Structural Analysis of the 17 Century Warship Vasa

STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Influence of the dowels on the stiffness of the hull

Master Degree Project in Applied Mechanics One year Level 30 ECTS Spring term 2014

Juan Carlos Hurtado Sierra Marina Muñoz García

Supervisor: Dr. Anders Biel Examiner: Prof. Ulf Stigh CERTIFICATE OF AUTHENTICITY

This dissertation work has been performed by Juan Carlos Hurtado Sierra and Marina MuñozGarc´ıaas final thesis in the Master Science program in Applied Mechanics at the University of Skövdeof Sweden.

We hereby certiﬁed that all the material of this dissertation work project belongs to our work and, and those part which are not, have been identiﬁed and referred to their authors.

Juan Carlos Hurtado Sierra Marina Mu˜nozGarc´ıa

II ACKNOWLEDGEMENTS

We would like to express our most sincere gratitude to all the people and entities that have enabled the realization of this dissertation work. Firstly, to the University of Sk¨ovde,for providing us the required knowledge and tools to achieve the development of this thesis. We would like to thank to our project supervisor Dr. Anders Biel for his dedication, support and all his valuable advices provided during all the work.

A non-less important mention to the department of Applied Mechanics from Upp- sala Universitet, especially to Iv´onHassel, for her boundless kindness, assistance and dedication. Her words have been always a source of inspiration, positive energy and encouragement to accomplish the next step in the project. An especial distinction to Nico Van Dijk, Alexey Vorobiev, Reza Afshar and Florian Bommier for all their help and material provided.

Moreover, we must oﬀer our profoundest thankfulness to our coordinator from VasaMuseet M. Sc. Anders Ahlgren for his time dedicated showing us the Vasa.

We are also indebted to our families for doing their best to provide us this opportunity. Their unconditional dedication and words of encouragement have been another cause of motivation.

Me, as Marina, I would like to make my last mention to my college, best friend and love Juan Carlos, who has been the most essential person during this period in Sweden. Thank you for your unlimited patient, comprehension, support and love.

Me, as Juan Carlos, I have to thank mainly to my love and college Marina who is my support every day. Without her help all of this work would not have been possible. Thank so much Marina. Lastly, but certainly not least, I have to thank my family who are the main reason I’ve become the person I am today, they have helped me every step of the way.

III ABSTRACT

After 333 years under depths of the Baltic Sea, the warship Vasa was salvaged and nowadays lies in a dry dock inside the Vasa Museum in Stockholm. Its current support system, which consists on eighteen cradle-stanchions pairs of steel, is not able to handle the present loads in a satisfactory manner. Experimental tests showed that the Vasa’s hull is gradually deforming mainly due to creep behavior.

Thus, in order to preserve the Vasa for future generations, a new support system has to be implemented in a foreseeable future. There are several factors to take into consideration for its construction, which are: the degradation of the oak, its current mechanical properties and its inhomogeneity in addition to the climatic conditions of the Museum and the impossibility of taking unlimited specimens for its analysis. Hence, it is crucial to investigate the areas where the stresses and deformations are critical in the ship and how aﬀected is the stiﬀness of the hull, its most important component.

In this dissertation work two Finite Element Analyses are accomplished. The first study consists on the creation of a superelement of a section of Vasa’s hull with the intention of investigating the influence of the dowels into the stiffness of the hull. In the second analysis a simplified model of the entire warship Vasa is created in order to analyze it and locate possible critical areas on the hull due to its own weight and the stresses originated by the support system. The software selected for these simulations are Abaqus and Creo Simulate 2.0.

From the first study it is concluded that that the dowels do not have a significant influence in the stiffness coefficients of the hull. The second analysis determines that the maximum stresses are located on the bottom part of the hull. This dissertation work concludes with a suggested future work.

IV Contents

CERTIFICATE OF AUTHENTICITY II

ACKNOWLEDGEMENTS III

ABSTRACT IV

1 INTRODUCTION 1 1.1INTRODUCTION...... 1 1.2BACKGROUND...... 1 1.3PURPOSE...... 2 1.4LIMITATIONS...... 2

2 LITERATURE REVIEW 3 2.1 HISTORICAL OVERVIEW OF THE WARSHIP VASA ...... 3 2.2 DIMENSIONS AND WEIGHT OF THE VASA ...... 5 2.3 THE RESTORATION AND THE PRESERVATION WORK OF THE VASA 6 2.4 IMPACT ON THE MECHANICAL PROPERTIES IN VASA OAK . . . . 7 2.5 OVERVIEW OF THE VASA HULL STRUCTURE ...... 9 2.5.1 Hull...... 9 2.5.2 Deckbeams...... 10 2.5.3 Knees...... 10 2.5.4 Futtock rider and second futtock rider ...... 11 2.5.5 Secondrider...... 11 2.5.6 Firstrider...... 12 2.5.7 Therider...... 12 2.6HULLSUPPORTSYSTEM...... 13 2.6.1 Supports for upper and main deck ...... 13 2.6.2 Another signiﬁcant supports ...... 14 2.7FINITEELEMENTANALYSIS...... 15

3 INFLUENCE OF THE DOWELS ON THE STIFFNESS OF THE HULL 17 3.1INTRODUCTION...... 17 3.2 CONSIDERATIONS FOR THE CONSTRUCTION OF THE MODEL . . 18 3.2.1 Determine the density of dowels per unit of area ...... 18 3.2.2 Dimensionsofthemodel...... 19 3.2.3 Material orientation for each layer and for the dowels ...... 20 3.2.4 Location of the dowels in the model ...... 21 3.2.5 Friction coeﬃcients between dowels and diﬀerent layers ...... 21 3.2.6 Boundaryconditions...... 22

V 3.2.7 Rigid plate to applied loads ...... 23 3.3 ABAQUS CAE SETTINGS ...... 24 3.3.1 Methodused...... 24 3.3.2 Material Properties ...... 24 3.3.3 Steps & increments ...... 24 3.3.4 Mesh...... 24 3.4RESULTS...... 25 3.4.1 BaseModel...... 26 3.4.2 Comparison of Base model, Random Pattern 1 and Random Pattern 2 27 3.4.3 Comparison of Base Model with friction and frictionless ...... 28 3.4.4 Comparison of Base Model with and without dowels ...... 28 3.4.5 Procedure to reach young modulus ...... 29 3.5CONCLUSIONS...... 31

4 GLOBAL ANALYSIS OF THE WARSHIP VASA HULL 32 4.1INTRODUCTION...... 32 4.2 CONSTRUCTION OF THE MODEL ...... 32 4.3 FINITE ELEMENT ANALYSIS ...... 35 4.4 ASSUMPTIONS AND CONSIDERATIONS ...... 35 4.5 STUDY OF THE CONVERGENCE ...... 40 4.6RESULTS...... 44 4.7CONCLUSION...... 46

5 DISCUSSION 48

6 FUTURE WORK 49

REFERENCES 50

APPENDIX I: Vasa’s Drawings 53

APPENDIX II: Results from the study of convergence 59

APPENDIX III: Results from global analysis - Critical area on the hull 63

VI List of Figures

2.1 Illustration of the warship Vasa [5]...... 3 2.2 Comparison of a 74 cannons French vessel from 1789 (left), and the Vasa (right).[5]...... 4 2.3 The sinking of the Vasa [5]...... 4 2.4 The salvage of the Vasa,1961.[4]...... 5 2.5 Up, a new stainless steel bolt. Down, the replaced epoxy-coated galvanized mildsteelbolt[4]...... 6 2.6 Degradation of the mechanical properties of the Vasa oak[10]...... 7 2.7 Principal axes of wood (1.Longitudinal - 2.Radial - 3.Tangential) ...... 8 2.8 Mechanical properties of Vasa oak ...... 8 2.9CrosssectionoftheVasa[5]...... 9 2.10 Lower part of the hull structure [10] ...... 10 2.11AdeckbeamfromtheVasa[7]...... 10 2.12 Knees attached to the hull structure of the Vasa [7] ...... 11 2.13 Futtock rider and second futtock rider [7] ...... 11 2.14Secondrider[7]...... 12 2.15FirstrideroftheVasa[7]...... 12 2.16RideroftheVasa[7]...... 12 2.17 Support system of the Vasa: Left, a focused view of one of it stanchions. Right, the complete support system [4]...... 13 2.18 Internal supports for the upper and main deck [5]...... 14 2.19 Steel wires support the weight of the ﬁgurehead [16]...... 14 2.20 Diﬀerence of displacement of a cantilever beam between the standard FEM (left), and adaptive hp-FEM(right).[23] ...... 16

3.1FEmodelandSuperelement...... 17 3.2 Area at the bottom part of the Vasa’s hull ...... 18 3.3 Area of the Vasa’s hull between two gun ports ...... 19 3.4Dimensionsofthedowel...... 19 3.5 Dimensions of planking and ceiling ...... 20 3.6Dimensionsofframe...... 20 3.7 Material orientations of every component ...... 21 3.8 Layouts of Regular, Random 1 and Random 2 Patterns ...... 22 3.9Frictionareaofdowelandhole...... 22 3.10 Boundary conditions of the model ...... 23 3.11Dimensionsoftherigidplate...... 23 3.12Meshesofthemodel...... 25 3.13 Forces and moments applied on the model ...... 25

