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ULTIMATESTRENGTHANDUNCERTAINTYANALYSISOFSTIFFENEDPANELS

CONFERENCEPAPER·OCTOBER2014

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MuratOzdemir

IstanbulTechnicalUniversity

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AhmetErgin

IstanbulTechnicalUniversity

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TEAM 2014, Oct. 13 - 16, 2014, Istanbul, Turkey

ULTIMATE STRENGTH AND UNCERTAINTY

ANALYSIS OF STIFFENED PANELS

Murat Ozdemir*, Ahmet Ergin

*Research Assistant, Istanbul Technical University, Faculty of Naval Architecture and Ocean

Engineering, Istanbul, Turkey.

E-mail: mozdemir@itu.edu.tr

Abstract

Correct calculation of ultimate strength is one of the most crucial aspects of ship structural design.

Ultimate strength of stiffened panels depends on geometry, material, loads and environmental aspects.

Due to the uncertainty of these parameters, ultimate strength of stiffened panels has inherent variability.

Under this point of view, nonlinear finite element analyses are performed in order to evaluate ultimate

strength of stiffened panels. Results are compared with those in literature. Artificial Neural Network

Model is developed considering nonlinear finite element results. Then, ANN based Monte Carlo

simulations are conducted to investigate the uncertainty of ultimate strength of panels.

Keywords: Ultimate Strength, Uncertainty Analysis, Stiffened Panel, Artificial Neural Network, Monte

Carlo Simulation.

1. INTRODUCTION

Collapse of structure must be identified from the safety point of view. Ultimate limit state of

structures is considered as failure of structure after reaching maximum load carrying capacity

(ultimate strength).

Ultimate strength of structures is relevant with loading condition, geometrical and material

properties. All of the parameters which affect the ultimate strength and collapse behavior of

panels have inherent uncertainties, due to this fact, ultimate strength shows variability. The

variability of structural capacity and reliability of structural system must be quantified.

Reliability techniques have been in development of years [1]. These methods first appeared in

a mathematical form in 1926 by Mayer [2], further developed by Streletzki [3] and Wierzbicki

[4]. With the structural reliability methods, all the uncertainties involved in the description of

the load and the structural capacity can be quantified through probabilistic models [5].

Probabilistic analysis of plate buckling was rare, Ivanov and Rousev [6] presented one of the

first studies of plate buckling problem from a probabilistic viewpoint. Guedes Soares [7]

modeled the uncertainties in plate buckling strength for the different cases. Special attention is

given to the implications of problem formulation and to the methodology of uncertainty

modeling.

A method was proposed by Kimiecik and Soares[8] to determine the cumulative distribution

function of the strength of compressed plates using a response surface approach, used the results

from the nonlinear finite element code to fit a response surface limit state function. Recently,

uncertainty and reliability analysis of stiffened panels have been carried out by [9,10] using

nonlinear finite element results.

2. ULTIMATE STRENGTH ANALYSIS OF STIFFENED PANELS

Nonlinear finite element analyses are conducted to evaluate the ultimate strength of stiffened

panels. Stiffened panels undergo several collapse modes depending on loading, plate and

stiffener properties. Collapse modes of stiffened panels can be classified into six modes [11].

If the stiffeners are rigid enough, local plate buckling occurs between the stiffeners. In case of

lower stiffener bending rigidity, overall buckling may be observed. Beside this, if the stiffeners

are very slender, tripping of stiffeners may be occurred.

Fig. 1. Overall collapse of stiffened panel. Fig. 2. Tripping of stiffener.

Overall collapse mode is considered in this study. Firstly, linear FEM eigenvalue analyses are

carried out to determine the buckling mode shapes of stiffened panels. Overall buckling mode

shape is imposed as initial imperfection to FEM models.

Fig. 3. Finite element model of stiffened panel.

Finite element model must capture all the mechanisms that could lead to collapse of the

structure. For inelastic analysis, 1-bay model cannot capture the collapse accurately because of

that inter-frame bay deflect in upward or downward half wave, while the next bay would deflect

in opposite sense. The boundary condition at the frame is intermediate between simply

supported and clamped, and cannot be accurately modeled as a loaded edge (see, for instance,

[12]). Because of these reasons, the finite element model is represented as a symmetric 12

1

as seen Fig. 3.

2.1 Boundary conditions

The long edges are considered as simply supported. The displacements in the z-

direction and rotations about y and z-axes, as in Fig. 4, are constrained to impose simply

supported boundary conditions.

The transverse frame is not modeled, but displacements in the z-direction along the

transverse frame are constrained.

