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PART-B: Spectral Fatigue Analysis of a ship using FEM

Yogendra Parihar, Amresh Negi, Suhas Vhanmane

(Indian Register of Shipping)

The present paper discussed about the Spectral Fatigue Analysis (SFA). Full stochastic spectral analysis is an

advance approach to evaluate fatigue response of ships. It involves the application of direct hydrodynamic loads and

structural analysis. In-house developed numerical codes used for loads evaluation while the structural analysis is

performed using FE model. The complexities of load application on FE model are discussed. Apart from SFA

methodology, a comparison of FE based SFA is also made with other Fatigue assessment approaches as discussed in

PART-A of this paper.

KEY WORDS Spectral fatigue, hydrodynamic loads, direct load application, fatigue damage.

INTRODUCTION

Fatigue is a failure of material due to cyclic loads. In case of ships, hull girder undergoes through various cyclic loads due wave actions which can induce the crack and fatigue failure. Estimation of fatigue damage is depending on prediction of these wave loads. Wave loads are calculated using numerical hydrodynamic analysis and accounting the actual environmental conditions. The nature of hydrodynamic loads is stochastic therefore prediction of wave loads itself is a complex task. Various numerical techniques have been developed for prediction of hydrodynamic loads can be found in Hess & Smith 1967; Salvesen et al. 1970. Apart from direct wave loads prediction method, if look on the conventional design philosophy like empirical relations based on classification society’s rules, all are well established techniques. These conventional methods typically presume a wave environment (North Atlantic) which provides highest wave loads, would typically results conservative estimates of fatigue life. Spectral analysis provides a platform to consider the actual sailing profile of the ship and therefore accuracy in estimation the fatigue damage of ship can be improved. Spectral fatigue analysis is quite popular for reliable prediction of fatigue life. Many papers/literature discussed about the spectral fatigue analysis (Wang 2010; Kukkanen & Mikkola 2004; Kim et al. 2002; Negi & Dhavalikar 2011). Also, classification societies (ABS 2016; DNV 2014; BV 2016; IRS 2018b) have published their detailed guidelines outlining spectral fatigue analysis procedures. Spectral fatigue assessment technique uses the Stress Transfer Function (STF) evaluated directly from structural analysis. STF actually defines the energy of wave estimated using the suitable wave environment (scatter data and wave distribution) for the selected critical fatigue location. Stress range is normally expressed in terms of probability density functions for different short-term intervals

corresponding to the individual cells of the wave scatter diagram. Linear addition of short term damages sustained over all the sea states gives the total fatigue damage for the structural detail. Total fatigue damage (TD) accumulated over operational service life can be estimated by accounting for all sea states encountered with the different wave directions and all possible load cases. Present papers discussed about the procedure of spectral fatigue analysis which is categorized into three parts namely sea-keeping analysis, structural analysis and spectral analysis. Evaluation of direct hydrodynamic loads can be performed using 2D strip or 3D panel methods. Strip method is one of the well known approaches for computation of wave induced loads. Details related to spectral fatigue analysis using 2D strip method for the load evaluation is discussed in Part A of the paper. usually, panel method is considered to be more advance as compared to 2D strip method (Li et al. 2014) and recommended if using finite element (FE) based analysis. Present paper (part B) is focused on the structural response based on the direct application of hydrodynamic load to FE model. Sea-keeping analysis in this part of the paper is performed using the 3D panel program, while general FE package is used for structural analysis. Finally, spectral method is employed to evaluate the fatigue damage of selected structural details. Current paper is also highlighting the detailed procedure of FE analysis as direct hydro load are applied to hull. Application of direct loads on FE model and balancing of model are very complex task which requires utmost care from users. This approach can predict accurate structural response compare to other conventional methods. But one must note that utilization of computational resources and time is high compare to other discussed methods. In present paper, efforts have been made to describe the methodology and complexity of FE based spectral fatigue. Overall, this paper described from the hydrodynamic analysis, load application and spectral fatigue analysis procedures.

METHODOLOGY The overview of spectral fatigue analysis is shown in Figure 1.

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Figure 1: Procedure for fatigue analysis

HYDRODYNAMIC LOAD COMPUTATION Estimation of sea-keeping loads is preferred using the potential theory based program for the computation of Response Amplitude Operators (RAOs) for motions and loads in regular wave. Frequency domain 3D panel methods are the popular numerical methods to estimate the direct sea-keeping loads for the FE structural analysis. These potential theory based methods can be distinguished as zero forward speed Green function and Rankine panel. Forward speed problem in ship sea-keeping is constitute of total potential that can be defined as the sum of steady contribution due to ship’s forward speed and unsteady potential due to ship motion. It is required to have solution for steady potential beforehand and then solving the unsteady potential around this potential which is very complicated mathematically and intense numerically. However, conventional cargo ships are having relatively low Froude number. Therefore, zero forward speed green function based panel method with higher order forward speed correction can be used to compute the loads/pressure for ships. In the present paper 3D Green function based panel method has been used to compute the hydrodynamic loads. The ship forward speed has been accounted using the speed correction factors.