VII 3.14 Initial cross-sectional area and original length of the model ...... 29

4.1 a) sketch of one cross section of the hull. b) Structure composed by the totality of sketches which comprises the hull structure...... 33 4.2 a) Sketch of a frame from the Vasa model. b) Sketch of some structural frames which comprise the Vasa...... 33 4.3 Process of construction of the keel of the Vasa...... 34 4.4 Final model of the Vasa...... 34 4.5 Symmetry applied to the model ...... 36 4.6 Stanchion drawing and boundary condition modeling of a stanchion[7] . . . 37 4.7 Boundary conditions: Stanchions and symmetry constraints ...... 38 4.8 Loads applied on the model of the Vasa warship ...... 39 4.9 Graphs of the number of elements, edges, faces and points ...... 41 4.10 Bad and good stress fringe plot [23] ...... 41 4.11 Graph of Not Converged Elements vs. P-pass ...... 42 4.12 Graph of Strain Energy vs. P-pass ...... 42 4.13 Graph of Maximum von Mises Stress vs. P-pass ...... 43 4.14 Graph of Maximum displacement vs. P-pass ...... 43 4.15 von Mises Stress on a section of Vasa’s hull [MPa] ...... 44 4.16 von Mises Stress on several sections of Vasa’s hull [MPa] ...... 45 4.17 Maximum Shear Stress on several sections of Vasa’s hull [MPa] ...... 45 4.18 Deformation on the hull ...... 46

VIII STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Chapter 1

INTRODUCTION

1.1 INTRODUCTION

This dissertation work is a part of a research project in collaboration with Vasamuseet and the department of Applied Mechanics of Uppsala Universitet. It has a scientific- engineering character, and it is focused on the structural and finite element analysis of the warship Vasa. It is intended for the entities which have certain influence with the results obtained. These are: the Museum, which encompasses technicians and other specialists, the research personnel from Uppsala Universitet and Högskolani Skövde.

1.2 BACKGROUND

Since the warship Vasa was salvaged in April 1961 after sinking on its maiden voyage in the harbor of Stockholm in August 1628, there have been an unceasing preservation and researching work which have been improved along the years. It is worth of mention the chemical preservation treatment to the wooden elements, the replacement of certain iron structural components, carpenter works, the maintenance of a stable climate at Vasamuseet where it disposed and opened to the public; and the creation of an external structure in which the ship is supported.

By focusing this study to the support system of the warship, different researches and tests concluded in the necessity of building an improved support system in order to preserve the ship for future generations. The current support, which was constructed in the 1960s and enhanced in the 1990s, is not able to handle the present loads in a satisfactory manner. Experimental tests showed that the Vasa is gradually deforming and distorting, chiefly due to creep deformations [1]. In addition, the chemical degradation of the wood is affecting its mechanical properties [2].

Due to this fact, the hull of the ship which is the responsible of transmitting the loads to the current support system, is deforming and losing strength. Therefore, in a foreseeable future, a new support system has to be designed taking into consideration the current support structure and the present mechanical behavior of the ship.

1 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

1.3 PURPOSE

Hence, the principal aim of this dissertation work consists of creating ﬁnite element models and analyze them in order to contribute with the development of a new external support system for the warship Vasa and avoid its progressive deformations. For this:

• A FE model of the hull of the ship is created in order to investigate how dowels aﬀect to the stiﬀness of the hull.

• The values of the stiﬀness obtained are used for the calculation of the Young’s modulus. These obtained Young’s modulus are used in the simpliﬁed model of the entire ship.

• A simpliﬁed model of the entire ship is simulated and the properties obtained in the previous study are introduced so as to identify the most critical area on the hull and its behavior.

1.4 LIMITATIONS

This dissertation work in Applied Mechanics aims at analyzing models of the warship Vasa with the intention of studying the eﬀect of the dowels on the stiﬀness of the hull as well as the behavior of the warship. In order to perform the study, some delimitations are required. These are to:

• Assume average values in addition to delimitate the areas of the oak where its properties change. For this, a ﬁne mesh will be created and diﬀerent material properties will be applied to the sandwich structure of the hull of the ship.

• Assume an average number of dowels to the section of the hull chosen for this study. In reality, there is not a patron or sequence established when the dowels were introduced in the construction of the Vasa. They are disposed randomly, avoiding the knots of the timbers.

• Assume the materials properties and weight of the Vasa from previous studies due to the fact that there is a limitation of specimens taken from the ship for testing.

• Assume average values in the material properties of the Vasa oak since its properties are not uniform through the entire ship. In addition, it is necessary to delimitate the areas of the sandwich structure of the hull where its properties varies due to its deterioration. For this, a ﬁne mesh will be created in the model and certain material properties will be applied to the sandwich structure.

• Avoid the variation of material orientation along the wood of the global model. For instance, the hull and the hull structure has diﬀerent material orientation. The material orientation established in the global model is the one which meets with the hull.

2 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Chapter 2

LITERATURE REVIEW

2.1 HISTORICAL OVERVIEW OF THE WARSHIP VASA

By the early 1620, Sweden was entangled in war with Poland. Simultaneously, the strong growth of the Thirty Years’ War, started in Germany, created some mistrust and tension in the country. Thus, Sweden was involved in a conflict in which the war did not progress and the state of the royal fleet was becoming worse [3]. Then, if the pretension of the King Gustav II Adolf of Sweden of being mightier by conquering the Baltic Sea wanted to be accomplished, he should act. As response of this situation, in 1625 the King Gustav II Adolf decide to enlarge his fleet and ordered the construction of four warships, and the Vasa would be the first of them [4]. An illustration of the warship is shown in figure 2.1.

Figure 2.1: Illustration of the warship Vasa [5]

Thus, during the following three years, the construction of the warship took place. The project was directed by the shipbuilders Henrik Hybertson and Arendt Hybertson [3]. Throughout the whole fabrication process, several setbacks entailed the Vasa to its catastrophic end. Worth of mention the frequent interruptions of the king to put pressure on the shipbuilders to make the ship longer and equipped with a second deck of guns. In addition, the master oﬃcial who managed the project died, so their inexperienced workers assumed his responsibility even though their lack of technical knowledge.

3 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Once the construction of the Vasa was fulﬁlled, it was expected to be the best equipped warship, but also it was surprisingly heavy compared with other constructed vessels. Furthermore, it was too long and high in relation to its width. This entailed an elevation of its center of mass, and therefore its instability increased [3]. In order to improve this, its weight was increased though a great part of its hull was submerged. In Figure 2.2, it is noticed the diﬀerence in the design between the stern of the Vasa and a 74 cannons French warship from 1780. They are drawn at the same scale and put at the same water line.

Figure 2.2: Comparison of a 74 cannons French vessel from 1789 (left), and the Vasa (right). [5]

On 10th August 1628, the royal warship Vasa sank at its maiden voyage after two nautical miles, still on the outskirts of Stockholm. The warship was tilted by a gust of wind, but the experienced captain lead to right it rapidly. Nevertheless, after a second gust of wind, the Vasa was tilted again and it began to ﬂood through the open gun- ports from the deck. At least, thirty victims were counted of two hundred passengers in the catastrophe [6]. In ﬁgure 2.3, a watercolor painting by the artist Graham Coton representing the disaster is exposed.

Figure 2.3: The sinking of the Vasa [5]

4 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

There is not a unique factor to determine the cause of the disaster. Several happenings lay behind its sinking, but the lack of stability of the warship was one of the most relevant causes. Some time after the catastrophe, experts realized that the ship was well built, but its proportions were not correct and its weight was excessive [6]. Therefore, there is not a unique responsible, and the King Gustav II Adolf can be mentioned as one of them. He ordered a larger warship which also included a second deck of cannons and he approved the changes. The shipbuilder Henrik Hybertson can be also considered partly responsible. He managed the construction of the ship even when he did not have enough experience. Finally, another responsible of the sinking of the Vasa can be ascribed to its captain. He sailed with the lower gun-ports opened even knowing about the instability of the ship and its possible risks [4].

After 333 years into the depths of Baltic Sea, the Vasa was salvaged in 24th April 1956 by the engineer and wreck researcher Anders Franz´en. The wreck was returned to the surface in rather good conditions due to the optimal state of the seawater for its preservation. Baltic Sea is characterized because of its brackish water, which means that its salinity is very low, and there is no presence of marine wood-boring mollusks. Furthermore, the constant low temperature into the depth of its water and the lack of oxygen also contributed with its conservation [6].

Figure 2.4: The salvage of the Vasa, 1961. [4]

The warship was raised up with the collaboration of an expert diving group. Six tunnels were dug under the hull of the vessel and steel cables passed through to the tunnels to be able to use two lifting pontoons, which would slowly raise the ship to the surface. In the ﬁgure 2.4, the salvage of the Vasa is shown. After the salvage, the vessel was moved to the Wasa Shipyard, in order repair it and start with its preservation work. Subsequently, VasaMuseet was built at that place and oﬃcially opened in 1990 [4].

2.2 DIMENSIONS AND WEIGHT OF THE VASA

The Vasa was the largest warship ever built of its time. Its estimated total length including bowsprit is 69 m, the maximum breadth is 11.7 m and the height of main truck above the keel is 52.5 m. Besides, the draught is 4.8 m and the displacement tonnage 1.2 · 106 kg. The Vasa ship has 10 sails which makes a total of 1275 m2 [4].

5 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

The weight of the Vasa ship was approximately 1.2 · 106 kg when it was constructed. However, nowadays theVasa has not so much items onboard, most of them were removed. Therefore, the only load which can be considered is its own weight. According to S¨orenson [7], the weight calculation can be estimated by the sum of the weight of the cross section of 1.6 meters width based on the density of fresh oak. Nowadays, the weight of Vasa is estimated approximately in 8·105 kg.

