.00505 0.02192 0.03127 0.03687 0.03874 0 2 E,G -0.00360 0.00266 0.01790 0.02702 0.03251 0.03434 ' .T —0.00360 0.00265 0.01788 0.02700 0.03250 0.03433 0 5 E,G -0.00504 -0.00069 0.01209 0.02068 0.02590 0.02765 ' T -0.00504 -0.00070 0.01208 0.02067 0.02589 0.02764 1 0 E,G -0.00611 -0.00297 0.00806 0.01632 0.02136 0.02306 ° T -0.00611 -0.00297 0.00806 0.01632 0.02136 0.02306 1.0 Exact 0.0231 2.0 E,G -0.00695 -0.00468 0.00494 0.01302 0.01799 0.01966 5.0 E,G —0.00769 -0.00606 0.00232 0.01035 0.01532 0.01700 .04- .03— 66° .02— 9. .014 0 -.01 KAN VAR. T .0 Exact .0 E,G .0 E,G -.06. 72 MOMENTS ALONG Y = ; Unit ——M¥§— pa X=0 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a -0.00005 -0.00315 -0.01481 -0.02296 —0.02720 —0.02851 -0.00004 -0.00332 -0.01517 -0.02318 -0.02733 —0.02861 -0.00011 -0.00416 -0.01769 -0.02744 -0.03266 -0.03432 -0.00010 -0.00422 —0.01780 -0.02750 -0.03269 -0.03416 -0.00022 -0.00624 -0.02190 -0.03432 -0.04162 -0.04404 -0.00022 -0.00622 -0.02191 -0.03432 -0.04162 -0.04403 -0.00029 -0.00761 —0.02436 -0.03890 —0.04815 —0.05128 -0.00029 -0.00761 -0.02436 -0.03890 -0.04815 -0.05128 -0.0513 -0.00033 -0.00836 -0.02561 —0.04209 -0.05332 -0.05719 -0.00033 -0.00886 -0.02583 -0.04430 -0.05775 -0.06245 0,1 012 013 0 4 015 X/a CX=0.1 CX=0.2 CX=0.5 CI: .0 Cl: .0 CX=5.0 Figure 3.140, Moments Along y=0 for /3 =1.0 ( 100 Elements Rectangular Model with .8a by .8a Center ) 73 b. Moments Figure 3.15. Moments and Deflection for Variation in CK 40- 3 3°“ 0 H 20- #43 10— II I I I l r ' 0 0.5 1.0 2.0 3.0 4.0 5.00< a. Center Deflection .06A __‘a—v" ”””””” _— ’,.—”’”' - Max. M // 05—I // ' / /' 044 / . . N ' / [3=1.0 Max1mum Pos1tive & Negative 0 for p‘ 03_-/ CX=0.2 Moments Are Equal 4 + Max. M .02 y .014 II I T l j I l 0.5 1.0 2.0 3.0 4.0 5.00( ( 100 Elements Rectangular Model with .8a by .8a Center ) 74 W = . - 4 DEFLECTION ALONG Y a/2 , Unit a 10-4 DC CX VAR. X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a 0 1 E,G 6.2782 14.2300 20.6100 24.6690 26.0560 ' T 6.2383 14.1870 20.5630 24.6200 26.0070 0 2 E,G 4.7622 11.2650 16.6960 20.2130 21.4230 ' T 4.7459 11.2470 16.6770 20.1940 21.4040 0 5 E,G 2.8877 7.5958 11.8540 14.7000 15.6910 ' T 2.8841 7.5921 11.8500 14.6960 15.6870 0 0.1 0.2 0.3 0.4 0.5 _j I l I 1 l X/a 10- CX==O-5 -7 20- O = 3 a. CX CL2 :24 0 C’ CX=0.1 30- Figure 3.16a, Displacements Along y=a/2 for /3=a.0 ( 100 Elements Rectangular Model with .9a by .9a Center ) 0.1 0.2 -001- -002- -.03‘ -.04- -.05— MOMENTS ALONG Y = VAR. E,G T C) ['11 .3. C) N «1 61.. Figure 3.16b. Moments Along y=0 for 113:1.0 X=O “0.00030 -0.00021 -0.00028 ’0000028 -0.00032 -0.00032 X=0.1a -Ol -00 -00 -00 -00 _Ol 00375 00338 00558 00539 00731 00727 O 75 X=0.2a -0. “O. -00 -00 -O 01559 01532 01915 01902 .0309? -0. 03095 Unit -AEL- 2 pa X=0.3a -00 '00 ”O. —Oo -00 “00 02170 02164 02766 02763 03499 03499 X=0.4a -0.02526 -0.02524 -0.03276 -0.03275 -0.04248 -0.04248 X=0.5a -0.02643 —0.02642 -0.03446 —0.03446 -0.04500 —0.04500 0.5 X/a ( 100 Elements Rectangular Model with .9a by .9a Center ) W61- “II-w { - . 0.1 0.2 0-5 MOMENTS ALONG Y = a/2 .045 .03- .02- .01.1 .00245 .00245 .00364 .00364 .00519 .00519 X=0.1a OO 00 -00 -0. .00714 .00711 .00383 .00382 00046 00046 76 X=0.2a CO OO 00 .01973 .01971 .01584 .01583 .01088 .01088 Unit -MX§- pa X=0.3a X=0.4a 0.02893 0.03452 0.02891 0.03450 0.02466 0.03002 0.02465 0.03002 0.01929 0.02443 0.01929 0.02442 X=0.5a CO CO CO .03638 -03637 .03178 .03181 .02615 .02615 X/a ‘001‘ 0.4 Figure 3.16c. Moments Along y=a/2 for 763:1'0 ( 100 Elements Rectangular Model with .9a by .9a Center ) T 0.5 77 401 “T 30- o H 3 20. \ O .316 10— X/_8._ U ” 0.'5 1.'0 2.'0 3.'0 4.'0 530a a. Center Deflection .06. .05- - Max. M / .04. / 2N0 / [3 =1.0 Maximum Positive 8c Negative for a. .03— I/ O1 =0.16 Moments Are Equal + Max. M .021 y .014 X/a. 0 " 0.'5 1.'0 2.'0 3.'0 4.'o 5.T0 (X b . Moments Figure 3.17. Moment and Deflection for Variation in OC ( 100 Elements Rectangular Model with .9a by .9a Center ) 78 the possibility Of reduction of the maximum moment and the effect on deflection. The 25 element model with .8a by .8a center portion was first studied. The center portion Of the plate will have a constant stiffness. The stiffness at the boundary region will vary according to the parabolic variation with fl3as the ceiling. First we study the case with O(= .5. As we defined before, this means that the stiffness at the center of the boundary region has a value of .5 that of the constant stiffness center region. The values for /3= .2, .5, 1.0, 2.0, 5.0 and 10.0 are used to do the analyses. The displacements are given in Figure 3.18a. This figure indicates that the deflections decrease as the boundary region is stiffened toward the corner; i.e., as the [3 value increases. For the moments, as the plate boundary region is stiffened at points away from the edge center lines, i.e., as the /3 value increases, more moment is attracted to the stiffer portions for both the positive and negative moments, as shown in Figures 3.19a and b. When./3= 5.0, the negative moment is nearly constant over the center part of the boundary region with a value Of — .0375 pa2 ; that is 73% of the maximum moment of the homogeneous clamped plate. The maximum positive moment occurs at the center of the plate. When /3= 5.0 the value is .0249 paz, which is 1.1 times the value for the homogeneous plate, as given in Figure 3.19b, but still less than the maximum negative moment. The maximum negative moment is about 1.506 times the maximum positive for fl: 5.0. If we consider the case when the fl value 79 increases to 10.0, the maximum negative moment increases to 3.19b. Witha = .5, the maximum negative moment is greater and the location is at X: .3a, as shown in Figure than the maximum positive moment in all the cases of )9 variation; thus, the boundary region stiffness should be reduced more to give a balanced condition. A study with CX= .1 was carried out and the results are given in Figure 3.20. It is still true that the deflections decrease as /3 increases as shown in Figure 3.20a. The maximum positive moment is no longer always smaller than the maximum negative moment for all the values of‘/3. For example, when [3=5.0 2 as shown in the maximum positive moment is .03532 pa Figure 3.20 . But the maximum negative moment has a value only - .02679 pa2 as in Figure 3.20 . This implies that an Optimal design for the internal moment is in the range between Gt: .1 to .5 and [3: 5.0 to 10.0. If we plot theO< versus M curves for [3: 5.0 and 10.0 we find the maximum positive and negative moments are equal at the following conditions. /3= 1.0 a: .2 tMmax= .0345 pa2 /3='5.0 (X: .18 iMmax= .0313 pa2 fl=10.0 CI: .13 iMmax= .0319 pa2 The values are plotted in Figure 3.21; the upper curve would indicate that the Optimum condition would be with/3‘? 7.0 and that the corresponding maximum moment values would be approximately equal to .0312 paz. The lower curve shows that corresponding to [3: 7.0, the value ofCX for which the 80- absolute values of the maximum positive moment and the maximum negative moment are equal, would be about .16. In Figures 3.22a and 3.22b the finite element results are shown for the case when.CX= .16 and [3=7.0. The maximum positive and negative moments each have a value of .0316 paZ. This is about 62% of the maximum moment for the homogeneous plate. 4 The center deflection is 20.6338><10-4-E%- which is 1 67 times that for the homogeneous case. The same analysis has also been carried out for the model of .9a by .9a center portion case. The results are given in Figures 3.23 and 3.24. If we also plot the CX versus M curves for j3=5.0 and 10.0, the maximum positive and negative moments are equal for the following conditions: [3: 1.0 Oz: .16 Mmax= .0343 pa2 [3: 5.0 a: .172 Mmax= .0303 pa2 f3=10.0 oz: .115 Mmax= .0315 pa2 These values would give an Optimum condition for internal moments at approximate (X: .17 and 3: 6.5. It would be possible by Optimization techniques to search out the values of (land [3 which give the exact minimum value for moment. This search was not carried out,since the purpose of this study is to indicate that a potential Optimization design for plate can be accomplished by varying the material prOperties and to show that the finite element technique would be able tO handle the problem. 81 W DEFLECTION ALONG Y=a/2 ; (X = 0.5 ; Unit 1* L1 .23.. 10' Dc /3 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a 0.2 2.71831 8.12213 12.86311 15.98044 17.05689 0.5 2.68711 8.03244 12.73067 15.82578 16.89511 1.0 -2.60222 7.78800 12.36444 15.39467 16.44533 2.0 2.47760 7.43067 11.83289 14.76889 15.79200 5.0 2.20151 6.64302 10.66578 13.39822 14.35911 10.0 1.90978 5.81653 9.44978 11.97333 12.87200 0 0.1 0.2 0.3 0.4 0.5 0_ I 1 11 1 1 X/a -? 10—. 0 =10.0 3 F' = 5.0 d- o = 1.0 4:: = 0.2 20-1 Figure 3.18. Displacements Along y=a/2 for CX =0.5 ( 25 Elements Rectangular Model with .8a by .8a Center ) 82 MOMENTS ALONG Y = 0 ; (X = 0.5 ; Unit —MX§ pa /3 x=0 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a 0.2 -0.00030 -0.00093 -0.01795 -0.03385 -0.04264 -0.04613 0.5 -0.00071 -0.00205 -0.01740 —0.03498 -0.04216 —o.04576 1.0 -0.00064 -0.00455 -0.02060 -0.03469 -0.04132 -0.04025 2.0 -0.00216 -0.00784 -0.02413 -0.03539 -0.04007 -0.0u211 5.0 -o.00365 -0.01328 -0.03211 -0.03732 -o.03757 -0.03724 10.0 -0.00460 -0.01658 -0.04018 -0 04387 -0.03515 -0.03206 c> :;;::::0]1 0 0.3 0 4 015 \ . X/a -... \\. -.02 \\ N . >513 \\ -.033 \\\ //]=10.0 \ f3=5 0 —.01+- ’% :1. =0.5 =0.2 Figure 3.19a. Moments Along y=0 for CX =0.5 ( 25 Elements Rectangular Model with .8a by .8a Center ) OU‘INl-‘OO OOOOU‘tN MOMENTS 83 ALONG Y =.