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CHAPTER 5

BUCKLING BEHAVIOUR OF THIN SHORT STAINLESS

STEEL CYLINDMCAL SHELLS WITH RANDOM

GEOMIETRICAL IMPERFECTIONS

Random modeling of imperfections is essential when a structure is designed

for the first time and no information is available about the initial geometrical

imperfections. The random modeling of imperfections is to be done in such a way

that the maximum allowable amplitude of imperfections at any point of the

structures (cylindrical shell) should be within the limit specified by a design code or

dictated by the manufacturing process used to build the structure or its various

members. The objective is to model the imperfections in a realistic manner, by

assuming the imperfections as random.

Arbocz and Hol (1991) and Chryssanthopoulos and Poggi (1995a,b) adopted

two approaches to generate the random, modeling of imperfections. The first

approach is based on dominant modes of imperfections present in the actual

cylindrical shell due to manufacturing process. Using these dominant modes of

imperfections obtained from the actual measurement of imperfections present in the

thin cylindrical shells the imperfections can be represented in Fourier form. By

varying the Fourier coefficients randomly, random imperfection models were

generated.

In the second approach, random imperfection models were generated by

adding eigen affine mode shapes and also it is suggested to use this approach in case

of non-availability of actual measured geometrical imperfections.

Warren (1997) in his work adopted the second approach for framed

structures td generate random geometrical imperfections, by linear combination of

selected eigen mode shapes using zk factorial design matrix of Design of

Experiments (DOE). In that work, the mean value of imperfections was maintained

as zero and variance of the model was maintained at the desired manufacturing

tolerance level. It was also suggested that six carefully selected eigen mode shapes

are enough to generate the random imperfection shapes to predict the reliability of

the struchrre using response surface methodology concept.

In most of the previous works, for example Amazigo and Budiansky (1 972),

Elishakoff and Arbocz (1982), Elishakoff et a1 (1987), Arbocz and Hol (1 991),

Chryssanthopoulos and Poggi (1995a,b), Bielewicz and . Gorski (2002),

Papadopoulus and Papadrakalus (2004) etc., random modeling of geometrical

imperfections were generated by different methods and to best of our knowledge no

study on comparison of buckling strength / behaviour of thin cylindrical shells with

mirror image random geomehical imperfections were reported in the available

literatures. Hence in this work, FE models of thin cylindrical shells with mirror

image random geometrical imperfections are generated and studied about the

buckling behaviour in detail.

To achieve both the aims of randomness i.e., radial amplitude of

imperfections at any nodal point of FE model (except the nodes at both top and

bottom edges of cylindrical shell model should be random) and generation of pairs

of models and its mirror image models, the six eigen affine mode shapes should be

combined linearly using 2k factorial design of Design of Experiments (DOE).

Generally, the random imperfections can be represented by two variables,

namely maximum amplitude of imperfections and RMS (Root Mean Square) value

of imperfections (Athiman and Palaninathan 2004) (Here, RMS value of

imperfections can be taken as standard deviation since RMS value is calculated

assuming zero mean). Hence, utilising the RMS values and maximum amplitudes

of actual measured geometrical imperfections in the above reference (Athiman and

Palaninathan 20033, in the first case, RMS value of all 64 random models are varied

as 0.45m.m and 0.93mm and in the second case, the maximum amplitude of

imperfections of all random models with its mirror image models are varied as

1.32mrn and 3.7mm.

In addition, buckling behaviour of other types of geometrical imperfection

shapes such as axisyrnmetric, asymmetric imperfection shapes with their mirror

image imperfection shapes on static buckling strength of thin cylindrical shells

subjected axial compression are also studied in detail. Further, buckling behaviours

of random geometrical imperfect thin flat plate models and random geometrical

imperfect thin cylindrical panel models with their mirror image imperfection models

subjected to in plane axial compression are also studied in detail.

In the later part of the section, using the distribution of buckling strengths

obtained from the random imperfection models, attempts are made to calculate the

reliability of the cylindrical shell for the given load and for either given amplitude of

imperfections or given RMS value of imperfections using Mean value First Order

Second Moment (MVFOSM) method.

5.1 MODELING OF INITIAL RANDOM GEOMETRICAL

IMPERFECTIONS

The modeling of the initial random geometrical imperfections is

accomplished using the following assumptions1 conditions.

A - radial imperfection mplitudes at all nodes except the nodes at

the boundary edges should follow independent normal distribution

Mean value of radial imperfection amplitude of a node fi-om all

random models should be made equal to zero.

Equal importance should be given for the eigen affine mode shapes

considered for random modeling.

The random imperfection shapes generated should be linear

combinations of the eigen affine mode shapes considered.

Based on the above assumptions, the nodal radial amplitude of imperfection

vector for the entire structure (except the edge nodes, where the radial displacements

are constrained) may be written as

A i x l = 4 i x M j x l

where, A - Nodal radial imperfection amplitude vector

$ - the matrix of eigen vectors containing the modal imperfection

amplitudes at all nodal points of selected eigen affine mode

shapes with equal maximum amplitude of imperfections

M - Modal imperfection magmtude vector

i - number ofnodes

j - number of eigen affine mode shapes

If the nodal amplitudes of imperfections are known, the modal imperfection

magnitudes can be obtained using the relation

- 4 * j x i x Aixl M j x l - (5.2)

where the matrix 4" is the pseudo-inverse of the matrix 4. The pseudo-inverse is

calculated using the Eqg .3)based on method of least squares

If the nodal imperfections Ai are independent normally distributed random

variables then the mean value and variance of each modal magnitude is given by

where, dA - variance of the nodal radial imperfection amplitude

dM - variance of the modal imperfection magnitude

Similarly, mean value and variance of each nodal amplitude is given by

Since it is required to have nodal amplitude Ai of any node i of the structure

to follow normal distribution withp* = 0 and as per Eq.lQ.4) p~ also becomes zero.

Hence, to get amplitude of imperfections of all nodes for each model, the modal

magnitude of each model has to be obtained by using Eq.(5.5) Using the modal

magnitudes obtained from previous step the nodal radial amplitudes of imperfections

can be obtained by using the Eq. (5.1) By varying the modal magnitudes of

imperfections randomly using 2k factorial design matnx of Design of Experiments,

random geometrical imperfection models can be generated.

5.1.1 Design of experiments

i) Factorial design

Many experiments involve the study about the effects of two or more factors

on the output response of a system. In general, Factorial design approach guides the

researchers to conduct the experiments efficiently. By a factorial design, in each trial

or the replication of the experiment, all possible combinations of the levels of the

factors are investigated. The effect of a factor is defined to be the change in response

produced by a change in the level of that factor. This is frequently called as main

effect because it refers to the primary factors of interest in the experiment. Different

types of factorial design are 2k full factorial, 2k fractional factorial and 3k factorial.

ii) Two-level factorial designs

Two-level factorial designs axe widely used in the research work, because

they form the basis of scientific method of conducting experiments to determine the

main effects of input parameters or factors (Montgomery 1991). In tlus factorial

design with k factors, each factors has to be maintained at two levels. These levels

may be 'high' or 'low' levels of factor or perhaps presence or absence of a factor. A

complete replicate of such a design requires 2*2*2 ....* 2 = 2k observations and it is

called a 2' factorial design.

The zk factorial design is particularly use l l in the early stages of

experimental work, when there are likely to be many factors to be investigated. It

provides the smallest number of m s with which k factors can be studied in a

complete factorial design. Consequently, it is used as a factor screening experiment.

The assumptions, which support the use of 2kfactorial design in this work, are

• The factor levels are fixed

The design is completely random.

The normality assumptions are satisfied. (i.e., to have mean equal to

zero and constant value of variance of input factors).

For example, to study about three independent variables or factors with two

levels, a total of 2"3 = 8 experiments. The resulting design matrix of 2"' factorial

design is:

Table 5.1 Design matrix of z3 factorial design

where, H indicates the high level and L indicates the low level of an independent

variable Xi. While using the 2k factorial design, the independent variables are often

transformed into coded variables and high and low levels of actual variables are

converted into +1 and -1 respectively. (i.e., H to +1 and L to -1) using the following

relationship,

where, the variables ti are referred to as coded variables. The quantity di is the

difference between the high and low values of Xi and is the average of the high

md low values for Xi. The various combinations of the coded variables for a z3 design are shown in Eqf5.9).