2.3 THE RESTORATION AND THE PRESERVA- TION WORK OF THE VASA

The restoration and conservation of the Vasa has been one of the greatest challenges undertaken ever by a restoration department. Originally, the structure of the Vasa was supported by more than 5,500 iron bolts; however, they were corroded away underwater [7]. Therefore, new bolts made of epoxy-coated galvanized mild steel were inserted instead after its salvage, despite the fact that the best recommendation was to use 16 stainless steel. This suggestion was not accomplished due to its high costs [4, 6]. Hence, nowadays this work still continues because the inserted bolts in the 1960s must be replaced again by high quality stainless steel bolts. In ﬁgure 2.5, it can be observed the new stainless bolts and the epoxy-coated galvanized bolts replaced.

Figure 2.5: Up, a new stainless steel bolt. Down, the replaced epoxy-coated galvanized mild steel bolt [4].

Secondly, and one of the most important challenges carried out, was the conservation process of the wooden elements. In order to preserve the ship in the best manner possible, the Vasa was sprayed with Polyethylene Glycol (PEG) mixed with water from April 1962 to 1979 [8]. During this time, it were used approximately 240 tons of PEG. The purpose of using this substance was to replace the water from the waterlogged wood and avoid its shrinkage. Saturated wood will not keep its shape unless its cellular structure is ﬁlled of water. If water evaporates, the oak start to shrink and ﬁnally cracks. In December 1988 [6], the ship was translated to the permanent VasaMuseet where it was performed some ultimate surface treatments with PEG until 1989.

Suddenly in the summer of 2000, it was noticed that the oak of the Vasa oozed white and yellow acidic salt particles on some areas. In order to avoid it, it was decided that the ship had to be more controlled. Hence, a better climatic control system was installed in the museum. The temperature was established in the range of 18-20oC and the

6 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA relative humidity at 55%. In addition, implementation of alkaline solutions were applied on the surface of Vasa with the intention of mitigating this problem [3]. Nowadays, the conservation of Vasa is still ongoing, having several researches under way.

2.4 IMPACT ON THE MECHANICAL PROPER- TIES IN VASA OAK

Owing to more than 300 years that the Vasa spent in the depths of the Baltic Sea in addition to its subsequent preservation treatment after its recovery, the deterioration of its wooden elements are plainly noticeable, especially on their external structural elements. The mechanical properties of the hull of the Vasa have been studied in recent years. Researchers from the Royal Institute of Technology came to the conclusion that the oak was slightly affected by PEG impregnation when it is loaded in longitudinal tension [9]. However, if it is loaded in radial direction, the properties of the wood, such as compressive modulus and yield stress, are heavily decreased. This reduction may reach in extreme cases, up to 50% [9]. Tangential stiffness can also be decreased up to 30% and tangential strength approximately 50% [9,4]. In figure 2.6, the deterioration of the external structure of the Vasa and how its mechanical properties are affected are shown.

Figure 2.6: Degradation of the mechanical properties of the Vasa oak [10]

During this years, several studies were accomplished taking into consideration diﬀerent variables of the oak, such as moisture content, density, and orientation of its ﬁbers. After comparing the results with other fresh oak specimens, it was concluded that the moisture content of Vasa oak was lower. If the moisture content is high, material properties of the oak decreases [9,11]. However, if swollen wood gets dry, shrinkage can be produced, and consequently cracks formation, which entails to the deterioration of the wood and an important reduction of its physical properties [9]. Therefore, in order to prevent this, the Vasa is sprayed with PEG. This impregnation agent replaces the water from the cellular walls of the oak and keep the wood in a swollen state in spite of reducing its mechanical properties [9,12].

After compressive testing of Vasa oak, its material properties are determined. Young’s modulus E, Poisson’s ratio ν, and shear modulus G along its longitudinal (1), radial (2) and tangential(3) directions (see ﬁgure 2.7). They are shown in table 1 [13].

7 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Table 1: Mechanical properties of Vasa oak

Figure 2.7: Principal axes of wood (1.Longitudinal - 2.Radial - 3.Tangential)

These materials properties will be used in future calculations and ﬁnite elements analysis, but taking into consideration the state of degradation of the oak. By focusing our study on the hull structure, which is considered as a sandwich structure, these material properties will be applied just in the frame and partially to the ceiling and planking. As seen in ﬁgure 2.6, the outer surface of the hull possess less mechanical integrity, and therefore, the resultant values from table 1 cannot be applied.

In addition, every layer has a different material orientation. The orientation of every layer is displayed in figure 2.8. The material orientation of the frame forms 90 degrees with respect to the ceiling and the planking. This entails a different behaviors, deformations and stiffness in every layer in the hull.

Figure 2.8: Mechanical properties of Vasa oak

8 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

2.5 OVERVIEW OF THE VASA HULL STRUCTURE

The hull structure consists of 25 main frames which are separated from each other 1.6 m approximately. The entire components are made of oak. The frames are composed of diﬀerent structural elements such as beams, knees and futtock riders. The bottom and walls of the hull are composed by three layers which are the frame, the planking and the ceiling. They are connected by treenails, which will be denoted as dowels [7] through this paper.

A layout of the transverse cross section is displayed in ﬁgure 2.9. From the ﬁgure, it can be seen that the bottom and side structure go from the keel to the gunwale.

Figure 2.9: Cross section of the Vasa [5]

These main frames are necessary to reinforce the entire structure. Aside from providing the required endurance to the whole structure, they work as frameworks for the diﬀerent decks. The diﬀerent parts of the hull structure are:

2.5.1 Hull In the hull, three section of interest can be distinguished which are part A; which consists of the bottom structure of the hull, part B which is the curved part, and part C; the wall of the hull. They are represented in figure 2.10. These parts are characterized by similar constructive characteristics; however, different load distributions, fiber orientation and working mechanism.

Thus, for future simulations and studies, the hull will be considered as a sandwich structure, composed by three layers (frame, planking and ceiling) of oak with diﬀerent

9 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Figure 2.10: Lower part of the hull structure [10]

ﬁber directions and thicknesses. In ﬁgure 2.10, is represented such composite material.

The hull structure transfers the load from each deck and from the load of the hull itself to the external support structure. Therefore, the load will vary along the hull structure from its upper part toward its lower part, issue that it is taken into consideration.

2.5.2 Deck beams

The deck beams are the responsible of receiving the loads from the deck structure and transmitting them to the side structure to the girders where they are resting. In ﬁgure 2.11 one beam is represented.

Figure 2.11: A deck beam from the Vasa [7].

2.5.3 Knees

These parts are placed inside the hull structure and symmetrically located at both sides of the hull structure as shown in ﬁgure 2.12. They are located on the upper and lower decks. They are bolted to the side structure in addition to the deck beams. Thus, these transfer loads from the deck beams to the side structure.

10 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Figure 2.12: Knees attached to the hull structure of the Vasa [7]

2.5.4 Futtock rider and second futtock rider Located inside the hull, the futtock riders run from the bilge to the upper deck, and second futtock riders go just to the lower deck. These are bolted to the side structure and to their respectively deck beams. Both can be seen in ﬁgure 2.13.

Figure 2.13: Futtock rider and second futtock rider [7]

2.5.5 Second rider These behave practically in the same manner than the futtock riders but they are larger. They are connected to the orlop deck and hold deck with bolts. This element is shown in ﬁgure 2.14.

11 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Figure 2.14: Second rider [7].

2.5.6 First rider The first riders are placed inside the hull structure at the bottom and the bilge part of the ship. The purpose of the first riders are to provide strength to the bilge part structure and to serve as the place where the orlop deck rests. In figure 2.15, a first rider of the Vasa is shown.

Figure 2.15: First rider of the Vasa [7]

2.5.7 The rider The riders are located in the hold at the bottom part of the ship. It is used as the ﬂoor of the hold and contributes to the strength of the hull structure. In ﬁgure 2.16, a rider of the Vasa is shown.

Figure 2.16: Rider of the Vasa [7]

12 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

2.6 HULL SUPPORT SYSTEM

When the Vasa was salvaged from the seabed in the 1960s, it was placed in a support made of wooden columns and blocks at the keel. However, the Vasa needed a permanent and reliable structure to hold such a valuable treasure in a properly way. Therefore, a support system had to be engineered. For this reason, the Swedish company Kockums Mekaniska Verkstad AB was asked to accomplished this challenge.

The support consisted in nine stanchions pairs of steel which transferred the weight of the vessel to the pontoon. Between the hull structure and the cradle, at the end of each stanchion, were placed wooden wedges, see ﬁgure 2.17. These wooden wedges are submitted to compression. On the other hand, the keel is supported by wooden blocks.

Some years ago, the numbers of stanchions were duplicated in order to distribute the local forces on the hull. Today, the distance between them is 2.3 meters. Furthermore, the stanchions are all connected by a steel girder which runs the entire ship along the bilge strake. Also, the stanchions are fastened to the pontoon with the help of steel wires.

Figure 2.17: Support system of the Vasa: Left, a focused view of one of it stanchions. Right, the complete support system [4].

2.6.1 Supports for upper and main deck In the orlop deck and lower deck there were wooden columns distributed along the ship. Their function was to avoid any deflection of the beams of every deck. However, nowadays the upper and main deck had not internal support for this. Due to this fact, deflections in the beams have been produced. In order to prevent future deformations, new aluminum adjustable stanchions were installed. In figure 2.18, the aluminum stanchions installed are shown.

