Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 UDK - UDC 624.074.4:539.3/.4 Izvirni znanstveni članek - Original scientific paper (1.01) Preskok sistema plitve osnosimetrične bimetalne lupine z uporabo nelinearne teorije Snap-through of the System for a Shallow Axially symmetric Bimetallic Shell using Non-linear Theory Marko Jakomin1 - Franc Kosel2 - Milan Batista3 - Tadej Kosel2 (1Primorski inštitut za naravoslovne in tehnične vede, Koper; 2Fakulteta za strojništvo, Ljubljana; 3Fakulteta za pomorstvo in promet, Portorož) V prispevku obravnavamo napetostne, deformacijske in stabilnostne razmere pri tankih, osnosimetričnih plitvih bimetalnih lupinah. Po teoriji drugega reda, ki upošteva ravnotežje sil na deformiranem telesu, podajamo model z matematičnim opisom geometrije sistema, premikov, napetosti in termoelastičnih deformacij. Enačbe temeljijo na teoriji velikih premikov. Kot primer predstavljamo rezultate za krogelne lupine, ki jih aproksimiramo s parabolično funkcijo. Poleg prosto položenih lupin obravnavamo tudi vrtljivo in konzolno vpete lupine ter lupine, ki so poleg segrevanja obremenjene tudi s silo v temenu. Deformacijsko krivuljo in temperaturo preskoka računamo numerično z nelinearno strelsko metodo. © 2006 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: lupine bimetalne, obremenitve toplotne, preskok sistema, teorija velikih premikov) The paper deals with the stresses, strains and buckling conditions in thin, axially symmetric, shallow, bimetallic shells. Based on third-order theory, which takes into account the equilibrium state of the forces and moments that are acting on the deformed system, the paper presents a model with a mathematical description of the geometry of the system, the stresses, the thermoelastic strains and the displacements. The mathematical formulation is based on the theory of large displacements. As an example, the results for spherical, shallow shells are shown, approximated by a parabolic function. Besides simple roller-supported shells, also simple bearing-supported shells and clamped shells are discussed. The shells are loaded with temperature and/or a concentrated load acting at the top. The displacement state and the snap-through temperature are calculated numerically using a non-linear method. © 2006 Journal of Mechanical Engineering. All rights reserved. (Keywords: bimetallic shell, thermal loads, stability, snap-through of the system, large displacement theory) 0 UVOD Pospešen razvoj strojnih znanosti v zadnjih stoletjih je omogočil izdelavo različnih naprav, od razmeroma preprostih mehanizmov pa vse do zelo zapletenih strojnih naprav, ki jih človeštvo uporablja v tehnično tehnološkem postopku preoblikovanja materialnih dobrin. Čeprav so sodobne naprave po obliki, namenu in zgradbi med seboj zelo različne, pa se zaradi pomembnosti nemotenega in zanesljivega delovanja ter njihove vrednosti izraža zahteva po njihovi zaščiti pred različnimi preobremenitvami. Posebej pri strojih, ki spreminjajo eno obliko energije v drugo ter se pri tem segrevajo, je potrebno poskrbeti za zanesljivo zaščito pred toplotno 0 INTRODUCTION The development of mechanical sciences over the centuries has enabled the production of various devices, from relatively simple devices to very complex mechanical appliances, which are used in the technical-technological process of remodelling the material goods. Although modern devices differ greatly in shape, purpose and structure, they – for the sake of their unobstructed and reliable functioning – all require protection from various forms of overloading. This is particularly so for machines that transform one type of energy into another, warming up during the process, thus requiring real protection against excessive temperature over-loading. For this 785 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 preobremenitvijo. S tem namenom v naprave vgrajujemo elemente, ki opravljajo funkcijo “toplotne varovalke”, tako da stroj izklopijo takoj, ko posamezen del doseže največjo še dopustno temperaturo. Zaradi zanesljivosti delovanja so se pri zaščiti pred toplotno preobremenitvijo uveljavili črtni in ploskovni bimetalni konstrukcijski elementi, katerih delovanje sloni na znanem fizikalnem dejstvu, da se telesa s povečevanjem temperature raztezajo. Idealno homogena telesa se širijo in krčijo izotropno. V primeru bimetalnih teles, ki so izdelana iz dveh materialov z različnima temperaturnima razteznostnima koeficientoma, pa deformacije zaradi temperaturnih sprememb ne bodo več izotropne. V prispevku želimo poiskati in matematično formulirati funkcijsko zvezo med temperaturo, napetostjo in premiki bimetala. Ta zveza je med drugim odvisna tudi od geometrijskih značilk bimetala, saj se, npr. črtni bimetalni konstrukcijski elementi v primerjavi s ploskovnimi, na temperaturne spremembe različno odzivajo. Za prakso so predvsem pomembne razlike v stabilnostnih razmerah. Plitve bimetalne lupine imajo lastnost, da pri določeni temperaturi pridejo v indiferentno stanje, kar vodi v pojav, ki je v literaturi znan pod pojmom preskok sistema. 1 GEOMETRIJA SISTEMA Na sliki 1 je predstavljena osrednja ploskev tanke bimetalne vrtilno simetrične lupine. Osnosimetrična oblika lupine nastane z vrtenjem neke funkcije okoli ordinatne osi. Obliko nedeformirane osnosimetrične lupine v Lagrangevem koordinatnem sistemu torej določa funkcija y = y(x). Zaradi spremembe temperature se lupina deformira v novo obliko, ki jo določa funkcija Y = Y(X) v Eulerjevem koordinatnem sistemu. Tudi ta oblika je osnosimetrična v predpostavljenem homogenem temperaturnem polju, zaradi česar se mehanske veličine v odvisnosti od kota j ne spreminjajo. Vektor premika u v naravnem koordinatnem sistemu (y,j,ry ) poljubne točke P na osrednji ploskvi določa točko P’. Torej: purpose, elements that function as “heat cut-out” devices are installed, disconnecting the appliance as soon as a particular part reaches the maximum permissible temperature. Because of their reliability in functioning as protection from heat over-loading, linear and plane bimetallic structural elements are often used in these devices. Their working is based on the known physical fact that bodies expand with increasing temperature. Ideally, however, homogeneous objects expand and shrink in an isotropic manner. In the case of bimetallic bodies, which are made of two materials with different linear expansion coefficients, however, the deformations due to temperature changes will no longer be isotropic. In this paper we try to find and mathematically formulate the functional connection between the temperature, the strain and the displacements of a bimetal. This connection, among other things, also depends on the bimetal’s geometrical characteristics as, for instance, the linear bimetallic structural elements in comparison with the plane elements react differently to temperature changes. For practical purposes, above all, the differences in the stability conditions are important. Shallow bimetallic shells have the property that at a certain temperature they change to an indifferent state, which leads to a phenomenon known in the literature as a “snap-through of a system”. 1 GEOMETRY OF THE SYSTEM Fig. 1 shows the flexible, middle plane, of a thin, bimetallic rotationally symmetrical shell. The axially symmetric form of the shell is a result of the rotation of a function around the ordinate axis. The shape of undeformed axially symmetric shells in the Lagrange coordinate system is thus determined by the function y = y(x). Because of the change in temperature, the shell will deform into new shape, determined by the function Y = Y(X) in the Euler coordinate system. This shape is axially symmetric too, providing that at each point the bimetallic shell is exposed to the same temperature change. The axially symmetric bimetallic shell within a homogeneous temperature field, however, represents an axially symmetric loading example, and because of this the physical magnitudes depending on the angle j remain unchanged. The displacement of an optional point P on the middle plane of the shell to the point P’ is determined by the r displacement vector u in the natural coordinate system. Therefore: ,j,ry) = (u,v,w) 786 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Ker je problem osnosimetrične narave, v = 0, lahko tudi pišemo: The displacement state is axially symmetric, so we can also write: r u = (u,0,w) Osrednja ploskev lupine je določena z enačbo r = r (v), zaradi cesar je vektor premika u funkcija kota V u = u (ip). Opazujmo premike na tanki bimetalni osnosimetrični lupini v homogenem temperaturnem polju (sl. 1), ki je v temenu obremenjena s silo F. Zaradi temperaturne obremenitve se točka P, ki ima na nedeformirani lupini lego P(x,y(x)), premakne v lego P’ s koordinatama P’(X,Y(X)). Premik točke P v točko P’ določimo z enačbo (1), ki jo v ravnini (^r) lahko tudi pišemo u = (u,w). Zveza med Eulerjevim (X,Y(X)) in Lagrange-vim (x,y(x)) koordinatnim sistemom je (sl. 1): X = (1). The middle plan is determined by the equation ry = ry(y). Consequently, the displacement vector r rr u is the function of angle y: u = u (y) . Let us observe the displacement state of a thin bimetallic axially symmetric shell in a homogeneous temperature r field, Fig. 1, loaded at the top with the force Fk . Because of the temperature load, the point P(x,y(x)) on the undeformed shell will move to the point P’(X,Y(X)). The shifting of point P to the point P’ is determined by Equation (1), which in the plane r (y,ry) can also be written as u = (u, w) . The relationship between the Euler (X,Y(X)) and Lagrange (x,y(x)) coordinates is, Fig. 1: >y,Y ty,Y Sl. 1. Osnosimetrična lupina v homogenem temperaturnem