5.1.2 Steps followed in random geometrical imperfections modeling

Step -1

Initially, substitute variance of modal imperfection magnitude vector as

Step -11

2 Using Eq.(5.7) the variance of nodal imperfection amplitude vector u A is

determined.

Step -111

2 Each element of the resulting cr A vector from Step-I is normalized with the

maximum value of element in that vector and multiplied with o2 value so as to

limit the maximum amplitude of imperfections to certain limiting value or to have

desired value of RMS value of imperfections for all the randomly generated

imperfection models.

Step -IV 2 Using the o A vector obtained fiom the Step-111, new a 2 M vector is found

using ~q6.3

Step -V

Since, = 0, PM = 0, using new 02nr vector determine the modal

imperfection magnitude vector M such that M= &DM.

Step -V

Using zk factorial design, design matrix is generated and each column of

design matrix is selected and is multiplied with corresponding element in the M

vector obtained from previous step. This new design matrix is used to generate 2k

(fir k=6, 26 = 64) random geometrical imperfection models.

i.e., A = 4 x new design matrix (5.1 1)

With the value of modal imperfection magnitude vector M, A nodal radial

imperfection vector is determined using the Eq. 5.1. But the 3. value of the modal

imperfection magnitude is decided by +1 or -1of design matrix obtained h m DOE.

The A matrix, thus formed has 64 rows, and each row corresponds to nodal radial

displacements of all nodes of one random imperfect cylinder model.

5.1.3 Thin cylindrical shell model

For all the analysis in the Sections 4 and 5, the thin cylindrical model taken

for study (Athiannan and Palaninathan 20M) is:

Radius (r) = 3 5 h

Height (h) = 340mm

Thickness (t) - - 1.25mm

l/d h) 1

r/t = 280

5.1.4 Material m o d e k g

The important properties of austenitic stainless steel (304) used in the

analysis are

Young's modulus (E) - - 193x1 o5 MPa

Yield stress (0,) = 205 Mya

Poisson's ratio (y) = 0.305

Strain hardening index = 6[Hautala].

Multi-linear kinematic hardening behaviour is considered for modeling the

material behaviour of stainless steel and the material behaviour is approximated by

Ramberg Osgood approximation ~q.(4.4)and it is shown in Fig. 4.6.

5.1.5 Boundary and loading conditions

Simply supported boundary conditions (only radially displacement restraints)

are applied on both the edges of the cylindrical shell and the uniform displacement

load is applied fjrom the top edge, and the bottom edge is restrained from moving

along load direction (Han et a1 2006).

5.1.6 Eigen affme mode shapes

In this work, She1193 element of ANSYS is used for modeling. Eight node

shell element she1193 is particularly well suited to model curved shells. The element

has six degrees of freedom at each node: translations in the nodal x, y, and z

directions and rotations about the nodal x, y, and z-axes. The deformation shapes are

quadratic in both in-plane directions. The element has plasticity, stress stiffening,

large deflection, and large strain capabilities.

In the previous section, to determine the mesh size for the perfect cylindrical

shell FE eigen buckling analysis solution of first eigen buckling strength with its

corresponding mode shape of perfect cylindrical shell is compared with analytical

solution given by Eq(3.4) Because of cluster of more number of almost equal

strength, buckling modes near the first bifurcation point, ANSYS requires more

number of elements along longitudinal and circumferential directions to predict

other higher strength buckling modes of the perfect cylindrical shell. This will

increase the computation time and efforts. Because of these above said reasons the

required eigen affine mode shapes are generated using the following [email protected]](which

is a slightly modified equation given in Ref. Huyan and Simities 1997) to generate

eigen affine modes with required number of circumferential and longitudinal half

lobes on the perfect cylindrical shell geometry.

where

ri - radius of the imperfect cylindrical shell at a particular

circumferential angular and longitudinal location.

r - radius of the perfect cylindrical shell geometry.

m - number of longitudinal half lobes

n - number of circumferential half lobes

t - thickness of cylindrical shell

a - amplitude imperfection scaling factor (max. amplitude of

imperfection / thickness of cylindrical shell.

And also to determine number of circumferential half lobes (n) and number

of longitudinal half lobes (m) of the higher strength mode shapes, the analytical

EqQ.4) can be used. The eigen mode shapes with its eigen buckling strengths

obtained from Eq0.4)for the perfect cylindrical shell taken for study are tabulated in

descending order is shown in Table 5.2. The lowest six eigen modes are considered

as eigen affine mode shapes to generate random geometrical imperfection models

and are shown in Fig 5.1.

Table 5.2 Eigen mode shapes (m,n) of perfect cylindrical shell taken for stuc

with its buckling strength in descending order

S.No.

Number of longitudinal half

lobes (m)

Number of

circumferential half lobes (n)

Buckling strength in N

Elgan @ine maris: I{ m=l, rP9) Eigen affine mode: 2t m=3, n=14)

y-axis ln mm a .* xlaxr~ in mm

Eigen amne mode: 3 ( m=4, n=15) Eigan Mnc mode: d( M, ~ 1 5 )

-dW -400 --- y-axis in mm x-axis in rnm y-axis in mrn -40'

x-axis in mm

Eigan nliins mods: 6f nr2, n-12) Eigen Hfne mode: $( ma?, n+l5j

Fig. 5.1 Six eigen affine mode shapes of perfect cylindrical shell considered

for random modeling of imperfections (amplitudes enlarged by

50 times)

5.1.7 Mesh convergence study

In case of random geometrical imperfections modeling, since there is no

geometrical symmetry in the model, the complete cylindrical shell should be

considered for analysis. For the cylindrical shell, taken for study, Athiaman and

Palaninathan (20049, in their work, to model the imperfect cylindrical shell based on

experimentally measured geometrical imperfections, used eight node higher order

quadrilateral shell element of ABACUS with a mesh of 125 elements along

circumferential direction and 25 elements along longitudinal direction respectively.

Table 5.3 Mesh convergence study for eigen affme mode shape of m=5, n=15

But here, since the random modeling of imperfections is generated using

linear combination of eigen affine mode shapes consisting of more number of

circumferential and longitudinal half lobes, more number of elements are required.

And hence mesh convergence study is carried using eigen affine mode shape of m=5

and n=15 and the numerical results obtained from non linear bucking analysis is

tabulated in Table 5.3.

From the Table 5.3 it can be seen that for a mesh size of 200 x 37 and above

variation in BSR values are noticed only in the fourth digit fi-om decimal point and

hence for computational economy 200x37 mesh is used for further Analysis.

5.1.8 FE random geometrical imperfection models

Following the step explained in section 5.1.2 the random geometrical

imperfection models are generated either keeping RMS (Root Mean Square) value

of imperfections as 0.45rnm and 0.93mm or keeping maximum amplitude of

imperfection as 1.32m.m and 3.7mm.

Since, the eigen affine mode shapes are generated using Eq.$.l2)with-out

phase shift, there may be some bias on imperfection pattern at O=OO. To overcome

h s problem the origin for the Eqg.l2)(i.e., 0=0°), each eigen affine mode is given a

phase shifl of approximately 60" with respect to lower strength eigen affine mode

shape before combining the eigen affine mode shapes linearly.

Ra.ldorn d~stnbd!on d the nsd* No 9aX] Coln ihe models of RMS 50 45m

amplitude of imperfections in mm

(a)

Rar+!Jorn d'kttibiatcm d the mde Pi3 lOma ftm:he models af 2WSa 45me : > a

(4 (dl Fig. 5.2 Normal distribution of radial amplitudes of imperfections from all

64 random imperfection models (keeping RMS = 0.45mm) of n node

(a) 9000 (b)10000 (c) 12000 and (d) 18157 on the surface of the

cylindrical shell

Fig.5.2 shows the normal distribution of radial amplitudes of imperfections

from all 64 random imperfection models of some nodal points on the surface of the

cylindrical shell. Thus, it is ensured here that the models generated are random in

nature.

Model number :%with Model RWlS =0,4Bmrn . . . . . . . . . ....... . . . , .. : . . . . ' I . . . , .

4

z - coordhats fn mm €he& in radians

MacIel number :$with Model RlulS =OAtimm 3 ..................................................................... . . . . .

