Fatigue analysis of welded joints in a forestry machinelnu.diva-portal.org/smash/get/diva2:825303/FULLTEXT01.pdf · Master's Thesis in Mechanical/Structural Engineering Fatigue analysis

Master's Thesis in Mechanical/Structural

Engineering

Fatigue analysis of welded

joints in a forestry machine - Utilizing the notch stress concept

Authors: Martin Nyström, Tainan Pantano

Tomaz

Surpervisor LNU: Andreas Linderholt

Examinar, LNU: Andreas Linderholt

Surpervisor company: Mattias König, Rottne

Industri AB

Course Code: 4BY05E/4MT01E

Semester: Spring 2015, 15 credits

Linnaeus University, Faculty of Technology

III

Abstract

Welding is one of the most applied technics in the world for joining steel. Welds are

liable to the phenomenon of fatigue, which is, primarily, the formation of a crack and

consequently reduction of strength due to the action of time varying loads. Fatigue is

one of the main causes of failure in steel structures. The aim of this thesis is to do

static and dynamic analyses of a forestry crane with the purpose of using the analyses

to determine the lifetime due to fatigue of welded components.

Two methods for fatigue assessment are used in this work, the Hot-Spot Method and

the Notch Stress Method. The first boom, which is a key component for the crane, is

analyzed in a Finite Element Method (FEM) software. The found principal stress in

accordance with the notch stress method in the first boom is ±165 MPa for the

analyzed load case, rendering in a stress range of 330 MPa. The fatigue strength class

FAT-225 (m=3), leads to an expected number of 6.33 ∙ 105cycles, with a probability

of survival of 97,7% for this case.

Key words: Weld, Fatigue, Finite Element Method, Hot-Spot Method, Notch Stress

Method, Structural Dynamics, Forestry Crane

IV

Sammanfattning

Svetsning är en av de vanligaste teknikerna för sammanfogning av stål. Svetsar är

känsliga för utmattning. Utmattningsfenomenet består primärt av en initial

dislokation som genom tidsvarierande belastning formar en spricka som växer och

därmed reducerar styrkan i konstruktionen. Utmattning är en av de vanligaste

orsakerna till skador i stålkonstruktioner. Målet med detta arbete är att genomföra

både statiska och dynamiska analyser av en skogsmaskins kran i avseende att

bestämma utmattningslivslängden för dess svetsade konstruktioner.

Två metoder för utvärdering används i detta arbete, hot-spot-metoden och

notch-stress-metoden. Kranens första bom (lyftarmen) som är en huvudkomponent i

kranen analyseras med hjälp av ett Finita Element program i enlighet med

notch-metoden. Högsta funna spänningsvariationen i första

huvudspänningsriktningen var ±165 MPa för ett av de analyserade lastfallen.

Utmattningsklass FAT 225 (m=3) ger en uppskattning om utmattningslivslängd på

6,33 ∙ 105 cykler med en sannolikhet för överlevnad på 97,7% i detta fall.

Nyckelord: Svets utmattningslivslängd, Finita-elementmetoden, Hot-spot,

Notch-stress, Notch-spänning, Strukturdynamik, Skogsmaskinskran

V

Acknowledgement

A special acknowledgement is given to Rottne Industri AB; Mattias König as

supervisor and Ulf Wilhelmsson and Daniel Sätermark for giving support on the

CAD- and FEM software.

We would also like to thank our supervisor, Andreas Linderholt, for helping us

during the whole thesis.

An acknowledgement is given to CNPq (the Brazilian National Council for Scientific

and Technological Development) and CAPES (Coordination for the Improvement of

Higher Education Personnel) for providing financial support through the Brazilian

mobility program - Science without Borders.

Martin Nyström & Tainan Pantano Tomaz

Växjö, Sweden, 27th of May 2015

VI

Table of contents

1. INTRODUCTION...................................................................................................... 1

1.1 BACKGROUND ................................................................................................................................... 2 1.2 AIM AND PURPOSE ............................................................................................................................. 2 1.3 HYPOTHESIS AND DELIMITATIONS ..................................................................................................... 3

1.3.1 Hypothesis ............................................................................................................................... 3 1.3.2 Delimitations ........................................................................................................................... 3

1.4 THESIS MOTIVATION .......................................................................................................................... 4 1.5 RELIABILITY, VALIDITY AND OBJECTIVITY ........................................................................................ 5

2. THEORY .................................................................................................................... 6

2.1 FATIGUE ............................................................................................................................................ 6 2.1.1 Definition ................................................................................................................................. 6 2.1.2 The S-N diagram and Palmgren-Miner rule ........................................................................... 6 2.1.3 Approaches for fatigue assessment ......................................................................................... 8 2.1.4 Influential factors on fatigue strength ................................................................................... 10

2.2 FINITE ELEMENT METHOD ................................................................................................................ 10 2.2.1 Definition ............................................................................................................................... 10 2.2.2 1D-elasticity beam ................................................................................................................. 11 2.2.3 The notch stress method applied to FE-analysis ................................................................... 12

2.3 STRUCTURAL DYNAMICS ................................................................................................................. 13 2.3.1 Eigenvalue problem (EVP) and modal matrices ................................................................... 13 2.3.2 Modal analysis ...................................................................................................................... 14 2.3.3 State-space ............................................................................................................................ 14 2.3.4 Free vibration for a single-degree of freedom system having viscous damping ................... 15

2.4 LITERATURE REVIEW ....................................................................................................................... 17

3. METHOD ................................................................................................................. 19

3.1 CRANE SETUP .................................................................................................................................. 19 3.2 DEFINITION OF LOAD CASES............................................................................................................. 20 3.3 CALCULATION MODEL ..................................................................................................................... 20

3.3.1 Static analysis ........................................................................................................................ 25 3.3.2 Dynamic analysis .................................................................................................................. 25 3.3.3 Finite Element analysis ......................................................................................................... 26

3.4 PREDICTION OF THE LIFETIME OF THE CHOSEN WELD ....................................................................... 32 3.5 DAMPING EXPERIMENT .................................................................................................................... 33

VII

4. RESULTS ................................................................................................................. 34

4.1 STATIC ANALYSIS ............................................................................................................................ 34 4.1.1 Load case 1............................................................................................................................ 36 4.1.2 Load case 2............................................................................................................................ 36 4.1.3 Load case 3............................................................................................................................ 37 4.1.4 Load case 4............................................................................................................................ 37 4.1.5 Load case 5............................................................................................................................ 38 4.1.6 Load case 6............................................................................................................................ 38 4.1.7 Load case 7............................................................................................................................ 39

4.2 DYNAMIC ANALYSIS ........................................................................................................................ 39 4.2.1 Load case 1............................................................................................................................ 42 4.2.2 Load case 2............................................................................................................................ 43 4.2.3 Load case 3............................................................................................................................ 44 4.2.4 Load case 4............................................................................................................................ 45 4.2.5 Load case 5............................................................................................................................ 46 4.2.6 Load case 6............................................................................................................................ 47 4.2.7 Load case 7............................................................................................................................ 48

4.3 FINITE ELEMENT ANALYSIS ............................................................................................................. 49 4.3.1 First step: hot-spot stresses ................................................................................................... 49 4.3.2 Second step: notch stress method .......................................................................................... 60

4.4 FATIGUE CALCULATIONS ................................................................................................................. 61 4.5 DAMPING EXPERIMENT .................................................................................................................... 61

5. ANALYSIS ................................................................................................................... 63

6. DISCUSSION ............................................................................................................... 65

7. CONCLUSIONS .......................................................................................................... 66

REFERENCES ................................................................................................................. 67

APPENDICES .................................................................................................................. 69

1

Nyström, M; Tomaz, T. P.

1. Introduction

The forest industry is known as a particularly strong cluster in Sweden.

From the economical perspective, it provides a direct employment to around

60,000 people and in several regions it accounts for at least 20% of the

employment (Swedish Forest Industries, 2013). On a global scale, around 10

million people are employed in forest management and conservation (FAO,

2010). There is a significant progress in developing forest policies, laws and

national forest programs worldwide. Also, there is an increase in the

European recreational forest area and more than 90% of European forests

are open for public access (FAO, 2010).

The most used way to harvest trees in Sweden is through thinning and clear-

cuts in the forest using harvester machines. These harvesters can measure

and cut-to-length logs, according to the market demand (Johansson, 2011).

Rottne Industri AB is a Swedish established company, located in the region

of Småland, which produces forestry machinery, such as harvesters and

forwarders, having 50% of its production exported to other countries in

Europe and North America (Rottne Industri AB, 2014).

2


1.1 Background

Rottne Industri AB was founded in 1955 and has, since then, been a supplier

of equipment and vehicles for the mechanized forestry work. Rottne takes a

role during all steps of forest machines fabrication, from design until

assembling of all parts. The company has, among other functions, its own

manufacturing of welded and machined components.

Welding is one of the most applied technics in the world for joining steel

and it is being used in innumerous engineering structures. In many cases,

other types of connections (for instance fasteners) are not suitable due to the

assembling procedure or to the geometry of the steel components.

Nevertheless, the welds are liable to the phenomenon of fatigue which is a

major challenge in welding design. Fatigue is, primarily, the formation of a

crack and consequently a reduction of strength due to the action of time

varying loads (Macdonald, 2011).

Cracks generated can increase in size and lead steel components to failure

(Witek, 2009). Many researchers, for instance (Marin & Nicoletto, 2009) and

(Ritchie, 1998), consider fatigue cracking as the principal failure mode in

welds. The last states that over 80% of all service failures are related to

mechanical fatigue, e.g. cyclic plasticity, elevated temperatures (creep

failure) and environmental damage (corrosion failure).

1.2 Aim and Purpose

The aim of this thesis is to do dynamic analyses of a forestry crane with the

purpose of using the analyses for further product development and to

determine fatigue lifetime of welded components.

In addition, this thesis aims to broaden the theoretical study in this field and

specifically to transfer scientific knowledge to Rottne Industri AB for use in

their design and modeling of structures.

3


1.3 Hypothesis and Delimitations

1.3.1 Hypothesis

It is possible to predict the lifetime of structural components in a forestry

crane by the use of finite element analyses and weld fatigue calculations of

the structure.

1.3.2 Delimitations

Engineering is considered to be a non-exact science because most of the

time, there is no single best solution to a problem (Chow, 2002). Based on

that, many approximations and simplifications are done in order to get a

result that comprises both quality and attend deadlines.

The delimitations for this thesis are presented in Table 1.

Table 1: Delimitations.

Delimitations Comments

The welds, including the heat-

affected zone, are assumed to be

lifetime limiting

The virgin steel used for the crane has

a higher reistance to fatigue than the

welds

The boom position is set to a

predefined configuration

In normal use the crane is constantly

changing configuration. The choice of

configuration will influence the result

significantly

Only the crane’s first boom is

analysed using the finite element

software

Other components may break before

any main component does. This is,

though, less negative for the customer

both from an economical and

productivity view

Only seven load cases originated

by the combination of three

different actuators will be

considered

In normal use there are infinitely many

possible load cases

4


1.4 Thesis motivation

Since Rottne is interested in improving design, increasing customer value

and minimizing weight through reducing material waste, a new study on

fatigue analysis of welded steel components, lifetime prediction, and

consequent product development was suggested.

The company recently developed a new harvester named H21D , which is

equipped with a crane RK-250 with a lifting torque of 310 kNm and a total

reach of 10 m distance, see Figure 1. This crane is the object of this study on

fatigue.

A harvester can work on an average productivity of 100 trees per hour, being

subjected to cyclically varying loads and having many welded connections.

Figure 1: Harvester H21D (Rottne Industri AB, 2014)

5


1.5 Reliability, validity and objectivity

All the measures of the crane and its components are obtained from

brochures and 3D solid CAD models provided by the company. The same

procedure can be applied for further studies in this area.

The boom configuration is chosen to one particular case. However, the crane

has a constantly changing configuration during its operation. The

configuration used during this work is chosen to represent the normal

working range (Rottne Industri AB, 2014) and therefore has enough validity.

The finite element analyses are made on the first boom’s welds, since they

are the key components for the crane’s operation. Practice, for example the

works Mechanisms of fatigue-crack propagation in ductile and brittle solids

(Marin & Nicoletto, 2009) and Fatigue design of welded joints using the

finite element method and the 2007 ASME Div.2 Master Curve (Ritchie,

1998), have shown that welds are the most critical parts for fatigue failure.

6


2. Theory

This chapter presents relevant theory of Fatigue Behavior, the Finite

Element Method and Structural Dynamics modeling.

2.1 Fatigue

2.1.1 Definition

Most steel structures are liable to repeated loadings and unloadings, which

can cause microscopic cracks. After many repeated loadings and unloadings,

the cracks can increase in size and then lead the metal to failure, as shown in

Figure 2. Fatigue can also be defined as the reduced strength originated by

the formation of cracks.

Figure 2: Locations of crack propagation in welded joints (Hobbacher, 2008)

2.1.2 The S-N diagram and Palmgren-Miner rule

The pioneer of fatigue studies was August Wöhler (1819-1914), who first

established a systematic way to identify and quantify fatigue on locomotive

axles (Timoshenko, 1983).

A simplified version of one of his experiments consisted of a 4-point

bending rotational beam. After imposing several bending moments, the

number of cycles until failure is counted and plotted versus the stress

variation. From the obtained points within the stress range it is possible to

draw an approximative relation between the stress (S) and the number of

cycles (N). This is known as the S-N diagram (S for strength and N for

number of cycles); or as a Wöhler Curve, which is shown in Figure 3.

7


Figure 3: A typical S-N diagram – Cycles to failure (European prestandard, 2002)

Each material has its own S-N curve and is classified according to the

“FAT” grade. A characteristic curve, FAT-225 for example, has a fatigue

strength of 225 MPa at 2 million cycles for a 2,3% failure probability

(Fricke, 2007).

The S-N diagram is obtained from tests stemming from constant loadings.

However, structures are usually subjected to many loading and unloading

cycles, for which the stresses varies in a more complicated manner. In order

to take these fluctuating stresses into account, the Palmgren-Miner theorem,

which consists of a damage summation, is applied.

According to this method (Eriksson, et al., 2002), failure occurs when,

∑𝑛𝑖𝑁𝑖

𝐼

𝑖=1

= 1

( 1 )

where 𝑛𝑖 is the number of loadings with stress value 𝜎𝑖 and 𝑁i is the fatigue

life for stress 𝜎𝑖. An example of a damage accumulation problem is shown in

Appendix 2.

8


2.1.3 Approaches for fatigue assessment

There are several approaches for fatigue assessment of welded joints. The

most relevant, due to large industry application in recent years (Hobbacher,

2008), are the:

Nominal stress method;

Structural hot-spot stress method;

Effective notch stress method.

Other approaches, such as fracture mechanics, are not addressed in this

thesis.

2.1.3.1 The nominal stress method

The Nominal Stress approach is based on the stresses in the cross-sectional

area of the component considering the effects of particular macro-geometric

shape in the vicinity of the joint (Figure 4). Eventual local stresses are

excluded (Hobbacher, 2008).

Figure 4: Examples of macrogeometric effects (Hobbacher, 2008)

The nominal stress method is recommended for evaluation of slender welded

frame structures, where shear has almost no impact on the stresses of the

cross-sectional area of the weld (Rother & Rudolph, 2010). However, in

more complex structures, this method is not recommended. Local concepts,

such as the hot-spot method (section 2.1.3.2) or effective notch method

(section 2.1.3.3) are applicable, since the fatigue process has a local

character and cannot fully be described by a global parameter, such as the

nominal stress (Radaj, et al., 2009).

2.1.3.2 The structural hot-spot stress method

This method is based on stresses at the hot-spots of the weld to be assessed.

Hot-spots can be defined as points where fatigue cracks are initiated (Radaj,

et al., 2009) and therefore, knowing their structural stresses are of great

importance for fatigue assessment.

9


There are eight different attempts (Rother & Rudolph, 2010) to obtain the

hot-spots’ structural stresses. Most of them are based on established

reference points from which, by extrapolation, the structural stresses are

obtained. This method is usually used together with the Finite Element

Method (section 2.2).

In one of these attempts, the stress, 𝜎𝐻𝑎𝑖𝑏𝑎𝑐ℎ, at a distance of 2 to 3 mm from

the weld toe is used as the hot-spot stress, see Figure 5. This is based on the

work done by Haibach (Haibach, 1968).

Figure 5: Hot-spot stress according to Haibach (Rother & Rudolph, 2010)

2.1.3.3 The effective notch stress method

The notch stress concept consists of modeling of the welds’ roots and toes

with a reference radius 𝑟𝑟𝑒𝑓 to obtain principal stresses or Von Mises

stresses. A sketch of the modeling is shown in Figure 6, in which 𝑃 is an

arbitrary load.

Figure 6: Fictitious radii used for notch stress calculations – Adapted from (Sonsino, et al., 2012)

The most common choices of the reference radius 𝑟𝑟𝑒𝑓 = 1.00 mm and

𝑟𝑟𝑒𝑓 = 0.05 𝑚𝑚. Besides the theoretical background of the two values (the

𝑟𝑟𝑒𝑓

P P

10


former is based on a micro-support theory and the latter on a stress-intensity

factor), there is also an empirical background, based on observations. The

larger radius could cause a weakening of cross-sections for thinner plates,

and therefore joints of structural steel with thickness 𝑡 < 5 𝑚𝑚 should

preferrably be modeled with 𝑟𝑟𝑒𝑓 = 0.05 mm (Sonsino et al., 2012).

The formula used to obtain the fatigue lifetime (Eriksson, et al., 2002),

𝑁𝑓 = 𝑁 (𝜎𝐹𝐴𝑇

𝜎𝑟𝑎𝑛𝑔𝑒)𝑚

( 2 )

𝜎𝑟𝑎𝑛𝑔𝑒 = 𝜎𝑚𝑎𝑥 − 𝜎𝑚𝑖𝑛 ( 3 )

where 𝑁𝑓 is the fatigue life length (number of cycles) for the analyzed weld,

N is the number of cycles before failure for a specific weld class, 𝜎𝐹𝐴𝑇 is the

weld strength according to the weld class, 𝜎𝑟𝑎𝑛𝑔𝑒 is the stress variation over

the analyzed weld and 𝑚 is the order, generally suggested as 𝑚 = 3 for

steel.

The notch stress method may be used together with the hot-spot method,

which is then called a two-step approach (Rother & Rudolph, 2010).

2.1.4 Influential factors on fatigue strength

Understanding influential factors on fatigue strength is essential for adequate

comprehension of fatigue behavior. There is a variety of factors that can

affect fatigue strength. Examples are:

Material characteristics

Stress concentrations, especially at weld toes

Detail configurations

Geometry defects, such as presence of discontinuities.

2.2 Finite element method

2.2.1 Definition

Many physical problems are described by differential equations. To find

exact solutions to these equations requires a lot of time and may even be

unsolvable. An alternative is to apply numerical methods which can give

approximate solutions. The Finite Element Method (FEM) is one of the most

recognized methods within industry (Radaj, et al., 2009).

The origin of finite elements date back to two different mathematical

methods developed by Rayleigh-Ritz and Galerkin (Assan, 2013). Both

methods are based on approximation functions.

11


Initially, the finite element analysis can be defined as a numerical method to

find the solution of a complex problem by substituting it by many simpler

ones. The solution region in this method consists of many small,

interconnected subregions, known as elements (Rao, 2004).