- a/2 ; X=0.1a .00180 .00186 .00150 .00108 .00011 .00008 X: OOOOOO 0.2a .01137 .01153 .01158 .01171 .01191 .01213 X=0.3a 000000 .02013 -01999 .01991 .01970 .01916 .01839 Unit -Mx§ pa X=0.4a X=0.5a 0.02585 0.02769 0.02572 0.02752 0.02533 0.02706 0.02474 0.02640 0.02339 0.02493 0.02194 0.02339 II II II P‘Knk‘O Figure 3.19b. Momenta Along y=a/2 for ( 25 Elements Rectangular Model with .8a by .8a Center ) CX==Oo5 OON 84 . W Unit 4 4 ON A ONG = 0 :: . ; DEFLECTI L Ya/2 . C1 0 1 a 10_ D C /3 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a 0.2 4.89013 13.76711 20.72978 25.10489 26.58933 0.5 4.85396 13.66756 20.58489 24.93511 26.41244 1.0 4.79644 13.50844 20.35467 24.62667 26.13156 2.0 4.68844 13.21067 19.92267 24.16089 25.60356 5.0 4.40578 12.43111 18.79289 22.83911 24.22311 10.0 4.03316 11.40716 17.30844 21.10489 22.41156 9; 0:1 oiz 0f3 0f4' 0f5 p\\\\\ X/a 10- “T 3: :3 201 =10. .3glcp =5.0 =1.0 =0.2 30‘ Figure 3.20a. Displacements Along y=a/2 for O(=0.1 ( 25 Elements Rectangular Model with .8a by .8a Center ) O MOMENTS ALONG Y = /3 X=o 0.2 -0.00011 0.5 -0.00027 1.0 -0.00026 2.0 -0.00099 5.0 -0.00212 0.0 -0.00341 X=0.1a -O. -O. -O. -0. .00505 -O. -0 00277 00221 00127 00055 01024 85 O : X=0.2a -0. -O. -0. -O. -O. -0 01148 01217 01316 01491 01942 .02553 (X = O. X=0.3a .02431 .02445 .02475 .02541 .02720 .02951 1 Unit X=0.4a .02700 .02691 .02675 .02644 .02576 .02514 _Mx. p82 X=0.5a .02948 .02928 .02897 .02839 .02679 .02450 "OOLL‘ cu 2m >10. Figure 3.20b, Moments Along y=0 for ( 25 Elements Rectangular Model with .8a by .8a Center ) a: 0.1 01“an NOO‘ pa 86 ( 25 Elements Rectangular Model with .8a by .8a Center ) MOMENTS ALONG Y=a/2 ; cx =0.1 ; Unit —M¥§ pa )3 x=0 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a 0.2 -0.00224 0.00213 0.02026 0.02955 0.03593 0.03783 0.5 -0.00222 0.00216 0.02024 0.02946 0.03577 0.03765 1.0 -0.00219 0.00219 0.02021 0.02931 0.03551 0.03736 2.0 -0.00214 0.00224 0.02015 0.02901 0.03502 0.03682 5.0 -0.00200 0.00240 0.01995 0.02820 0.03373 0.03532 10.0 -0.00181 0.00275 0.01952 0.02715 0.03198 0.03352 .04— =0.2 =1.0 =5.0 =10. .0} .02- 00k 01———”’r 0:1 32 033 034 0f5 x/a Figure 3.20c- Moments Along y=a/2 for (X =0.1 O 87 _M_ 2 pa 0,034.. 0.033— 0.032- 0.031 0 1‘.0I 51.0 10.0 [3 a. M Versusjg with Maximum +M Equal Numerically Maximum -M O( 0.20- 0.10- 0 1 l I I 1.0 5.0 10.0 /3 b. V sus with Maximum +M E ual Numericall Maximum -M Figure 3.21. Curves to Approximate Optimum Value of /3 &:CX 88 Ct =0.16 8. [3=7.0 X=0 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a w DEFLECTION ALONG Y=a/2 ; Unit pa4DC 10—4 0.00 3.61582 10.34756 15.84000 19.40267 20.63378 MOMENTS ALONG Y:0 ; Unit -M1§ pa -0.00168 -0.01059 -0.02636 -0.03118 -0.02824 -0.02803 MOMENTS ALONG Y=a/2 ; Unit —M¥§ pa -0.00130 0.00188 0.01774 0.02532 0.03008 0.03163 0 0.1 0.2 0.3 0.4 0.5 _J 1 1 I 1 1 X/a 10.. '3 O v-I 3 ffiflc? 20.. Figure 3.22a.Displacement for O(=0.16 & /3=7.0 ( 25 Elements Rectangular Model with .8a by .8a Center ) 89 .03- .02. My Along Y=a/2 .01‘ X/a gig? 0T1 0.2 0.3 0.4 0.5 -.01"‘ -002. My Along Y=O "003'4 b. Moments Figure 3.22b. Moments for O(=0.16 & /3=7.0 ( 25 Elements Rectangular Model with .8a by .8a Center ) 90 w DEFLECTION ALONG Y=a/2 ; Unit 4 4 _Pé— 10— CX =0.1 It /3 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a 0.2 6.4559 14.614 21.133 25.267 26.677 0.5 6.3859 14.463 20.927 25.031 26.432 1.0 6.2782 14.230 20.610 24.669 26.056 2.0 6.0844 13.812 20.040 24.018 25.380 5.0 5.6177 12.808 18.675 22.457 23.759 10.0 5.0746 11.645 17.095 20.652 21.884 Figure 3.23a, Displacements Along y=a/2 fOr CXU=O.1 ( 100 Elements Rectangular Model with .9a by .9a Center ) OONU‘U" ' O NUIOO' O My /'pa2 OKJ'INHOO OOOOU‘IN 91 MOMENTS ALONG Y=0 CX x=001a .00124 .00237 .00375 .00596 .01052 .01433 0.1 X=0.2a -0. -O. -O. -O. -0. -0. 01329 01416 01559 01823 02451 03141 Unit 4% X=O o 33 .02067 .02105 .02170 .02294 .02268 .03084 M pa X=0.4a -O. -O. -O. -O. -0. -0. 02559 02546 02526 02491 02424 02373 X=0.5a -0. -O. -0. -O. -O. -0. 02714 02686 02643 02566 02377 02150 -.O4— Figure 3.23b. Moments Along y=0 for O(=0.1 ( 100 Elements Rectangular Model with .9a by .9a Center ) Xia II II II II II OOHNKA NUIOOOO 92 OU‘NHO MOMENTS ALONG Y=a/2 ; Unit -41§ a (x = 0.1 p [3 x=0 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a 0.2 -0.002518 0.007016 0.019870 0.029312 0.035044 0.036962 0.5 -0.00249 0.007063 0.019816 0.029164 0.034836 0.036734 1.0 -0.002446 0.007136 0.019729 0.028934 0.034516 0.036384 2.0 -0.002367 0.007264 0.019564 0.028516 0.033938 0.035753 5.0 -0.002177 0.007546 0.019125 0.027478 0.032538 0.034229 10.0 -0.001956 0.007850 0.018541 0.026222 0.03089 0.032454 ;1 .04-i 0034 .02— . X/a N“ 0" / / 1— | I F l s n. / 0.1 0.2 0.3 0.4 0.5 Figure 3.23c. Moments Along y=a/2 for 0:0.1 ( 100 Elements Rectangular Model with .9a by .9a Center ) OOOON 93 w DEFLECTION ALONG Y=a/2 ; Unit 34 10-4 a =0.5 C j? X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a 0.2 3.0055 7.8781 12.250 15.155 16.163 0.5 2.9579 7.3637 12.089 14.971 15.972 1.0 2.8877 7.5958 11.854 14.700 15.691 2.0 2.7718 7.3204 11.468 14.258 15.233 5.0 2.5321 6.7593 10.688 13.363 14.304 10.0 2.2902 6.2063 9.926 12.491 13.397 0.1 0.2 0.3 0.4 0.5 05 1 l 11 1 I X/a d' 10-— lo f1 3 "' .:. in” : 20.. Figure 3.24a. Displacements Along y=a/2 for' CK=0.5 ( 100 Elements Rectangular Model with .9a by .9a Center ) OHNUXO NOOOO 94 M MOMENTS ALONG Y=O ; Unit 3’2 pa CX = 0.5 fl X=O X=0. 1a X=0 . 2a X=0 . 3a X=0. 4a X=0. 5a 0.2 -0.00009 -0.004129 -0.01958 -0.03398 -0.04378 -0.04684 0.5 -0.00020 -0.00554 -0.02096 -0.03435 -0.04324 -0.0461 1.0 -0.00032 -0.00731 -0.03097 -0.03499 -0.04248 -0.0450 2.0 -0.00045 -0.00955 -0‘.02615 -0.03634 -0.04131 -0.04316 5.0 -0.00049 -0.01214 -0.03183 -0.04014 -0.03934 -0.03926 10.0 -0.00040 -0.01166 -0.03579 -0.04519 -0.03813 -0.03514 014 0.15 X/a ./ —10.0 /. x. ‘ ~70= 5-0 -\‘\ =: 2.0 \\ = 1.0 \ = 0.5 /3= 0.2 Figure 3.24b. Moments Along y=0 for O(=O.5 ( 100 Elements Rectangular Model with .9a by .9a Center ) 95 MOMENTS ALONG Y=a/2 ; Unit 1115— pa 0( = 0. 5 l3 X=0 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a 0.2 -0.005426 -0.000912 0.010744 0.019459 0.024782 0.026568 0.5 -0.005332 ~0.000729 0.010803 0.019395 0.02464 0.026398 1.0 -0.005193 -0.000460 0.010879 0.019293 0.024426 0.026146 2.0 -0.004962 -0.000023 0.010982 0.019109 0.024068 0.025732 5.0 -0.004482 0.000857 0.011091 0.018663 0.023314 0.024875 10.0 -0.003991 0.001704 0.011045 0.018120 0.022532 0.024017 ;1 .on .014 , x N X/a 3* O l T’ l g 0.2 0.3 0.4 0.5 5.. “\M——’/ Figure 3.24c. Moments Along y=a/2 for O(=0.5 ( 100 Elements Rectangular Model with .9a by .9a Center ) OU‘lNHO OOOON 96 3.3.2. IsotrOpic Homogeneous Plate Effect of Varying the Thickness of the Plate For the thin plate theory the stiffness of the plate actually is a function of E,l/and t and is called the flexural rigidity. The relation can be eXpressed as Et3 D: . . 1211-)! 2) as given in equation (3.1). In 3.3.1(A) we studied the Optimization of a clamped square plate by varying the stiffness prOperties at each element in the plate. This was done by using different values for E and L’. Similar results should be obtained if we keep E and L1 constant and vary the thickness t in each element. If E and 2/ are constants for all the elements the plate is an isotrOpic and homogeneous one. By varying the thickness of each element we can Optimize the design of a clamped square plate. The same approach as in 3.3.1 (A) has been carried out for varying the thickness in each element instead of E and L’. Almost identical results have been obtained. The displace- ments, moments along y=0 and y=a/2 are listed in the Figures 3.7 through Figure 3.16 together with the E and l/ variation. Because of similarity to previous results only results for E and 1/ variation are plotted there. Comparing the outputs between E and l/ , and t variations we find that the finite element stiffness formulation does follow the thin plate theory. 97 3.3.3. OrthotrOpic Nonhomogeneous Plate-——-—- Effect of Varying the PrOperties in Different Directions An extended study has been made of the internal moment Optimum design for the clamped orthotrOpic nonhomoge- neous square plate. The stiffness prOperties were varied differently in the x and y directions in each element and the moment distribution was studied. From the previous study, the Optimum condition for the internal moment for a plate with .8a by .8a center portion idealization is with (X31 .2 and 13937.0. A particular structural modeling for the orthotrOpic nonhomogeneous plate with.CX==.2 and [3=7.0 is given in Figure 3.25a. The maximum positive and negative moments are plotted in Figure 3.25b. It can be seen that the maximum negative moment has a value of - .03256 pa2 and the maximum positive moment is .03123 pa2. The maximum negative moment is smaller at x: .5a, center of the plate, ‘but the maximum occurred at x: .3a. If we stiffen the eelements in the region between the center and the edge of the plate, the maximum moment will shift to -another location and a better distribution may Occur. Figures 3.26 through 3.28 show the results of three different trials Conducted to study the possibility of reducing the maximum anments. The best Of the three is shown in Figure 3.26b with a value of .031 pa2. Next, orthotrOpic elements with different prOperties in the x and y directions were used. We tried (X = .1, /3 = 14.0 for x-direction and a: .2, .63 = 7.0 for y-direction. The results are shown in Figures 98 3.30 and 3.31. It indicates that the results are not very favourable. Then, a trial with a: .2, [3: 7.0 for x- direction and (X: .1, [3: 14.0 for y-direction with symmetric about the diagonal was run. The results for each different idealization are given in Figures 3.29, and 3.32 through 3.37. The best result in this group is the one shown in Figure 3.37b with a maximum positive moment equal to 0.0312 pa2 -0.0295 pa2. This is larger than the case with (X: .2, and maximum negative moment equal to j? = 7.0 for both x and y directions as in Figure 3.26. SO the Optimum condition is with (I: .2 and /3= 7.0 as Obtained in the isotrOpic non-homogeneous plate study in Section 3.3.2. The above study and its results provided two major achievements : (1). Introduction of the finite element application and demonstration of the variety of uses. (2). The results for the Optimum design of maximum moment leads us to a better understanding about the square plate with complex material prOperties. The principle Of the Optimum design may also suggest practical uses in engineering applications. 