13 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Figure 2.18: Internal supports for the upper and main deck [5].

2.6.2 Another significant supports The figurehead at the stem of the Vasa is not able to support its own weight. Steel wires connected to the ceiling of the museum supports this part. Moreover, the citadel at the stern lies on a support which also is connected to the ceiling [4]. In figure 2.19, it it shown how the wires connected to the ceiling support the figurehead.

Figure 2.19: Steel wires support the weight of the ﬁgurehead [16].

14 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

2.7 FINITE ELEMENT ANALYSIS

Finite element analysis (FEA), also denoted as finite element method (FEM), is a numerical technique based on the study of algorithms that use numerical approximation for the solution of an engineering problem. It is characterized by the use of differential equations and a set of additional constraints denoted boundary conditions. The resolution of the differential equations uses variational methods to minimize the error committed and obtain a reliable solution. This method states that a complicated domains can be sub- divided into a series of smaller regions where differential equations can be solved. Every small region is referred to a specific type of element and they are connected by means of nodes. The set of this finite elements comprises the total solution [15].

Let us now deﬁne and diﬀerentiate between two types of FEM used in the FEA program CREO Simulate 2.0 and Abaqus CAE: a) Standard FEM used in Abaqus CAE: This method uses non-adaptive h-element technology, what implies that shape functions of the elements are described by using quadratic equations [16]. This entails several consequences:

• The set of elements consisting of nodes is called mesh. A reﬁned mesh, which implies a mesh with large quantity of elements, is required for accurate results. Reducing the element size and increasing the density of the mesh in critical areas is the correct way of performing an analysis. However, the computational time increases signiﬁcantly. • This type of method is used in whatever type of geometry, but in cases where the geometry is complex the computational time is larger. • In the case of stress concentration, the values of stress never converge. This is a type of mistake produced in areas such as corners, where the stresses can never be obtained [15]. • When a convergence study is required, its solution has to be performed manually by decreasing the element size and this way of procedure is very time consuming. b) Adaptive p-element method, also denoted as hp-FEM used in CREO Simulate 2.0 : This method overcomes the analysis by the use of adaptive p-technology, which increases the order of the polynomial equations used. This method is characterized by:

• The use of higher polynomial order equations to follow more precisely the geometry of the model. • The highest polynomial order equation to describe functions can go up to 9th (defined by the software), what means that high stress, displacement and strain gradients can be followed more accurately. • Reliable results are obtained with a mesh composed by fewer and larger elements, what means that its stiffness matrix is much smaller as well as its computing time. • A model can the refined by increasing the number of elements in the mesh and by increasing the polynomial degree.

15 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

• Its automatic convergence. With this method, the polynomial order is increased sequentially, starting by a low p-order for a rapid response and eﬃciency of the software. Then, in areas of high stress gradient in which accuracy is required, the p-order increases.

Moreover, this way of computing tends the displacement to be more accurate along the element as expressed in figure 2.20. From the figure, it is possible to see that the displacement produced by the cantilever beam can be approximated much more efficiently by using adaptive hp-FEM instead of using small and linear elements.

Figure 2.20: Diﬀerence of displacement of a cantilever beam between the standard FEM (left), and adaptive hp-FEM(right).[23]

Thus, with this kind of method it is possible to run more complicated geometry, obtaining reliable results in a lower computational time.

16 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Chapter 3

INFLUENCE OF THE DOWELS ON THE STIFFNESS OF THE HULL

3.1 INTRODUCTION

The purpose of this study is to analyze how the dowels affect the stiffness of the Vasa’s hull. In order to do this, a superelement of Vasa’s hull section (figure 3.1) is created and its stiffness coefficients are calculated.“A super-element is a grouping of finite elements that, upon assembly, may be regarded as an individual element for computational purposes” [17].

The hull of the Vasa has a complex structure and it is quite hard to introduce the material properties of an entire model of the ship in a finite element software, as well as, consume too much computational time. Therefore, these stiffness coefficients are used as a help for the calculation of the engineering constants of a section of Vasa as simple as shown figure 3.1. Thus, all of these outputs are referring to a simple single body, denote as ’superelement’ (figure 3.1).

Young’s modulus is the only parameter which will be calculated in this analysis. Shear modulus and Poisson’s ratios will be avoided inasmuch they have no much relevance in the stiﬀness of the superelement. It is noted that by changing the shear modulus and the Poisson’s ratios, the values for the simpliﬁed model of the entire structure of the warship leads to slight changes in the output values.

Figure 3.1: FE model and Superelement

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3.2 CONSIDERATIONS FOR THE CONSTRUCTION OF THE MODEL

In order to create the FE model, several considerations must be taken into account; these are: An average of the number of dowels, the dimensions of the model, its material orientation for each layer, the location of the dowels of the model, its friction coeﬃcient, boundary conditions and loads. Let us detailed them:

3.2.1 Determine the density of dowels per unit of area For the simulation of a section of Vasa’s hull, it is necessary to determine an average number of dowels per unit of area. An approximation of the density of dowels must be calculated due to the fact that the dowels are positioned without following any special rule, symmetry or sequence. For this, two pictures were taken from diﬀerent areas of the hull.

These two areas are analyzed in order to establish this density. The ﬁrst area is located at the bottom part of the ship and is placed between two stanchions. The density is set by knowing its area and counting the number of dowels as it is shown in ﬁgure 3.2.

The red circles determine where the dowels are located, the blue circles are bolts which will be avoided for this study. The area is ca 1.84 m2. It is calculated by knowing the distance between stanchions (2.3 m) and the width of a board (0.4 m). In addition, number of dowels from this area is 39. Thus,

39 dowels ρ = = 21.19 ≈ 21 dowels/m2 (3.1) 1 1.84 m2

Figure 3.2: Area at the bottom part of the Vasa’s hull

The same procedure is performed for a second area of the hull, which is located between two gun ports. The delimited area can be seen in ﬁgure 3.3. The distance between these gun ports is 2,35 m and the height is 0,85 m. The area of this part is taken from some drawings provided from Vasa Museum and it can be seen in Appendix I: Vasa’s Drawings.

60 dowels ρ = = 30.04 ≈ 30 dowels/m2 (3.2) 2 1.9975 m2

From these two results, it can be established an average of 25 dowels/m2.

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Figure 3.3: Area of the Vasa’s hull between two gun ports

3.2.2 Dimensions of the model The dimensions are set taking into account the density of dowels calculated in the previous section. The FE model has to have an enough quantity of dowels to simulate the reality as best as possible. However, it should not have too many dowels which lead to a very complicated model and a high computational time for the simulation. Therefore, it is decided to select an area of 0.6 x 0.6 meters and then, 9 dowels are set. The dimensions of each component of the model are detailed below: a. Dowel has a diameter of 30 mm and a length of 500 mm as it is shown in ﬁgure 3.4.

Figure 3.4: Dimensions of the dowel b. Planking and ceiling have a width and a length of 600 mm. The height is 100 mm. It is displayed in ﬁgure 3.5. c. Frame has a width and a length of 600 mm. The height is 100 mm. It is exposed in ﬁgure 3.6.

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Figure 3.5: Dimensions of planking and ceiling

Figure 3.6: Dimensions of frame

3.2.3 Material orientation for each layer and for the dowels Every component in the model has a different material orientation. This entails to a different behavior and different stiffnesses in every direction. This is a matter which has to be taken into consideration.

The planking and the ceiling have the same layout, longitudinal direction fibers follow the X -axis and tangential direction fibers follow the Y -axis. On the other hand, the longitudinal direction fibers for the frame follow the Y -axis and tangential direction fibers follow X -axis.

For the dowels, longitudinal direction fibers are placed coincidently to Z -axis, tangential direction fibers coincident to Y -axis and radial direction fibers coincident to X -axis, as it can be seen at figure 3.7.

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Figure 3.7: Material orientations of every component

Figure 3.7 shows the three layers of the model (ceiling, planking and frame) and a dowel. The blue lines represent the longitudinal direction, the yellow ones represent radial direction and the red ones the tangential direction. The tangential direction of the three layers goes upwards in the Z direction. Its representation was omitted with the intention of clarifying the ﬁgure.

3.2.4 Location of the dowels in the model A priori, it could be thought that the results could change for different disposition of the dowels, and therefore the stiffness of the model would be dependent of the dowels localization. In order to know how much this pattern affects to the stiffness of the FE model, different cases are studied, which are:

A. REGULAR PATTERN This pattern will be symmetric with respect Y-Z and X-Z planes as it is shown in ﬁgure 3.8. This will be compared with the cases B and C to see the dependency of the disposition of the dowels in the results obtained.

B. RANDOM PATTERN 1 In addition, two more irregular patterns are used in this analysis. These are achieved taking into account the pictures taken from the Vasa. The intention was to distrubute the dowels in a similar manner as in the real Vasa. In this disposition, the dowels are located randomly and manually as shown in ﬁgure 3.8.

C. RANDOM PATTERN 2 As in the previous case, the dowels are located in another random pattern. This second irregular location is selected in order to assure that the results are consistent. This pattern is displayed in ﬁgure 3.8.