Some of the common problems that can be solved by using the finite

element method includes structural mechanics, heat conduction, fluid

dynamics, electric and magnetic fields and many others. Each of these

problems is described by a differential equation, which is then called a

strong formulation. From the strong formulation it is possible to obtain the

weak formulation, and from that, the finite element formulation. An example

of this procedure is presented for supporting the understanding of the Finite

Element Analysis.

2.2.2 1D-elasticity beam

The following example consists of a cantilever beam (clamped in one side

and free in the other) subjected to an axial force 𝑃 as can be seen in Figure 7.

The beam has a cross sectional area (𝐴) and modulus of elasticity (𝐸).

Figure 7: 1D-Elasticity model (Ekevid, 2014)

The displacement at a certain point 𝑥 is 𝑢(𝑥), the normal force function is

𝑁(𝑥) and 𝑏 is the body force. The strong formulation consists of a

differential equation,

𝑑

𝑑𝑥(𝐴𝐸

𝑑𝑢

𝑑𝑥) + 𝑏 = 0; 0 ≤ 𝑥 ≤ 𝐿

( 4 )

Together with the boundary conditions according to the model,

𝑢(0) = 0;𝑁(𝐿) = 𝑃

𝑁(𝑥) = 𝐴𝐸

𝑑𝑢

𝑑𝑥

( 5 )

Multiplying the weighting function 𝑣 (also called test function) to the strong

form and integrating over the entire region,

∫ 𝑣(

𝑑

𝑑𝑥(𝐴𝐸

𝑑𝑢

𝑑𝑥) + 𝑏)𝑑𝑥

𝐿

0

= 0 ( 6 )

12


∫ 𝑣(

𝑑

𝑑𝑥(𝐴𝐸

𝑑𝑢

𝑑𝑥))𝑑𝑥

𝐿

0

+∫ 𝑣 ∙ 𝑏 𝑑𝑥𝐿

0

= 0 ( 7 )

Applying partial integration to the last equation, gives the weak formulation:

∫𝑑𝑣

𝑑𝑥(𝐴𝐸

𝑑𝑢

𝑑𝑥)𝑑𝑥

𝐿

0

= [𝑣 ∙ 𝑁]0𝐿 −∫ 𝑣 ∙ 𝑏 𝑑𝑥

𝐿

0

= 0 ( 8 )

To obtain the Finite Element formulation, the Galerkin Method is applied to

the weighting function 𝑣, where �� is the shape function vector over the

element’s boundaries and 𝐶 is an unknown (arbitrary) vector,

𝑣 = �� ∙ 𝐶 = 𝐶𝑇 ∙ ��𝑇 = 𝑣𝑇 ( 9 )

𝑑𝑣

𝑑𝑥=𝑑��

𝑑𝑥∙ 𝐶 = 𝐶𝑇 ∙

𝑑��

𝑑𝑥

𝑇

= 𝐶𝑇 ∙ 𝐵𝑇 ( 10 )

Introducing the approximation formula

𝑢(𝑥) = �� ∙ 𝑎 ( 11 )

𝑑𝑢

𝑑𝑥=𝑑��

𝑑𝑥∙ 𝑎 = 𝐵 ∙ 𝑎

( 12 )

where 𝑎 is the displacement vector. Substituting equations [( 9 ) - ( 12 )] into

the weak formulation ( 8 ), and assuming that all the points are valid for 𝐶𝑇,

it is possible to obtain the Finite Element Formulation,

∫ 𝐵𝑇𝐴𝐸𝐵𝑑𝑥𝐿

0

∙ 𝑎 = [��𝑇𝑁]0𝐿 −∫ 𝑁𝑇 ∙ 𝑏 𝑑𝑥

𝐿

0

= 0 ( 13 )

𝐾 = ∫ 𝐵𝑇𝐴𝐸𝐵𝑑𝑥

𝐿

0

( 14 )

𝑓𝑙 = [��𝑇𝑁]0

𝐿 ( 15 )

𝑓𝑏 = ∫ 𝑁𝑇 ∙ 𝑏 𝑑𝑥

𝐿

0

( 16 )

where K is the stiffness matrix, fl is the load vector and fb is the body force

vector.

The same procedure can be applied for 2D and 3D models, which are more

commonly seen in daily engineering problems. Reference reading (Ottosen

& Petersson, 1992).

2.2.3 The notch stress method applied to FE-analysis

Models for FE-analyses must be made with good precision in order to

represent the real components. The variables that influence the result

13


directly are the element size (mesh precision), element type and

approximation order (quadratic, cubic, etc).

An example of the notch stress method using FE-modeling is presented in

Figure 8. The first step consists of setting a reference fictive radius, then the

element size is defined near to the weld toe and the mesh is created. The

results are given as principal stresses or von Mises stresses, and by using the

SN-diagram, a limit number of cycles with 2,3% probability of failure is

obtained.

Figure 8: Notch stress approach (Pedersen, et al., 2010)

Models for analysis can be built in several types of software, such as

MATLAB®/CALFEM, ABAQUS and SOLIDWORKS®. One important

tool is that CAD-models from SOLIDWORKS® can be exported to

ABAQUS and vice versa.

2.3 Structural dynamics

The equation of motion that represents the dynamic behavior of a damped

system with N degrees-of-freedom is shown below, where 𝑀, 𝐶 and 𝐾, are

the mass, damping and stiffness matrices (all of them with size NxN), 𝑃(𝑡) is

the load vector, 𝑢 is the displacement vector, �� the velocity vector and

finally u represents the acceleration vector (size Nx1).

𝑀�� + 𝐶�� + 𝐾𝑢 = 𝑃(𝑡) ( 17 )

2.3.1 Eigenvalue problem (EVP) and modal matrices

For an undamped system, the following holds:

𝑀�� + 𝐾𝑢 = 𝑃(𝑡) ( 18 )

For an unloaded situation and assuming that the motion is harmonic,

𝑢 = 𝑈 ∙ 𝑐𝑜𝑠 (𝜔𝑡 − 𝛼) ( 19 )

where ω is the driving frequency and α is a delay variable.

�� = −𝜔2𝑢 ( 20 )

𝑀(−𝜔2𝑢) + 𝐾𝑢 = 0 ( 21 )

14


A non-trivial solution for this eigenvalue problem, where 𝜔 is the natural

circular frequency (rad/s) and 𝜙𝑖 is a mode shape, fulfills

(𝐾 − 𝜔𝑖2𝑀)𝜙𝑖 = 0 ( 22 )

𝑑𝑒𝑡(𝐾 − 𝜔𝑖2𝑀) = 0 ( 23 )

The following relationship holds and:

𝑓𝑛 = 𝜔

2𝜋 [𝐻𝑧] ( 24 )

where 𝑓𝑛 is the natural frequency in Hz

2.3.2 Modal analysis

Modal analysis consists of creating modal mass, damping and stiffness

matrices using the data collected from the mode shapes 𝜙 and the matrices M

and K, aiming to use the modal matrices to calculate the generalized modal

damping matrix.

𝜙𝑇𝑀𝜙⏟ 𝑀𝑜𝑑𝑎𝑙 𝑚𝑎𝑠𝑠 𝑚𝑎𝑡𝑟𝑖𝑥

�� + 𝜙𝑇𝐶𝜙⏟ 𝐺𝑒𝑛𝑒𝑟𝑎𝑙𝑖𝑧𝑒𝑑 𝑑𝑎𝑚𝑝𝑖𝑛𝑔𝑚𝑎𝑡𝑟𝑖𝑥

�� + 𝜙𝑇𝐾𝜙⏟ 𝑀𝑜𝑑𝑎𝑙 𝑠𝑡𝑖𝑓𝑓𝑛𝑒𝑠𝑠𝑚𝑎𝑡𝑟𝑖𝑥

𝑢 = 𝜙𝑇𝑃(𝑡)⏟

𝑀𝑜𝑑𝑎𝑙 𝑙𝑜𝑎𝑑𝑣𝑒𝑐𝑡𝑜𝑟

( 25 )

𝑀𝑚𝑜𝑑𝑎𝑙 = 𝜙𝑇𝑀𝜙 ( 26 )

𝐾𝑚𝑜𝑑𝑎𝑙 = 𝜙𝑇𝐾𝜙 ( 27 )

The modal damping matrix 𝐶𝑚𝑜𝑑𝑎𝑙 can be calculated using the following

formula, where ξ is the modal relative viscous damping.

𝐶𝑚𝑜𝑑𝑎𝑙 = √𝑀𝑚𝑜𝑑𝑎𝑙 ∙ 𝐾𝑚𝑜𝑑𝑎𝑙 ∙ 2 ∙ 𝜉 ( 28 )

𝑃𝑚𝑜𝑑𝑎𝑙 = 𝜙𝑇𝑃(𝑡) ( 29 )

𝐶 = (𝜙𝑇)−1𝐶𝑚𝑜𝑑𝑎𝑙(𝜙)−1 ( 30 )

2.3.3 State-space

There are many ways to mathematically represent the dynamic behavior in

an N degree-of-freedom (DOF) model. One of the most used ways to do that

is through the use of state-space representation, which consists of reducing

the order of the equation of motion from second-order to twice as many first-

order differential equation.

15


The procedure is here shown by an example for a single-DOF, having the

initial conditions 𝑢(0) = 0; ��(0) = 0.

𝑥1 = 𝑢 ( 31 )

𝑥2 = �� ( 32 )

𝑥 = [𝑥1𝑥2] = [

𝑢��] ( 33 )

𝑥1 = 𝑥2 ( 34 )

M𝑥2 + C𝑥2 + K𝑥1 = P(t) ( 35 )

[−1 00 𝑀

] [𝑥1𝑥2] + [

0 1𝐾 𝐶

] [𝑥1𝑥2] = [

0𝑃(𝑡)

] ( 36 )

[𝑥1𝑥2] = − [

−1 00 𝑀

]−1

[0 −1−𝐾 −𝐶

] [𝑥1𝑥2]

+ [−1 00 𝑀

]−1

[0P(t)

]

( 37 )

[𝑥1𝑥2] = [

0 −1

−𝐾

𝑀−𝐶

𝑀

]⏟

𝐴

[𝑥1𝑥2] + [

01

𝑀

]⏟𝐵

𝑃(𝑡) ( 38 )

x = Ax + B ∙ P(t) ( 39 )

2.3.4 Free vibration for a single-degree of freedom system having viscous

damping

A system is considered to be under free vibration when it oscillates after an

initial disturbance, with no external forces applied after the disturbance. A

damped mass-spring system is shown in Figure 9.

Figure 9: Damped mass-spring SDOF system

The equation of motion for this system can be written as

�� +

𝐶

𝑀�� +

𝐾

𝑀𝑢 = 0

( 40 )

16


The characteristic equation, where 𝑡 is the time and 𝑆1,2 are the two

solutions,

𝑆2 +

𝐶

𝑀𝑆 +

𝐾

𝑀= 0

( 41 )

The solution becomes

𝑢(𝑡) = 𝐴𝑒𝑆1∙𝑡 + 𝐵𝑒𝑆2∙𝑡 ( 42 )

𝑆1,2 = −𝐶

2 ∙ 𝑀±√(−

𝐶

2 ∙ 𝑀)2

−𝐾

𝑀⏟ 𝑆𝑒𝑐𝑜𝑛𝑑 𝑡𝑒𝑟𝑚

( 43 )

Different solutions appear depending on the second term. When the second

term disappears, then the damping 𝐶 = 𝐶𝑐𝑟𝑖𝑡 (critical damping). The

undamped natural frequency is ωn,

(−

𝐶𝑐𝑟𝑖𝑡2 ∙ 𝑀

)2

−𝐾

𝑀= 0

( 44 )

𝐶𝑐𝑟𝑖𝑡2 ∙ 𝑀

= √𝐾

𝑀= 𝜔𝑛

( 45 )

𝐶𝑐𝑟𝑖𝑡 = 2 ∙ 𝑀 ∙ 𝜔𝑛 ( 46 )

Another important concept for dynamic models (Lussanet, et al., 2002) is the

relative damping ξ,

𝜉 =

𝐶

𝐶𝑐𝑟𝑖𝑡

( 47 )

𝐶 = 2 ∙ 𝜉 ∙ 𝑀 ∙ 𝜔𝑛 ( 48 )

There are three different cases for ξ, which are shown in Table 2.

Table 2: Relative damping cases.

Relative damping level Comments

Case 1: 0 < 𝜉 < 1 Underdamped (oscilattory)

Case 2: 𝜉 > 1 Overdamped

Case 3: 𝜉 = 1 Criticially damped

The solutions 𝑆1,2 can be written as

𝑆1,2 = −𝜉 ∙ 𝜔𝑛 ±√𝜉

2𝜔𝑛2 − 𝜔𝑛

2 ( 49 )

𝑆1,2 = −𝜉 ∙ 𝜔𝑛 ± 𝜔𝑛 ∙ √𝜉2 − 1 ( 50 )

17


For underdamped systems, where i = √−1, and ωd is the damped circular

frequency,

𝑆1,2 = −𝜉 ∙ 𝜔𝑛 ± 𝑖 ∙ 𝜔𝑛 ∙ √1 − 𝜉2⏟ 𝜔𝑑

( 51 )

The solution for the equation of motion is (Craig & Kurdila, 2011),

𝑢(𝑡) = 𝑒−𝜉∙𝜔𝑛∙𝑡 ∙ (𝐴 ∙ 𝑐𝑜𝑠(𝜔𝑑𝑡) + 𝐵 ∙ 𝑠𝑖𝑛(𝜔𝑑𝑡)) ( 52 )

This can also be written as (Rajasekaran, 2009),

𝑢(𝑡) = 𝑒−𝜉∙𝜔𝑛∙𝑡 ∙ (𝐴 ∙ 𝑠𝑖𝑛 (𝜔𝑛 ∙ √1 − 𝜉2𝑡 + 𝜑)) ( 53 )

A typical displacement response for harmonically damped systems is shown

in Figure 10.

Figure 10: Typical displacement graphic for damped system

2.4 Literature review

Recently, the notch stress method was applied to obtain fatigue lifetime of

different steel structures such as diggers (Pedersen, et al., 2010), offshore

K-nodes, sandwich panels for ship decks, automotive doors and trailing links

(Sonsino, et al., 2012), which shows a vast application field and an ongoing

development process of the method. Developments done in the past decade,

such as studies done with both 𝑟𝑟𝑒𝑓 = 1.00 mm and 𝑟𝑟𝑒𝑓 = 0.05 𝑚𝑚 for the

notch stress concept, were discussed in (Radaj, et al., 2009).

e−𝜉∙ω𝑛∙t ∙ |𝑄| Exponential decay

e−𝜉∙ω𝑛∙t ∙ Q ∙ sin(ωdt) Displacement function

18


The hot-spot method has been intensively applied, for instance in bridge

designs (Akhlaghi, 2009) and ship industry (Hyun Kim, et al., 2009). The

work done by Rother & Rudolph (2010) synthetizes the many types of hot-

spot subtypes that were created, and gives advantages and disadvantages of

each one. For example, the proposal by Haibach is considered to yield better

results when compared to extrapolation methods, and also requires relatively

low effort in order to obtain the structural stress, being therefore suitable for

large structures. However, it has the limitation of not considering potential

weld root failure.

To summarize, the guideline of IIW (Fricke, 2008) and the recommendation

report of IIW (Hobbacher, 2008) give a rather complete background of

fatigue design and approaches, including nominal stress method, structural

hot spot stress method and effective notch stress method used in the industry

nowadays.

A wider introduction to Structural Dynamics for single and multiple-DOF

systems, including continuous models and advanced topics on the theme, is

given in the book “Introduction to Structural Dynamics” (Craig & Kurdila,

2011).

19


3. Method

3.1 Crane setup

The crane is a structure consisting of different components, such as actuators

(hydraulic cylinders), booms and a column. The static model of a crane is

interpreted as a truss consisting of bars, beams and actuators, all connected

at momentum free connections (pin or ball joints). A sketch of the crane can

be seen in Figure 11.

Figure 11: Crane components sketch

The crane has four independently maneuverable actuators, out of which

three are used as loading devices for the structure, as seen in Figure 12. The

crane configuration was chosen to represent the normal working range.

Figure 12: Crane setup

Lifting actuator

Parallel actuator

Turntable

actuator

Crane tip

Crane tip

First boom

20


3.2 Definition of load cases

During this step, the load cases are defined. Three different actuators are

combined, which gives seven different load cases (LCs). P𝑚𝑎𝑥 is the

maximum loading for each actuator according to Table 3, where 0 means

that the actuator is deactivated, and 1 means that the actuator has maximum

loading. For the lifting and parallel actuator, a positive normal force means

that the actuator is pulling. The actuators taken into account are the

hydraulic motor (turntable), lifting actuator and parallel actuator.

Table 3: Load cases

Actuator LC 1 LC 2 LC 3 LC 4 LC 5 LC 6 LC 7

Rotational actuator

P𝑚𝑎𝑥 = 65 kNm 0 0 0 1 1 1 1

Lifting actuator

P𝑚𝑎𝑥 = 177 kN 0 1 1 0 0 1 1

Parallel actuator

P𝑚𝑎𝑥 = 115 kN 1 0 1 0 1 0 1

3.3 Calculation model

There are two types of modeling used in this work. The first one is done

with the software MATLAB®, and includes a Static Analysis (section 3.3.1)

and a Dynamic Analysis (3.3.2). A more detailed scheme using MATLAB®

is presented in Figure 13, where bc are the boundary conditions.

Figure 13: MATLAB® modeling flow chart

Lcct.m

•Input: load case (LC) to be analyzed

•Output: load forces in the crane tip

Static.m

•Input: element coordinates, properties, DOFs, load forces from lcct.m, bc

•Output: K and 𝑀 matrices, static displacement 𝑎𝑠𝑡, element forces 𝑒𝑠𝑠𝑡

Dynamic.m

•Input: 𝐾 and 𝑀 matrices, damping coefficient z, timesteps ℎ•Output: C matrix, Dynamic response 𝑋(𝑇)

Energy.m

•Input: Dynamic response 𝑋(𝑇), coordinates file

•Output: Dynamic element forces (𝑒𝑠𝑑), potential energy 𝐸 for the LC

21


The model has 168 degrees-of-freedom and 26 Euler-Bernoulli beam

elements, as seen in Figure 14, Figure 15 and Figure 16. The beam elements

are implemented by the CALFEM toolbox. Pictures from the MATLAB

model are shown in Figure 17, Figure 18 and Figure 19. The element

coordinates, connections and degree-of-freedom numbering can be seen in

Appendices 3, 4 and 5, respectively.

Figure 14: Element numbering

Figure 15: Element numbering – Detail (1/2)

Detail (2/2)

Detail (1/2)

Turning axis

22


Figure 16: Element numbering – Detail (2/2)

Figure 17: MATLAB® plot of the model (1/3)

23




24


The second type of model is a 3D solid CAD-model provided by Rottne.

The model is created with the software SOLIDWORKS®. In that software

more specific tasks, such as individual assessment of the first boom, can be

performed as described in the Finite Element analysis (section 3.3.3)

In the Finite Element stress analysis the 3D CAD models of the crane

components are used. They are shown in Figure 20 and Figure 21.

Figure 20: First boom FEM model

Figure 21: Element numbering + FEM model

Node 2

Node 6

Node 2 Node 2

Node 2

Node 8 Node 7 Node 7

X

Y

Z

X

Y

Z

Node 2 Node 2

Node 6

Node 2 Node 2

Node 7 Node 7

Node 7 Node 8

25


3.3.1 Static analysis

For the static analysis, a system of equation is used to transform actuator

loads into crane tip loads by the use of static equilibrium. The crane tip

forces for each load case are shown in Table 4.