3.3.4. Simply-supported Square Plate A simply-supported square plate with uniform loading condition is next to be considered. Due to the characterist- ics Of zero moments at the boundary lines the structural idealization is different from that of the clamped edge 99 r-_X X CD CD ® 6) C9 X 1' 2’ny E CDCDCDCDCQSD E'=1_5U :9 y x y ”C3 UXEy UyEx ooooom g‘ 1 bkzvy 1 bk b3 G) CD CD C9 C3 5 658]— “EXEy “HIM/X Uy) © ® ® ® ® ® ELEMENT MATERIAL é Clamped Edge TYPE NO' _STIFFNESS 0F MATERIAL TYPE N0. 6 2 (X ‘STIFFNESS 0F MATERIAL TYPE N0. 1 ' /3 _STIFFNESS 0F MATERIAL TIRE N0. 2 7 0 ‘ STIFFNESS 0F MATERIAL TYPE N0. 6 ' @ E U U E' E' E" E x y x y x y 1 10920. 10920. 0.3 0.3 12000. 12000. 3600. 4200. 2 15288. 15288. 0.3 0.3 16800. 16800. 5040. 5880. 3 9555. 9555- 0-3 0-3 10500. 10500- 3150- 3675- 4 5460. 5460. 0.3 0.3 6000. 6000. 1800. 2100. 5 300 . 3003. 0.3 0.3 3300. 3 00. 990. 1155. 6 218 . 2184. 0.3 0.3 2400. 2 00. 720. 840. Figure 3.25a. Central Portion Idealization of Plate with Homogeneous 100 MOMENTS : Unit My/pa2 X=0 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a @ y=0 -0.00189 -0.01224 —0.0287O -0.03256 -0.02952 -0.02895 @ y=-%- 0.00179 0.00144 0.01705 0.02484 0.02963 0.03123 M@y=a/2 .034 —M@y=0 o 02“ .01- N ‘3. \:>. 2 O X/a ___,,,/’b;& 0.2 0 5 0.4 0.5 Figure 3.25b. Moment Results 101 r“ E Figure 3.26a. I X Y ‘"1§ E = — O o (8 o O 1 ”ny E E' = Y CD®®®®§0 y l'UXUY b’E L’E 'U E": X37 or yX @ ® ® (9% 1'Uny l-Uny @0000S E="EXEY 2(1+\/Uny) @ © @ @ @ @ ELEMENT MATERIAL TYPE N0. g Clamped dge CX aSTIEPNESS 0F MATERIAL TYPE N0. 6 z 0 2 STIFFNESS 0F MATERIAL TYPE N0. 1 ' _STIFFNESS 0F MATERIAL TYPE N0. 2 _ 0 /3 'STIFFNESS 0F MATERIAL TYPE N0. 6 ‘ 7' @ Ex Ey Z/X Z/y E); Ey E" G 1 10920. 10920. 0.3 0.3 12000. 12000. 3600. 4200. 2 15288. 15288. 0.3 0.3 16800. 16800. 5040. 5880. 3 9555. 9555. 0.3 0.3 10500. 10500. 3150. 3675. 4 5460. 5460. 0.3 0.3 6000. 6000. 1800. 2100. 5 3003. 3003. 0.3 0.3 3300. 3300. 990. 1155. 6 2184. 2184. 0.3 0.3 2400. 2400. 720. 840. 7 21840. 10920. 0.3 0.15 22869. 11435. 3430. 6370. 8 21840. 21840. 0.3 0.3 24000. 24000. 7200. 8400. 9 10920. 21840. 0.1 0.3 11435. 22869. 3430. 6370. Idealization of Plate with Homogeneous Central Portion Except for 3 rd. Set Elements From Center 102 MOMENTS ; Unit —415— pa X=0 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a @ y=0 -0.00176 -0.01136 -0.02815 -0.03101 -0.02923 -0.02745 @ y=-%— 0.00169 0.00167 0.02079 0.03077 0.02683 0.02827 .03“ \ My@y:a/2 AIIm-> —My@y=0 ,on O 01— N S. \\ :Q' 0 X/a \\__,,//6.1 032 033 034 035 Figure 3.26b. Moment Results 103 [_— _ E —'é' E3: X Y G) CO C9 C9 C9 X 1'”X ”Y E CD CD CD C9 ® 3’0 E' =1-y 8 21ny 6 LIME 1D’E Q) G CD (9 CD A E"=1-" X (B ('2 C?) Q 3 a; z VExyE 2(1+\/UX Uy) @ G C9 (3 ® ® ELEMENT MATERIAL TYPE NO. 2 Clamped Edge CX _STIFFNESS 0F MATERIAL TYPE N0. 6 _ 0 2 ‘STIFFNESS OF MATERIAL TYPE N0. 1 - ' /3__§TIFFNESS 0F MATERIAL TYPE NO. 2 _ 7 O ‘S IFFNESS 0F MATERIAL TYPE N0. 6 _ ® EX Ey ux yy E' E' E" G 1 10920. 10920. 0.3 0.3 12000. 12000. 3600. 4200. 2 15288. 15288. 0.3 0.3 16800. 16800. 5040. 5880. 3 9555. 9555. 0.3 0.3 10500. 10500. 3150. 3675. 4 5460. 5460. 0.3 0.3 6000. 6000. 1800. 2100. 5 3003. 3003. 0.3 0.3 3300. 3300. 990. 1155. 6 2184. 2184. 0.3 0.3 2400. 2400. 720. 840. 7 21840. 10920. 0.3 0.15 22869. 11435. 3430. 6370. 8 21840. 21840. 0.3 0.3 24000. 24000. 7200. 8400. 9 10920. 21840. 0.15 0.3 11435. 22869. 3430. 6370. Figure 3.27a . Idealization of Plate with Homogeneous Central Portion Except 4 th. Set of Elements From Center 104 MOMENTS ; Unit —M¥§— p8. X=0 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a @ y=0 —0.00189 -0.00947 —0.02944 —0.03289 -0.02756 -0.02784 @ y=-%- -0.00163 0.00265 0.02110 0.02221 0.02739 0.02899 My@y=a/2 .03- -My@y=0 .02-1 .01.. N 8. E E X/a O I I I l I ~e__,///0.1 0.2 0.3 0.4 0.5 lFigure 3.27b. Moment Results 105 {—33 E "13 E' = —5 (1) Q) Q) Q) @ X 1'UX ”Y E E' = y (3) 3‘33 (C) (:) (:) E? 1- b&:D& lfl o o o o o 8 = .35wa 01.1254 C81. UX y 2/X Dy (0 ® @ G G) @ IS a : “ExEy O 0 GD 0 O ““3””? . Cl d Ed " ELEMENT MATERIAL g ampe ge ® TYPE N0. 0( ESTIFFNESS 0F MATERIAL TYPE N0. 6 = 0 2 STIFFNESS 0F MATERIAL TYPE N0. 1 ° _STIFFNESS 0F MATERIAL TYPE N0. 2 _ 0 “STIEFNESS 0F MATERIAL TYPE N0. 6 ‘ 7' ® Ex Ey yx z/y E); Ey E" G 1 10920. 10920. 0.3 0.3 12000. 12000. 3600. 4200. 2 15288. 15288. 0.3 0.3 16800. 16800. 5040. 5880. 3 9555. 9555. 0.3 0.3 10500. 10500. 3150. 3675. 4 5460. 5460. 0.3 0.3 6000. 6000. 1800. 2100. 5 3003. 3003. 0.3 0.3 3300. 3300. 990. 1155. 6 2184. 2184. 0.3 0.3 2400. 2400. 720. 840. 7 21840. 10920. 0.3 0.15 22869. 11435. 3430. 6370. 8 21840. 21840. 0.3 0.3 24000. 24000. 7200. 8400. 9 10920. 21840. 0.15 0.3 11435. 22869. 3430. 6370. Figure 3-283' Idealization of Plate with Homogeneous Central Portion Except 2 nd. Set of Elements From Center 106 MOMENTS ; Unit —415 pa X=0 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a @y:0 -0.00183 -0.01204 -0.02810 -0.03139 -0.02868 -0.02759 @y=-%- -0.00168 0.00184 0.01496 0.02991 0.03541 0.02605 -My@y=0 .03‘ "IL My@y=a/2 .02— o 01.1 N 8. E O X/a r I 1 I 1 -—"”’30.1 0 2 0.3 0.4 0 5 Figure 3.28b- Moment Results 107 TX Ex 0 0 ® 0 O ‘3 "EXMU o o (9 o (93.0 E&=1-5xvy O O o 8:: = 135:3, 145:9, o O o 0 @§ .— = V”— e O 8 O 6 WW) . CIampeTE g'e (i) ELEMENT MATERIAL TYPE é N0. CX _STIFFNESS OF MATERIAL TYPE NO. 'STIFFNESS OF MATERIAL TYPE NO. HO\ 1%} _STIFFNESS OF MATERIAL TYPE NO. 2 _STIFFNESS OF MATERIAL TYPE NO. 6 cx = 0.2 & /3 = 7.0 X-DIRECTION PROPERTIES CK = 0.1 & /9 =14.0 Y-DIRECTION PROPERTIES @ EX Ey yx yy E Ey E G 1 10920. 10920. 0.3 0.3 12000. 12000. 3600. 4200. 2 15288. 15288. 0.3 0.3 16800. 16800. 5040. 5880. 3 9555. 9077. 0.3 0.285 10448. 9926. 2978. 3602. 4 5460. 4641. 0.3 0.255 5912. 5025. 1508. 1972. 5 3003. 1979. 0.3 0.198 3192. 2104. 6 1. 980. 6 2184. 1092. 0.3 0.15 2287. 1144. 3 3. 637. 7 21840. 10920. 0.3 0.15 22869. 11435. 3430. 6370. 8 21840. 21840. 0.3 0.3 24000. 24000. 7200. 8400. 9 10920. 21840. 0.15 0.3 11434. 22869. 3430. 6370. Figure 3.29a. Idealization of Plate with OrthotrOpic Homogeneous Central Portion Except 3 rd. Set of Elements From Center '=O @110 -0.00: . a '” @‘JIT '000C. .03 (D (\J 108 MOMENTS 3 Unit -M¥§ pa X=O X=0.1a X=0.2a X=0.3a X=O.ua X=0.5a @ y=o -0.00189 -0.01202 ~0.02788 —0.02780 -0.02338 -0.02065 @ y=-%— -0.00082 0.00252 0.02142 0.03157 0.02748 0.02896 .03- My@y=a/2 '02“ -My@y=O .01- N S. E. E O X/a 0.i 0.2 0.5 0.4 0.5 Figure 3.29b. Moment Results 6170' 1 CL’ PT: a : -:: QT:- NJ - 1 10920 2 15289 ? 9077. i 4641. 2 1979 U 1092 ‘30 ‘QHE 3. 109 X --f§ E' = Ex 6) © © © @ X 1‘ny Y Y E o o o o o E‘ “d III-1 UE UE © © © © (9 U En : 1..ny U or 1-YUX U a x y x Y E © © ® ® @ g “G— _ "EXEy @ o o 0 <2) “MW/y) é Clamped Edge (:) ELEMENT MATERIAL TYPE NO. CZ STIFFNESS OF MATERIAL TYPE NO. 6 =STIFENESS OF MATERIAL TYPE NO. 1 /3 _STIFFNESS OF MATERIAL TYPE NO. 2 "STIFENESS OF MATERIAL TYPE NO. 6 CI = 0.1 & =14.0 x-DIREOTION PROPERTIES CX = 0.2 & = 7 0 Y-DIRECTION PROPERTIES @ Ex Ey yx yy E); Ey E" G 1 10920. 10920. 0.3 0 3 12000. 12000. 3600. 4200. 2 15288. 15288. 0.3 0 3 16800. 16800. 5040. 5880. 3 9077. 9555. 0.285 0.3 9926. 10448. 2978. 3603. 4 4641. 5460. 0.255 0.3 5025. 5912. 1508. 1972. 5 1979. 3003. 0.198 0.3 2104. 3192. 631. 980. 6 1092. 2184. 0.15 0 3 1144. 2287. 343. 637. Figure 3.30a. Idealization of Plate with Homogeneous Central Portion and OrthotrOpic Nonhomogeneous Boundary Portion @ y=O @ y: 2 .03- o 02"1 .01— 2 My/Pa 0d :==’I* 0.& 0.5 0.5 0.4 0 110 MOMENTS ; Unit —M1§ pa X=O X:O.1a X=0.2a X=0.3a X=O.#a X=0.5a -0.00203 -0.01297 -0.03020 -0.03383 -0.03014 -0.02935 -§— -0.00091 0.00325 0.01928 0.02675 0.03131 0.03284 My@y=a/2 X/a Cd Figure 3.30b. Moment Results H @ QQ 111 ax [g EH ®®®® Clamped Edge (0363666 @®®@® @®®®® C9 ®@@@@ 4 Clampe ‘Edge 6) 7:3 $2 CNOCDxJowntn»roF* .Figure 3.31a. M E y 10920. 15288. 9555- 5460. 3003. 2184. 9077. 4641. 1979. 1092. 14.0 7.0 Q: annwx OOOOOOOOOO Kn\OUICI) wwwu HHN mmw X-DIRECTION PROPERTIES Y—DIRECTION PROPERTIES aSTIFFNESS OF MATERIAL TYPE NO. 6 STIFFNESS OF MATERIAL TYPE NO. 1 _STIFFNESS OF MATERIAL TYPE NO. 2 _STIFFNESS OF MATERIAL TYPE NO. 6 LE. EX 0.3 12000. 0.3 16800. 0.3 9926. 0.3 5025. 0.3 2104. 0.3 1144. 0.285 10448. 0.255 5912. 0.198 3192. 0.15 2287. 2(1+\/ UX Z/y) ELEMENT MATERIAL TYPE NO. PrOperties Symmetric About The Diagonal. El y 12000. 16800. 10448. 5912. 3192. 2287. 9926. 5025. 2104. 1144. EH 3600. 5040. 2978. 1508. 631. 343. 2978. 1508. 631. 343. Idealization of Plate with Homogeneous Central Portion and OrthotrOpic Nonhomogeneous Boundary Portion ml 4200. 5880. 3603. 1972. 980. 637. 3603. 1972. 980. 637. @ y=O 112 MOMENTS ; Unit —M1§ pa X=O X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a -0.00l91 -0.01234 -0.02872 -0.03245 -0.02917 -0.02846 @ y=—§- —0.00084 0.00231 0.01728 0.02501 0.02982 0.03141 2 My@y=a/2 ~03-1 -My@y=0 .02.. .01- N 2. \\ >2. 5 () X/a -_.,/"011 012 013 014 015 Figure 3.31b. Moment Results 113 7. A Y o o o o 01"”:va o o o o 0);} Eh-ziu. U xEy I/yEX 9 9 9 9 Rumor @0000)3 v—E; @ o O o @ 2‘1+W’ 4’ Clamped Edge (:) ggEMENT MATERIAL TYPE CX _STIFFNESS OF MATERIAL TYPE NO. ‘STIFFNESS OF MATERIAL TYPE NO. HO\ PrOperties Symmetric About The Diagonal. _STIFFNESS OF MATERIAL TYPE NO. 2 _STIFFNESS OF MATERIAL TYPE NO. 6 CX== 0.2 & = 7.0 X-DIRECTION PROPERTIES CX== 0.1 & =14.0 Y-DIRECTION PROPERTIES @ EX Ey UK My EX Ey E 0 1 10920. 10920. 0.3 0.3 12000. 12000. 3600. 4200. 2 15288. 15288. 0.3 0.3 16800. 16800. 5040. 5880. 3 9555. 9077. 0.3 0.285 10448. 9926. 2978. 3603. 4 5460. 4641. 0.3 0.255 5912. 5025. 1508. 1972. 5 3003. 1979. 0.3 0.198 3192. 2104. 631. 980. 6 2184. 1092. 0.3 0.15 2287. 1144. 343. 637. 7 9077. 9555. 0.285 0.3 9926. 10448. 2978. 3603. 8 4641. 5460. 0.255 0.3 5025. 5912. 1508. 1972. 9 1979. 3003. 0.198 0.3 2104. 3192. 631. 980. 10 1092. 2184. 0.15 0.3 1144. 2287. 343. 637. Figure 3.32a- Idealization of Plate with Homogeneous Central Portion and OrthotrOpic Boundary Portion 11h MOMENTS ; Unit ——M1§- pa X=O X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a @ y=O -0.00217 —0.01364 -O.02992 -0.O3051 -O.024O9 -0.02242 @ y=a/2 -O.OOO95 0.00329 0.01970 0.02733 0.03199 0.03354 ~03~ My @ y=a/2 cu m E 04 .02— - M @ :0 y y .01_ X/a O l l 1 l T 0.1 0.2 0.3 O.