Figure XX.1.1 Plane truss structure
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1 Truss Eements Formution. TRUSS ELEMENT.1 INTRODUTION ne truss struture is ste struture on the sis of tringe, s shown in Fig The end of memer is pin juntion whih does not trnsmit moment. As for the truss memers tht onstitute pne truss struture, ony xi fores t, ending moments nd sher fores do not ppy. In ddition, midde ods wi not t on the truss memers. i i i x i U i e, ε N Figure.1.1 ne truss struture y θ The hrteristis of the truss eement n e summrized s foows: 1) A truss eement is sender memer (ength is muh rger thn the ross-setion). ) It is two-fore memer i.e. it n ony support n xi od nd nnot support ending od. Memers re joined y pins (no trnstion t the onstrined node, ut free to rotte in ny diretion). ) The ross-setion dimensions nd esti properties of eh memer re onstnt ong its ength. 4) The eement my interonnet in -D or -D onfigurtion in spe. The eement is mehniy equivent to spring, sine it hs no stiffness ginst ppied ods exept those ting ong the xis of the memer 5-1 Shoji Nzw (nzw@e.tut..jp)
2 Truss Eements Formution. LOCAL AND GLOBAL COODINATES..1 Retionship etween Lo nd Go Dispements We strt y ooing t the truss eement shown in Figure.1. This eement tthes to two nodes, i nd j. In the figure we re showing two oordinte systems. One is one dimension oordinte system tht igns with the ength of the eement. We wi this the o oordinte system. The other is two dimension oordinte system tht does not ign with the eement. We wi this the go oordinte system. The [ x, y ] oordintes re the o oordintes for the eement nd [, ] re the go oordintes. We n onvert the dispements shown in the o oordinte system y ooing t the foowing figure. u i, v i, u j nd v j represent dispements in the o oordinte system nd U i, i, U j nd j represent dispements in the [, ] (go) oordinte system. Lo oordinte system [ x, y ] Nod dispement of memer in the x diretion:u i,u j Nod dispement of memer in the y diretion:v i,v j Nod fore of memer in the x diretion:p i,p j Nod fore of memer in the y diretion:q i,q j Go oordinte system [, ] Nod dispement of memer in the diretion:u i,u j Nod dispement of memer in the diretion: i, j Nod fore of memer in the diretion: i, j Nod fore of memer in the diretion: i, j y j x y v j x i i U i θ j U j v i u i θ u j () Nod dispements of Lo oordinte system y i i j x y () Nod dispements of Lo oordinte system q i q j p j x θ j p i θ () Nod fores of Lo oordinte system () Nod fores of Lo oordinte system Figure..1.1 Nod dispements nd nod fores of Lo nd Go oordinte systems 5 - Shoji Nzw (nzw@e.tut..jp)
3 Truss Eements Formution y i x v i u i i θ U i Figure..1. Nod dispements nd nod fores of Lo nd Go oordinte systems Let us onsider the retion etween the nod dispements U i, i in the go oordinte system nd the nod dispements u i, v i in the o oordinte system. From geometri retion s shown in Fig...1., the governing equtions reting the two oordinte vues (U i, i, u i, nd v i ) re given s u = U osθ + v = U sinθ + sinθ i i i osθ i i i Simiry, for the nod dispements of Node j, we n otin the foowing retions. u = U osθ + v = U sinθ + In mtrix form sinθ j j j osθ j j j ui v i = u j v j osθ sinθ Ui sinθ osθ i osθ sinθ U j sinθ osθ j Simiry, for the nod fores, we so n otin the foowing retions. p = osθ + q = sinθ + sinθ i i i osθ i i i p = osθ + q = sinθ + In mtrix form sinθ i i i osθ i i i pi osθ sinθ i q i sinθ osθ i = p j osθ sinθ j q j sinθ osθ j (..1.1) (..1.) (..1.) (..1.4) (..1.5) (..1.6) Aordingy, the retionship etween the vues in the go oordinte nd the vues in the o oordinte is expressed in the mtrix form. 5 - Shoji Nzw (nzw@e.tut..jp)
4 Truss Eements Formution { p } = [ T ]{ } (..1.7) { d } = [ T ]{ D } (..1.8) i pi Ui ui i q i i v i { } =, { p} =, { D} =, { d} = j p j U j u j j q j j v j osθ sinθ sinθ osθ [ T ] = osθ sinθ sinθ osθ (..1.9) (..1.1) in whih [T ] is the trnsformtion mtrix. {D }, { } re nod dispement nd fore vetor in the go oordinte, nd {d }, {p } re nod dispement nd fore vetor in the o oordinte.. FINEIT ELMENT EUATION IN LOCAL COODINATE SSTEM Now, we wi derive the finite eement eqution in o oordinte system. Consider the truss eement shown in Fig...1.1, with nodes i nd j, dispements v i, u i, u j nd v j, nd fores p i, q i, p j nd q j. y q i q j p i Δθ p j N N i u i v i () j u j v j x Figure Strin-dispement retion: The strin-dispement retion in the truss eement is uted y interpotion of defetion vues shred y nodes of the eement. The retionship strin ε nd defetion e is given y ' ( + e ) e ε = = = (..1.1) in whih is memer ength efore deformtion, nd ' is memer ength fter deformtion, nd e is eongtion of the truss eement. The strin ε is expressed y nod dispements nd the memer ength. ε + u j ui + v j vi u j ui v j vi ( ) ( ) = = Shoji Nzw (nzw@e.tut..jp)
5 Truss Eements Formution u j ui u j ui v j vi = (..1.) u j u When the terms, n e pproximted s i nd v j v i u u u u u u ε = in ove the eqution re sm enough, the ove eqution j i j i j i (..1.).. Stress strin retion (onstitutive eqution) The stress strin retion of the truss eement is dopt the hoo s w, σ = E ε (...1) where, E is young s moduus or moduus of estiity.. Finite eement eqution in Lo Coordinte System From the ove retionship, n xi fore, N, of the truss eement is given y N = ε = e = ( u j ui ) = ( u j ui ) (...1) where A is ross-setion re of the truss eement, nd is n xi stiffness of the truss eement. = E A (...) As iustrted in Fig...1.1, the ppied fores p i, q i, p j nd q j re expressed y the xi fore, nd the finite eement eqution in o oordintes is given s, p = N os θ N = ( u u ) = u u q i j i i j i = N sin θ p = N os θ N = ( u u ) = u + u q j j i i j j = N sin θ in whih the foowing equtions re ssumed in infinitesim deformtion theory. os θ 1, sin θ As mtrix form of Eq.(...), (...) { p } = [ ]{ d } (...4) 1 1 [ ] = ; = 1 1 E A (...5) Here, Eq.(...4) is finite eement eqution in o oordinte system, [ ] is stiffness mtrix of truss eement in o oordintes. This eqution wi e onverted to go oordinte 5-5 Shoji Nzw (nzw@e.tut..jp)