polju in zveza med Eulerjevim in Lagrangevim koordinatnim sistemom Fig. 1. The axi-symmetric shell in the homogeneous temperature field and the connection between Euler and Lagrange coordinate system Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 787 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 kjer je: iz česar izhaja: r u (x,y) = where: cosy sin y siny -cos y following that: X = x +w siny +u cosy Y = y -w cosy +u siny (2) (3). Nadalje določimo ukrivljenost nedeformirane The curvature of the undeformed shell in the lupine v meridialni smeri y in obročni smeri j. meridian y and circular j direction is determined. An Infinitezimalno majhen del loka na krivulji je: infinitesimally small part of the arc of the curve in V obročni smeri: the circular direction is: dsj = r -siny-dj = x-dj v meridialni smeri : and in the meridian direction: dsy = ry •dy Geometrija sistema na sliki 1 določa naslednja The geometry of the system in Fig. 1 leads to razmerja: the following relations: y =arctan(y/) 1 ky == y " kj== r ry V(1 + y'2)3 sinarctan(y ) 1 kjer je s ky in k označena ukrivljenost lupine v meridialni oziroma obročni smeri, z eno oziroma dvema črticama pa prvi oziroma drugi odvod po Lagrangevi koordinati x: 1 (4) (5) (6), where ky and kj are the curvatures of the shell in the meridian and circular directions, respectively, while the one and two apostrophes, respectively, mark the first and second derivative with respect to the independent variable x: dx ( ) = ( )', T2( ) = ( )" Zaradi deformacije lupine se kot y spremeni v kot y, ukrivljenost lupine v meridialni smeri r oziroma obročni smeri r smeri pa v ry oziroma rj. Veljajo naslednje zveze, določene z geometrijo sistema na sliki 1: dx Due to the deformation of a shell the angle y y changes into the angle . The curvatures of the deformed shell are ry and rj in the meridian and circular directions, respectively. From the geometry of the system in Fig. 1 the following relations are determined: _ - • - - 1 Y X = r siny, Y = tany, ky= — = , & 1 siny sinarctanY 1 == = X & Y X X Z eno piko oziroma dvema pikama nad koordinato Y(X) sta označena prvi oziroma drugi odvod po Eulerjevi koordinati X: dY Y& = in/and Y& & dX One and two points above the coordinate Y(X) mark the first and the second derivatives, respectively, with respect to the variable X d2Y dX (7). 788 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Ker je koordinata Y zaradi enačbe (2) posredno funkcija Lagrangeve koordinate x, torej Y = Y(X) = Y[X(x)] izračunamo oba odvoda v enačbah (7) z uporabo t.i. verižnega pravila. Po vstavitvi teh odvodov v enačbo za koordinato X in enačbi za ukrivljenost lupine v deformiranem stanju sledi: Due to Equation (2) the coordinate Y is indirectly a function of the variable x , i.e., Y = Y(X) = Y[X(x)]. The derivatives in Equations (7) can be calculated by means of the so-called Chain Rule. After inserting into the Equation for the coordinate X and the Equations for the curvature of the shell in the undeformed state, it follows: y = arctan {Y' X} Y' X" Y' J{X'2+Y'2f Y' X X2 + Yl (8) (9) (10). Diferencial dolDine dsy v meridialni smeri deformirane lupine je: The differential of the length dsy in the meridian direction of the deformed shell is: dsy JdX2 + dY2 = dxX2 + Y' ryd (11). 2 DOLOČITEV TENZORJEV DEFORMACIJ IN NAPETOSTI TER VEKTORJA PREMIKA Deformacijo lupine podamo z elementi vektorja premika na osrednji oziroma primerjalni ploskvi, in sicer s premikom u v meridialni ter s premikom w v prečni smeri. Osrednja ploskev se zaradi upogibnih napetosti ne deformira. Elemente deformacijskega tenzorja v krivočrtnem pravokotnem koordinatnem sistemu določimo po pravilih za spremembo tenzorjev iz enega v drugi koordinatni sistem ali pa z neposredno postavitvijo na podlagi deformiranega stanja elementarnega dela lupine, kjer upoštevamo premike na ukrivljeni ploskvi ([1] do [3]): du dy/ dw dz 2 DEFINING THE STRAIN AND STRESS TENSORS AS WELL AS THE DISPLACEMENT VECTOR The shell’s deformation state is shown by the displacement vector in the middle, i.e., the reference plane, and that, by the displacement of u in the meridian, and the displacement of w in the radial direction. Due to bending, the middle plane does not deform. The elements of the strain tensor in the curvilinear orthogonal coordinate system are determined by rules for tensor transformation from one into another coordinate system or by direct forming on the basis of the deformed state of the elementary part of the shell, where displacements on the curved plane are taken into account ([1] to [3]): 1 2ry2 g yj g yr dw dy dv 1 +--------- dj sin y dv 1 du dy ry dj -u du du dy, 2r2 dw dy / 1 tanyJ r [ tan y/ ------cot y = 0 rjsiny ry dw 1 -u dw 1 ry dr dy ry dy ry (12) (13) (14) (15) (16) Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 789 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 -v dv \ dr V enačbah od (12) do (17) smo upoštevali predpostavko, da je lupina tanka in osnosimetrična tudi v deformiranem stanju, kakor sledi: du dw V enačbi (13) smo za pravokotno specifično deformacijo s v meridialni smeri upoštevali tudi nelinearni člen. Izkazalo se je namreč, da je upoštevanje nelinearnega člena in s tem teorije velikih premikov nujno za pravilnost rezultatov. Pri tem smo upoštevali, da je v nelinearnem členu komponenta u v primerjavi s komponento w zanemarljivo majhna. Z uvedbo krajevne koordinate z, z izhodiščem na osrednji ploskvi lupine, so zaradi ukrivljenosti lupine elementi tenzorja specifičnih deformacij tudi funkcija koordinate z: +—-----------= 0 (17). dp r ¦ sin tp In Equations (12) to (17) we have assumed that the shell is also thin and axially symmetric in the deformed state, as follows: ±( ) = 0, v = 0 (18). dp According to the third-order theory we also take into consideration the non-linear term in for the strain s in the meridian direction. Furthermore, we take into account that the displacement u in the nonlinear term is negligible compared with the displacement w. By introducing the local coordinate z with the centre of origin in the middle plane of the shell, the elements of the strain tensor are, due to the shell’s curvature, also a function of the coordinate z, as follows: L. = in eyz 1 + L* 1 1 r V ) = ey +z and = ej +z { 1 ( 1 r 1 1 r ?) zz 1+ 1+ ryry ejz zz 1+ 1+ = Li,+z-^ z-T y(19) j(20), kjer smo upoštevali, da je pri tankih lupinah: where it has been taken into account that for thin shells: @ @y@j@0 Deformacijski in napetostni tenzor sta: ryr The stress and strain tensors are: L, = Ll 0 ez yr sl 0 tz yr 0 ejz 0 , sij = 0 sjz 0 ipr 0 0 ipr 0 0 kjer je zveza med deformacijskim in napetostnim tenzorjem [3]: (21), where the relationship between the stress and strain tensors [3] is: E 1-m2 E 1-m2 (22) (23) (24) Z oznako T je v enačbah (22) in (23) označena sprememba temperature glede na primerjalno temperaturo T, pri kateri je napetostno stanje v lupini povsod enako nič: where T in Equations (22) and (23) denotes the change in temperature with respect to the reference temperature T0 at which the stress state in the shell is throughout equal to zero: syz (T0,y,z)= sjz (T0,y,z)= tyz r (T0,y,z)= 0 790 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 3 SILE, MOMENTI IN MEHANSKO RAVNOVESJE 3 EQUILIBRIUM STATE OF THE FORCES AND MOMENTS Slika 2 prikazuje elementarno majhen del Fig. 2 shows an elementary small part of the bimetalne lupine z napetostmi, ki se pojavijo na bimetallic shell, with stresses that appear on the prereznih ploskvah lupine. Zaradi napetostnega cross-section of the shell’s plane. The forces dNy, stanja, delujejo na ploskvah elementa ABCD sile dNy, dNj, dTy and the bending moments dMy, dMj on the dNj in dTyr ter upogibna momenta dMy in dMj: element ABCD planes are: dNy = J syz -h1 r +z dT dNj = J s -h1 r+z ry +z h2 yr yr f -h rj r +z dMy = J zs -h1 dMj =- 2 zs -h r siny ¦ dz ¦ dj = ny ¦ r sin y ¦ dj = nyX ¦ dj ry-dz-dy = njry ¦ dy = njdL rj sin y ¦ dz -dj = tyrrj sin y ¦ dj = tyrX ¦ dj r sin y ¦ dz ¦ dj = my ¦ r sin y ¦ dj = myX ¦ dj ry +z ry -dz-dy = m • ry • dy = m dL (25) (26) (27) (28) (29) kjer so n , n in t enotske sile in m , m enotska where n , n , ty denote unit forces and m , m unit mo- momenta, ki se pr yr tankih lupinah poenostavijo: ments, which are simplified in the case of thin shells: ny = 2 sydz h nj = 2 sjz¦ dz -h 2 my = J zs -h1 ¦ dz (30) (31) (32) (33) fzcr'-dz (34). j J j Po definiciji je osrednja ali primerjalna -h According to the definition, the middle, i.e., the ploskev tista ploskev na oddaljenosti h od reference plane is the one that lies at the distance h1 spodnjega roba lupine, ki se zaradi upogibnih from the lower edge of a shell. Because of the bending napetosti ne deformira. Lego osrednje ali primerjalne stresses the middle plane does not deform. The position ploskve dobimo iz pogoja ravnovesja notranjih osnih of the middle plane is obtained from the condition that sil, ki se pojavijo zaradi delovanja upogibnih takes into account that all inner forces, as a result of napetosti: the bending moment, must be in an equilibrium state: Jsyz-dA= 11 E1eyzxj- dz+ 2 E2eyzxj-dz = 0 (35). A -h1 d1 -h1 Specifično deformacijo ez računamo po The normal strain ez is calculated using enačbi (19), ki v primeru tanke lupine, ob upoštevanju, Equation (19), which, in the case of a thin shell and da je specifični raztezek s osrednje ploskve enak taking into account that the normal strain s of the nič, preide v obliko: y plane is equal to zero, takes the form: Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 791 Osrednja ravnina The middle plane Sl. 2. Element na deformirani lupini Fig. 2. Element on deformed shell z my+ dmy ny+ dny Y(X) ny + dny dmy Sl. 3. Enotske sile in momenti na elementu deformirane lupine Fig. 3. Unit forces and moments in the element of deformed shell 792 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 11 (36). Po integraciji enačbe (35) z uporabo enačbe After integration of Equation (35) and by using (36) je lega h1 osrednje ploskve : Equation (36), the position h1 of the middle plane is: E3+2E2SA+E2S22 (37). 