. I ( , . . . . .

. . . . . . ' : . ,

. . . . E f . 9 : . : ..... $. ' ... .:,,, .... : . . . . . . ' . . ,

, . ! . . .................................. . j.. ...... ............... ....,..... I . . . . .

, _ _ . , . . . . . - 4 - 9 - 2 - 1 a I z s 4

theta in radians

Modal number :Iwlth Modef WS =0.45mrn 2 r..,., ..................................................................................

Fig. 5.3 Development of amplitudes of imperfections in (a) isometric view @)

f xont view and (c) side view for random imperfection model No. 1 with RMS value of imperfections = 0.45mm

Model number :l5with Model RMS =ORfrrnm

2; - coordinate In mm theta in radians

Model number :Iswith Model RMS =0.45mm ..... ,... ..........................................................................

I I , I 8 , 4 - 3 - 2 - 0 1 2 3 1

theta in radians

Mode1 number tl5with Madial RlVlS t0~45mrn f.Sr .... ..'. .............................. ..................... ...........................

Fig. 5.4 Development of amplitudes of imperfections in (a) isometric view

(b) front view and (c) side view for random imperfection model

No. 15 with RMS value of imperfections = 0.45mm

Madel number :32wfth Model RMS =0,45rnm . . . . . . . . . . . . . .

E . ." . . .' . E

. , , , . . . . . . . . . . . ,

Model number :3Zwl& Model RliRS =#A!imm ........................................ .-. ....................................... $ . a

"4 -3 -2 -t O Z 2 3 4 th& in radians

. - .: ' -f.5i , . : + , I I I 0 m f W I J O 2 P b m m 3 ? 1 0

r - coordinate irr mm


(b) front view and (c) side view for random imperfection model

No. 32 with RMS value of imperfections = 0.45mm

Model number :33with Model RMS =OASrnrn

E E

z - coordinate in mm theta in radians

Nladsrl number r33wWt Modd M S =OASrnm .; ............................................................................

: : , i .

ModeX rmrnbsir : B W Model RMS =UASrnrn ..................... ....................................*............ . . , I. .


(b) $kont view and (c) side view for random imperfection model

No. 33 with RMS value of imperfections = 0.45mrn

Model number :5Owittr Model WS =.0.45rnrn

2 - coordinate in rnm 5 '4 theta in radians

Model number :50wrlth Madel W S =OASmm Zr

Modal number :5Ddtb Modal RMS =U.#rnm 2 r......,..,..,..,,,.., ..................................................................

-1.31 " f I ..1 0 5 0 I w I w 2 0 6 m o 3 0 0 m

r - coordinate in mm


@) front view and (c) side view for random imperfection model

No. 50 with R M S value of imperfections =0.45mrn

Model number : 6 4 W Model RMS =OA5mm

z - coordinate in mrn theta in radians

Model number :fi4with Model W S =0.45mrn .... 2r""""" "."." " . " " ' " " " ' " ",""""."" ............. " " ... "".

. . : ' . , , 1 . :


@) front view and (c) side view for random imperfection model

No. 64 with RMS value of imperfections = 0.45m

Madel number :IWh Model RMS =0,45mm Model number :A 5with Model RMS =0.45mm

Model number :32with Model RMS =0.45rnm Madel number :33wlth Model RMS =0.45mrtl

Model number :50with Model RMS =&45mrn Model number :64wHh Modef RMS =0.45rnrn

Fig. 5.9 Images of some random imperfect cylindrical shell models

(amplitudes enlarged by 50 times)

Figs. 5.3 to 5.8, show the development of amplitudes of geometrical

imperfections in isometric view, £i-ont view and side view of random geometrical

imperfection cylindrical shell models (called as random imperfection models)

numbered as 1, 15, 32, 33, 50 and 64 with mean of imperfections equal to zero and

the RMS value of imperfections equal to 0.45xnrn. From these figures it can be

clearly seen that random imperfection models 1 and 64, random imperfection

models 15 and 50 and random imperfection models 32 and 33 are mirror image

models to each other.

Fig.5.9 shows the images of random imperfection cylindrical shell models I,

15,32,33, 50 and 64 with RMS value of imperfections equal to 0.45mm.

5.1.9 Results and discussion

Table 5.4 compares the BSR values of pairs of mirror image random

imperfect cylindrical shell models keeping RMS value of imperfections equal to

0.45rnm and it is found that maximum deviation of 5.37% between BSR values of a

pair of mirror image random imperfect cylindrical shell models 24 and 41.

Fig. 5.10(a) shows the variation of BSR values with respect to model

numbers. The maximum and minimum values of BSR are 0.242349 and 0.229672

respectively for constant RMS value of 0.45mm. Fig.S.lO(b) shows variation of

maximum amplitude of imperfections in each model and only at two models

maximum amplitudes of imperfections are maintained as 1.59rnm and -1.59rnrn

respectively with respect perfect cylindrical shell geometry and all the other models

almost same maximum amplitudes of imperfections are maintained as shown in

Fig. 5.10(b). It can also be seen that the maximum amplitudes of imperfections

between models 1 and 32 are exactly minored between models 33 and 64. But no

such mirroring of BSR values is noticed in Fig.5.1 O(a).

0.25 BSR vs. Model No.

0.245

E 0.24

m 0.235

0.23

0.225 0 10 20 30 40 50 60

Model No.

Max.amplitude of imperfections vs. Model No. 1.7

1.2

0.7

PII 0.2 V) * -0.3

-0.8

-1.3

-1.8 0 10 20 30 40 50 60

Model No.

Fig. 5.10 (a) BSR value vs Model Number (b) Maximum amplitudes of

imperfections vs Model Number of all 64 random imperfect

cylindrical shell models keeping RMS value of imperfections =

0.45mm

The Fig. 5.1 1 (a) and (b) show the stifiess curves of two pairs of mirror

image random imperfect cylindrical shell models 21 and 41 and models 28 and 37

respectively. These stiffness curves indicates that there will be some variation on

load carrying capacity of the cylindrical shells even though the models have exact

mirror image imperfections. This is due to fact that change in stiffhess of cylindrical

shell occur on loading because of initial geomebical imperfections present in the

cylindrical shell. Further, it can be seen that the failure of cylindrical shell happens

only on reaching the limit load condition i.e., the slope of the stiffkess curve

becomes zero. Figs.5.12 (a) and (b) show von-Mises stress contours superimposed

on deformed (fictitious rnagnttude) cylindrical shell geometry at limit load condition

of a pair of mirror image random imperfect cylindrical shell models. It can be noted

that the plastic zones can be seen at multiple locations and also mostly on inward

projections on surface of cylindrical shell due to initial random geometrical

imperfections. And also it can be noted that the multiple plastic zones can be seen on

the sunface area of the cylindrical shell in between supporting edge and just above

half the height of the cylindrical shell. Further, it can be noted that no formation of

plastic zones near the loading edge.

In .the Ref. Athiman and Palaninathan (2004), the experimental and

numerical buckling strength of the test cylindrical shell (taken for study in the

present work) with RMS value of imperfections equal to 0.45mm were 321W and

398.9kN respectively. The numerical buckling strength predicted was higher than

the experimental buckling strength by 24.2%. But in the present work of random

modeling of imperfections the maximum and minimum of the buckling strength

predicted from all random imperfect cylindrical shell models are 277.21dV and

262.61kN respectively. Both values are lower than the experimental buckling

strength by 13.64% and 1 8.19% respectively. The predicted buckling strength are

lower than the experimental buckling strength, because of the material model data of

304 stainless steel material whch is used in all the analyses is taken fiom Ref.

Hautala, since the actual material model data of the test cylindrical shell is not given

in the Ref. Athiman and Palaninathan (2004).

Table 5.4 Comparison of BSR values of pairs of mirror image random

imperfect cylindrical shell models keeping RMS value of

imperfections = O.45mm

Model No.