6 Truss Eements Formution system, whih n e used to generte go strutur eqution for the given struture..4 FINEIT ELMENT EUATION IN GLOBAL COODINATE SSTEM Using the retionships etween o nd go defetions nd fores, we n onvert n eement eqution from o oordinte system to go system. Sustituting Eqs.(..1.7) nd (..1.8) into the eement o equiirium, Eq.(...4), we otin [ T ]{ } = [ ][ T ]{ D } (.4.1.1) Mutipying oth sides of this mtrix eqution y the inverse of the trnsformtion mtrix, tht is, y [T ] -1, we otin 1 { } [ T ] [ ][ T ]{ D} = (.4.1.) For this nd most ses, the trnsformtion mtrix [T ] T hs the property of orthogonity. Therefore, the inverse is the sme s the trnspose [T ] T. 1 [ T ] [ ] T T = (.4.1.) Finite eement eqution of the truss eement in go oordintes is expressed s { } = [ K ]{ D } (.4.1.4) in whih [ K ] is stiffness mtrix of truss eement in go oordintes. The stiffness mtrix is expressed s [ K ] = [ T ] T [ ][ T ] = T osθ sinθ 1 1 osθ sinθ sinθ osθ E sin os A θ θ = osθ sinθ 1 1 osθ sinθ sinθ osθ sinθ osθ E θ θ θ θ θ θ os sin os os sin os sin os sin sin os sin A θ θ θ θ θ θ os θ sinθ osθ os θ sinθ osθ sinθ osθ sin θ sinθ osθ sin θ (.4.1.5) Eh of its terms hs physi signifine, representing the ontriution of one of the dispements to one of the fores. The go system of equtions is formed y omining the eement stiffness mtries from eh truss eement in turn, so their omputtion is entr to the method of mtrix strutur nysis. This mtrix hs sever noteworthy hrteristis: The mtrix is symmetri Sine there re 4 unnown dispements (DOFs), the mtrix size is 4 x 4. The mtrix represents the stiffness of singe eement. 5-6 Shoji Nzw (nzw@e.tut..jp)
7 Truss Eements Formution.5 ASSEMBLING THE STIFFNESS MATRICES The next step is to onsider n ssemge of mny truss eements onneted y pin joints. Eh eement meeting t joint, or node, wi ontriute n extern fore. To mintin stti equiirium, eement fore ontriutions { }t given node must sum to the fore tht is externy ppied t tht node: (.5.1.1) { } = { } = [ K ]{ D } = [ K ]{ D} eem eem Here, { } is extern fore, { D } is dispement vetor, nd [ K ] is over, or go stiffness mtrix. Eh eement stiffness mtrix [ K ] is dded to the pproprite otion of the over stiffness mtrix [ K ] tht retes of the truss dispements nd fores. This proess is ed ssemy. Eq.(.5.1.1) is simutneous iner equtions. Soving the Eq.(.5.1.1), the dispement vetor n e uted s 1 { D} [ K ] { } = (.5.1.).6 CALCULATION OF THIAL FORCES OF TRUSS ELEMENT To evute n xi fore of eh memer, the nod dispements of eh memer re uted sed on the resut of dispement vetor in Eq.(.5.1). The eongtion e of truss memer is expressed with the nod dispements of the go oordinte system. e = u u j i ( U j osθ j sinθ ) ( Ui osθ i sinθ ) = + + = osθ U sinθ + osθ U + sinθ i i j j Ui i = [ osθ sinθ sinθ osθ ] U j j By using the nod dispements, the xi fore N of truss memer is expressed s N Ui E A E A i = e = [ osθ sinθ sinθ osθ ] U j j (.6.1.1) (.6.1.) In gener, sustituting Eq.(..1.8) into Eq.(...4), the nod fore vetor {p } in the o oordinte is expressed s { p } = [ ]{ d } = [ ][ T ]{ D } (...4) pi q i p j q j = E A 1 1 osθ sinθ Ui sin os θ θ i 1 1 osθ sinθ U j sinθ osθ j 5-7 Shoji Nzw (nzw@e.tut..jp)
8 Truss Eements Formution osθ sinθ osθ sinθu i E A i = (...4) osθ sinθ osθ sinθ U j j in whih the vue of third row, p j, of the vetor {p } orresponds to the xi fore N. 5-8 Shoji Nzw (nzw@e.tut..jp)
