On the elastoplastic stability problem of the cylindrical panels subjected to the complex loading wi...

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VNU JOURNAL OF SCIENCE, Mathematics - Physics T.XIX, No3 - 2003

ON THE ELASTOPLASTIC STABILITY PROBLEM OF THE CYLINDRICAL PANELS SUBJECTED TO THE COMPLEX LOADING WITH THE SIMPLY SUPPORTED AND CLAMPED BOUNDARY CONSTRAINTS

Dao Van Dung Department of Mathematics, College of Science, VNU

Abstract In this paper, an elastoplastic stability problem of the cylindrical panels under the action of the compression force along the generatrix and external pressure has been investigated By the Bubnov-Galerkin method, we have established the expression for determining the critical loads The sufficient condition of extremum for a long cylindrical panels was considered Some numerical results have been also given and discussed

1 Formulation of the stability problem and fundamental equations Let us consider a round cylindrical panel of thickness h and radius of the middle surface equal to R We choose a orthogonal coordinate system Ozxyz so that the plane Oxy coincides with the middle surface and the axis Ox lies along the generatrix of cylindrical panel while y = RO, with 6,-the angle circular arc and z in the direction of the normal

to the middle surface Denote the sides of cylindrical panel by a and b respectively to the axis Ox and Oy

Suppose that the cylindrical panel is simultaneously subjected to the comp force of intensity p(t) along the generatrix and external pressure of intensity q (/) increasing monotonously and depending arbitrarily on any loading parameter ¢ We have to find the

critical values t = t,, Pe = p(ts), Ge = q(t.) at which an instability of the structures

appears In order to investigate the proposed we will use the criterion of bifurcation

of equilibrium states and dont take into account the unloading in the cylindrical panel Afterhere we will present the fundamental equations of stability problem

lon

1.1 Pre-buckling process

Suppose that at any moment ¢ in the pre-buckling stage, there exists a membrane plane stress state

Pr Fy =—n(t) O12 = 013 = 023 = 033 = 0;

R

1

The material is assumed to be incompressible

&33 = —(€11 + €22)

Typeset by 4A46-TEX

The components of the strain velocity tensor determined according to the theory of elasto-

plastic proc are of the form [1]

"Nà ~b+ s4) - Us - 5)

= —(211 +222); 612 = €13 = 23 = 0,

where

23% be des

1 1 PP + 44 ~ 5P ~ 304

) P-pg+ge `

‘The are-length of the strain trajectory is respectively calculated by the formula

ds 2

dt

ciated with the relations (1.2), (1.3) and boundary con- and strain state at any point M in the structure at any

So the equilibrium equations ass

ditions entirely define the stre

moment of pre-buckling process

1.2 Post-buckling process

The system of stability equations of the thin cylindrical panel established in [5] is written in the form

» ow es tow “ise aw - 9 7 Pow 7 Pow 1 Las UAT TSAR TS Hy + TN (° 2m 74 Øy? Hôz2

where the coefficients a,, 3; (i = 1,3, 5) are calculated as follows

ma1-G(-§)s on =2-5 (1-5) 8

f=32+(5~I)ÈP 04T), i= 147 (3 - th,

For solving the stability problem of cylindrical panel, we consider two types of kinematic constraints following

* The cylindrical panel is simply supported at the four edges x = 0, r = a, y= 0, y=b

* The cylindrical panel is simply supported at the edges y = 0, y = 6 and clamped

at the edges = 0,2 =a

2 The solving method for the simply supported cylindrical panel at four edges

We find the increment of deflection 6w in the form

ne, 2wru

ôn = ` > mn Amn sin —— sin (2:1) 2.1

it is easy to see that this solution satisfies the kinematic boundary conditions Substituting the expression of dw into (1.5) we receive the particular solution y as follows

g= > › Brn sin 2 - sin > (2.2)

where

Bon = (ME) helo (SE)+04(%) CE) HACE) 0

It is seen that the system of functions

(m,n =1,2, ,M)

na, 2n 6Umn = sin —— sin C50 a b

is linearly independent Therefore we can apply the Bubnov-Galerkin method for estab- lishing an expression of critical forces

First of all, substitute the expressions of dw and y from (2.1), (2.2) into (1.4),

afthward multiply both sides of the just received equation by éwiy = sin = sin Zin and integrate that equation following x and y Finally we get

25980 L How + How + 9 OP 5w + Piw 1p Dea +84 Ox? dy? os Oy? ˆ hÀN Paạz T4 Ø2 — H0x2

0

37

sin

wn sin Yardy =0 (i,j = 1,2, -,M) (24)

