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Vietnam Journal of Mechanics, VAST, Vol. 27, No. 2 (2006), pp. 111 - 119 INFLUENCE OF THE HARDENING CHARACTERISTICS OF MATERIAL ON THE CRITICAL LOAD IN THE ELASTO - PLASTIC STABILITY PROBLEM OF CONICAL SHELLS DAO 1 Huy BrcH 1 AND Vu K HAC BAY 2 Vietnam National University, H anoi 2 University of Forestry A bst ract. In this paper by using the t heory of elasto-plastic processes and adjacentequilibrium criterion the governing equations of the elasto-plastic stability problem of conical shells are derived. The Bubnov-Galerkin's method combined with the loading parameter method are applied in solving the mentioned problem. The influence of the hardening characteristics of material on the critical load is investigated. 1. INTRODUCTION Analysis of the elasto-plastic stability problem of shells with homogeneous membrane stress of the prebuckling state was considered by some authors [1 + 4], but in the case when the prebuckling state is non- homogeneous many difficulties arise in solving the problem, because now stability equations are a set of partial differential equations with variable . coefficients. Otherwise we can get more difficulties in determination of material funct ions occured in the constitutive relations, for example the secant modulus and tangent modulus which become functions of point coordinates. Furthermore in the shell occur elastic and plastic zones, the boundary of which is unknown, it must be determined simultaneously in the solution process. In t his paper the governing equations of the elasto-plastic stability problem of conical shells are developed based on the theory of elasto-plastic processes and the adjacentequilibrium criterion. The Bubnov-Galerkin's method combined with the loading parameter method can be applied in solving considered problem. A piece-wise linearization procedure of the material function figured in constitutive relations is demonstrated for material with general hardening characteristics and t he influence of this characteristics on the critical load is investigated. 2. PREBUCKLING STATE OF A CONICAL SHELL. Points in the middle surface of a conical shell may be referred to coordinates (x, B), where x is a coordinate taken from the shell top to t he considered point in the ·generatrix direction, e- a circumferential coordinate, a- an open angle at the shell top, l- the length of shell anh h- the shell thichness. The stress occurs in the prebuckling state depending on loading process, here we restrict ourselves the applied load is axisymmetric and the linear bending equations are used for the prebuckling· deformation. If the shell is acted on by external pressme with intensity p, the prebuckling stress state is of the form o CT() = px -htga, / CT~()= 0, o CTu = vf3px 2htga; 112 Dao Huy Eich and Vu Khac Bay px Nox = - - 2 tga, Ng= -pxtga, N~o = O. It is clear that along the shell generatrix the stress intensity increases linearly with respect to x. Thus at points 0 ~ x ~ Xs elastic state occurs, while Xs ~ x ~ l the shell is in plastic state, the boundary of two zones is determined by 2as h Xs = - - }3ptga' where as- yield stress (when a~ = as), Xs- elastic- plastic zones boundary. This boundary will be determined simultaneously in the solving process. Putting x t= l' 10p l X 0 t auo -- v_ 2<>h tg a -l = a M ' where o - )3pl tga, 2h aM - one can get 0 ~ t ~ 1, as= a~ts, z- - a~as X ts= s - ~-. 