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NTNU-Trondheim

Department of Marine Technology

TMR-4320 Simulation Based Design Final Report

Autumn 2015

Optimum Geometry Design for a Point-Loaded Cruise Ship Balcony Made from Normal

Steel using Finite Element Method ANSYS Program

Muhammad Mukhlas

Student Number : 763599

November 2015

Abstract

This report describes the analysis and optimization of ship’s balcony structure for ship with length

100 m and above. In thin study case, Norwegian Gem cruise ship is taken as an example for

applying DNV-GL Rules for Ships Pt. 3 as a part of optimization. The balcony is modeled with 6

meter length and 3 meter breadth, and consists of minimum 2 transversal bar and 1 longitudinal

stiffener. At the beginning, the position and number of transversal bars and stiffeners are assumed.

The plate is loaded with distributed water pressure along the balcony, and also it is assumed that

there are 3 human weight load in 3 points on the balcony as the concentrated loads. By comparing

the minimum criteria given by DNV-GL rules with the FEM analysis using ANSYS program, it

shows that the safety factor criteria governed by this rules are too big. The optimization part have

been done by using Particle Swarm Optimization (PSO) method. By considering the weight,

maximum stress and maximum displacement of the structure, the optimized geometry of the

balcony structure is determined. PSO gives a very good result and all the objective is accomplished

without violating any constraints. Result of PSO gives Von Mises stress below the yield criteria

(233 MPa) and small vertical displacement (0.051 m). Total weight of the structure also decreasing

about 461.6 kilogram.

Table of Contents

1. Introduction…………………………………………………………………………………..1

2. Design Parameters…………………………………………………………………………...2

2.1. Given Parameters………………………………………………………………………..2

2.2. Assumed Parameters……………………………………………………………………2

3. Modeling of Geometry Structure……………………………………………………………3

3.1. Modeling using Autodesk Inventor……………………………………………………..3

3.2. ANSYS Workbench……………………………………………………………………..4

3.3. Joint Fixing in ANSYS Mechanical…………………………………………………….6

4. Finite Element Analysis (FEA) using ANSYS Mechanical APDL…………………………7

4.1. Plate Theory……………………………………………………………………………..7

4.2. Introduction of FEA…………………………………………………………………….7

4.3. Element Types in ANSYS……………………………………………………………….9

4.4. Von Mises Stress…………………………………………………………………………9

4.5. Geometry Reduction in ANSYS (HP Profile Bulb Section)…………………………..10

4.6. Mesh Convergence Test………………………………………………………………..11

4.7. ANSYS Initial Model Analysis………………………………………………………...12

5. Applying DNV-GL Classification Rules as a Part of Optimization………………………13

6. Particle Swarm Optimization………………………………………………………………17

6.1. Introduction…………………………………………………………………………….17

6.2. PSO Input Parameter………………………………………………………………….17

6.3. PSO Script Way of Works……………………………………………………………..17

6.4. PSO Output…………………………………………………………………………….18

6.5. PSO Conclusion and Discussion………………………………………………………19

7. Conclusion…………………………………………………………………………………..21

8. Suggestion and Improvement………………………………………………………………22

9. References…………………………………………………………………………………...22

10. Attachment………………………………………………………………………………….22

1

1. Introduction

The best way to design marine or another structure is to have a minimum criteria that can

withstand the external forces with boundary conditions applied. The only way to actualize that

is to do the Optimization in the structure.

There are many methods to test the design, for example Physical Test and Numerical Test.

Comparing to the physical test, numerical test usually consumes less time and budget. Another

advantage of this test, numerical test gives flexibility to the user to change the parameter that

wanted to be included in the test.

Finite element method is one of the methods that used for numerical test and it is well

established, especially for structural analysis. In this study case, ANSYS is used to solve the

finite element calculation and then the structure will be optimized by using Particle Swarm

Optimization. The objective of this study case is to determine the optimal structure with

minimum parameter as possible, and it also can resist the external forces applied.

2. Design Parameters

In this section, the parameters used in the study case are described. Parameters are divided in

to Given Parameters and Assumed Parameters and it is showed in the next two sub-sections.

The illustration of initial model (first assumption) used for the analysis is given below.

Wide

Length

Figure 2.1 : Structure Geometry Definition

Information :

= Point load applied

= Transverse beam

= Stiffener

Constrained side

2

2.1. Given Parameters

The given parameters given in this study case are described below.

Table 2.1 : Given Parameters

Parameter Value Unit

Balcony Length 6 meter

Balcony Width 3 meter

Water height above balcony 1 meter

Number of transverse beam 2 unit

Boundary condition Clamped in one of longitudinal side

Assumption Balcony behavior is uncoupled from the global ship

2.2. Assumed Parameters

The assumed parameters given in this study case are described below.

Table 2.2 : Assumed Parameters

Parameter Value Unit

Steel Density 7850 kg/m3

Young’s Modulus of Steel 200 GPa

Steel’s Yield Stress (Design) 235 MPa

Plate thickness 0.01 meter

Stiffener type HP 180x8

Number of Stiffener 5

Stiffener Spacing 0.5 meter

Transverse beam type 200x200

Transverse beam web thickness 0.001 meter

Transverse beam flange thickness 0.001 meter

Transverse beam spacing 2 meter

Person 1 weight 120 kilogram



3

3. Modeling of Geometry Structure

Before doing the finite element method analysis, the determination of structure geometry must

be completed. There are two ways of drawing the structure geometry, using the CAD

(Computer-Aided Design) software or directly draw the structure in CAE (Computer-Aided

Engineering).

3.1. Modeling using Autodesk Inventor

Autodesk Inventor is one of the CAD software that available in the market. By using this

software, the wanted structure geometry can simply be drawn with graphical interface.