“ 0.5 Figure 3.32b. Moment Results 115 r“cocoo—E 00000;; 00000;, 000008 @0000 ' Clamped Edge \/EXE STIFFNESS OF MATERIAL TYPE NO. 6 ”STIFFNESS OF MATERIAL TYPE NO. 1 _STIFFNESS OF MATERIAL TYPE NO. 2 _STIFFNESS OF MATERIAL TYPE NO. 6 Figure 3.33a- C1k= 0.2 & ,/3= 7.0 a = 0.1 86 fl=1l~h0 @ Ex Ey VX uy 1 10920. 10920. 0.3 0.3 2 15288. 15288. 0.3 0.3 3 9555' 90770 003 0.28 4 5460. 4641. 0.3 0.25 5 3003. 1979. 0.3 0.19 6 2184. 1092. 0.3 0.15 7 9077- 9555- 0-285 0-3 8 4641. 5460. 0.255 0.3 9 1979. 3003. 0.198 0.3 10 1092. 2184. 0.15 0.3 11 21840. 10920. 0.3 0.15 12 21840. 21840. 0.3 0.3 13 10920. 21840. 0.15 0.3 5 5 8 1 1 1 2 El X 2000. 6800. 0448. 5912. 3192. 2287. 9926. 5025. 2104. 1144. 2869. 24000. 11035. ,y 2(1+\/I/X Uy) ELEMENT MATERIAL TYPE NO. PrOperties Symmetric About The Diagonal X-DIRECTION PROPERTIES Y-DIRECTION PROPERTIES E§ E" G 12000. 3600. 4200. 16800. 5040. 5880. 9926. 2978. 3603. 5025. 1508. 1972. 2104. 631. 980. 1144. 343. 637. 10448. 2978. 3603. 5912. 1508. 1972. 3192. 631. 980. 2287. 343. 637. 11435. 3430. 6370. 24000. 7200. 8400. 22869. 3430. 6370. Idealization of Plate with Homogeneous Central Portion Except 3 rd. Set of Elements From Center and OrthotrOpic Nonhomogeneous Boundary Portion 116 M MOMENTS ; Unit ——1§ p8. X=O X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a @ y=0 -0.00201 -0.01263 -0.02926 -0.02892 -0.02411 -0 02121 @ y=a/2 -0.00089 0.00340 0.02377 0.03366 0.02883 0.03020 .4 (U 94 .03« M @ =a 2 y y / .Ofd X/a O r I I I *1 “~——”’ 0.1 0.2 0.3 0.4 0.5 Figure 3.33b. Moment Results 117 ®®®® 0":STIPENESS OF MATERIAL TYPE NO. _STIFFNESS OF MATERIAL TYPE NO. _STIFFNESS OF MATERIAL TYPE NO. @ I‘FAPHA meoomwmmtme Figure 3.34a. [ooooo C o o o o 9 C0 ®©Q®® ®©®®© lamped Edge STIFFNESS OF MATERIAL TYPE NO. E! X 0) 8“ E' “J y 'd 8. E EH (0 H U E 10920. 15288. 9555- 5460. 3003. 21840 9077- 4641. 1979- 1092. 16380. 16380. 10920. & = 7.0 & =14.0 Ey L; 10920. 0.3 15288. 0.3 9077. 0.3 4641. 0.3 1979. 0-3 1092. 0.3 9555. 0.2 5460. 0.2 3003. 0.1 2184. 0.1 10920. 0.3 16380. 0.3 16380. 0.2 U1\O\JI CI) 00mm X-DIRECTION PROPERTIES Y-DIRECTION PROPERTIES V: wwmwwuu HHN NWUJ c< OOOOOOOOOOOOO UI\O\J‘I (I) CDUIUI O\l\) HO\ E. X 12000. 16800. 10448. 5912. 3192. 2287. 9926. 5025 o 2104. 1144. 17426. 18000. 11617. UXEy 1- Uny N/EXE 1L 2(1+\/1/X My) ELEMENT MATERIAL TYPE NO. PrOperties Symmetric About The Diagonal. EU Y 12000. 16800. 9926. 5025. 2104. 1144. 10448. 5912. 3192. 2287. 11617. 18000. 17426. E" 3600. 5040. 2978. 1508. 6 1. 3 3- 2978. 1508. 631. 343. 3485. 5400. 3485. Idealization of Plate with Homogeneous Central Portion Except 3 rd. Set of Elements From Center and OrthotrOpic Nonhomogeneous Boundary Portion Q l 4200. 5880. 3603. 1972. 980. 637. 3603. 1972. 980. 637. 5371. 6300. 5371. 118 M MOMENTS ; Unit ——12 pa X=O X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a @ y=0 -0.00208 -0.01308 -0.02956 —0.02968 -0.02411 -0.02182 @ y=a/2 -0.00092 0.00333 0.02185 0.03059 0.03028 0.03174 N <0 .4. / '03“ M @ y=a/2 y .02_ -My @ y=0 .01— X/a 0.1 0.2 0.3 0.4 0.5 Figure 3.34b- Moment Results MO“: @ r21 \£>(jo\10\mkw’\)"“ k.) 0 7O \.\ Flgure 3 119 TX '3? y ©®O®© E); @®O@®8 , 'U E .a y "@®®'§ EE" @8888] a 4 Clamped Edge _STIFFNESS OF MATERIAL TYPE NO. 6 ’STIFFNESS OF MATERIAL TYPE NO. 1 ‘fl3‘STIFFNESS 0F MATERIAL TYPE NO. 2 ~ 'STIFFNESS OF MATERIAL TYPE NO. 6 @Ex Ey Ux Uy E}; 1 10920. 10920. 0.3 0.3 12000. 2 15288. 15288. 0.3 0.3 16800. 3 9555- 9555- 0-3 0-3 10500. 4 5460. 5460. 0.3 0.3 6000. 5 300 . 3003. 0.3 0.3 3300. 6 218 . 2184. 0.3 0.3 2400. 7 8736. 8736. 0.3 0.3 9600. 8 6552. 6552. 0.3 0.3 7200. 9 4368. 4368. 0.3 0.3 4800. E K 1-LQUy E V 1-Uny UxEy U YEX l—vxuy °r 1-vxu VEXEy 2(1+‘\/. Ux Uy) y ELEMENT MATERIAL TYPE NO. E& E" G 12000. 3600. 4200. 16800. 5040. 5880. 10500. 3150. 3675. 6000. 1800. 2100. 3300. 990. 1155. 2400. 720. 840. 9600. 2880. 3360. 7200. 2160. 2520. 4800. 1440. 1680. Figure 3.35a- Idealization of Plate with Nonhomogeneous Central Portion 120 M MOMENTS ; Unit —% pa X=O X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a @ y=0 -0.00192 -0.01385 -0.03285 —0.03468 -0.03316 -0.03144 @ y=a/2 -0.00232 0.00062 0.01260 0.02332 0.03085 0.03390 N 2% a My @ y=a/2 .0 -M @ =0 3‘ y y .02‘ .01- X/a O I I ’T I I J4 0.2 0.3 0.4 0.5 Figure 3.35b. Moment Results 121 “—1 ®®®©® ®©®®© Q) ® ® C9 C9 QQQQQ . Clamped CX= _STIFFNESS 0F MATERIAL TYPE -STIFFNESS 0F MATERIAL TYPE HHHH WNHOOCIJVO‘skn-PWNH Figure 3.36a. STIFFNESS 0F MATERIAL TYPE dge STIFFNESS 0F MATERIAL TYPE 1 15288. 1 . 1 . 1 . 1 0920. 5288. 9077- 4641. 1979- 1092. 9555- 5460. 3003. 2184. 0920. 9110. 9110. 7.0 14.0 CIDUIUI OOOOOOOOOOOOO UI\OU1CD wa PH wmmwwmww V H Central - é <:9 E' = EX x 1- LQI/y (D E ® g E! = 1 y (B U Y UXVy (D g E" = UXEY or UyEX 0 H 1-1/XIJy 1-2/XUy O _ th'E‘ G = X y C9 2(1+\/ uxuy) (:) ELEMENT MATERIAL TYPE N0. N0. 6 NO. 1 PrOperties Symmetric About The Diagonal. N0. 2 N0. 6 X-DIRECTION PROPERTIES Y-DIRECTION PROPERTIES 17y EX Ey E G 0.3 12000. 12000. 3600. 4200. 0.3 16800. 16800. 5040. 5880. 0.285 10448 9926. 2978. 3603. 0.255 5912. 5025. 1508. 1972. 0.198 3192. 2104. 631. 980. 0.15 2287. 1144. 343. 637. 0.3 9926. 10448. 2978. 3603. 0.3 5025. 5912. 1508. 1972. 0.3 2104. 3192. 631. 980. 0.3 1144. 2287. 343. 637. 0.171 20146. 11512. 3454. 5888. 0.3 21000. 21000. 6300. 7350. 0.3 11512. 20146. 3454. 5888. Idealization of Plate with Homogeneous Portion Except 3 rd. Set of Elements From Center and OrthotrOpic Nonhomogeneous Boundary Portion .03- .021 .014 04 7T Figure 122 M MOMENTS ; Unit -—¥§ pa X=0 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a @ y =0 -0.00205 -0.01284 -0.02941 -0.02929 -0.02411 -0.02151 @ y=a/2 -0.00090 0.00336 0.02283 0.03215 0.02953 0.03094 M 2 pa My @ y=a/2 .03- .02_ -My @ y=O .01— X/a 0" r I I I I 0.1 0.2 0.3 0.4 0.5 Figure 3.36b. Moment Results 123 Figure 3.37a- 11 E X y 1-1/X l/Y ELEMENT MATERIAL TYPE 4200. 5880. 3603. 1972. 980. 637. 3603. 1972. 980. 637. 5582. 6720. 5582. Idealization of Plate with Homogeneous Central Portion Except 3 rd. Set of Elements From Center and OrthotrOpic Nonhomogeneous Boundary Portion I“ "i Y o o c o o E. _ Ex _ 1_ o o c o o X :ny Ed E' = E; 'd y 1- UXVy @ ® ® C9 CB 2, UE 5 E" = X.X. or G C) CD CD (7 8 “VXVY _ \fETiT’ C5 C3 C9 C9 (9 G = "y , __ 2M+VLQL/) I Clamped Edge y l 9 _STIFFNESS OF MATERIAL TYPE NO. 6 ‘STIFFNESS OF MATERIAL TYPE NO. 1 PrOperties Symmetric About The Diagonal. _STIFFNESS OF MATERIAL TYPE NO. 2 ‘STIFPNESS 0F MATERIAL TYPE NO. 6 CY: 0.2 &: [3: 7 0 X-DIRECTION PROPERTIES o<= 0.1 8. [3:14 0 Y-DIRECTION PROPERTIES ® EX Ey ux Uy E}; Ey E" 1 10920. 10920. 0.3 0.3 12000. 12000. 3600. 2 15288. 15288. 0.3 0.3 16800. 16800. 5040. 3 9555. 9077. 0.3 0.285 10448. 9926. 2978. 4 5460. 4641. 0.3 0.255 5912. 5025. 1508. 5 3003. 1979. 0.3 0.198 3192. 2104. 631. 6 2184. 1092. 0.3 0.15 2287. 1144. 343. 7 9077. 9555. 0.285 0.3 9926. 10448. 2978. 8 4641. 5460. 0.255 0.3 502 . 5912. 1508. 9 1979. 3003. 0.198 0.3 210 . 3192. 631. 10 1092. 2184. 0.15 0.3 1144. 2287. 343. 11 17472. 10920. 0.3 0.188 18513. 11571. 3471. 12 17472. 17472. 0.3 0.3 19200. 19200. 5760. 13 10920. 17472. 0.188 0.3 11571. 18513. 3471. 124 M MOMENTS ; Unit ——¥§ pa X=O X=0.1a X=0.2a - X=0.3a X=0.4a X=0.5a @ y=O -0.00207 -0.01298 -0.02950 -0.02953 -0.02411 -0.02169 @ y=a/2 -0 00091 0.00334 0.02225 0.03123 0.02997 0.03141 N .12. My @ y=a/2 .021 -My @ y=0 001.4 0 X/a *:==’f I I I T I 0.1 0.2 0.3 0.4 0.5 Figure 3-37b. Moment Results 125 plate. The maximum positive moment is always at the vvnfvr of the plate for the isotrOpic homogeneous uniform thickness simply-supported plate. As shown in Figure 3.38 the maximum bending moment is at the center for the square simply—supported plate and has a value of 0.04807 pa2. Corresponding to the positive bending moment, there is a 2 twisting moment Mx with a maximum value of 0.029552 pa y at the edge of the plate. To Optimize these two moments in the simply—supported plate we established a model with 7 ratio varying along both x and y directions of the plate as shown in Figure 3.6b. When 1r: .5, the stiffness prOperty of the edge element has a value of 0.5 to that of the center element with the others varying by following a parabolic variation; the bending moment has increased and the twisting moment decreased as given in Figure 3.39. As 7r: 2.0 both the bending and twisting moments have a maximum value of 0.0397 pa2 as shown in Figure 3.40. It is the Optimum condition for the simply-supported square plate with a uniform loading. In Figure 3.41 through 3.43, results for ir==5.0, 7.5, and 10.0 are given. 3.3.5. Results Comparison of T—18 and Simpler Elements The T-18 plate bending finite element program was used to study a clamped square plate. Due to the limitation on the memory core that can be used in the University computer facility, a relatively small number of elements Figure 126 @@®® ©©©© @@@® @@@@ Simply-support Edge @®@®® GD @ @ @ * Simply-supported Edge (:) ELEMENT MATERIAL TYPE N0. _STIFFNESS OF MATERIAL TYPE NO._§ "STIFFNESS OF MATERIAL TYPE NO. 1 (:) E 2/ G 1 1092.0 0.3 420.0 2 1092.0 0.3 420.0 3 1092.0 0.3 420.0 4 1092.0 0.3 420.0 5 1092.0 0.3 420.0 Figure 3.38a. Idealization of Nonhomogeneous Simply- supported Square Plate with 7’=1.0 127 M MOMENTS ; Unit -—L§ X20 M xy pa ]’=1.0 X=0.1a X=0.2a My@y=a/2 0.00229 0.01636 0.03006 0 02‘ 001‘ Figure 3.38b. 0.1 032 Moment Results X=0.3a X=0.4a X=0.5a @ y=0 0.02955 0.02748 0.02228 0.01546 0.07886 0.00229 0.04002 0.04605 0.04807 M @ =a 2 y y / M @ y=0 X/a 013 044 (L5 128 NR @©®® @©©@ 6909036969 (9 (9696369 @@©®® @ Simply-supported Edge (9 Simply-supported Edge (:) ELEMENT MATERIAL TYPE N0. _STIFFNESS 0F MATERIAL TYPE N0. 45 "STIFFNESS OF MATERIAL TYPE N0. 1 '7': 0.5 (:) E 2/ G 1 1092.0 0.3 420.0 2 1057.9 0.3 406.9 3 955.5 0.3 367.5 4 784.9 0.3 301.9 5 546.0 0.3 210.0 Figure 3.39a- Idealization of Nonhomogeneous Simply- supported Square Plate with ']'=0.5 129 M MOMENTS ; Unit '2 pa ]'= 0.5 X=0 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a Mxy@ y=0 0.02030 0.01862 0.01492 0.01024 0.00518 0.00239 My @ y=a/2 0.00239 0.01522 0.03197 0.04506 0.05284 0.05510 .0 51 My @ y=a/2 .04. .03. N G! '2 Q4 .02. 