3.2.5 Friction coefficients between dowels and different layers The dowels are inserted into the holes which are made through the three layers. However, it is hard to know how tight the joint hole-dowel is. Hence, two different experiments will

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Figure 3.8: Layouts of Regular, Random 1 and Random 2 Patterns be performed. The ﬁrst one will be established with a friction coeﬃcient of oak-oak of 0.48 [18]. The second one will be done without friction. Figure 3.9 displays one friction zone.

Figure 3.9: Friction area of dowel and hole

Any friction coeﬃcient is set between the layers frame-planking and the frame-ceiling of the hull due to its deterioration along the years. Nowadays, there is a gap in between.

3.2.6 Boundary conditions In order to perform an analysis in a FE software, some constraints are necessary. Thus, in the opposite side of the model in which the force or moment will be applied, the model is ﬁxed. This means that any displacement neither rotation are allowed along any direction. In ﬁgure 3.10, the boundary conditions are represented with orange triangles.

22 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Figure 3.10: Boundary conditions of the model

It is worth of mention that this type of boundary condition does not correspond com- pletely with the reality, because Poison ratio is slightly inﬂuenced for this issue. However, this matter will be taken into consideration in the results.

3.2.7 Rigid plate to applied loads

Later on this study, different types of loads will be applied into the model in order to obtain its stiffnesses. Thus, due to the complexity of the model and to guarantee that the force/moment is applied correctly, a plate infinitely rigid is tied in the face of the model where the forces are applied. The dimensions of the rigid plate (600x500x10 mm) are displayed in figure 3.11.

Figure 3.11: Dimensions of the rigid plate

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3.3 ABAQUS CAE SETTINGS

In this chapter, the definition of the model explained in the previous section is detailed for the FE software Abaqus [19]. The method used in the program is presented, its material properties are defined and the type of element for the mesh applied to the model is defined.

3.3.1 Method used

The method used for this study is the Standard FEM, and its procedure type is Static, General. This procedure allows an implicit solution for problem such as static or low-speed dynamic problems where stress solutions are rather important.

3.3.2 Material Properties

Since Vasa oak is considered as an orthotropic material due to its wood ﬁber orientation, the selected material type is Engineering Constants, where nine parameters have to be introduced in every element in the model. There are three Young modulus, the three shear modulus and the three Poisson’s ratios, one per principal direction. All these parameters are taken from table 1 of section 2.4.

3.3.3 Steps & increments

Two steps are deﬁned. The ﬁrst one, denoted Contact, is in charge of establishing in a proper manner the contact between all the elements of the model. During this step, the forces are not applied yet. The second step is called Load and it is the responsible, per se, of applying the loads. The time period for both steps are 1 and the initial increment time is set in 0.2.

3.3.4 Mesh

The element type used is quadratic hexahedral using reduced integration (C3D8R). Dif- ferent element sizes are used. The mesh is ﬁner for the dowels and the areas of the three layers which are close to the holes. This means that the mesh is reﬁned in areas where the geometry is more complicated and in areas where it is necessary more precise results. However, a coarse mesh is applied for the rest of the model to reduce the computational time.

In order to reduce the mesh just in this areas, four partitions are performed in the model. The diﬀerence in the construction of the mesh can be seen in ﬁgure 3.12. Thereby, the number of elements are reduced until reach ca 30 % of the initial elements. As illustration, the number of elements in the planking without any partition is 10536. When the partition is done, the number of elements in the planking is 3297.

As shown in the figure 3.12, the adjoining area to the dowels is not affected of this partitions, hence it is a good way to reduce the mesh and the computational time without affect other components.

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Figure 3.12: Meshes of the model

3.4 RESULTS

In order to analyze the influence of the dowels in the stiffness on the hull of the Vasa, the model detailed in the previous section is subjected to several tests. Specifically, twelve tests are accomplished. Since it is necessary to know the stiffness of the model in all the directions, six test are performed with the rigid body attached perpendicular to the X -direction, and six test are performed with the rigid body fixed perpendicularly to the Y -direction.

Figure 3.13: Forces and moments applied on the model

In every test, a force or moment is applied, so the behavior and the deformation of the model is analyzed and then, its stiﬀness. In ﬁgure 3.13, the direction of the forces

25 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA and moment of the twelve tests are represented. With green arrows, it is represented the direction of every force applied and with red arrow the moments applied are displayed. It is worth of mention that every force and moment are applied into the rigid body indepen- dently. Thus, twelve stiﬀness coeﬃcient are obtained for each layout that are compared with each other. The numerical values for forces and moments are displayed in the table 2.

Table 2: Values of forces and moments in the analysis

By applying these loads, a deformation in the model is obtained. When a force is applied, the used output is a displacement in the direction of that force. When a moment is applied, the used output is a rotation over the axis in which is applied that moment. Abaqus CAE provides these values with the name of U, translation and UR, rotation.

3.4.1 Base Model In the following tables the numerical values for a model analyzed with a regular pattern of dowels is provided. This model is set with friction between the dowels and the diﬀerent layers. This ﬁrst layout is in somehow the base-model for future comparisons.

Tables 3 and 4 : Displacements and rotations for the Base Model

With these numerical values, the stiffness coefficients are reached because it is known the force or moment applied for each case and the displacement/rotation obtained. The stiffness coefficients for linear translation are calculated by using the equation 3.3 [20]:

 i = x, y F  k = k j = 1, 2, 3 (3.3) ij δ k  k = x, y, z The stiffness coefficients for rotation is called rotational stiffness and it is calculated by using the equation 3.4 [20]:

 i = x, y M  k = k j = 4, 5, 6 (3.4) ij θ k  k = x, y, z

26 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

By using these two equations, the twelve stiﬀness coeﬃcients for the base model are:

Tables 5 and 6 : Stiﬀness coeﬃcients for the Base Model

To summarize, it is reached twelve stiﬀness coeﬃcients kx1, kx2, kx3, kx4, kx5, kx6, ky1, ky2, ky3, ky4, ky5, ky6 for each layout.

3.4.2 Comparison of Base model, Random Pattern 1 and Ran- dom Pattern 2 This comparison is performed in order to check how the layout of the disposition of the dowels in the model affects to the stiffness coefficients. As it is detailed in the section c of chapter 2.1, two random patters are created. In the following tables the results of the aforementioned stiffness coefficients of the model with two different random disposition are shown.

Tables 7 and 8 : Stiﬀness coeﬃcients for the Random model 1

Tables 9 and 10 : Stiﬀness coeﬃcients for the Random model 2

This results can be compared with the values from tables 5 and 6 which corresponds to the Base Model outputs.

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3.4.3 Comparison of Base Model with friction and frictionless This comparison is accomplished in order to study how the friction between the dowels and the others three layers affect to the stiffness coefficient. The friction in the model is detailed in the section 3.2.5. The results of the stiffness coefficient of the base model frictionless are compared with a model with friction. The results can be seen in tables 5, 6, 11 and 12. The procedure to reach this coefficient is the same explained in the previous sections.

Tables 5 and 6 : Stiﬀness coeﬃcients for the Base Model

Tables 11 and 12 : Stiﬀness coeﬃcients for the Frictionless Model

3.4.4 Comparison of Base Model with and without dowels A third study is performed in order to check how much contribution in the stiﬀness of the model is due to the dowels. Hence, the base model which contains the dowels is compared with a model without them. This comparison is shown in tables 5, 6, 13 and 14 .

Tables 13 and 14 : Stiﬀness coeﬃcients for the Without Dowels Model

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This results can be compared with the values from tables 5 and 6 which corresponds to the Base Model outputs. The conclusions obtained from every comparison are shown in section 3.5.

3.4.5 Procedure to reach young modulus In this section, the Young’s modulus for the sandwich structure of the hull of the Vasa is obtained. The purpose of reaching these values is to use them in future studies and facilitate the analysis. Instead of creating a model composed by a sandwich structure, a unique layer with the resultant engineering constants would be enough to simulate the material properties and the behavior of the hull.

In this paper, Young’s modulus in x and y direction are obtained. The other engineering constants are not calculated because its relevance in the behavior of the hull is lower or because a deeper study is required to reach a reliable value.

Hence, in order to obtain the values of Young’s modulus mentioned, a certain axil force is applied along x and y direction and then the axial displacement in those directions are got. The value of the displacement for the direction X and Y can be taken from section 3.4.1, table 3 and 4. Pursuant to these data, Young’s modulus can be reached with the equation 3.5 [20]: F · l E = ; i = 1, 2, 3 (3.5) i δ · A

where F is the applied force, l is the original length of the model, δ is the displacement along the axis which is applied the force and A is the initial cross-sectional area through which the force is applied. The dimensions are displayed in ﬁgure 3.14:

Figure 3.14: Initial cross-sectional area and original length of the model

The length of the model and the initial cross-sectional area is deﬁned at the beginning of this chapter. The applied force is established previously and the displacement is obtained from the tests performed in Abaqus. Therefore, let us apply the equation 3.5 with the proper values:

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1000000 · 0.6 E = = 2.77 GP a (3.6) 1 7.21 · 10−1 · 103 · 0.3

1000000 · 0.6 E = = 3.70 GP a (3.7) 2 5.41 · 10−1 · 103 · 0.3

Where E1 is the Young’s modulus in X direction, and E2 is the Young’s modulus along Y direction. These engineering constants obtained in this section will be used in the following study.

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3.5 CONCLUSIONS

After comparing all the tests, it can be established that the stiffness coefficients obtained from the model are not significantly affected by the layout of the dowels. Thus, the difference in the results is insignificant when the disposition of the dowels in the model is symmetric or if they are placed randomly. Furthermore, the stiffness coefficients are slightly influenced when the model is tested without dowels. However, this could be ob- viate taking into account the rate of variation.