Figure 2: Panel distribution ship geometry of bulk carrier in

homogeneous loading condition

The wave pressure and motions are calculated by the program with respect to CG of vessel. These loads and motions are transferred to the FE model using appropriate technique as discussed in present paper. The complete methodology and essential details are explained in the following sections.

INTERACTION OF LOADS Sea-keeping programs provide wetted hull pressures, motions and hull girder loads such as shear forces (SF) and bending moments (BM) as outputs. But, application of these direct loads on finite element (FE) structural model is not a straight forward. In regards to panel pressure, Zhao et al. (2013) has demonstrated the utility of panel based methods. As transfer of the hydrodynamic panel pressures is convenient to 3D FE structural models, panel pressure is considered as a regular approach in while using direct loads. Apart from application of direct panel pressure, Parihar et al. 2014 and Parihar 2014 were discussed about the application of direct shear force and bending moments. The pro and cons of alternative method compare to panel pressure were also discussed by the Parihar et al. (2017). In context of hydro-structure interaction, complexities related to load application have been discussed in Parihar et al. 2017; Malenica & Derbanne 2014; Ma et al. 2012. Present study is using the panel pressure for structural analysis. Procedure of same is discussed below. Considering the load application as most critical task, the following procedure needs to be performed to transfer the direct hydrodynamic loads on FE model:

• Weight distribution of light weight, cargo weight, ballast water, bunkering and other appendages etc.

• Consideration of inertial loads (acceleration to FE model as shown in Figure 8)

• Mapping of panel pressure onto FE mesh

• Balancing of the structural model (inertia relief method)

• Solution of the numerical problem

Weight Distribution The lightship weights are simulated by providing the density of material or using nodal mass elements. Minor differences in the lightship weight of ship can be adjusted by providing lumped mass elements or by changing the steel density (special cases). The cargo weights are modeled considering the loading condition as provided in the trim and stability booklet. The bulk cargo mass are simulated using point/lumped mass element at the CG of respective cargo holds. The rigid body elements are used to connect the mass element to relevant hold surfaces as shown in Figure 3 and Figure 4. Ballast and bunker weights are simulated as tank pressures. The appropriate pressures acting on internal surfaces of liquid tanks are calculated and applied to selected entities in FE model as shown in Figure 5 and Figure 6. With above discussed task, procedure of weight distribution is to be performed such a way that CG of structural model remains as given in loading manual/stability booklet. The external fluid pressure is applied on FE model using the suitable pressure mapping technique as shown in Figure 7.

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Figure 3: Distribution of cargo hold mass in hold no3

Figure 4: Distribution cargo hold mass in all holds

Figure 5: Distribution of ballast pressure (normal ballast

condition)

Figure 6: Distribution of ballast pressure in starboard, port

side tanks and one cargo hold (heavy ballast condition)

Inertial Loads Each type of loads discussed above consist the static and dynamic load components. The dynamic loads can be decomposed into quasi-static and inertial loads. The quasi-static components are induced due to ship’s roll and pitch inclination. The directions of resultant gravity loads in ship’s fixed coordinate system are shown in Figure 8 with the variation of roll and pitch motion respectively. The inertial loads in FE model are to be simulated by using the Equation (1)-(4). In case of liquid tank pressure, inertia loads are calculated using the following (5)-(6). All the relevant inputs (ship motions and accelerations) for Equation (1) are to be obtained from sea-keeping analysis. IRS guideline for direct application of loads (IRS 2018a) can be referred for detailed procedure of weight distribution, pressure mapping, inertial and quasi-static loads distribution.