I 2 3 4 5 6 7 8 9 10 11 12 13

BSR

0.240828 0.239674 0.2391 87 0.240144 0.242335 0.241264 0.239317 0.241 71 7 0.230866 0.229672 0.230786 0.23 1595 0.236946

Model No. 64 63 62 61 60 59 58 57 56 55 54 53 52

BSR

0.237152 0.236058 0.236646 0.234715 0.23 1842 0.232559 0.233168 0.232094 0.240457 0.239025 0.240868 0.240846 0.23653

Difference

0.003676 0.00361 6 0.002541 0.005429 0.01 0493 0.008705 0.006149 0.009623 -0.009591 -0.009353 -0.01 0082 -0.009251 0.000416

% deviation

1.550060721 1.53 1826924 1.073755736 2.313017915 4.52592714 3.74313615

2,637154326 4.146 1649 16 -3.98865494 -3.91297981 -4.1 8569507 -3.841 04365 0.17587621

Dlmk'?& vs Load DLgalecmeRt Vs Load

DsspLacrmem rn mm

Fig. 5-11 Comparison of stiffness curves of pairs of mirror image random

imperfection models keeping RMS = 0.45mm a) model 24 with its

mirror image model 41 (b) model 28 with its mirror image model 37

~ / m r n ~

Fig. 5.12 Comparison of von Mises stress contours of (a) model 24 with (b) its

mirror image random imperfection model 41 keeping RMS value of

imperfections = 0.45mm


imperfect cylindrical shell models keeping RMS value of imperfections equal to

0.93 mm. The maximum difference between BSR values of a pair of mirror image

random imperfect cylindrical shell models 2land 44 is 5.37%.


imperfect models keeping RMS value of imperfections = 0.93mm

28 29 30 31 32 -

0.1971 89 0.197496 0.191988 0.198091 0.190478

Maximum %deviation of BSR with respect to its mirror image model

37 36 35 34 33

5.3674419

0.194085 0.198023 0.195874 0.1 98991 0.194396

0.003 104 -0.00053 -0.00389 -0.0009

-0.00392

1.5992993 -0.266 13 1

1 -1.983928 -0.452282 -2.01 5474

- --

Fig. 5.13(a) shows the variation of BSR values of different models. The

maximum and minimum values of BSR are 0.203 886 and 0.1 86755 respectively for

constant RMS value of. 0.93mm. Fig 5.1 3(b) shows the variation of maximum

amplitudes of imperfections for different models. It can be seen that maximum

amplitude of imperfection in each model is almost same but only in two models

maximum amplitudes of imperfections are exactly maintained as 3 . 2 m and

-3.2mrn respectively. It can also be seen that the maximum amplitudes of

imperfections between models 1 and 32 is exactly mirrored between models 33 and

64. But no such mirroring of B SR values is noticed in Fig. 5.1 3 (a).

BSR vs. Model No. 0.205

0.2 Ri g0.195

0.1 9 Y I 0.185 ! I I I I I I

0 10 20 30 40 50 60 Model No.

Max.amplitude of imperfections vs Model No. 4

'5; 3 4 . E 2 = @ I .z c

- w

I I I 1 I I

0 10 20 30 40 50 60

Model No.

Fig. 5.13 (a) BSR value vs. Model number (b) Maximum amplitude of

imperfections vs. Model Number of all 64 random imperfect

cylindrical shell models keeping RMS value of

imperfections=0.93mm

The Figs.5.14 (a) and (b) show the stiffness curves of two pairs of mirror

image random imperfect cylindrical shell models 14 and 51 and models 21 and 44

From these graphs it can be noted on reaching the limit load condition

of stiffness of the shells becomes zero.

D t k p l a c m Ys Load

2 w r ~ -model 14 I

Fig. 5.14 Comparison of stiffness curves of pairs of mirror image random imperfect models keeping RMS = 0.93mm (a) model 14 with its mirror image model 51 (b) model 21 with its mirror image model 44

f l Supporting edge

0 25 50 75 100 125 150 205 260

~ / r n r n ~

Fig. 5.15 Comparison of von Mises stress contours of (a) model 21 with (b) its mirror image random imperfect model 44 keeping RMS value of imperfections = 0.93mm

Fig.5.15 shows the von Mises stress contours superimposed on deformed

(fictitious magnitude) cylindrical shell geometry of a pair of mirror image random

imperfection models at limit load condition. It can be noted that the plastic zones can

be seen at multiple locations and also mostly on inward projections on surface of

cylindrical shell due to initial random geometrical imperfections. And also it can be

noted that the multiple plastic zones can be seen on the surface area of the

cylindrical shell in between supporting edge and just above half the height of the

cylindrical shell. Further, it can be noted that no formation of plastic zones near the

loading edge.

In the Ref. Athiannan and Palaninathan (2004), the experimental and


present work) with RMS value of imperfections equal to 0.93rnm were 309.8kN and

42 1.6kN respectively. The numerical buckling strength predicted was higher than

the experimental buckling strength by 36.09%. But in the present work of random

modeling of imperfections the maximum and minimum of the buckling strength

predicted are 2 13.62k.N and 233.22kN respectively. Both values are lower than the

experimental buckling strength by 3 1.05% and 24.72% respectively.


imperfect cylindrical shell models keeping maximum amplitudes of imperfections of

the all the models equal to either 1 . 3 2 m or -1.32 nun with respect to perfect

cylindrical shell geometry. The maximum difference between BSR values of a pair

of mirror image random imperfect cylindrical shell models 22 and 43 is 5.9%.

Fig.5.16 (a) shows the variation of BSR values of different models. The

maximum and minimum values of B SR are 0.253446 and 0.234542 respectively for

maximum amplitudes of imperfection of all the models equal to either 1.32mm or - 1.32 mm as shown in Fig 5.16@). It can also be seen that the maximum amplitudes

of imperfections between models 1 and 32 is exactly mirrored between models 3 3

and 64. Similarly the Fig 5.1 6(c) shows the variation of RMS value of imperfections

for different models. It can also be seen that the RMS of imperfections between

models 1 and 32 is exactly mirrored between models 33 and 64. The maximum and

minimum values of RMS value of imperfections are 0.412n-m and 0.372mm

respectively. But no such mirroring of BSR values is noticed in Fig. 5.16(a).

Table5.6 Comparison of BSR values of pairs of mirror image random

imperfect models having maximum amplitude of imperfections

either +1.32mm or -1.32mm

Model Model 1 No. 1 BSR 1 1 BSR 1 Difference 1 % deviation 1

32 1 0.242867 1 33 1 0.247128 1 -0.004261 Maximum %deviation of BSR with respect to its

mirror image model

-1.724207698

5.895514841

BSR vs Model No.

0 10 20 30 40 50 60 Model No.

Maximum amplitude af imperfections vs.Model No.

0 10 20 30 40 50 60 Model No.

RMS vs. Model No. 1 0.43

0.35

0 I 0 20 30 40 50 60 Model No.

Fig. 5.16 (a) BSR vs. Model number (b) M a d u r n amplitude of imperfections vs. Model Number (c) RMS value of imperfections vs. Model Number of all 64 random imperfect cylindrical shell models having maximum amplitude of imperfections either + 1.32mm or -1.32mm.

The F i g 5 17 (a) and (b) shows the stiffness curves of two pairs of mirror

image random geometrical imperfect cylindrical shell models 14 and 51 and models

21 and 44 respectively. From these graphs at can be noted on reaching the limit load

condition the stiffness of the shells becomes zero.

Di.place~#ent Yr Load

1;" /

% 0.05 0.f 0.75 0.2 0.25 0.3 035 0.4 I

Displacement in mm

Fig. 5.17 Comparison of stiffness curves of pairs of mirror image random imperfect models keeping maximum amplitude imperfections either cl.32mm or -1.32mm (a) model 11 with its mirror image model 54 (b) mode1 22 with its mirror image model 43

0 25 50 75 100 125 150 205 260

~ / m m ~

Fig. 5.18 Comparison of von Mises stress contours of (a) model 22 having maximum amplitude of imperfections = 1.32mm with (b) its mirror image random imperfect model 43 having maximum amplitude of imperfections = -1.32mm


(fictitious magnitude) cylindrical shell geometry of a pair of mirror image models at

limit load condition. It can be noted that the plastic zones can be seen at multiple

locations and also mostly on inward projections on surface of cylindrical shell due to

initial random geometrical imperfections. And also it can be noted that the multiple

plastic zones can be seen on the surface area of the cylindrical shell in between

supporting edge and just above half the height of the cylindrical shell. Further, it can

be noted that no formation of plastic zones near the loading edge.

