CIVL 7/8111 Chapter 1 - Introduction to FEM 1/21

Transcription

1 CIV 7/8 Chapter - to FEM / The nte element method has become a powerl tool or the nmercal solton o a wde range o engneerng problems. Applcatons range rom deormaton and stress analyss o atomotve, arcrat, bldng, and brdge strctres to eld analyss o heat l, ld low, magnetc l, seepage, and other low problems. Pecewse lnear ncton n one dmensons. Wth the advances n compter technology and CAD systems, comple problems can be modeled wth relatve ease. Several alternatve congratons can be tred ot on a compter beore the rst prototype s blt. All o ths sggests that we need to eep pace wth these developments by nderstandng the basc theory, modelng technqes, and comptatonal aspects o the nte element method. Pecewse lnear ncton n two dmensons. Dscretzaton o two dmensonal doman Orgnal two dmensonal doman In ths method o analyss, a comple regon denng a contnm s dscretzed nto smple geometrc shapes called nte elements. The materal propertes and the governng relatonshps are consdered over these elements and epressed n terms o nnown vales at element corners. An assembly process, dly consderng the bondary condtons, reslts n a set o eqatons. Solton o these eqatons gves s the appromate behavor o the contnm.

2 CIV 7/8 Chapter - to FEM / A magnetc problem sng FEM sotware FEM solton to the problem Colors ndcate that the analyst has set materal propertes or each zone, n ths case a condctng wre col n orange; a erromagnetc component (perhaps ron) n lght ble; and ar n grey. The color represents the ampltde o the magnetc l densty, as ndcated by the scale n the nset legend, red beng hgh ampltde.

3 CIV 7/8 Chapter - to FEM 3/

4 CIV 7/8 Chapter - to FEM / Ths eample dplcates a benchmar problem or tme-dependent boyant low n poros meda. Known as the Elder problem, t ollows a laboratory eperment to stdy thermal convecton. Ths model eamnes the Elder problem or concentratons throgh a -way coplng o two physcs nteraces: Darcy s aw and Solte Transport. Ths eample models the radaton o an nose rom the annlar dct o a trboan aeroengne. When the jet stream ets the dct, a vorte sheet appears along the etenson o the dct wall de to the srrondng ar movng at a lower speed. The near eld on both sdes o the vorte sheet s calclated. Chemcal vapor deposton (CVD) allows a thn lm to be grown on a sbstrate throgh molecles and moleclar ragments adsorbng and reactng on a srace. Ths eample llstrates the modelng o sch a CVD reactor where trethyl-gallm rst decomposes, and the reacton prodcts along wth arsne (AsH3) adsorb and react on a sbstrate to orm GaAs layers. Dampng elements nvolvng layers o vscoelastc materals are oten sed or redcton o sesmc and wnd ndced vbratons n bldngs and other tall strctres. The common eatre s that the reqency o the orced vbratons s low. Ths model stdes a orced response o a typcal vscoelastc damper. The analyss nvolves two cases: a reqency response analyss and a tmedependent analyss. A typcal atomotve ehast system s a hybrd constrcton consstng o a combnaton o relectve and dsspatve mler elements. The relectve parts are normally tned to remove domnatng low-reqency engne harmoncs whle the dsspatve parts are desgned to tae care o hgher-reqency nose. The mler analyzed n ths model, s an eample o a comple hybrd mler n whch the dsspatve element s created completely by low throgh perorated ppes and plates. Ths model treats the ree convecton and heat transer o a glass o cold water heated to room temperatre. Intally, the glass and the water are at 5 C and are then pt on a table n a room at 5 C. The nonsothermal low s copled to heat transer sng the Heat Transer modle.