Table 4: Loads for static analyses

DOF LC1 LC2 LC3 LC4 LC5 LC6 LC7 Unit

73 16,34 1,23 17,57 0,00 16,34 1,23 17,57 [kN]

74 0,19 -16,83 -16,64 0,00 0,19 -16,83 -16,64 [kN]

75 0,00 0,00 0,00 -8,22 -8,22 -8,22 -8,22 [kN]

The stiffness matrix 𝐾 is created. Then the static equation

𝐾 ∙ 𝑢 = 𝑓𝑙 + 𝑓𝑏 ( 54 )

is solved, where u is the displacement vector, fl is the load vector (loads

from Table 4) and fb is the boundary vector. From this step, static

displacements and section forces are obtained as output for each load case.

3.3.2 Dynamic analysis

In this step, the mass matrix 𝑀 and stiffness matrix 𝐾 are calculated for each

element according to the general 3D Euler-Bernoulli beam formula, as can

be seen in Appendix 3. The element properties are obtained from the 3D-

model of the crane are obtained using the software SOLIDWORKS®.

Then, using MATLAB®, the eigenvalue problem is solved for a non-

damped system, thus obtaining mode shapes, 𝜙. The mode shapes are used

to obtain the modal matrices for the state-space representation, which is then

created. Time response is calculated, using each of the seven static

displacements as initial displacement condition.

The damping coefficient 𝜉 to be used in the model is chosen to obtain a fast

decay in order to shorten the calculation time:

𝜉 = 0.10 ( 55 )

The timesteps ℎ, the time limit, 𝑡𝑚𝑎𝑥, and the number of timesteps, 𝑛, are

here:

ℎ = 10−6 [s] ( 56 )

𝑡𝑚𝑎𝑥 = 2 [𝑠] ( 57 )

𝑛 =

𝑡𝑚𝑎𝑥

ℎ= 2000000

( 58 )

26


A reduced number of timesteps 𝑛0 is stored in order to save computational

time and yet to obtain a relevant quantity of timesteps.

𝑛𝑜 =𝑛

100+ 2 = 20002

( 59 )

Then, the potential energy E for each load case is calculated, to compare

against each other and to obtain the worst case (largest energy), where:

𝐸𝑛𝑗,𝑖 is the energy corresponding to one timestep/one load case/one DOF.

𝐹𝑛𝑗,𝑖 is the section force (normal forces and moment forces) and 𝑥𝑛𝑗,𝑖 is

displacement (translation and rotation) at DOF i for the timestep 𝑛𝑗, where

𝑛𝑗 ranges from 0 to 𝑛𝑜.

𝐸𝑛𝑗,𝑖 = ∫ 𝐹𝑛𝑗,𝑖 𝑑𝑥𝑛𝑗,𝑖

𝑥𝑛𝑗,𝑖

0

=𝐹𝑛𝑗,𝑖 ∙ 𝑥𝑛𝑗,𝑖

2

( 60 )

The energy Enj that corresponds to the timestep 𝑛𝑗 is the sum of the energy

for all DOFs for that timestep.

𝐸𝑛𝑗 =∑𝐸𝑛𝑗,𝑖

𝑛𝑜

0

( 61 )

The maximum energy for load case X (LC X) is Emax,LC X.

𝐸𝑚𝑎𝑥,𝐿𝐶 𝑋 = 𝑚𝑎𝑥(𝐸𝑛1 , 𝐸𝑛2 , … , 𝐸𝑛𝑜) ( 62 )

3.3.3 Finite Element analysis

The Finite Element Method is applied to the first boom. Rottne has a

requirement that their simulation tool should be used for this task. Rottne

uses SOLIDWORKS® Simulation (version 2014) for finite element

analysis. The FE-mesh close to the welds needs to be designed so that

principal hot-spot stresses can be evaluated with good accuracy within the

distance of 3 mm from the weld toe or root (Haibach, 1968).

The finite element modeling will consist of a two-step approach, based on

the study done by (Rother & Rudolph, 2010).

First step: global and quick assessment using the hot-spot stress

method;

Second step: local assessment with finer mesh using the notch stress

method.

27


3.3.3.1 Boundary conditions

The boundary conditions (Figure 22) applied at the node 2 represents an

elastic support with extremely high stiffness (Figure 23). Node 6 is rigidly

connected in the actuator’s direction (Figure 24).

Figure 22: Boundary condition - General

Figure 23: Boundary condition applied to node 2

X

Y

Z

X

Y

Z

Boundary

condition

Boundary

condition

Boundary

condition

Boundary

condition

Boundary

condition

28


Figure 24: Boundary condition in the node 6

3.3.3.2 External loads

Loading is defined according to the element forces in the timestep having

the highest potential energy (see section 3.3.2). The external loads are

applied to nodes number 7 and 8 over the plate’s thicknesses in these nodes,

according to the load cases. An example of the loading area is shown in

Figure 25 and Figure 26.

Figure 25: External load in the node 8

X

Y

Z

X

Y

Z

Node 7 Node 8

29


Figure 26: External load in the node 7

3.3.3.3 Element mesh

The element type used in the FE mesh consists of a ten-node tetrahedral,

having four corner nodes and six mid-nodes, see Figure 27. Element edges

can be curved or straight.

Figure 27: Ten-node tetrahedral element

A part of the mesh of the first boom is shown in Figure 28. Global element

size corresponds to the average length of the edges of the tetrahedral

elements.

X

Y

Z

Node 7

30


Figure 28: First boom mesh

3.3.3.4 First step: apply the hot-spot method

This step follows the method done in the previous work (Haibach, 1968) for

determination of hot-spot stresses for a large structure and is used since it

requires relatively low effort in finding structural stresses and therefore is

suitable for analysis of large structures and it is not limited to any specific

kind of weld.

The result from the FE-analysis is used to choose stress exposed welds for

further investigation. The selection of welds to further investigate is

performed together with experienced Rottne designers. This ensure that the

evaluated welds are chosen both on a theoretical base as well as empirical

knowledge.

3.3.3.5 Second step: apply the notch stress method

The FE-model used in the previous stress study (1st step) is updated in the

stress exposed areas according to the IIW recommendations (Hobbacher,

2008) and the journal article (Fricke, 2007), as shown in Figure 29. The

analysis is done for the load case (LC) 5, since this case showed an elevated

level of energy. The cross section of the notch is shown in Figure 30.

The effective notch element size and fictive radius are:

Fictive radius 𝑟𝑟𝑒𝑓 = 1 𝑚𝑚;

Notch element size <𝑟𝑟𝑒𝑓

4= 0.25 𝑚𝑚;

Plate thickness > 5 𝑚𝑚.

X

Y

Z

Element

size =

14 mm

31


Figure 29: FEM - View of the notch

Figure 30: A-A cross-section, normal to the weld direction

The notch stress method requires a very detailed mesh of the critical welds

in order to obtain more accurate information about the stress distribution,

which is shown in Figure 31 and Figure 32.

X

Y

Z

Notch detail

A

A

32


Figure 31: FEM Notch mesh

Figure 32: FEM Notch mesh - Detail

3.4 Prediction of the lifetime of the chosen weld

In this section, the fatigue life is calculated using the stress values at weld

toes and weld roots obtained by the notch stress method. The IIW guideline

(Fricke, 2008) is used for interpreting the stress results into fatigue life. The

fatigue lifetime is calculated according to previous work (Eriksson, et al.,

2002).

X

Y

Z

X

Y

Z

Standard

element size =

14 mm

Notch

element size =

0.25 mm

33


3.5 Damping experiment

An experiment is run with the real crane to verify the damping behavior in

one of the DOFs. The DOF chosen is the y-axis displacement in the crane tip

(DOF 74, end of the element 22). The experiment consists of applying an

initial disturbance in the crane tip, and then, let it move freely in free

vibration. A camera over a tripod records the crane’s movement and a table

is placed in front of it as a reference point to get height measuraments. From

the video recording, the delta displacement is counted using the tape fixed in

the crane tip. The steps needed to execute this experiment are shown in

Figure 33, where Step 1 represents the initial disturbance done in the crane.

Step 2 represents the situation when the crane’s head moves freely (free

vibration). In the Step 3, a displacement vs time plot is done, and an

approximation curve is obtained using Microsoft Excel.

Figure 33: Damping experiment

1

2

X

Y

X

Y

3

Crane head

Tape

34


4. Results

4.1 Static analysis

The results for the first boom static analyses are shown below. The

displacement matrix ast is shown in Table 5 and the local forces esst are

shown in Table 6. Complete results for the displacements are presented in

Appendix 7. Also, the deformed shape for each load case is presented from

Figure 34 to Figure 40.

Table 5: Static displacement 𝑎𝑠𝑡 – First boom

Node Global

DOF LC1 LC2 LC3 LC4 LC5 LC6 LC7 Unit

2

7 2,0E-05 2,9E-04 3,1E-04 0,0E+00 2,0E-05 3,1E-04 2,9E-04 [m]

8 -2,6E-05 -6,3E-04 -6,5E-04 0,0E+00 -2,6E-05 -6,5E-04 -6,3E-04 [m]

9 0,0E+00 0,0E+00 0,0E+00 -2,7E-04 -2,7E-04 -2,7E-04 -2,7E-04 [m]

10 0,0E+00 0,0E+00 0,0E+00 -8,1E-04 -8,1E-04 -8,1E-04 -8,1E-04 [rad]

11 0,0E+00 0,0E+00 0,0E+00 1,5E-03 1,5E-03 1,5E-03 1,5E-03 [rad]

143 -1,5E-03 -2,7E-03 -4,3E-03 0,0E+00 -1,5E-03 -4,3E-03 -2,7E-03 [rad]

6

31 2,3E-03 3,5E-03 5,7E-03 0,0E+00 2,3E-03 5,7E-03 3,5E-03 [m]

32 -2,6E-03 -4,7E-03 -7,3E-03 0,0E+00 -2,6E-03 -7,3E-03 -4,7E-03 [m]

33 0,0E+00 0,0E+00 0,0E+00 -4,5E-03 -4,5E-03 -4,5E-03 -4,5E-03 [m]

34 0,0E+00 0,0E+00 0,0E+00 -2,4E-04 -2,4E-04 -2,4E-04 -2,4E-04 [rad]

35 0,0E+00 0,0E+00 0,0E+00 3,2E-03 3,2E-03 3,2E-03 3,2E-03 [rad]

36 -1,9E-03 -2,3E-03 -4,2E-03 0,0E+00 -1,9E-03 -4,2E-03 -2,3E-03 [rad]

7

37 5,4E-03 4,8E-03 1,0E-02 0,0E+00 5,4E-03 1,0E-02 4,8E-03 [m]

38 -7,9E-03 -8,1E-03 -1,6E-02 0,0E+00 -7,9E-03 -1,6E-02 -8,1E-03 [m]

39 0,0E+00 0,0E+00 0,0E+00 -1,1E-02 -1,1E-02 -1,1E-02 -1,1E-02 [m]

40 0,0E+00 0,0E+00 0,0E+00 5,7E-04 5,7E-04 5,7E-04 5,7E-04 [rad]

41 0,0E+00 0,0E+00 0,0E+00 4,6E-03 4,6E-03 4,6E-03 4,6E-03 [rad]

42 -3,8E-03 -1,1E-03 -4,9E-03 0,0E+00 -3,8E-03 -4,9E-03 -1,1E-03 [rad]

8

43 5,6E-03 4,8E-03 1,0E-02 0,0E+00 5,6E-03 1,0E-02 4,8E-03 [m]

44 -9,0E-03 -8,4E-03 -1,7E-02 0,0E+00 -9,0E-03 -1,7E-02 -8,4E-03 [m]

45 0,0E+00 0,0E+00 0,0E+00 -1,2E-02 -1,2E-02 -1,2E-02 -1,2E-02 [m]

46 0,0E+00 0,0E+00 0,0E+00 6,9E-04 6,9E-04 6,9E-04 6,9E-04 [rad]

47 0,0E+00 0,0E+00 0,0E+00 4,7E-03 4,7E-03 4,7E-03 4,7E-03 [rad]

48 -3,9E-03 -9,7E-04 -4,9E-03 0,0E+00 -3,9E-03 -4,9E-03 -9,7E-04 [rad]

35


Table 6: Local forces and moments – Static analysis - First boom

Element Node

Element

Coor-

dinate

LC1 LC2 LC3 LC4 LC5 LC6 LC7 Unit

8 2

x -13,0 -143,1 -156,1 0,0 -13,0 -143,1 -156,1 [kN]

y 10,2 -8,9 1,2 0,0 10,2 -8,9 1,2 [kN]

z 0,0 0,0 0,0 -8,2 -8,2 -8,2 -8,2 [kN]

mx 0,0 0,0 0,0 30,6 30,6 30,6 30,6 [kNm]

my 0,0 0,0 0,0 50,8 50,8 50,8 50,8 [kNm]

mz 0,0 0,0 0,0 0,0 0,0 0,0 0,0 [kNm]

11 6

x 71,4 -98,4 -27,0 0,0 71,4 -98,4 -27,0 [kN]

y -95,9 123,1 27,1 0,0 -95,9 123,1 27,1 [kN]

z 0,0 0,0 0,0 -8,2 -8,2 -8,2 -8,2 [kN]

mx 0,0 0,0 0,0 30,6 30,6 30,6 30,6 [kNm]

my 0,0 0,0 0,0 15,4 15,4 15,4 15,4 [kNm]

mz 0,0 0,0 0,0 0,0 0,0 0,0 0,0 [kNm]

12 7

x 22,4 0,0 22,4 0,0 22,4 0,0 22,4 [kN]

y 112,8 0,0 112,8 0,0 112,8 0,0 112,8 [kN]

z 0,0 0,0 0,0 0,0 0,0 0,0 0,0 [kN]

mx 0,0 0,0 0,0 0,0 0,0 0,0 0,0 [kNm]

my 0,0 0,0 0,0 0,0 0,0 0,0 0,0 [kNm]

mz 0,0 0,0 0,0 0,0 0,0 0,0 0,0 [kNm]

10 8

x 83,7 -132,0 -48,3 0,0 83,7 -132,0 -48,3 [kN]

y -28,4 44,7 16,4 0,0 -28,4 44,7 16,4 [kN]

z 0,0 0,0 0,0 0,0 0,0 0,0 0,0 [kN]

mx 0,0 0,0 0,0 0,0 0,0 0,0 0,0 [kNm]

my 0,0 0,0 0,0 0,0 0,0 0,0 0,0 [kNm]

mz 0,0 0,0 0,0 0,0 0,0 0,0 0,0 [kNm]

36


4.1.1 Load case 1

Figure 34: Static displacement – LC 1

4.1.2 Load case 2


37


4.1.3 Load case 3


4.1.4 Load case 4


38


4.1.5 Load case 5


4.1.6 Load case 6


39


4.1.7 Load case 7


4.2 Dynamic analysis

The results for the dynamic analyses are shown in this section. The dynamic

response for DOFs 73, 74 and 75 (displacement in the x, y and z-axis

directions) in the crane tip are shown. Also, the energy response, excluding

the time interval 0 < t < 0.08 [s], is shown for each load case.

This time interval is excluded from the energy calculations since these initial

timesteps contain a lot of disturbance information stemming from high

frequencies due to high axial stiffness compared to low mass for short

elements in the model.

The displacements for the timestep when the potential energy is at its

maximum is shown in Table 7. Complete displacement results can be found

in Appendix 9.

The forces and moments at node ends for the first boom are shown in Table

8 together with information about the maximum energy for each load case

and the time when the maximum occurs.

40


Table 7: Displacement for timestep with maximum energy – First boom

Node DOF LC1 LC2 LC3 LC4 LC5 LC6 LC7 Unit

2

7 8,6E-05 -7,6E-05 4,0E-05 0,0E+00 8,5E-05 6,5E-05 3,9E-05 [m]

8 -1,2E-04 1,1E-04 -5,1E-05 0,0E+00 -1,2E-04 -9,7E-05 -5,0E-05 [m]

9 0,0E+00 0,0E+00 0,0E+00 9,9E-04 7,5E-04 -1,6E-03 6,6E-04 [m]

10 0,0E+00 0,0E+00 0,0E+00 7,8E-03 5,7E-03 -1,3E-02 5,0E-03 [rad]

11 0,0E+00 0,0E+00 0,0E+00 -1,2E-03 -1,3E-03 1,1E-03 -1,3E-03 [rad]

143 -2,8E-03 1,8E-03 -2,2E-03 0,0E+00 -2,8E-03 -1,7E-03 -2,2E-03 [rad]

6

31 3,5E-03 -2,2E-03 3,0E-03 0,0E+00 3,5E-03 2,0E-03 3,0E-03 [m]

32 -4,5E-03 2,9E-03 -3,7E-03 0,0E+00 -4,4E-03 -2,7E-03 -3,6E-03 [m]

33 0,0E+00 0,0E+00 0,0E+00 1,4E-02 1,1E-02 -2,1E-02 1,0E-02 [m]

34 0,0E+00 0,0E+00 0,0E+00 7,8E-03 5,5E-03 -1,3E-02 4,8E-03 [rad]

35 0,0E+00 0,0E+00 0,0E+00 -2,5E-03 -2,7E-03 2,0E-03 -2,7E-03 [rad]

36 -2,6E-03 1,5E-03 -2,4E-03 0,0E+00 -2,6E-03 -1,4E-03 -2,4E-03 [rad]

7

37 6,0E-03 -2,8E-03 6,4E-03 0,0E+00 5,9E-03 2,8E-03 6,4E-03 [m]

38 -9,6E-03 4,8E-03 -9,7E-03 0,0E+00 -9,4E-03 -4,7E-03 -9,6E-03 [m]

39 0,0E+00 0,0E+00 0,0E+00 2,6E-02 2,2E-02 -3,7E-02 2,0E-02 [m]

40 0,0E+00 0,0E+00 0,0E+00 7,5E-03 5,1E-03 -1,3E-02 4,3E-03 [rad]

41 0,0E+00 0,0E+00 0,0E+00 -3,4E-03 -3,7E-03 2,5E-03 -3,8E-03 [rad]

42 -2,6E-03 3,9E-04 -4,0E-03 0,0E+00 -2,5E-03 -6,1E-04 -4,0E-03 [rad]

8

43 6,2E-03 -2,8E-03 6,7E-03 0,0E+00 6,1E-03 2,8E-03 6,6E-03 [m]

44 -1,0E-02 4,9E-03 -1,1E-02 0,0E+00 -1,0E-02 -4,9E-03 -1,1E-02 [m]

45 0,0E+00 0,0E+00 0,0E+00 2,8E-02 2,3E-02 -3,9E-02 2,1E-02 [m]

46 0,0E+00 0,0E+00 0,0E+00 7,5E-03 5,0E-03 -1,3E-02 4,2E-03 [rad]

47 0,0E+00 0,0E+00 0,0E+00 -3,5E-03 -3,8E-03 2,6E-03 -3,9E-03 [rad]

48 -2,6E-03 3,1E-04 -4,1E-03 0,0E+00 -2,5E-03 -5,4E-04 -4,0E-03 [rad]