0 X/a 0T1 032 013 014 015 Figure 3.39b. Moment Results ”4'1 >< Figure 130 “‘1 .4; @@@@ @©@@ @@@@ Simply—supported Edge @@@® @ @ @ @ Simply-supported Edge 'oeooo (:) ELEMENT MATERIAL TYPE N0. _STIFFNESS OF MATERIAL TYPE N0. 5 _STIFFNESS 0F MATERIAL TYPE N0. 1 ]'= 2.0 (:> E 2/ G 1 1092.0 0.3 420.0 2 1160.3 0.3 446.3 3 1365.0 0.3 525.0 4 1706.3 0.3 656.3 5 2184.0 0.3 840. Figure 3-403. Idealization of Nonhomogeneous Simply- supported Square Plate with 7’=Z.O 'ure Ac // 1 AC 1 v v I 3 .... a... .... A . . . Q ,, .-.. L. I MAM : 1 «1.... Will ......6 E 131 M MOMENTS ; Unit 2 pa 7’: 2.0 X=0 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a Mxy@ y=0 0.03979 0.03728 0.03040 0.02130 0.01097 0.02177 My @ y=a/2 0.00217 0.01777 0.02839 0.03461 0.03820 0.03974 .04- M @ y=a/2 y .03— a: m a. 002—1 Mxy@ y=0 .01- X/a O I I I I l O 1 O 2 0.3 O 4 O 5 Figure 3.40b. Moment Results .—< "AA fié’ur‘e 132 @©®® @@©© @®@® @@@@@ Simply-supported Edge . @®©®® G) SimpIy-supported Edge @ @ (:) ELEMENT MATERIAL TYPE N0. _STIFFNESS OF MATERIAL TYPE NO. 5 _STIFFNESS OF MATERIAL TYPE N0. 1 (:) E 2/ G 1 1092.0 0 3 420.0 2 1365.0 0.3 525.0 3 2184.0 0.3 840.0 4 3549.0 0 3 1365.0 5 5460.0 0 3 2100.0 Figure 3.41a- Idealization of Nonhomogeneous Simply- supported Plate with 2V=5.0 (’3 \f‘ .01 Q 4. Sure 133 MOMENTS ; Unit 7:5.0 X=0 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a p8. Mxy@ y=0 0.05230 0.04927 0.04036 0.02853 0.01483 0.02039 My @ y=a/2 0.00204 0.01957 0.02695 0.02823 0.02796 0.02837 -05_ .04... pa .O3_ .02._ .01._ ' 011 0.'2 Figure 3.41b. Moment Results H: :1. ‘lgure 134 H: (9696969 (969696») @@@@ @@@@ Simply-supported Edge“;L @@©®® @ @@ @ Simply—supported Edge (:) ELEMENT MATERIAL TYPE N0. _STIFFNESS 0F MATERIAL TYPE NO. 5 _STIFFNESS 0F MATERIAL TYPE NO. 1 7: 7.5 (:> E L/ G 1 1092.0 0.3 420.0 2 1535.6 0.3 590.6 3 2866.5 0.3 1102.5 4 5084.6 0.3 1955.6 5 8190.0 0.3 3150.0 Figure 3.42a. Idealization of Nonhomogeneous Simply- supported Square Plate with 2r=7.5 135 MOMENTS ; Unit “M—§ pa 7: 7.5 X=0 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a Mxy@ y=0 0.05685 0.05364 0.04401 0.03118 0.01626 0.01990 My@ y=a/2 0.00199 0.02023 0.02661 0.02601 0.02403 0.02373 .05_ .04_, N E (U a. .03_ M @ = 2 y Y a/ .02-‘ Mxy @ y=0 .01 _ X/a O r T I l 1 0.1 _ 0.2 0.3 0.4 0.5 Figure 3.42b- Moment Results [_X A @@®® @0399) @©@@ ®®@@@ @®@®© Q) (D @ é Simply-supported Edge (:) ELEMENT MATERIAL TYPE N0. '7-_STIFFNESS 0F MATERIAL TYPE N0. 5 _ 10 0 “STIFFNESS 0F MATERIAL TYPE N0. 1 ‘ ' 1 1092.0 0.3 420.0 2 1706.3 0.3 656.3 3 3549.0 0.3 1365.0 4 6620.3 0.3 2546.3 5 10920.0 0.3 4200.0 Figure 3.43a. Idealization of Nonhomogeneous Simply- supported Square Plate with 7=10.0 137 MOMENTS ; Unit 7’: 10.0 X=0 X=0.1a X=0.2a X=0.3a X=0.4a X=0.5a Mxy@y=0 0.05963 0.05631 0.04624 0.03282 0.01714 0.00196 My @y=a/2 0.00196 0.02063 0.02647 0.02468 0.02159 0.02070 .06-: .05.- .04.J (\I E: «3 Q4 .03‘ M @ y=a/2 .02.; Mxy@ y=0 .01.. X/a O l I I I I I 0.1 0.2 0.3 0.4 0.5 Figure 3.43b. Moment Results 138 can be handled at this time. Also the internal moment for T-18 program is not a constant within each element but varies from point to point in the element. The T-18 program was develOped to print out only the internal moments at the centroid of each element. So it is difficult to make a comparison between T—18 and ELAS outputs on moment print- outs. Only the displacements at each nodal point can be used to compare the two. A very simple model of T—18 has been used. As shown in Figure 3.44, an idealization with 5 nodes and 4 elements is used to represent the complete clamped square plate. With the uniform load applied on the plate a center displacement of 1.3366 is obtained for this simple T-18 model; this is 2.86 % error compared with the exact solution of 1.37592. As shown in Figure 3.45, a comparison was made with 5 different sizes of ELAS models. For the same accuracy, the ELAS plate bending program would need about 100 nodal points and 96 elements in comparison with the 5 nodal points and 4 elements for T-18 program. From the above comparison, we can see that the T-18 finite element program has a good potential use in the future when a large core becomes available. During the develOpment, we have found that a professional programmer can save up to 10 % to 15 % core locations. Combining the saving from the programming technique and the rapid imorovement in computer hardware the T-18 program should have wider application in the future. This study is just finite elem Hcpefullb' t Erection- L- 139 an initial step toward the use of better elements in the finite element method for the practical engineering field. H0pefully this study may stimulate the develOpment in that direction. 140 I Square Plate Clamped All 4 Edges Uniformly Loaded IsotrOpic Case X a=b=10.0 E=1000.0 t=1.0 U =0.3 1L a 'I p=10.0 T=18 4 Elements & 5 Nodes Mesh. @ Node Point No. @ 0w 0w 82w 52w 02w w ——— ——— bx by 0 x2 0x03! 93'2 1 O O O O O O 2 O O O O O O 3 -1.3366 0 0 0.1934 0 0.1934 4 O O O O O O 5 O O O O O O b, _ a _ (j exact— -0.00126EE— — -1.37592 Error = 2.94 % Figure 3.44. T-18 Results for Clamped Square Plate 141 ELAS 1 ELAS 2 ELAS 3 T-18 ELAS Square Plate Clamped All 4 Edges Uniformly Loaded IsotrOpic Case DISPLACEMENT COMPARISON EngggT ELAS 1 ELAS 2 ELAS 3 ELAS 4 ELAS 5 T-18 Center Displ. 0.21667 1.27906 1.32503 1.3452 1.37427 1.3366 Exact 1.37592 Error % 84.25% 7.04% 3.70% 2.23% 0.12% 2.86% Figure 3.45. Results Comparison IV. Conclusions and Recommendations 4.1. Conclusions The results show the effects of varying prOperties in many different ways for a clamped square plate with uniform load applied. By using a uniform boundary region with stiffness of about .2 of the center region, the maximum moment is reduced to about 66.8% of that for the homogeneous plate for the case with the boundary region having a width of .2a. If the boundary region decreases to a width of .1a, the maximum moment is reduced to 64.8% of the homogeneous one when the uniform stiffness of the boundary region becomes .16 of the center region. If, in addition, the stiffness of the boundary region is varied parabolically, it is found that the maximum moment is further reduced to a value of 62% for the 20% boundary region case. The same results can also be found by varying the thickness of the plate instead of the material prOperties. The same reduction on the maximum moment is also achieved by varying the prOperties in different directions within the plate. In the uniformly loaded square plate with simply supported edge conditions, the maximum moment can also be reduced to about 82% by allowing the edge of the plate to be twice as stiff as at the center of the plate. 142 143 Practically, both the variation of element stiffness and thickness to reduce the maximum moment have value in engineering design applications. For example, in the reinforced concrete structural design, a composite section formed by reinforcement steel and concrete can always, by varying the ratio of reinforcement steel to concrete, be made to resist the applied moment. In areas of large moment, we use more reinforcement steel; in the sense of element stiffness this just increases the stiffness in that local high moment region and leads to a further increase in moment. By using the results of this study, we should be able to reduce the maximum moment by the pr0per arrangement of the steel. The rearrangement of moment can also be attained by varying element thickness. For example, in the prestressed concrete containment building of a nuclear power plant we use i " thick carbon steel plate as liner inside the prestressed concrete face for radioactive shielding. Whenever there is an Opening- such as for pipes or cables— or at some area where a large local moment is produced by a postulated pipe break or equipment failure, the thickness of the liner plate is increased to transfer the additional moment. The standard way to handle this case is to assume a uniform thickness and stiffness in the area and apply the thrust force onto the plate; then, the moment distribution is determined by conventional or computer analysis. From the resultant moment we design 144 the plate thickness which is sufficient to transfer the moment or force. Actually an analysis should be done by using the variation of the thickness as an input to calculate the actual moment distribution. This moment is different from the one obtained by assuming uniform thickness. Furthermore, by using the approach of this study we should be able to vary the element thickness to reduce the maximum moment and Optimize the design. T-6 plane stress and T—18 plate bending finite element programs were develOped and examples show that more accurate results can be obtained by using these improved element programs rather than the commonly used programs such as ELAS. 4.2. Recommendations Since the primary goal of this study is to pave the road for develOping a suitable finite element computer program to handle the folded plate structures in general, the following suggestions are made for further investigation and develOpment. (1). Resolve the theoretical difficulty on assembling the T-6 plane stress and T-18 plate bending programs by develOping a suitable transformation matrix for the assembly. (2). Assemble the T-6 plane stress and the T—18 plate bending programs to form a general finite program for folded plate structures. 145 (3). By using the results for the clamped and simply support square plates, study the possibility of reducing the stress concentration and Obtaining a redistribution for the thin plate, folded plate and thin shell structures. BIBLIOGRAPHY BIBLIOGRAPHY Lowe, P. C., and Melchers, R. E., "On the Theory of Optimal, Constant Thickness, Fibre—Reinforced Plates," International Journal of Mechanical Sciences, Vol. 14, 1972. Kirsch, U., "Optimum Design of Prestresse Plates," Journal of the Structural Division, ASCE, Vol. 99, ST 6. June 1973. Zienkiewicz, 0. C., and Irons, B., "Triangular Element in Plate Bending Conforming and Non-Conforming Solution" lst Conference on Matrix Methods in Structural Mechanics, Wright—Patterson AFB, Ohio, Oct. 1965. Melosh, R., "A Flat Triangular Shell Element Stiffness St Conference on Matrix Methods in Structural Matrix," 1 Mechanics, Wright—Patterson AFB, Ohio, Oct. 1965. Clough, R. W., and Tocher, J. L., "Finite Element Stiffness Matrices for Analysis of Plate Bending," Proceedings of the 1st Conference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, Ohio, 1965. Clough, R. W., and Felippa, C.A., "A Refined Quadrilateral Element for Analysis of Plate Bending," Proceedings of d the 2n Conference on Matrix Methods in Structural Mechanics, Wright—Patterson Air Force Base, Ohio, 1968. 