2 E1d1+E2d2 () Iz pogoja h + h = d se določi tudi vrednost From the condition h1 + h = d the value for h h2. Ker je element na sliki 2 v ravnovesju, lahko is determined. Because the element in Fig. 2 is in an zapišemo enačbe ravnovesja sil in momentov equilibrium state we can write Equations of (sl. 3). equilibrium of the forces and moments, Fig. 3. Ravnovesje sil v meridialni smeri y: The equilibrium Equation of the forces in the meridian direction y: (dNy +d(dNy))-dNy cosdy -dTy sindy -2dN dip sin-----cos 2 y in v prečni smeri: and in the radial direction: (dTy +d(dTyr ))-dTyr cosdy +dNy sindy +2dNj dj sin-----sin 2 y dy dy (38). (39). Ravnovesna enačba v obročni smeri je zaradi Due to axially symmetric deformation, the osnosimetričnega napetostno-deformacijskega equilibrium Equation in the circular direction is equal stanja enaka nič. Zapišemo še upogibno momentno to zero. The equilibrium Equation for the bending ravnovesje: moments is: (dMy +d (dMy))-dMy -2dMj sin dj [2) dT,costp-dX Upoštevajmo, da je: ter dT^ sin^ ¦ dY - dn^ sin^ -dX + dn^ cos^ • dY + 2dn,fi sin Note that: sin( dip) @ dtp, sin( dp) @ dip, cos dtp @ 1 dj dY (40). d(dN y Tako imamo po ureditvi: d(dN ) and -------- dtp = dnX ¦ dip + ndX ¦ dip dtp d(dT, ) = ( dTr ) d^j = dt X¦ dip + t.dX¦ dip dtp d(dM,)= ( d-----) dtp = dmX ¦ dip + mdX ¦ dip dtp After rearranging: ( naX) -nL'cos^-t^X^) =0 ( taX") + nvL' sin $ + n<,X • (a) = 0 ( m^X ) - m^L' + n^X ( Y' cos tp - sin tp) - t^X ( Y' sin tp + cos tp) = 0 (41) (42) (43). 4 REŠEVANJE ENAČB ZA PRIMER PLITVIH LUPIN 4 SOLUTION OF THE EQUATIONS OF A SHALLOW SHELL Mešani sistem enačb (2), (3), (8) do (10), (13), The mixed system of Equations (2), (3), (8) to (14), (16), (19), (20), (22) do (24), (30) do (34) in (41) (10), (13), (14), (16), (19), (20), (22) to (24), (30) to (34) Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 793 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 do (43) z neznankami crz , ¦ y @ siny @ tany = y , cosy @ 1, y @ 0 Y' y < 1 =>¦ y @ sin y @ tan y = —j @ Y', cosy @ 1, Y 2 @ 0 Eulerjevi koordinati (X,Y) v enačbi (2) sta s Considering these simplifications, Euler’s co- temi poenostavitvami sedaj: ordinates (X,Y) in Equation (2) are now: X = x + wy' + u, Y = y-w + uy' (48). Iz enačb (5), (6), (9) in (10) izhaja: From Equations (5), (6), (9) and (10) it follows that: — @ y, — @ —, —@Y @y -w , — @ — @--------- ter zato: ry r x ry and thus: r x 11 ±y r y. = 1 - 1 @ -w/ 45 r r x p p in iz enačb (13) in (14): and from Equations (13) and (14): 11 1 1 / i\2 = y w + u + (w ey 2v ) (49) (50) (51) 794 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 e =-(y'w + u) (52). * x V enačbah (48) upoštevamo tudi, da je premik In Equations (48) we have taken into account u majhen v primerjavi s premikom w, ta pa je majhen that the displacement u is small in comparison with v primerjavi z Lagrangevo koordinato x, tako da sta the displacement w, and the latter is small in com- Eulerjevi koordinati X in Y približno: parison with the Lagrange coordinate x, so that the Euler coordinates X and Y are approximately: X @ x (53) Y@y-w (54). Ravnovesne enačbe (41) do (43) so sedaj: Thus the equilibrium Equations (41) to (43) are: nx'-n,-trxY" = 0 (55) ( rx ) +nY' + n,xY" = 0 (56) (mi,x)-mv-t^x = 0 (57). Pomnožimo sedaj enačbo (55) z Y’ ter ji Let us now multiply Equation (55) with Y’ and prištejmo enačbo (56). Tako imamo: add Equation (56). Thus, we have: (n^x) Y' + n^xY" + (t+rx) - t^xY'Y" = 0 Zadnji člen v enačbi zanemarimo, saj je po We disregard the last term in the equation, predpostavki: assuming that: Y'Y" = -(Y'2)'@0 Tako imamo po ureditvi zvezo: In this way we obtain the relationship: (trx )^-( nxY")' ter po integriranju: and after integration: c x tr=-nY + : (58), kjer je c integracijska stalnica, ki je odvisna od zunanje where c is a constant of integration, which depends on sile Fk v temenu lupine. Zvezo (58) vstavimo v enačbi the outer force Fk in the top of the shell. The (56) in (57): connection (58) is inserted into Equations (56) and (57): ( n*x ) -n =0 (59) ( mi>x ) -mv+ni>xY'-c = 0 (60). Sedaj imamo sistem enajstih enačb (44) do Now we obtained a system of 11 Equations (44) (47), (49) do (52), (54), (59) in (60) z enajstimi to (47), (49) to (52), (54), (59) and (60) and with eleven neznankami ny , n, m , m, u, w, e , ej, Y,? ,? . Naprej variables ny , n, m , m, u, w, ey , ej, Y,Y, Y . The next postopamo takole. Izrazimo premik u iz enačbe (52), steps are as follows. The displacement u is j expressed ga odvajamo po koordinati x ter vstavimo v enačbo from Equation (52). The derivative of the displacement (51). Nastane zveza: u with respect to the variable x is then obtained and inserted into Equation (51). In this way we get: (xe) +-(w')2=e<,+w'y' (61). Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 795 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Iz enačb (44) in (45) izrazimo ey in e ter ju vstavimo v enačbo (61). T in T nadomestimo z enačbama (49) in (59), enotsko silo n pa s prvim členom v enačbi (59). S tem dobi enačba (61) obliko: From Equations (44) and (45) are expressed ey and e, which are inserted into Equation (61). Next/ and Y are substituted by Equations (49) and (59), while the unit force n is substituted by the first term in Equation (59). Thus, Equation takes the form: -(x3n) A2 -A2 A y'w' AB -AB A (62). Sedaj v enačbo (60) vstavimo enačbi (46) in (47) ter spet izrazimo specifični deformaciji ey in e z enotskima silama n , n in razlikama ukrivljenostiY in Y . Tako kako/prej nadomestimo Y in Y z enačbama (49) in (50), enotsko silo n pa s prvim členom v enačbi (59). Če iz enačbe (61) izrazimo še n' postane enačba (60): Now, Equations (46) and (47) are inserted into Equation (60). The normal strains ey and e are expressed by the unit forces ny, n and the differences in curvature Y andY . Also, Y and Y are substituted by Equations (49) and (59), and the unit force n by the first term in Equation (59). If from Equation (61) is expressed too, Equation (60) becomes [4]: (y AB -AB A y'w' wi AC -B2 A (63). Prvotni sistem, ki ga je sestavljalo 21 enačb in prav toliko neznank, smo naposled prevedli v sistem dveh diferencialnih enačb (62) in (63), ki opisujeta najbolj splošen primer, ko sta m1 * m2 in & ^ <52. Enačbi (62) in (63) poenostavimo, če imata obe plasti lupine enak Poissonov količnik^ = M2, ker je_ takrat: A = mA; B = mB; C = mC; AB-AB = 0. Zato sta v tem primeru enačbi (62) in (63): The initial system, originally consisting of 21 equations and as many variables, has been finally converted into a system of two differential equations, (62) and (63), which outline the commonest example where m1 ^ m2 and S1 ^ S2. Equations (62) and (63) can be simplified if both layers of a shell have an equal Poisson’s coefficient M1 = ^because then: A = mA; B= mB; C = mC; AB - AB = 0 . Hence, in this case Equations (62) and (63) are: x 3 / = A(1-m in )\y'w'--(w')\ and xn^ (y - w) AC -B2 A (64) (65). Z uvedbo brezrazsežne vodoravne koordinate %. By introducing a dimensionless horizontal coordinate c: (66) in Wittrickovih funkcij G, F0 in F: no = -Y' = xa prevedemo problem v brezrazsežno obliko [6]. Iz enačbe (66) izhajajo namreč razmerja: and Wittrick’s functions G, F0 and F: (67) (68) (69) A2 - A2 the problem is converted into the dimensionless form [6]. Because of the introduction of a dimensionless coordinate ^we can write: G(c) a2 (AC-B2 A j F0(c) J2(AC-B2) a2 V A2-A2 F(c) J2(AC-B2) 796 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 x = ac, — = ~c, Ox a 'dc .dx, 2 c, „2 2 a ox a zato so posamezni členi v enačbah (64) in (65) po therefore, the individual terms in Equations (64) and spremembi: (65), after the transformation, are: ( xn) = 3x + x2 = 4c dx ii i( i i) c y w = y y - Y = — AC -B 2 x a '2 ( AC-B2 ) A d2 dc (c-G(c)) [A2(1-^2)) xw" + 1 wy] =x ( y'-Y')" + ( y'-Y')' - x1( y'-Y') = 4a c2 (F0(c)-F(c))F0(c) d2 2 AC-B2 A2(1-fj,2 2 c-(F0(c)-F(c) (y-w)' =Y' Diferencialni enačbi (64) in (65) v brezrazsežni obliki sta tako: Thus, the differential equations (64) and (65) in the dimensionless form are: 4(c(F-F0)) =FG 4(cG)" =F02-F2 c-a2 A2 1-S c AC-B2\2(AC-B2 (70) (71), pri čemer smo z dvema črticama označili drugi odvod where two apostrophes mark the second derivative po koordinati Z: with respect to the coordinate Z: — ( ) = ( )', d2 ( ) = ( )ff dc dc Odvisni brezrazsežni spremenljivki, po The dependable, dimensionless variables katerih rešujemo diferencialni enačbi (70) in (71), with which we are solving the differential equations sta torej oblikovna funkcija trenutne oblike (70) and (71) are thus the formative function F() of bimetalne lupine F(Z) ter napetostna funkcija the present form of the bimetallic shell and the stress G(x). Integracijsko stalnico c v enačbi (71) function G(Z). The constant of integration c in določimo z upoštevanjem ravnovesja sil na robu Equation (71) is determined by taking into account lupine. the equilibrium of forces at the edge of a shell. Če je lupina prosto položena, je sila podpore If the shell is simply roller supported, the force V na robu lupine nasprotno usmerjena in po r vrednosti enaka sili Fk v temenu lupine (sl. 4): V at the edge of a shell is pointing in the opposite Fk r direction and is equal to the force Fk in the top of the shell, Fig. 4: 1 Ve Sl. 4. Obremenitev prosto poloUene lupine s silo Fk Fig. 4. Loading of simply-roller supported shell with the force F Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 797 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 rrr r Fk +V = Fk +(2pa)Ve = 0 (72), kjer je z V označena enotska navpična sila na robu lupine. Enačba (72) je v komponentni obliki: r where Ve denotes the vertical unit force at the edge of the shell. Equation (72) in component form is: F tyr +ny dY dx 2pa = 0, -tyr dY dx +n =0 (73). Rešitev sistema enačb (73) je potem, ko upoštevamo tudi plitvost: When also taking into account the shallowness, the solution of the system of Equations (73) is: ny(a)= ta yr() 2pa dx 2pa (74) (75). Vstavimo enačbi (74) in (75) v enačbo (58) ter izrazimo stalnico c: Če na lupino ne deluje zunanja sila Fk je integracijska stalnica c v enačbi (76) enaka nič, diferencialni enačbi (70) in (71) pa se poenostavita v obliko, ki jo je v [6] zapisal W. H. Wittrick: Equations (74) and (75) are inserted into Equation (58) and the constant c is expressed by: -Fk 2p (76). r If the external force Fk does not act upon the shell, the integrating constant c in Equation (76) is equal to zero, whilst the differential Equations (70) and (71) are simplified into a form, according to W. H. Wittrick in [6]: 4(cG)"=F02-F2 4(c(F-F0))"=FG (77) (78). 