The Finite Strip Method (FSM) 1. Introduction

Transcription

1 The Fnte Stp ethod (FS). ntoducton Ths s the ethod of se-nuecal and se-analtcal natue. t s sutale fo the analss of ectangula plates and plane-stess eleents o stuctues eng the conaton of oth. Theefoe, the follong tpes of cvl engneeng stuctues can e dealt th: dge slas, o gdes, oofs consstng of plane eleents, etc. Hoeve, the ethod has one potant daac, hch lts ts vesatlt the analsed eleents ust e ectangula and spl-suppoted at to opposte edges. dge slas oofs of ndustal halls. Analss of plates FS The plate s consdeed as thn,.e. t s assued that the vaaton of dsplaceents as the plate thcness n neglgle and the plate s eplaced th a plane suface coespondng to the d-suface of the eal od. Also the n-plane defoatons ae neglected. Hence, at each pont of the d-suface the deflecton and ts devatves th espect to to n-plane co-odnates and ae enough to defne the defoaton state. (,) (,) (,) The co-odnate sste s ntoduced th the as along to opposte spl-suppoted edges. The to edges paallel to the as a have an tpe of suppot. n the FS the plate s dvded nto the fnte nue of stps unnng along the as, thus each of the stps s spl-suppoted. The stps ae connected along the so called nodal lnes. All the stps a have an dth, not necessal the sae one. The statng pont fo the devaton of the ethod pocedues s the appoaton of dsplaceent functon (,) fo the ponts on a gle stp. et us denote the nodal lnes along ths stp as and.

2 odal lne Fnte stp Sne sees olnoal ϕ ϕ The deflecton of the stp s appoated as a conaton of the e haonc sees n the longtudnal decton (analtcal aspect) and the polnoal functon f n the tansvese decton (nuecal aspect) (, ) f ( ) ( A...) π π hee: s the adopted nue of haonc functons n the sees and A,, etc. ae the coeffcents of the polnoal functon f, hch ust e found fo the ounda condtons coespondng to the deflectons and tansvese slopes at the nodal lnes and. The contnut condtons along the nodal lnes eue the adacent stps to have the sae deflectons and slopes. Fo an ata stp and the -th haonc functon e have fou pesced nodal values, ϕ,, ϕ, so the eued polnoal ust e of the thd ode f ( ) A The nodal values of dsplaceents and slopes ae π ϕ ϕ ϕ π ϕ π π The unnon coeffcents n the polnoal ae otaned fo the ounda condtons and afte soe eodeng one gets: hee the coeffcents ae f ( ) and f ( ) and f f f ( ) ( ) ( ) ϕ ϕ ϕ ϕ ote, that these functons ae dentcal th the shape functons,, 5 and n the clapedclaped ea.

3 The deflecton appoaton can e gven n the at fo ( ), π th the follong defntons of the vectos [ ] ϕ ϕ The supescpt denotes the -th stp and the suscpt the endng state. ote, that the nodal paaetes, ϕ,, ϕ, ae actuall the apltudes of -th haonc functons descng the deflectons and tansvese slopes along the nodal lnes. Havng specfed the deflecton appoaton n tes of nodal paaetes, ϕ,, ϕ, e can consde the eneg and deve the eulu condtons fo the stp. The total eneg fo one stp conssts of the stan eneg and the eneg of loadng s The stan eneg can e epessed n tes of endng and tosonal oents,,, and the coespondng cuvatues s dd hle the loadng eneg nvolves the deflectons ( ) dd, n the at fo the stan eneg can e tten as T s dd κ hee the vectos of oents and cuvatues κ ee ntoduced. Afte the susttuton of deflecton the cuvatues vecto taes the fo κ hee the at s

4 and π et us assue that the plate ateal s sotopc. Then fo the theo of sotopc plates the oents can e found hee the plate stffness, the coupled stffness and the tosonal stffness ( ) Eh ν ν ν fo the plate of the thcness h ee ntoduced. The vecto of oents can e no epessed n the at fo κ th the at of endng stffness coeffcents fo the sotopc plate The oent vecto can also e epessed n tes of the nodal dsplaceent paaetes and ts tanspose s T T T Hence, the stan eneg taes the fo n T n T n s dd The doule ntegal ll nvolve the follong epessons

5 n d fo fo n n fo nd fo Hence, the gle su s suffcent n ths epesson and e get The loadng eneg taes the fo n n T T s dd T T (, ) dd The eneg of the ente plate p s the su of eneges fo all the -stps p n the pesented eneg appoach the theoe s used sang, that the sste s n the eulu hen the ente potental eneg has the nal value. Ths leads to the condtons of vanshng patal devatves of p calculated th espect to the nodal dsplaceent paaetes n fo eve -th haonc functon p p p p p p,,,,, ϕ ϕ ϕ Afte ths dffeentaton the set of euatons s otaned fo eve hee the stffness at fo a gle stp and -th haonc functon T dd and the loadng vecto fo ths stp and ths haonc functon T ee ntoduced. The eplct fo of the stp stffness at s th: s (, ) 5 dd