After the drawing is finished, the structure will be transferred into CAE for the further

analysis. There are several steps for drawing the specified structure observed in this study

case.

i. Create the parts

The structure consists of three main parts, plate, transversal beam and stiffener. The

dimension is drawn and defined manually by the user. In Inventor, the drawn structure

can easily transformed into 3D geometry by applying extrusion on the drawn 2D plane.

Figure 3.1 is the example of stiffener drawing in Inventor.

Figure 3.1 : Stiffener Part Drawing in Inventor

ii. Parts Assembly

The advantage of using Inventor is the flexibility of drawing the structure. The

structure can be drawn separately and then assembled as a full structure. Figure 3.2 is

the assembly made for the study case.

4

Figure 3.2 : Parts Assemble in Inventor

3.2. ANSYS Workbench

The assembled structure will be transformed to line, area and joint so that ANSYS APDL

can read and process the geometry. This work can be done using ANSYS Workbench.

CAD model can be easily transferred from Inventor into ANSYS Workbench. The aim of

this step is to create the mid-surface of structure thickness and also create the joint in the

intersecting line on the structure.

i. Creating the Mid Surface

In this step, volume of the structure will be transferred into mid-surface plane using

ANSYS Workbench. Illustration about mid-surface is described in Figure 3.3 and

Figure 3.4 below.

Figure 3.3 : Mid Surface Definition

Figure 3.4 : Mid Surface Geometry in ANSYS Workbench

Mid Surface Thickness

5

ii. Connecting the plate with Transversal Beam and Stiffener

Mid-Surface generation can caused a gap between the plate and the beams above the

structure. For the analysis in ANSYS, it is necessary for the structure to be connected.

Therefore, the mid-surface of the plate should be moved half of the plate thickness

into the beams. Figure 3.5 illustrated the problem described in this step.

Figure 3.5 : Plate Movement Illustration

iii. Creating the Joints

This step is to make sure that all the parts connected, so when the finite element

analysis using ANSYS conducted, all the parts are interacting each other. Figure 3.6

shows the joint created using ANSYS workbench (green line).

Figure 3.6 : Creating the Joints

Final Position

Initial Position

6

3.3. Joint Fixing in ANSYS Mechanical

After transferring the geometry from ANSYS Workbench into ANSYS Mechanical,

usually the model is not ready yet to be analyzed. Some geometry error might appear and

need to be fixed. In this case, the correct joint and surface area is needed before conducting

finite element analysis. Figure 3.7 describes the error that needs to be fixed.

Figure 3.7 : Geometry Error After Transferring into ANSYS Mechanical

It can be seen from figure above that the joint between parts does not make any specific

area between them. It is indicating that the line and area is not well connected and need to

be repaired. Some graphical repairing and addition command in ANSYS such as AGLUE

or AOVLAP needed to make sure the division of joint and area. The result after fixing this

problem can be seen in Figure 3.8 below.

Figure 3.8 : Geometry After Fixing the Problem

7

4. Finite Element Analysis (FEA) using ANSYS Mechanical APDL

4.1. Plate Theory

This theory is based on Beam Theory, just the extension from 2D into 3D. The

characteristic of this theory is also the same, that the lateral loads to the plate are carried

by hear forced and bending moments. According to the behavior of plates, it can be

distinguished by three method of analysis :

i. Thin plate theory (Kirchhoff theory) for t/L<1/10

This theory is analogous to Euler-Bernoulli Beam Theory in 2D problem

ii. Moderately Thick plate theory (Mindlin-Reissner theory) for 1/3>t/L>1/10

This theory is analogous to Timoshenko Beam Theory in 2D problem

iii. Thick Plate theory (3D FEA) for t/L>1/3

4.2. Introduction of FEA

Finite Element Analysis is approximation from Beam and Plate theory by numerical

analysis method that may be used to solve problems in many engineering disciplines

especially in structural mechanics. There are three basic steps that FEA usually has :

i. Preprocessing phase: in this phase, the problem will be created and discretized into

finite elements (meshing)

ii. Solution Phase : in this phase, the problem will be solved by a set of linear algebraic

equations to obtain nodal results

iii. Postprocessing Phase: in this phase, the strains and stressed will be computed from

nodal results. It also have to be verified for its accuracy by the user.

The illustration of FEA approximation can be seen in figure below.

Figure 4.1. Illustration of FEA

Meshing

8

As we can see from figure above, the structure will be modeled into discretized model

(meshing). Each mesh (element) will consist of number of nodes depending in the choosen

element. The choice of the element will affects the accuracy of finite element calculation.

In FEA, the displacement is assumed by polynomial function with n order (shape function)

with the equation below.

v=N1v1+N2v2+N3v3+…+Nmvm

with :

v = displacement within the element

Ni = shape function of node i (i=1,2,3) with nth order polynomial function

vi = displacement of node i

m = number of nodes

To ensure that the FE solution converges to the exact solution of the mathematical model

as the FE mesh is indefinitely refined, the following requirements must be satisfied :

Completeness Criterion : the element must represent all rigid body and constant

strain modes => the polynomial used in the shape function N have to be complete

the mth order, where m defines the order of the strain-displacement differential

operator Δ

Compatibility Criterion : the assumed displacements should be such that the

strains at the interface between elements are finite (even though they may be

discontinuous). Which means that the displacement within the element v have to

be chosen such that Cm-1 continuity is obtained for all nodes and across all

interelement boundaries.

Since FEA is approximation, the result is not exactly the same with the reality. But since

continuity is already assured by compatibility criterion, then the displacement result will

exactly the same with the real problem (accurate).On the other hand, the stress result is

based on averaged sense of equilibrium which will give the stress result as average of

exact stress within element. The typical FEA result can be seen from figure below.

9

Figure 4.2 Example of FEA Result (Left : Displacement, Right : Stress)

4.3. Element Types in ANSYS

In this study case, there are two types of element that have been used to conduct the

simulation using ANSYS, Beam 188 and Shell 181.