For taking this integral, it needs to use the result

Nam _ an ine nmy _ 2yjmy { 0 with m#i; nA Jj

sin sin — sin sin = 4 ab

After series of calculations, the relation (2.4) gives us

we) +0(F) Ge) +e) ~ eee) GY tape) CZ) +a) EY +E) Tam =

Because of the condition on the existence of non-trivial solution i.e Amn # 0 then we receive the expression for determining critical loads

2Á) +(22) (CE) 0)

tm() TE) +) CE)

Putting X = (=) Y¥ =n?, i= —; the relation (2.6) can be rewritten in the other

J

(2.6)

AN? (aX tay +22) (8.x ++ 2

ức 5

Y(pX +4)(8X + + Ở) — Tang

Minh ininizing 7?, it means ax tn ĐỂ — 0, ĐỂ — 0, that yield 1 By =O that yields

ĐỀN

>»e(y+Ÿ)fex ta +Ÿ)

(5=)6x+»+#)=(5-)fnz+a+# 2q ‘i F Ps\ _ ,

x

Substituting the expressions (2.8) and (2.9) into (2.7) we obtain

2= _ re(p+ 4)" {aux toast Shox ++ 2} (2.10)

where X is found from the equation (2.9)

Applying the loading parameter method [1], and solving simultaneously the equation

(1.3) and (2.10) we can find the critical values t., ps = p(ts) de = @(ts)

For long cylindrical panel, i.e Y = 1, X < 1 we have from (2.7)

4 2 = * DX + q)Bs — CoX?' 4Nr?as Co= 55 ' oO APR? (2.11)

Calculating ax = 0, leads us X = 2Q = X, In addition

⁄0

OW - 8CoN Tass >0,

ôX?Ìx=x (65+ tết 20) By!

So the sufficient condition of extremum is verified

Substituting the values of as, G5 and X = X, into (2.11) we obtain

= 12 { N Gp-a)? pH zg BN } (2.12) Pie; (-Ủp Nrai † pm

3 The solving method for the simple supported cylindrical panel at y = 0,

y = b simultaneously clamped at the sides x = 0, x =a

The kinematic boundary constraints of stability problem are satisfied completely

by choosing

M M bw= D> D> Cmn (1 = 008 SE) dn “rẽ (3.1)

Using the equation (1.5) and the expression of dw we can find the particular solution # in the form

g= > > Dyn COs sin te (3.2)

nạ) In(ŒP)'+a(9JŒP)+»(PJT an

It is possible to prove that the system of functions dWnn = ( — cos 27") sin SAY

is linearly independent Then we can use the Bubnov-Galerkin By the same method presented in the above part we change the equation (1.4) into a relation as follows

b

/ a 280 ow lượn C2 9 ow fi củng Gat + Margy =.nG: (p Oa? ` Tay

“PRzazjú~e — b drdụ=0 (¡,j = 1,2, , Äf) (3.4)

For taking this integral above all we substitute éw and ¢ represented by (3.1) and (3.2)

into (3.4), afterwards integrate that received expression We will obtain a system of linear

algebraic equations with the unknowns C,; which is written in the matrix form

la;][Œu¿] =

Because of the condition on the existence of non-trivial solution i.e Cj; # 0 then the determinant of the coefficients of C;; must be equal to zero

detla,;] =0; i,j =1,2, ,M (3.6)

Associating this expression with (1.3) we can determine the critical values t., p = p(t.),

qe = Gta)

Note that a development of the determinant (3.6) in general case is complicated mathematically therefore we will take the solution in the first approximation

In this case we choose 6w and ¿ in the form

2mza+ 3um

ðtt = Cnn ( = cos wat din a b 8

=

Substituting dw and ¿ into (3.4), integrating that relation and taking into account the condition Cian # 0, leads us

fos 2) om 2) CE) to 2S) — pe EY eZ)

+ Bs

9 /2mm ủ 2mz 3m2 —— /2nz 2nzy\4)-! — =0 gì

Using notations £ = ?È?; 7 = (=) ;i= m the equation (3.8) is rewritten as follows

AN? (ain tay + 8) (int Bo + Be

đ(p+Š st) (int +2 Tả” sân

An? R?

ai? a?

Minimizing this relation i.e rd =0, on = 0, gives us

t?

