MAT-VET-F 20002 Examensarbete 15 hp June 2020
Investigating strains on the Oseberg ship using photogrammetry and finite element modeling
Andreas Eriksson Erik Thermaenius Abstract Investigating strains on the Oseberg ship using photogrammetry and finite element modeling Andreas Eriksson, Erik Thermaenius
Teknisk- naturvetenskaplig fakultet UTH-enheten The Oseberg ship is known as one of the finest surviving artifacts from the Viking age, with origins dated back to the 800s. The ship has been displayed in the Viking ship Besöksadress: museum in Oslo since 1926. The nearly 100 years on museum display along with the Ångströmlaboratoriet Lägerhyddsvägen 1 over 1000 years it was buried has weakened the structure of the ship. To slow down Hus 4, Plan 0 the deterioration, several research projects has been initiated, among them the project ''Saving Oseberg''. A part of ''Saving Oseberg'' is contributing to the planning Postadress: of a new museum for the ship. As a basis for the planning, the ship has been Box 536 751 21 Uppsala monitored with photogrammetry. This is intended as a way to visualise the deformation and displacements of the ship due to seasonal changes in indoor Telefon: temperature and humidity. 018 – 471 30 03
Telefax: In this thesis the photogrammetry data from the hull of the ship was used to make a 018 – 471 30 00 finite element model, and through this model calculate the average strain on each element. The method was based on a previous research project conducted on the Hemsida: Swedish warship Vasa by a research group at the Division of Applied Mechanics at http://www.teknat.uu.se/student Uppsala University. The measurements of the ship was formed into a hull by Delaunay triangulation. The strain was approximated as a Green strain and evaluated using isoparametric mapping of the elements. Through the nodal displacements, the strain was evaluated by approximating the elements as tetrahedrons and calculating the average strain from these elements between the measurements. The result showed an oscillating behavior of the displacements, proving the proposal of seasonal depending displacements. The measured principal strains also matched to the corresponding relative humidity fluctuation during the year. The strain magnitude was relatively even throughout the ship, mostly varying between ±0.4% but certain areas were more subjected than others. A few elements on the starboard side showed very large strains through most of the measurements, this seemed very unusual and was probably the result of inaccuracies or errors in the data. Though the ship is subjected to relative small strains and permanent displacements after annual cycles, the mechano-sorptive strains may lead to accumulated deformation and eventually failure in the weak parts of the wood or at the high stress concentraion parts. In addition, the cyclic strain even in elastic range may cause fatigue failure in any material which could pose a large threat for the future conservation of the ship.
Handledare: Kristofer Gamstedt, Reza Afshar Ämnesgranskare: Johan Gising Examinator: Martin Sjödin ISSN: 1401-5757, MAT-VET-F 20002 Tryckt av: Uppsala Populärvetenskaplig sammanfattning
Osebergsskeppet är känd som en av de bäst bevarade och mest åtråvärda artefakterna som överlevt från Vikingatiden. Det stora långskeppet med en omkrets på över 40 meter hittades i en gravplats utanför staden Tønsberg i Norge år 1904 och grävdes upp följande år av arkeologerna Haakon Shetelig och Gabriel Gustafson. Till en början förvarades skeppet i en av Oslo Universitets lokaler till dess att Vikingaskeppsmuséet slog upp sina dörrar år 1926, där det befunnit sig sedan dess. Trots att skeppet inte brytits ned under de 1000 år det varit begravet har denna process nu ökat takten. Ett flertal forskningsprojekt har initierats för att sakta ned nedbrytningen och möjliggöra skeppets förefintlighet, bland annat det pågående ”Saving Oseberg”. Då museet som huserar Osbergsskeppet är en gammal byggnad som saknar en modern klimatanläg- gning vilken kan reglerar inomhustemperatur och luftfuktighet till ideala förhållanden så ingår det i ”Saving Oseberg” att bidra med