Beam 188 is based on Timoshenko beam theory which includes shear-deformation on its

calculation. This element includes two nodes and consists of 6 degree of freedom in each

node (3 translation and 3 rotation). The provided stress-stiffness term enable the element

to analyze flexural, lateral and torsion stability problem. This element is used in bulb

section of stiffener, while assuming its section with rectangular form. The discussion

about this will be represented in Chapter 4.5.

Shell 181 represents Reissner-Mindlin theory (first-order shear deformation). This

element consist of 4 nodes and each node consists of 6 degree of freedom. Shell 181 can

represent linear elastic, elastoplastic, creep or hyperelastic material. However, only

isotropic, anisotropic and orthotropic linear elastic can be input for elasticity. Shell 181 is

used for modeling area of plate, transverse beam and also web of stiffener.

4.4. Von Mises Stress

Von Mises stress is a criterion which tells us the maximum stress that can be handled by

the material due to a certain load case within its elastic limit. Von Mises stress criteria

works well especially when the material is ductile in nature.

𝜎𝑣 = √(𝜎1 − 𝜎2)2 + (𝜎2 − 𝜎3)2 + (𝜎3 − 𝜎1)2

2

Criteria : 𝜎𝑣 > 𝜎𝑦𝑖𝑒𝑙𝑑

10

The illustration of Von Mises stress best described in figure below.

Figure 4.3 : Illustration of Von Mises Stress

4.5. Geometry Reduction in ANSYS (HP Profile Bulb Section)

Since ANSYS reads our geometry based on line and area, there was some change of the

balcony geometry that already modeled in CAD software. In this case, the bulb profile in

ANSYS is only modelled with line, and some correction must be made to overcome this

problem.

Figure 4.4 : Bulb Section Problem in ANSYS

The problem can be overcome by using Parallel Axis Theorem and applying it then

comparing the Area and Second Moment of Inertia of bulb section with created new

section that using rectangular Beam 188 element with the same number of those two

parameters. The result and illustration can be seen from table and figure below.

Table 4.1 : Defined Analogue Rectangular Section

Bulb Section (HP Type 180x8) Analogue Rectangular Section

Area of Bulb Section 2.17x2.04 Cm2 Area of Rectangular Section 4.43 Cm2

Second Moment of Inertia

Ix 610.4956 Cm4 Ix 610.5 Cm4

Iy 7.2491 Cm4 Iy 7.43 Cm4

11

Figure 4.5 : Stiffener Problem Correction Illustration

4.6. Mesh Convergence Test

To get better result in FEA, there are two ways to improve the result. First is H-

Refinement, which the mesh size is decreased hence the number of degree of freedom is

increased. Second is P-Refinement, where the order of interpolation polynomial is

increased.

While it is better to do all of those refinement, but nothing is for free, it supposed to come

with a higher computing price. Mesh Convergence Test is to find a mesh size which gives

accurate enough result with reasonable computing time. The way to do it is by decreasing

the mesh size in certain range and spacing between values, and the solution results is

observed (you can choose between Displacement and Stress). Since displacement is

exactly approximated by FEA in nodal point, we can observe the result in one certain

nodal point. There must be significant change of result between coarse mesh and finer

mesh, and in finer mesh region the value will converge into one point. The aim of this

Mesh Convergence Test is to take the largest mesh size in that Convergence Line. It is

best to see Figure 4.6 for better picture of the test.

Figure 4.6 : Convergence Test

Based on figure above, 0.04 m mesh size will be used in this study case.

2.32E-02

2.34E-02

2.36E-02

2.38E-02

2.40E-02

2.42E-02

2.44E-02

2.46E-02

00.050.10.150.20.250.3

Dis

pla

cem

ent

(m)

Mesh Size (m)

Stiffener Height

Analogue Area

Convergence

Line

12

4.7. ANSYS Initial Model Analysis

After completing all the steps above, analysis using ANSYS can be proceed to get the Von

Mises Stress and also the Displacement. Those two parameters is used because we want

to design durable structure with considering aesthetics when it is being used (not too high

displacement). The initial result of the structure can be seen from figure and table below.

Figure 4.7 : Vertical Displacement of Structure (Initial Assumption)

Figure 4.8 : Von Mises Stress of Structure (Initial Assumption)

13

Table 4.2 : Result of Assumption Model

Variables Results

Von Mises Stress 323.09 MPa

Displacement 0.04546 m

Structural Weight 2044.8 kg

As we can see from the results above, Von Mises Stress still far above the requirement

(Yield Stress), but it gives a very good result in displacement. It shows that the biggest

stress is located in transverse beam area, while it seems like the stiffeners just give small

contribution in resisting the external force. However, stiffeners retain the displacement in

the area that is not detained by transverse beam.

Some optimization needed to decrease the Von Mises Stress and also decrease the

structural weight if it is possible. Some improvement must be made in T-Bar strength in

withstanding the vertical force, decreasing the number of stiffeners (No need to change

the stiffener type) and also decreasing the plate thickness (which gives very large

contribution in total structure mass). More light the structure weight is better and will

simplify the work of installation.

5. Applying DNV-GL Classification Rules as a Part of Optimization

Because of the assumed structure fail, the optimization must be made to ensure the objective

of durable and aesthetic structure fulfill. One of the methods of optimization in this study case

is by using DNV-GL rules Part 3 Chapter 1 “Hull Structural Design – Ships with Length 100

meters and above”. There are several additional assumption has been made, to fulfill the

requirement of variable for calculation using this rule.

Table 5.1 : Basic Assumption for DNV-GL Rule

Basic Assumption

Ship type Cruise Ship

Ship name Norwegian Gem

Size 93530 GT

Draft 8.2 m

Length 294.13 m

14

Table 5.2 : Basic Assumption for DNV-GL Rule (cont.)