Figure 7: Distribution of real component of complex pressure in frequency domain (Homogeneous condition, head.=180˚)

Figure 8: Inertial load computation

CG

φ

Z

Y

gsinφ cosθ

g cosφ cosθ g

-gSinθ Z

X

θ

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������� = �� + Θ��� × �� (1)

where:

���� : acceleration vector at ship’s CG can be estimated as follows in respective directions using Equation (2), (3), and (4):

� = −� sin � + � (2)

�� = � sin � cos � + �� (3)

�� = � cos� cos � +�� (4)

��� : roll, pitch and yaw acceleration vector

�� : distance vector from ship’s CG to calculation point ax, ay, az : surge, sway, heave accelerations due to ship motions

θ, φ : pitch and roll angle (by sea-keeping program)

−� sin �, � sin � cos �, � cos� cos � are the quasi-static components in x, y and z direction respectively with reference to ship’s fixed coordination system due to combined motions of roll and pitch. Liquid pressure in tank is calculated using following equations:

� = �� + ��ℎ����� (5)

where

�� = !"��#2 + ��%2 + ��&2'

(6)

ρl : density of liquid

hi : internal pressure head at CG of element measured from the top of tank to the load point

Ael,Aet,Aev: accelerations in longitudinal, transverse and vertical direction respectively are to be calculated using

Equation (1).

CASE STUDY A bulk carrier is taken for the analysis purpose. The main particulars of ships are as follows:

Table 1: Main particulars of ships

Ship Particulars Bulk carrier

Length overall [m] 287.50

LBP [m] 279.00

Breadth (moulded) [m] 45.00

Depth (moulded) [m] 24.10

Design Draught [m] 16.50

Scantling Draught [m] 17.60

Max Service speed [knots] 14.60

Motion and load calculation The convention used in sea-keeping and throughout in analysis is as follows: AP 270˚ FP Following sea 0˚

Head sea 180˚

Beam Sea 90˚

Figure 9: Direction of wave headings considered for sea-keeping analysis

As loads interact from hydro model to FE model, therefore, there should be good correlation between these two models. The two models should be geometrically similar. To maintain the same, one IGES file was used to create the hydro and FE models. The considered hydrodynamic model is represented by the total 1992 panels till mean waterline in homogeneous loading condition as shown in Figure 2. The dimension of panel can be decided by the considered wave lengths for analysis, atleast 6 to 10 panel points should present over one wave length. Total 27 frequencies (λ/L ranges from 0.1 to 5) and 12 wave headings from 0 to 330 with an interval of 30 are considered in present analysis. Another important aspect in sea-keeping analysis is mass distribution, as inertia matrices are required for hydrodynamic analyses and derived from the mass model, therefore mass model must reflect the actual weight distribution of hull, cargo, ballast, bunkering appendages etc. for considered loading conditions. Any inaccuracy in the mass matrix may result in an unbalanced hydro model and thus incorrect end shear forces and bending moments. There can be small difference in the calculated mass by hydrodynamic analysis and total mass given in loading manual. An iteration process for tuning the mass and CG locations of ship can be carried out till the difference reaches to negligible quantity.

Structural Model A full length structural model of container ship is created using FEM software. All primary and most secondary structural members are modeled in order to simulate actual stiffness of the hull girder. All applicable weights are simulated in FE model as discussed above. Simulation of weights is to be done accurately. The consequent vertical, transverse and longitudinal centers of gravity (VCG, TCG and LCG) as obtained from the model are compared and verified with the corresponding values listed in stability booklet.

Type of Elements and Model Idealization Based on the usage, the following elements are used in the analysis.

Table 2: Structural idealization

Element type Idealization

Beam Stiffeners

Shell Girder, plates, bracket

Mass Lumped mass e.g. machinery components, container weight

Rigid link Connection of node-node, node-mass

Material Properties Standard material properties are given in Table 3.

Table 3: Material properties

Material Young’s Modulus (N/mm2)

Poisson Ratio

Density (t/mm3)

Steel 2.05E+05 0.3 7.8E-09

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Boundary Conditions The right-handed co-ordinate system, with the x-axis positive forward, y-axis positive to port and z-axis positive vertically from baseline to deck are taken for the analysis as shown in Figure 10. The origin was located at the intersection between aft perpendicular (AP), baseline and centerline. The boundary conditions are applied on the FE model as listed in Table 4 and shown in Figure 10.

Table 4: Boundary conditions for global model

Location Direction

Engine Room Front Bulkhead

SB & PS Z

CL Y

Collision Bulkhead CL X, Y, Z

Figure 10: Boundary conditions

Model Balancing Hydrodynamic model is perfectly balanced as equilibrium is implicitly imposed in solution of equation of motion. The critical issue is balancing of the FE model when the loads are transferred from hydro model to FE model. Problem is raised due to difference in mesh size of these two models. Hydro model is having the 1500 to 2000 panels in ship’s geometry while outerhull geometry of FE model can have 20000 to 30000 elements. Mapping of pressure from coarser panels to finer elements can lead to unbalanced forces at artificial supports. Appropriate interpolation scheme is to be used for the pressure application though model cannot be balanced perfectly. Present study used the 3D interpolation technique to map the pressure onto hull surface. Unbalanced non-zero reaction forces at the artificial supports are corrected using the inertia relief method. Inertia relief is a technique in which applied forces and moments are balanced by counter forces induced by accelerating the body. The application of these accelerations is performed such a way that it precisely cancels or balances the additional forces. However one should be cautious that application of inertia relief can alter the response profile of structure. Application of inertia relief method in ship structures is limited to cases where unbalanced forces are within engineering limits.