In the Ref Athiman and Palaninathan (2004), the experimental and


present work) with maximum amplitude of imperfections equal to 1.32rnm were

32 1 kN and 3 98.9kN respectively. The numerical buckling strength predicted was

higher than the experimental buckling strength by 24.27%. But in the present work

of random modeling of imperfections, the maximum and minimum of the buckling

strength predicted keeping maximum amplitude of imperfections either 1.32mrn and

-1.32mrn are 289.9kN and 268.28kN respectively. Both values are lower than the

experimental buckling strength by 9.06% and 16.42% respectively.


imperfect cylindrical shell models keeping maximum amplitudes of imperfections of

the all the models equal to either 3 . 7 m or -3.7mm with respect to perfect

cylindrical shell geometry. The maximum difference between BSR values of a pair

of mirror image random imperfect cylindrical shell models 2 1 and 44 is 5.3 1 %.

Fig.5.19 (a) shows the variation of BSR values of different models. The

maximum and minimum values of B SR are 0.205688 and 0.1 90757 respectively for

constant maximum amplitude of imperfection of all the models equal to 3.7mm. Fig

5.19(b) shows that the maximum amplitudes of imperfections of the all the models

equal to either 3 . 7 ~ or -3.7mxn. It can also be seen that the maximum amplitude

of imperfections between models 1 and 32 is exactly minored between models 33

and 64.

Table5.7 Comparison of BSR values of pairs of mirror image random

imperfect models having maximum amplitude of imperfections

either +3.7mm or -3.7mm

BSR vs. Model Number 0.21

0.205

PL 0.2

V)

0.195

0.19 -

0.185 1

0 10 20 30 40 50 60

Model Number

Maximum amplitude of imperfection v s Model Number

5 4

L 3

z Z 2 'C, = a u r ' C .- 5 5 0 6 2 - 1 't i g-2

= ,E-3 -4

-5

0 10 20 30 40 50 60

Model Number

I RMS vs. Model Number 1

Model Number

Fig. 5.19 (a) BSR vs. Model number @) Maximum amplitude of imperfections

vs. Model Number (c) RMS value of imperfections vs. Model Number of all 64 random imperfect cylindrical shell models having

magimum amplitude of imperfections either + 3.7mm or -3.7mm.

Similarly the Fig 5.19(~) shows the variation of RMS value of imperfections

for different models. It can also be seen that the RMS of imperfections between

models 1 and 32 is exactly mirrored between models 33 and 64. The maximurn and

minimum values of RMS value of imperfections are 1.16mm and 1 . 0 4 m

respectively. But no such mirroring of BSR values is noticed in Fig. 5.19(a).The

Fig.5.20 (a) and @) show the stiffness curves of two pairs of mirror image random

geometrical imperfect cylindrical shell models 20 and 45 and models 21 and 44

respectively. From these graphs it can be noted on reaching the limit load condition

the stiffhess of the shells becomes zero.

DispEQeemult Vs Load

I

0.1 0.2 0.3 0.4 0.5

Fig. 5.20 Comparison of stiffness curves of pairs of mirror image random

imperfect models keeping maximum amplitude imperfections either

+3.7mm or -3.7mm (a) model 17 with its mirror image model 58

(b) model 13 with its mirror image model 52


(fictitious magnitude) cylindrical shell geometry of a pair of mirror image random

geometrical cylindrical shell models 21 and 44 at limit load condition. It can be

noted that the plastic zones can be seen at multiple locations and also mostly on

inward projections on surface of cylindrical shell due to random geometrical

imperfections. And also it can be noted that the multiple plastic zones can be seen on

the surface area of the cylindrical shell in between supporting edge and just above

middle of the cylindrical shell. Further, it can be noted that no formation of plastic

zones near the loading edge.

Fig. 5.21 Comparison of von Mises stress contours of (a) model 21 having

maximum amplitude of imperfections = 3.7mm with (b) its mirror

image random imperfect mode1 44 having maximum amplitude of

imperfections = -3.7mrn

Fig. 5.22 BSR vs. RMS value of imperfections of random geometrical

imperfect cylindrical shell models

BSR vs RMS value of imperfections

0.25 -

0-24 - 0.23

0.22 - V) m 0.21 -

0.2 -

0.19 -

0.18

I

,

! 0.4 0.5 0.6 0.7 0.8 0.9 1

RMS value of imperfections

Fig 5.22 shows BSR vs RMS value of imperfections and from this figure, it

can be seen that as the RMS value of imperfections increases the range over which

h e BSR value distributed also increases.

Fig.5.23 BSR vs. Maximum amplitude of imperfections of random

geometrical imperfect cylindrical shell models

BSR vs Amplitude of imperfections

In the Ref. Athiannan and Palaninathan (2004), the experimental and


present work) with maximum amplitude of imperfections equal to 3.7mm were

309.8kN and 421.6kN respectively. The numerical buckling strength predicted was

higher than the experimental buckling strength by 36.09%. But in the present work

of random modeling of imperfections the maximum and minimum of the buckling

strength predicted keeping maximum amplitude of imperfections either 3.7rnm or

-3.7mm are 235.28kN and 218.2kN respectively. Both the values are lower than the

experimental buckling strength by 24.05% and 29.57% respectively.

0.26

0.24 -

PI: m 0.22 - m

0.2 -

0.18

Fig. 5.23 shows BSR vs maximum absolute amplitude of imperfection and

from this figure it can be seen that as the maximum absolute amplitude of

imperfection increases the range over which the BSR value distributed decreases.

i I

I I 1 ,

1 1.5 2 2.5 3 3.5 4

Amplitude of imperfections in mm

5.2 STUDY %ITa, BUCKLING BEHAVIOR OF THIN WIDE PANEL

STRUCTURES WITH MIRROR IMAGE RANDOM GEOMETRICAL

IMPERFECTIONS

In case of h n cylindrical shells with random geometrical imperfections, a

pair of mirror image random imperfection models predicts different buckling loads

(or behaviors). The reason for this effect is initially thought as due to

(1) circumferential interactions of imperfections present in the cylindncal shell

because of the closed structural form of cylindrical shells and (2) radius of curvature

of thin cylindncal shell structure. Hence, in this section, to verify whether h s effect

is due to circumferential interactions of imperfections present in the cylindrical shell

because of the closed structural form of cylindrical shells, a thin wide panel structure

is taken for study. The wide panel with simply supported boundary conditions along

the longitudinal edges can be considered as widely supported stringer stiffened

cylindrical shell (Spagnoli and Chryssanthopoulos 1999). This panel model can

isolate the circumferential interactions due to imperfections present in the cylindrical

shell due to its closed structural form.

The dimensions of thin wide panel structure taken for study are radius

= 350mrn, length =340rnrn, thickness= 1.25mm and angle subtended at the centre of

curvature /3=90° (this wide panel can be considered as widely stiffened cylindrical

shell with four stringer stiffened cylindrical shell placed 90" apart). Only one quarter

of the cylindrical shell is taken as panel for study so as to have same radius of

curvature for cylindrical shell and panel taken for studies. Stainless steel material

properties are assumed. The simply supported boundary conditions and loading

conditions applied on the models are shown in Fig.5.24.

loading

Fig. 5.24 Boundary conditions and loading conditions applied on thin wide

panel models

Here also She1193 elements are used for analysis. From the mesh

convergence study by comparing the buckling strength obtained fiom analytical

solution using Eq.(3.17)and FE eigen buckling strength of thin wide panel taken for

study, a mesh of size 50 x 40 elements along circumferential and longitudmal

directions respectively is selected for all analysis. Table 5.8 shows the comparison

of buckling strength of perfect thin wide panel fiom analytical solution and

F.E. eigen buckling analysis.

Table 5.8 Comparison of analytical solution with FE eigen buckling result of

perfect thin wide panel

Buckling strength in N / % deviation with i respect to analytical

Analytical First eigen buckling solution

solution analysis

The six F.E eigen affine mode shapes shown in Fig.5.25 obtained from eigen

buckling analysis are used to generate pairs of thin panel models of mirror image

random geometrical imperfections by adopting the procedule explained

section 5.1.2.