41


Table 8: DOF forces and moments for timestep with maximum energy - First boom

Element Node

Element

Coor-

dinate

LC1 LC2 LC3 LC4 LC5 LC6 LC7 Unit

8 2

x -140,3 127,3 -56,0 0,0 -139,6 -112,0 -54,8 [kN]

y -3,0 8,1 6,6 0,0 -3,2 -6,2 6,7 [kN]

z 0,0 0,0 0,0 10,7 9,9 -12,5 9,5 [kN]

mx 0,0 0,0 0,0 -15,3 -20,0 3,9 -21,2 [kNm]

my 0,0 0,0 0,0 -52,3 -52,1 51,7 -51,4 [kNm]

mz 0,0 0,0 0,0 0,0 0,0 0,0 0,0 [kNm]

11 6

x 13,6 0,0 23,4 0,0 13,1 2,0 23,3 [kN]

y 70,3 -2,9 120,0 0,0 67,9 10,5 119,4 [kN]

z 0,0 0,0 0,0 0,0 0,0 -0,6 0,0 [kN]

mx 0,0 0,0 0,0 0,0 0,0 0,0 0,0 [kNm]

my 0,0 0,0 0,0 0,0 0,0 0,0 0,0 [kNm]

mz 0,0 0,0 0,0 0,0 0,0 0,0 0,0 [kNm]

12 7

x -67,8 114,1 51,3 0,0 -69,3 -92,9 51,9 [kN]

y 22,9 -38,7 -17,4 0,0 23,4 31,5 -17,6 [kN]

z 0,0 0,0 0,0 0,0 0,0 0,0 0,0 [kN]

mx 0,0 0,0 0,0 0,0 0,0 0,0 0,0 [kNm]

my 0,0 0,0 0,0 0,0 0,0 0,0 0,0 [kNm]

mz 0,0 0,0 0,0 0,0 0,0 0,0 0,0 [kNm]

10 8

x -46,8 85,9 46,5 0,0 -48,1 -69,7 46,9 [kN]

y 54,6 -106,8 -63,5 0,0 56,3 85,8 -63,9 [kN]

z 0,0 0,0 0,0 8,2 8,4 -7,7 8,3 [kN]

mx 0,0 0,0 0,0 -15,4 -20,0 4,1 -21,3 [kNm]

my 0,0 0,0 0,0 -9,8 -11,7 5,0 -12,2 [kNm]

mz 0,0 0,0 0,0 0,0 0,0 0,0 0,0 [kNm]

Time_max 0,2556 0,3486 0,2497 0,2789 0,2591 0,5309 0,2527 [s]

Energy_max 927,71 649,68 1004,5 288,19 1186,8 789,82 1245 [Nm²]

42


4.2.1 Load case 1

Figure 41: Dynamic response for the crane tip – LC 1

Figure 42: Energy response in the main boom - LC 1

43


4.2.2 Load case 2



44


4.2.3 Load case 3



45


4.2.4 Load case 4



46


4.2.5 Load case 5



47


4.2.6 Load case 6



48


4.2.7 Load case 7



49


4.3 Finite Element analysis

The results for the Finite Element analyses are shown in this section, both

for the first step (hot-spot method) and second step (notch stress method).

4.3.1 First step: hot-spot stresses

In this section, three results for each load case are shown. The first figure of

each load case corresponds to a general view from the highest principal

stresses using the tool isoclipping from the software. The second one is a

close up on the hot-spots. Finally, the Von Mises stresses are shown.

4.3.1.1 Load case 1

Figure 55: Principal stress – Values above 10 MPa – LC 1

X

Y

Z

50


Figure 56: Principal stress hot-spots – LC 1

Figure 57: Von Mises stress – LC 1

X

Y

Z

X

Y

Z

51


4.3.1.2 Load case 2



X

Y

Z

X

Y

Z

52



4.3.1.3 Load case 3


X

Y

Z

X

Y

Z

53




X

Y

Z

X

Y

Z

54


4.3.1.4 Load case 4



X

Y

Z

X

Y

Z

55



4.3.1.5 Load case 5


X

Y

Z

X

Y

Z

56




X

Y

Z

X

Y

Z

57


4.3.1.6 Load case 6



X

Y

Z

X

Y

Z

58



4.3.1.7 Load case 7


X

Y

Z

X

Y

Z

59




X

Y

Z

X

Y

Z

60


4.3.2 Second step: notch stress method

The results from the Notch Stress Method analyses are presented in Figure

76 and Figure 77. The maximum principal stress obtained from the LC 5 is

shown, together with the stress distribution around the notch.

Figure 76: Stress distribution in the notch (1/2) – LC 5

Figure 77: Stress distribution in the notch (2/2) – LC 5

X

Y

Z

X

Y

Z

Maximum

principal stress

𝜎𝑚𝑎𝑥 = 165 𝑀𝑃𝑎

Analyzed

weld

61


4.4 Fatigue calculations

The results for the fatigue calculations are shown in this section. The stress

range 𝜎𝑟𝑎𝑛𝑔𝑒 is obtained based on the principal stresses from the notch stress

method. 𝑅 is the ratio of the minimum and maximum stresses. 𝑁 is the

number of cycles before failure for a weld class FAT-225 (𝑚 = 3) with a

probability of survival of 97,7% (Fricke, 2007) and 𝑁𝑓 is the number of

cycles for the analyzed weld.

𝜎𝑚𝑎𝑥 = 165 𝑀𝑃𝑎 ( 63 )

𝜎𝑚𝑖𝑛 = −165 𝑀𝑃𝑎 ( 64 )

𝜎𝑟𝑎𝑛𝑔𝑒 = 𝜎𝑚𝑎𝑥 − 𝜎𝑚𝑖𝑛 = 330 𝑀𝑃𝑎 ( 65 )

𝑅 =𝜎𝑚𝑖𝑛

𝜎𝑚𝑎𝑥= −1 ( 66 )

𝑁𝑓 = 𝑁 (𝜎𝐹𝐴𝑇

𝜎𝑟𝑎𝑛𝑔𝑒)𝑚

= 2 ∙ 106 (225

330)3

= 633921 𝑐𝑦𝑐𝑙𝑒𝑠 ( 67 )

4.5 Damping experiment

The displacement response from the crane’s experiment for the DOF 74 is

shown in Figure 78, together with the exponential decay curve and the

adjusted curve, in order to show how the experiment relates to the theory.

The adjusted curve is obtained by trial, using the experimental curve as

reference and correlating it with the theoretical proposal of the equation of

motion for a system in free vibration having viscous damping (Rajasekaran,

2009).

Figure 78: Damping experiment curve

62


The adjusted/approximation curve ua(t) and its derivative with respect to

time are,

𝑢𝑎(𝑡) = −𝑒−0,45𝑡 ∙ 0,055𝑠𝑖𝑛(8,3𝑡 + 1,6) + 0,0022𝑡 − 0,014⏟

𝑂𝑓𝑓𝑠𝑒𝑡 𝑎𝑑𝑗𝑢𝑠𝑡𝑚𝑒𝑛𝑡

( 68 )

𝑑

𝑑𝑡(𝑢𝑎(𝑡)) = (0,02475𝑒

−0,45𝑡 ∙ 𝑠𝑖𝑛(8,3𝑡 + 1,6)

− 0,4565𝑒−0,45𝑡 ∙ 𝑐𝑜𝑠(8,3𝑡 + 1,6) + 0,0022)

( 69 )

Evaluating the derivative for each timestep it is possible to determine when

the slope changes, and then, the period 𝑇𝑛 is obtained,

𝑇𝑛 = 0.38 [𝑠] ( 70 )

The natural circular frequency ωn can be written as

𝜔𝑛 =

2𝜋

𝑇𝑛= 16.53 [𝑟𝑎𝑑/𝑠]

( 71 )

Comparing the adjusted curve 𝑢𝑎(𝑡) with the formula suggested by the book

“Structural Dynamics for Earthquake Engineering” (Rajasekaran, 2009), it

is possible to show that the damping coefficient ξ is

𝑢(𝑡) = 𝑒−𝜉∙𝜔𝑛∙𝑡 ∙ (𝐴 ∙ 𝑠𝑖𝑛 (𝜔𝑛 ∙ √1 − 𝜉2𝑡 + 𝜑)) ( 72 )

𝜉 ∙ 𝜔𝑛 = 0.45 ( 73 )

𝜉 =

0,45

𝜔𝑛= 0.027 = 2.7%

( 74 )

63


5. Analysis

The load case resulting in the highest energy was the LC 7, as expected,

when all the actuators are fully loaded. It is possible to observe a correlation

between potential energy and hot-spot stresses, as shown in Table 9.

Table 9: Potential energy and Hot-spot stress correlation

Variable LC1 LC2 LC3 LC4 LC5 LC6 LC7 Unit

Hotspot_stress 50 40 20 40 65 45 70 [MPa]

Energy_max 927,71 649,68 1004,5 288,19 1186,8 789,82 1245 [Nm²]

Figure 79: Potential energy x stress correlation

From the hot-spot method analysis, it is possible to observe that there are

common hot-spots for more than one load case. This behavior can be seen in

Figure 80, Figure 81 and Figure 82. In Figure 80, LC 5 corresponds to the

hot-spot analyzed in previous sections. This is an indication that these spots

are candidates for possible weld failures.

64


Figure 80: Common hot-spot stresses – Case 1

Figure 81: Common hot-spot stresses- Case 2

Figure 82: Common hot-spot stresses- Case 3

65


6. Discussion

It is possible to conclude, from the presented results and from the analyses,

that both the Euler-Bernoulli beam FE model and the a 3D solid

CAD-model behaves in a compatible way with the real life crane from some

perspectives, such as the deformed shape and stress levels. This is indicating

that the calculations done for the fatigue life length are valid.

Our suggestions for future studies are:

Adjust the damping coefficient in the model according to

experimental data

Use of the damage summation theory (Palmgren-Miner) to quantify

different loadings from the load cases

Perform the analyses for other hot-spots and other crane components

Study the relation between the varying stress direction in the notch

and the fatigue life

Eliminate the axial stiffness from the beam element, in order to

remove the high eigenfrequencies associated with axial vibrations.

66


7. Conclusions

The results from the thesis were obtained according to the standards and the

recent literature about the subject. The aim and purpose of the work were

contemplated.

Progress on the research of the assessment methods for fatigue calculations

is concluded but there is more to be done in further studies.

The methods used in this thesis fulfilled the company's requirements and the

results are expected to contribute in further product development, since the

crane model works fine and the results revealed important crane properties.

Also, the method is general and can be applied to the design procedure for

the company’s wide product range.

67


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69


Appendices

Appendix 1: Beam type element according to CALFEM

Appendix 2: Palmgren-Miner example

Appendix 3: Mass and stiffness matrices

Appendix 4: Model element coordinates

Appendix 5: Model element node-number and connections

Appendix 6: Model element degree-of-freedom

Appendix 7: Model element properties

Appendix 8: Static displacement

Appendix 9: Displacement for dynamic analysis

Appendix 10: MATLAB codes

Appendix 1: page 1/1


APPENDIX 1: Beam type element according to CALFEM

The beam type element used by the CALFEM toolbox is based on the Euler-

Bernoulli beam. A basic representation with signal rule is presented in Figure 83 and

Figure 84.

Figure 83: Beam element orientation (Austrell, et al., 2004)

Figure 84: Beam element section forces orientation (Austrell, et al., 2004)



APPENDIX 2: Palmgren-Miner example

A short example of Palmgren-Miner is given: assume that Figure 3 represents a

material’s structure and that this structure has the following loading cycles: 10

loadings of stress type 1, 500 loadings of type 2, 103 loadings of type 3 and 1500

loadings for type 4. Suppose maximum number of cycles for each type is N1 =102; 𝑁2 = 10

3; N3 = 5000; N4 = 104.

∑

𝑛𝑖𝑁𝑖

𝐼

𝑖=1

=10

102+500

103+103

5000+1500

104= 0,95

( 75 )

Since:

∑𝑛𝑖𝑁𝑖

𝐼

𝑖=1

< 1

( 76 )

The structure has enough strength to withstand to fatigue.



APPENDIX 3: Mass and stiffness matrices

The mass and stiffness matrices for an Euler-Bernoulli 3D-beam element can be seen

in Figure 85 and Figure 86, where 𝐴 is the element’s cross-sectional area, 𝐼𝑦 and 𝐼𝑧

are the area moments of inertia of the cross section, 𝐼𝑝 is the area polar moment of

inertia about the “x” axis, 𝐿 is the element’s length, 𝐸 is the modulus of elasticity, 𝐺𝐽 is the torsional stiffness and ρ is the mass density

.

Figure 85: Mass matrix – Euler-Bernoulli 3D-beam element – Source (Craig & Kurdila, 2011)



Figure 86: Stiffness matrix – Euler-Bernoulli 3D-beam element – Source (Craig & Kurdila, 2011)



APPENDIX 4: Model element coordinates

The crane coordinates and element length is shown in Table 10.

Table 10: Element coordinates table

Element

number

X start

(mm)

X end

(mm)

Y start

(mm)

Y end

(mm)

Z start

(mm)

Z end

(mm)

Length

(mm)

1 0 696 698 698 0 0 696,0

2 696 696 698 923 0 0 225,0

3 62,19 100,41 1377,73 1350,31 0 0 47,0

4 100,41 251,49 1350,31 1241,92 0 0 185,9

5 251,49 696 1241,92 923 0 0 547,1

6 100,41 420,71 1350,31 1796,74 0 0 549,4

7 251,49 477,11 1241,92 1556,4 0 0 387,0

8 696 2395 923 1973,42 0 0 1997,5

9 2395 4130,58 1973,42 3046,46 0 0 2040,5

1 4130,58 4353,43 3046,46 3184,24 0 0 262,0

11 2395 2244,6 1973,42 2216,69 0 0 286,0

12 4130,58 4085,88 3046,46 3118,76 0 0 85,0

13 420,71 3651,35 1796,74 3600,32 0 0 3700,0

14 477,11 2244,6 1556,4 2216,69 0 0 1886,8

15 3651,35 3953,86 3600,32 3583,25 0 0 303,0

16 3953,86 4365,69 3583,25 3560,67 0 0 412,4

17 3953,86 3958,71 3583,25 3669,26 0 0 86,1

18 4085,88 3958,71 3118,76 3669,26 0 0 565,0

19 4365,69 4550,21 3560,67 3796,84 0 0 299,7

20 4353,43 4734,84 3184,24 3643,51 0 0 597,0

21 4550,21 4734,84 3796,84 3643,51 0 0 240,0

22 4734,84 6896,59 3643,51 1848,26 0 0 2810,0

23 6896,59 7909,24 1848,26 1007,29 0 0 1316,3

24 -25 62,19 698 1377,73 0 0 685,3

25 -25 0 698 698 0 0 25,0

26 7909,24 7909,24 1007,29 -1492,71 0 0 2500,0



APPENDIX 5: Model element node-number and

connections

Some elements are connected by pin or ball joint. Pin joints are free to move in one

rotational DOF around “y” axis. Ball joints are free to move around the three

rotational axis. Pin joints are represented by number 1 and ball joints by number 3, as

seen in Table 11.

Table 11: Element node and connections type

Element

number

Node

start

Connection

start

Node

end

Connection

end

1 22 14

2 14 2 1

3 3 3 15

4 15 16

5 16 2 1

6 15 4 3

7 16 5 3

8 2 1 17

9 17 18

1 18 8 1

11 17 6 3

12 18 7 3

13 4 3 9 3

14 5 3 6 3

15 9 3 19

16 19 11 1

17 19 10 3

18 7 3 10 3

19 11 1 20

20 8 1 21

21 20 21

22 21 12

23 12 13 3

24 1 3 3 3

25 1 3 22

26 13 3 23



APPENDIX 6: Model element degree-of-freedom

The crane DOF’s division is shown in Table 12, according to a beam-type element.

The six first values represent the DOF’s of the starting node of the element.

Table 12: Element degree-of-freedom

Element

number

Starting node DOF’s Ending node DOF’s

x y z mx my mz x y z mx my mz

1 127 128 129 130 131 132 79 80 81 82 83 84

2 79 80 81 82 83 84 7 8 9 10 11 12

3 13 14 15 16 17 18 85 86 87 88 89 90

4 85 86 87 88 89 90 91 92 93 94 95 96

5 91 92 93 94 95 96 7 8 9 10 11 142

6 85 86 87 88 89 90 19 20 21 22 23 24

7 91 92 93 94 95 96 25 26 27 28 29 30

8 7 8 9 10 11 143 97 98 99 100 101 102

9 97 98 99 100 101 102 103 104 105 106 107 108

10 103 104 105 106 107 108 43 44 45 46 47 48

11 97 98 99 100 101 102 31 32 33 34 35 36

12 103 104 105 106 107 108 37 38 39 40 41 42

13 19 20 21 147 148 149 49 50 51 52 53 54

14 25 26 27 150 151 152 31 32 33 153 154 155

15 49 50 51 160 161 162 109 110 111 112 113 114

16 109 110 111 112 113 114 61 62 63 64 65 66

17 109 110 111 112 113 114 55 56 57 58 59 60

18 37 38 39 156 157 158 55 56 57 163 164 165

19 61 62 63 64 65 166 115 116 117 118 119 120

20 43 44 45 46 47 159 121 122 123 124 125 126

21 115 116 117 118 119 120 121 122 123 124 125 126

22 121 122 123 124 125 126 67 68 69 70 71 72

23 67 68 69 70 71 72 73 74 75 76 77 78

24 1 2 3 4 5 6 13 14 15 144 145 146

25 1 2 3 139 140 141 127 128 129 130 131 132

26 73 74 75 167 77 168 133 134 135 136 137 138



APPENDIX 7: Model element properties

The element properties were taken from the FEM-model. Cross-sectional area (A),

area moment of inertia (Iy, Iz) and torsional constant (Kv) were taken based on the

main cross-section of the element. Polar mass moment of inertia (J) and mass per

unit length (m) were taken from the complete FEM-model. Shear modulus of

elasticity G = 79000 MPa and modulus of elasticity E = 210000 MPa for alloy

steel. Actuators (elements 14 and 24) have a specific modulus of elasticity (E14 =49950 MPa and E24 = 24920 MPa).