projekteringen till det nya muséet. En forskare har som underlag till projekteringen för den nya muséebyggnaden gjort punktmätningar över skrovet på skeppet med jämna mellanrum. Målet med dessa mätningar är att se hur de uppmätta punkterna förflyttas mellan mättillfällena vilket ger möjlighet att analysera deformationen av skeppet. Detta var syftet med det presenterade kandidatarbetet, som innebar att skapa en modell av skeppet från all mätdata och utifrån förflyttningarna i mätdatan översätta det till töjningar i skrovet på skeppet i %. Mätdatan på skeppet analyserades med programvaran Matlab, där det sammanfogades och tri- angulerades till ett tvådimensionellt skal av skrovet. Utifrån förflyttningarna av mätpunkterna, beräknades den genomsnittliga töjningen på varje triangel (även kallat element) som bygger upp skrovet för varje mätning. Töjning visualiserades med hjälp av en färgskala, där negativ och positiv töjning färgade elementen blå respektive röda. Detta gjorde det enkelt att överblicka skeppet och identifiera s.k. ”hotspots” som utsätts för större töjningar än övriga delar på skeppet. Dessutom visualiserades medelvärdet av mätpunkternas förflyttningar vid varje mättillfälle för de två sidorna på skeppet i en graf. Resultatet visade att det fanns delar på skeppet som var mer utsatta än andra, särskilt utsatt var relingen runt båten och delar av kölen. Detta förklarades enklast av att relingen saknar samma stöd som resten av skrovet och eftersom ungefär halva tyngden av skeppet vilar på kölen medför detta en högre belastning och därmed större töjningar. Halva tyngden eftersom den andra halvan är fördelat på stöttor runt sidorna av skeppet. När medelvärdet av förflyttningarna studerades uppmärksammades ett cykliskt beteende av förflyttningarna på babordsidan vilket troligen beror på årstidsvariationer i temperatur och luftfuktighet, då förflyttningar återfanns på ungefär samma positioner ett år senare. Styrbordsidan var dock mycket mer utsatt och hade ett delvis cyklisk beteende men att utslagen var betydligt högre. För alla skeppets delar syntes även flera element vars töjning var flera storleksordningar högre än elementen runt om. Detta var väldigt orimligt, och vad det berodde på blev inte klarlagt. Troligen något fel i den ursprungliga mätdatan. Slutsatsen blev att skeppet utsätts för töjningar, att dessa skiljde sig på olika delar av skeppet och att förflyttningarna varierade cykliskt under året. Med viss osäkerhet inkluderat så var resultatet relativt väntat och rimligt.
i Acknowledgements
We, the authors of this thesis, would first of all like to express our gratitude towards our supervi- sors, professor Kristofer Gamstedt and researcher Reza Afshar from the Division of Applied Mechanics at Uppsala University. Both for introducing the project to us and for their guidance and valuable input throughout the project. Along with their previous work on a similar project which has been a helpful inspiration for the general structure of this thesis. Secondly, we want to thank doctoral research fellow David Hauer from the Department of Col- lection Management at Oslo University who is the researcher from ”Saving Oseberg” behind this project. His help with supplying us with the data and answering all our question regarding the data and ”Saving Osberg”-project in general was much appreciated. We would also like to thank our project mentor, researcher Johan Gising from the Division of Drug Design and Discovery and the Division of Nanotechnology and Functional Materials at Uppsala University. His effort in managing the stage-gates, and his positive feedback and support when the project was falling behind the planned time schedule was a great help in keeping us on track. Finally, we would like to acknowledge our dear classmates who kept us company during the whole time span of the project. From the lunches in the sun, to the activities on the nights and weekends was truly delightful highlights which kept our moral high considering the unexpected pandemic which fully broke out just days prior to the start of the project. Without their contributions, this thesis would not had been made possible.