On the other hand, it can be set that the friction coefficient, applied between holes and dowels, is not a decisive issue on the outcomes. The results obtained from the model with friction and the model without friction does not differ significantly; therefore, its effect on the stiffness coefficients is irrelevant.

Regarding to the Young’s Modulus calculations in section 3.4.5., it can be mentioned that the values obtained are consistent with respect to the Vasa Oak properties. As expected, the two Young’s modulus calculated for the model are located between the range of longitudinal and radial young modulus of the Vasa Oak (5.81 and 0.93 MPa).

Finally, one more thing worthy of mention is that, as might be expected, the value of E2 is larger than E1. This is due to the fact that the area corresponding to the longitudinal ﬁbers (Ey = 5.81 MPa) perpendicular to Y axis is larger.

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Chapter 4

GLOBAL ANALYSIS OF THE WARSHIP VASA HULL

4.1 INTRODUCTION

In this chapter, the creation of a simplified model of the entire structure of the warship is detailed and it is subsequently analyzed with the intention of defining the most critical area on the hull of the Vasa. The software selected in the construction of the model is Autodesk Inventor 2014 [21]; however, its finite element analysis is performed with the help of CREO Simulate 2.0 [22]. Due to the complexity and the large quantity of components which comprise the Vasa, some simplifications are assumed during the creation of the model. They are established taking into consideration their possible effect on the hull structure, which is the focus of this study. In addition, it is worth of mention that the results obtained are only focused in the hull of the warship.

4.2 CONSTRUCTION OF THE MODEL

In order to shape the Vasa, dimensions from Appendix I: Vasa’s Drawings were used; however, the geometry was partially simpliﬁed due to the complexity of its elements and the limited time available for the realization of the work. Hence, in order to understand the construction process and describe all the assumptions taken into account, the process is divided into three principal steps detailed as follows:

1. The hull structure is the first part to be created and the most complex one due to its singular shape. Its procedure starts by sketching every transverse cross section along the warship with the help of technical drawings of different cross sections of the Vasa as shown figure 4.1-a. It is worth of mention that in reality the warship is not symmetric along its longitudinal axis due to the fact that when it was constructed, it was not available specialized and precise techniques of drawing, manufacturing and construction. However, in the case of this study, the model is considered as symmetric. Furthermore, every cross sectional sketch is created with an approximate separation of 3 m, and the thickness of the hull is considered as the average value of 400 mm through the entire structure. Thereby, there is a total number of 16 sketches created

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Figure 4.1: a) sketch of one cross section of the hull. b) Structure composed by the totality of sketches which comprises the hull structure.

to shape the hull of the Vasa taking into consideration the variation in the width, height, possible changes on its curvature and the elevation of the keel along the entire ship. The totality of the sketches are displayed in ﬁgure 4.1-b. Thereafter, the sketches are connected and swept blended to create a solid hull of the Vasa.

2. Secondly, the internal frames of the warship are created. They comprise the structural components of the ship such as deck-beams, knees, riders and columns. They are therefore, the skeleton of the vessel and responsible of providing the required strength to the ship. In figure 4.2-a, the procedure of construction of a frame is represented. The set of the components of a frame are simplified and designed as a solid unity. Thus, a total quantity of 25 frames are created by the same procedure but adjusting its geometry to the hull of the warship, as represented in figure 4.2-b.

Figure 4.2: a) Sketch of a frame from the Vasa model. b) Sketch of some structural frames which comprise the Vasa.

In addition, the riders from the bottom part of the warship are created in order to provide the necessary strength at that part of the ship. These are also extruded and disposed according to the geometry of the Vasa.

3. Thirdly, the keel is constructed and extruded taking into account its total length, width and other relevant dimensions. In ﬁgure 4.3, it is represented how the keel is sketched, extruded and thereafter the excess part of the prow is removed. Thus, in

33 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA order to ﬁnish the model, gun ports are created and disposed along the hull structure and extra reinforcement components are added at the stern of the warship.

Figure 4.3: Process of construction of the keel of the Vasa.

It is worth mentioning the type of joint established between the components in the model. In the case of this study and due to the fact that just a simpliﬁed analysis is required, every joint is considered as totally ﬁxed although in the real Vasa the members are connected by means of bolts and dowels, what allow more freedom to their members. Hence, the entire model behaves as a unique entity and solid.

Figure 4.4: Final model of the Vasa.

In figure 4.4, the final model is displayed. However, it can be studied half of the model, which is a cut view performed according to its longitudinal symmetry plane. This simplification will be taken into consideration for future studies and calculations.

34 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Finally, the elements of the Vasa which were omitted from the model are the gallery room, the three masts, the prow pole and the ﬂooring of every deck, which will be substitute by its weight for future studies.

4.3 FINITE ELEMENT ANALYSIS

In this study, the method selected for analyzing the hull structure of the Vasa is the adaptive hp-FE method, detailed in the previous chapter, and the software selected is CREO Simulate 2.0 [22]. At this moment, CREO Simulate 2.0 oﬀers three choice of solution strategies, which are[23]:

• Quick check: It is a single-pass adaptive with all the elements set to the polynomial order 3. It can be a way of check that it is possible to run the model. Results of stress and displacement can be obtained, but they may not be very reliable.

• Single-Pass Adaptive (SPA). This method is characterized by its fast runs, and it is usually selected for analyses where it is not required to know the accuracy of the results, or when it is necessary a fast check of the geometry. In this solution strategy only two solution passes are performed. For the first pass the elements edges are set to a polynomial order 3. Then, when the first solution pass is done and in order to get a good solution, the algorithm performs an error estimation through the entire model. In the areas where more accuracy is required, the second and final pass solution is made with a minimum polynomial degree of 5 and a maximum of 8.

An important pros of using this technique is the rapid run and the possibility of providing a soon preliminary result. However, it is not possible to have control over the accuracy desired.

• Multi-Pass Adaptive (MPA): In this case, a series of passes are conducted and the results obtained in every pass are compared with the results from the previous pass with the intention of determining where a further increase in the polynomial order may be required. The process continues until the requested percentage of convergence is reached. The most important advantages of using this technique is the great accuracy achieved in the results and the possibility of having the convergence curve for future studies. However, its computational time is much elevated than in SPA. In the case of this study, SPA and MPA techniques are used.

4.4 ASSUMPTIONS AND CONSIDERATIONS

With the intention of deﬁning the model and its properties in CREO-Simulate, some assumptions and considerations are established.

1. Symmetry: Since the model is subjected to symmetry along its longitudinal axis, the model selected for this study is half of the ship, as shown in ﬁgure 4.5.

35 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Figure 4.5: Symmetry applied to the model

2. Mechanical properties: Firstly, a density of Vasa Oak has to be established. This density is taken from a currently research which is being performed by the Uppsala University. Several specimens are taken from the bottom part of the hull structure of Vasa, and its properties are shown in table 15:

Table 15 : Diﬀerent samples of densities from Vasa Oak [24]

The average density is calculated in order use it as an input value for the FEA. This value is:

ρ = 731.37 kg/m2

Other parameters from Vasa oak are taken from table 1 and the results of Young’s Modulus obtained in the previous chapter in order to deﬁne the materials properties

36 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

of the model. The material is considered as orthotropic, according to the behavior of the entire hull. Thus, the material parameter used are exposed in table 16. Table 16 : Material parameter for the model.

3. Material orientation: Furthermore, the material orientation has to be set since wood is an orthotropic material. Inasmuch as the model is a unique body, a same material orientation has to be established for all the ship. However, since the most important part of this study is the hull of the warship, the material orientation is set by satisfying it.

4. Boundary conditions: Since the Vasa warship is supported by the system support in the VasaMuseet, some constraints has to be set into the model. The area occupied for every stanchion of the system support is delimited in the model and the boundary conditions are applied there. There is restriction of displacement in the normal direction to the external surface of the hull. This type of boundary condition is illustrated in ﬁgure 4.6.

Figure 4.6: Stanchion drawing and boundary condition modeling of a stanchion[7]

Besides, since the model is symmetric, a symmetry constraint has to be set in the symmetry plane. All the boundary conditions are shown in ﬁgure 4.7. Thus, in this boundary condition due to symmetry, there is a restriction of rotation along all directions and there is restriction of movement in the perpendicular direction to the plane of symmetry of the model.

37 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Figure 4.7: Boundary conditions: Stanchions and symmetry constraints

5. Applied loads:The Vasa is submitted to several loads. Due to the complexity of the involved acting forces, these have to be simpliﬁed in the model. Firstly, the weight of the model of the ship is determined taking into consideration the average density of the Vasa oak and the gravitational acceleration which has a value of −9.81 m/s2. Thus, the total weight of the entire model is 5.35 · 105 kg, and the weight of the half of the model is 2.68 · 105 kg.

Furthermore, due to the fact that the ﬂooring of every deck is omitted in the model, its weight has to be applied into the model. Its weight is stablished according to Lindberg [25]. Then, a total weight of 5.30 · 104 kg is applied to every deck of the model.

The gallery, placed in the stern of the ship, was also excluded during the construction of the model. Hence, its load is introduced in the model taking into account the symmetry with a total value of 1.55 · 103 kg. In ﬁgure 4.8, the green arrows represent the load applied due to the weight of the gallery.