Table 13: Element properties

Element

number E G

A

[m2]

Iy

[m4]

Iz

[m4]

Kv

[m4]

m

[kg/m]

J

[kg/m2]

1 E G 5,0E-02 4,6E-04 1,6E-04 6,1E-04 2,7E+02 3,3E+00

2 E G 3,0E-02 2,3E-06 1,6E-04 1,6E-04 2,3E+02 2,7E-01

3 E G 1,7E-02 5,7E-07 2,7E-04 2,7E-04 2,1E+02 6,6E-02

4 E G 1,7E-02 5,7E-07 2,7E-04 2,7E-04 2,1E+02 2,6E-01

5 E G 1,1E-02 3,6E-07 6,8E-05 6,9E-05 1,1E+02 3,4E-01

6 E G 1,7E-02 5,7E-07 2,7E-04 2,7E-04 1,0E+00 1,0E-03

7 E G 1,7E-02 5,7E-07 2,7E-04 2,7E-04 1,0E+00 1,0E-03

8 E G 1,3E-02 3,5E-04 2,1E-04 5,6E-04 2,1E+02 3,4E+00

9 E G 1,3E-02 3,5E-04 2,1E-04 5,6E-04 2,1E+02 2,0E+00

10 E G 1,3E-02 3,5E-04 2,1E-04 5,6E-04 1,8E+02 2,9E-01

11 E G 1,3E-02 3,5E-04 2,1E-04 5,6E-04 1,0E+00 1,0E-03

12 E G 1,3E-02 3,5E-04 2,1E-04 5,6E-04 1,0E+00 1,0E-03

13 E G 2,5E-03 4,9E-06 3,7E-06 8,6E-06 3,2E+01 5,3E-01

14 E14 G 3,8E-03 1,2E-06 1,2E-06 2,4E-06 1,1E+02 1,1E+00

15 E G 1,0E-02 5,2E-07 3,3E-05 3,4E-05 1,2E+02 1,6E-01

16 E G 1,0E-02 5,2E-07 3,3E-05 3,4E-05 1,1E+02 1,1E-01

17 E G 1,0E-02 5,2E-07 3,3E-05 3,4E-05 1,0E+00 1,1E-01

18 E G 5,0E-03 2,6E-07 4,2E-06 4,4E-06 3,9E+01 3,0E-02

19 E G 5,8E-03 5,0E-05 6,5E-05 1,1E-04 1,0E+00 2,7E-01

20 E G 5,8E-03 5,0E-05 6,5E-05 1,1E-04 1,0E+00 5,4E-01

21 E G 9,1E-03 9,1E-05 3,5E-04 4,4E-04 4,1E+02 9,0E-01

22 E G 5,8E-03 5,0E-05 6,5E-05 1,1E-04 1,7E+02 3,4E+00

23 E G 4,9E-03 3,0E-05 4,0E-05 7,0E-05 1,1E+02 4,6E-01

24 E24 G 5,0E-03 2,0E-06 2,0E-06 4,0E-06 2,6E+02 1,2E+00

25 E G 1,5E-02 1,1E-06 7,8E-05 7,9E-05 1,2E+02 1,4E-01

26 E G 8,3E-02 5,7E-04 5,7E-04 1,1E-03 6,4E+02 2,3E+01



A specific calculation was done in order to obtain the modulus of elasticity for the

actuators, based on “Formelsamling i Hydraulik och pneumatic” (Linköping

universitetet, 1995), where β is the average value for the compressibility of the

hydraulic oil (bulk modulus of elasticity). L14, L24 are the element lengths in the

model and Lc14, Lc24 are the oil lengths in the actuators.

𝛽 = 1,6𝑒10 ( 77 )

𝐿14 = 1,767 𝑚 ( 78 )

𝐿24 = 0,6853 𝑚 ( 79 )

𝐿𝑐14 = 0,566 𝑚 ( 80 )

𝐿𝑐24 = 0,44 𝑚 ( 81 )

The modulus of elasticity for each actuator is shown below.

𝐸14 =

𝛽 ∙ 𝐿14

𝐿𝑐14= 49950 𝑀𝑃𝑎

( 82 )

𝐸24 =

𝛽 ∙ 𝐿24

𝐿𝑐24= 24920 𝑀𝑃𝑎

( 83 )



APPENDIX 8: Static displacement

The complete results for static displacement is shown in Table 14.