5 ANALIZA RAZMER PRI KROGELNIH LUPINAH V enačbah (70) in (71) se pojavlja oblikovna funkcija začetne oblike lupine F(X). Ta je odvisna od funkcije y = y(x), ki opisuje osrednjo ploskev začetne, nedeformirane oblike lupine. V primeru krogelnih lupin, katerih teme je postavljeno v izhodišče koordinatnega sistema, je ta funkcija v posredni obliki: x2 + y2 = 2yR. Zaradi plitvosti lupine zanemarimo drugi člen ter dobimo po zamenjavi spremenljivke x v brezrazsežno spremenljivko^: 5 ANALYSIS OF THE CIRCUMSTANCES IN SPHERICAL SHELLS In Equations (70) and (71) occurs the formative function F0(c) of the initial shape of the shell. It depends on the function y = y(x), which describes the middle plane of the initial, undeformed shape of the shell. However, in the case of spherical shells, whose crowns are set at the beginning of the coordinate system, this function is in the implicit form x2 + y2 = 2yR. Because of the shallowness of the shell, the second term is neglected, so that after the substitution of the variable x into the dimensionless variable c, we obtain: x ac = 2R 2R h0c (79). Oblikovna funkcija začetne oblike lupine je po enačbi (68): According to Equation (68) the formative function of the initial shape of the shell is: 2hA 1-m2 2 AC-B2 konst (80). Če upoštevamo, da je začetna oblikovna funkcija F0 nespremenljiva, sta diferencialni enačbi When taking into account that the initial formative function F0 is a constant, the differential 798 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 (70) in (71): Equations (70) and (71) are: 4G"x + 8G' 4F"X + 8F' = FG- F02 -F2 A2 1-t? X AC-B2\2(AC-B2 (81) (82). Diferencialni enačbi (81) in (82) najlažje rešimo za primer, ko lupina ni obremenjena z zunanjo silo F , ker je takrat v enačbi (82) stalnica c enaka nič. k e je lupina prosto položena, so napetosti in momenti na njenem robu enaki nič, v temenu lupine pa imajo napetosti in momenti le končne vrednosti. Za napetostno funkcijo G(c) veljata torej robna pogoja: The easiest way to solve Equations (81) and (82) is in the case where the shell is not loaded by an external force Fk because then the constant c in Equation (82) is equal to zero. If the shell is simply roller supported, the stresses and moments at its edges are equal to zero, while the stresses and moments in the top of the shell have only limited values. For the stress function G(c) the following boundary conditions thus hold: G(1)=0, G(0)^oo Robna pogoja za oblikovno funkcijo F(c) dobimo iz enačbe (46), ki jo zapišemo v brezrazsežni obliki. Iz enačbe (44) in (45) izrazimo ey in ej ter vrednosti vstavimo v enačbo (46) za enotski moment m . Spremembo ukrivljenosti T in T nadomestimo z enačbama (49) in (50) prvi in drugi odvod premika w pa izrazimo z odvodom enačbe (54). Če nadalje spet vzamemo enak Poissonov koeficient za obe plasti bimetalne lupine se enotski moment my izraža: (83). The boundary conditions for the formative function F(c) are obtained from Equation (46), written in dimensionless form. From Equation (44) we express ey and insert the value into Equation (46) for the unit moment my . The curvature differences Y andY are then replaced by Equations (49) and (50), while j the first and second derivatives of the displacement w are expressed by the derivative of Equation (54). In the case of an equal Poisson’s ratio for both layers of the bimetallic shell, the unit moment m runs as follows: C 2C A(1-f) (F0(1 + fi) - F(X)(1 + ^) -2XF'(X)- kadar velja tudi: d1 =d2 in the special case where: in/and E1 = E2 = E QT(84), (85). Ker je na robu lupine poleg enotske sile n ničen tudi enotski moment m , izrazimo iz (84) povezavo med temperaturo lupine T in trenutno oblikovno funkcijo F: At the edge of the shell, besides the unit force ny, also the unit moment my is equal to zero. So, the relationship between the shell’s temperature T and the present formative function F can be derived from Equation (84): F F(1) 1+ m F (1) (86), kjer je t brezrazsežna funkcija temperature, T pa nespremenljiva: where t is the dimensionless function of the temperature and Tm is a constant: 2h0(1+m)(AC -B2) Tm = 2 (87). a (AQ -BP) Za oblikovno funkcijo F sta torej robna Thus, the boundary conditions for the pogoja: formative function F are: F F(1) 1 +m F (1) F(0) (88). Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 799 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Kadar je bimetalna lupina takšna, da veljajo enačbe (85), potem je iz enačbe (37): h = h = d1 = d2 = d/2. Če slednje upoštevamo pri izračunu stalnic A,A, B, B, C, C, P in Q, se enačbi (80) in (87) za nespremenljivi F0 in T poenostavita: F 2y/6-h 1 T 2d Let us take into consideration the case where Equations (85) are fulfilled. Consequently, from Equation (37) it follows that h1 = h2 = d1 = d2 = d/2. If what was stated above is considered in the calculation of the constants A, A , B, B , C, C , P and Q, Equations (80) and (87) for the constants F0 and Tm are simplified: 2d2F (89). 3R(a1-a2) 3V6-a21~ Za numerično reševanje problema pa robna pogoja G(0) ^ oo in F(0) ^ oo nista primerna, zato uvedemo novi spremenljivki g(c) in f(c): f = c-F, Po spremembi diferencialnih enačb (81), (82) in robnih pogojev (83) in (88) ter upoštevanju, da sta funkciji G in F v c = 0 omejeni, imamo naslednji problem robnih vrednosti: 4g = F02-g(0) = g(1) = f(0) = 0, t = 1 f c2 For a numerical solution of the problem, however, the boundary conditions G(0) ^ oo andF(0) ^ oo are not suitable; hence new variables g(c) and f(c) have been introduced: g = c-G (90). After transforming the differential equations (81) and (82), the boundary conditions (83) and (88), and taking into consideration that functions G and F in c = 0 are limited, we obtain the following boundary-value problem: f-g 2 4f" = (2f'(1) f(1)•(1-m )) (91). Prevedemo ga v sistem navadnih diferencialnih enačb prvega reda: y1 = y2, y2 y1(0) = y1(1) = y3(0) če uvedemo zamenjavo: F2 2 c , t = 1- F0(1 + m) It is then converted into a system of ordinary differential equations of the first order: 4 c2 (2y4(1)-y3(1)-(1-)) y3 =y4, y4 (92), dg dc g = y1, g = — = y2, Sistem enačb (92) s temperaturo t v območju 0 < t < 2 smo rešili z uporabo nelinearne strelske metode. Izbrali smo približni vrednosti y2 (1) in y4 (1) ter izračunali približne vrednosti funkcij g in f po običajni enokoračni metodi Runge Kutta 4. reda. Določanje natančnejših vrednosti y2 (1) in y4 (1) je potekalo po Newtonovi metodi reševanja nelinearnih enačb. Ker je sistem enačb (92) v točki c = 0 singularen, smo odstopanje približnih vrednosti g in/od danih robnih pogojev v c = 0 računali v točki c = c = 1010. Deformacijo lupine smo zapisali z razmerjem x med trenutno višino deformirane lupine in začetno višino nedeformirane lupine: F0(1 + m) when the substitution is introduced: f f i df = ž/3, = — = y4 The system of Equations (92) with a temperature t in the interval 0 < t < 2 was solved using the non-linear shooting method. We took approximate values for y2 (1) and y4(1) and calculated rough values for the functions g and f using the classical one-step Runge Kutta method of the fourth order. For defining more exact values of y2 (1) and y4(1), the Newton method of solving non-linear equations was used. And since the system of Equations (92) at the point c = 0 is singular, the digressions from the approximate values g and f from the given boundary conditions in c = 0 were calculated at the point c = c = 10-10. The shell’s deformation was recorded with the ratio x between the present height of the deformed shell and the initial height of the undeformed shell: h h Y (1) y(1) 1 1 1 -f-f(c)-dc F0c0 c (93). 