(a) Mode 1 (b) Mode 2

(c) Mode 3 (d) Mode 4

(e) Mode 5 (0 Mode 6

Fig. 5.25 Six eigen affiie mode shapes considered for random modeling of

imperfections of a curved thin panel taken for study

(a) model 1 (b) model 64

Fig. 5.26(a)&(b) A pair of thin panel models with mirror image random

geometrical imperfections

Max.amplitude vs model No. 1.5 s

-1.5 4 I 0 10 20 30 40 50 60

Model number

BSR vs Model No. 1 om

Model Nuwer

Fig. 5.27 (a) M a h u m amplitude of imperfections vs. model number (b) BSR

vs. Model number of thin panel models with random geometrical

imperfections

A sample of a pair thin panel models with random geometrical imperfections

are shown in Fig.5.26. The Fig 5.26 (a) shows that maximum amplitudes of

imperfections of the models generated are maintained with in 1.32mm with RMS

value of imperfections as 0.3685mm. Further, in Fig 5.27 (a) it can be fiuther noted

that the maximum amplitudes of imperfections between models 1 and 32 is exactly

mirrored between models 33 and 64. But no such mirroring of BSR values is

noticed in Fig. 5.27@). Table 5.9 and Fig 5.27(b) compare the buckling strengths of

the pairs of mirror image random imperfection models. From the Table 5.9 it can be

seen that the maximum deviation in BSR value noticed between model 10 and

model 55 is 5.41931%. Here also, different buckling strengths or behaviours can be

noticed between a pair of mirror image random imperfect thin wide panel models.

Hence, fi-om the analysis of th.m panel structure with random geometrical

imperfections it can be concluded that the different buckling strengths or behaviours

noted between a pair of mirror image random imperfect cylindncal shell models

may not be primarily due to circumferential interactions of imperfections present in

the cylindrical shell because of closed structural form of the cylindrical shell.

Table 5.9 Comparison of BSR values of pairs of thin panel models with their

mirror image random imperfect models

Model No. 1 2 3 4 5 6 7 8 9 10

I-

11 12 13 14 15 16 17 18 19 20 2 1 22 2 3 24 25

BSR 0.258714 0.261433 0.246458 0.252233 0.252233 0.246458 0.26 1433 0.258714 0.2574 1 6 0.260022 0.248006 0.250994 0.250994 0.248006 0.26001

0.25741 6 0.261433 0.258714 0.25223 3 0.246458 0.246458 0.252233 0.25 87 14 0.261433 0.25991

Model No. 64 63 62 6 1 60 5 9 58 5 7 5 6 55 54 5 3 52 5 1 50 49 48 47 46 45 44 43 42 41 40

26 27 2 8 29 30 3 1 32

BSR 0.258865 0.256745 0,254914 0.250363 0.250471 0.2549 14 0.256745 0.258865

- - -

0.250293 0.246655 0.255607 0.2581 16 0.2581 16 0.255607 0.246655 0.250293 0.256745 0.258865 0.250471 0.2549 14 0.254914 0.250363 0.258865 0.256745 0.246655

% deviation -0.05833 1.825936 -3.3172

0.746915 0.703475 -3.3172

1.825936 -0.05833 2.845865 5.41931 -2.97371 -2.75922 -2.75922 -2.97371 5.414445 2.845865 1.825936 -0.05833 0.703475 -3.3172 -3.3 172

0.74691 5 -0.05833 1.825936 - 5.373903

0.2574 16 0.250994 0.248006 0.248006 0.250994 0.25741 6 0.2598 12

Maximum %deviation of BSR with respect to its mirror image model

39 38 37 36 35 34 3 3

5.41931

0.250293 0.258 11 6 0.255607 0.255607 0.258 11 6 0.250293 0.246655

2.845865 -2.75922 -2.97371 -2.97371 -2.75922 2.845865 5.334171

5.3 STUDY ON BUCKLING BEHAVIOR OF THIN PLATE

STRUCTURES WITH MIRROR IMAGE RANDOM GEOMETRICAL

IMPERFECTIONS

In the previous section, buckling behaviour of thin wide panel models with

mirror image random geometric imperfections are studied in detail and it is also

proved that the variation of buckling strengths between a pair of thin cylindrical

shell models with random geometrical imperfections may not be primarily due to

circumferential interactions of impexfections present in the cylindrical shell because

of closed structural form of the cylindrical shell. Hence, in this section, to verify

that whether this effect is due to radius curvature of cylindrical shell, a thin plate

structure is taken for study even though thin plate structures are insensitive for

geometrical imperfections (Bushnell 1985).

Thickness = 1.25 mm \ uniform edge ROTZ=O (at all nodal points) displacement loading

Fig. 5.28 Geometry, boundary conditions, and loading conditions used in

buckling analysis of a thin plate (not to scale)

The dimensions of the thin wide square plate taken for study are length (lm)

x width (lm) x thickness (1.25mm). Here also same stainless steel material

properties used for analysis. m e boundary conditions and loading conditions used

for analysis of thin plate structure are shown in Fig.5.28.

5.3.1 F.E. eigen buckling analysis of perfect thin plate

Here also she1193 element is used for analysis. After conducting the mesh

convergence study, a mesh of 50 x 50 elements is used along longitudinal and

transverse directions of plate. To validate numerical results, F.E eigen buckling

analysis results are compared with analytical solution with its mode shapes obtained

from ECJ@. 14 Fig 5.29 shows the buckling mode shapes and in Table. 5.10 both FE

eigen buckling analysis results and analytical solutions are compared. These six

affine mode shapes are used to generate b n plate models with random geometrical

imperfections.

5 ElGEN MODE SHAPES FOR PERFECT PLATE

Fig. 5.29 Eigen buckling mode shapes of perfect thin plate structure taken for

study

Table 5.10 Comparison of F.E. eigen analysis results with analytical solutions

of perfect thin plate

I ~uckliug strength in N I I

5.3.2 F,E nonlinear buckling analysis of imperfect planes with random

geometrical imperfections

By adopting the procedure explained in section 5.1.2, 32 pairs of thin plate

models with mirror image random geometrical imperfections are generated. A

sample of a pair of thin plate models with mirror image random imperfections are

shown in Fig.5.30.

% deviation Mode No.

(a) model I (b) model 64

Fig. 530(a)&@) A pair of thin plate models with mirror image random

FE buckling eigen analysis

imperfections (50 times enlarged)

Analytical solutions

The Fig 5.3 1(a) shows that maximum amplitudes of imperfections of all the

models generated are maintained with in & 1.25rnm. But the RMS value of

imperfections varies between 0.3327mrn and 0.3064mm. This variation in RMS

values is because of the first affine mode selected does not possess zero mean.

Further, in Fig 5.31(a) it can be further noted that the maximum amplitudes of

imperfections between models 1 and 32 is exactly mirrored between models 33 and

64. The numerical results obtained from the nonlinear buckling analysis are shown

in Table 5.1 1. Here BSR values are calculated with respect to sixth eigen buckling

strength value of perfect thin plate. Since th.m plates are insensitive to geometrical

imperfections, BSR values are greater than one (Featherston 2003). Unlike the

buckling behaviour of a pair of mirror image geometrical imperfect cylindrical

shells, a pair of mirror image geometrical imperfect thin plates give out same

buckling strength. This can be verified in the Fig 5.3 1 (b).

Max.ampliiude of imperfections vs Model No.

From Table 5.1 1 and Fig. 5.3 1 @), it can be seen that pairs of mirror image

random geometrical imperfection thin plate models have same BSR values. Hence it

can be concluded that different buckling loads predicted by a pair of cylindrical shell

models with mirror image random imperfections are not significantly due to closed structural form of the cylindrical shell and this may be due to radius of curvature of

thin cylindrical shell.

BSR vs Model NO.

Model Number Model No.

(a) (b)

Fig. 5.31 (a) Maximum amplitude of imperfections vs. Model number (b) BSR

vs. Model number of models of a thin plate with random geometrical

imperfections


imperfect thin plate models

From the analysis of thin plate structure and thin panel structure with random

geometrical imperfections it can be concluded that the different buckling strengths

or behaviours between a pair of mirror random imperfection models are significantly

due to curvature present in the cylindrical shell and are not significantly due to

closed structural form of cylindrical shell.