Basic Assumption

Width 32.31 m

Block Coefficient 0.71

Speed 28 knot

Source http://transport.vestforsk.no/Dokumentasjon/pdf/Skip/Cruise.pdf

Because of our problem is very theoretical, the rules cannot be followed blindly. Section

modulus of transverse beam and stiffeners must be found by using Beam Theory because of

clamped end in one of the longitudinal side, and also it is assumed that modulus of section of

stiffeners can be found by the same method (stiffeners end clamped in transverse beam). Load

that used in this calculation is the same as the given parameter in section 2. Illustration of the

assumption can be seen from figure below.

Figure 5.1 : Structural Basic Assumption

All the formulas used in the calculation can be found in DNV-GL Rules digital copy, and also

the calculation sheet is attached in Attachment section. The result shows requirement that must

be fulfilled based on the rule and it shows in Table 4.4 below.

Table 5.3 : Requirement Based on DNV-GL Rules

DNV-GL Requirement

Plate Thickness 12.883 mm

Stiffener’s Section Modulus (Zstiff) 86.221 cm3

Stiffener’s Thickness 6.5 mm

Transverse Beam’s Section Modulus (ZTbar) 818.708 cm3

Transverse Beam’s Thickness 12.433 cm3

And several changes has been made to fulfill the requirement above, Table 4.5 describes all

the changes that made.

http://transport.vestforsk.no/Dokumentasjon/pdf/Skip/Cruise.pdf

15

Table 5.4 : Requirement Based on DNV-GL Rules

Structure Parameter Changes

Plate Thickness 12.883 mm

Stiffener Type HP 180x8 (the same as before)

Number of Stiffener 9

Zstiff Required After Changes 51.732 cm3

Transverse Beam Type T 350x300 (Section Modulus = 447.3 cm3)

Transverse Beam’s Section Thickness Flange = 24 mm, Web = 13 mm

Number of Transverse Beam 5

ZTbar Required After Changes 409.354 cm3

New structure have to be analyzed by using ANSYS to make sure that the rule give the good

result for optimization. Figure 5.2, Figure 5.3 and Table 5.5 shows the result of analysis in Von

Mises Stress and Displacement.

Figure 5.2 : Vertical Displacement of Structure (DNV-GL Rules Optimization)

16

Figure 4.8 : Von Mises Stress of Structure (DNV-GL Rules Optimization)

Table 5.5 : Result of DNV-GL Rules Model

Variables Results




From the result we can have several conclusions :

1. Our objective to have only 2 T-bar in the structure is violated. Even though bigger

dimension of T-bar can be used, but availability in the market also have to be considered.

So the only way to fulfill the DNV-GL rules standard is to increase the number of T-bar.

2. Increment of number of stiffener has been made since the thickness requirement already

fulfilled but not for section modulus. The other way is to change the stiffener’s type, but to

increase stiffener’s number is much preferable in this study case.

3. Safety factor is around 5.438 and it is considerably high. This is due to factors that DNV-

GL Rules account for :

Acceleration (effect of motion)

Corrosion thickness addition

17

Another safety factor

Since the safety factor is too high in this case, some other optimization must be conducted to

fulfill the objective with efficient result.

6. Particle Swarm Optimization

6.1. Introduction

Particle Swarm Optimization (PSO) is a computational method that optimizes a problem by

iteratively trying to improve a candidate solution with regard to a given measure of quality.

The method is originally attributed to Kennedy, Eberhart and Shi and was first intended for

simulating social behavior. A basic variant of the PSO algorithm works by having a population

(called swarm) of candidate solutions (called particles). These particles are moved around in

the search-space according to a few simple formulas. When improved positions are being

discovered these will then come to guide the movements of the swarm. The process is repeated

and by doing so it is hoped to hit the satisfactory solution.

6.2. PSO Input Parameter

Since the only interest is to reduce the Von Mises stress that located in T-bar section, all the

parameters related to T-bar will be assigned as the input of PSO process. Plate thickness also

included to reduce the maximum weight of structure. Table 6.1 describes the interval and range

of each parameter that will be optimized.

Table 6.1 : PSO Input Parameter

Parameter Range (minimum : interval : maximum)

Plate Thickness 0.003 : 0.001 : 0.02 m

Transverse Beam Height 0.2 : 0.05 : 0.6 m

Transverse Beam Flange 0.2 : 0.05 : 0.6 m

Transverse Beam Web Thickness 0.005 : 0.001 : 0.012 m

Transverse Beam Flange Thickness 0.005 : 0.001 : 0.012 m

6.3. PSO Script Way of Works

PSO process in this study case is conducted by using Matlab script that provided by TMR 4320

– Simulation Based Design course and developed by the lecturer Prof. Sören Ehlers. As a

18

student, the only thing that need to understand is how to input, set the objective and constraint

function, and how to read the output.

After set the input by assigning the value in previous sub-section, the objective and constraint

function must be established. Because our objective is to decrease the total weight of structure

and Von Mises stress that below yield stress, we can assign total weight as an objective

function and stress problem as a constraint function.

Objective = Mass of the plate + Mass of total stiffeners + Mass of total transverse beams

Constraint = 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑆𝑡𝑟𝑒𝑠𝑠

𝑌𝑖𝑒𝑙𝑑 𝑆𝑡𝑟𝑒𝑠𝑠− 1, if Constraint > 0 => Constraint=1.

Feasible Solution => -1 < Constraint < 0

Infeasible Solution => 0 < Constraint < 1

Modified Matlab script is attached in Attachment section.

6.4. PSO Output

After modifying and running the given script, it will present the output graphically as can be

seen in figure below.