Qn? (p + =) (am +83 + `

(a _ 7) (ai + As + a „ (5 = 2) (an + ag-+ a

6,

tạ ẤM — (suy sai vật q 722) (61n + on + 2 )=0 (3.11)

(p+ a)"

Putting the just found values of € and 77 into (3.9) we have

i = taint ant 2} {Bint t+ & } ' (3.12)

IÈ(p+ ty’

where 7) is deterrnined by the equation (3.11)

For finding the critical value ¢, of loading parameter t, we need to solve simultane-

ously the equation (1.3) and (3.12)

After determining t we can obtain the critical forces as follows

Px = p(te), Ge = a(ta)

Now consider the case of a long cylindrical panel Based on {2] leads us

= TC cóc C THỦ ý a ccc 3.15

€=1 7<l, 7 (pn + 3q)55 — Cuế Co Tere (3.13)

ee a oe es a aay OP Ổ: The minimization of the expression i* in (3.13), i.e On = 0, yields » = ; =1

On? In=n (30 p+ 2B)’

So the sufficient condition of extremum is satisfied Taking into account a5, (5, 7., the

relation (3.13) becomes

_

Aba l)p“m+õ|† san"

2 _— 122W?

=

Remarks

1) If the cylindrical panel has a very small curvature i.e KR — +00; g = 0 and

m= 1,n=1 then the expression (2.7) coincides with the result of {1, 5, 7]

2) If b = 27 that means the cylindrical panel becomes a closed round cylindrical shell, then the expressions (2.10), (2.12), (3.12), (3.15) return respectively to the previous well-known results

4 Numerical calevlations and discussion

A numerical analysis is considered on the long cylindrical panel made of the steel 30XICA with an elastic modulus 3G = 2.6- 105 MPa, an yield point ¢, = 400 MPa and

the material function ¢(s) presented in [1]

The relations for determining the critical loads are given in the form:

* Formulae (2.12) and (1.3) for the part a) of the examples

* Formulae (3.15) and (1.3) for the part b) of the examples

The numerical results are realized by the program of MATLAB

Example 1 The complex loading law is given in the form

P=P(t)=potmt’; q=q(t)=g+ qt

where po = 2 MPa, p; = 0.1 MPa; go = 2 MPa, gq; = 0.1 MPa

a) Numerical results for the simply supported cylindrical panel

Table 1: : = bi R35

b) Numerical results for the clamped cylindrical panel

Table 2: 2 = Table 2: Ro

100 8.44 4.392° 508.3 2.843 506.9

ˆ

et MPa

Fig 1 I - Simply supported, II - Clamped

Example 2 Suppose that the complex loading law is of the form

p= p(t)=po+pil®; po=2MPa, pi =0.1MPa

q=4()=qo+4@if; qo= 1MPa, qị =0.1MPa

a) Results of numerical calculation for the simply supported cylindrical panel

1

b

Table 3: == ‘able 3 R75

100 16.90 2.581 484.7 30.563 470.2

200 16.47 1.775 448.5 29.117 434.7

300 16.25 1.609 431.1 28.408 417.6

400 15.83 1.479 399.1 27.075 386.2

500 14.54 1.145 309.5 23.146 298.6

b) Results of numerical calculation for the clamped cylindrical panel

b1

Table G4 R8 4: ===

te s:100 p.MPa q.MPa otMPa

17.34 - 4.278 523.1 32.059 507.9

1682 2.373 478.1 30.301 463.7

300 16.52 1.831 452.6 29.282 438.7

400 1637 1.689 440.8 28.804 427.2

500 16.21 1.588 428.0 28.279 414.6

we so Bg

Sỹ MP,

400 200 yoo heo Fig 2 1 - Simply supported, II - Clamped

‘The above received results leads us to some conclusions

1 By using the Bnbnov-Galerkin we have solved the elastoplastic stability problem

of the cylindrical panels with two types of kinematic boundary constraints

2 We have shown, for long cylindrical panel, the sufficient conditions of extremum

3 The critical loads of the simply supported cylindrical panels are always stnaller than critical loads of the clamped cylindrical panels (see tables 1, 2, 3, 4 and figures 1, 2)

4 The more the cylindrical panel is thin the more the value of critical stress intensity

7, is small (see table 1, 2, 3, 4)

‘This paper is completed with financial support from the National Basic Research Program in Natural Sciences

References

Dao Huy Bich, Theory of clastoplastic processes, Vietnam National University Pub-

lishing House, Hanoi 1999 (in Vietnamese)

Volmir A S Stability of deformable systems, Moscow 1963 (in Russian) Ulo Tepik, Bifurcation analysis of elastic-plastic cylindrical shells, Int Journal of

Non-linear Mech, 34(1999), 299-311,

4, W 'T Koiter, Buckling and postbuckling behaviour of a cylindrical panel under axial compression, Nat Luchtvaart labort Rep., No 476(1956), Amsterdam Dao Van Dung, On the stability problem of cylindrical panels according to the theory of elastoplastic processes Proceeding of the Seventh National Congress on Mechanics, Hanoi, 18+20 December 2002, pp 141-150 (in Vietnamese)

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shells with compressible material subjected to complex loading processes, Vietnam

Journal of Mechanics, NCST of Vol 23, No 2(2001), pp 69-86

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