800 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Integral v (93) smo računali numerično. Zaradi singularnosti v = 0 je integracija potekala od = 1 do X= Z0. Tako smo izračunali razmerje višin Lpri različnih temperaturah t. Problem robnih vrednosti (91) lahko rešimo tudi po metodi, ki jo je v [6] predlagal W. H. Wittrick. Ker sta funkciji g in f v točki x= 0 singularni, ju v okolici te točke zapišemo v obliki potenčne vrste, tako da zadostimo robnim pogojem v = 0: The integral (93) was solved numerically. Due to the singularity in^= 0 the integration occurred from = 1 to X=X0. Consequently, we have calculated the values of ratio L at various temperatures t. The boundary value problem (91) could be solved too using the method proposed by W. H. Wittrick in [6]. Because the functions g and f at the point x = 0 are singular, they have been written in the form of power series about the point = 0 in such a way as to meet the boundary conditions in = 0: CO J2gnXn+1, f = J2fnXn+1 (94). n=0 n=0 Po vstavitvi potenčnih vrst (94) v diferencialni enačbi robnega problema (91) ter primerjavi koeficientov pri enakih potencah spremenljivke X na obeh straneh enačb, dobimo razmerja med koeficienti: F0-f0 After inserting the power series (94) into the boundary-value problem (91) and comparing the coefficients in equal exponents of the variable c on both sides of the equations, we obtain the relations between the coefficients: g1 f1 g2 -2f0f1 g4 g3= 4 -2f0f3-2f1f2 242 -f1 f0g1 +f1g0 = 24 f0g2 +f1g1 +f2g0 f4 , f2 f3 48 f0g3 +f1g2 +f2g1 +f3g0 (95). itn./etc. Sistem enačb (95) je rekurziven. Pri izbranem g0 in f0 so določeni vsi nadaljnji koeficienti g in fn funkcije g in f. Koeficienta g0 in f0 seveda izberemo tako, da funkciji g in f na robu lupine v * = 1 zadoščata robnima pogojema v enačbi (91). Za izbrani koeficient g0 smo izbrali neko začetno vrednost koeficienta f0 ter v območju 0 < v < 0 05 določili člene potenčne vrste funkcij g in f. S potenčnima vrstama smo izračunali funkcijske vrednosti g(0,05), g’(0,05), f(0,05) in f’(0,05) ter s temi začetnimi vrednostmi numerično izračunali vrednosti funkcij g in f v območju 0,05 < x < 1 . Ker je izbrana začetna vrednost koeficienta f0 le približek, funkcija g v točki x = 1 odstopa od robnega pogoja v enačbi (91). Po Newtonovi metodi smo zato določili novo, natančnejšo vrednost koeficienta f0, ter postopek ponovili tolikokrat, da je postala absolutna vrednost funkcije g(1) natančna do vnaprej predpisane vrednosti. Prednost tega numeričnega postopka pred prej opisanim je v tem, da z Newtonovo metodo ali bisekcijo računamo vrednost samo ene neznanke, in sicer vrednost koeficienta f0 oziroma y3 (0,05), medtem ko je treba pri strelski metodi izračunati vrednosti dveh The system of Equations (95) is recursive. By selecting g0 and f0 all further coefficients g and fn of the function g and f have been determined. Clearly the coefficients g0 and f0 are selected in such a way that the functions g and f at the edge of a shell in^= 1 meet the boundary conditions in Equation (91). For the selected coefficient g0 we have taken an initial value for the coefficient f0 and in the interval 0 < y < 0 05 determined the elements of the power series’ , for functions g and f. Then we have calculated, using power series, the functional values g(0,05), g’(0,05), f (0,05) and f’(0,05). These initial values were used to compute the numerical values of the functions g and f in the interval 0,05 < y < 1. Because the selected initial value for the coefficient f0 is only an approximation, the function g at the point^= 1 deviates from the boundary condition in Equation (91). Using Newton’s method we have then determined a new, more exact value for the coefficient f0 and kept repeating the procedure until the absolute value of the function g(1) was as precise as set up initially. The advantage of this numerical procedure over the one described before is that by using Newton’s method or bisection we calculate the value of only one variable y3(0,05), while with the shooting method it is necessary to calculate the values of two variables, y2(1) Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 801 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 FQ = 12 -0.5 Sl. 5. Funkcija t = t(0 za primer osnosimetrine lupine s F = 12 in M = 1/3, ki izkazuje pojav preskoka sistema med točkama AB ob segrevanju in točkama CD ob ohlajanju Fig. 5. The function t = t(® as an example of axi-symmetric shell for F = 12 and M = 1/3 expressing the phenomenon of a snap-through system between points AB in the process of heating up and the points CD in the process of cooling spremenljivk y2 (1) in y4 (1). Po obeh numeričnih postopkih smo dobili enake rezultate. Slika 5 prikazuje razmere v lupini s Poissonovim koeficientom M = 1/3 in začetno oblikovno funkcijo F = 12. Graf funkcije brezrazsežne temperature t v odvisnosti od razmerja višin 4 predstavlja stabilnostne razmere ob temperaturnem obremenjevanju bimetalne lupine . V začetnem, temperaturno neobremenjenem stanju t = 0, v točki O(1,0) je razmerje višin ^enako ena. S povečevanjem brezrazsežne temperature t se to razmerje zmanjšuje. Kakor je razvidno s slike 5, je območje na krivulji med točko O in točko A(p1, tp1) = A(0,360;1,195), kjer ima funkcija t(^ lokalni vrh, območje stabilnega ravnovesja. Do preskoka lupine bo torej prišlo v točki A pri temperaturi tp1 = 1,195, ker je korak med točko A in točko C(^2, A = C(-0,360;0,805), kjer ima funkcija lokalni dol, območje nestabilnega ravnovesja. Po preskoku bo lupina zavzela novo ravnovesno lego v točki B(-0,752;1,195) pri temperaturi t = 1,195. Pri nadaljnjem segrevanju lupine razmerje L še naprej zmanjšuje. Pri ohlajanju lupine imamo nasproten pojav in v točki C pri temperaturi tp2 = 0,805, ponoven preskok. Tokrat lupina preskoči v ravnovesno lego v točki D(0,752;0,805) pri temperaturi t = 0,805. S and y4(1). Nevertheless, with both of these numerical methods we obtained the same results. Fig. 5 shows the condition for the shell with Poisson’s ratio m = 1/3 and the initial formative function F0 = 12. The graph of the function of dimensionless temperature t depending on the ratio of heights x represents the stability circumstances during the shell’s temperature load. In the initial state with temperature t = 0, at point O(1,0) the ratio of heights x is equal to one. By increasing the dimensionless temperature t this ratio is decreasing. As shown in Fig. 5, the portion of the curvature between point O and point A(xp1, tp1) = A(0.360,1.195) where the function t(x) has the local maximum is the range of stable equilibrium. Hence, the snap-through of the shell will happen at point A at temperature tp1 = 1.195 because the interval between point A and point C(xp2, t p 2 ) = C(-0.360,0.805)where the function has the local minimum is the range of unstable equilibrium. After the snap-through, the shell will take a new equilibrium state at point B(-0.752,1.195) at temperature t = 1.195. In the course of the shell’s further heating up, the ratio of heights x continues to decrease. However, in the cooling down of the shell the reverse situation occurs and at point C at temperature tp2 = 0.805, another snap-through happens. This time the shell snaps into the equilibrium state at point 802 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Sl. 6. Stabilnostne razmere pri lupinah z različnimi vrednostmi funkcije F0 in M = 1/3 Fig. 6. Snap-through behaviour for shells with various values of function F0 and M = 1/3 Preglednica 1. Temperatura in lega preskoka lupine za različne vrednosti funkcije F0 in M = 1/3 Table 1. Temperature and position of the shell's snap-through for different values of function F0 and M= 1/3 F0 8, 93 10 12 14 16 S.T.1 tp1 =1 xp1 =0 tp1 = 1,0 43 xp1 = 0, 248 tp1 = 1,195 xp1 = 0, 361 tp1 = 1, 381 xp1 = 0, 411 tp1 = 1, 567 xp1 = 0, 445 S.T.2 tp2 =1 xp2 =0 t p2 = 0.957 xp2 = -0, 248 tp2 = 0, 805 xp2 = -0, 361 tp2 = 0, 619 x p2 = -0, 41 1 tp2 = 0, 433 xp2 = -0,445 ponovnim segrevanjem lupine do temperature prvega preskoka tp = 1,195 lahko celoten krog preskokov lupine ponovimo. Stabilnostne razmere pri lupinah z drugačnimi vrednostmi začetne oblikovne funkcije F0 so prikazane na sliki 6, preglednične vrednosti za temperaturo preskoka tp in razmerje višin L v trenutku preskoka lupine pa so zapisane p v preglednici 1. Kritična vrednost začetne oblikovne funkcije F0 izpod katere preskok lupine ni mogoč, znaša za lupine s Poissonovim količnikom H= 1/3, F0 = Fk = 8,93. Krivulja s F0 = 0 ponazarja razmere pri okrogli bimetalni plošči. Potek krivulje za brezrazsežno temperaturo t v odvisnosti od razmerja višin ^ je asimetrična glede na premico F0 = 0. D(0.752,0.805) at the temperature t = 0.805. By heating the shell up to the temperature of the first snap-through tp1 = 1.195, the whole cycle of the shell’s snaps is repeated. The snap-through behaviour of the shells with different values of the initial formative function F0 is shown in Fig. 6, while the tabulated values of the snap-through temperature tp and the ratio of height xp at the moment of the shell’s snap-through are presented in Table 1. The critical value of the initial formative function F0 under which the shell’s snap-through is not possible amounts to F0 = Fkr = 8.93 for the shells with Poisson’s ratio m = 1/3.The curve with F0 = 0 shows the conditions for the round bimetallic plate. The curve’s line for dimensionless temperature t relative to ratio x, is asymmetrical with respect to the straight line F0 = 0. Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 803 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Na sliki 7 sta prikazani krivulji za temperaturo prvega preskoka tp 1 in razmerje višin L v odvisnosti od začetne oblikovne funkcije F0. p1 Posledica temperaturne obremenitve prosto položene lupine je tudi vodoravni premik, določen z razliko med Eulerjevo in Lagrangevo koordinato X-x. Enačba (48) za Eulerjevo koordinato X je, potem ko izrazimo komponento u vektorja premika u iz enačbe (52), specifično deformacijo s pa iz enačb (44) in (45): Fig. 7 shows the curvatures for the temperature of the first snap-through tp1 and the ratio of heights xp1 depending on the initial formative function F0. The consequence of the temperature loading of the simply roller-supported shell is also a horizontal displacement, determined by the difference between the Euler and Lagrange coordinates X-x. After expressing the component u of the dis-r placement vector u from Equation (52) and the normal strain ej from Equations (44) and (45), Equation (48) for the Euler coordinate X is: X = aVx mny PT A(1-m2) A(1+ m) (96). Enotsko silo n izrazimo iz enačbe (67), enotsko silo n pa z zvezo med obema enotskima silama iz enačbe (59). Vodoravni premik X(,t) je torej pri danih parametrih lupine odvisen od napetostne funkcije G(et): The unit force ny is expressed by Equation (67) and the unit force nj by the connection between both unit forces by means of Equation (59). Hence, the horizontal displacement X(c,t), for the given parameters of the shell, depends on the stress function G(c,t): X(X,t) = ay/X C(2G'x + G(1-fi)) P-t-Tm (97). a2A(1-n2) A(1 + fi) Za primer smo izračunali brezrazsežni As an example, we have calculated the di- vodoravni premik (X(a) - a)/a v odvisnosti od mensionless horizontal displacement (X(a) - a)/a temperature t za lupino s parametri : depending on the temperature t for the shell with the parameters: F =12, h0= 0,78mm, a=15mm, <5=0,3mm, a1 = 3,41 • 10-5 / K, a2 = 1,41 • 10-5 / K Kakor je razvidno s slike 8, se vodoravni premik bimetalne lupine s temperaturo veča. Največji vodoravni premik je v trenutku preskoka v As shown in Fig. 8, the horizontal displacement of the bimetallic shell increases with temperature. The biggest horizontal displacement is at the beginning of z, "C 1.5 J_-^-~""""" 1.25 _ 0.75 0.5 _^L.____—------- 0.25 ?; 10 13 14 .1 IS Sl. 7. Temperatura in lega preskoka v odvisnosti od oblikovne funkcije F0 za lupine z m = 1/3 Fig. 7. Temperature and snap-through position depending on the formative function F0 for the shells with m = 1/3 804 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 F0=V2: h, = 0,78mm, a=16mm, <5=0:3mm. Ql = 3.41 ¦ 1CT° / K. a, = 1,41 ¦ 10"s / K 0.003 - Sl. 8. Vodoravni premik na robu lupine v odvisnosti od temperature t za prosto položeno lupino Fig. 8. Horizontal displacement at the edge of the simply-roller supported shell relative to temperature t FO = Br93: 10; 12: 14; 16 0.005 ¦ 3 . u j 4 ¦ : .o: 3 3.032 ¦ 3.031 F0=1L Sl. 9. Vodoravni premik na robu lupine za prosto položene lupine različnih vrednosti začetne oblikovne funkcije F0 Fig. 9. Horizontal displacement at the edge of the simply-roller supported shell of various values of the initial formative function F0 točki A. Po preskoku zavzame lupina novo the snap-through process, at the point A. After the snap- ravnovesno lego, vodoravni premik pa preide v točko B. Na sliki 9 je prikazan vodoravni premik na robu lupine za lupine z debelino d = 0,3 mm, tlorisnim polmerom a = 15 mm, razteznostnima koeficientoma a1 = 3,41 . 10- /K in a2 = 1,41 . 105 /K ter začetno oblikovno funkcijo F0, kar izhaja s slike. S povečevanjem začetne višine h0 in s tem through, the shell assumes a new equilibrium position, while the horizontal displacement passes to the point B. Fig. 9 shows the horizontal displacement at the edge of the shell with thickness d = 0.3 mm and horizontal radius a = 15 mm, coefficients of linear temperature .. expansion a1 = 3.41 10-5 /K and a2 = 1.41 10-5 /K and initial formative function F0, as follows from the graphical presentation. By increasing the initial height h0 and Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 805 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 funkcije F0 poleg temperature preskoka se zvečuje tudi vodoravni premik na robu lupine v x = a. Obravnavali bomo tudi preskok sistema lupine z vrtljivim vodoravno nepomično vpetim robom. Takšen je primer, če lupino vstavimo v valj ter tako preprečimo širjenje lupine v smeri osi X. V tem primeru vpetja je premer lupine a med temperaturnim obremenjevanjem stalen: a(T) = a = konst. Veljata robna pogoja: s (1) = 0; m(1) = 0. y Specifična deformacija s ima v primeru, ko sta debelini slojev enaki d1 = d2 = d/2 naslednjo obliko: subsequently the function F0, as well as an increase in the snap-through temperature the horizontal displacement at the edge of the shell at x = a is increased too. The next example we will discuss is the snap-through of the simply bearing-supported shell. This is the case when the shell is inserted into a cylinder, preventing in this way the expansion of the shell in the direction of the X axis. In this type of support, the radius of the shell a remains constant: a(T) = a = const. during the temperature loading. The boundary conditions are: ej(1) = 0; my(1) = 0. The normal strain ej has, in the case where the thicknesses of layers are equal, the following form: L, = (98). (ny-n)+PT(1-//) Ker se ny in n v (98) izražata z enačbama (67) Because n and n in (98) are expressed by in (59), zapišemo problem robnih vrednosti za vrtljivo Equations (67) and (59) we can write the boundary- vodoravno nepomično vpeto lupino: value problem for simply bearing-supported shells: 4g F2 f2 f 4 fll = ' g g(0) = f(0) = 0, g'(1)-g(1)[ c 1+m -72/3-F0(a1+a2)1-~ (99). t =1- F0(1 + M) Sistem enačb (99) smo rešili numerično s predhodno opisano nelinearno strelsko metodo. (2f'(1)-f(1).(1-M)) The system of Equations (99) was solved numerically using the above-described non-linear 15mm, 1/3. č = 0. Snm 1.75 n2=3.41xl{TtKr1 afcl.aixiar^KT1 a2=1.41»;l(r5K-1 :[k; Sl. 10. Razmerje višin $v odvisnosti od temperature T za vrtljivo vodoravno nepomično vpeto lupino Fig. 10. Ratio of heights L relative to the temperature T for simply bearing-supported shell 806 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Izbrali smo približni začetni vrednosti za funkciji g in f v točki x = 1 ter problem začetnih vrednosti rešili po metodi Runge Kutta 4. reda. Bolj natančne funkcijske vrednosti g(1) in f(1) smo spet računali z Newtonovo metodo za reševanje nelinearnih enačb. Na sliki 10 so predstavljene razmere pri temperaturnem obremenjevanju vrtljivo vodoravno nepomično vpete lupine za številčni primer F = 12; a = 15 mm;^1/3; d = 0,3 mm. Pri takšnem vpetju bimetalne lupine se razmerje višin L s povečevanjem temperature povečuje. Zaradi segrevanja se lupina razteza. Ker je raztezanje lupine v vodoravni smeri onemogočeno, se lupina razteza v smeri navpičnice. S temperaturo T se povečuje trenutna višina lupine Y in s tem razmerje LKer funkcijamT) nima lokalnega ekstrema sklepamo, da vrtljivo vpeta lupina za ta primer nima preskoka. Podobne razmere opažamo tudi pri enoslojnih lupinah oziroma lupinah z nespremenljivim koeficientom linearnega temperaturnega raztezka a1 = a2. Kakor je razvidno s slike 10, je deformacija bimetalne lupine s temperaturnim raztezkom slojev a1 > a2 nekje med deformacijama enoslojnih lupin z nespremenljivim temperaturnim raztezkom a1 in a2. Deformacijske krivulje, ki prikazujejo obliko lupine glede na temperaturno obremenitev, so razvidne s slike 11. shooting method. The approximate initial values for the functions g and f at the point c = 1 were selected and the problem of the initial values solved by the Runge Kutta method of the fourth order. More precise functional values g(1) and f(1) were again calculated by means of Newton’s method for solving non-linear equations. Fig. 10 shows the conditions in the temperature loading of simply bearing-supported shells for the numerical sample F0 = 12; a = 15 mm; m = 1/3; d = 0.3 mm. In this example of the bimetallic shell, the ratio x increases with increasing temperature. Because of the heating, the shell is expanding. And since the expansion of a shell in the horizontal direction is not possible, the shell is expanding in the vertical direction. Along with the temperature T the height of the shell Y is also expanding, and with this the ratio x. Since the function x(T) does not have a local extreme we can conclude that shells with a simple bearing-support do not have a snap-through. Similar results were also observed in the single-layer shells with a constant coefficient of linear temperature expansion a1 = a2. As seen from Fig. 10, the deformation of the bimetallic shell with a temperature expansion of layers a1 > a2 is somewhere between the deformations of single layer shells with constant temperature expansion a1 and a2. The deformation states of the shell relative to the temperature are shown in Fig. 11. Sl. 11. Deformacijske krivulje za vrtljivo vodoravno nepomično vpeto lupino s F = 12 in M = 1/3 Fig. 11. Displacement states for simply bearing-supported shell with F0 = 12 and M = 1/3 Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 807 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Sl. 12. Razmerje višin x v odvisnosti od temperature T za vrtljivo vodoravno nepomično vpeto lupino z m = 1/3, (a + a2)/(a1 - a) = 1 in F = 1,86 Fig. 12. Ratio of heights x relative to temperature T for simply bearing-supported shell with m = 1/3, (a1 + a2)/(a1 - a) = 1 and F0 = 1.86 Sl. 13. Razmerje višin x v odvisnosti od temperature T za vrtljivo vodoravno nepomično vpeto lupino z m = 1/3, (a + a2)/(a1 - a) = 1 in F = 1,87 Fig. 13. Ratio of heights x relative to the temperature T for simply bearing-supported shell with m = 1/3, (a1 + a2)/(a1 - a) = 1 in F0 = 1.87 Analizirali smo tudi razmere pri zelo plitvih 2 We have also analyzed the conditions in very lupinah z majhno začetno višino h0. Izkazalo se je, da se pri takšnih lupinah razmerje višin x(T) z višanjem temperature T zmanjšuje, če je le začetna oblikovna funkcija F0 dovolj majhna glede na shallow shells with a small initial height h0. It turned out that in such shells the ratio x(T) decreased with increasing temperature T only if the initial formative function F0 was small enough compared to the ratio 808 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 razmerje (a1 + a2)/(a1 - a2) v sistemu enačb (99). Največja vrednost za začetno oblikovno funkcijo, pri kateri se razmerje višin RT) z višanjem temperature Tše vedno zmanjšuje, znaša F = 1,86 (sl. 12). Takrat ima razmerje med vsoto in razliko