6 The loadng vectos depend on the tpe of loadng. Fo nstance: - the pont load - the patch load c ( ) c n n n fo n,, and The assel of the stffness at p and the loadng vecto p fo the ente plate dvded nto -stps s caed out accodng to the follong schee, ϕ, ϕ, ϕ, ϕ, ϕ

7 p ϕ ϕ ϕ ϕ p ϕ ϕ ϕ ϕ ϕ ϕ ote, that the dectons of aes of local co-odnates n the stps concde th gloal co-odnates and no tansfoaton s necessa. As fo the ounda condtons, the sple suppoted opposte edges ae alead nheent n the sste n the fo of the e sees functons, hch fulfl the condtons of vanshng dsplaceents and endng oents (second devatves of th espect to ). On the othe hand, the suppot condtons on to eanng edges ust e ntoduced. f an of the edge paaetes s vanshng, then the coespondng os and coluns n p and the coespondng eleent of p can e eoved o eplaced th zeos. Fo nstance, f the edge along the -nodal lne s spl suppoted, then the dsplaceent vanshes and the fst o and the fst colun n p ust e odfed as ell as the fst eleent of p. n ths a the gloal set of eulu euatons fo the ente plate fo the -th haonc functon s otaned p p The soluton of these euatons povdes the vecto of apltudes of e functons fo deflectons and tansvese slopes along all the nodal lnes fo the -th haonc functon. The value of the dsplaceent at an ata pont of the plate s otaned a suaton of esults fo all the assued -haonc functons accodng to the foula (, ) p π The ethod ensues the contnut of deflectons and slopes eteen the stps along the nodal lnes. Hoeve, due to the appoate fo of the dsplaceent functon n the stps, the endng and tosonal oents calculated ug the second devatves of dsplaceents ae not contnuous. The appoate values of oents along the nodal lnes can e otaned as ean values coputed fo the oents eldng fo to adacent stps. Fo nstance, fo the nodal lne lng eteen the stps and e get ( ) ght hee: ght left left ( ) ( )

8 and the appopate atces have the fo ( ) ( ) The dffeences eteen the oents otaned fo the adacent stps decease th the nceag nue of stps. ote, that geneall the accuac of the esults otaned ug the FS depends on to paaetes: the nue of stps and the nue of the haonc functons.. Analss of plane stess eleents (plate-le eleents loaded n the plane) FS We consde thn plane eleents,.e. t s assued that the vaaton of dsplaceents as the eleent thcness n neglgle and the eleent s eplaced th a plane suface coespondng to the d-suface of the eal od. ue to the estence of onl n-plane loadng the dsplaceents ae also onl n-plane. Thus, e have to dsplaceent functons u and v. ese n the plate analss, the eleent s dvded nto a fnte nue of stps, hch span the ente length of the eleent eteen to spl-suppoted opposte edges. et us ntoduce the stan and stess vectos v u v u γ ε ε ε τ σ σ σ and te don the phscal la fo the plane stess eleent ε σ p p() p() u(,) v(,) u, σ u, σ

9 hee the plane stess stffness at has the geneal fo Fo the sotopc ateal E p ( ν ν ) E ν E ν E E E, ν ν ν E, ( ) ν ν E ( ν ) E E G The ounda condtons fo the spl suppoted opposte edges and ae u, σ The appoaton of dsplaceents n a gle stp has the fo u v π f π ( ) ( E F...) π π hee E, F, etc. ae the unnon coeffcents. The assued fo of dsplaceent functons fulfls the ounda condtons at the spl suppoted edges. The pesence of the functon (π/) n the dsplaceent u ensues the fulflent of the dsplaceent ounda condtons. As fo the stess ounda condtons e have u v σ ηε ξε η ξ The dffeentaton th espect to n the fst te does not change the functon (π/), hle the dffeentaton th espect to n the second te tansfos the functon (π/) nto (π/). Hence, oth the tes contan the desed functon (π/) and the condtons fo the vanshng σ ae also autoatcall fulflled. The calculaton of dsplaceents fo the stp lted the nodal lnes and nvolves the follong nodal paaetes: u and u fo u as ell as v and v fo v. Thus, each of the to appoatons of dsplaceents ll eue a lnea polnoal functon th to coeffcents E and F. The ae found fo the ounda condtons, hch fo nstance fo u ead: Ths leads to the follong at elaton ( ) f u f ( ) u hee the at of coeffcents s u v p p p π π π and the nodal paaetes fo the gle stp ae asseled nto the vecto π