The short edge, which is the mid-length of the mid-bay of the full three bay model, has

symmetric boundary condition. The symmetric boundary condition is satisfied by

constraining displacements in the x-direction and rotation about y-axis.

The displacements in the y-direction are constrained at the mid-width node in each of

two short edges to prevent rigid body motion.

2.2 Initial imperfections

In the nonlinear collapse analysis, bifurcation buckling is not a critical phenomenon. The

problem has to be considered as a continuous response problem [12]. Therefore, initial

imperfections are introduced for the stiffeners and plating. The imperfect geometry is assumed

as overall buckling mode shape obtained from the eigenvalue analysis. The selected mode shape

has an upward deflection (plate induced) in full bay and a downward deflection in half bay as

seen in Fig. 4. The scaling factor for initial imperfection of the stiffened panel is 0 0.0025w a

where, a is the length of one bay.

Fig. 4. Overall buckling mode shape of stiffened panel.

Material properties are same for all the panels as modulus of elasticity 205800 MPa, Poisson

ratio 0.3 and yield stress 352.8 MPa. Material is assumed as elastic-perfectly plastic.

Fig. 5. Stress-strain diagram for the panel material.

3. UNCERTAINTY ANALYSIS OF STIFFENED PANELS

Several parameters affect the ultimate strength of stiffened panels. These parameters are

considered as deterministic in classical ultimate strength analyses. Whereas, the parameters

have inherent uncertainty; since, ultimate strength value of panels shows variability. Practical

aspects affecting ultimate strength behavior of panels can be classified into three groups as

physical aspects, model uncertainties and ageing effects [13].

3.1 Artificial Neural Network

Artificial Neural Networks (ANNs) can be applied to many areas, in this study ANN is applied

to estimate ultimate strength of stiffened panels based on nonlinear FEM results. General

structure of ANN is given in Fig. 6. First layer is input layer where raw input data is prepared

for training process. Second layer is hidden layer where most of the neurons are located and;

third layer is output layer where output of the ANN is given.

Models are constructed as discreetly for 3 stiffened and 5 stiffened panel cases. 270 Nonlinear

FEM analyses are performed to train network for each model.

ANN is simple and relatively accurate method to conduct uncertainty analyses; since, it gives

ultimate strength estimation in a good manner. Also, we can conduct uncertainty simulation by

ANN model in a short CPU time.

Fig. 6. General Structure of ANN[16]. Fig. 7. Single neuron of the model.

3.1.1 Mathematical Background

From the mathematical point of view, developing an empirical formula from experimental or

numerical data is to find an approximate function, which can best represent the relation between

input and output variables [14].

Assume that input and output variables are given Eq. 1 and Eq. 2, respectively.

1 2 3( , , ,... )nx x x xTX = (1)

1 2 3( , , ,... )Ly y y yTY = (2)

Output of the i-th neuron in the hidden layer is expressed as in Eq. 3.

1

nk

i i k i

k

f wih x b

X (i=1,…,m) (3)

k

iwih is the weight of the k-th input variable in the input layer to the i-th neuron in the hidden

layer, and bi is constant.

Mathematical expression of j-th output variable is given as follow;

1

ni

j j i j

k

y f who c

X (j=1,…,L) (4)

Eq. 3 and 4 are expressed in matrix form as given;

2

1

Tf Wf W

I

Y I

X

(5)

In Eq. 5; I is one by one unit matrix, and the W1 and W2 are weight matrices.

3.2 Monte Carlo Simulation

In reliability analysis of structures Monte Carlo Simulation (MCS) is particularly applicable

when an analytical solution is not attainable and the failure domain cannot be expressed or

approximated by an analytical form.

In reliability analysis of structures limit state function G(x) must be defined principally; the

limit state function contains vector of random variables which define the capacity and load

characteristics. In classical reliability analysis, probability of failure of structure can be

expresses as in Eq. 6.

( ) 0

( )f x

G x

p f x dx

(6)

In eq. 6 fx(x) is joint probability distribution function of random variables. Analytical

integration of Eq. 6 is practically compelling issue; since, probability of failure of structure can

be examined by MCS as given in Eq. 7.

1

1( )

N

f j

j

p I xN

(7)

It is assumed that all variables have normal distribution and standard deviation is %3 of design

value for the Monte Carlo Simulation.

4. RESULTS

A series of nonlinear FEM analyses are performed to estimate ultimate strength of stiffened

panels. Main purpose of this study was to investigate variability of ultimate strength of panels.