Table 14: Static displacement

DOF LC1 LC2 LC3 LC4 LC5 LC6 LC7

1 -3,6E-12 1,8E-07 1,8E-07 0,0E+00 -3,6E-12 1,8E-07 1,8E-07

2 -1,1E-12 5,6E-08 5,6E-08 0,0E+00 -1,1E-12 5,6E-08 5,6E-08

3 0,0E+00 0,0E+00 0,0E+00 1,1E-20 1,1E-20 1,1E-20 1,1E-20

4 0,0E+00 0,0E+00 0,0E+00 7,4E-02 7,4E-02 7,4E-02 7,4E-02

5 0,0E+00 0,0E+00 0,0E+00 5,8E-01 5,8E-01 5,8E-01 5,8E-01

6 -6,0E-05 -1,8E-03 -1,9E-03 0,0E+00 -6,0E-05 -1,9E-03 -1,8E-03

7 2,0E-05 2,9E-04 3,1E-04 0,0E+00 2,0E-05 3,1E-04 2,9E-04

8 -2,6E-05 -6,3E-04 -6,5E-04 0,0E+00 -2,6E-05 -6,5E-04 -6,3E-04

9 0,0E+00 0,0E+00 0,0E+00 -2,7E-04 -2,7E-04 -2,7E-04 -2,7E-04

10 0,0E+00 0,0E+00 0,0E+00 -8,1E-04 -8,1E-04 -8,1E-04 -8,1E-04

11 0,0E+00 0,0E+00 0,0E+00 1,5E-03 1,5E-03 1,5E-03 1,5E-03

12 -8,9E-05 -1,3E-03 -1,4E-03 0,0E+00 -8,9E-05 -1,4E-03 -1,3E-03

13 4,1E-05 1,4E-03 1,4E-03 0,0E+00 4,1E-05 1,4E-03 1,4E-03

14 -5,3E-06 8,0E-04 8,0E-04 0,0E+00 -5,3E-06 8,0E-04 8,0E-04

15 0,0E+00 0,0E+00 0,0E+00 3,2E-04 3,2E-04 3,2E-04 3,2E-04

16 0,0E+00 0,0E+00 0,0E+00 -8,1E-04 -8,1E-04 -8,1E-04 -8,1E-04

17 0,0E+00 0,0E+00 0,0E+00 1,5E-03 1,5E-03 1,5E-03 1,5E-03

18 1,2E-04 -2,5E-03 -2,4E-03 0,0E+00 1,2E-04 -2,4E-03 -2,5E-03

19 -4,7E-05 2,5E-03 2,4E-03 0,0E+00 -4,7E-05 2,4E-03 2,5E-03

20 5,0E-05 -1,2E-04 -6,6E-05 0,0E+00 5,0E-05 -6,6E-05 -1,2E-04

21 0,0E+00 0,0E+00 0,0E+00 -5,6E-04 -5,6E-04 -5,6E-04 -5,6E-04

22 0,0E+00 0,0E+00 0,0E+00 -8,1E-04 -8,1E-04 -8,1E-04 -8,1E-04

23 0,0E+00 0,0E+00 0,0E+00 1,5E-03 1,5E-03 1,5E-03 1,5E-03

24 2,2E-04 -2,7E-03 -2,4E-03 0,0E+00 2,2E-04 -2,4E-03 -2,7E-03

25 6,9E-05 1,8E-03 1,9E-03 0,0E+00 6,9E-05 1,9E-03 1,8E-03

26 1,3E-05 -2,1E-04 -2,0E-04 0,0E+00 1,3E-05 -2,0E-04 -2,1E-04

27 0,0E+00 0,0E+00 0,0E+00 -4,5E-04 -4,5E-04 -4,5E-04 -4,5E-04

28 0,0E+00 0,0E+00 0,0E+00 -8,1E-04 -8,1E-04 -8,1E-04 -8,1E-04

29 0,0E+00 0,0E+00 0,0E+00 1,5E-03 1,5E-03 1,5E-03 1,5E-03

30 -5,4E-05 -2,4E-03 -2,5E-03 0,0E+00 -5,4E-05 -2,5E-03 -2,4E-03

31 2,3E-03 3,5E-03 5,7E-03 0,0E+00 2,3E-03 5,7E-03 3,5E-03

32 -2,6E-03 -4,7E-03 -7,3E-03 0,0E+00 -2,6E-03 -7,3E-03 -4,7E-03

33 0,0E+00 0,0E+00 0,0E+00 -4,5E-03 -4,5E-03 -4,5E-03 -4,5E-03

34 0,0E+00 0,0E+00 0,0E+00 -2,4E-04 -2,4E-04 -2,4E-04 -2,4E-04

35 0,0E+00 0,0E+00 0,0E+00 3,2E-03 3,2E-03 3,2E-03 3,2E-03

36 -1,9E-03 -2,3E-03 -4,2E-03 0,0E+00 -1,9E-03 -4,2E-03 -2,3E-03

37 5,4E-03 4,8E-03 1,0E-02 0,0E+00 5,4E-03 1,0E-02 4,8E-03

38 -7,9E-03 -8,1E-03 -1,6E-02 0,0E+00 -7,9E-03 -1,6E-02 -8,1E-03



39 0,0E+00 0,0E+00 0,0E+00 -1,1E-02 -1,1E-02 -1,1E-02 -1,1E-02

40 0,0E+00 0,0E+00 0,0E+00 5,7E-04 5,7E-04 5,7E-04 5,7E-04

41 0,0E+00 0,0E+00 0,0E+00 4,6E-03 4,6E-03 4,6E-03 4,6E-03

42 -3,8E-03 -1,1E-03 -4,9E-03 0,0E+00 -3,8E-03 -4,9E-03 -1,1E-03

43 5,6E-03 4,8E-03 1,0E-02 0,0E+00 5,6E-03 1,0E-02 4,8E-03

44 -9,0E-03 -8,4E-03 -1,7E-02 0,0E+00 -9,0E-03 -1,7E-02 -8,4E-03

45 0,0E+00 0,0E+00 0,0E+00 -1,2E-02 -1,2E-02 -1,2E-02 -1,2E-02

46 0,0E+00 0,0E+00 0,0E+00 6,9E-04 6,9E-04 6,9E-04 6,9E-04

47 0,0E+00 0,0E+00 0,0E+00 4,7E-03 4,7E-03 4,7E-03 4,7E-03

48 -3,9E-03 -9,7E-04 -4,9E-03 0,0E+00 -3,9E-03 -4,9E-03 -9,7E-04

49 3,4E-03 7,7E-03 1,1E-02 0,0E+00 3,4E-03 1,1E-02 7,7E-03

50 -7,4E-03 -7,5E-03 -1,5E-02 0,0E+00 -7,4E-03 -1,5E-02 -7,5E-03

51 0,0E+00 0,0E+00 0,0E+00 -7,1E-03 -7,1E-03 -7,1E-03 -7,1E-03

52 0,0E+00 0,0E+00 0,0E+00 -2,1E-02 -2,1E-02 -2,1E-02 -2,1E-02

53 0,0E+00 0,0E+00 0,0E+00 -9,8E-03 -9,8E-03 -9,8E-03 -9,8E-03

54 -2,2E-03 -2,4E-03 -4,6E-03 0,0E+00 -2,2E-03 -4,6E-03 -2,4E-03

55 3,6E-03 7,7E-03 1,1E-02 0,0E+00 3,6E-03 1,1E-02 7,7E-03

56 -8,3E-03 -7,5E-03 -1,6E-02 0,0E+00 -8,3E-03 -1,6E-02 -7,5E-03

57 0,0E+00 0,0E+00 0,0E+00 -8,9E-03 -8,9E-03 -8,9E-03 -8,9E-03

58 0,0E+00 0,0E+00 0,0E+00 2,0E-03 2,0E-03 2,0E-03 2,0E-03

59 0,0E+00 0,0E+00 0,0E+00 6,5E-03 6,5E-03 6,5E-03 6,5E-03

60 -2,8E-03 -4,8E-04 -3,3E-03 0,0E+00 -2,8E-03 -3,3E-03 -4,8E-04

61 3,3E-03 7,7E-03 1,1E-02 0,0E+00 3,3E-03 1,1E-02 7,7E-03

62 -9,3E-03 -7,9E-03 -1,7E-02 0,0E+00 -9,3E-03 -1,7E-02 -7,9E-03

63 0,0E+00 0,0E+00 0,0E+00 -1,2E-02 -1,2E-02 -1,2E-02 -1,2E-02

64 0,0E+00 0,0E+00 0,0E+00 2,0E-03 2,0E-03 2,0E-03 2,0E-03

65 0,0E+00 0,0E+00 0,0E+00 6,5E-03 6,5E-03 6,5E-03 6,5E-03

66 -2,3E-03 -1,3E-03 -3,6E-03 0,0E+00 -2,3E-03 -3,6E-03 -1,3E-03

67 2,3E-02 -1,6E-02 7,0E-03 0,0E+00 2,3E-02 7,0E-03 -1,6E-02

68 1,8E-02 -4,1E-02 -2,3E-02 0,0E+00 1,8E-02 -2,3E-02 -4,1E-02

69 0,0E+00 0,0E+00 0,0E+00 -4,2E-02 -4,2E-02 -4,2E-02 -4,2E-02

70 0,0E+00 0,0E+00 0,0E+00 5,9E-03 5,9E-03 5,9E-03 5,9E-03

71 0,0E+00 0,0E+00 0,0E+00 1,1E-02 1,1E-02 1,1E-02 1,1E-02

72 1,4E-02 -1,6E-02 -2,7E-03 0,0E+00 1,4E-02 -2,7E-03 -1,6E-02

73 3,5E-02 -3,0E-02 4,7E-03 0,0E+00 3,5E-02 4,7E-03 -3,0E-02

74 3,2E-02 -5,8E-02 -2,6E-02 0,0E+00 3,2E-02 -2,6E-02 -5,8E-02

75 0,0E+00 0,0E+00 0,0E+00 -5,9E-02 -5,9E-02 -5,9E-02 -5,9E-02

76 0,0E+00 0,0E+00 0,0E+00 6,6E-03 6,6E-03 6,6E-03 6,6E-03

77 0,0E+00 0,0E+00 0,0E+00 1,2E-02 1,2E-02 1,2E-02 1,2E-02

78 1,5E-02 -1,8E-02 -2,9E-03 0,0E+00 1,5E-02 -2,9E-03 -1,8E-02

79 1,1E-06 -1,4E-06 -3,3E-07 0,0E+00 1,1E-06 -3,3E-07 -1,4E-06

80 -2,6E-05 -6,2E-04 -6,5E-04 0,0E+00 -2,6E-05 -6,5E-04 -6,2E-04

81 0,0E+00 0,0E+00 0,0E+00 -1,6E-04 -1,6E-04 -1,6E-04 -1,6E-04

82 0,0E+00 0,0E+00 0,0E+00 -3,6E-05 -3,6E-05 -3,6E-05 -3,6E-05

83 0,0E+00 0,0E+00 0,0E+00 4,5E-04 4,5E-04 4,5E-04 4,5E-04



84 -7,6E-05 -1,3E-03 -1,4E-03 0,0E+00 -7,6E-05 -1,4E-03 -1,3E-03

85 4,4E-05 1,3E-03 1,3E-03 0,0E+00 4,4E-05 1,3E-03 1,3E-03

86 -5,5E-07 7,1E-04 7,1E-04 0,0E+00 -5,5E-07 7,1E-04 7,1E-04

87 0,0E+00 0,0E+00 0,0E+00 2,9E-04 2,9E-04 2,9E-04 2,9E-04

88 0,0E+00 0,0E+00 0,0E+00 -8,1E-04 -8,1E-04 -8,1E-04 -8,1E-04

89 0,0E+00 0,0E+00 0,0E+00 1,5E-03 1,5E-03 1,5E-03 1,5E-03

90 1,2E-04 -2,5E-03 -2,4E-03 0,0E+00 1,2E-04 -2,4E-03 -2,5E-03

91 5,5E-05 1,0E-03 1,1E-03 0,0E+00 5,5E-05 1,1E-03 1,0E-03

92 1,1E-05 3,4E-04 3,5E-04 0,0E+00 1,1E-05 3,5E-04 3,4E-04

93 0,0E+00 0,0E+00 0,0E+00 1,5E-04 1,5E-04 1,5E-04 1,5E-04

94 0,0E+00 0,0E+00 0,0E+00 -8,1E-04 -8,1E-04 -8,1E-04 -8,1E-04

95 0,0E+00 0,0E+00 0,0E+00 1,5E-03 1,5E-03 1,5E-03 1,5E-03

96 3,2E-05 -2,4E-03 -2,4E-03 0,0E+00 3,2E-05 -2,4E-03 -2,4E-03

97 1,8E-03 2,9E-03 4,7E-03 0,0E+00 1,8E-03 4,7E-03 2,9E-03

98 -2,9E-03 -5,1E-03 -8,0E-03 0,0E+00 -2,9E-03 -8,0E-03 -5,1E-03

99 0,0E+00 0,0E+00 0,0E+00 -4,9E-03 -4,9E-03 -4,9E-03 -4,9E-03

100 0,0E+00 0,0E+00 0,0E+00 -2,4E-04 -2,4E-04 -2,4E-04 -2,4E-04

101 0,0E+00 0,0E+00 0,0E+00 3,2E-03 3,2E-03 3,2E-03 3,2E-03

102 -2,0E-03 -2,3E-03 -4,3E-03 0,0E+00 -2,0E-03 -4,3E-03 -2,3E-03

103 5,1E-03 4,7E-03 9,8E-03 0,0E+00 5,1E-03 9,8E-03 4,7E-03

104 -8,1E-03 -8,2E-03 -1,6E-02 0,0E+00 -8,1E-03 -1,6E-02 -8,2E-03

105 0,0E+00 0,0E+00 0,0E+00 -1,2E-02 -1,2E-02 -1,2E-02 -1,2E-02

106 0,0E+00 0,0E+00 0,0E+00 5,7E-04 5,7E-04 5,7E-04 5,7E-04

107 0,0E+00 0,0E+00 0,0E+00 4,6E-03 4,6E-03 4,6E-03 4,6E-03

108 -3,8E-03 -1,1E-03 -4,9E-03 0,0E+00 -3,8E-03 -4,9E-03 -1,1E-03

109 3,3E-03 7,7E-03 1,1E-02 0,0E+00 3,3E-03 1,1E-02 7,7E-03

110 -8,3E-03 -7,5E-03 -1,6E-02 0,0E+00 -8,3E-03 -1,6E-02 -7,5E-03

111 0,0E+00 0,0E+00 0,0E+00 -9,1E-03 -9,1E-03 -9,1E-03 -9,1E-03

112 0,0E+00 0,0E+00 0,0E+00 2,0E-03 2,0E-03 2,0E-03 2,0E-03

113 0,0E+00 0,0E+00 0,0E+00 6,5E-03 6,5E-03 6,5E-03 6,5E-03

114 -2,8E-03 -5,0E-04 -3,3E-03 0,0E+00 -2,8E-03 -3,3E-03 -5,0E-04

115 1,3E-03 1,0E-02 1,1E-02 0,0E+00 1,3E-03 1,1E-02 1,0E-02

116 -7,8E-03 -9,8E-03 -1,8E-02 0,0E+00 -7,8E-03 -1,8E-02 -9,8E-03

117 0,0E+00 0,0E+00 0,0E+00 -1,3E-02 -1,3E-02 -1,3E-02 -1,3E-02

118 0,0E+00 0,0E+00 0,0E+00 2,0E-03 2,0E-03 2,0E-03 2,0E-03

119 0,0E+00 0,0E+00 0,0E+00 6,5E-03 6,5E-03 6,5E-03 6,5E-03

120 7,9E-03 -9,7E-03 -1,9E-03 0,0E+00 7,9E-03 -1,9E-03 -9,7E-03

121 2,5E-03 8,6E-03 1,1E-02 0,0E+00 2,5E-03 1,1E-02 8,6E-03

122 -6,4E-03 -1,2E-02 -1,8E-02 0,0E+00 -6,4E-03 -1,8E-02 -1,2E-02

123 0,0E+00 0,0E+00 0,0E+00 -1,4E-02 -1,4E-02 -1,4E-02 -1,4E-02

124 0,0E+00 0,0E+00 0,0E+00 2,0E-03 2,0E-03 2,0E-03 2,0E-03

125 0,0E+00 0,0E+00 0,0E+00 6,5E-03 6,5E-03 6,5E-03 6,5E-03

126 7,8E-03 -9,6E-03 -1,8E-03 0,0E+00 7,8E-03 -1,8E-03 -9,6E-03

127 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00

128 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00

129 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00



130 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00

131 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00

132 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00

133 -9,3E-02 3,3E-02 1,5E-02 0,0E+00 -1,8E-02 1,5E-02 3,3E-02

134 3,2E-02 -5,8E-02 -2,6E-02 0,0E+00 3,2E-02 -2,6E-02 -5,8E-02

135 0,0E+00 0,0E+00 0,0E+00 -4,7E-02 -4,7E-02 -4,7E-02 -4,7E-02

136 0,0E+00 0,0E+00 0,0E+00 -4,7E-03 -4,7E-03 -4,7E-03 -4,7E-03

137 0,0E+00 0,0E+00 0,0E+00 1,2E-02 1,2E-02 1,2E-02 1,2E-02

138 -5,1E-02 2,5E-02 4,2E-03 0,0E+00 -2,1E-02 4,2E-03 2,5E-02

139 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00 0,0E+00

140 0,0E+00 0,0E+00 0,0E+00 4,8E-19 4,8E-19 4,8E-19 4,8E-19

141 6,8E-11 -3,3E-06 -3,3E-06 0,0E+00 6,8E-11 -3,3E-06 -3,3E-06

142 -1,5E-04 -2,1E-03 -2,2E-03 0,0E+00 -1,5E-04 -2,2E-03 -2,1E-03

143 -1,5E-03 -2,7E-03 -4,3E-03 0,0E+00 -1,5E-03 -4,3E-03 -2,7E-03

144 0,0E+00 0,0E+00 0,0E+00 7,4E-02 7,4E-02 7,4E-02 7,4E-02

145 0,0E+00 0,0E+00 0,0E+00 5,8E-01 5,8E-01 5,8E-01 5,8E-01

146 -6,0E-05 -1,8E-03 -1,9E-03 0,0E+00 -6,0E-05 -1,9E-03 -1,8E-03

147 0,0E+00 0,0E+00 0,0E+00 -2,1E-02 -2,1E-02 -2,1E-02 -2,1E-02

148 0,0E+00 0,0E+00 0,0E+00 -9,8E-03 -9,8E-03 -9,8E-03 -9,8E-03

149 -2,2E-03 -2,4E-03 -4,6E-03 0,0E+00 -2,2E-03 -4,6E-03 -2,4E-03

150 0,0E+00 0,0E+00 0,0E+00 4,4E-02 4,4E-02 4,4E-02 4,4E-02

151 0,0E+00 0,0E+00 0,0E+00 1,9E-02 1,9E-02 1,9E-02 1,9E-02

152 -1,7E-03 -2,5E-03 -4,3E-03 0,0E+00 -1,7E-03 -4,3E-03 -2,5E-03

153 0,0E+00 0,0E+00 0,0E+00 4,4E-02 4,4E-02 4,4E-02 4,4E-02

154 0,0E+00 0,0E+00 0,0E+00 1,9E-02 1,9E-02 1,9E-02 1,9E-02

155 -1,7E-03 -2,5E-03 -4,3E-03 0,0E+00 -1,7E-03 -4,3E-03 -2,5E-03

156 0,0E+00 0,0E+00 0,0E+00 4,3E-03 4,3E-03 4,3E-03 4,3E-03

157 0,0E+00 0,0E+00 0,0E+00 -9,0E-05 -9,0E-05 -9,0E-05 -9,0E-05

158 3,2E-03 -5,3E-03 -2,1E-03 0,0E+00 3,2E-03 -2,1E-03 -5,3E-03

159 6,3E-03 -7,6E-03 -1,4E-03 0,0E+00 6,3E-03 -1,4E-03 -7,6E-03

160 0,0E+00 0,0E+00 0,0E+00 2,0E-03 2,0E-03 2,0E-03 2,0E-03

161 0,0E+00 0,0E+00 0,0E+00 6,5E-03 6,5E-03 6,5E-03 6,5E-03

162 -3,1E-03 -2,8E-05 -3,1E-03 0,0E+00 -3,1E-03 -3,1E-03 -2,8E-05

163 0,0E+00 0,0E+00 0,0E+00 4,3E-03 4,3E-03 4,3E-03 4,3E-03

164 0,0E+00 0,0E+00 0,0E+00 -9,0E-05 -9,0E-05 -9,0E-05 -9,0E-05

165 3,2E-03 -5,3E-03 -2,1E-03 0,0E+00 3,2E-03 -2,1E-03 -5,3E-03

166 8,2E-03 -1,0E-02 -2,1E-03 0,0E+00 8,2E-03 -2,1E-03 -1,0E-02

167 0,0E+00 0,0E+00 0,0E+00 -4,7E-03 -4,7E-03 -4,7E-03 -4,7E-03

168 -5,1E-02 2,5E-02 4,2E-03 0,0E+00 -2,1E-02 4,2E-03 2,5E-02



APPENDIX 9: Displacement for dynamic analysis

The complete results for displacement when the potential energy is maximum in the

dynamic analysis is shown in Table 15.