5 CIV 7/8 Chapter - to FEM 5/ The complete analyss conssts o two dstnct bt copled procedres: a lddynamcs analyss wth the calclaton o the velocty eld and pressre dstrbton n the blood (varable n tme and n space) and the mechancal analyss wth the deormaton o the tsse and artery. The materal s assmed to be nonlnear and a hyperelastc model s sed. Hstorcal Bacgrond Basc deas o the nte element method orgnated rom advances n arcrat strctral analyss. In 9, Hreno presented a solton o elastcty problems sng the rame wor method. Corant s paper, whch sed pecewse polynomal nterpolaton over tranglar sbregons to model torson problems, appeared n 93. Trner et al. (956) derved stness matrces or trss, beam, and other elements. The term nte element was rst coned and sed by Clogh n 96. Ths model stdes the ld low throgh a bendng ppe n 3D or the Reynolds nmber 3. Becase o the hgh Reynolds nmber, the -epslon trblence model s sed. Calclatons wth and wthot corner smoothng are perormed. The reslts are compared wth epermental data. Hstorcal Bacgrond In the early 96s, engneers sed the method or appromate solton o problems n stress analyss, ld low, heat transer, and other areas. A boo by Argyrs n 955 on energy theorems and matr methods lad a ondaton or rther developments n nte element stdes. The rst boo on nte elements by Zenewcz and Chng was pblshed n 967. In the late 96s and early 97s, nte element analyss was appled to nonlnear problems and large deormatons. Ths model smlates the tme-dependent low past a cylnder. The velocty eld magntde at derent tme steps s shown. Hstorcal Bacgrond Mathematcal ondatons were lad n the 97s. New element development, convergence stdes, and other related areas all n ths category. Today, developments n manrame compters and avalablty o powerl mcrocompters have broght ths method wthn reach o stdents and engneers worng n small ndstres.

6 CIV 7/8 Chapter - to FEM 6/ Role o Compters n Fnte Element Methods Untl the early 95s, matr methods and the assocated nte element method were not readly adaptable or solvng complcated problems becase o the large nmber o algebrac eqatons that reslted. Hence, even thogh the nte element method was beng sed to descrbe complcated strctres, the resltng large nmber o eqatons assocated wth the nte element method o strctral analyss made the method etremely dclt and mpractcal to se. Wth the advent o the compter, the solton o thosands o eqatons n a matter o mntes became possble. Prmary lne elements consst o bar (or trss) and beam elements. They have a cross-sectonal area bt are sally represented by lne segments. The smplest lne element (called a lnear element) has two nodes, one at each end, althogh hgher-order elements havng three nodes or more (called qadratc, cbc, etc. elements) also est. Step nvolves dvdng the body nto an eqvalent system o nte elements wth assocated nodes and choosng the most approprate element type. The basc two-dmensonal (or plane) elements are loaded by orces n ther own plane (plane stress or plane stran condtons). They are tranglar or qadrlateral elements. The total nmber o elements sed and ther varaton n sze and type wthn a gven body are prmarly matters o engneerng jdgment. The elements mst be made small enogh to gve sable reslts and yet large enogh to redce comptatonal eort. Small elements (and possbly hgher-order elements) are generally desrable where the reslts are changng rapdly, sch as where changes n geometry occr, whereas large elements can be sed where reslts are relatvely constant. The smplest two-dmensonal elements have corner nodes only (lnear elements) wth straght sdes or bondares althogh there are also hgher-order elements, typcally wth md-sde nodes (called qadratc elements) and crved sdes.

7 CIV 7/8 Chapter - to FEM 7/ The most common three-dmensonal elements are tetrahedral and heahedral (or brc) elements; they are sed when t becomes necessary to perorm a threedmensonal stress analyss. The basc three dmensonal elements have corner nodes only and straght sdes, whereas hgher-order elements wth md-edge nodes (and possble md-ace nodes) have crved sraces or ther sdes Water phase satratons (top gres) and CO concentraton (bottom gres) proles at 5 and days. The CO moves n complcated and nepected ways.

8 CIV 7/8 Chapter - to FEM 8/ The asymmetrc element s developed by rotatng a trangle or qadrlateral abot a ed as located n the plane o the element throgh 36. Ths element can be sed when the geometry and loadng o the problem are asymmetrc. As a conseqence o or changng clmate, large eorts have been made to nderstand the socal rss o storm srges (hypotheszed to ncrease n reqency n warmer clmate scenaros) and sea level rse n coastal areas. O partclar nterest s the role that wetlands and coastal marshes play n storm srges and loodng events.