razteznostnih koeficientov najmanjšo mogočo vrednost (a1 + a2)/(a1 - a2) = 1. Če vrednost začetne oblikovne funkcije F0 le malenkostno povečamo, se bo razmerje višin RT) z višanjem temperature T zvečalo, kar je razvidno s slike 13. V primeru konzolno vpete lupine sta robna pogoja: s (1) = 0; F(1) =/(1) = konst. Funkciji Y’ in y’ izrazimo iz enačb (69) in (68). Po krajšanju enakih členov je robni pogoj: F(1) = F0. Problem robnih vrednosti za konzolno vpeto lupino je s tem: 4g F2 (a1 + a2)/(a1 - a2) in the system of equations (99). The highest value for the initial formative function, in which the ratio of heights x(T) with increasing temperature T is still decreasing, is F0 = 1.86, Fig. 12. In this case the relation between the sum and the difference of the expansion factor has the smallest possible value (a1 + a2)/(a1 - a2) = 1. If we only slightly increase the value of initial formative function F0, the ratio x(T) will increase with the increase of the temperature T, as is clear from Fig. 13. In the case of the clamped shell, the boundary conditions are: ej(1) = 0; Y’(1) = y’(1) = const. The functions Y’ and y’ are expressed from Equations (69) and (68). After reducing the same terms, the boundary condition is: F(1) = F0. Then, the boundary-value problem for the clamped shell is: f 2 4f" g(0) = f(0) = 0, f(1) = F0, g'(1)-g(1) Tokrat smo izbrali približni vrednosti za g(1) in f’(1). Ob vsakem koraku smo preverili, kolikšno je odstopanje od robnih pogojev v točki x = 0 ter z Newtonovo metodo izračunali boljša približka za g(1) in f’(1). Rezultati so razvidni s slik 14 in 15 in so podobni razmeram pri vrtljivo vodoravno nepomično vpeti lupini. Razlika v stopnji deformacije različno vpetih lupin z enakimi snovno geometrijskimi značilkami je razvidna s slik 16 in 17. S C0 smo označili obliko obeh lupin v nedeformiranem stanju, s C1 obliko za vrtljivo vodoravno nepomično vpeto lupino in s C2 obliko za konzolno vpeto lupino. Pri tem pomenita zgornji deformacijski krivulji obliko lupin pri temperaturi t = 1, spodnji krivulji pa obliko pri temperaturi t = -1. Kakor je razvidno s slike 16 je razlika v deformaciji različno vpetih lupin očitnejša šele pri večji temperaturni obremenitvi. Pri konzolno vpeti lupini se razmerje višin RT) z višanjem temperature T povečuje ne glede na vrednost začetne oblikovne funkcije F0. Oglejmo si še razmere pri analizi preskoka sistema pri prosto položeni lupini, ki je v temenu obremenjena z zunanjo silo Fk. Ta obremenitveni primer opisujeta diferencialni enačbi (81) in (82). Iz primerjave enačb (74) in (75) izhaja, da je normalna enotska sila np v primerjavi s strižno enotsko silo V majhna, zato jo zanemarimo: ny (1) @ 0 . f g 2 c )1 (100). We chose approximate values for g(1) and f’(1). In each step we checked how far the digression from the boundary conditions can go at point c = 0 and by means of Newton’s method, better proximities for g(1) and f’(1) are computed The results can be seen in Figures 14 and 15; they are similar to the conditions found in the simple bearing-supported shell. Displacement states for different kinds of boundary conditions of the shells of the same material and geometrical characteristics can be seen in Figures 16 and 17. C0 denotes the shape of both shells in the undeformed state, C1 denotes the simple bearing-supported shell and C2 the clamped shell. In such case the upper displacement curves represent the shell’s shape at temperature t = 1 and the lower ones the shape at the temperature t = -1. As is clear from Figures 16 and 17, the difference in the displacement of the various boundary conditions of the shells is clearly expressed only in the case of higher temperature loads. In the clamped shells, the ratio of heights x(T) is increasing with increasing temperature T, regardless of the value of the initial formative function F0. The object of discussion is also the snap-through phenomenon in the simple roller-supported shell, wrhich is additionally loaded with an external force Fk in its top. This loading example is outlined by the differential equations (81) and (82). When comparing Equations (74) and (75) it follows that the normal unit force nj compared to the tangential force ty is small, hence it can be neglected: ny(1) @ 0 . Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 809 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Fli = 12, a = lEcnrc, ji = 1/3, 5 = 0.3mm r ol^.^IxlOT^ItL Sl. 14. Razmerje višin L v odvisnosti od temperature T za konzolno vpeto lupino Fig. 14. Ratio of heights {relative to the temperature T for clamped shell Sl. 15. Deformacijske krivulje za konzolno vpeto lupino s F0 = 12 in m = 1/3 Fig. 15. Displacement states for clamped shell with F0 = 12 and m = 1/3 Če ponovno upoštevamo, da sta funkciji G in F v točki = 0 omejeni, zapišemo problem robnih vrednosti: g(0) = g(1) = f(0)=0 t = 1 Considering that functions G and F at point c = 0 are limited, the boundary-value problem can be expressed as: 2 2 X 1 F0(1 +m) 2f'(1)-f(1).(1-/*)) (101), 810 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 F0=12, H=l/3, ; 1.4 1.2 -'"Cl 3- -: —0 . 5 0.8 0.6 o.4 0.5 i Sl. 16. Razmerje višin {pri vrtljivo vodoravno nepomično vpeti lupini (C1 ) in konzolno vpeti lupini (C2 ) v odvisnosti od temperaturnega obremenjevanja za lupine s F = 12 in M = 1/3 Fig. 16. Ratio of heights t in simply bearing-supported shell (C1 ) and clamped shell (C2 ) depending on temperature loading for shells with F0 = 12 and M = 1/3 ro = 12, f*= 1/3 St/a L-i ^VToi' ¦cl / 0.06 i = t) T = l f/ s °-04 cb^ ff / 0-03 T = -L if/ _^~~------"___Q - °^ If/J^^' 0-01 '--' j% Sl. 17. Deformacijske krivulje pri vrtljivo vodoravno nepomično vpeti lupini (C ) in konzolno vpeti lupini (C 2 ) za temperaturne vrednosti t = 0, t = -1, t = 1 Fig. 17. Displacement states in simply bearing-supported shell (CJ and clamped supported shell (C2 ) for . temperature values t = 0, t = -1, t = 1 kjer je fk, po analogiji z brezrazsežnotemperaturo t, where f is a dimensionless force, and FM is a con- brezrazsežna sila, FM pa nespremenljiva: stant: Fk ~ fk' FM ; F 2tt 2C3 E pd4 a2A(1-M2) a26J6(1-M2) 3 Reševanje sistema enačb (101) je potekalo To solve the system of Equations we use the po prej opisani numerični metodi. Zaradi nazornosti above-described numerical method. For the sake of Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 811 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Preglednica 2. Temperatura in lega preskoka lupine za različne vrednosti sile Fk Table 2. Temperature and shell's snap-through position for different force values Fk Fk[N] 0N 10N 20N 30N Fkr = 34,44N 40N S.T.1 tp1 = 1,19 5 x1 = 0, 361 tp1 = 0.879 xp1 = 0.366 t p1 = 0.535 xp1 = 0, 360 t p1 = 0.166 xp1 = 0, 335 tp1 =0 xp1 = 0, 312 t p1 = -0, 214 xp1 = 0, 269 S.T.2 tp2 = 0, 805 x p2 = -0, 361 tp2 = 0.510 xp2 = -0.348 t p2 = 0.231 xp2 = -0, 327 t p2 = -0, 04 0 xp2 = -0, 293 t p2 = -0.159 xp2 = -0, 2 74 t p2 = -0, 3 08 x p2 = -0, 24 0 Sl. 18. Stabilnostne razmere pri različnih vrednostih sile Fk Fig. 18. Snap-through behaviour for different force values Fk predstavljamo grafične in preglednične rezultate za clarity, the graphical and tabular results for the shell lupino z naslednjimi podatki: are shown below, with the following data: F 12, h0 = 0,78mm, a = 15mm, d = 0.3mm, E = 1, 7 • 105 MPa (102). S slike 18 je razvidno, da se temperatura preskoka lupine tp znižuje z večanjem sile Fk. Pri sili Fk = 0 smo za temperaturo tp in lego xp preskoka lupine dobili enake rezultate, kakor smo jih predhodno izračunali za samo temperaturno obremenjeno lupino. Temperaturo preskoka tp 1 za vmesne vrednosti sile Fk v območju [0,40] smo določili z interpolacijskim polinomom 4. stopnje (sl. 19): From Fig.18 it follows that the snap-through temperature tp is decreasing with increasing force Fk. For the force Fk = 0 we get the same results for the temperature tp and position xp of shell’s snap-through as those previously calculated for the shell that was loaded only by temperature. The snap-through temperature tp1 for the intermediate values of the force Fk in the interval [0,40] was determined by the interpolating polynomial of the fourth degree, Fig. 19: tp1 (Fk) = 1, 195 - 3,03 • 10-2Fk -1,045 • 10-4F2 - 2,25 • 10-6Fk3 + 4,583 • 10-8F4 (103). S polinomom (103) smo določili kritično silo Fkr, pri kateri lupina preskoči pri temperaturi t = 0: Using the polynomial (103) the critical force Fkr, at which the shell snaps-through at temperature t = 0, is determined: 812 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 F0=12, a=15mn, 5=0. arms, E=lr V^lO^/ami,2 ^=1/3 T 1.2 1 o.e 0.6 0.4 0.2 "^ 0.2 10 L '. 