Figure 6.1 : Objective Function PSO Output

19

Figure 6.2 : Constraint Function PSO Output

And the Figure 6.3, Figure 6.4 and Table 6.2 below shows the output from FEA using ANSYS

with the optimized parameter by PSO.

Figure 6.3 : Vertical Displacement of Structure (PSO)

20

Figure 6.3 : Vertical Displacement of Structure (PSO)

Table 6.2 : Result of PSO

Variables Results




6.5. PSO Conclusion and Discussion

After doing the optimization using PSO, there are several improvement in the structure

comparing from the first initial assumption.

1. From Figure 6.1 and Figure 6.2 above, it shows that the iteration process fulfill the set

objective and constraint function works correctly (constraint function graph varied in

the range of feasible solution). Iteration to find minimum mass with feasible solution

due to allowable stress (yield criteria) is reached.

2. Objective to find Von Mises stress and vertical displacement below the allowable is

accomplished.

21

3. Our allegation proved, the most vital part to be optimized is transverse beam section.

The result of optimized parameter (including plate thickness) can be seen in Table 6.2

below.

Table 6.2 : Result of PSO

Parameter Range (minimum : interval : maximum)

Plate Thickness 0.007 m

Transverse Beam Height 0.4 m

Transverse Beam Flange 0.2 m

Transverse Beam Web Thickness 0.005 m

Transverse Beam Flange Thickness 0.006 m

7. Conclusion

This report presents a systematic approach to optimize structure under given assumption

parameters and boundary conditions. Since the problem is in 3D, FEA helps to calculate the

result with better approximation on condition that the user understand the theory behind and

how to perform the analysis. In this study case, ANSYS software is being used since it is

already well established. Problem like geometry reduction and convergence test has been made

to make sure the analysis perform with good input, hence it will gives better approximation of

the problem. The element choice also have to be decided based on beam and plate theory,

ANSYS Help gives explanation about the element provided in the program.

For the first time, several assumption of structure have to be made. Result shows failure in Von

Mises stress, and the optimization must be made to reach the objective of yield stress criteria,

vertical displacement and also lighter structural weight.

From optimization using DNV rule, it shows a not very satisfactory result due to the

considerably high safety factor assigned in that rule. Number of maximum transverse beam

violated and also increment of number of stiffener has been made to fulfill the requirement.

While PSO gives a very good result and all the objective is accomplished without violating

any constraints. Total structure weight decreasing about 461.6 kilogram which is a good thing

for easiness of installation.

22

8. Suggestion and Improvement

Since PSO does not consider about safety factor, there are two ways to improve the result of

the optimization :

1. Adding safety factor such as corrosion thickness, etc.

2. Detailed load case and extreme load analysis, so the result will be based on the maximum

load that might be happened in the structure.

9. References

[1] Moan, Torgeir. Kompendium, TMR 4190 Finite Element Modeling and Analysis of Marine

Structures. Department of Marine Technology. 2003

[2] Mathisen, Kjell Magne. Lecture Notes, TMR 4190 Finite Element Modeling and Analysis of

Marine Structures. Department of Marine Technology. 2015

[3] DNV-GL. DNV-GL Rules Part 3 Chapter 1 : Hull Structural Design – Ships with Length 100

meters and above. DNV-GL. 2015

[4] Madenci, Erdogan & Guven, Ibrahim.The Finite Element Method and Applications in

Engineering Using ANSYS. Springer. 2015

10. Attachment

10.1. Matlab Exercise Result

Exercise 1

1. a. Topology Matrix I

% Topology matrix T(node1,node2,propno),

T = [ 1 2 1

2 3 1 ]; %propno defines the element property on each element.

b. Topology Matrix II % Topology matrix T(node1,node2,propno),

T = [ 1 7 1;2 7 1;8 7 1;1 8 1;2 8 1;3 8 1;4 8 1;5 8 1;6 8 1;9 8 1;

5 9 1;6 9 1 ]; %propno defines the element property on each element.

2. a. assmk.m code

function K = assmk(K,Ke,Te,dof)

%**********************************

*****************

% AssmK:

% Assembles system matrix by

adding element

% matrix to existing global

matrix.

% Syntax:

% K = assmk(K,Ke,Te,dof)

% Input:

% K : global matrix.

% Ke : element matrix.

% Te : element topology vector.

% dof: degrees of freedom per

node.

% Output:

% K : updated global matrix.

%**********************************

****************

%%% Initial K Matrix defined in

topology.m %%%

%%%%%%Building Topology

Matrix%%%%%%

23

% Element Numbering Component %

a=zeros(rows(Te),2*(cols(Te)-1));

for j=1:(cols(Te)-1);

if Te(1,j)==1

a(1,((2*j)-1):2*j)=[1 2];

else

a(1,((2*j)-

1):2*j)=[((2*Te(1,j))-1)

(2*Te(1,j))];

end

end

% Element Topology Matrix %

ae1=zeros((dof*2),cols(K));

ae2=ones((dof*2),cols(K));

%%%%%%Building Stiffness

Matrix%%%%%%

% Assembling Topology Matrix for

Stiffness Matrix Calculation

for i=1:cols(a)

ae1(i,a(i))=ae1(i,a(i))+ae2(i,a(i))

;

end

% Assembling Stiffness Matrix %

K=K+(ae1'*Ke*ae1);

b. calculate global stiffness matrix

c. write setbc.m

function [K] = setbc(K,C,dof)

%**********************************

*****************

% SetBC:

% Syntax:

% [K] = setbc(K,C,dof)

% Input:

% K : original global

stiffness matrix.

% C : constraint matrix, C = [

node1 dof1 u1

%

node2 dof2 ...].

% or for dof=1, C = [

node1 u1

%

node2 ...].

% dof : number of dof pr. node.

% Output:

% K : new global stiffness

matrix.