147 10. 11. 12. 13. 14. 15. 148 Utku, 8., "0n Derivation of Stiffness Matrices with C rd Rotation Fields for Plates and Shells," 3 Conference on Matrix Methods in Structural Mechanics, Wright- Patterson AFB, Ohio, Oct. 1971. Utku, S., "Stiffness Matrices for Thin Triangular Elements Of Nonzero Gaussian Curvature," AIAA Journal, Vol. 5, NO. 9, Sept. 1967. Bell, K., "A Refined Triangular Plate Bending Finite Element," International Journal for Numerical Method in Engineering, Vol. 1, No. 1, Jan.-March 1969. Butlin, G. A., "A Compatible Triangular Plate Bending Finite Element," International Journal of Solids & Structures, 1970. Whang, B., "Elasto-Plastic OrthotrOpic Plates and Shells." Proceedings of the Symposium on Application of Finite Element Methods in Civil Engineering,Nashville, Tenn. Nov. 1969. Zienkiewicz, O. C.,The Finite Element Method in Engineer- ing Science. McGraw-Hill, 1971. Desai, C. & Abel, J.,Introduction to the Finite Element Method-A Numerical Method for Engineering Analysis , Van Nostrand Reinhold, 1972. Martin, H. & Carey, C.,Introduction to Finite Element Analysis-—-Theory and Application , Mcgraw-Hill, 1973. Utku, S., "ELAS 75 Computer Program—-—User's Manual," Structural Mechanics Series NO. 10, School of Engineering, Duke University, Dec. 1971. 16. 17. 18. 19. 20. 21. 22. 23. 24. 149 Wilson, E., Bathe, K. & Peterson, F.,"SAP IV--- A Structural Analysis Program For Static and Dynamic Response of Linear System," College of Engineering, University of California, Berkeley, Calif. June 1973. Control Data Corporation,"MRI/STARDYNE Static and Dynamic Structural Analysis Systems," Minneapolis, 1973. Argyris, J. H.,"Energy Theorems and Structural Analysis," Aircraft Engineering, 26, 1954 and 27, 1955. Turner, M. J., Clough, R. W., Martin, H. C. and TOpp, L. J.,"Stiffness and Deflection Analysis of Complex Structures," Journal of the Aeronautical Sciences, Vol. 23. NO. 9. 1959- Langefors, B.,"Analysis of Elastic Structures by Matrix Transformation with Special Regard to Semimonocoque Structure," Journal of the Aeronautical Sciences, Vol. 19, NO. 10, July 1952. Denke, P. H.,"A Matrix Method of Structural Analysis," nd U.S. National Congr. Appl. Mech., Americam Proc. 2 Society of Mechanical Engineering, June 1954. Proceedings, 1St Conference on Matrix Methods in Structural Mechanics, Wright—Patterson AFS, Ohio, Oct. 1965. Proceedings, 2nd Conference on Matrix Methods in Structural Mechanics, Wright-Patterson AFS, Ohio, Oct. 1968. Proceedings, 3rd Conference on Matrix Methods in Structural Mechanics, Wright—Patterson AFS, Ohio, Oct. 1971. 25. 26. 27. 28. 29. 150 Adini, A.,"Analysis of Shell Structures by the Finite Element Method," Ph.D. Dissertation, Department of Civil Engineering, University of California, Berkeley, 1961. Roland, I. & Bell, K., Finite Element Methods in Stress Analysis , Tapir, 1970. Tocher, J. L.,"Analysis of Plate Bending Using Triangular Elements," Ph.D. Dissertation, Department of Civil Engineering, University of California, Berkeley, 1962. Szilard, R., Theoryrand Analysis of Plates Classical and Numerical Methods , Prentice-Hall, 1974. Tocher, J. L. & Hartz, B. J.,"Higher-Order Finite Element for Plane Stress," Proceedings ASCE, EM4, Aug. 1967. APPENDI CES APPENDIX A—1 Simple Example To Demonstrate The Formulation Of The Overall Stiffness Matrix K Consider a pin—jointed bar with length 21. Two ends and one middle point will be chosen as nodal points which separate two elements (1) and (2) as shown in Figure A—l. y A A J—~4——-IBI l x 1 V 2 V 3 l i 1 1 Figure A-1 Idealization for the Pin-jointed Bar The displacement function uX along the longitudinal axis of the bar is given by uX = u1 @ x = O uX = u2 @ x = l and for element (2) are 152 153 uX = u2 @ x = l ------------------------ (A-2) 21 u = u @ x x 3 where ul, u2 and u3 are the nodal displacements. Substituting equation (A—2) into equation (A-l), the displacement function for each element can be expressed in terms of the nodal displacements as 11:1) : ul '1' (112—U1)% --------------------- (A_38‘) 42> = u ) 11L ___________________ (A—3b) +(u l 2 3’u2 The strain vector can be related to the displacement by a simple differentiation. Thus, (1) I (1) bu 1 Ful X 2‘ _5_X—X-—: T [-1 1] L12 -------------- (Ii-48.) (2) Z i (2) bux 1 u2 E X = "'8;— = T [-1 1] 1.13 -------------- (A-4b) where -%’[-1 1] is the matrix B in equation (2-4) of Chapter II . The element stiffness Kg can be formulated by following equation (2-7) of Chapter II. Kg =_[- BT D B dV Vol. Where D=E ( Modulus of Elasticity ) for the one—dimensional Pin-jointed bar. SO 154 —1 (I) _ .1. J; Kg 1 1E1[-11] dV i=1,2 Vol. 1 -f 1 :f __:1L_ El[—11]Adx 0 1.1 1 1 -1 =-£%? --------------------- ( Au5) —1 11 where K(l) is the element stiffness matrix relating g i=1 or 2 the element nodal displacements to the element nodal forces which can be represented by S(i) = Kéi) u(i) __________________________ ( A—6) To form the overall stiffness matrix, K, for the entire structure, a transformation matrix, A, should be formulated first. Let A represent the relationship between the element nodal displacement and the structure displacement as u = A U ----------------------------------- ( A-7) T _ (1) (2) T _ where u — [u u 1 , U — [ u1 u2 u3] The transformation matrix A is in the simple form for pin—jointed bar as A = A(1) A(2) 1 0 0 O 1 0 where A(1) = & A(2) = - ( A-8) 0 1 0 0 0 1 The equation (A-7) can be expressed eXplicitly as 155 T ' / \ 0 11(1) U1 [1 O 1 Fu\ u2 0 1 O 1 () ==0 1 o u """"""" (A'9) 2 u 0 0 1 Lu \u3/ \ j ~ 3; where I . 1 0 0 O 1 0 A = 0 1 0 0 0 lj Then, the overall stiffness matrix becomes T1 -1 0 0 r1 0 0 T 1 O O 0 AE -1 1 0 0 0 1 0 K =A KgA = 0 1 1 O l 0 0 1 _1 1 0 1 O O 1 0 0 -1 11 0 0 U 1 -1 O =—§5L‘-E—-1 2 -1 ------------------------- (A-10) 0 —1 1 As we discussed in Chapter II, this multiplication of ATKgA is never carried out. We can formulate the overall stiffness matrix, K, by just placing the elements of each element stiffness matrix, Kg, in their correct position and summing all the overlapping terms. It can be easily seen from this simple example as <1) <2> <3> 1 —1 0 <1) 1 —1 0 K 2 {LE— -1 1+1 -1 <2) = {E— -1 2 -1 0 -1 1 <3) 0 -1 1 APPENDIX Integration Formulas for a Plane Triangle rder n=r+s J4 xryS dA A x1 y1 I‘S X3y3) I‘ S x13’1 I‘S x3y3) I' S lel + I'S x3y3) \n-PKJDN X1V1 + I‘S X3y3) P60:14A P40P 20 + P. HP4OJ + 1 TOMA (6P40P02 +24P 2 11A +75P21 ) 840 P42: 20 3 ”Ext A (25P +225P P=33 28 30 P03 A _ 1 P51"I4§‘(2P40P118+5P30P x8 1 +2P P 21 20 3 P 22 +3OP 30 2 yi 21 P12 +36P _— 1)“ 168 +48P P12 31 20 P13 +36P IMO. y. Hwfi P11 02P31 +108P P 11 22 )‘ IE XEY§ 1:1 ER? P15 and P06 are found by interchanging x and P51 and P60 P24' y in the expressions for P42’ respectively. xi and y1 refer to the local Cartesian Coordinate System with the origin located at the centroid of the triangle. 156 APPENDIX A-3 Nonzero Terms Of The Integration J[ Qrl D Q dA For The IsotrOpic Material A (2,2)=A (6,6)=2(1—Z/)PO2 (2.9)=Z/A (6,10)=2(1-2/)P11 (3 3): (-1--'—--2 UM (6,11)=(1-2/)P 02 (3.8):(1—54m (8 8): (1; ”>11 (4.4)=4P20 (9.9)=A (4,5)=2P11 (10,10)=2(1-l/)P20 (4,11)=22/P20 (10,11)=(1-Z/)P11 (4,12)=42/P11 (11,11)=p20+(1—gl)p02 (5 5): PO 2+%-<1— 2/)P2 (11,12)=2P11 (5.6)=(1-2/)P11 (12,12)=4P02 1+Z/ (5,11)=( )P11 (5,12)=2l/P02 Matrix is symmetric, so only diagonal and upper triangular elements are shown. All eXpressions must be multiplied by a E factor of 1_212. For Prs the Appendix A-2 should be referred to. 157 APPENDIX A-4 Nonzero Terms Of The Integration -/~ QT D Q dA For The A OrthotrOpic Material (2,2)=nA (5,11)=nZ/2P11+TP11 (2.9)=nU2A (5,12)=2n 2/2P02 (3.3)=TA (6,6)=4TP02 (3.8)=TA (6,10)=4TP11 (4.4)=4nP20 (6,11)=2TP02 (4,5)=2nP11 (8,8)=TA (4,11)=2n2/2P20 (9,9)=A (4,12)=4nZ/2P11 (10,10)=4TP20 (5.5)=nP02+TP20 (10,11)=2TP11 (5.6)=2TP11 (11,11)=P20+TP02 (5.10)=2TP20 (11,12)=2P11 (12,12)=4P02 Matrix is symmetric, so only diagonal and upper triangular elements are shown. T=m(1-nZ/§). For Prs’ the Appendix A—2 should be referred to. 158 APPENDIX A-5 Nonzero Terms Of The Multiplication 0f P: D Pg For The IsotrOpic Material (4,4)=4 (4.7)=12x (4,9)=4Z/x (4,11)=24x2 (4,13)=4y2+42/x2 (4,15)=242/y2 (4,17)=24x2y (4,19)=4y3+122/x2y (4,21)=40Uy3 (5,8)=4(1-2/)x (5,12)=6(1-2/)x2 (5,14)=6(1-2/)y2 (5,18)=12(1-Z/)x2y (5.2o)=8(1-2/)y3 (6,7)=122/x (6,9)=4x (6,11)=242/x2 (4,6)=4l/ (4,8)=4y (4310)=122/y (4,12)=12xy (4,14)=12ny (4,16)=4Ox3 (4,18)=12xy2+42/x3 (4,20)=24ny2 (5.5)=2(1-L/) (5.9)=4(1-U)y (5.13)=8(1-Z/)Xy (5.17)=8(1-2/)x3 (5,19)=12(1-2/)xy2 (6,6)=4 (6,8)=4l/y (6,10)=12y (6,12)=12l/xy 159 160 (6.13)=4Uy2+4x2 (6,14)=12xy (6,15)=24y2 (6,16)=402/x3 (6,17)=24L/x2y (6,18)=122/xy2+4x3 (6,19)=4Z/y3+12x2y (6.20)=24xy2 (6,21)=40y3 (7,7)=36x2 (7,8)=12xy (7.9)=12Z/x2 (7.10)=362/xy (7,11)=72x3 (7.12)=36X2y (7.13)=12xy2+12z/x3 (7.14)=36Z/X2y (7.15)=722/xy2 (7,16)=120x“ (7.17)=72x3y (7,18)=36x2y2+122/xu (7,19)=12xy3+36Z/x3y (7.20)=72Ux2y2 (7,21)=1202/xy3 (8,8)=4y2+8(1-U)x2 (8,9)=4ny+8(1—2/)xy (8,10)=122/y2 (8,11)=24x2y (8,12)=12xy2+12(1-2/)x3 (8,13)=4y3+4z/x2y+16(1-z/)x2y (8,14)=12xy2 (8,15)=24Z/y3 (8,16)=40x3y (8,17)=24x2y2+16(1—l/)xu (8,18)=12xy3-20z/x3y+24x3y (8,19)=4y”+24x2y2-122/x2y2 (8,20)=16xy3+8Z/xy3 (8,21)=4OL/yu (9.9)=4x2+8(1-2/)y2 (9,10)=12xy (9,11)=24Z/X3 (9,12)=12xy2 (9.13)=4x3+16xy2-12Z/xy2 (9,15)=24xy2 (9,17)=16x3y+8Z/x3y (9.19)=12x3y+24xy3—2oz/xy3 (9.21)=40xy3 (1O,11)=722/x2y (10,13)=12L/y3+12x2y (10,15)=72y3 (10,17)=722/x2y2 (10,19)=12Z/yu+36X2y2 (10,21)=120yu (11,12)=72x3y (11,14)=722/x3y (11,16)=240x5 (11,18)=72x3y2+242/x5 (11,20)=1442/x3y2 (12,12)=36x2y2+18(1-2/)x4 (12,14)=36x2y2+182/x2y2 (12,16)=120xuy (12,18)=36x2y3-24z/x“y+36x (12.2O)=24x2y3+482/x2y3 4 y 161 2y+12(1-2/)y3 (9.14)=12x (9.16)=40 2A.