30 J [K] r Sl. 19. Temperatura preskoka tp1 v odvisnosti od zunanje sile Fk r Fig. 19. Snap-through temperature tp1 depending on the external force Fk Pri kritični sili Fk = 34,44 N lupina preskoči, ne da bi jo bilo potrebno segrevati. Če se po preskoku lupine sila zmanjša, se zmanjša tudi razmerje višin x. Kadar je lupina dovolj plitva, se po prenehanju sile vrne v izhodiščno lego. Obravnavana lupina je že takšna, saj je pri mehansko neobremenjeni lupini Fk = 0 možno samo eno ravnovesno stanje pri temperaturi t = 0 in sicer pri razmerju višin x = 1. Da bi lupina po prenehanju kritične sile F ne preskočila v izhodiščno lego, mora krivulja za temperaturo t v odvisnosti od razmerja višin x sekati negativni del abscisne osi ali se je vsaj dotakne. Samo v tem primeru sta mogoči dve stabilni ravnovesni stanji in razmerje višin ^ 1 pri temperaturi t = 0. Približno najmanjšo vrednost začetne oblikovne funkcije F0 i izračunamo z interpolacijskim polinomom n Interpoliramo preglednico 1, in sicer temperaturo tp2 drugega preskoka (S.T.2) z začetno oblikovno funkcijo F0 v območju [8,93 < F0 < 16]: At the critical force Fk = 34.44 N the shell snaps-through without being heated up. If after the shell’s snap-through the force decreases, the ratio x is decreased too. When the shell is shallow enough, on removal of the force it returns to the initial position. The shell we are discussing is of this type, because only one equilibrium state is possible at temperature t = 0 at x = 1 if there is no external force in the top of the shell. To prevent the shell snapping-through into the initial position on removal of the critical force Fkr the curve for the temperature t depending on the ratio x should be crossing the negative part of the abscissa axis or at least touch it. Only in this way are two stable equilibrium conditions and a ratio of heights L * 1 possible at the temperature t = 0. As an approximation, the lowest value of the initial formative function F0 is calculated using the interpolating polynomial In Table 2 we interpolate the temperature tp2 of the second snap-through (S.T.2) with the initial formative function F0 in the interval [8,93 1 za T < 0, razen pri vrtljivo vodoravno nepomično vpeti lupini z majhno začetno oblikovno funkcijo F0. Če na lupino poleg spremembe temperature T deluje tudi sila Fk v temenu lupine, se pojavi preskok lupine pri nižji temperaturir. Pri dovolj veliki sili Fk lupina preskoči, ne da bi p jo bilo treba dodatno segrevati. Velikost kritične sile Fkr je odvisna od snovno geometrijskih lastnosti lupine v enačbah In thin-walled, shallow bimetallic shells a snap-through into a new equilibrium state occurs when a certain temperature is reached. The snap-through temperature Tp depends on the material and the geometrical properties of the shell. As a special case, the stability conditions for spherical shells whose two layers have equal thickness d1 = d2 = d/2 and the same Poisson’s ratio m1 = m2 = m were analyzed. From Equation (89) it follows that the position of a snap-through xp, in shells with equal thickness d and Poisson’s number m, depends only on the initial value of the shell’s height h0. The curvature of the shell 1/R and the difference in the coefficients of the linear expansion a1 - a2 affect only the snap-through temperature Tp and have no influence on the snap-through xp position. In shallow, single-layer shells with a constant coefficient of the linear temperature expansion a(z) = a = const. the ratio of heights x(T) remains constant, regardless of the temperature loading. With increasing temperature T the shell’s horizontal radius a increases while the vertical component of the deformation Y at the edge of the shell always remains the same: = konst (104). From Equation (104) it follows that single-layer shells have no snap-through. Also, very thin bimetallic shells with the initial formative function F0 < 8.93 and Poisson’s ratio m = 1/3 have no snap-through. In order that snap-through can occur in bimetallic shells, it is necessary, besides a high enough temperature, to ensure that the edge of the shell can freely expand in the horizontal direction. The greatest horizontal displacement occurs at the edge of a shell at the moment of the shell’s snap-through. Bimetallic shells with a greater initial height h0 have, for the same material and geometrical characteristics, a greater horizontal displacement. Clamped bimetallic shells cannot expand in the horizontal direction. Because the expansion in the horizontal direction of such shell is prevented, the shell is expanding vertically. Consequently, the ratio of heights x(T) increases along with the temperature increase x(T) >1 for T < 0, except in simple bearing-supported shells with a small initial formative function F0. When in addition to the temperature T, also a force Fk is acting on the shell’s top, the snap-through occurs at a lower temperature tp. When the force Fkr is high enough, the shell snaps-through without the need to be additionally loaded with temperature. The value of the critical force Fkr depends on the material and the geometrical =y(a) 814 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 (102). Po prenehanju kritične sile se lupina vrne v začetno ravnovesno lego, razen pri manj plitvih lupinah, pri katerih je vrednost začetne oblikovne funkcije F0 i > 21 66 za lupine z značilkami v enačbah (102) n properties of the shell in Equations (102). When the force is removed, the shell returns to its initial equilibrium state, except in the case of shallower shells, whose value of initial formative function is F0 i > 21 66 for shells with the characteristics in EquationT(102). tlorisni polmer nedeformirane lupine Youngov elastični modul materiala 1 in 2 začetna oblikovna nedeformirana funkcija trenutna oblikovna deformirana funkcija sila v temenu lupine kritična sila v temenu lupine členi potenčne vrste trenutne oblikovne funkcije napetostna funkcija členi potenčne vrste trenutne napetostne funkcije začetna višina nedeformirane lupine razdalji osrednje ploskve od spodnje (h1) oziroma zgornje (h2) ploskve bimetalne lupine ukrivljenosti nedeformirane lupine ukrivljenosti deformirane lupine notranja momenta v lupini notranja enotska momenta v lupini notranji sili v smeri normale v lupini notranji enotski sili v smeri normale na lupino polmer nedeformirane krogelne lupine polmera ukrivljenosti nedeformirane lupine polmera ukrivljenosti deformirane lupine dol Qna na nedeformirani lupini dol Qna na deformirani lupini primerjalna temperatura temperatura, temperatura preskoka notranja striCna sila na lupini notranja enotska striaa sila v lupini vektor premika členi vektorja premika sila podpore, enotska sila podpore Lagrangev koordinatni sistem Eulerjev koordinatni sistem temperaturna koeficienta dolinskega raztezka materiala 1 in 2 7 OZNAKE 7 SYMBOLS a E1,E2 F0 (c) F (c) Fk F kr fi G (c) gi h0 h1,h2 ky,kj ky,kj My,Mj Ny,Nj ny,nj R s s T0 T,Tp T yr t yr r u u,v,w V,Ve (x,y,z) (X,Y,Z ) horizontal radius of undeformed shell Young’s elastic modulus of materials 1 and 2 initial formative function present formative function force acting at the top of a shell critical force acting at the top of a shell elements of power series of the present formative function stress function elements of power series of the stress function initial height of the undeformed shell middle plane distance from the lower (h1) and upper (h2) plane of the bimetallic shell the curvatures of the undeformed shell the curvatures of the deformed shell internal moments in the shell internal unit moments in the shell internal forces in the direction normal to the shell internal unit forces in the direction normal to the shell radius of the undeformed spherical shell radii of the undeformed shell radii of the deformed shell the length on the undeformed shell the length on the deformed shell reference temperature temperature, snap-through temperature internal tangential force in the shell internal unit tangential force in the shell displacement vector elements of the displacement vector reaction force, unit reaction force Lagrange coordinates Euler coordinates linear temperature expansion coefficients of material 1 and material 2 Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 815 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 debelina lupine debelini slojev iz materiala 1 in 2 deformacijski tenzor Poissonovi števili za material 1 in 2 napetostni tenzor brezrazse Cna temperatura, temperatura preskoka brezrazse Cna neodvisna spremenljivka razliki ukrivljenosti kota na nedeformirani lupini kot na deformirani lupini [1] [2] [3] [4] [5] [6] d d1,d2 e ij m1,m2 s ij c y shell’s thickness thickness of layers made of material 1 and 2 strain tensor Poisson’s ratios of materials 1 and 2 stress tensor dimensionless temperature, snap-through temperature dimensionless independent variable curvature differences angles of the undeformed shell angles of the deformed shell 8 LITERATURA 8 REFERENCES Timoshenko S. (1959) Theory of plates and shells. McGraw-Hill Book, New York. Alfutov N. A. (2000) Stability of elastic structures. Springer-Verlag. Reddy J. N. (1999) Theory and analysis of elastic plates. Taylor & Francis. Drole R., Kosel F. (1993) Stability analysis of shallow axi-symmetric bimetallic shells in a homogeneous temperature field. The 14th Canadian Congress of Applied Mechanics, Kingston, Ontario, June 1993. Drole R., Kosel F. (1994) Analysis of stress-strain state in shallow spherical bimetallic shells by non-linear theory, Z. angew. Math. Mech. Wittrick. W. H. (1953) Stability of bimetallic disk, The Quarterly Journal of Mechanics and Applied Mathematics. Naslovi avtorjev: dr. Marko Jakomin Univerza na Primorskem Primorski inštitut za naravoslovne in tehnične vede, Pristaniška cesta 14 6000 Koper markojakomin@pint.upr.si prof. dr. Franc Kosel doc. dr. Tadej Kosel Univerza v Ljubljani Fakulteta za strojništvo Aškerčeva 6 1000 Ljubljana franc.kosel@fs.uni-lj tadej.kosel@fs.uni-lj prof. dr. Milan Batista Univerza v Ljubljani Fakulteta za pomorstvo in promet Pot pomorščakov 4 6320 Portorož milan.batista@fpp.edu Prejeto: Received: 12.5.2005 Sprejeto: Accepted: Authors’ Addresses:Dr. Marko Jakomin University of Primorska Primorska Institute for Natural Sciences and Technology Pristaniška cesta 14 6000 Koper , Slovenia marko.jakomin@pint.upr.si Prof. Dr. Franc Kosel Doc. Dr. Tadej Kosel University of Ljubljana Faculty of Mechanical Eng. Aškerčeva 6 1000 Ljubljana, Slovenia franc.kosel@fs.uni-lj tadej.kosel@fs.uni-lj Prof. Dr. Milan Batista University of Ljubljana Faculty of Maritime studies and Transport Pot pomorščakov 4 6320 Portorož, Slovenia milan.batista@fpp.edu Odprto za diskusijo: 1 leto 22.6.2006 Open for discussion: 1 year 816 Jakomin M. - Kosel F. - Batista M. - Kosel T.