Table 15: Displacement for dynamic analysis when energy is maximum


1 1,6E-07 -1,6E-07 4,9E-08 0,0E+00 1,6E-07 1,4E-07 4,8E-08

2 5,5E-08 -5,3E-08 1,7E-08 0,0E+00 5,5E-08 4,6E-08 1,6E-08

3 0,0E+00 0,0E+00 0,0E+00 3,8E-08 3,8E-08 -3,5E-08 3,8E-08

4 0,0E+00 0,0E+00 0,0E+00 8,0E-02 7,8E-02 6,1E-02 7,7E-02

5 0,0E+00 0,0E+00 0,0E+00 5,7E-01 5,7E-01 5,8E-01 5,7E-01

6 -9,7E-04 9,9E-04 -3,6E-04 0,0E+00 -9,6E-04 -7,7E-04 -3,5E-04

7 8,6E-05 -7,6E-05 4,0E-05 0,0E+00 8,5E-05 6,5E-05 3,9E-05

8 -1,2E-04 1,1E-04 -5,1E-05 0,0E+00 -1,2E-04 -9,7E-05 -5,0E-05

9 0,0E+00 0,0E+00 0,0E+00 9,9E-04 7,5E-04 -1,6E-03 6,6E-04

10 0,0E+00 0,0E+00 0,0E+00 7,8E-03 5,7E-03 -1,3E-02 5,0E-03

11 0,0E+00 0,0E+00 0,0E+00 -1,2E-03 -1,3E-03 1,1E-03 -1,3E-03

12 -3,8E-04 3,4E-04 -1,8E-04 0,0E+00 -3,8E-04 -2,9E-04 -1,8E-04

13 7,8E-04 -7,5E-04 2,7E-04 0,0E+00 7,8E-04 6,5E-04 2,6E-04

14 7,8E-04 -7,6E-04 2,3E-04 0,0E+00 7,8E-04 6,6E-04 2,3E-04

15 0,0E+00 0,0E+00 0,0E+00 4,4E-03 2,8E-03 -8,6E-03 2,3E-03

16 0,0E+00 0,0E+00 0,0E+00 8,5E-03 6,0E-03 -1,5E-02 5,1E-03

17 0,0E+00 0,0E+00 0,0E+00 -2,6E-04 -9,1E-04 -1,7E-03 -1,1E-03

18 -1,6E-03 1,6E-03 -3,6E-04 0,0E+00 -1,6E-03 -1,4E-03 -3,4E-04

19 1,5E-03 -1,5E-03 3,9E-04 0,0E+00 1,5E-03 1,3E-03 3,8E-04

20 2,1E-04 -1,7E-04 1,1E-04 0,0E+00 2,1E-04 1,5E-04 1,1E-04

21 0,0E+00 0,0E+00 0,0E+00 8,2E-03 5,7E-03 -1,4E-02 4,8E-03

22 0,0E+00 0,0E+00 0,0E+00 8,7E-03 6,1E-03 -1,5E-02 5,2E-03

23 0,0E+00 0,0E+00 0,0E+00 -3,8E-04 -9,7E-04 -1,4E-03 -1,1E-03

24 -1,6E-03 1,7E-03 -3,0E-04 0,0E+00 -1,6E-03 -1,5E-03 -2,8E-04

25 1,1E-03 -1,0E-03 3,8E-04 0,0E+00 1,1E-03 8,8E-04 3,7E-04

26 1,4E-04 -1,2E-04 5,9E-05 0,0E+00 1,3E-04 1,1E-04 5,8E-05

27 0,0E+00 0,0E+00 0,0E+00 6,1E-03 4,3E-03 -1,1E-02 3,6E-03

28 0,0E+00 0,0E+00 0,0E+00 8,6E-03 6,0E-03 -1,5E-02 5,2E-03

29 0,0E+00 0,0E+00 0,0E+00 -3,5E-04 -9,4E-04 -1,4E-03 -1,1E-03

30 -1,6E-03 1,5E-03 -5,3E-04 0,0E+00 -1,6E-03 -1,3E-03 -5,1E-04

31 3,5E-03 -2,2E-03 3,0E-03 0,0E+00 3,5E-03 2,0E-03 3,0E-03

32 -4,5E-03 2,9E-03 -3,7E-03 0,0E+00 -4,4E-03 -2,7E-03 -3,6E-03

33 0,0E+00 0,0E+00 0,0E+00 1,4E-02 1,1E-02 -2,1E-02 1,0E-02

34 0,0E+00 0,0E+00 0,0E+00 7,8E-03 5,5E-03 -1,3E-02 4,8E-03

35 0,0E+00 0,0E+00 0,0E+00 -2,5E-03 -2,7E-03 2,0E-03 -2,7E-03

36 -2,6E-03 1,5E-03 -2,4E-03 0,0E+00 -2,6E-03 -1,4E-03 -2,4E-03

37 6,0E-03 -2,8E-03 6,4E-03 0,0E+00 5,9E-03 2,8E-03 6,4E-03

38 -9,6E-03 4,8E-03 -9,7E-03 0,0E+00 -9,4E-03 -4,7E-03 -9,6E-03

39 0,0E+00 0,0E+00 0,0E+00 2,6E-02 2,2E-02 -3,7E-02 2,0E-02




40 0,0E+00 0,0E+00 0,0E+00 7,5E-03 5,1E-03 -1,3E-02 4,3E-03

41 0,0E+00 0,0E+00 0,0E+00 -3,4E-03 -3,7E-03 2,5E-03 -3,8E-03

42 -2,6E-03 3,9E-04 -4,0E-03 0,0E+00 -2,5E-03 -6,1E-04 -4,0E-03

43 6,2E-03 -2,8E-03 6,7E-03 0,0E+00 6,1E-03 2,8E-03 6,6E-03

44 -1,0E-02 4,9E-03 -1,1E-02 0,0E+00 -1,0E-02 -4,9E-03 -1,1E-02

45 0,0E+00 0,0E+00 0,0E+00 2,8E-02 2,3E-02 -3,9E-02 2,1E-02

46 0,0E+00 0,0E+00 0,0E+00 7,5E-03 5,0E-03 -1,3E-02 4,2E-03

47 0,0E+00 0,0E+00 0,0E+00 -3,5E-03 -3,8E-03 2,6E-03 -3,9E-03

48 -2,6E-03 3,1E-04 -4,1E-03 0,0E+00 -2,5E-03 -5,4E-04 -4,0E-03

49 7,1E-03 -5,0E-03 5,1E-03 0,0E+00 7,0E-03 4,6E-03 5,0E-03

50 -8,9E-03 4,5E-03 -9,0E-03 0,0E+00 -8,8E-03 -4,4E-03 -9,0E-03

51 0,0E+00 0,0E+00 0,0E+00 2,8E-02 2,1E-02 -4,3E-02 1,9E-02

52 0,0E+00 0,0E+00 0,0E+00 -1,8E-02 -1,8E-02 -2,4E-02 -1,8E-02

53 0,0E+00 0,0E+00 0,0E+00 -1,6E-02 -1,5E-02 -5,0E-03 -1,5E-02

54 -2,5E-03 1,4E-03 -2,3E-03 0,0E+00 -2,4E-03 -1,4E-03 -2,2E-03

55 7,2E-03 -5,0E-03 5,2E-03 0,0E+00 7,2E-03 4,6E-03 5,2E-03

56 -9,3E-03 4,4E-03 -9,9E-03 0,0E+00 -9,2E-03 -4,4E-03 -9,9E-03

57 0,0E+00 0,0E+00 0,0E+00 2,9E-02 2,3E-02 -4,4E-02 2,1E-02

58 0,0E+00 0,0E+00 0,0E+00 6,9E-03 4,2E-03 -1,3E-02 3,3E-03

59 0,0E+00 0,0E+00 0,0E+00 -3,5E-03 -4,5E-03 9,0E-04 -4,8E-03

60 -1,5E-03 -1,4E-04 -2,8E-03 0,0E+00 -1,4E-03 -7,9E-05 -2,8E-03

61 7,1E-03 -5,1E-03 4,9E-03 0,0E+00 7,0E-03 4,6E-03 4,9E-03

62 -1,0E-02 4,5E-03 -1,1E-02 0,0E+00 -9,9E-03 -4,6E-03 -1,1E-02

63 0,0E+00 0,0E+00 0,0E+00 3,0E-02 2,5E-02 -4,4E-02 2,2E-02

64 0,0E+00 0,0E+00 0,0E+00 6,8E-03 4,2E-03 -1,3E-02 3,3E-03

65 0,0E+00 0,0E+00 0,0E+00 -4,4E-03 -5,0E-03 2,8E-03 -5,2E-03

66 -1,9E-03 5,3E-04 -2,5E-03 0,0E+00 -1,8E-03 -6,2E-04 -2,5E-03

67 -1,4E-03 1,4E-02 1,9E-02 0,0E+00 -1,8E-03 -1,1E-02 1,9E-02

68 -2,2E-02 3,1E-02 8,2E-03 0,0E+00 -2,2E-02 -2,6E-02 8,4E-03

69 0,0E+00 0,0E+00 0,0E+00 3,5E-02 3,7E-02 -3,0E-02 3,7E-02

70 0,0E+00 0,0E+00 0,0E+00 5,0E-03 1,8E-03 -1,3E-02 7,6E-04

71 0,0E+00 0,0E+00 0,0E+00 -6,6E-03 -7,9E-03 3,4E-03 -8,2E-03

72 -6,0E-03 1,3E-02 9,5E-03 0,0E+00 -6,3E-03 -1,1E-02 9,5E-03

73 -6,7E-03 2,6E-02 2,7E-02 0,0E+00 -7,4E-03 -2,0E-02 2,7E-02

74 -2,8E-02 4,6E-02 1,8E-02 0,0E+00 -2,9E-02 -3,7E-02 1,8E-02

75 0,0E+00 0,0E+00 0,0E+00 3,8E-02 4,4E-02 -2,3E-02 4,5E-02

76 0,0E+00 0,0E+00 0,0E+00 4,7E-03 1,4E-03 -1,3E-02 3,7E-04

77 0,0E+00 0,0E+00 0,0E+00 -7,0E-03 -8,3E-03 3,5E-03 -8,7E-03

78 -6,5E-03 1,4E-02 1,0E-02 0,0E+00 -6,8E-03 -1,2E-02 1,0E-02

79 -9,3E-07 1,3E-06 3,3E-07 0,0E+00 -9,5E-07 -1,1E-06 3,3E-07

80 -1,2E-04 1,1E-04 -4,9E-05 0,0E+00 -1,2E-04 -9,1E-05 -4,8E-05

81 0,0E+00 0,0E+00 0,0E+00 6,8E-05 7,1E-05 -6,1E-05 7,1E-05

82 0,0E+00 0,0E+00 0,0E+00 1,7E-04 1,3E-04 -2,7E-04 1,1E-04

83 0,0E+00 0,0E+00 0,0E+00 -2,7E-04 -2,8E-04 2,4E-04 -2,8E-04



84 -3,9E-04 3,5E-04 -1,7E-04 0,0E+00 -3,9E-04 -3,0E-04 -1,7E-04

85 7,4E-04 -7,0E-04 2,6E-04 0,0E+00 7,4E-04 6,1E-04 2,5E-04

86 7,2E-04 -7,0E-04 2,2E-04 0,0E+00 7,2E-04 6,1E-04 2,1E-04

87 0,0E+00 0,0E+00 0,0E+00 4,2E-03 2,7E-03 -8,2E-03 2,2E-03

88 0,0E+00 0,0E+00 0,0E+00 8,5E-03 6,0E-03 -1,5E-02 5,1E-03

89 0,0E+00 0,0E+00 0,0E+00 -2,6E-04 -9,1E-04 -1,7E-03 -1,1E-03

90 -1,6E-03 1,6E-03 -3,6E-04 0,0E+00 -1,6E-03 -1,4E-03 -3,4E-04

91 5,7E-04 -5,3E-04 2,1E-04 0,0E+00 5,6E-04 4,6E-04 2,1E-04

92 4,9E-04 -4,7E-04 1,6E-04 0,0E+00 4,9E-04 4,1E-04 1,6E-04

93 0,0E+00 0,0E+00 0,0E+00 3,3E-03 2,2E-03 -6,3E-03 1,8E-03

94 0,0E+00 0,0E+00 0,0E+00 8,5E-03 6,0E-03 -1,5E-02 5,1E-03

95 0,0E+00 0,0E+00 0,0E+00 -2,9E-04 -9,2E-04 -1,6E-03 -1,1E-03

96 -1,6E-03 1,5E-03 -4,4E-04 0,0E+00 -1,5E-03 -1,3E-03 -4,2E-04

97 2,9E-03 -1,8E-03 2,5E-03 0,0E+00 2,9E-03 1,7E-03 2,4E-03

98 -4,9E-03 3,1E-03 -4,0E-03 0,0E+00 -4,8E-03 -2,9E-03 -4,0E-03

99 0,0E+00 0,0E+00 0,0E+00 1,2E-02 1,0E-02 -1,8E-02 9,2E-03

100 0,0E+00 0,0E+00 0,0E+00 7,8E-03 5,5E-03 -1,3E-02 4,8E-03

101 0,0E+00 0,0E+00 0,0E+00 -2,5E-03 -2,7E-03 2,0E-03 -2,7E-03

102 -2,7E-03 1,5E-03 -2,5E-03 0,0E+00 -2,6E-03 -1,4E-03 -2,5E-03

103 5,8E-03 -2,8E-03 6,1E-03 0,0E+00 5,7E-03 2,8E-03 6,1E-03

104 -9,7E-03 4,9E-03 -9,9E-03 0,0E+00 -9,5E-03 -4,8E-03 -9,8E-03

105 0,0E+00 0,0E+00 0,0E+00 2,6E-02 2,1E-02 -3,6E-02 2,0E-02

106 0,0E+00 0,0E+00 0,0E+00 7,5E-03 5,1E-03 -1,3E-02 4,3E-03

107 0,0E+00 0,0E+00 0,0E+00 -3,4E-03 -3,7E-03 2,5E-03 -3,8E-03

108 -2,6E-03 3,9E-04 -4,0E-03 0,0E+00 -2,5E-03 -6,1E-04 -4,0E-03

109 7,1E-03 -5,1E-03 5,0E-03 0,0E+00 7,0E-03 4,6E-03 4,9E-03

110 -9,3E-03 4,4E-03 -9,9E-03 0,0E+00 -9,2E-03 -4,4E-03 -9,8E-03

111 0,0E+00 0,0E+00 0,0E+00 2,9E-02 2,3E-02 -4,3E-02 2,1E-02

112 0,0E+00 0,0E+00 0,0E+00 6,9E-03 4,2E-03 -1,3E-02 3,3E-03

113 0,0E+00 0,0E+00 0,0E+00 -3,5E-03 -4,5E-03 9,0E-04 -4,8E-03

114 -1,5E-03 -1,2E-04 -2,8E-03 0,0E+00 -1,4E-03 -9,3E-05 -2,8E-03

115 7,9E-03 -7,0E-03 3,6E-03 0,0E+00 7,9E-03 6,1E-03 3,5E-03

116 -1,1E-02 6,0E-03 -9,9E-03 0,0E+00 -1,1E-02 -5,8E-03 -9,8E-03

117 0,0E+00 0,0E+00 0,0E+00 3,3E-02 2,7E-02 -4,7E-02 2,4E-02

118 0,0E+00 0,0E+00 0,0E+00 6,8E-03 4,2E-03 -1,3E-02 3,3E-03

119 0,0E+00 0,0E+00 0,0E+00 -4,4E-03 -5,0E-03 2,9E-03 -5,2E-03

120 -3,4E-03 7,8E-03 5,7E-03 0,0E+00 -3,5E-03 -6,2E-03 5,7E-03

121 7,4E-03 -5,8E-03 4,4E-03 0,0E+00 7,4E-03 5,2E-03 4,4E-03

122 -1,1E-02 7,5E-03 -8,9E-03 0,0E+00 -1,1E-02 -6,9E-03 -8,8E-03

123 0,0E+00 0,0E+00 0,0E+00 3,3E-02 2,7E-02 -4,6E-02 2,5E-02

124 0,0E+00 0,0E+00 0,0E+00 6,8E-03 4,2E-03 -1,3E-02 3,3E-03

125 0,0E+00 0,0E+00 0,0E+00 -4,4E-03 -5,0E-03 2,9E-03 -5,2E-03

126 -3,3E-03 7,7E-03 5,6E-03 0,0E+00 -3,5E-03 -6,1E-03 5,6E-03

127 4,4E-11 -1,4E-11 6,7E-11 0,0E+00 4,2E-11 9,3E-12 6,6E-11

128 6,9E-09 -6,6E-09 2,4E-09 0,0E+00 6,9E-09 5,7E-09 2,3E-09

129 0,0E+00 0,0E+00 0,0E+00 -3,0E-09 -3,1E-09 2,7E-09 -3,1E-09



130 0,0E+00 0,0E+00 0,0E+00 3,6E-09 2,7E-09 -5,9E-09 2,4E-09

131 0,0E+00 0,0E+00 0,0E+00 -2,6E-08 -2,7E-08 2,4E-08 -2,7E-08

132 -6,3E-08 6,0E-08 -2,1E-08 0,0E+00 -6,2E-08 -5,2E-08 -2,1E-08

133 2,8E-03 4,9E-03 4,0E-03 0,0E+00 3,2E-03 2,8E-02 4,0E-03

134 -2,8E-02 4,6E-02 1,8E-02 0,0E+00 -2,9E-02 -3,7E-02 1,8E-02

135 0,0E+00 0,0E+00 0,0E+00 -9,5E-02 -9,8E-02 -6,5E-02 -9,9E-02

136 0,0E+00 0,0E+00 0,0E+00 5,3E-02 5,7E-02 1,7E-02 5,8E-02

137 0,0E+00 0,0E+00 0,0E+00 -7,0E-03 -8,3E-03 3,5E-03 -8,7E-03

138 3,8E-03 -8,4E-03 -9,2E-03 0,0E+00 4,2E-03 1,9E-02 -9,2E-03

139 0,0E+00 0,0E+00 0,0E+00 -9,1E-07 -6,8E-07 1,5E-06 -5,9E-07

140 0,0E+00 0,0E+00 0,0E+00 1,9E-06 1,9E-06 -1,7E-06 1,8E-06

141 -3,0E-06 2,9E-06 -9,1E-07 0,0E+00 -3,0E-06 -2,5E-06 -8,8E-07

142 -1,4E-03 1,2E-03 -5,3E-04 0,0E+00 -1,3E-03 -1,1E-03 -5,2E-04

143 -2,8E-03 1,8E-03 -2,2E-03 0,0E+00 -2,8E-03 -1,7E-03 -2,2E-03

144 0,0E+00 0,0E+00 0,0E+00 8,0E-02 7,8E-02 6,1E-02 7,7E-02

145 0,0E+00 0,0E+00 0,0E+00 5,7E-01 5,7E-01 5,8E-01 5,7E-01

146 -1,0E-03 9,0E-04 -3,2E-04 0,0E+00 -1,0E-03 -8,6E-04 -3,2E-04

147 0,0E+00 0,0E+00 0,0E+00 -1,8E-02 -1,8E-02 -2,4E-02 -1,8E-02

148 0,0E+00 0,0E+00 0,0E+00 -1,6E-02 -1,5E-02 -4,0E-03 -1,5E-02

149 -3,3E-03 1,7E-03 -3,2E-03 0,0E+00 -3,2E-03 -1,6E-03 -3,2E-03

150 0,0E+00 0,0E+00 0,0E+00 4,6E-02 4,6E-02 4,2E-02 4,6E-02

151 0,0E+00 0,0E+00 0,0E+00 1,2E-02 1,3E-02 2,4E-02 1,3E-02

152 -4,2E-03 2,2E-03 -4,2E-03 0,0E+00 -4,1E-03 -1,8E-03 -4,1E-03

153 0,0E+00 0,0E+00 0,0E+00 4,6E-02 4,6E-02 4,3E-02 4,6E-02

154 0,0E+00 0,0E+00 0,0E+00 1,4E-02 1,4E-02 1,9E-02 1,4E-02

155 -1,2E-03 1,1E-03 -3,6E-04 0,0E+00 -1,2E-03 -1,3E-03 -3,3E-04

156 0,0E+00 0,0E+00 0,0E+00 5,7E-03 2,8E-03 -1,2E-02 1,8E-03

157 0,0E+00 0,0E+00 0,0E+00 2,3E-04 -4,5E-04 -3,9E-03 -6,7E-04

158 -2,2E-03 4,0E-03 2,1E-03 0,0E+00 -2,3E-03 -3,3E-03 2,1E-03

159 -2,5E-03 6,0E-03 4,6E-03 0,0E+00 -2,6E-03 -4,7E-03 4,7E-03

160 0,0E+00 0,0E+00 0,0E+00 6,9E-03 4,2E-03 -1,3E-02 3,3E-03

161 0,0E+00 0,0E+00 0,0E+00 -3,4E-03 -4,5E-03 5,9E-04 -4,7E-03

162 -1,2E-03 -5,3E-04 -3,0E-03 0,0E+00 -1,2E-03 2,4E-04 -3,0E-03

163 0,0E+00 0,0E+00 0,0E+00 5,7E-03 2,8E-03 -1,2E-02 1,8E-03

164 0,0E+00 0,0E+00 0,0E+00 2,2E-04 -4,6E-04 -3,9E-03 -6,8E-04

165 -2,2E-03 4,0E-03 2,1E-03 0,0E+00 -2,3E-03 -3,3E-03 2,1E-03

166 -3,7E-03 8,2E-03 5,9E-03 0,0E+00 -3,8E-03 -6,6E-03 5,9E-03

167 0,0E+00 0,0E+00 0,0E+00 5,3E-02 5,7E-02 1,7E-02 5,8E-02

168 3,8E-03 -8,4E-03 -9,2E-03 0,0E+00 4,2E-03 1,9E-02 -9,2E-03



APPENDIX 10: MATLAB codes

The complete MATLAB codes for the thesis are presented below. For more

information about the coding procedure, check Figure 13. The codes are presented in

the following order:

Code 1: lcct.m

Code 2: static.m

Code 3: statespace.m

Code 4: energy.m

Code 5: beam3e_mod_3.m*

*Adapted version of the CALFEM function beam3e, originally written by Serrano

(1995). The output for this function are the local mass and stiffness matrices.



Code 1: lcct.m

% lcct.m %---------------------------------------------------------------- % PURPOSE % Create the load case for the static analysis (static.m). % 26/05/2015 %---------------------------------------------------------------- % REFERENCES % Martin Nyström, Tainan Pantano Tomaz %---------------------------------------------------------------- % CHANGING LOAD CASE: % The matrix Switch should be changed according to the load case % The load case table can be seen in the thesis document % % LEGEND FOR Switch MATRIX: % “0” -> actuator is deactivated % “1” -> actuator has maximum loading % % 1st value corresponds to the turntable (turning actuator) % 2nd value corresponds to the actuator 1 (lifting actuator) % 3rd value corresponds to the actuator 2 (parallel actuator) % %---------------------------------------------------------------- % TRANSFORMATION MATRICES ARE CREATED %---------------------------------------------------------------- Lc=[-65000 0 0 0 123 7031 0 -10507 512]; Ct=[-8218 0 0 0 1000 0 0 0 1000]; Lci=inv(Lc); T=Lci*Ct %---------------------------------------------------------------- % Swith IS CREATED: % Here the load cases are turned on and off. %---------------------------------------------------------------- Switch=[0 0 1]; %---------------------------------------------------------------- % Ct IS CREATED: % This matrix contains the load forces to be added to the static % analysis. The values are added, respectively in the DOFs 75, 73 % and 74. %---------------------------------------------------------------- Ct=Switch*[-65000 0 0 0 177000 0 0 0 115000]*T %---------------------------------------------------------------- % OUTPUT: % The file 'lcct_matrix1.mat' will contain the load forces % for Load Case 1 to be added to the Static Analysis. %---------------------------------------------------------------- save('lcct_matrix1.mat','T','Ct')

%---END---



Code 2: static.m

% Static.m %---------------------------------------------------------------- % PURPOSE % Static analysis of Rottne's crane RK250 - part of the Thesis Work. % 26/05/2015 %---------------------------------------------------------------- % REFERENCES % Martin Nyström, Tainan Pantano Tomaz %----------------------------------------------------------------

echo off clear all

%-- 1.1) Element properties and topology ----------------- %% %--- DOFs numbering is created

Edof=[ 1 127 128 129 130 131 132 79 80 81 82 83 84 2 79 80 81 82 83 84 7 8 9 10 11 12 3 13 14 15 16 17 18 85 86 87 88 89 90 4 85 86 87 88 89 90 91 92 93 94 95 96 5 91 92 93 94 95 96 7 8 9 10 11 142 6 85 86 87 88 89 90 19 20 21 22 23 24 7 91 92 93 94 95 96 25 26 27 28 29 30 8 7 8 9 10 11 143 97 98 99 100 101 102 9 97 98 99 100 101 102 103 104 105 106 107 108 10 103 104 105 106 107 108 43 44 45 46 47 48 11 97 98 99 100 101 102 31 32 33 34 35 36 12 103 104 105 106 107 108 37 38 39 40 41 42 13 19 20 21 147 148 149 49 50 51 52 53 54 14 25 26 27 150 151 152 31 32 33 153 154 155 15 49 50 51 160 161 162 109 110 111 112 113 114 16 109 110 111 112 113 114 61 62 63 64 65 66 17 109 110 111 112 113 114 55 56 57 58 59 60 18 37 38 39 156 157 158 55 56 57 163 164 165 19 61 62 63 64 65 166 115 116 117 118 119 120 20 43 44 45 46 47 159 121 122 123 124 125 126 21 115 116 117 118 119 120 121 122 123 124 125 126 22 121 122 123 124 125 126 67 68 69 70 71 72 23 67 68 69 70 71 72 73 74 75 76 77 78 24 1 2 3 4 5 6 13 14 15 144 145 146 25 1 2 3 139 140 141 127 128 129 130 131 132 26 73 74 75 167 77 168 133 134 135 136 137 138 ];

%--- Total number of elements Enum=size(Edof,1) ; Dnum=max(max(Edof))

%--- System matrices K=zeros(Dnum,Dnum); M=zeros(Dnum,Dnum); f=zeros(Dnum,1); g=zeros(Enum*12,24);

Ep=zeros(Enum,8);



E=210000e6; G=79000e6; %--- Shear modulus A=1; Iy=1; %--- Moment of inertia with respect to the y-axis Iz=1; %--- Moment of inertia with respect to the z-axis Kv=1; %--- St Venant torsional stiffness Kv m=1; %--- Mass per length J=1 %--- Polar mass moment of inertia

%--- Ep=[E G A Iy Iz Kv];

%--- Actuator loads and stiffness: %--- Actuator nr.1 (element 24) = lifting %--- Actuator nr.2 (element 14) = parallell

F1=0; Ef24=1;%---Elasticity factor

F2=0; Ef14=1;%---Elasticity factor

%--- Change E modulus for actuators:

Beta=1.6e10; %---- Bulk modulus elasticity - compressibility - hydraulic

oil

L14=1.767; %--- Actuator 14 length Lc14=0.566; %--- Oil length L24=0.6853; %--- Actuator 24 length Lc24=0.44; %--- Oil length

%--- Formula according to Linköping universitetet, 1995. Hydraulvätska. %--- In "Formelsamling i hydraulik och pneumatik".

E14=Beta*L14/Lc14*Ef14 E24=Beta*L24/Lc24*Ef24

%--- Element properties for each element - Ep=[E G A Iy Iz Kv m J] %--- Properties taken from 3D-CAD solids provided by the Company

Ep=[ E G 4.99680E-02 4.56360E-04 1.57433E-04 6.13792E-04 2.67800E+02

3.28944E+00 E G 3.00000E-02 2.25000E-06 1.56250E-04 1.58500E-04 2.31200E+02

2.74601E-01 E G 1.72000E-02 5.73333E-07 2.65023E-04 2.65000E-04 2.09500E+02

6.55000E-02 E G 1.72000E-02 5.73333E-07 2.65023E-04 2.65000E-04 2.09500E+02

2.59090E-01 E G 1.09400E-02 3.64636E-07 6.81774E-05 6.85308E-05 1.10600E+02

3.35304E-01 E G 1.72000E-02 5.73333E-07 2.65023E-04 2.65000E-04 1.00000E+00

1.00000E-03 E G 1.72000E-02 5.73333E-07 2.65023E-04 2.65000E-04 1.00000E+00

1.00000E-03 E G 1.28000E-02 3.54347E-04 2.10398E-04 5.64745E-04 2.09700E+02

3.39180E+00



E G 1.28000E-02 3.54347E-04 2.10398E-04 5.64745E-04 2.09700E+02

1.99967E+00 E G 1.28000E-02 3.54347E-04 2.10398E-04 5.64745E-04 1.82800E+02

2.92251E-01 E G 1.28000E-02 3.54347E-04 2.10398E-04 5.64745E-04 1.00000E+00

1.00000E-03 E G 1.28000E-02 3.54347E-04 2.10398E-04 5.64745E-04 1.00000E+00

1.00000E-03 E G 2.47700E-03 4.90025E-06 3.69564E-06 8.59589E-06 3.19000E+01

5.31706E-01 E14 G 3.84845E-03 1.17859E-06 1.17859E-06 2.35718E-06 1.11300E+02

1.09887E+00 E G 1.00000E-02 5.20829E-07 3.33333E-05 3.38542E-05 1.20400E+02

1.55176E-01 E G 1.00000E-02 5.20829E-07 3.33333E-05 3.38542E-05 1.07000E+02

1.08216E-01 E G 1.00000E-02 5.20829E-07 3.33333E-05 3.38542E-05 1.00000E+00

1.08216E-01 E G 5.00000E-03 2.60417E-07 4.16667E-06 4.42708E-06 3.85000E+01

2.96100E-02 E G 5.76100E-03 4.97346E-05 6.52223E-05 1.14957E-04 1.00000E+00

2.72448E-01 E G 5.76100E-03 4.97346E-05 6.52223E-05 1.14957E-04 1.00000E+00

5.44890E-01 E G 9.06100E-03 9.11397E-05 3.48267E-04 4.39407E-04 4.05000E+02

8.99655E-01 E G 5.76100E-03 4.97346E-05 6.52223E-05 1.14957E-04 1.68800E+02

3.38193E+00 E G 4.86100E-03 2.98625E-05 3.96542E-05 6.95167E-05 1.06400E+02

4.64455E-01 E24 G 5.02700E-03 2.01062E-06 2.01062E-06 4.02124E-06 2.60470E+02

1.18821E+00 E G 1.50000E-02 1.12500E-06 7.81250E-05 7.92500E-05 1.15600E+02

1.37301E-01 E G 8.29000E-02 5.73300E-04 5.73300E-04 1.14662E-03 6.40000E+02

2.26457E+01 ]; %%

%--- Coordinates

%%

%--- Element coordinates

Ex=1e-3*[ 0 696 696 696 62.19 100.41 100.41 251.49 251.49 696 100.41 420.71 251.49 477.11 696 2395 2395 4130.58 4130.58 4353.43 2395 2244.6 4130.58 4085.88 420.71 3651.35 477.11 2244.6 3651.35 3953.86



3953.86 4365.69 3953.86 3958.71 4085.88 3958.71 4365.69 4550.21 4353.43 4734.84 4550.21 4734.84 4734.84 6896.59 6896.59 7909.24 -25 62.19 -25 0 7909.24 7909.2401 ];

Ey=1e-3*[ 698 698 698 923 1377.73 1350.31 1350.31 1241.92 1241.92 923 1350.31 1796.74 1241.92 1556.4 923 1973.42 1973.42 3046.46 3046.46 3184.24 1973.42 2216.69 3046.46 3118.76 1796.74 3600.32 1556.4 2216.69 3600.32 3583.25 3583.25 3560.67 3583.25 3669.26 3118.76 3669.26 3560.67 3796.84 3184.24 3643.51 3796.84 3643.51 3643.51 1848.26 1848.26 1007.29 698 1377.73 698 698 1007.29 -1492.71 ];

Ez=[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0



0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ];

%--- Element z-axis orientation in global coordinates Eo=[ 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 ];

Eq=[0 0 0 0]; %% %-- 1.2) Initial load matrix from weight forces ---------------- %%

f=[ 0 0 0 0 0 0 0 -4306.59 0 0 0 0 0 0 0



0 0 0 0 -579.771 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -124.587 0 0 0 0 0 -8220.78 0 0 0 0 0 -579.771 0 0 0 0 0 -1182.105 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -10123.92 0 0 0



0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -7848 0 0 0 0 0



0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]; f=f*0;%--- Switching off "f"

%--- Externaly added forces

%--- From actuator L1=((Ex(24,2)-Ex(24,1))^2+(Ey(24,2)-Ey(24,1))^2)^0.5;%--- Actuator

length F1x=F1*(Ex(24,2)-Ex(24,1))/L1;%--- Actuator force F1y=F1*(Ey(24,2)-Ey(24,1))/L1;%--- Actuator force L2=((Ex(14,2)-Ex(14,1))^2+(Ey(14,2)-Ey(14,1))^2)^0.5;%--- Actuator

length F2x=F2*(Ex(14,2)-Ex(14,1))/L2;%--- Actuator force F2y=F2*(Ey(14,2)-Ey(14,1))/L2;%--- Actuator force %--- Attributing forces components to global force vector f(1,1)=F1x;f(13,1)=-F1x; f(2,1)=F1y;f(14,1)=-F1y; f(25,1)=F2x;f(31,1)=-F2x; f(26,1)=F2y;f(32,1)=-F2y;