9 CIV 7/8 Chapter - to FEM 9/ Dscrete means essentally that we are wllng to accept a model that wll yeld normaton abot the dependent varables at a nte nmber o ponts, reerred to as nodes, wthn the nterval. Each node s assgned a dsplacement, = to 5. The problem has been converted rom a contnos model o nnte degrees o reedom to one wth a nte nmber o degrees o reedom, n ths case n = 5. () q() 3 5 P Consder the problem o the aal deormaton o a lnearly elastc bar nder an aal load P at = and dstrbted eternal load q(). The cross-sectonal area, A(), the modls o elastcty, E, and the mass densty, (), are gven. () P = eternal load q() = dstrbted load () = aal dsplacement The elastc eects o the dscrete parts o the bar may be represented as elements. In or problem, the elongaton o an aal bar nder an aal load s represented by: Pl e A E () avg A E l avg P e e P q() P q() 3 5 et s assme that the varaton o the loads, P() and q(), and the cross-sectonal area, A(), are complcated and the eact solton to the above eqaton cannot be ond. The basc concept o FEM s to ct the problem p nto a seres o smpler dscrete problems and relate the parts to each other to model the contnos materal. A possble eample o a dscrete model o the bar s: Thereore, an elastc bar o length l s eqvalent to a smple lnear sprng. The stness assocated wth each element wll be a derent vale snce A avg vares rom node to node. et s appromate the stness,, by tang: A A A E A A avg l () () AE AE AE3 AE q() P q() P l l l 3 l

10 CIV 7/8 Chapter - to FEM / Eqvalent systems o sprngs connectng each set o nodes are reerred to as elements. An element generally descrbes some basc physcal property o the system. In the case o the aal bar, the relatonshp between orce and dsplacement s: F e Keepng the concept o the element we have developed so ar, let s consder the mass o the porton o the bar between nodes and + denng element. () + l l l 3 l Another mportant physcal parameter assocated wth the element s the mass. There are several ways to dstrbte the mass. Keepng the concept o the element we have developed so ar, let s consder the mass o the porton o the bar between nodes and + denng element. The sm o the masses shold appromately satsy the ollowng relatonshp: M A d Keepng the concept o the element we have developed so ar, let s consder the mass o the porton o the bar between nodes and + denng element. () Keepng the concept o the element we have developed so ar, let s consder the mass o the porton o the bar between nodes and + denng element. () + + l l l 3 l l l l 3 l One method o dstrbtng the mass s to average the mass over the element and dvde t eqally between the two nodes denng the element. The average mass ntensty s: A A m m m * Identcal to the lmpng technqe sed or mass, we wll tae the average o the loadng ntensty: q q q q q * Keepng the concept o the element we have developed so ar, let s consder the mass o the porton o the bar between nodes and + denng element. () Keepng the concept o the element we have developed so ar, let s consder the mass o the porton o the bar between nodes and + denng element. () + + l l l 3 l l l l 3 l Thereore the dscrete lmped mass system s: Thereore the dscrete lmped loadng s: M M M 3 M M 5 Q Q Q 3 Q Q 5 Nodal loads Q m ml M m ml m m3l M m m3l m3 ml3 M3 m3 ml3 m m5l M m m5 l M5 q q l Q q ql q q3l Q q q3l q3 ql3 Q3 q3 ql3 q q5l Q q q5l Q5

11 CIV 7/8 Chapter - to FEM / Keepng the concept o the element we have developed so ar, let s consder the mass o the porton o the bar between nodes and + denng element. () Eqlbrm o a Sprng Mass System - Vectoral Approach Consder a typcal sprng-mass system, where each sprng s assmed to behave n a lnear way ( F = ) and the loads P are appled slowly to the system so that the problem s statc. l + l l 3 l The nodal dsplacements and the correspondng nternal orces or an element are: + The sm o the nodal loads shold appromately satsy the ollowng relatonshp: Q q d + or e e e Keepng the concept o the element we have developed so ar, let s consder the mass o the porton o the bar between nodes and + denng element. The nal dscrete model or ths system wth sprngs, masses, and loads wold be: Q Q Q 3 Q Q 5 l () + l l 3 l Eqlbrm o a Sprng Mass System - Vectoral Approach e e e where e s called the element stness matr, e s the element orce, and e s the element dsplacement vector. Ths eqaton s a statement o the sprng relatonshp F = on the elemental level. The ndvdal e can be assembled nto the global stness matr whch represents the physcal natre o the entre system. M M M 3 3 M M 5 Step - Select a Dsplacement Fncton Ths completes the process o convertng the contnos system nto what s hoped to be a eqvalent dscrete system. Eample - Consder a norm sqare bar nder a dstrbted loadng. Use ve eqally-spaced nodes to dscretze the ollowng problem. Solve or the dsplacement at each node. The dscretzaton shold be mplct n the representaton o the mass, elastc propertes, and loads. () ps E 9, s A n. Whether the aal model s contnos or dscrete, eqlbrm o the system (Newton s second law) mst be satsed. t. The remanng steps o assembly, constrants, solton, and comptaton o derved varables can be best llstrated n an eample. The dscretzaton o the bar s: () t..5 t..5 t..5 t. ps e e e