10 p u v u Agan, slal as n the plate analss, to deve the stp stffness at and the loadng vecto the eneg appoach s used. The stan eneg fo a stp n the plane stess eleent can e epessed as s h T h σ ε dd ( σ ) ε σ ε τ γ The potental eneg of the loadng s The vecto of stans can e epessed as ( p( ) u p( ) v ) dd dd th the at of devatves p ε p p π and. Wth ths n hand the phscal la can e epessed n the at fo Thus, the eneg pats can e put as: s h σ T p p p p T p p p dd ( ) ( ) T p p p dd p onsdeng the fact, that the eneg fo the ente plane stess eleent conssts of the eneges fo all the stps e and ug the sla condtons of eulu as n the plate case p

11 e get the set of eulu euatons fo the ente eleent fo each -th haonc functon p e p p The stffness at fo the gle stp s otaned fo p and ts eplct fo s hee: p p h p s T p p p p p p5 p dd p p5 p p. p p α δ, p β δ, ( γ δ ) p and p α δ, p5 ( γ δ ), p β δ he α, ν ν he β, γ h ν α hν β, δ he ν ν The follong steps of the soluton ae analogous to the plate analss.. Analss of copound stuctues et us consde a o gde stuctue and ts dscetsaton nto the fnte stps. The s-secton loos le ths Such stuctues can e consdeed as copound of ectangula eleents, hch ae sultaneousl suected to endng and plane stess acton. The stffness at of such an eleent (stp) can e otaned as an appopate assel of stffness atces fo stps n the endng state and n the plane stess state. The eulu fo a gle stp n ts local co-odnates can e epessed the at euaton et us epesent the stp atces n endng and plane stess ug the ( )-suatces

12 s. 5 5 p s. p p p p5 p p5 p p p p p p p p o the stffness at fo a stp n a copound stuctue can e gven n the follong fo p s. Ths at coesponds to the follong vectos of nodal dsplaceent paaetes (apltudes) and nodal foces fo the -th stp and the -th haonc functon u v ϕ u v ϕ p p X Y Z X Y Each nodal lne has fou nodal paaetes thee fo dsplaceents and one fo tansvese slope. splaceents and foces n local co-odnates z z ϕ, v, Y, Z u, X, z z splaceents and foces n gloal co-odnates ϕ, v, Y, Z u, X,, α z z

13 The elatons eteen the foces ae hat leads to the follong tansfoaton ule X X α Z α Y Y Z X α Z α α α The decton coe at can e denoted as and the tansfoaton at s α α α α T X Y Z α α X Y Z o the tansfoaton of vectos of the dsplaceent and foce nodal paaetes can e gven as T T T T and fo the stp stffness at e have T T T T T The follong steps of the soluton ae analogous to the plate analss. 5. Analss of ult-span and colun suppoted plates the FS The plates th nte-span suppots, oth pont tpe o nfe-edge tpe, can e solved ug a conaton of the FS and the flelt ethod. Reactons n the addtonal nte-span suppots ae consdeed as edundant foces and the gle span plate fo hch the soluton can e found ug the pue FS s taen as a odfed (detenate) sste. Thus the schee of the flelt ethod s set. The contnuous eacton n the nfe-edge suppot can e appoated as a set of pont loads, fo nstance at the ponts hee the suppot lne ntesects the nodal lnes of the FS dscetsaton.

14 X X X X The dentt of the odfed sste th eoved suppots and the ognal sste th suppots pesent s ensued the neatcal condtons eung the deflectons at the eoved suppots to e zeo δ δ Afte consdeaton of the asc states: X, X, X and and the applcaton of the supeposton ule the canoncal euatons of the flelt ethod ae otaned δ X δ X... δ δ X δ X δ X X... δ... δ X... δ... δ The flelt coeffcents δ, hch ae the appopate negatve deflectons (the edundant foces vectos have upads oentaton, opposte to the as z) at the ponts hee foces X ae appled ae otaned fo the FS analss of the odfed one-span plate loaded an appopate loadng state X. Fo nstance, the set of dsplaceents δ, δ, δ follos fo the state X. X X δ δ δ δ δ δ δ δ X Thus, asc states and the state ust e solved to foulate the set of canoncal euatons. These euatons ae then solved and the values of edundant foces ae found. Then the last stage of the soluton follos, hee the odfed one-span plate s loaded sultaneousl th the etenal loadng and all the edundant foces. Altenatvel, the supeposton ule can e used ut ths eues the copute stoage of the coplete esults fo all the asc states, ncludng deflectons and oents.