“ (9,18)=4x”+24x2y2-12Z/xzy2 (9,2O)=24x2y2+16(1-l/)yu (10,10)=36y2 (10,12)=362/xy2 (10,14)=36xy2 (10,16)=1202/x3y (10,18)=36Z/xy3+12x3y (10,20)=72xy3 (11,11)=144x4 (11,13)=24x2y2+24Z/x4 (11,15)=144Z/x2y2 (11,17)=144x”y (11,19)=24x2y3+722/x4y (11,21)=24OL/x2y3 (12,13)=12xy3-122/x3y+24x3y (12,15)=722/xy3 (12,17)=72x3y2+24(1-Z/)x5 (12,19)=12xy1++36x3y2 (12.21)=1202/xy4 4 4 (13, 13): 4y +4x +32x2 y2 -24L/xy (13,14)=12x3y+24xy3-12Z/xy3 (13,16)=40x3y2+402/x5 162 22 (13,15)=24Z/yu+24x2y2 (13,17)=24x2y3+32x yEBZ/xuy (13'18 )=12qu+4x5+48x3y2-32 2/x3y2 (13.19)=4y5+12xuy+48x2y3—322/X2y3 (13, 20): —24x3y2 +32xy“ —8Z/xy4 (14,14)=36x2y2+18(1-Z/WAL (14,16)=12oz/x”y (14,18)=—12xuy+36x2y3 2y3+24(1- U)y5 4 (14,20)=72x (15.15)=144y (15.17)=144z/x2y3 (15.19)=24Z/y5+72X2y3 (15.21)=240y5 (16.17)=240x5y (16:19)=4OX3y3+1202/x5y (16.21)=4ooz/x3y3 (17'18)=72X3y3-24l/x5y+48x5y (17,20)=32x3y3+1122/x3y3 (13,21)=402/y5+40x2y3 E (14.15)=72xy3 1 (14,17)=24x3y2+482/x3y2 (14,19)=36x3y 2+36xyLL -24Z/xyu (14,21)=120xyu (15,16)=24OZ/x3y2 (15,18)=722/xy”+24x3y2 (15,20)=144xyu (16,16)=400x6 42 (16,18)=120xy +4OZ/x6 (16.20)=2402/x“y (17,17)=144xuy2+32(1-U)x6 24 42 (17,19)=24xy +48xy +24Z/xuy2 (17,21).:2402/x2y1+ (18,18)=36x2yu+4x6+72xuy 2-482/Xuy2 (18.19)=12xy5+12x5y+72x3y 3 —322/x3y3 163 (18,20)=24xuy2+48X2yu+242/X2y4 (18,21)=12OZ/xy5+40x3y3 (19,19)=4y6+36x4y2+72x‘2yu—48Z/xzyl‘L (19,20)=72x3y3+48xy5-242/xy5 (19,21)=4ol/y6+120x2y4 (20,20)=144x2y4+32(1-Z/)y6 (20,21)=240xy5 (21,21)=400y6 Matrix is symmetric, so only the diagonal and upper triangu- lar elements are shown. All the elements must be multiplied Etgt by a factor of 2 12(1-2/ )0 APPENDIX A—6 T Nonzero Terms Of The Multiplication Of Pg D Pg For The OrthotrOpic Material (4,4):40X (4,7)=12xDX (4,9):4x01 (4,11)=24x2DX (4,13)=4y2DX+4X2D1 (4,15)=24y2D1 (4,17)=24x2ny (4,19)=4y3DX+12x2yD1 (4,21)=40y3D1 (5v 8)=8Xny 2 (5,12)=12x DX y 20 xy _ 2 (5,20)=16y3DXy (6,7)=12xD1 6, = ( 9) 4ny (6.11)=24x2D1 (4.20)=24xy2D (4,6)=4D1 (4,8)=4yDX (4,10)=12y01 (4,12)=12xyDX (4.14)=12xyD1 (4,16)=40x3D X _ 2 3 (4.18)—12xy DX+4x D1 1 (5'5)=L|'ny (5.9)=8yDXy (5.13)=16XYD xy _ 3 (5,17)—16x ny (5,19)=24xy20Xy (6,6)=4Dy (6,8)=4yD1 (6,10)=12y0y (6,12)=12xyD1 164 _ 2 2 (6,13)—4y D1+4x Dy _ 2 (6,15)—24y Dy (6,17)=24x2yD1 (6,19)=4y3D1+12x2 yDy (6,21)=40y3Dy (7,8)=12xyDX (7,10)=36xy01 (7,12)=36x2yDX (7.14)=36X2yD (7,16)=120x40X 1 (7.18)=36x2y20X+12x“D1 20 (7.2o)=72x2y 1 _ 2 2 (8,8)—4y Dx+16x ny 20 (8,10)=12y 1 2 _ 3 (8,12)—12xy DX+24x DX y 2 _ 2 (8,14)—12xy D1+24xy ny (8,16)=40x3yDX (8,18)=12xy3DX+4x3yD1+48x3yD (8,19)=4yuDX+12x2y2D1+48x2y2 _ 3 3 (8,20)—24xy D1+32xy ny _ 2 2 (9,9)—4x D&+16y ny 165 (6,14)=12xyDy (6,16)=40x3D1 _ 2 3 (6,18)—12xy D1+4x Dy (6.20)=24xy20y (7.7)=36x20X (799):12X2D1 _ 3 (7.11)-72X DX (7.13)=12xy2DX+12x3D1 (7.15)=72xy2D1 (7.17)=72x3yDX (7.19)=12xy3DX+36x3yD1 (7.21)=120xy3D1 (8.9)=4XyD1+16xyDXy (8,11)=24x2yDX (8,13)=4y3DX+4x2yD1+32x2yDX (8.15)=24y3n1 4 _ 2 2 (8,17)—24x y DX+32x ny xy D XY 4 (8,21)=40y D1 = D y 166 (9.11)=24x301 _ 2 3 2 (9,13)—4xy D1+4x Dy+32xy ny (9,15)=24xy20y _ 3 3 (9,17)-24x yD1+32x nyy _ 3 3 3 (9.19)—4xy D1+12x yDy+48xy ny (9.21)=40xy30y (10,11)=72x2yD1 (10.13)=12y3D1+12x2yDy (10,15)=72y3Dy 20 (10.17)=72x2y 1 A (10,19)=12y D1+36x2y2Dy “D y (11,12)=72x3yDX (10.21)=120y (11,14)=72x3y01 (11,16)=240x5DX (11,18)=72x3y2DX+24x5D1 (11,20)=144x3y2D1 _ 2 2 4 (12,12)_36x y DX+36x DXy (12,14)=36x2y2D1+36x2y2D (12,16)=120x”y0X xy (12.18)=36x2y3DX+12xuyD1+72xuyDX _ 2 2 (9,12)—12x yD1+24x yDXy 2 _ 3 (9.14)—12x yDy+24y DX y (9,16)=40x”D1 y2D1+4quy+48x2y2D _ 2 2 4 (9.20)—24x y Dy+32y DX (9,18):12x2 y (10,10)=36y20y (10,12)=36xy2D1 (10,14)=36xy2Dy (10,16)=120x3yD1 (1O,18)=36xy3D +12x3yDy 1 (10,20)=72xy3Dy (11,11)=144xl+DX (11,13)=24x2y2DX+24qu1 (11,15)=144x2y2D (11,17)=144xuny 1 (11,19)=24x2y3nx+72x“y01 (11,21)=240x2y301 (12,13)=12xy3DX+12x3yD1+48x3yDX (12,15)=72xy3D1 (12.17)=72x3y2DX+48x5DXy y xy A13 167 (12, 19): -12xy3D X+36x3y2 D 1+72x3y2 D xy 3 3 4 (12, 20): 72x2 y D1 +48x2 y D xy (12,21)=120xy D A 1 (13,13)=4y DX+8x2 y 2D1 +4x3D y+64x2y 2D xy = 3 3 3 (13,14) 12xy D1+12x yD y+48xy D A XY 2 2D (13, 16): —40x3y2 D X+40x3D A D1 D y +64X y Xy (13, 15): 24y D1+24xy 1 (13,17)=24x2y3DX+24xL‘L (13,18)=-12xy3DX+16x3y2D1+4x3Dy+96x3y2DXy (13,19)=4y3DX+16x2 y3D1+12x3yD:+96x2 y3DXy (13,20)=24xy3D1+24x3y2Dy+64xy3DXy ' (13,21)=4Oy3D1+40x2y3Dy (14,14): 36x2 y 20 H+36y3D Xy (14,15L72xy30y (14,16)=120x“yn1 (14,17)=72x3y2D1+48x3y2DXy (14,18)=36x2y3D1+12x3yDy+72x2y3DXy (14,19)=-12xy3D1+36x3y2Dy+72xy3DXy (14 20): 72x 2y3Dy +48y3DX Xy (14,21)=120xy‘*0y (15,15L144yL’Dy (15,16)=24Ox3y2D1 (15.17)=144x2y3D1 (15,18)=72xy3D1+24x3y2Dy (15.19)=24y3D1+72x2y3Dy (15,20)=144xy3Dy (15.21L240y50y (16,16)=4OOX6DX (16,17)=24Ox3yDX (16,18)=120x3y2D X+4Ox6D1 4 2D (16,19)=40x3y3DX+120x3yD (16, 20): -240xy 1 D1 168 (16,21)=400x3y3D (17,17)=144x”y2ox+64x6nXy 1 (17,18)=72X3y3DX+24x5yD1+96x5ny (17,19)=24x2yuDX+72xuy2D1+96xuy2D y xy (17,20)=144x3y3D1+64x3y3DXy (17,21)=240x2y“D _ 2 4 4 2 6 4 2 (18,18)—36x y DX+24X y D1+4x Dy+144x y ny 1 (18,19)=12xy5DX+4OX3y3D +12x5yDy+144X3y3DX 4D xy 1 y (18,20)=72x2yuD1+24xuy2Dy+96x2y (18,21)=120xy5D1+4OX3y3D y _ 6 2 4 4 2 2 4 (19,19)—4y DX+24X y D1+36x y Dy+144x y ny (19,20)=24xy5D1+72x3y3Dy+96xy5D 6 Xy _ 2 4 6 (20,20)—144X y Dy+64y DXy 6D y D +12OX 2 4 1 y D (19,21)=40y (20,21L240xy5Dy (21,21)=400y Matrix is symmetric, so only the diagonal and the upper triangular elements are shown. APPENDIX A-7 Nonzero Terms Of The Generalized Element Stiffness K For The IsotrOpic Material (4,4)24A (4,11)=24P20 (4,13)=4P22+4l/P20 (4,15)=24L’P02 (4,17)=24P21 (4,19)=4P03+12L/p21 (4,21)=u02,/P03 (5.12)=12,LLP20 (5,14)=12LLP02 (5.18)=2L+MP21 (5.2o)=16,L(P03 (6,11)=2uL/P20 (6,13)=1+U P02+4P20 (6.15)=24P02 (6.17)=24UP21 (6.19)=br?/PO3+12P21 g21 UhéfiuUA (4,12)=12P11 (4,14)=1221P11 (4,16)=40P30 (n.18)=12P12+uL/P30 (4.20)=24l/P12 (5.5)=WA (5,13)=16LLP11 (5.19)=24# P12 (6,6)24A (6,12)=12L/P11 (6,14)=12P11 (6,16)=qu/P30 (6,18)=12L/P12+4P30 (6,2o)=2up12 169 ? ..' '7’: \u' .'W (6,21)=40P03 (7.8)=12P11 (7,1o)=36UP11 (7,12)=36P21 (7,1u)=36Z/P21 (7,16)=120P40 (7'18)=36P22+121/P40 (7,20)=72UP22 (8,8)=uP02+16A£P20 (8,1o)=122,/P02 (8,12)=12P12+24fiLP30 (8,1u)=12P12 (8,16)=40P 31 (8,18)=12P13+(24—2oz/)P31 (8,2o)=(16+81/)P13 (9.9)=4P20+16TLP02 (9,11)=24UP30 (9.13)=4P30+(16-122/)P12 (9.15)=24P12 (9.17)=(16+82/)P21 (9.19)=12P31+(24-202/)P13 170 (7.7)=36P20 (7.9)=12L/P20 (7,11)=72P30 (7.13)=12P12+12Z/P30 (7.15)=72UP12 E: _ q (7.17)—72P31 . (7.19)=12P13+36Z/P31 (7,21)=120L/P13 ' (8.9)=(8-uZ/)P11 (8,11)=24P21 (8,13)=4PO +(16—12l/)P21 3 (8,15)=2uZ/PO3 (8.17)=24P22+32fi{Pu0 (8,19)=4P04+(24-12Z/)P22 (8,21)=40UP04 (9.10)=12P11 (9,12)=12P21 (9,1u)=12P21+2u¢LP03 (9,16)=4ol/Pu0 (9,18)=uP40+(24-122/)P22 (9.20)=24P22+32filPOu 171 (9,21)=40P13 (10,11)=722,/P21 (10,13)=122/P03+12P21 (10,17)=7ZZ/P22 (10,19)=122/Pou+361>22 (10.21)=120P04 (11,12)=72P31 (11,1u)=721/P31 (11,16)=240P5O (11,18)=72P +24L/P5 32 O (11,2o)=14u2/P32 (12,12)=36P22+36L‘P40 (12,14)=(18+182/)P22 (12,16)=120P41 (12,18)=36P23+(36-242/)P41 (12,2o)=(2u+482/)p23 (13.13)=4Pu0+4P0u+(32-2HZ/)P22 (13.15)=242/p04+2up22 (13.17)=24P23+(32-82/)P41 +12P41+(#8-322/)P2 (10,10)=36P02 (1o,12)=362/P12 (10,1u)=36P12 (10,16)=120UP31 (1o,18)=362/P1 +12? 3 31 E f- T (10,20)=72P13 (11,11)=144P40 (11,13)=24P22+24L/P40 ' ' (11,15)=1442/P22 (11,17)=144P41 (11,19)=24P +722/P41 23 (11,21)=240L/P23 (12,13)=12P13+(2u-122/)P31 (12.15)=72UP13 (12,17)=72P32+48,L1PSO (12,19)=12P14+36P32 (12,21)=1201/P14 (13,14)=12P31+(24-122/)P13 (13,16)=40P 2+uol/P5O 3 (13,18)=4P O+12P1u+(48-322/)P32 5 3 (13,20)=24P32+(32-82/)P14 (14,14)=36P22+36LLP04 (14.16)=120UP41 (14,18)=12P41+36P23 (14,2o)=72P23+48ALP05 (15,15)=1L+L+P04 (15,17)=1442/P23 (15.19)=24UP05+72P23 (15,21)=240P05 (16,17)=240P51 (16,19)=40P +1202/1351 33 (16,21)=4ool/P33 (17,18)=72P33+(48-24 U )P51 (17,20)=(32+112 U )P33 172 (13,21)=qu/P05+40P23 (14,15)=72P13 (1u,17)=(2u+482/)P32 (1h,19)=36P32+(36—242/)Plu (14,21)=120P1u (15,16)=24OL’P32 (15,18)=72L/P14+24P32 (15,2o)=1me14 (16,16)=uOOP60 (16,18)=120P42+402/P60 (16,2o)=2uol/P42 (17,17)=144P42+64fl1P60 (17,19)=24P2u+(48+242/)P42 (17,21)=24OZ/P24 (18,18)=36P24+L+P60+(72-u82/)P42 (18,19)=12P15+12P51+(72-322/)P33 (18,20)=24Pu2+(u8+242/)qu (18,21)=1202/P15+40P33 (19,19)=4P06+36P42+(72-482/)P24 (19,20)=72P33+(u8-242/)P15 (20,2o)=1uuP24+6up(P06 (19,21)=4OZ/P06+120P2u (20,21)=2u0P15 173 Matrix is symmetric, so only the diagonal and upper triangu~ lar elements are shown. t.Et3 12(1-—2/2) All expressions must be multiplied by a factor of For details of Prs APPENDIX A—2 should be referred to. LL=-l%JZ—' and A=Area of the triangle. ’2. ,V" f“ ._A-, p APPENDIX A-8 Nonzero Terms Of The Generalized Element Stiffness K The OrthotrOpic Material (4,4):uDXA (4,13)=4DXP02+uD1P20 (4,15)=2401P02 (4,17)=24DXP21 (4,19)=4DXP03+12D1P21 (4.21)=u0D1P03 (5.12)=12nyP20 (511Q)=12nyP02 (5.18)=2uDXyP21 (6,11)=24D1P20 (6.13)=4D P +4DyP 1 02 20 6 = ( .17) 21ml?21 6, 2 ( 19) 401P03+12DyP21 (6,2 = 1) uonypo3 g21 (4,6)24D1A (4,12)=12DXP11 (n.1u)=12D1P11 (4,16)=40DXP30 (4,18)=12DXP12+4D1P30 (4,20)=24D1P12 (595)24DXyA (5,13)=16DXyP11 (5:17)=16nyP30 (5919)=24nyP12 (6,6)=4DyA (6,12)=12D1P11 (6,14)=12DyP11 (6,16)=40D1P30 (6,18)=12D P12+4Dyp 1 30 (6,20)=2L+Dyp12 (7,7)=36DXP20 174 o . _‘.