Andreas Eriksson Erik Thermaenius Uppsala, May 2020
ii Contents
1Introduction 1 1.1 Background ...... 1 1.2 Objective ...... 2 1.3 Aims ...... 2
2Theory 3 2.1 Strainanddeformation...... 3 2.2 Finiteelements ...... 7 2.3 Isoparametricmapping...... 10 2.4 Delaunaytriangulation...... 13 2.5 Mapping between vector spaces ...... 14
3Method 17 3.1 Calculatingthestrain ...... 17 3.2 Verifyingthestraincalculation ...... 18 3.3 Measurementsoftheship ...... 20 3.4 Utilisingthemeasurementdata ...... 22
4Results 25
5Discussion 38
6 Conclusions 39
Appendix A Strain analysis A.1 A.1 strain.m ...... A.1 A.2 elementInOnePlane.m ...... A.2 A.3 colourFromStrain.m ...... A.2 A.4 assemble.m ...... A.4 A.5 centerDataInSphere.m ...... A.7
iii Glossary aft The direction towards the stern, or rear of a ship. 26, 28, 38 bow The direction towards the stem, or front of a ship. 26, 28, 38 engineering strain Describes strain during a deformation as the ratio between the change of length from the initial dimension and the original length. 3, 6 functional A mathematical construction that maps a vector space into a field of scalars. 7, 8 gunwale The top edge of the hull of a ship. 38, 39 port The left facing side of a ship relative the direction the ship is heading. 20, 21, 25–27, 30, 32, 34, 35, 38 shear strain Describes the strain as the change of angles between two points on an element during a deformation. 6 starboard The right facing side of a ship relative the direction the ship is heading. 20, 25, 28–30, 32, 36–38 stem The forward-facing part of a ship. 1, 20, 25, 32, 33, 38 stern The back part of a ship. 1, 20, 21, 25, 30, 31, 38 strake A strip of planking running alongside the bottom and sides of a ship, making up its’ hull. 18, 22
Stem
Stern
Gunwale -
6 Port Starboard Strake
An illustration of the nautical terms described in the glossaries. Note that the gunwale is marked in blue and a strake is marked in red.[1]
iv 1 Introduction
1.1 Background The Oseberg Ship is known to be one of the finest and most well-preserved artefacts from the Viking Era. The over 20 meters long and 5 meters broad ship of the type Karve was found in 1903 at a large burial mound near the Norwegian town of Tønsberg and excavated the following years 1904-1905 by the Norwegian archaeologist Haakon Shetelig and Swedish archaeologist Gabriel Gustafson, see Figure 1.1a. The age of the ship has not been determined exactly, researchers have seen that timber found in the grave with the ship is from the year 834 AD but the ship is thought to be older.[2] Since the excavation, the most deteriorated artefacts of the find were treated with alum as a conser- vation process. However, the oak built ship was in better condition and could be slowly air dried, it was only surface treated with a mixture of creosote and linseed oil. [3] Completely destroyed parts were reconstructed with new wood, among them were the posts on the stem and stern. The ship was then stored at the University of Oslo temporally until the first halls of the Viking Ship Museum in Oslo were finished in 1926 and the ship was moved where it has been ever since, see Figure 1.1b.[2][4] Being displayed in the museum hall in over 90 years, the ship has slowly been degrading and shown signs of deformation and fracture formations, it was not until the 2000s that the extent of the degradation became apparent.[4]
(a) (b)
Figure 1.1: (a) The Oseberg Ship being excavated outside Tønsberg.[5] (b) The ship displayed in the Viking Ship Museum in Oslo.[6]
In 2014 a research project was launched called ”Saving Oseberg” with the aim of preventing and if possible stop the deterioration of the ship.[4] One part of the ”Saving Oseberg”-project is contributing to the planning of a new museum and the future move of the ships to the new building. The group has been monitoring the ship through photogrammetric surveillance, making point-wise position measuring of the ship on a regular basis. The researchers have observed that the measured positions of the ship are not consistent throughout the year, but instead are displaced from the original position due to seasonal changes in the indoor climate. The Viking Ship Museum housing the ship is an old building, and have not been modified with a modern air conditioning system making the indoor humidity and temperature of the museum irregular. The changes in humidity makes the wood of the ship swell and shrink, possibly causing the displacements of the measured positions.