9 Truss Eements Formution.7 COMUTAIONAL ROCEDURE.7.1 Geometry of the struture There re nodes nd eements ming up the truss struture. We re going to do two dimension nysis so eh node is onstrined to move in ony the or diretion. We these diretions of motion degrees of freedom or dof for short. There re nodes nd degrees of freedom (two degrees of freedom for eh node). Nod dispements re ssumed to e D 1, D nd D s shown in Fig Appied extern fores re ssumed to e 1, nd orresponding to D 1, D nd D, respetivey. 1 D D 1 1 () () 45 () D Fig We n ote eh node y its oordintes. Te.7.1 shows the oordintes of the nodes in the proem we re soving. We n use these oordintes to determine the engths nd nges of the eements. Te Coordintes of the nodes in the truss. Node 1 Te.7.1. Eements nd Nodes they onnet in the truss. Eement From Node To Node Length Cosine Sine () 1 1/ 1/ () 1 () 1 1 Eh eement n e desried s extending from one node to nother. This so n e defined in Te From these two tes we n derive the engths of eh eement nd the osine nd sine of their orienttion. This is shown in the te eow. Axi stiffness re ssumed to e expressed s = E A in whih E, A nd ; =, nd (.7.1.1) re the young s moduus, the ross-setion re nd ength of the memer. 5-9 Shoji Nzw (nzw@e.tut..jp)
10 Truss Eements Formution.7. Find the stiffness mtrix for eh eement In the previous setions we deveoped the stiffness mtrix for n eement. (1) Eement () θ = 45 deg., = = θ = θ = θ = θ = θ θ = sin, os, sin, os, sin os U = = (.7..1) U D D () Eement () θ = 9 deg., = = θ θ θ θ θ θ sin = 1, os =, sin = 1, os =, sin os = U D = = 1 U1 D D () Eement () θ = deg., = = θ θ θ θ θ θ sin =, os = 1, sin =, os = 1, sin os = 1 1 U 1 1 = = 1 1 U 1 1 D (.7..) (.7..).7. Asseming the go stiffness mtries Sine there re dispements (or DOFs), D 1 through D, the mtrix is x. Now, we wi pe the individu mtrix eement from the eement stiffness mtries into the go mtrix ording to their position of row nd oumn memers. (1) Eement () The nod fores, 1 nd 1 orresponding to D 1 nd D diretions re expressed s 1 = D1 + D (.7..1) 5-1 Shoji Nzw (nzw@e.tut..jp)
11 Truss Eements Formution 1 = D1 + D (.7..) In the mtrix form, 1 / / D1 1 / / = D D (.7..) () Eement () The nod fores,, 1, nd 1 orresponding to D, D 1 nd D diretions re expressed s = (.7..4) 1 = (.7..5) = D (.7..6) 1 In the mtrix form, 1 D1 1 = D D () Eement () The nod fore orresponding to D diretion is expressed s In the mtrix form, (.7..7) = D (.7..8) D1 = D D (.7..9) Asseming the terms for eements (), () nd (), we get the ompete mtrix eqution of the struture. 1 = = D1 + D + = D1 + D = + = D + D + D = D + + D = = D In the mtrix form, 1 / / ΟD1 / / = + Ο D D Ο Ο In other word, dding the Eqs.(.7..), (.7..7) nd (.7..9), we get (.7..1) (.7..11) 5-11 Shoji Nzw (nzw@e.tut..jp)