%**********************************

*****************

%%% Define Global Number of Nodes

Based on C Matrix %%%

%%% zp = [(Global Node Number)

(Displacement)] %%%

zp=zeros(rows(C),(cols(C)-1));

for i=1:rows(C)

zp(i,1)=((dof*C(i,1))-

dof+C(i,2));

zp(i,2)=C(i,3);

end

%%% Define K Matrix Unreduced

Dimension %%%

for i=1:rows(zp);

K(zp(i,1),:)=K(zp(i,1),:)*zp(i,2);

K(:,zp(i,1))=K(:,zp(i,1))*zp(i,2);

end

%%% Define K Matrix Reduced

Dimension %%%

% Define Retained Node %

in=1:(rows(K));

ns=zeros(rows(C),(rows(K)));

for i=1:rows(zp);

for j=1:(rows(K));

if in(j)==zp(i)

ns(i,j)=in(j);

else

ns(i,j)=0;

end

end

end

ns=sum(ns);

fn=in-ns;

fn=fn(fn~=0);

% Define Reduced Dimension

Stiffness Matrix %

Kt=zeros(size(fn,2));

for i=1:size(fn,2);

24

for j=1:size(fn,2);

Kt(i,j)=K(fn(i),fn(j));

end

end

K=Kt

d. find the diplacements (assumed force with magnitude 10 in node 2 (x-dir)

10.2. DNV-GL Rules Calculation Sheet

10.3. Parametric Modeling Script

/input,parameter,inp

!MATERIAL MODEL

/PREP7

MPTEMP,,,,,,,,

MPTEMP,1,0

MPDATA,EX,1,,200000000000

MPDATA,PRXY,1,,0.3

MPDATA,DENS,1,,7850

NUMSTR,KP,1001

K,,0,0,0

K,,l_plate,0,0

K,,l_plate,b_plate,0

K,,0,b_plate,0

NUMSTR,AREA,1001

A,1001,1002,1003,1004

ASEL,S,AREA,,1001,1999,,,

CM,PLATEDUMMY,AREA

ASEL,ALL

NUMSTR,KP,2001

*if,n_tbeam,ge,1,then

*Do,i,1,n_stiff

K,,0,i_stiff+((i-1)*s_stiff),h_stiff

K,,0,i_stiff+((i-1)*s_stiff),0

*Do,j,1,n_tbeam

t0 (mm) 5.5

k 0.02

f1 1 (normal steel)

tk (mm) 1.5 (corrosion addition)

t (mm) 12.8826

Plating

a 0.8 (assumed : weather deck elsewhere)

ks 2 (assumed : Between 0.2L and 0.7L from AP)

Cw (Wave Coeff) 10.73577812 (300<=L<=350)

f 8.588622492 (Vertical distance from the waterline to the top)

Kf 8.2 (Smallest between Draft and f)

V/sqrt(L) 1.632632246

pl 31.00365591

y (m) 8.0775 (assumed : minimum, cause don't know load point)

z (m) 8.2 (assumed : maximum , cause don't know load point)

pdp (kN/m2) 41.1654535

h0 (m) 3 (assumed distance from waterline to deck with draft T)

p1 (kN/m2) 22.3723628

g0 (m/s2) 9.81

kv 0.7 (assumed between 0.3L and 0.6L from A.P.)

Cv 0.2

Cvi 1.632632246

a0 0.436026787

av (m/s2) 4.217177388

rho water (kg/m3) 1025

Water height (m) 1

pwater (ton/m2) 1.025

phuman (ton/m2) 0.014777778

q (ton/m2) 1.039777778

p2 (kN/m2) 12.39268367

Design Load

l (m) 2 (stiffener span, considering the clamped sect is intersect tbar-stiff)

q (kN/m2) 12.39268

N Longitud 9

Spacing 0.45 (assumed the spacing between longitudinal is equal)

M (kNm) 11.15342

yield (Mpa) 235

tkw 1.5 mm

wk 1.09

Z 51.73286 cm3

k 0.5

tk 1.5 mm

t 6.5 mm

Longitudinal (stiffener)

l (m) 3

q (kN/m2) 12.39268

N Longitud 5

s (m) 1.5

M (kNm) 83.65061

yield (Mpa) 235

tkw 1.5 mm

tkf 1.5 mm

wk 1.15

Z 409.3541 cm^3

k 5.8826 mm

tk 1.55 mm

t 12.4326 mm

Transversal

25

K,,i_tbeam+((j-1)*s_tbeam),i_stiff+((i-

1)*s_stiff),h_stiff

K,,i_tbeam+((j-1)*s_tbeam),i_stiff+((i-

1)*s_stiff),0

*enddo

K,,l_plate,i_stiff+((i-1)*s_stiff),h_stiff

K,,l_plate,i_stiff+((i-1)*s_stiff),0

*enddo

*else

*Do,i,1,n_stiff

K,,0,i_stiff+((i-1)*s_stiff),h_stiff

K,,0,i_stiff+((i-1)*s_stiff),0

K,,l_plate,i_stiff+((i-1)*s_stiff),h_stiff

K,,l_plate,i_stiff+((i-1)*s_stiff),0

*enddo

*endif

NUMSTR,AREA,2001

*if,n_tbeam,ge,1,then

*Do,i,1,n_stiff

*Do,j,1,n_tbeam+1

A,2000+(2*j-1)+((i-

1)*((2*n_tbeam)+4)),2001+(2*j-1)+((i-

1)*((2*n_tbeam)+4)),2003+(2*j-1)+((i-

1)*((2*n_tbeam)+4)),2002+(2*j-1)+((i-

1)*((2*n_tbeam)+4))

*enddo

*enddo

*else

*Do,i,1,n_stiff

A,2000+((4*i)-3),2001+((4*i)-3),2003+((4*i)-

3),2002+((4*i)-3)