Structural Analysis Static structural analyses are performed for given number of load cases corresponding to each loading condition. For each loading condition, 12 headings, 27 frequencies and two parts of frequency (real and imaginary) are taken for analysis at given speed. Total number of 648 (1�12�27�2�1) load cases are considered in single loading condition. Four loading conditions

(homogeneous, alternate, heavy ballast and normal ballast) are analysed. Equilibrium check for each load case is mandatory. The applied hydrodynamic panel pressure must be in equilibrium with the counter loads induced due to motion of loaded (distributed weights) FE model of ship. To estimate the imbalance forces, all the forces and moments are to be summed up in global direction. In case of unbalanced forces, a suitable method to balance the FE model is employed as discussed above before performing the structural analysis. Before preceding the structural response evaluation, analysis is verified with the following aspects as given in loading manual or stability booklet:

• Total weight and CG of each tank/hold

• Shear force distribution along the length of ship

• Moment distribution along the length of ship

Finally, hot-spot stress is evaluated using the recommended method as given in IIW recommendation (IIW 2008) for fatigue

design of welded joints. Fine mesh of size 50 mm × 50 mm is created at given locations as shown in Figure 11.

Figure 11: Representative mid-ship section showing the butt plate joints (DK1, DK2 and SS1)

SPECTRAL ANALYSIS The fatigue damage at selected locations is predicted using a spectral approach. The spectral fatigue analysis is based on the Palmgren-Miner’s linear damage summation rule and used the appropriate S-N curves. The procedure for spectral fatigue is well documented (ABS 2016; BV 2008; IRS 2018b). Here only the basic explanation of spectral method is given. The detailed methodology can be referred from the literature (Xiang-chun et al. 2006; Nguyen et al. 2013; Wang 2010; Negi & Dhavalikar 2011; Parihar 2014; ABS 2016).

Closed Form Damage Expression The closed form expression for total cumulative fatigue damage D, where the long term stress range distribution is defined through a short term Rayleigh distribution within each short term sea state, the fatigue damage for bi-linear S-N curves, is given as in Eq. 1 (ABS 2016).

DK1

DK2 SS1

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(= )� *2√2,

-. /02+ 12334*0, 6�7,8�7 ��79�*:;:<:-=>�7,-

?

7@A

B

�@A

(1)

Where 0C = D ECFG"E|I= , )�'JEK� (2)

Where FG"E|I= , )�, �' = |IG"E|�'|LFM"E|I= , )�' (3)

Where, M and D denotes the total number of sea states and

direction respectively. >�7 = !0��7 is the standard deviation of

the stress response in the sea state and direction respectively and µi is the endurance factor having its value between 0 to 1 and measuring the contribution of lower branch to the damage. Γ is complete gamma function with the argument (m/2+1) and λ is rainflow factor of Wirsching, and f0i is zero-up-crossing frequency of stress response, and pi is joint probability distribution of Hs and Tz. kh, kt and kms are factors for high tensile steel, thickness effect and for mean stress effect respectively. The spectral moments is calculated Eq. 2. Sσ(ω|Hs,Tz,θ) is stress spectrum generated by scaling the wave energy spectrum Sη(ω|Hs,Tz,θ) and complex stress transfer function Hσ(ω|θ) using Eq. 3, where Hσ(ω|θ) is the structural response for sinusoidal wave of unit amplitude computed. Total fatigue damage accumulated over operational service life (T) is estimated by accounting for all sea states encountered with the different wave directions and given loading condition.

RESULT AND DISCUSSION Spectral fatigue analysis is performed by following the procedure is given in Figure 1. D class of S-N curves (IACS 1999; UK Department of Energy (DEn) 1990) is considered for selected weld location considering that hotspot stress range is being computed and used for evaluation of number of cycles. Wave data has been taken from the IACS recommendation 34 (IACS 2001). Spectral analysis is performed considering Pierson Moskovitz (P-M) wave spectrum. Stress response is obtained for various sea states and respective directions in given loading conditions. The complex stress transfer function is evaluated from FE analysis. Figure 12 to Figure 14 shows the STF in alternate loading condition. Total fatigue damage is evaluated using spectral moments and STF as shown in Eq. (1), (2) and (3). Mean stress effects also considered for the details investigated in the present study.