1 1.2 Objective The deformation of the ship can pose a potential threat for the future conservation of it, depending on the magnitude and character of the deformations. Elastic deformations are completely reversible, meaning that the displaced points of the ship will return to their original position after a time cycle. But even if the deformations are elastic, cyclic deformation as fatigue in elastic range may cause failure in any material. Plastic deformations are irreversible, slowly deforming the ship until inevitable fracture. Concerned with the displacements over time, the research team monitoring the ship wants to determine if the ship is subjected to irreversible deformations or if the displacements are strictly composed of elastic deformations as well as if there are ”hot spots” where the strain is larger than in other places. A research group at the Division of Applied Mechanics at Uppsala University have conducted a similar study on the Swedish warship Vasa. Using geodetic measurements of the ship over a 12- year period, they computed the average strain of different sections on the hull of the ship through the measured position data from the original measurement to the final measurement. The building housing the ship have an advanced air conditioning system installed monitoring, among other things, the humidity in the museum which is kept constant. Their result was that the Vasa ship was slowly deformed during the monitored time and subjected to a constant stress called creep deformation. The displacements varied in different regions and directions on the hull.[7] The objective of this thesis is to conduct a similar study as the one conducted on the Vasa Ship. Using the same technique as the Vasa Ship project, this project aims to visualise the displacements of the Oseberg ships hull during the measured time. The data will in this project instead be measured using photogrammetry. Furthermore, using the same technique as the Vasa Ship project, calculate the average strain of the hull, determine the strain in percentages with respect to the reference position along the different directions on the hull and the different sections of the ship.
1.3 Aims We the authors of this thesis aim to produce a finite element model of the ship using the gathered position data with Matlab and its triangulation functions to visualise the hull. This model will be easy to implement on general position data for later use on similar projects. If the ship is further studied and more position data is added it should be possible to implement the code on this data as well. The model will calculate the average strain on finite elements composed from the data. The result will then be plotted with colourised triangular elements where the colour is decided from the scale of the strain. The plots for each time point will then be used in an animation to give a better view of the deformation over time. From the result it can be discussed where the strains are most tangible and possibly determine the cause of these strains.
2 2 Theory
2.1 Strain and deformation Consider a point in an arbitrary body B in R3, P :(x, y, z). Then look at the same point in 3 the deformed body B⇤ in R , P ⇤ :(x⇤,y⇤,z⇤). Assuming the coordinate functions (x⇤,y⇤,z⇤) are continuous and differentiable in the (x, y, z) variables. This implies that
x = x(x⇤,y⇤,z⇤), 8y = y(x⇤,y⇤,z⇤), <>z = z(x⇤,y⇤,z⇤).
Note that (x⇤,y⇤,z⇤) is considered as independent:> variables. It can be shown that when dealing with small displacements it is not necessary to distinguish between the two points (x, y, z) and (x⇤,y⇤,z⇤). Now lets add the point Q :(x +dx, y +dy, z +dz) to B to make a line element ds = PQ. This is deformed in the body B⇤ into ds⇤ by adding the point Q⇤ :(x⇤ +dx⇤,y⇤ +dy⇤,z⇤ +dz⇤). See Figure 2.1.
z z⇤
B⇤ Q⇤ :(x⇤ +dx⇤,y⇤ +dy⇤,z⇤ +dz⇤) • ds⇤ B
• (u, v, w) P ⇤ :(x⇤,y⇤,z⇤)
y⇤ x Q :(x +dx, y +dy, z +dz) • ds x⇤ •P :(x, y, z) y
Figure 2.1: The line segments ds between the points P , Q in the undeformed body B and ds⇤ between the points P ⇤, Q⇤ in the deformed body B⇤.Notethe distance between the points P and P ⇤ is (u, v, w).