3. STABILITY EQUATIONS Applying stability equations [5] for conical shells subjected to external pressure yields R2 ____, oo, r.p = 7r / 2- a, lim R2dr.p = dx, sin r.p = cos a, ~( fJN) _l_88Nxo _ fJN = O £::. x x + . (} ' ux sin a u£::.() 1 88No 1 8 2 -.-~() + --;=J(x 8Nxo) = 0, Sln a u X uX 2 2 8 ~M 2 (8 8Mxo 188Mxo) -X u +-8x 2 ( x) sin a 8x8() +x 8() 88Mo - -a;- - r Rr-•oo =x sina, cos r.p = sin a, (3.1) 2 1 8 8Mo sin2 a 8() 2 2 2 2 ( 88w x 8 8w 1 8 8w) 8No cotga + ptga x ax + 2 ax2 + sin2 a 8()2 = 0, +x with the following boundary conditions: - at the fixed point 0 of the shell top bu = <5v = <5w = 0 with x = O; (3.2) - the end cross section of the shell is simply supported such that Ow = 0, <5v = 0, <5Mx = 0, <5Nx = 0. (3.3) Influence of The Hardening Characteristics of Material on ... . 113 Remark: • With a = 7r /2 equations (3.1) reduce to stability equations for a circular plate. • With x sin a = a = canst and a = 0, equations (3.1) become stability equations for a circular cylindrical shell of radius a. According to the elasto-plastic process theory [3] the expressions for internal forces increments and internal moments increments of a conical shell subjected to external pressure are obtained as follows where N = a~/ s = ( s) / s plays a role of a secant modulus and <1> s) - tangent modulus of the hardening material. In this case N and <1> are functions of x alone, ' au a' , u x(3N +)-+ (3N + + 3x- + x - ) - + (2x- - 4<1> )-+ 2 8x ox ox ox ax x 1 2 2 I 1 8 v I 8<1> 1 av N 8 u + + (N + 2<1> ) - - - (N +4<1> - 2 x - ) - - + x sin2 a o()2 sin a oxo() ox sin a 8() 1 OW ( I + 2<1> cotga--;:;;- - 4<1> I uX 0<1> 2x~ uX ) W cotga- = O; X 1 8 2v 8N av 8N v <1> 8 2v I 1 8 2u x N ~ 2 + ( N + x ~) n - (N + x ~) - + 4 . 2 ~e 2 + (N + 2<1> ) - . - ~ ~e + ux ux ux ux x x sin a u sin a uxu 1 I 0N 1 OU <1> aw + (N + 4<1> + x~ ) - . - ~e + 4-.-cotga ~e = O; ux xsma u xsma u 1 4 I 8 w . I 8N 8<1> 8 3 w x ( 3N + ) ox4 + (6N + 2<1> + 6x ax + 2x ax ) ax3 + 2N oN ' o' o 2' a 2w 4 , 4 a' a 2' aw · + ( 3 x2- + 6 - - 4 - + 4 - + x 2- ) -2+ (-P + 2 -)-+ ax ax x ax ax ax x2 x ax ax 2 ax a 114 Dao Huy Eich and Vu Khac Bay 84w 8N o' 1 8 3w -4(N+P - x - - x - ) --+ 2 2 2 2 x sin a 8x 8() ax ox x2 sin a 8x8() 2 aN 8 x 2 8 2 1 8 2w 84w +4(N+3P - x - - 2 x - + - -2) + 4 + ax ax 2 8x x3 sin2 a 8() 2 x3 sin4 a 8() 4 I +4(N+P) 1 I 1 1 1 I 1 481> 1 au 1 av u w . + -h + . ne + - + -cotga)+ 2 cotga(-2 !.'l ux xsina u x x 36p aw x 2 8 2w 1 8 2w - h3 t9 a(x ax + 2 8x2 + sin2 a f)()2) = o. (3.5) 4. SOLVING METHOD a) In the case of elastic conical shells I N=P = 3G, equations (3.5) reduce to ones of elastic stabili.ty equations considered in [7]. b) In the case of conical shells made of material with linear hardening characteristics 1 P (s) where b = a 8 N = 9 =canst, = 0 = au s 0 a u0 - 9CTu (a s - 9€ s ) b = 9 + _9__ a u0 - b' gc 8 , the set of equations (3.5) can be written as - I 8 2u I a N au Iu N 8 2u x(3N + P) n 2 + (3N +P + 3x-;::i-)!.'l -41> - + . 