Table 5: Fraction of time for considered loading conditions

Loading conditions Fraction of time

Homogeneous 0.25

Alternate 0.25

Normal ballast 0.20

Heavy ballast 0.30

The fatigue damage for each loading condition is shown in Figure 15, where, homogeneous (max draught) and normal ballast (min draught) conditions makes the larger contribution in fatigue damage. Although, the combined fatigue damage is

calculated considering the fraction of each loading condition as shown in Table 5 and it is assumed that ship is sailing 85% of its lifetime. The present FE based approach for the fatigue assessment has been compared with the other approaches as discussed in part-A of the paper. Table 6 summarizes the methods which have been followed for the load evaluation in Part A of the paper. Current approach is added as Fatigue Assessment Method-5 (FAM-5). The loads for FAM-1 and FAM-3 are calculated using the closed loop formulation based semi-analytical approach. Similarly, FAM-2 and FAM-4 have used the 2D strip theory based loads. The two approaches (FAM 1 and 3 and FAM 2 and 4) are differed by the estimation of stress range distribution like long term approach based on the Weibull Distribution for FAM-1 and FAM-2 while long term distribution defined through short-term Rayleigh distribution for FAM-3 and FAM-4. Current method FAM-5 is using the 3D panel based loads application to FE model to estimate the STF. Fatigue damage is evaluated using the long term stress range distribution accumulated using short term Rayleigh distribution for each sea-state.

Figure 12: STFs at deck location DK1

Figure 13: STFs at deck location DK2

Figure 14: STFs at side shell location SS1

0

5

10

15

20

25

0 1 2 3 4 5

Str

ess

RA

O [

MP

a/m

m]

λ/L

Location ID:DK10

30

60

90

120

150

180

0

5

10

15

20

25

0 1 2 3 4 5

Str

ess

RA

O [

MP

a/m

m]

λ/L

Location ID: DK20

30

60

90

120

150

180

0

5

10

15

20

25

0 1 2 3 4 5

Str

ess

RA

O [

MP

a/m

m]

λ/L

Location ID: SS1

0

30

60

90

120

150

180

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Table 6: Summary of the fatigue assessment approaches

Method

- IDs

Load Evaluation

Method Fatigue Damage Approach

FAM-1 Semi analytical formulation

Closed form approach based on long-term response

FAM-2 2-D Strip theory Closed form approach based on long-term response

FAM-3 Semi analytical formulation

Spectral approach Based on short term response

FAM-4 2-D Strip theory Spectral approach Based on short term response

FAM-5 3D Panel method Spectral approach Based on short term response

Figure 15: Fatigue damage in various loading conditions and

combined for considered locations

Figure 16: Total fatigue damage at given location using specified methods

To compare the above approaches, Figure 16 showed the total fatigue damage at specified locations estimated using all five approaches. FAM-5 is based on the comprehensive structural analysis for the evaluation of structural response. Therefore the following points are discussed and summarized with respect to FAM-5:

• Closed loop based semi-analytical approach is under predicting the fatigue damage but time and computational resources required by this method are very less.

• Strip theory based fatigue damage prediction is more conservative, although required time and computational resources are moderate compared to FE based approach.

• 3D panel and FE based fatigue damage values for considered location is coming in-between of above two methods. This method is considered to be more realistic. However, time and computational resources required to perform such analysis are very high.

CONCLUSION Present paper showed the spectral fatigue analysis for the butt-weld joints for a bulk carrier. It detailed about the direct application of sea-keeping loads on FE model. Issues like the practical difficulties in implementation of direct loads and balancing of FE model were discussed. Processes, where the number of load cases are huge, can be automated as shown in present analysis. Among three locations, location-2 (DK2) is more critical. Similar to 2D strip theory based damage assessment using FAM-2 and FAM-4, the detailed structural analysis based fatigue damage (FAM-5) shows the moderate influence of the HBM and torsion. Beam theory and FE based structural response approaches can be adopted for the fatigue assessment based on the design requirement as discussed though these two papers. Semi analytical load evaluation method (FAM-1 and FAM-3) does not possess a feasible solution for the fatigue damage assessment in ship design.

ACKNOWLEDGEMENT The authors express their sincere thanks to the Indian Register of Shipping (IRS) for providing the support to perform this study.

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0.9

20

0.9

92

0.7

93

0.3

84

0.4

59

0.3

86

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54

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73

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