12 Truss Eements / / D1 1 1 / / = + + = + D D Formution (.7..1).7.4 Soving the mtrix eqution Writing the mtrix eqution into geri iner equtions, we get D = (.7.4.1) / / D1 1 / / = + D Soving Eqs.(.7.4.1) nd (.7.4.), we get D / (.7.4.) = (.7.4.) D1 1 1 = = = D (.7.4.4) Aordingy, the dispements of the struture re D1 = + 1, D = 1 +, D = (.7.4.5).7.5 Axi fore of eh eement The xi fore of eh truss eement is uted y Eq.(.6.1.). (1) Eement () N = D D = + D 1 D ( ) 1 () Eement () = + + = = D 1 1 N = [ 1 1] = D = + = D D () Eement () (.7.5.1) (.7.5.) 5-1 Shoji Nzw (nzw@e.tut..jp)
13 Truss Eements N = [ 1 1 ] = D = = D Formution (.7.5.).7.6 Comprison It disputes out omprison etween xi fores of truss memers sed on stti equiirium eqution, nd the xi fores uted from Finite Eement nysis. 1 1 () () 45 () H As shown in ove figure, the retnt fores H, nd re uted sed on stti equiirium eqution s Σ = : H = + 1 Σ = : + + = Σ M = : + = () 1 (.7.6.1) in whih the rd eqution in Eq.(.7.6.1) expresses the equiirium of moment t Node. From the resut of ove equtions, we n get the retnt fores H, nd. = 1 = = + = 1 1 H = + 1 From the resut of ove equtions, the xi fore of eh memer is uted s N + = N = = + N = 1 1 N sin 45 + = N = N = o 1 1 (.7.6.) (.7.6.) It is onfirmed tht the xi fore of eh truss memer sed on the stti equiirium eqution is in greement with the xi fore uted y Finite Eement nysis. 5-1 Shoji Nzw (nzw@e.tut..jp)
14 Truss Eements Formution [ Exmpe BB.1] Compute go stiffness mtrix of the truss struture shown in Fig.BB.1. There re 4 nodes nd 4 eements ming up the struture. The young s moduus E nd the setion re A of eh truss memer re onstnt. Two dimension nysis is onsidered so eh node is onstrined to move in ony the or diretion. There re 4 degrees of freedom. Nod dispements re ssumed to e D 1, D, D nd D 4 s shown in Fig.BB.1. Appied extern fores re ssumed to e 1,, nd 4 orresponding to D 1, D, D nd D 4, respetivey. As oding ondition, extern od t downwrd t Node. D D 1 1 () (d) () D 4 4 () D BB.1 Truss struture (1) Stiffness mtrix for eh eement () Eement () [ Node 1 Node ] θ = deg., sinθ =, osθ = 1, sinθ osθ =, = = U = = (EE.1) 1 1 U 1 1 D1 D = D, = (EE.) 1 () Eement () [ Node Node ] θ = 9 deg., sinθ = 1, osθ =, sinθ osθ =, = = U D D 4 = = (EE.) U D D =, = D + D, =, = D D (EE.4) 4 4 () Eement () [ Node 4 Node ] 5-14 Shoji Nzw (nzw@e.tut..jp)
15 Truss Eements Formution θ = deg., sinθ =, osθ = 1, sinθ osθ =, = = U = = (EE.5) 1 1 U 1 1 D D 4 = D, = (EE.6) (d) Eement (d) θ = 45 deg., sin θ d =, os θd =, sinθd osθd =, 4 d U 4d d 4d d d = = d U d D1 d d D d d d = = d (EE.7) d d d d d = D1 + D, d = D1 + D (EE.8) () Go stiffness mtries Sine there re 4 dispements (or DOF), D 1 through D 4, the mtrix is 4x4. Asseming the terms for eements (), (), () nd (d), we get the ompete mtrix eqution of the struture. From the retion etween extrn fores nd nod fores, the equiiurium equtions of fores re expressed s = d = + + d = + = + 4 Sustituting Eqs.(EE.), (EE.4), (EE.6) nd (EE.8) into ove eqution, we get d d d d 1 = D1 + + D1 + D = + D1 + D = + D D + D + D = D + + D D = + D = D d d d d = D + D + = D + D In mtrix form (EE.9) (EE.1) 5-15 Shoji Nzw (nzw@e.tut..jp)
16 Truss Eements d d d D1 D = = d d d + D 4 + D 4 Considering the onditions of extern fores ( 1 = = = nd 4 = ), d d + D1 D d d + = D D4 Formution (EE.11) (EE.1) 5-16 Shoji Nzw (nzw@e.tut..jp)
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