12 CIV 7/8 Chapter - to FEM / Snce the area o the bar does not vary, the vale o stness or each element s constant: A A E n. 9,s l n..5t. t ps/ n. The eqlbrm eqatons are: Element : or e e e Element : Element 3: Element : These eqatons can be wrtten n matr orm as: Element P Element P Element 3 3 P3 Element P 5 P5 Applyng the vales or the geometry, materal propertes, and the bondary condtons gven or ths problem reslt n: 3 5 These eqatons can be wrtten n matr orm as: K G T G T P G K P G G G P P P P P 3 5 where K G s called the global stness matr, G s the global dsplacement vector, and P G s the global load vector. The solton o these eqatons s: Sbstttng or the nmercal vales or the dsplacement are:.3 n..7n. 3.3 n..n. 5 The eact solton may be determned rom the ollowng epresson: P P ( ) d EA EA n. (9, s ).3n..7n. 3.3n. 5.n. A carel nspecton o the global eqlbrm eqatons reveals that each elemental stness matr, e, s present n the global stness matr. Thereore the global stness matr can be wrtten as: G G G G Eample - Consder a norm sqare bar nder a dstrbted loadng. Use ve eqally-spaced nodes to dscretze the ollowng problem. Solve or the dsplacement at each node. q() () t. The dscretzaton o the bar s: q() () t..5 t..5 t..5 t. P q E 9,s A n P ps/ n e e e

13 CIV 7/8 Chapter - to FEM 3/ To handle the dstrbted load, we wll lmp the loads nto each node. The average loadng ntensty s compted as: * q q q q q The sm o the nodal loads shold appromately satsy the ollowng relatonshp: Q o q( ) d Applyng the vales or the geometry, materal propertes, and loadng dstrbton condtons reslts n: Element 8 Element 8 Element Element The ndvdal vales or the dstrbted lmped loads are: q q l Q q P P q.5 q.5 t. P.75P.75 q q l P 7 8 Q 6 56 Applyng the vales or the geometry, materal propertes, loadng dstrbton, and the bondary condtons reslts n: The solton o these eqatons s: 3 5,AE,AE,AE,AE The ndvdal vales or the dstrbted lmped loads are: Q q q l 8 56 q q l q q3 l 8 Q 56 q q3 l q3 q l3 3 Q3 56 q3 q l3 q q5 l 6 Q 56 Q 5 q q5 l 56 Sbstttng the nmercal vales or P,, and reslts n :.89n..737n. 3.83n n. The eact solton may be determned rom the ollowng epresson: ( ) q( ) d ' d EA( ) AE P AE( ) P d ' P C AE C

14 CIV 7/8 Chapter - to FEM / Sbstttng the nmercal vales or P,, and reslts n :.89n..737n. 3.83n n. The eact solton may be determned rom the ollowng epresson: 3 ( ) AE 6.78n..7n. 3.87n n. The ndvdal vales or the dstrbted lmped loads are: Q q q l 56 q q l q q3 l 8 Q 56 Q3 Q Q Q6 Q7 8 Q8 Q Eample - Repeat the prevos problem sng nne eqally-spaced nodes (8 elements) to dscretze the problem. Solve or the dsplacement at each node. q() () t. The dscretzaton o the bar s: q() () P q E 9,s A n P ps/ n e e e Applyng the vales or the geometry, materal propertes, and loadng gven n ths problem reslts n: Element Element 8 Element 3 3 Element Element Element Element Element 8 To handle the dstrbted load, we wll lmp the loads nto each node. The average loadng ntensty s compted as: * q q q q q The sm o the nodal loads shold appromately satsy the ollowng relatonshp: Q o q( ) d Applyng the bondary condton reslts n:

15 CIV 7/8 Chapter - to FEM 5/ The solton o these eqatons s: ,8AE,8AE,8AE,8AE PROBEM #3 - Consder a norm sqare bar nder a dstrbted loadng. Use ve eqally-spaced nodes to dscretze the ollowng problem. Solve or the dsplacement at each node. () q P E 9,s A n. P 5 ps/n. Sbstttng the nmercal vales or P,, and reslts n :.8n..77n n n. The eact solton may be determned rom the ollowng epresson:.78n. 3.7n..87n n. q() n. PROBEM # - Consder a sqare bar sbjected to a seres o concentrated loads. Use ve eqally-spaced nodes to dscretze the ollowng problem. Solve or the dsplacement at each node and compare to the eact solton. l () P P 3P l l l P E 9,s l 5 n P 5 ps A 5 n From yor eperence n strctral analyss yo are aware o strctral elements or members called two orce members. These elements are pn connected and transmt only an aal orce. There s no shear, bendng, or torsonal loads transmtted by these members n a strctre. A strctre composed o two-orce members whch behaves elastcally may be replaced by a system o connected sprngs. Consder a sngle two-orce member: y P P P P PROBEM # - Repeat PROBEM # sng twce the nmber o elements. Compare yor reslts wth those obtaned n PROBEM # and the eact solton. Eplan any derences n the soltons. l () P P 3P l l l P E 9,s l 5 n P 5 ps A 5 n The sprng stness constant s (AE/ ), where A s an area, E s the modls o elastcty, and s the length o the member. Consder a plane trss wth or bars or members or elements: P P Althogh each member n the trss wll elongate (or contract) and transmt a tensle (or compressve) load, the dsplacements and the orces are n derent drectons.

16 CIV 7/8 Chapter - to FEM 6/ Y, v y F, V Y, v F, V Y XYF,,, F global coordnates X, F, U X Y FX, U y,,, element coordnates y, X y y In matr orm these qanttes can be epressed as: X FY y F R U R F cos sn F R sn cos U U V v The global orce and global dsplacement vectors and R s a transormaton matr or rotaton o an as ( R - = R T ). A set o smlar qanttes can be wrtten or the other end o the element Althogh each member n the trss wll elongate (or contract) and transmt a tensle (or compressve) load, the dsplacements and the orces are n derent drectons. y, v y F, V Y, F, U X The stness matr or the aal element n the elemental or local coordnates s: Y, v y FX, U y,,, element coordnates y, X F, V Y XYF,,, F global coordnates The global orce components may be related to the elemental orce components by: FX cos ysn FY sn ycos X Y Rewrtng the elemental orces-dsplacement relatonshp or both and y components: y v y v Notce the second and orth eqatons relect the act that only aal loads, n the -drecton locally, are possble n the absence o bendng, shear, or torson. y Y, v y F, V Y, v F, V X The dsplacements may be related n a smlar ashon: XYF,,, F global coordnates U cos v sn V sn v cos Y X, F, U X Y FX, U y,,, element coordnates y, y These eqatons may be wrtten n parttoned orm as: To convert these relatonshps to global coordnates (X, Y) we apply the coordnate transormaton R. R F R U R F R U R U R F R U R U Mltply both sde by R: F R R U R R U F R R U R R U

17 CIV 7/8 Chapter - to FEM 7/ Snce R - = R T = T T F RR RR U T T F RR RR U In a more convenent orm: T F R R U = T F R R U Wrtng these eqatons n stll a more compact orm gves T R F TT U KU T R where K s the global stness matr or a sngle two-orce member or element. Consder the ollowng two ways to nmber the nodes o the same trss: From these dealzatons, t s clear that the second nmberng scheme prodces a global matr that has a smaller band wdth. Generally, ths type o symmetry reslts n qcer soltons and a redcton n the reqred memory or storage capacty. The hal-band wdth o a symmetrc set o eqatons or row and colmn j o the last non-zero entry may be compted as: nb j where NB (hal the band wdth) s the mamm o the (nb) over all rows. Sbstttng the vales o R and and perormng the mltplcaton gves: cos K sn In ths case, K s the global stness matr or a sngle trss element. In a strctre composed o two-orce elements, say a trss, we wold have to assembly the element global matrces nto a global matr or the entre system. Beore we dscss any problems or wor any eamples, let s loo at the eect o dscretzaton on the orm o the system stness matr. SOUTION PROCEDURE. Dene a dscretzaton o the trss (recall the node nmberng scheme we dscssed above). Assemble the elemental stness and load matrces. Each element matr shold be transormed nto the global system as prevosly descrbed. 3. Apply bondary condtons or constrants to the system eqatons. Solve the system eqatons 5. Compte the orces n the members. Recall the orce dsplacement relatonshp T T U UUcos VV sn y UUcos VV sn y Consder the ollowng two ways to nmber the nodes o the same trss: Eample - Develop the element stness matrces and system eqatons or the plane trss below. Assme the stness o each element s constant. Use the nmberng scheme ndcated. Solve the eqatons or the dsplacements and compte the member orces. 3 Nmber Scheme # Nmber Scheme # 3 All elements have a constant STEP. The node nmberng s gven n the dagram above (Note that ths s the optmm nmberng congraton).