Other components of the model follow the same procedure, and its reaction force are gathered in table 17. It is worth of mention the reaction obtained from the prow

38 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Figure 4.8: Loads applied on the model of the Vasa warship

pole. In this case, after symmetry, it is applied a total load of 2.23 · 103 kg and also its moment reaction created is applied to the prow of the ship. Table 17 : Loads from other components of Vasa

The data of the weight of every component, is provided by [25].

6. Difference of weight: It is worth of mention the significant difference of weight between the model and the real Vasa. This is due to the simplification in the geometry performed during the construction of the model, the variation of the density though the entire ship and the omission of secondary elements which did not con- form a structural element of the ship. However, since the purpose of this study is

39 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

to identify the loading mechanisms and the critical areas on the hull, this can be accomplished with the characteristics of this model.

7. Type of model: The analysis is carried out considering the model as a solid, due to the fact that the structural element of the ship analyzed in this paper is the hull. Is is necessary to investigate the diﬀerent states of stresses and strains along the width of the hull, which has an average value of 400 mm.

8. Type of finite element: Two different options of element type for solids is given from CREO Simulate 2.0. The first one is “Wedge and Tetra” and the second one is “Brick, Wedge and Tetra”. The first option is chosen for this analysis in order to avoid the brick element type. Since the use of brick elements increases the computational time [22]. It is also important to notice that tetrahedral elements fit very well in any arbitrarily shaped volume, especially with curves.

4.5 STUDY OF THE CONVERGENCE

This study of convergence is performed with the intention of guaranteeing the reliability of the results. In case that the analysis would not converge during the study, the results are not reliable so the model should be modiﬁed to get better results [26].

In this study, the reﬁnement of the mesh used, the polynomial order of the equations and the percent of convergence is controlled in order to reach more accurate results. It can be ascertained how the element size and the polynomial order aﬀect to the convergence.

There are four diﬀerent element sizes which are selected to mesh the model:

• MESH 1000: maximum element size 1000 mm

• MESH 800: maximum element size 800 mm

• MESH 400: maximum element size 600 mm

• MESH 200: maximum element size 200 mm

In ﬁgure 4.9, the graphs display the number of points, edges, faces and elements that the model has, depending of the maximum element size.

For each element size, it is ran a simulation using Single-Pass Adaptive (SPA) and Multi-Pass Adaptive (MPA). A percent of convergence of 10% is used when Multi-Pass Adaptive method is used. The minimum polynomial order is set at 1 and maximum at 6.

The study of the convergence may be checked by studying the convergence plots and see if they converge to a certain value. However, there are other ways to see if the results are enough accurate. One of these is checking the “Summary report file”. In this report file, there is a value called “RMS Stress Error estimate” which provide a good view of the error committed. If this value is greater than 10% probably someplace in the model has a significant error [23].

40 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Figure 4.9: Graphs of the number of elements, edges, faces and points

Another way to check the convergence is to look at the un-averaged stress fringe plot. It has to be checked that the stress fringes line up across element boundaries. The better solution is that which present a more continuous stress ﬁeld [23]. In ﬁgure 4.10, it can be seen a good and bad solution regarding to the averaged stress fringe.

Figure 4.10: Bad and good stress fringe plot [23]

To conclude, it can be seen that the measure convergence plots have an asymp- tote. It is analyzed the convergence for von Mises Stress, maximum displacement and strain energy. It is worthy to mention that the strain energy is one of the best proof that you have a good convergence [23].

41 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Another thing worthy of mention is that ﬁner meshes lead to the percentage of the total number of elements decrease in an early P-pass. For instance, from ﬁgure 4.11, at pass 3, MESH 400 reaches a 5% of convergence and on the other hand MESH 1500 reaches 30%. Therefore, there are more elements which have converged in MESH 400.

Figure 4.11: Graph of Not Converged Elements vs. P-pass

CONVERGENCE RESULTS In this section, the graphs which show the evidence of a good convergence are displayed. These graphs are plotted for a MESH 800. The other results of convergence are shown in Appendix II: Results from the study of convergence. Thus, as stated before, the plots of convergence of strain energy, von Mises (VM) stress and maximum displacement are studied. They are shown in ﬁgures 4.12, 4.13 and 4.14.

Figure 4.12: Graph of Strain Energy vs. P-pass

42 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Figure 4.13: Graph of Maximum von Mises Stress vs. P-pass

Figure 4.14: Graph of Maximum displacement vs. P-pass

As it is seen from these graphs, strain energy and maximum displacement plots converge before and better than the Maximum Stress VM plot. This is due to the concentration of stresses which the model has in certain areas.

43 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

4.6 RESULTS

The results from this study should be interpreted in a qualitative manner, rather than a quantitative, since the model created contains certain generalities and simpliﬁcations.

The results plotted in this section correspond with a mesh of a maximum element size of 800 mm, due to the fact that their accuracy and computational time are acceptable.

Figures 4.15 and 4.16 show the plotted stresses according to von Mises. They are originated by the own weight of the Vasa and the other reaction forces actuating on the warship. The colored stresses displayed represent the stresses on the hull of the Vasa. The stresses of the other structural components are removed because the focus of this study is the hull of the ship. Nevertheless, the entire stresses can be seen in APPENDIX III: Results from global analysis - Critical area on the hull.

Figure 4.15: von Mises Stress on a section of Vasa’s hull [MPa]

In the ﬁgures, some cross-sectional cuts are made in the hull in order to see the stresses through it. The dark-blue areas represent the region of lowest stresses and the red ones represent the areas of higher stresses. A concentration of high stresses can be detected at the bottom part of the hull and at its curved part. The maximum value registered is 13 MPa. The reason for which the highest stresses are located at the bottom part of the hull is because the lower part of the ship supports more weight that the higher part of it.

44 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Figure 4.16: von Mises Stress on several sections of Vasa’s hull [MPa]

In ﬁgure 4.17, the values of shear stresses obtained in the analysis are represented. They are displayed in the same manner as before. Some cross-sectional cuts are made in order to see the shear stresses in the hull of the Vasa. In this case, the areas of high stresses are also located at the bottom of the hull and at its curved part, though its numerical value is much lower. Thus, it can be stated that shear stress is less inﬂuent in the results and it is not a critical type of stress.

Figure 4.17: Maximum Shear Stress on several sections of Vasa’s hull [MPa]

45 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

4.7 CONCLUSION

The measurements and results obtained from this analysis reveal that the Vasa is slowly deflecting and deforming due to its own weight. The deformation occurs in the cross sections of the ship which entails deformation on its longitudinal section as well. As shown in figure 4.18, the upper part of the hull is warping inwards, deforming the hull and other structural components such as deck beams. The curved part of the hull of the warship is contented by the cradle-stanchions of the system support; however, swelling outwards occurs between the stanchions. Thus, the model deflects in the same manner as the real ship, and therefore the results of deformation provided from this study confirm that in order to preserve the Vasa for future generations, a new system support must be implemented.

Figure 4.18: Deformation on the hull

The outputs of stress according to von Mises reveal that the critical area on the hull is located in the bottom part of the ship. This is due to the fact that in this part, its weight is higher. The outcomes of shear stress obtained are located in the same area, but its inﬂuence and eﬀect on the hull is lower.

It is recommended for the future support system that the stanchions, which holds the hull, should cover a bigger area, speciﬁcally, the area which is located at the bottom part. In that way, these concentrations of stresses would be reduced along the hull. However, the visibility of the lower part of the Vasa will be decreased slightly, and this would be a negative point for the Museum.

46 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

In order to reduce the outwards curvature shown in ﬁgure 4.18, an internal structure could be implemented. Previous studies as stated by Ljungdahl [27], propose an inner structure composed by wires which reduce such deformation.

Another recommendation could be to change the current wood wedges for other elements made from another elastic material such as synthetic rubber or any hard polymer composite. By improving this, the indentation produced in the hull could be reduced.

Furthermore, the Vasa was constructed to support lateral pressure caused by the sur- rounding seawater and not punctual loads which are produced by the system support. These point loads entail stress concentrations in the vicinities of the stanchions, which contribute to the deformation of the hull structure. In addition, this deformation is also aﬀected by the current condition of the Vasa oak, which is deteriorated and weakened. The FE analysis performed in this study conﬁrms this matter. The areas of the hull which are in contact with the cradle-stanchions, present higher stresses.

It can be concluded that the values of E reached in the previous study, the other engineering constants and material properties introduced in the model represent in a good manner the behavior of the ship.

47 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Chapter 5

DISCUSSION

From the work achieved during this dissertation work it is possible to determine that the main objectives have been satisﬁed. The results obtained from the studies performed are in accordance with the speciﬁcations established on this paper. Nevertheless, these results and conclusions must be compared with previous researches in order to assure the consistency of this work.

The main motivation why this thesis was developed is focused in the study of the influence of the dowels in the stiffness of the hull of the Vasa. As stated in [2, 6, 8, 9], it is known that the degradation and variation of the Vasa’s oak is one of the factors that contributes to the progressive deformation of the warship. Thus, it was essential to know the stiffness coefficients of the hull with the current conditions of the wood and ascertain if the disposition of the dowels, the lack of one or some of them or its different possible configurations were crucial factors to take into consideration in the stiffness of the hull of the Vasa. Finally, the results obtained reveal that the dowels do not affect significantly to the stiffness of the hull.