*enddo

*endif

ASEL,S,AREA,,2001,2999,,,

CM,STIFF,AREA

ASEL,ALL

NUMSTR,KP,3001

*if,n_stiff,eq,1,then

*Do,i,1,n_tbeam

K,,i_tbeam+((i-1)*s_tbeam),0,h_tbeam

K,,i_tbeam+((i-1)*s_tbeam),0,0

K,,i_tbeam+((i-1)*s_tbeam),i_stiff,h_tbeam

K,,i_tbeam+((i-1)*s_tbeam),i_stiff,0

K,,i_tbeam+((i-1)*s_tbeam),b_plate,h_tbeam

K,,i_tbeam+((i-1)*s_tbeam),b_plate,0

*enddo

*else

*Do,i,1,n_tbeam


K,,i_tbeam+((i-1)*s_tbeam),0,0


K,,i_tbeam+((i-1)*s_tbeam),b_plate,0

*enddo

*endif

NUMSTR,KP,4001

*Do,i,1,n_tbeam

K,,i_tbeam+((i-1)*s_tbeam)-

(f_tbeam/2),b_plate,h_tbeam

K,,i_tbeam+((i-1)*s_tbeam)-

(f_tbeam/2),0,h_tbeam



K,,i_tbeam+((i-

1)*s_tbeam)+(f_tbeam/2),b_plate,h_tbeam

K,,i_tbeam+((i-

1)*s_tbeam)+(f_tbeam/2),0,h_tbeam

*enddo

NUMSTR,AREA,3001

*If,n_stiff,gt,1,then

*Do,i,1,n_tbeam

A,3000+((4*i)-3),3001+((4*i)-

3),2002+(2*i),2001+(2*i),2001+(2*i)+((n_stif

f-1)*((2*n_tbeam)+4)),2002+(2*i)+((n_stiff-

1)*((2*n_tbeam)+4)),3003+((4*i)-

3),3002+((4*i)-3)

*enddo

*elseif,n_stiff,eq,1,then

*Do,i,1,n_tbeam

A,2995+(6*i),2996+(6*i),2998+(6*i),2997+(6

*i)

A,2997+(6*i),2998+(6*i),3000+(6*i),2999+(6

*i)

*enddo

*else

*Do,i,1,n_tbeam

A,3000+((4*i)-3),3001+((4*i)-3),3003+((4*i)-

3),3002+((4*i)-3)

*enddo

*endif

*if,n_stiff,gt,1,then

NUMSTR,AREA,5001

*Do,i,1,n_tbeam

A,2001+(2*i),2002+(2*i),2002+(2*i)+((n_stiff

-1)*((2*n_tbeam)+4)),2001+(2*i)+((n_stiff-

1)*((2*n_tbeam)+4))

*enddo

*endif

NUMSTR,AREA,4001

*Do,i,1,n_tbeam

A,3995+(6*i),3996+(6*i),3998+(6*i),3997+(6

*i)

A,3997+(6*i),3998+(6*i),4000+(6*i),3999+(6

*i)

*enddo

ASEL,S,AREA,,3001,3999,,,

CM,TBASE,AREA

ASEL,ALL

ASEL,S,AREA,,4001,4999,,,

CM,TTOP,AREA

ASEL,ALL

ASEL,S,AREA,,5001,5999,,,

CM,TBASECUT,AREA

ASEL,ALL

*if,n_stiff,GT,2,then

NUMSTR,AREA,6001

ASBA,TBASECUT,STIFF,,DELETE,KEEP

ASEL,S,AREA,,6001,6999,,,

CM,TBASECUTN,AREA

*elseif,n_stiff,EQ,0,then

ASEL,S,AREA,,3001,3999,,,

CM,TBASECUTN,AREA

*else

ASEL,S,AREA,,5001,5999,,,

CM,TBASECUTN,AREA

*endif

CMGRP,TBASEN,TBASE,TBASECUTN

ASEL,ALL

CMGRP,VERT,STIFF,TBASEN

NUMSTR,AREA,7001

*if,n_stiff,GT,0,and,n_tbeam,GT,0,then

ASBA,PLATEDUMMY,VERT,,DELETE,KE

EP

ASEL,S,AREA,,7001,7999,,,

CM,PLATE,AREA

ASEL,ALL

NUMMRG,ALL

*else

ASEL,S,AREA,,7001,7999,,,

CM,PLATE,AREA

ASEL,ALL

NUMMRG,ALL

*endif

SECT,1,SHELL,,PLATE

SECDATA,t_plate,1,0.0,3

SECOFFSET,MID

SECCONTROL,,,, , , ,

SECT,2,SHELL,,TB-W

SECDATA,t_tbase,1,0.0,3

SECOFFSET,MID


SECT,3,SHELL,,TB-F

SECDATA,t_ttop,1,0.0,3

SECOFFSET,MID


SECT,4,SHELL,,HP-W

SECDATA,t_stiff,1,0.0,3

SECOFFSET,MID


SECTYPE,5,BEAM,RECT,HP-F,0

SECOFFSET,USER,(b_bulb/2)+(t_stiff/2),h_

bulb/2

SECDATA,b_bulb,h_bulb,0,0,0,0,0,0,0,0,0,0

26

ET,1,SHELL181

ET,2,SHELL181

ET,3,BEAM188

CMSEL,S,PLATE

AESIZE,PLATE,e_plate

CMSEL,S,TBASEN

AESIZE,TBASEN,e_tbase

CMSEL,S,TTOP

AESIZE,TTOP,e_ttop

CMSEL,S,STIFF

AESIZE,STIFF,e_stiff

LSEL,S,LOC,Z,h_stiff-0.01,h_stiff+0.01

LSEL,R,LOC,Y,i_stiff,i_stiff+((n_stiff-

1)*s_stiff)

*do,i,1,n_tbeam

LSEL,U,LOC,X,i_tbeam+((i-1)*s_tbeam)