Results of FEM analyses are used for training of ANN model. Developed ANN model showed

good agreement with FEM results; since, ANN based Monte Carlo Simulations are conducted

to investigate uncertainty of ultimate strength estimations. Results are given in Table 1 and

Table 2 for 3 stiffened and 5 stiffened panels, respectively.

Table 1. Results for 3 stiffened panels. PANEL FEM ANN MEAN STD MIN MAX PANEL FEM ANN MEAN STD MIN MAX

P1 0.39 0.38 0.39 0.02 0.28 0.50 P52 0.72 0.73 0.72 0.03 0.59 0.82

P2 0.33 0.31 0.31 0.02 0.22 0.39 P53 0.62 0.64 0.64 0.03 0.53 0.74

P3 0.31 0.36 0.36 0.02 0.25 0.46 P54 0.48 0.46 0.46 0.03 0.35 0.61

P4 0.66 0.64 0.64 0.04 0.48 0.79 P55 0.40 0.40 0.40 0.03 0.30 0.54

P5 0.50 0.48 0.48 0.03 0.35 0.63 P56 0.56 0.58 0.58 0.03 0.46 0.69

P6 0.61 0.59 0.59 0.04 0.41 0.79 P57 0.36 0.35 0.35 0.02 0.26 0.45

P7 0.93 0.91 0.90 0.03 0.76 1.01 P58 0.26 0.26 0.26 0.02 0.19 0.38

P8 0.78 0.80 0.80 0.04 0.64 0.94 P59 0.22 0.21 0.21 0.01 0.16 0.29

P9 0.76 0.76 0.76 0.04 0.61 0.90 P60 0.74 0.75 0.75 0.03 0.63 0.88

P10 0.34 0.33 0.33 0.02 0.26 0.44 P61 0.67 0.69 0.68 0.04 0.51 0.85

P11 0.28 0.28 0.28 0.02 0.21 0.36 P62 0.70 0.69 0.69 0.03 0.55 0.84

P12 0.35 0.34 0.34 0.02 0.22 0.43 P63 0.62 0.61 0.61 0.02 0.52 0.73

P13 0.48 0.45 0.46 0.03 0.33 0.63 P64 0.48 0.46 0.47 0.03 0.35 0.60

P14 0.43 0.42 0.42 0.03 0.29 0.54 P65 0.58 0.55 0.55 0.02 0.45 0.66

P15 0.39 0.39 0.39 0.03 0.27 0.52 P66 0.52 0.51 0.51 0.03 0.39 0.66

P16 0.80 0.77 0.77 0.04 0.61 0.92 P67 0.28 0.26 0.27 0.01 0.20 0.33

P17 0.79 0.74 0.74 0.03 0.62 0.87 P68 0.23 0.22 0.22 0.02 0.15 0.30

P18 0.70 0.69 0.68 0.03 0.55 0.83 P69 0.73 0.73 0.73 0.03 0.62 0.89

P19 0.25 0.24 0.24 0.01 0.18 0.31 P70 0.67 0.66 0.66 0.04 0.49 0.85

P20 0.26 0.25 0.25 0.02 0.18 0.33 P71 0.69 0.68 0.68 0.04 0.55 0.86

P21 0.22 0.24 0.23 0.02 0.15 0.32 P72 0.62 0.60 0.60 0.02 0.51 0.71

P22 0.46 0.48 0.48 0.02 0.38 0.59 P73 0.50 0.48 0.48 0.03 0.35 0.63

P23 0.47 0.46 0.46 0.02 0.36 0.58 P74 0.58 0.54 0.54 0.03 0.44 0.65

P24 0.45 0.45 0.45 0.03 0.33 0.61 P75 0.79 0.80 0.80 0.04 0.65 0.95

P25 0.27 0.25 0.25 0.02 0.18 0.31 P76 0.30 0.30 0.30 0.02 0.22 0.40

P50 0.74 0.76 0.76 0.02 0.65 0.85 P77 0.25 0.24 0.24 0.03 0.14 0.36

P51 0.68 0.67 0.67 0.04 0.49 0.81

Table 2. Results for 5 stiffened panels. PANEL FEM ANN MEAN STD MIN MAX PANEL FEM ANN MEAN STD MIN MAX