%--- Loading force case from "lcct.m" and adding it to loading vector load('lcct_matrix1.mat','T','Ct') f(73,1)=Ct(1,2);f(74,1)=Ct(1,3);f(75,1)=Ct(1,1); save('f_file1.mat','f')

%% %-- 1.3) Draw the fe-mesh as a check of the model -------------- %% figure(1), clf; eldraw2(Ex,Ey,[1 2 0],Edof(:,1)), xlabel('x (m)'), ylabel('y (m)'),

title('Model shape'),xlim([-1 9]),ylim([-3 5])



%%

%-- 1.4) Create and assemble global stiffness (K) and mass matrix

(M) -

%%

%--- Create mapping for stiffness matrix % %--- Value 13 for second column means MANUAL stiffness matrix %--- Value 1 for second column means AUTOMATIC stiffness matrix % %--- Mapping function for local stiffness matrix %--- All set to automatic % map1=[1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 11 1 12 1 13 1 14 1 15 1 16 1 17 1 18 1 19 1 20 1 21 1 22 1 23 1 24 1 25 1 26 1];

for j=1:Enum

%--- Creates ELEMENT K and M matrices

[Ke1,Mec, Me1, fe1]=beam3e_mod_3(Ex(j,:), Ey(j,:), Ez(j,:), Eo(j,:),

Ep(j,:), Eq); g([j*12-11:j*12],[1:12])=Ke1;

%------------------------------------------------------------------- Ke1; Kr=Ke1-Ke1';

%--- Transforms ELEMENT K and M matrices to GLOBAL according to map1

[K,f]=assem(Edof(j,:),K,g([j*12-

11:j*12],[map1(j,2):map1(j,2)+11]),f,fe1); [M]=assem(Edof(j,:),M,Me1);



save('MKfile.mat','M','K') end

%%

%-- 1.5) Set boundary conditions (bc) at each DOF --------------

%%

%--- Restrained in all DOFs in the crane base bc=[ 127 0 128 0 129 0 130 0 131 0 132 0]; %%

%-- 1.6) Calculate node displacement and element forces (es) for

STATIC

%% %--- Solve static equation K*a=f+r, where: %--- global displacement (a), reaction forces (r) [a,r]=solveq(K,f,bc);

%--- Extract element displacements (Ed) Ed=extract(Edof,a);

%--- Extract local forces for each element start/end node

for j=1:Enum es([j*2-1

j*2],:)=beam3s(Ex(j,:),Ey(j,:),Ez(j,:),Eo(j,:),Ep(j,:),Ed(j,:)); end save('a_es_file1.mat','a','es') %% %-- 1.7) Draw displaced figure for the crane for STATIC -------- %% figure(2) clf eldraw2(Ex,Ey,[1 2 0]), title('Structure and displacement - LC 1') hold on legend('Original shape','Location','SouthWest') eldisp3(Ex,Ey,Ez,Ed,[1 3 0]),xlabel('x (m)'),ylabel('y (m)'),zlabel('z

(m)'), xlim([-1 9]),ylim([-3 5]),zlim([-1 1]), grid on

%% %-- 1.8) Component forces at nodes for STATICS ----------------- %%

%--- Boom loads matrix from es_matrix Boom_es=zeros(4,6); Boom_es(1,:)=es(15,:); Boom_es(2,:)=es(22,:); Boom_es(3,:)=es(24,:);



Boom_es(4,:)=es(20,:) %--- Jib loads matrix from es_matrix Jib_es=zeros(3,6); Jib_es(1,:)=es(39,:); Jib_es(2,:)=es(37,:); Jib_es(3,:)=es(44,:) %--- Extension loads matrix from es_matrix Ext_es=zeros(2,6); Ext_es(1,:)=es(45,:); Ext_es(2,:)=es(46,:) %--- Linkage loads matrix from es_matrix Link_beam_es=zeros(3,6); Link_beam_es(1,:)=es(29,:); Link_beam_es(2,:)=es(34,:); Link_beam_es(3,:)=es(32,:) Link_bar_es=es([35 36],1) %--- Triagle Link loads matrix from es_matrix Tria_es=zeros(4,6); Tria_es(1,:)=es(5,:); Tria_es(2,:)=es(12,:); Tria_es(3,:)=es(14,:); Tria_es(4,:)=es(10,:) %--- Crane base loads matrix from es_matrix C_base_es=zeros(2,6); C_base_es(1,:)=es(49,:); C_base_es(2,:)=es(4,:) %--- actuator normal force or load from es_matrix Actuator_es=zeros(2,1); Actuator_es(1,:)=es(48,1); Actuator_es(2,:)=es(28,1)

%---END---



Code 3: statespace.m

% statespace.m %---------------------------------------------------------------- % PURPOSE % Static analysis of Rottne's crane RK250 - part of the Thesis Work. % April/2015 %---------------------------------------------------------------- % REFERENCES % Martin Nyström, Tainan Pantano Tomaz %---------------------------------------------------------------- echo off clear all %---------------------------------------------------------------- clear figure %% %--- Mass and stiffness matrices for the structure load('MKfile.mat','M','K') load('f_file1.mat','f') f=f*0; z=0.10;%---damping coefficient;

%--- Extract modal information [fi,lambda]=eig(K,M);%--- Solve EVP problem lambda=diag(lambda);%--- Transform lambda to a column vector om=sqrt(lambda);%--- Natural circular frequencies [rad/s] freq=abs(om/(2*pi));%--- Natural frequencies [Hz]

%--- Create modal matrices Mm=fi'*M*fi; Km=fi'*K*fi; Mmd=diag(Mm); Kmd=diag(Km); Cmd=sqrt(Mmd.*Kmd)*2*z; Cm=abs(diag(Cmd)); %--- Create damping matrix c c=inv(fi')*Cm*inv(fi); %--- Boundary conditions (same as "static.m") bc=[ 127 0 128 0 129 0 130 0 131 0 132 0];

%% %--- Define system matrices dimensions Dnum=(size(M,1)); bc_num=size(bc,1);

%--- Define timesteps in seconds (h) and time limit (tmax) h=1e-6; tmax=2; n=floor(tmax/h);



%--- Define time vector-

ti=[0:h:tmax].';

[tnum,t_aux]=size(ti);

%%

%-- 2.2) Define and build continuous A, B, C and D matrices

%% %--- Define matrices according to State Space theory Mei=inv([zeros(Dnum) M;eye(Dnum) zeros(Dnum)]); Kei=[K c;zeros(Dnum) -eye(Dnum)]; A=-Mei*Kei; B=Mei*[f;zeros(Dnum,1)]; C=[(eye(Dnum)) zeros(Dnum)]; D=zeros(Dnum,1); %--- Build continuous system sys=ss(A,B,C,D); %%

%--- Create discrete matrices from continuous using c2d

%% sysd=c2d(sys,h,'foh');%---First order hold--- %--- Extract discrete A B C D matrices ---------------------------------

---- Ad=sysd.A; Bd=sysd.B; Cd=sysd.C; Dd=sysd.D; %% %-- 2.4) Create load vector controller / 1 (ON) and 0 (OFF) %--- U vector for zero load U=zeros(n,1); U(1:(floor(n/2)),1)=zeros; U((1:10),1)=zeros;

%--- Displacement vector---Initial displacement--- a=zeros(Dnum,1); load('a_es_file1.mat','a') init=(zeros(2*Dnum,1)); init([1:Dnum],1)=1*a; X0=init;Y=X0;

%------------------------------------------------------------------- %---Loop for State space simulation using discrete matrices l=100;%---number of iterations between storing output X(:,1)=X0(1:Dnum,1);%---X is output matrix tic for j=1:n n1=floor((j/l)+2); %n1=floor(sqrt(j)+1); Uo(1,n1)=U(j,1); X(:,n1)=Y(1:Dnum,1); T(1,n1)=j*h; for k=1:bc_num;%---Loop for setting and keeping boundary

conditions X0(bc(k,1),1)=bc(k,2); X0(bc(k,1)+Dnum,1)=0; end X0=Ad*X0+Bd*U(j,:); Y=Cd*X0+Dd*U(j,:);



end toc save('XT_file1.mat','X','T') save('Y_file1.mat','Y') %------------------------------------------------------------------- %-------------------------------------------------------------------

%-------------------------------------------------------------------

n n1 close all figure(1)

plot(T(1,:),X(73,:),'b',T(1,:),X(74,:),'r',T(1,:),X(75,:),'g') grid on legend('u73 crane tip x','u74 crane tip y','u75 crane tip z','pulse

duration','Location','SouthEast') xlabel(['Time [s]']) ylabel(['Displacement [m]']) title(['Release from initial displacement - LC 1'])

%---END--- %-------------------------------------------------------------------



Code 4: energy.m

% energy.m %---------------------------------------------------------------- % PURPOSE % Create forces distribution over the time and potential energy for

the % specific load case % April/2015 %---------------------------------------------------------------- % REFERENCES % Martin Nyström, Tainan Pantano Tomaz %---------------------------------------------------------------- % % % Coord_file.mat - Coordinate and element properties. % Origin: static.m % % XT_file.mat - DOF displacement and time progress % Origin: statespace.m %

tic load('XT_file1.mat','X','T') load('Coord_file.mat','Ex','Ey','Ez','Eo','Eq','Edof','Ep','Enum')

[dof,tnum]=size(X);

%-----------Create element displacement for dynamics--------------------

---

Edd=zeros(Enum,12*tnum);

for t=1:tnum Edd(:,[(t*12-11):(t*12)])=extract(Edof,X(:,t)); end

%-------------Create element forces matrix for dynamics-----------------

---

for j=1:Enum

for t=1:tnum esd([j*2-1 j*2],[t*6-

5:t*6])=beam3s(Ex(j,:),Ey(j,:),Ez(j,:),Eo(j,:),Ep(j,:),Edd(j,[(t*12-

11):(t*12)])); end

end

%--Transforms element forces matrix for each time step into a column

vector

%esd_vec=zeros(2*Enum*6,tnum);

for g=1:tnum



for j=1:(Enum*2)

esd_vec([j*6-5:j*6],g)=esd(j,[g*6-5:g*6]);

end

end

clc close all tic

figure(1) plot(T(1,:),esd_vec(85,[1:tnum]),'b',T(1,:),esd_vec(86,[1:tnum]),'r',T(1

,:),esd_vec(87,[1:tnum]),'g') grid on legend('Fx','Fy','Fz','Location','SouthEast') xlabel(['Time [s]']) ylabel(['Force [N]']) title(['Force [N] distribution over the time - Node 2 - Element 8S'])

figure(2) plot(T(1,:),esd_vec(88,[1:tnum]),'b--',T(1,:),esd_vec(89,[1:tnum]),'r--

',T(1,:),esd_vec(90,[1:tnum]),'g--') grid on legend('Mx','My','Mz','Location','SouthEast') xlabel(['Time [s]']) ylabel(['Moment [N*m]']) title(['Moment [N*m] distribution over the time - Node 2 - Element 8S'])

figure(3) plot(T(1,:),esd_vec(127,[1:tnum]),'b',T(1,:),esd_vec(128,[1:tnum]),'r',T

(1,:),esd_vec(129,[1:tnum]),'g') grid on legend('Fx','Fy','Fz','Location','SouthEast') xlabel(['Time [s]']) ylabel(['Force [N]']) title(['Force [N] distribution over the time - Node 6 - Element 11E'])

figure(4) plot(T(1,:),esd_vec(130,[1:tnum]),'b--',T(1,:),esd_vec(131,[1:tnum]),'r-

-',T(1,:),esd_vec(132,[1:tnum]),'g--') grid on legend('Mx','My','Mz','Location','SouthEast') xlabel(['Time [s]']) ylabel(['Moment [N*m]']) title(['Moment [N*m] distribution over the time - Node 6 - Element

11E'])


(1,:),esd_vec(141,[1:tnum]),'g') grid on legend('Fx','Fy','Fz','Location','SouthEast')



xlabel(['Time [s]']) ylabel(['Force [N]']) title(['Force [N] distribution over the time - Node 7 - Element 12E'])



12E'])


(1,:),esd_vec(117,[1:tnum]),'g') grid on legend('Fx','Fy','Fz','Location','SouthEast') xlabel(['Time [s]']) ylabel(['Force [N]']) title(['Force [N] distribution over the time - Node 8 - Element 10E'])



10E']) toc

%tic tnum=size(esd_vec,2);

BoomEnergy=zeros(1,tnum);

% -------- S refers to start node, and E refers to end node ----------- % % --Element 8S - element DOF's (85:90) – global DOF’s (7,8,9,10,11,143) % --Element 11E - element DOF's (127:132) – global DOF’s (31 a 36) % --Element 12E - element DOF's (139:144) – global DOF’s (37 a 42) % --Element 10E - element DOF's (115:120) – global DOF’s (43 a 48) %

BoomEdof=[linspace(85,90,6) linspace(127,132,6) linspace(139,144,6)

linspace(115,120,6)]'; Boom_num=size(BoomEdof,1);

tinit=800; %Energy counter starting at tinit [s]

for t=tinit:tnum %Time loop - Each column corresponds to 1 energy value



for i=1:Boom_num %Energy calculation for each DOF

N=BoomEdof(i,1);

elem=ceil(N/12); int=mod(N,12); if int==0 int=12; end GlobalDOF=Edof(elem,[int+1]);

EnergyStored=abs(esd_vec(N,t)*X(GlobalDOF,t))/2; BoomEnergy(1,t)=BoomEnergy(1,t)+EnergyStored;

end

end

[MAX_Energy, idx]=max(BoomEnergy) %--Step when the maximum energy occurs

MAX_Time=T(1,idx) %--Time when the maximum energy occurs

figure(9) plot(T(1,:),BoomEnergy(1,:),'r') grid on legend('|Energy|') xlabel(['Time [s]']) ylabel(['|Energy| [N*m^2]']) title(['Energy [N*m^2] distribution - Boom'])

esd_max_boom=zeros(Boom_num,1);

for i=1:Boom_num %Energy calculation for each DOF

N=BoomEdof(i,1); esd_max_boom(i,1)=BoomEdof(i,1); esd_max_boom(i,2)=esd_vec(N,idx);

end

format short g esd_max_boom save('esd_max_boom1.mat','esd_max_boom','MAX_Time','idx') toc

%---END



Code 5: beam3e_mod_3.m

function [Ke,Mec,Mel,fe]=beam3e(ex,ey,ez,eo,ep,eq) % Ke=beam3e(ex,ey,ez,eo,ep) % [Ke,fe]=beam3e(ex,ey,ez,eo,ep,eq) %---------------------------------------------------------------- % PURPOSE % Calculate the stiffness matrix for a 3D elastic Bernoulli % beam element. % % INPUT: ex = [x1 x2] % ey = [y1 y2] % ez = [z1 z2] node coordinates % % eo = [xz yz zz]; orientation of local z axis % % ep = [E G A Iy Iz Kv m J]; element properties % E: Young's modulus % G: Shear modulus % A: Cross section area % Iy: moment of inertia,local y-axis % Iz: moment of inertia,local z-axis % Kv: Saint-Venant's torsion constant % m: mass per length % J: polar moment of inertia % % eq = [qx qy qz qw]; distributed loads % % OUTPUT: Ke : beam stiffness matrix (12 x 12) % % Me : beam mass matrix (12 x 12) % % fe : equivalent nodal forces (12 x 1) %----------------------------------------------------------------- % LAST MODIFIED: E Serrano 1995-09-21 % Copyright (c) Division of Structural Mechanics and % Department of Solid Mechanics. % Lund Institute of Technology %-------------------------------------------------------------

b=[ ex(2)-ex(1); ey(2)-ey(1); ez(2)-ez(1) ]; L=sqrt(b'*b); n1=b/L;

lc=sqrt(eo*eo'); n3=eo/lc;

%

if nargin==5; eq=[0 0 0 0]; end

qx=eq(1); qy=eq(2); qz=eq(3); qw=eq(4); % E=ep(1); Gs=ep(2); A=ep(3); Iy=ep(4); Iz=ep(5); Kv=ep(6);%Ix m=ep(7);



j=ep(8);

a=E*A/L ; b=12*E*Iz/L^3 ; c=6*E*Iz/L^2; d=12*E*Iy/L^3 ; e=6*E*Iy/L^2 ; f=Gs*Kv/L; g=2*E*Iy/L ; h=2*E*Iz/L ; cj=j/m/L;

Kle=[ a 0 0 0 0 0 -a 0 0 0 0 0 ; 0 b 0 0 0 c 0 -b 0 0 0 c ; 0 0 d 0 -e 0 0 0 -d 0 -e 0 ; 0 0 0 f 0 0 0 0 0 -f 0 0 ; 0 0 -e 0 2*g 0 0 0 e 0 g 0 ; 0 c 0 0 0 2*h 0 -c 0 0 0 h ; -a 0 0 0 0 0 a 0 0 0 0 0 ; 0 -b 0 0 0 -c 0 b 0 0 0 -c ; 0 0 -d 0 e 0 0 0 d 0 e 0 ; 0 0 0 -f 0 0 0 0 0 f 0 0 ; 0 0 -e 0 g 0 0 0 e 0 2*g 0 ; 0 c 0 0 0 h 0 -c 0 0 0 2*h]; % % % consistent mass matrix:

Mlec=(m*L/420)*[140 0 0 0 0 0 70 0 0 0 0 0; 0 156 0 0 0 22*L 0 54 0 0 0 -13*L; 0 0 156 0 -22*L 0 0 0 54 0 13*L 0; 0 0 0 140*cj 0 0 0 0 0 70*cj 0 0; 0 0 -22*L 0 4*L^2 0 0 0 -13*L 0 -3*L^2 0; 0 22*L 0 0 0 4*L^2 0 13*L 0 0 0 -3*L^2; 70 0 0 0 0 0 140 0 0 0 0 0 ; 0 54 0 0 0 13*L 0 156 0 0 0 -22*L; 0 0 54 0 -13*L 0 0 0 156 0 22*L 0; 0 0 0 70*cj 0 0 0 0 0 140*cj 0 0; 0 0 13*L 0 -3*L^2 0 0 0 22*L 0 4*L^2 0; 0 -13*L 0 0 0 -3*L^2 0 -22*L 0 0 0 4*L^2]; % % lumped mass matrix: mtot=(140+70+(156+54)*2)*2; s=(140+156+156)*2; Identity12=eye(12); Mlel=Identity12*Mlec*mtot/s; % fle=L/2*[qx qy qz qw -1/6*qz*L 1/6*qy*L qx qy qz qw 1/6*qz*L -

1/6*qy*L]';

% n2(1)=n3(2)*n1(3)-n3(3)*n1(2); n2(2)=-n1(3)*n3(1)+n1(1)*n3(3); n2(3)=n3(1)*n1(2)-n1(1)*n3(2); % An=[n1'; n2; n3]; % G=[ An zeros(3) zeros(3) zeros(3); zeros(3) An zeros(3) zeros(3); zeros(3) zeros(3) An zeros(3); zeros(3) zeros(3) zeros(3) An ]; %



% Ke1=G'*Kle*G; Me1c=G'*Mlec*G; Me1l=G'*Mlel*G; fe1=G'*fle; %---------------------------------------------------------- if nargout==0 disp('Element stiffness matrix: '); disp(Ke1);

if nargin==6 disp('Element load vector: '); disp(fe1); end return end Ke=Ke1; Mec=Me1c; Mel=Me1l;

if nargin==6 fe=fe1; end %-------------------------- end -------------------------------

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