*enddo

CM,BULB,LINE

LESIZE,BULB,e_bulb

!PLATE MESH SELECTION

TYPE,1

MAT,1

REAL,

ESYS,0

SECNUM,1

CHKMSH,'AREA'

CMSEL,S,PLATE

AMESH,PLATE

ESLA,S

CM,PLATEE,ELEM

ALLSEL

!T-BEAM WEB MESH SELECTION

TYPE,2

MAT,1

REAL,

ESYS,0

SECNUM,2

CHKMSH,'AREA'

CMSEL,S,TBASEN

AMESH,TBASEN

ESLA,S

CM,TBASENE,ELEM

ALLSEL

!T-BEAM FLANGE MESH SELECTION

TYPE,1

MAT,1

REAL,

ESYS,0

SECNUM,3

CHKMSH,'AREA'

CMSEL,S,TTOP

AMESH,TTOP

ESLA,S

CM,TTOPE,ELEM

ALLSEL

NUMMRG,NODE

!STIFFENER WEB MESH SELECTION

TYPE,2

MAT,1

REAL,

ESYS,0

SECNUM,4

CHKMSH,'AREA'

CMSEL,S,STIFF

AMESH,STIFF

ESLA,S

CM,STIFFE,ELEM

ALLSEL

!STIFFENER FLANGE MESH SELECTION

*if,n_stiff,GT,1,then

TYPE,3

MAT,1

REAL,

ESYS,0

SECNUM,5

CMSEL,S,BULB

LMESH,BULB

ESLL,S

CM,BULBE,ELEM

ALLSEL

*else

ALLSEL

*endif

/PREP7

NSEL,S,LOC,Y,0

D,ALL,,0,,,,UX,UY,UZ,ROTX,ROTY,ROTZ

NSEL,ALL

/PREP7

ALLSEL

SFA,PLATE,1,PRES,press

ACEL,,,9.81

Fk,2004,FZ,force1

Fk,2022,FZ,force2

Fk,2038,FZ,force3

/SOLU

SOLVE,,

!POST PROCESSOR

/POST1

/EFACET,1

PLNSOL,S,EQV,0,1.0

*GET,max_stress,PLNSOL,0,MAX,,,,

*cfopen,out_stress.inp

*vwrite,max_stress

(f15.2)

*cfclos

PLNSOL,U,Z,0,1.0

*GET,max_disp,PLNSOL,0,MAX,,,,

*cfopen,out_disp.inp

*vwrite,max_disp

(f10.5)

*cfclos

10.4. Particle Swarm Optimization Matlab Script

StructOpt.m

function StructOpt

% How to get initial population?

% 1 = random or 2 = read from file.

source = 1; %source

% Number of continous variables

continous_variables = 0;

% Upper and lower bound for continous

variables

LowerB = [];

UpperB = [];

% PSO parameters.

Algorithm = [5 -1 10 1.4 0.8 3 2 1];

% PSO-parameter

% swarm_size = 5;

% feasibles_initial_population = -1;

% Feasible designs in intial population (-1

dont want to use).

% generations = 10;

% Calculation rounds.

% inertia = 1.4;

% Intertia at start.

% beta = 0.8;

% Factor for dynamic inertia reduction

% beta_k = 3;

% Number of rounds when it should improve,

otherwise make inertia smaller

% penalty_factor = 2;

% Penalty factor for violated constraints

% print_results = 1;

% 0 not printed, 1 is printed (results)

27

% Set of discrete variables

Feasible_Set=[];

nvariables=3;

Feasible_Set=inf(nvariables,31);

% % Plate Thickness

Feasible_Set(1,1:21)=[0.008:0.0001:0.01];

% % T-Bar Web Thickness

Feasible_Set(2,1:11)=[0.006:0.001:0.016];

% % T-Bar Flange Thickness

Feasible_Set(3,1:11)=[0.006:0.001:0.016];

% Carry out the optimization.

if source == 1

[best_f,best_x,best_g,history_f,history_x,hi

story_g,iterations,...

particle,particle_history,t_history]

=

PSO('particle_fun',continous_variables,Feasi

ble_Set,LowerB,UpperB,Algorithm,[]);

elseif source == 2

[best_f,best_x,best_g,history_f,history_x,hi

story_g,iterations,...

particle,particle_history,t_history]

=

PSO('particle_fun',continous_variables,Feasi

ble_Set,LowerB,UpperB,Algorithm,'initial_fea

sible_population.txt');

end

%Best feasible particletion

format compact;

disp(' ')

disp(' ')

disp('Results:')

best_f,best_x,best_g

disp(' ')

disp(' ')

particle_fun.m

function [f,g] = particle_fun(Xxx)

%parameters

t_plate=Xxx(1);

t_tbase=Xxx(2);

t_ttop=Xxx(3);

fid=fopen('parameter2.inp','w')

fprintf(fid,'t_plate=%0.4f\n',t_plate);

fprintf(fid,'t_tbase=%0.4f\n',t_tbase);

fprintf(fid,'t_ttop=%0.4f\n',t_ttop);

fclose(fid);

!modeling.bat

fid2=fopen('out_stress.inp','r')

line=fgetl(fid2);

stress_max=str2num(line);

fclose(fid2);

yield=250000000 ;

objective=(l_plate*b_plate*t_plate*rho_steel)+(n_tbeam*((h_tbeam*t_tbase*b_plate*rho_steel)+(f_tb

eam*t_ttop*b_plate*rho_steel)))+(n_stiff*((h_stiff*t_stiff*l_plate*rho_steel)+(b_bulb*h_bulb*l_pl

ate*rho_steel)));

constraint=stress_max/yield-1;

% objective and constraint function value

f=objective; %Objective

g=constraint; %Constraint feasible -1 to 0 an infeasible is from >0 to 1

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