18 CIV 7/8 Chapter - to FEM 8/ STEP. Develop the element normaton Member Node Node Elemental Stness 3 3/ 3 3 / Compte the elemental stness matr or each element. The general orm o the matr s: cos K sn The nconstraned (no bondary condtons satsed) eqatons are: U V 3 U P V P U P 3 3 V3 For element : U V U V U V K U V For element : U V U V 3 3 U V K U3 V3 cos K sn For element 3: U V U3 V3 U V K U3 V3 STEP 3. The dsplacement at nodes and 3 are zero n both drectons. Applyng these condtons to the system eqatons gves: U V 3 U P V P U3 V 3 Assemble the global system matr by spermposng the elemental global matrces. U V U V U3 V3 Element U V 3 U K V U3 3 V3 Element 3 Element STEP. Solvng ths set o eqatons s arly easy. The solton s: P P P 3P U V U V U V 3 3 STEP 5. Usng the orce-dsplacement relatonshp the orce n each member may be compted. Member (element) P P P P y P P P P y

19 CIV 7/8 Chapter - to FEM 9/ STEP 5. Usng the orce-dsplacement relatonshp the orce n each member may be compted. Member (element) Eample - Develop the element stness matrces and system eqatons or the plane trss below. Assme the stness o each element s constant. Use the nmberng scheme ndcated. Solve the eqatons or the dsplacements and compte the member orces. P P P 3P P y Element nmber Node nmber P P P 3P P 3 y 3 STEP. A node nmberng congraton s gven (note that ths s the optmm nmberng congraton). STEP 5. Usng the orce-dsplacement relatonshp the orce n each member may be compted. Member (element) 3 3 y3 y 3 P Y Element 3 X P P 3 Element Element P P P STEP. Develop the element normaton Element Node Node Elemental Stness / 3 3/ 3 3 7/ 5 3 Compte the elemental stness matr or each element. The general orm o the matr s: cos K sn Eample - Develop the element stness matrces and system eqatons or the plane trss below. Assme the stness o each element s constant. Use the nmberng scheme ndcated. Solve the eqatons or the dsplacements and compte the member orces. All elements have a constant vale o STEP. A node nmberng congraton s gven (note that ths s the optmm nmberng congraton). For elements and : U V U V U V U V 3 3 U U V V K K U U3 V V For elements 3 and : U V U V U V U V 3 3 U U V V K K U3 U V 3 V 3

20 CIV 7/8 Chapter - to FEM / U3 V3 U V For element 5: U3 V3 K U V Assemble the global system matr by spermposng the elemental global matrces. U V U V U V3 U V 3 3 U V U V K U 3 V3 3 U V STEP. Solvng ths set o eqatons s arly easy. The solton s: P P U V U V P U3 V3 U V STEP 5. Usng the orce-dsplacement relatonshp the orce n each member may be compted. Member (element) P P P P P P The nconstraned (no bondary condtons satsed) eqatons are: 3 U V U P V U3 V P 3 3 U V STEP 5. Usng the orce-dsplacement relatonshp the orce n each member may be compted. Member (element) P P P P 3 Member (element) 3 3 Member (element) 3P P P 3P P P STEP 3. Apply the bondary condtons to the system eqatons: U V U P V U3 V P 3 U V STEP 5. Usng the orce-dsplacement relatonshp the orce n each member may be compted. Member (element) 5 3 Element Node Node U node U node V node V node P/ -P/.77P (C) 3 P/ -P/ -P/ P (T) 3 3 -P/ P/ -P/.P (C) 5 3 -P/

21 CIV 7/8 Chapter - to FEM / PROBEM # - Develop the element stness matrces and system eqatons or the plane trss below. Assme the stness o each element s constant. Use the nmberng scheme ndcated. Solve the eqatons or the dsplacements and compte the member orces. End o All elements have a constant vale o Element nmber Node nmber PROBEM #5 - Develop the element stness matrces and system eqatons or the plane trss below. Assme the stness o each element s constant. Use the nmberng scheme ndcated. Solve the eqatons or the dsplacements and compte the member orces. P P P P 3 5 o 5 o P P Element nmber Node nmber PROBEM #6 - Consder the ollowng two-dmensonal plane trss. For the gven node nmberng scheme, determne the dsplacements o each node and the member orces. Chec yor reslts by sng the method o sectons and the method o jonts rom statc analyss. For comptatonal prposes assme a P = ps, E = 9, s, = t., and A = n..