2 ne 2 + ux ux ux x x sin a u 2 I) 1 8 v ( I) 1 0V I OW I W + (N + 21> - . - - - - N + 41> - . - - + 21> cotga- - 41> cotga-'- = O; sm a 8x8() sm a f)() ox x aN av v 1 82v 1 82u 82v xN n 2 + (N + x-;::i-) ( !.'l - - ) + 41> . ne + (N + 21> )-.- n ne+ ux ux ux x x sin2 a u 2 sin a uxu 8N 1 au 1 aw + (N + x-;::i- + 4P ) - . - ne + 41> -.-cotga ne = O; ux xsina u xsina u 84 w aN a 3w 82N 8N p' 8 2 w x(3N + P) ~ + (6N + 2P + 6x-;::i-) ~ + (3x!l2 + 6-;::i- - 4-) !l2+ uX uX uX uX ux X ux p' aw aN 1 8 3w 1 8 4w + 4 2 -;::;- - 4 ( N + P - x-;::i-) 2 . 2 n ne 2 + 4 ( N + P ) _ " ~ " ~ ~-- + x ux ux x sin a uxu aN 1 8 2w 1 84w + 4 ( N + 31> - x-;::i-) 3 . 2 ne2 + 41> 3 . 4 .ne4 + ux x sin a u x sin a u 1 481> 1 au 1 av u w + --cotga(-- + - . - - +-+ -cotga)+ h2 28x xsmao() x x aw x 2 8 2 w 1 8 2w 36p - h3 tga(x ax+ 2 8x 2 +siri'2a ·ae2 ) = o. (4 ·1) I I I I I I I I I I By using these equations the stability problem has been investigated in [6]. Later equations (4 .1) will be used as basic equations for solving step - by -step the stability problem of conical shell made of material with general hardening characteristics. 115 Influence of The Hardening Characteristics of Material on .. .. c) In the case of conical shells made of general hardening material, relation a-e = ' a2 s) of material results in greater critical load. - With the same material and the same ratio l/h the critical loads decrease when the open angle a at shell top increases, i.e for a flatter conical shell the critical load is smaller. In conclusion, an analysis procedure has been developed for solving the elastoplastic stability problem of conical shells. 1 ( Acknowledgement. The present publication is supported by the National Council for Natural Sciences. REFERENCES l. J . Lubliner, Plasticity Theory, Macmillan publishing company, 1990. 2. Hill R, Plastic deformation and instability in thin- walled tubes under combined loading: a general theory, Journal of Mech. Phys of Solids 47 (1999) 921 - 933. 3. Dao Huy Bich, Theory of Elastoplastic Processes, VNU Publishing House, 1999. 4. Ulo Lepik, Bifurcation analysis of elastic plastic cylindrical shells, Int. Journal of Non - linear Mechanics 34 (1999) 299 - 311. 5. Dao Huy Bich, On the elasto-plastic stability problem of shells of revolution, Vietnam Journal of Mechanics 25 (1) (2003) 9 - 18. 6. Dao Huy Bich, Tran Thanh Tuan, Vu Khac Bay, On the elasto-plastic stability problem of conical shells, Proc. of 7-th Nat. Conj. on Solid Mechanics 2004, pp. 22 30. 7. D . 0. Brush, B. 0 . Almorth, Buckling of Bars, Plates and Shells, Mc Graw - Hill, 1975. Received November 11, 2005 ANH HUONG CUA DAC TRUNG TA.I BEN CUA VAT LIEU DEN TAI TRONG T6'i HAN TRONG BA.I TOAN DINH . DAN-DEO CUA VO NON . ON Trong bai bao nay da thiet l~p cac phucmg trlnh ca ban cua bai toan 6n d.inh ngoai gi&i h~n dan hoi cua v6 n6n d\fa tren ly thuyet qua trlnh dan deo va tieu chuan ton t~ cac di;tng can bang Ian c~n, khi tri;tng thai mang tru&c khi mat 6n d.inh la khong thuan nhat. Lai giai cO.a bai toan nh~n duqc nha ap di.mg phuang phap Bubnov-Galerkin va phuang phap tham s6 tai, qua d6 khao sat anh hucmg cO.a d~c trnng tai ben cua v~t li~u den tai tn;mg t&i h~n cO.a v6 n6n ch!u tac d\mg cO.a ap suat ngoai.

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