5.4 STUDY ' - ON ' BUCKLING BEHAVIOR OF THIN CYLIM)RICAL

SHELLS WITH MIRROR IMAGE AXISYMMETRIC

IMPERFECTIONS

5. \ In sectiog it is found that a pair of cylindrical shells with mirror

image random geometrical imperfections showed different buckling strengths or

behaviours. To check whether such behaviour is possessed by cylindrical shells of

other types of imperfections such as axisymmetric and asymmetric imperfection

patterns or not, in this section ' cylindrical shells with axisymmetric imperfection

patterns are studied in detail and in the next section the cylindrical shells with

asymmetric imperfection patterns are studied in detail. The aim of this part work is

to compare the buckling behaviour of a pair of cylindrical shells with misymmetric

imperfections and their mirror image axisylnmetric imperfections. For this study

also the same short stainless cylindrical shells of diameter 700mrn, length 340rnm

and thickness 1.25mm is used for analysis. Seven pairs of mirror image

axisyrnmetric imperfection models are generated and their non linear buckling

malysis results are tabulated in Table 5.12,

In all these cases, the maximum amplitudes of imperfections are made equal

to the thickness of cylindrical shell and also stainless steel material behavior is

assumed for analysis. The simply supported boundary conditions are applied on both

top and bottom edges of the cylindrical shell and uniform axial displacement loading

is applied on one edge and the other edge is restrained to move along axial direction.

A mesh of 200x37 elements along circumferential and longitudinal directions

respectively is used model this axisymmetric imperfection patterns.

Longitudinal half lobes

(a) m=4 & Max. amplitude = t

ongitudinal

Reference surface -b

half

V (c) m-5 & Max. amplitude = t

half

(b) m= 4 &Max. amplitude = -t

(d) m= 5 &Max. amplitude = -t

Fig. 5.32 Mirror image axisymmetric imperfection patterns formed on the

surface of the cylindrical shell

Fig. 533 FE model of cylindrical shell with axisymmetric imperfections (m=6,

n=O) (50 times enlarged)

From th~s Table 5.12, it can be noted that when even numbers of half lobes

are present on the cylindrical shell, both models and its mirror image imperfection

models give out same buckling strength because the number of inward half lobes is

equal to number of outward lobes as explained in the Fig.5.32 (a) & (b). '

Fig.5.33 shows a sample of cylindrical shell model with axisymmetric

imperfections. For the models represented in Fig 5.32(c), the nurnber of inward half

lobes is one number less than that of number of outward half lobes and where as in

its mirror image axisymmetric imperfection model represented in Fig.5.32(d), the

number of inward half lobes is one number more than that of number of outward

half lobes. When odd number of half lobes except single lobe (which satisfies the

Koiter's theory with 0.16% deviation with its mirror image model) are present on

the cylindrical shell both models and its mirror image imperfection models give out

different buckling strengths as shown in Table 5.12 and also it can be further noted

that whenever the middle half lobe is of inward half lobe the buckling strength of

models are less than its mirror image imperfection model. For example models

(i) m=3, a = +t, (ii) m=5, a = -t (iii) m =7, a = +t, have buckling strength less than

its mirror image imperfection models.

Table 5.12 Comparison of BSR values of pairs of mirror image axisymmetric

imperfection models

Number of longitudinal lobes

Maximum amplitudes of imperfections (a)

BSR

When even numbers of longitudinal half lobes are present in the cylindrical

both the models and its mirror image models give out same buckling strength.

~ n d when odd numbers of longitudinal half lobes are present in the cylindrical

shells, both the models and its mirror image models give out different buckling

Further when the central half lobe is of inward half lobe, the reduction in

buckling strength is more than that of outward central half lobe.

5.5 STUDY ; ON ' BUCKLING BEHAVIOR OF THIN CYLINDRICAL

SHELLS WITH MIRROR IMAGE ASYMMETRIC

IMPERFECTIONS

In this section, buckling behavior of thin cylindrical shells with

mirror image asymmetric imperfection patterns are studied in detail. Here also,

material (stainless steel with kinematics strain hardening), size of cylindrical shell

(dia.7001nm x length 340mm x thickness 1.25mm), boundary conditions (simply

supported boundary conditions at both the edges) and loading conditions (uniform

axial edge displacement loading at one edge and the other edge is restrained for axial

movement) are assumed for this study as assumed or taken for study in the case

random geometrical imperfection modeling of thin cylindrical shell.

Here, the eigen affine mode shapes used for modeling of random geometrical

imperfections are selected for study, since these mode shapes are of asymmetric

geometrical imperfection forms. In all these cases, the maximum amplitude of

imperfections is made and equal to the thickness of cylindrical shell. Mesh size of

200 x 37 elements (she1193) along circumferential and longitudinal directions of

cylindrical shell are used in nonlinear buckling analysis. The numerical results

obtained from nonlinear buckling analysis are tabulated in Table 5.13.

Table 5.13 Comparison of buckhg strength ratios of cylindrical shells with

mirror image asymmetric imperfections

From the above results, it can be concluded that cylindrical shells with mirror

image asymmetric imperfection patterns will have same buckling strength because

the mirroring of asymmetric imperfection pattern is nothing but rotating the model

with imperfections by an angle of (90/n), where n= number of circumferential half

lobes. And also from the Table 5.1 3, it can be seen that as number of longitudinal

half lobes increases, buckling strength decreases.

Longitudinal half lobes (m)

1

2

3

4

5

7

5.6 RELIABILITY ANALYSIS

- .. The main aim of this -

' , -;, section is to determine the reliability of the

cylindrical shells with random distributed geometrical imperfections using the

simple reliability method described in the following paragraphs.

Circumferential half lobes (n)

9

12

14

15

15

13

Reliability is defined as the probabilistic measure of assurance of

performance of a design in its intended environment. Various methods have been

Maximum amplitude of

imperfections (a) a = t

a = -t

a = t

a = -t

a = t

a = -t

a = t

a = -t

a = t

a = -t

a = t

a = -t

BSR

0.32522

0.32522

0.22486

0.22486

0.21714

0.21714

0.2 1874 ".. 0.21 874

.- 0.21 360

0.21366

0.20607

0.20606

proposed for calculating the reliability of a system. Some of the more interesting

methods are the computational methods and in particular the second-moment

The most common of the second-moment methods is the first-order

second-rnoment method, which models the response of a system at a point using a

first order surface, or plane. In this work, Jirst-order second-moment method is

adopted. The underlying meaning of each part of the name is as follows. First order

means only first-order terns in the Taylor series expansion are used. Second

moment means only the first and second moments of a random variable are needed

which are more commonly referred as the mean and variance. Mean value means the

Taylor series is expanded about the mean values

5.6.1 Concepts of reliability

The concept of probability of failure is best described by considering the

specific example of strength (S) versus load (L). Failure happens when load is

greater than or equal to strength, or when S-L 5 0. In reliability analysis, the

variables S and L are treated as continuous random variables with probability

density functions as shown in Fig. 5.34 the strength and load are statistically

independent, then the expression for the probability of failure is

where FL(s)is the cumulative distribution of L at s and fs(s) is the probability

density of S at s.

A Performancefitnetion or Limit statefunction can be defmed as

G(x)= s - L (5.14)

where S and L are independent normally distributed random variables.

Here, the limit state corresponds to boundary between desired and undesired

performance, i.e., at g=O. If g 2 0, the structural system will be safe (desired

p&xmxince); if g < 0 the structure is unsafe (undesired performance). The

probability of failure, Pf , is equal to the probability that the undesired performance

will occur, mathematically, this can be expressed in terms of performance filnction

as

S o r L

Fig. 5.34 Probability distributions of strength and load

Since the safety margin G(X) is a h d i o n of two independent normally

distributed random variables, G(X) itself is a normally distributed random variable

with probability density function fG(g). The mean value of G(X) is given by

The standard deviation of G(X) is given by

The probability of failure is given by

is represented by the cross-hatched area shown in Fig. 5.35. If P is the number of

standard deviations CJG from the mean value p~ to the failure region, then failure

occurs when

or when

Substituting ~q.6.19)into E&. 12 the probability of failure becomes

where Q(P) is the standard normal cumulative density h c t i o n evaluated at P. The

quantity P is often referred to as the reliability or safety index.