‘\._—w (7,8)=12DXP11 (7,1O)=36D1P11 (7,12)=36DXP21 (7,14)=36D1P21 (7,16)=120DXP40 (7.18)=36DXP2 +12D1P40 2 (7.20)=72D1P22 (8,8)=4DXP02+16nyP20 (8,1O)=12D1P02 (8,12)=12DXP 2+2413X P y 30 (8.1u)=12an12+2uDXyP12 1 (8,16)=4ODXP31 (8,18)=12DXP1 +uD P +48DXyP 3 1 31 +3213X P (8,2o)=2uD1P y 13 13 (9.9)=4DyP20+16DXyP 175 (7.9)=12D1P20 (7,11)=72DXP30 12+12D1P3o (7.15)=72D1P12 (7,13)=12DXP (7.17)=72DXP31 (7.19)=12DXP +36D1P 13 (7.21)=120D1P13 31 (899)2L1'D P 1 11+16DX P y 11 (8,11)=24DXP21 (8,13)=4DXPO +uD P 3 1 21+32D xyP21 (8,15)=2L+D1P03 (8,17)=24DXP22+32DXyP40 (8,19)=4DXP04+12D1P22+48nyP22 (8,21)=40D1Pou (9.10)=12D P 02 y 11 (9.11)=24D1P30 (9.12)=12D1P21+24DXyP21 (9,13)=4D1P12+4DyP30+32DXyP12 (9,14)=12DyP21+2quyPO3 (9,15)=2L+DyP12 (9,16)=1+0D1P40 (9.17)=24D1P31+3213XyP31 (9,18)=12D1P22+uDyP40+48DXyP22 (9.19)=4D1P13+12DyP31+h8DXyP13(9,20)=24DyP22+32DXyPOu (9.21)=£+0DyP13 (10,1O)=36DyP02 176 (10,11)=72D P (10,12)=36D1P 1 21 12 (10,13)=12D1P03+12DyP21 (10,1u)=36DyP12 (10,15)=72DyP03 (10,16)=—120D1P31 . (10,17)=72D1P22 (10,18)=36D1P13+12Dy P31 (10,19)=12D1P04+36DyP22 (10,20)=72DyP13 (10,21)=120DyPOu (11,11)=1uuDXP40 (11,12)=72DXP31 (11,13)=2L+DXP22+2L+D1PLLO (11,14)=72D1P31 (11,15)=144D1P22 (11,16)=2L+oDXP5O (11,17)=14hDXP41 (11,18)=72DXP32+24D1P50 (11, 19): —2L+DX P 23+72D1P41 (11,2o)=—-1P1u4DP32 (11, 21): 2uoD1P P23 (12,12)=36DXP22+36DXyP40 (12,13)=12DXP13+12D1P31+L+8DXyP31 (12, 1L1)=—36D1P22+36nyP22 (12,15): 72D1P13 (12,16)=120DXP41 (12,17)=72DXP32+L+8DXyP50 (12,18)=36DXP23+12D1PM+72DXyP41 (12,19)=1ZDXP14+36D1P32+72DXyP32 (12,2o)=72D1P23+u8DX yP23 (12,21)=12OD1P1u (13,13)=4DXP04+8D1P22+MD yP40+64DXyP22 (13 1L1)=_12D1p13+12Dyp31+uanxyp13 (13,15)=24D1P0u+24DyP22 (13,16)=L+0DXP32+40D1P50 (13,17)=24DXP23+2L+D1P41+64DXyP41 177 (13, 18): 12D x1P14+16D P32 +4D yP50+96nyP32 (13, 19): -4DXP O5—1-16D1P23+12Dy P41+96DXyP 23 (13,20)=24D1P14+2#D y2P3 +6uDX yP14 (13,21Lu0D1P05+40DyP23 (14,14)=36DyP22+36DXyP04 (14,15L72DyP13 (14,16L120D1PLL1 (14,17)=72D1P32+48Dx yP32 (14,18)=36D1P23+12DyP41+72D (14,19L12D1P14+36DyP32+72DXyP14 (14,20)=72DyP23+48DXyP05 (14,21L120DyP14 (15,15L1L1L1DyP04 (15,16L240D1P32 (15,17)=1L14D1P23 (15,18)=72D1P14+2#DyP32 (15.19L 24D1P 05+72DyP23 (15,20L1L1L1DyP14 (15,21)=2uoDyP05 (16,16)=400DXP60 (16,17L2L10DXP51 (16,18)=120DXP42+L10D1P6O (16,19)=40DXP33+120D1P51 (16,2oL2uoD1P42 (16,21LuooD1P33 (17,17)=1HuDXP42+6uDXyP6 (17, 18): 72D XP33+24D1 P51+96nyP51 (17,19)=24DXP24+72D1P42+96DXyP42 (17,20)=1L14D1P33+64nyp33 (17,21)=240D1P2u (18,18)=36DXP24+24D1P42+4DyP60+1u4nyP42 1 5+40D1 P33+1 2DyP51+1 HquyPBB (18,20)=72D1P24+24DyP42+96nyP24 (18,19)=12DXP 178 (18,21L120D1P15+40DyP33 (19,19)=thPO6+24D1P24+36DyP42+1uquszu (19,2o)=24D1P15+72DyP33+96nyP15 (19,21)=4OD1P06+120DyP2u (20,20)=144DyP24+64DXyP06 (2o.21)=240DyP15 (21.21L400DyP06 Matrix is symmetric, so only the diagonal and upper triangu— lar elements are shown. For detail of Prs APPENDIX A—2 should be referred to. All expressions must be multiplied by a factor t. APPENDIX A-9 Complete Listing of Square Matrix G (:) <:> (:) ®®Q®©®®®®®®®®Q®®@®®®® I OO O O O O O O HO O OO O H O O O O OH® >4 0 O O 0 HP >4 m OOOOH N w 3 y1X1 X13’1 y1 X1 y1 2 O 2x.1 y1 0 3x1 2x1y1 2 1 0 x1 2y1 0 x1 0 2 O 0 6x1 2yl O O 1 O 0 2x1 0 O O 2 O O 2 2 3 2 y2 X2 X23’2 y2 X2 X232 2 0 2x2 y2 0 3x2 2x2y2 2 1 0 x2 2y2 0 x2 0 2 o 0 6x2 2y2 O O 1 O 0 2x2 0 O O 2 O O 2 2 3 2 y3 X3 X3y3 y3 2; X3y3 0 2x3 y3 0 3x3 2x3y3 1 0 x3 2y3 0 xi 0 2 O O 6 x3 2y3 O O 1 O 0 2x3 0 O O 2 O O C -ZS x C x —S 2C -BS x2 CXZ-ZS x y u u a u a uyu 4Y4 a a Cu a u u u 2 C C - C - S 5 22525 5X5 Ssys 2 5y5 325232 2 5X 179 <2 ® 180 4 3 © X1Y1 y1 1 X13’1 X13’1 <2) 5’2 0 4"? 3X23’1 2213’2 @ 2X1y1 ”f O 2% 2X23’1 @ o o 12x2 6x1y1 2y2 @ 2y1 o ' 0 3x1 4xly1 2 6) 2x1 6y1 o 0 2x1 2 3 4 3 2 2 ® X23’2 y2 X2 X2 3' 2 X2 3’ 2 2 2 2 y2 0 2X2 3X23’2 2X23’2 Q) 2X2y2 ”2 0 X2 2X23’2 ® 0 O 12x: 6x23!2 2y: ® 2y2 O 0 3x3 4x2y2 2 @ 2x2 6y2 O 0 2x2 2 3 4 3 2 2 © X33’3 ya X3 X3y3 23% @ yg 0 4x3 3x§y3 2x3y§ @ 2x y 3y 0 x3 2x2y 3 3 3 3 3 3 ® 0 o 12x§ 6x3y3 23% @ 2y3 O 0 3x3 4x3y3 ® 2x3 6y3 o o 2x§ 2 3 2 2 2 @ 2C4X4y4‘s43'4 3243’4 ‘234X4 C4X4‘354X4y4 204X4y4‘254X4y4 ._ 2 2 _ 3 3_ 2 2 _ 2 29 2252533 Ssys 3253’s ”3525 C525 3Ssxsy5 225253’5 2352533 2 3 3 2 2 2 @226X6y6‘36y6 3263’s “236226 06X6‘336X6y6 2C6X6Y6‘236X6Y6 ®®®®©®®®©®©®®®®®®©®®® 252 2 3 304X4y4‘suyu 2_ 3 3Csxsy5 Ssy5 2 3 326X6y6‘36y6 181 Q§> Kn N Kn >4 PAC-H 20x I"‘\.o.JCD O >< %‘ 0 Nb.) 0 N4: NKJI O 20x X E‘ 0 new 0 ko-P'KJOUI O 20x ‘554X4 ”52525 ‘536X6 12x 12x 12x 4 3 Cuxu-uSuxuyu 4 3 4. 3 C6x6—4S6x6y6 3X§Y§ 222y3 623y3 6x3y3 2x; 3 2 2 2C4X4y4‘3s4x4y4 3 2 2 22525y5‘32525y5 3 2 2 2C6X6y6’336X6y6 3C5X 5y5 3 326X6y6‘256X6y6 182 2% X13’1 2X1y2 3xfyf 2y? 6X1Y1 6x1y1 2 3 X23’2 3 2x2y2 2 2 3X232 2y; 6x2y2 6x2y2 X§Y3 2x3y3 3X§y§ 2y; 6x3y3 6x3y3 2 3 3C4X4y4‘2s4x4y4 3 285x5y5 12x 3 4 ”C4X4y4‘54y4 40 x 3 s 5 5y5‘ 5y5 3 4 ”Csxsyé‘ssys 20y; SCQYQ 5C5y5 5C6y6 (Z) G?) Cf) €31<§D <23 <§D (53 <53 <29 <29 {ED'<;D <29 (C) (Z) (Z) (Z) 183 Where Si=Sin CXi , Ci=Cos C11 and xi , yi coordinates of the nodal points. <:) is the ith row or column of the matrix. are the APPENDIX A-lO The Transformation Matrix A Relating The Mid-side Normal SlOpe To The Corner Parameters A : [A1. A25 A3] 3x21 * l (4) (4) (4) (4) (4) 0 -K1 K2 —K3 K4 K3 A1: 0 o o o o o (6) (6) (6) (6) (6) 0 —K1 K2 —K3 K4 K3 ' l (4) (4) (4) (4) (4) 0 -K1 K2 K3 —K4 -K3 _ (5) (5) (5) (5) (5) A2— 0 -K1 K2 K3 -K4 —K3 0 o o o o o 1 1 ”o o o o o o 1 _ (5) (5) (5) (5) (5) A _ 0 -K1 K2 -K3 K4 K3 (6) (6) (6) (6) (6) 0 -K1 K2 K3 K K _ 4 3 Where Kim): % Sm . Kém)= 2 C m ’ 3 Kim)=-%-l (C2 2 _ - _ m m -Sm ) and Sm— Sin am’ C — CosCXm m and 1m is the length of the element side on which node number m lies ( m=4,5,6 ). 184 APPENDIX A—11 User's Manual For T-6 & T—18 Programs I. Variable Definition NP NE NB NLD NDF NMAT NSZF ORT(N,I) CORD(N,I) NOP(N,I) IMAT(N) T(N) NBC(I) NFIX(I) NFIY(I) NONP PRESS(N) Total number of nodal points Total number of elements Total number of boundary nodal points Total number of concentrated loading conditions Number of degrees of freedom per node Total number of different isotrOpic and orthotrOpic materials Total number of equations in the system Element type material prOperties Nodal point coordinate array Element connection array Element material type array Element thickness array Restrained boundary node numbers Boundary condition type in X-direction Boundary condition type in Y-direction Total number of nodal points that have concentrated load applied to them Element uniform load 185 186 X-coordinate Y-coordinate II. 187 User's Manual---- T-6 Plane Stress Program Title Card (12A6) Col. 1--72 Title to be printed with output Control Card (715) Col. 1—— 5 Col. 6--10 Col.11--15 Col.16-—20 Col.21--25 Col.26——30 Col.31--35 Total No. Total No. Total No. Total No. Number of Number of Number of ?E of nodal points I of elements of boundary nodal points u» of loading conditions ' degrees of freedom per node different isotrOpic materials different orthotrOpic materials Material Cards (IlO,2F10.2) For IsotrOpic Material Elements -—one card per each different material Col. 1-—1O Material type number Col.11--20 Modulus of elasticity Col.21-—30 Poisson's ratio Material Cards (110,4F10.3) For OrthotrOpic Material Elements --one card per each different material (If Control Card Col. 31-35 equals zero do not enter) Col. 1--1O Material number Col.11--20 Modulus of elasticity E 1 Col.21-—30 Poisson's ratio 2/1 Col.31--40 Col.u1--5O Coordinate point Col. 1--1O Col.11--20 Col.21--30 188 Modulus of elasticity E2 Poisson's ratio Lé Cards (IlO,2F10.3) --one card per each node Node number X-coordinate Y-coordinate Element Cards (915,F10.3) ——one card per each element Col. 1-- 5 Element number Col. 6-—10 1St corner node -—- Col.11--15 Intermediate node Col.16--20 2nd corner node Counterclockwise Col.21--25 Intermediate node Col.26—-30 3rd corner node Col.31-—35 Intermediate node ——~ Col.36--40 Material type No. for isotrOpic material Col.41-—h5 Material type No. for orthotrOpic material Col.#6--55 Element thickness Boundary Cards (215) --one card per each boundary condition Col. 1-— 5 Boundary node number Col. 6--1O Boundary condition index (01=Fixed in Y-direction,10 Fixed in X-direction) Total Number Of Nodes Having Concentrated Load Applied 189 Col. 1-- 5 Total No. of node Load Cards condition Col. 1--1O Col.11--20 Col.21--30 (I10,2F10.2) -—one card per each loading Node number X-direction loads Y-direction loads III. 190 User's Manual ----- T-18 Plate Bending Program Title Card (12A6) Col.1—-Col.72 Title to be printed with output h Control Card (615) r! Col. 1—- 5 Total No. of nodal points Col. 6——1O Total No. of elements Col.11--15 Total No. of boundary nodal points b Col.16--20 Total No. of concentrate load conditions Col.21-—25 Number of degrees of freedom per node Col.26-—30 Total No. of material types Material Cards (110,5F10.3) -—one card per each different material type Col. 1-—10 Col.11--20 Col.21--30 Col.31--40 Col.u1--5O Col.51-—6O Coordinate point Col. 1—-1O Col.11——20 Material number Modulus of elasticity in X-direction Poisson's ratio in X-direction Modulus of elasticity in Y-direction Poisson's ratio in Y-direction Shear modulus Cards (IlO,2F10.3) —-one card per each nodal Node number X-coordinate 191 Col.21--30 Y-coordinate Element Cards (5I5,2F10.3) —-one card per each element Col. 1-- 5 Element number Col. 6—-10 ISt corner node \ Col.11-—15 2nd corner node ) counterclockwise F Col.16--20 3rd corner node I j Col.21——25 Material type number Col.26--35 Element thickness W Col.36--45 Element uniform load ' Boundary Cards (3I5) —-one card per each boundary condition Col. 1—- 5 Boundary node number Col. 6--1O Boundary condition in X-direction (11=simply supported,10=clamped edge) Col.11--15 Boundary condition in Y-direction (22=simply supported,20=clamped edge) Total Number of Nodes Having Concentrated Load Applied Col. 1-- 5 Total No. of nodes having concentrated load Load Cards (I10,6F10.3) —-one card per each loaded point Col. 1--1O Node number Col.11--20 Concentrated load value 192 MA .41.” 0N (v b-X (u— 22 O O ’7‘ ZN Zn. -uil' ozu o:m tzu—zou mmwmhm 4440 mummmmpn “~4430440 AXD43m 024.1101 3404 4440 MO404 0401 042.4n—4 com on mazoazumNmz 4b4oot4440 mu—hmuaoza 324 >mbwtowo hDQZ— D401 ¢bZ ozuzua ouZUZ -nchz wooz mommlomz oz< muzzou no 103134. 924 .02 014p u-44upnz~ muhmz 44mm .oomvp\h\zoz100 4N~.N~.L—hmz\11z\zozzo0 .omavx34..cco0044\mm4\zm1200 .N.om.4hzo..oc~.4h4t~..coaoocuvzm .oo .«1.~ .032..N.mN.bzoo.oc~0p4z~..o.oo_01mz..m.oo~vozo0\ zmzzmu .uNmz.h4tZ.O4z.z0z.uoz.mz.mzoaz.. ~.w4h~h\mh200\ 2 It 0 z4mooxa z~- (4 Z.‘ " ’ o o-4( a r—t— K 0 AA u. ~0- r— <4 .. 22 Q I-‘Z m It AN “V Z (’3 I 22 N 0 P'- o z o o o 0 ca— (d 2 0° '49 (4 III-0 A v-u-o . ...c I: a H 0 Am 0 0 Lu c—o #- 22 AN 0A N 0 AA 2 ow D O O N 0 AH 0—4 00 o °< Q ILL. < o—a u-n o o-au .- 0- CL OO— 2 DO 0- mu 0! u}: AA A Z o-a< h-I ZZ ( "NO X! 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