P26 0.49 0.49 0.50 0.02 0.40 0.61 P41 0.93 0.92 0.92 0.02 0.81 1.05

P27 0.39 0.40 0.40 0.02 0.30 0.50 P42 0.89 0.88 0.88 0.02 0.78 0.99

P28 0.45 0.46 0.46 0.02 0.33 0.57 P43 0.81 0.79 0.78 0.03 0.64 0.92

P29 0.79 0.75 0.74 0.03 0.63 0.87 P44 0.35 0.36 0.36 0.02 0.28 0.47

P30 0.61 0.61 0.61 0.03 0.48 0.72 P45 0.35 0.36 0.35 0.02 0.23 0.43

P31 0.70 0.69 0.69 0.03 0.56 0.84 P46 0.35 0.36 0.35 0.02 0.19 0.42

P32 0.96 0.96 0.97 0.03 0.88 1.11 P47 0.72 0.72 0.71 0.03 0.56 0.84

P33 0.87 0.86 0.86 0.02 0.76 0.98 P48 0.64 0.66 0.66 0.03 0.51 0.77

P34 0.86 0.83 0.84 0.03 0.70 0.95 P49 0.63 0.62 0.62 0.04 0.43 0.75

P35 0.44 0.44 0.45 0.02 0.36 0.54 P78 0.92 0.91 0.91 0.04 0.75 1.03

P36 0.35 0.35 0.35 0.02 0.25 0.43 P79 0.92 0.91 0.91 0.02 0.81 1.03

P37 0.40 0.41 0.40 0.02 0.29 0.48 P80 0.91 0.90 0.90 0.03 0.79 1.03

P38 0.61 0.60 0.60 0.03 0.47 0.73 P81 0.90 0.90 0.90 0.03 0.78 1.02

P39 0.54 0.53 0.53 0.03 0.40 0.65 P82 0.86 0.85 0.85 0.04 0.73 1.00

P40 0.49 0.49 0.49 0.02 0.36 0.59 P83 0.80 0.77 0.77 0.03 0.63 0.93

Table 2. Results for 5 stiffened panels.(Continue) PANEL FEM ANN MEAN STD MIN MAX PANEL FEM ANN MEAN STD MIN MAX

P84 0.84 0.83 0.83 0.03 0.75 0.97 P96 0.55 0.52 0.52 0.03 0.41 0.62

P85 0.62 0.60 0.61 0.02 0.52 0.74 P97 0.43 0.44 0.43 0.03 0.26 0.55

P86 0.56 0.54 0.54 0.02 0.45 0.64 P98 0.90 0.89 0.89 0.01 0.80 1.01

P87 0.42 0.44 0.44 0.03 0.33 0.57 P99 0.89 0.87 0.87 0.03 0.74 0.96

P88 0.91 0.88 0.89 0.02 0.79 1.02 P100 0.89 0.88 0.88 0.03 0.75 0.98

P89 0.91 0.89 0.90 0.02 0.80 1.01 P101 0.89 0.88 0.88 0.03 0.76 0.99

P90 0.90 0.88 0.88 0.03 0.77 1.01 P102 0.84 0.81 0.81 0.03 0.72 0.95

P91 0.89 0.89 0.89 0.02 0.79 1.01 P103 0.77 0.77 0.76 0.02 0.64 0.88

P92 0.85 0.82 0.82 0.03 0.72 0.96 P104 0.78 0.78 0.77 0.02 0.64 0.89

P93 0.78 0.77 0.77 0.03 0.66 0.89 P105 0.81 0.79 0.79 0.02 0.71 0.91

P94 0.82 0.80 0.80 0.02 0.72 0.94 P106 0.63 0.61 0.62 0.03 0.51 0.75

P95 0.63 0.61 0.61 0.03 0.51 0.74 P107 0.55 0.53 0.52 0.03 0.39 0.63

As can be seen from Table 1 and Table 2, ANN results are in good agreement with FEM results.

Monte Carlo Simulation results indicate that uncertainty of ultimate strength is higher than the

individual uncertainty of each design parameter.

Fig. 8. Ultimate Strength Distribution of P6. Fig. 9. Ultimate Strength Distribution of P28.

5. CONCLUSIONS

Ultimate strength of stiffened panels are estimated by nonlinear FEM and results are used for

ANN simulation as target data. ANN simulations have shown good agreement with FEM

results. Also, ANN based Monte Carlo Simulations are performed to estimate uncertainty in

ultimate strength of panels. Following findings have been obtained from this study:

- ANN is a powerful tool for the estimation of ultimate strength and uncertainty

simulation.

- Uncertainty of design parameters (C.O.V: %3) caused variability of ultimate strength

which level is up to %11 of mean value.

- Uncertainty of ultimate strength and probability of failure of the structure must be

quantified from the safety point of view.

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