Notwithstanding, working on this project we came to the conclusion that unifying and simplifying in a reliable manner the material properties of the sandwich structure of the hull of the Vasa is a crucial point for studying its behavior. To reach the engineering constant of the totally of the hull would simplify the future FE analysis signiﬁcantly, obtaining well-founded results. This is the reason for which E along the three orthogonal directions were calculated. The other engineering constants were suggested as future work because of the lack of time for the constructions of this thesis. According to previous studies [10], experimental tests [13] and comparing the results obtained, it is concluded that the values of E can be considered as valid.

With the intention of using the general value of E reached and identify possible critical areas in the hull of the Vasa, a FE model of the entire ship was created. The results conﬁrms that the most critical part is located at the bottom part of the ship. Moreover, the curved part of the hull and the vicinities areas close to the stanchions of the support system are also parts to take into account. This can be compared with previous studies from [7, 10, 27]. The results of deformation can be also compared from [1, 4, 27]. Thus, both results can be considered as valid.

48 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

Chapter 6

FUTURE WORK

The studies accomplished in this dissertation work has provided diverse research paths to continue analyzing and developing the future support system for the Vasa. There are several potential researches which can be conducted from the realization of this thesis, such as:

• Improve and reﬁne the model of the section of the Vasa’s hull in order to reduce the computational time of the simulation and obtain more accurate results.

• Improve the global model of the warship Vasa in order obtain better the results of stress, strain, displacement and convergence.

• Deﬁne all the material properties (engineering constants) of the superelement which represent the sandwich structure of the hull, and use them in the global model of the ship. In this paper, a numerical value of Young’s modulus on the three orthogonal directions it was obtained; however it is necessary to obtain the values of Poisson’s ratio and Yield strength as well. Once the engineering constants are obtained, they could be compared with the experimental test performed previously (table 1), and ascertain the validity of the results. The values of Young’s modulus were compared and they are in accordance with the experimental data.

• Study the influence of the hull sandwich structure into the dowels. Several ongoing researches about the dowels and its insertion into the hull of the Vasa leaded to the conclusion that it is essential to consider the effect of the friction between the dowels and the timbers of the hull. The original dowels of the Vasa are nowadays degraded and they are not fitted into their holes. However, the new ones are designed about 1 mm larger than its pre-drilled hole, so the interference stress created in this area must be calculated. As future work, this matter should be verified.

• Perform experimental tests and compare the results of stiﬀness with the FE model of the section of the Vasa’s hull accomplished with Abaqus CAE.

49 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

REFERENCES

[1] Lechner, T., Bjurhager, I. and I Kliger, R., (2013). Strategy for developing a future support system for the Vasa warship and evaluating its mechanical properties. Heritage Science, vol. 1 p. 1-11.

[2] Ljungdahl, J., (2006). Structure and Properties of Vasa Oak, Licentiate Thesis, Royal Institute of Technology, Stockholm.

[3] Landstr¨om, B., (1988). The Royal Warship Vasa, Stenstr¨omInterpublishing, Stockholm.

[4] VasaMuseet, (2014). The ship. [WWW] Available from: http://www.vasamuseet.se /en/The-Ship/ [Accessed: 27/03/2014].

[5] Wasa (2006). Wasa. [WWW]. Available from: http://www.wasadream.com/Index/ indexenglish.html [Accessed: 27/03/2014].

[6] Cabrera, C., (2010). Re-conservation of wood from the seventeenth-century Swedish warship the vasa with alkoxysilanes: A re-treatment study applying thermosetting elas- tomers. M. Sc. Thesis, Oﬃce of Graduate Studies of Texas A& M University, United States.

[7] S¨orenson,M., (1999). Hull strength of the warship Vasa, M.Sc. Thesis, Royal Institute of Technology, Stockholm.

[8] Hocker, E., Almkvist, G. and Sahlstedt, M., (2012). The Vasa experience with polyethylene glycol: A conservator’s perspective. Elsevier vol. 13 p. S175-182.

[9] Ljungdahl, J. and Berglund, L.A., (2008). Transverse mechanical behavior and moisture absorption of waterlogged archaeological wood from the Vasa ship, Holzforschung vol. 61 p. 279-284.

[10] Garza, C. and Dabbagh, A., (2011). Finite Element Analysis of the Vasa’s Bot- tom structure. M.Sc. Thesis in Applied Mechanics, University of Sk¨ovde,Sweden.

[11] Ljungdahl, J., Berglund, L.A. and Burman, M., (2006). Transverse anisotropy of compressive failure in European oak – A digital speckle photography study, Holzforschung vol. 60 p. 190-195.

[12] Bjurhager, I., (2008). Mechanical behavior of hardwoods - Eﬀects from cellular

50 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA and cell wall structure. Licentiate Thesis, Royal Institute of Technology, Stockholm.

[13] Vorobyev, A., Longo, R., Laux, D. and Arnould, O., (2013). Elastic charac- terization of the Vasa wood by resonant ultrasound spectroscopy and comparison with compression test. Uppsala University.

[14] Ship and harbor photos, (2009). Vasa, the ﬁgurehead. [WWW] Available from: http://www.shipsandharbours.com/picture/number10525.asp [Accessed: 27/03/2014].

[15] Cook, R., Malkus, D. Plesha, M and Witt, R., (2002). Concepts and applications of ﬁnite element analysis . 4 ed. New York: Wiley.

[16] NC State University, (2013). H-Method and P- Method [WWW] Available from: http://www.mae.ncsu.edu/klang/courses/mae533/Reference/Methods.htm [Accessed: 26 /08/2014].

[17] University of Colorado, (2013). 10: Superelements and Global-Local Analysis [WWW] Available from: http://www.colorado.edu/engineering/cas/courses.d/IFEM.d/I FEM.Ch10.d/IFEM.Ch10.pdf [Accessed: 26/08/2014].

[18] Applied Industrial Technologies (2014). Coeﬃcients of friction “F”. [WWW] Available from: http://web.applied.com/assets/attachments/492ACC9E-E5C2-2D43-0B8CCD A72ACE3361.pdf [Accessed: 26/08/2014].

[19] Dassault Syst`emes,(2009). Abaqus/CAE User’s Manual.

[20] Sundstr¨om,B. (2010). Handbook of solid mechanics. Stockholm: Department of Solid Mechanics, KTH.

[21] Hansen, L. Scott, (2013). Autodesk Inventor 2013. New York, NY: McGraw-Hill.

[22] Rider, Michael J., (2013). Designing with CREO Parametric 1.0. 1st ed. New York, NY: McGraw-Hill.

[23] Mechanical design simulation and optimization, (2003). Tips and Tricks [WWW] Available from: http://www.tsdengineering.com/ [Accessed: 26/08/2014].

[24] Vorobyev, A., (2014). Unpublished material; Data taken from diﬀerent specimens from the hull of the Vasa. Department of Applied Mechanics of Uppsala University, Swe- den.

[25] Lindberg, J., (2008). Massf¨ordelningsanalys av skeppet Vasa. Statens Maritima Museer Vasa Museet, Stockholm.

[26] Multi-Pass Adaptive convergence, (2003). Percent convergence. [WWW] Avail- able from: http://ww3.eng.cam.ac.uk/DesignOﬃce/cad/proewild3/usascii/proe/promec/anal ysis/struct/reference/stat convper.html [Accessed: 26/08/2014].

51 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

[27] Ljungdahl, J., (2004). Computer based FE-model for evaluation of the support system to the Vasa ship, M.Sc. Thesis, Royal Institute of Technology, Stockholm.

52 STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

APPENDIX I: Vasa’s Drawings

STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

APPENDIX II: Results from the study of convergence

59 CONVERGENCE RESULTS WITH A MAXIMUM ELEMENT SIZE OF 1000 mm

Graph of Strain Energy vs. P-Pass

1,38

1,28

1,18

1,08

0,98

0,88

0,78 Strain Energy (KJ) Strain 0,68

0,58

0,48

0,38 1 2 3 4 5 6 7 P Loop Pass

Graph of Maximum displacement vs. P-Pass

16,51

14,51

12,51

10,51

8,51

6,51 Maximum displacemment (mm) Maximum displacemment 4,51

2,51 1 2 3 4 5 6 7 P Loop Pass CONVERGENCE RESULTS WITH A MAXIMUM ELEMENT SIZE OF 600 mm

Graph of Strain Energy vs. P-Pass

1,31

1,21

1,11

1,01

0,91

0,81 Strain Energy (KJ) Strain

0,71

0,61

0,51 1 2 3 4 5 6 7 P Loop Pass

Graph of Maximum displacement vs. P-Pass

15,13

13,13

11,13

9,13

7,13 Maximum Displacement (mm)Maximum Displacement 5,13

3,13 1 2 3 4 5 6 7 P Loop Pass

CONVERGENCE RESULTS WITH A MAXIMUM ELEMENT SIZE OF 400 mm

Graph of Strain Energy vs. P-Pass

1,35

1,25

1,15

1,05

0,95 Strain Energy (KJ) Strain 0,85

0,75

0,65 1 2 3 4 5 6 7 8 P Loop Pass

Graph of Maximum displacement vs. P-Pass

14,93

13,93

12,93

11,93

10,93

9,93

8,93

7,93

Maximum displacement (mm) Maximum displacement 6,93

5,93

4,93 1 2 3 4 5 6 7 8 P Loop Pass STRUCTURAL ANALYSIS OF THE 17th CENTURY WARSHIP VASA

APPENDIX III: Results from global analysis - Critical area on the hull

63 MESH OF THE MODEL:

MAXIMUM ELEMENT SIZE 800 mm

ELEMENT TYPE: “BRICK, WEDGE AND TETRA”