Fig. 5.35 Probability density function for the safety margin G

This first-order reliability method offers the best approach to surmount

deterministic inherent deficiencies. It is the simplest, most expedient, and the most

developed and familiar of all reliability methods. Because, the fist-order reliability

is restricted to normal probability distributions, the approach of normalizing all

skewed distributions, leads to the adoption of the first-order reliability method by

Verderaime (1994). This is described below with a simple example. In this

technique, only half of the distribution data is used to construct a symmetrical

(normal) distribution. After converting the skewed strength distribution into normal

distribution, the reliability of the given load can be obtained using ~qf.t.19) and

~.20],taking p ~ = given load and a~ = 0.

This method'has both advantages and disadvantages in structural analysis.

Among its advantages,

1. It is easy to use.

2. It does not require knowledge of the distributions of random

variables,

Among its disadvantages,

1. Results are inaccurate if the tails of the distributions functions cannot

be approximated by normal distribution.

2. The value of reliability index depends on the specific form of the

limit state knction.

5.6.2 Conversion from skewed distribution to normal distribution

For example, let us consider a tapered round shaft subjected to torsional

loading. The actual load variation on the shaft can be plotted into frequency

distribution, or probability histogram, as shown in Fig.5.36. The base of the

histogram is bounded by successive and equal ranges of measured values, and the

heights represent the number of observations (frequency) in each range.

Distribution mean, p = 14 #%mm

Fig. 5.36 Actual distribution of load

Because the greater torque side (right side) defines the worst load case, only

data from the shaded right side are used to calculate the normalized distribution

variables. Using the ~ ~ ~ 6 . 2 2 ) .

the square of deviation of each column is determined, where n is £i-equency in that

c~lumn of histogram, xi is value of random variable in that column.

Therefore, Total Square of deviation in the shaded region is

1 x 8 (14.0-14.0) = 0

2 x 7 (14.5-14.0)~= 3.5

2 x 4 (15.0-14.0) = 8.0

2 x2 (15.5-14.0) * = 9.0

2 x1 (16.0-14.0) 2 = 8.0

2 = 28.5

sample size; Cn = 8+2 (7+4+2+1) = 36

.*.variance 2 = 28.5 /35 = 0.81

and standard, deviation a = 0.90

Similarly, for strength variation, the lower strength side (left side) defines the

worst strength case; only data fiom the shaded left side are used to calculate the

normalized distribution variables. Fig 5.37 shows an example of strength frequency

distributions of the material of a structural member and this skewed distribution is

converted into normal distribution as explained above taking the mode of the

strength distribution as the mean of the equivalent normal distributiongTo obtain the

right side of the normal distribution, the left side distribution is mirrored about the

mode. Thus, the equivalent normal distribution of strength is obtained.

Fig. 5.37 Actual distribution of strength

5.6.3 Reliability calculations

By adopting the procedue explained above, the reliability calculations are

carried out using the BSR values of cylindrical shells with random geometrical

imperfections given in Table 5.6 i.e., for the case of maxirnum amplitude of

imperfections = i. 1.32m.m.

0% BSR Mean of distribution = 0.2443 Mean of distribution = 0.2467 Mode of distribution = 0.2467 Mode of distribution = 0.2467 S. D of distribution = 0.0046 S.D of distribution = 0.0059

Fig. 5.38 (a) Actual skewed strength distribution (b) Equivalent normal

distribution of strength ratio for maximum amplitude of

imperfections = 132 mm

From this population of 64 models, the actual skewed strength distribution is

shown in Fig 5.38 (a). Since the normal distribution shape is the simplest, best

developed, most known and expedient (Verderaime 1994), the skewed strength

distribution is converted into a equivalent normal distribution using the method

suggested in Verderaime (1 994). According to this method, the mode of the strength

distribution is taken as the mean of the equivalent normal distribution. The left side

of the skewed distribution is alone considered for the equivalent normal distribution.

TO obtain the right side of the distribution, the lefi side distribution is mirrored about

the mode. Thus, the equivalent normal distribution of strength is obtained as shown

in Fig.5.38 (b).

According to the mean value first order second moment method, the

reliability index is given by ~q.6.l $and the probability of failure is also given by,

PI = @(-PI

where, Q, = cumulative normal distribution function

Then, reliability of the structure is given as,

R = l - P f

In this case, the load applied is assumed as a deterministic single value.

Hence, OL = 0 and now P is defined as,

p, -Load applied P =

0 s

By varying the load applied, the reliability of the structure at each load is

obtained. The failure probability at different loads is shown in Fig. 5.40 (a), @) (c).

The variation of reliability with respect to the buckling strength ratio is shown in Fig

5.40 (d). From this grape it can be seen that reliability is maximum of 100% for BSR

value of 0.23 and minimum of 0% for BSR value of 0.265 if the random imperfect

cylindrical shell models have maximum amplitude of imperfections .t 1.3 2mm.

ProbabilQ offallure+ OB2823 fur 13SR%2;# 0 . 4 ~ iZ

Probability of failure a0.4988 for BSRaOlldfil

Critical Vdue Weal Value

Prababili& affiillure*0= kr BSR 4285 FELlABlLlN Y s.BSR IMax.amp = +I .32mm) 0.4 *

0.35 -

1.1 -

4 - 9 - 2 4 0 1 2 3 1 0 . ' " Crytical Value 0.22 0.23 0.24 0.15 0.26 0.27 U.28 0%

BSR

Fig.5.39 (a) (b) (c) Failure probability at different loads and (d) Reliability Vs

buckling strength ratio for maximum amplitude of imperfections =

=t 1.32 mm.

Similar reliability calculations are also carried out using BSR data in

Table 5.7 i.e., for the case of maximum amplitude of imperfections = 13.71n.m.

Fig. 5.40 (a) shows histogram of actual skewed strength distribution and Fig. 5.40

(b) shows histogram of equivalent normal distribution of strength. The failure

probability at different loads is shown in Fig. 5.41 (a), (b) (c). The variation of

reliability with respect to the buckling strength ratio is shown in Fig 5.41 (d). From

this graph, h- it can be seen that reliability is maximum of 100% for BSR value of

0.187 and minimum of 0% for BSR value of 0.208 if the random imperfect

cylindrical shell models have maximum amplitude of imperfections 3.7mm.

Histogram for actual distribution of BSR value for max.amp=t3.7mm

7 4

Histogram of Equivalent strength disbibutim for Max.arnp t3.7mm 14 I I 1

BSR (b)

Mean of distribution = 0.1970 Mean of distribution = 0.1976 Mode of distribution = 0.1976 Mode of distribution = 0.1976

S. D of distribution = 0.0034 S.D of distribution = 0.0036

Fig. 5.40 (a) Actual skewed strength distribution (b) Equivalent normal

distribution of strength ratio for maximum amplitude of

imperfections = d= 3.7 mm.

In the above reliability analysis using small samples of size of 64 random

imperfection models, it can be seen that only two cases are presented for discussion

that is using BSR values obtained from random geometrical imperfection models

with maximum amplitude of imperfections equal to k 1.32 mm and 2 3.7 mm

respectively. Out of these two cases, the first case of maximum amplitude of

imperfections equal to f 1.32 mm, the actual distribution of strength and converted

equivalent distribution of strength matches better with the normal distribution.

Whereas in the other case of random imperfection models with maximum amplitude

of imperfections equal to k 3.7 mm, both the actual distribution of strength and the

converted equivalent distribution of strength fairly matches the normal distribution.

The actual distribution of BSR values fi-om other two cases i.e., random geometrical

imperfect cylindrical shell models with RMS value of imperfections equal to 0.45

mm and 0.93 rnm respectively are very poorly distributed (i.e., the left side

distribution of strength poorly matches normal distribution) when applied to this

simple reliability method.

Probabiiiy b~lu~aOX1181d for BSR= OA3 Prcbabilb of failure ~DJC@tfor BSRa81976

Probabillly offallm = 09T948 For BSRgOSE5

Critical Value

(b)

Fig. 5.41 (a) @) (c) Failure probability at different loads and (d) Reliability Vs

buckling strength ratio for maximum amplitude of imperfections =

=t 3.7 mrn

Hence these cases are not presented here. The reason for poor distribution of

strength is due to the fact that not enough BSR values are available to get good

distribution of BSR values to adopt this reliability method. Hence, it can be

concluded that for better reliability estimation using this simple reliability method,

more number of BSR values are required which can be obtained from numerical

of more number of random imperfect cylindrical shell models using more

than six eigen affine mode shapes.

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