The deformation of ds can be described by the engineering strain "E
ds⇤ ds " = . (2.1) E ds
3 Since there are two different bodies, B and B⇤ which do not necessarily have the same coordinate axes, each line element variable can be expressed as
@x⇤ @x⇤ @x⇤ dx⇤ = dx + dy + dz. (2.2) @x @y @z
The same follows for dy⇤ and dz⇤. Noting from Figure 2.1 the displacement of the point P to P ⇤ as (u, v, w), the relation between the spaces is described as
x⇤ = x + u, 8y⇤ = y + v, (2.3) <>z⇤ = z + w.
The length of the line element ds can be described:> as
ds 2 =dx2 +dy2 +dz2, | | 2 2 2 2 ds⇤ =dx⇤ +dy⇤ +dz⇤ . | | There are several ways to calculate strain, among them are Greens strain which is used to calculate finite strains. The expression for Greens strain "G is
1 ds 2 ds2 " = ⇤ . G 2 ds2 This can also be written as the magnification factor M which is just another expression for Greens strain, see [8] for the full derivation of M.
1 ds 2 M = ⇤ 1 2 ds ✓ ◆ ⇣ 1 ⌘ = " + "2 E 2 E (2.4) 2 2 2 = l "xx + lm"xy + ln"xz + ml"yx + m "yy + mn"yz + nl"zx + nm"zy + n "zz 2 2 2 = l "xx + m "yy + n "zz +2lm"xy +2ln"xz +2mn"yz where the parameters l, m and n are defined as dx dy dz l = ,m= ,n= (2.5) ds ds ds
4 and the strain terms as
@u 1 @u 2 @v 2 @w 2 " = + + + , xx @x 2 @x @x @x ✓ ◆ ✓ ◆ ✓ ◆ ! @v 1 @u 2 @v 2 @w 2 " = + + + , yy @y 2 @y @y @y ✓ ◆ ✓ ◆ ✓ ◆ ! @w 1 @u 2 @v 2 @w 2 " = + + + , zz @z 2 @z @z @z ✓ ◆ ✓ ◆ ✓ ◆ ! (2.6) 1 @v @u @u @u @v @v @w @w " = " = + + + + , xy yx 2 @x @y @x @y @x @y @x @y ✓ ◆ 1 @w @u @u @u @v @v @w @w " = " = + + + + , xz zx 2 @x @z @x @z @x @z @x @z ✓ ◆ 1 @w @v @u @u @v @v @w @w " = " = + + + + . yz zy 2 @y @z @y @z @y @z @y @z ✓ ◆
Note that when the line element ds is along for example the x-axis (ds =dsx) then equation 2.5 will output l =1and m = n =0. In turn equation 2.4 will output
M = Mx = "xx.
Similar to equation 2.5, the deformed forms can be written as
dx⇤ dx⇤ ds dy⇤ dy⇤ ds dz⇤ dz⇤ ds l⇤ = = ,m⇤ = = ,n⇤ = = . ds⇤ ds ds⇤ ds⇤ ds ds⇤ ds⇤ ds ds⇤ Combining equation 2.2 and 2.3, it is known that dx @u @u @u ⇤ = 1+ l + m + n, ds @x @y @z dy ⇣@v ⌘ @v @v ⇤ = l + 1+ m + n, ds @x @y @z dz @w ⇣@w ⌘ @w ⇤ = l + m + 1+ n. ds @x @y @z ⇣ ⌘ Rewriting equation 2.1, it is possible to express ds 1 = . ds⇤ 1+"E From the equations above, the expressions for the deformed line elements is given as @u @u @u (1 + " )l⇤ = 1+ l + m + n, E @x @y @z ⇣@v ⌘ @v @v (1 + " )m⇤ = l + 1+ m + n, E @x @y @z @w ⇣@w ⌘ @w (1 + " )n⇤ = l + m + 1+ n. E @x @y @z ⇣ ⌘ 5 This will be useful in the following. The goal is to explain how a shear strain can be expressed. Considering two line elements ds1 and ds2, the product of these vectors can be expressed as
ds ds = cos (✓), ds⇤ ds⇤ = cos (✓⇤). 1 · 2 1 · 2 From this the shear strain between the two deformed line elements can be written as