Sebastiaan Marynissen beams according to EC4 Development of a calculation software for ...

103 Pages·2015·1.51 MB·English

Development of a calculation software for composite
beams according to EC4
Sebastiaan Marynissen
Supervisor: Prof. ir. Rik Debruyckere
Counsellor: Dr. ir. Delphine Sonck
Master's dissertation submitted in order to obtain the academic degree of
Master of Science in Civil Engineering
Department of Structural Engineering
Chairman: Prof. dr. ir. Luc Taerwe
Faculty of Engineering and Architecture
Academic year 2014-2015
Preface
Ever since I was a little kid I was interested in everything that had to do with construc-
tion, witness my creations in Duplo or at the beach in Wenduine. Through the years
this interest has become a true passion for structural analysis. At the age of 13 however,
I became interested in computer programming as well, especially web development.
The present master’s dissertation allowed me to merge my passion and my interest into
something I could never have imagined before: Composite Beam District. The develop-
mentofthissoftwarehasproventobesomuchmorethansimplyprogrammingformulas
intothecomputer. Thepresentdocumenthasbecomeageneraldocumentonhowtode-
velop constructional software and what obstacles and pitfalls can be encountered along
it’s way.
The elaboration of this master’s dissertation never had been such an instructive expe-
rience without a few people who I would like to thank. First of all I want to thank my
supervisor prof. ir. Rik Debruyckere for providing me the necessary information every
time I requested it and for the useful feedback, my counsellor dr. ir. Delphine Sonck
and dr. ir. Iveta Georgieva for sharing thoughts on the development of software for
composite beams.
I would like to thank my family as well, especially my mother and father for giving me
all the opportunities I have received, my nephew Jeroen for introducing me to website
programming and my grandmother Jos´ee.
Another thank you goes to my friends of my student association Poutrix. You made my
last two years as a student a memorable period. A particular thank you goes to Laura.
She knows why.
At last I would like to thank Della Mae, to whose beautiful music I wrote a major
part of this master’s dissertation. Music has always been an important part of my life
and especially Della Mae’s music has guided me through the tough hours of hunting
down bugs which were buried deeply in the dungeons of the software’s source code. I
recommend them to anyone reading this master’s dissertation to experience the joyous
feeling their music induces.
Permission of use on loan
The author gives permission to make this master dissertation available for consultation
and to copy parts of this master dissertation for personal use. In the case of any other
use, thecopyrighttermshavetoberespected, inparticularwithregardtotheobligation
to state expressly the source when quoting results from this master dissertation.
Sebastiaan Marynissen
May 22, 2015
Disclaimer
The author, nor the supervisor, nor Ghent University can be held liable for direct or
indirect damage as a result of any imperfections of the software and/or this document.
i
Overview
Master’s dissertation submitted in order to obtain the academic degree of Master of
Science in Civil Engineering
Title: Development of a calculation software for composite beams according to EC4
Author: Sebastiaan Marynissen
Supervisor: Prof. ir. Rik Debruyckere
Counsellor: Dr. ir. Delphine Sonck
Research group: Laboratory for Research on Structural Models
Director: Prof. ir.-arch. Jan Belis
Department: Department of Structural Engineering
Chairman: Prof. dr. ir. Luc Taerwe
Faculty of Engineering and Architecture
Ghent University
Academic year 2014-2015
Summary
In the context of the present master’s dissertation the software Composite Beam Dis-
trictwasdeveloped. CompositeBeamDistrictisasoftwarewhichcalculatestheinternal
forces and the deflections of a composite beam. The composite beam can be either sim-
ply supported or continuous with in theory an unlimited amount of spans.
Based on the internal forces and the deflections Composite Beam District will perform
a a number of verifications which are in accordance with EN 1994-1-1:2004. The verifi-
cations carried out in ultimate limit state are the resistance of the cross-section, which
includes a verification of moment resistance and the resistance to vertical shear, a ver-
ification for lateral torsional buckling and a verification of the shear connection. In
ultimate limit state both the total deflection due to permanent and line loads as well as
the additional deflection due to live loads are verified.
The only load types that can be specified in Composite Beam District are uniformly
distributed surface loads on the concrete flange, uniformly distributed line loads acting
directly on the steel profile and point loads.
Keywords
Composite Beam District - Composite beams - Software - Eurocode 4
ii
Development of a calculation software for
composite beams according to EC4
Sebastiaan Marynissen
Supervisor: Prof. ir. Rik Debruyckere

Abstract- Composite Beam District is a software tool which III. VERIFICATIONS
allows the designer to quickly enter a composite beam definition
Composite Beam District effectuates the following
and perform a set of basic verifications. The internal forces are
calculations and verifications in Ultimate Limit State:
calculated using the Direct Stiffness Method and the verifications
Steel profile Class— The Class of the steel profile is
are carried out according to EN 1994-1-1.
calculated in accordance with EN 1993-1-1 5.5.
Keywords- Composite Beam District, composite beam, software,
Plastic resistance moment— For composite beams
Eurocode 4
containing a steel profile of Class 1 or 2, the plastic resistance
moment is calculated in accordance with EN 1994-1-1 6.2.1.2.
I. INTRODUCTION Elastic resistance moment— For composite beams
containing a steel profile of Class 3, the elastic resistance

moment is calculated in accordance with EN 1994-1-1 6.2.1.3.
Composite beams typically consists out of a steel profile, a
Vertical shear resistance— The vertical shear resistance of
concrete flange and a connection between both. A typical
a composite beam is calculated in accordance with EN 1994-
composite beam is depicted in figure 1.
1-1 6.2.2.2.
Bending-shear interaction— Interaction between bending
and vertical shear is taken into account in accordance with EN
1994-1-1 6.2.2.4.
Lateral torsional buckling— Lateral torsional buckling of
composite beams is calculated in accordance with EN 1994-1-
1 6.4.2 using the inverted U-frame method.
Shear connection— The effects of a partial shear
connection on the resistance moment are taken into account in
accordance with EN 1994-1-1 6.6.
In Serviceability Limit State Composite Beam District

effectuates the following calculations and verifications:
Figure 1 Typical composite beam Calculation of the maximum total deflection δ —
max
While the use of composite beams yields a number of Composite Beam District calculates for each span the
advantages, their use is not widely adopted yet in Belgium. A maximum total deflection δmax and verifies that it is smaller
good and easy-to-use calculation software could stimulate than L/400.
their use. It is in this context that the software Composite Calculation of the maximum additional deflection δ2—
Beam District was developed. Composite Beam District calculates for each span the
Composite Beam District does not aim to replace fully maximum additional deflection δ2 due to live loads and
featured finite element software, but aims to be an aid in the verifies that it is smaller than L/500.
design process by allowing to quickly calculate and
subsequently adapt in an iterative way composite beam IV. RESTRICTIONS
definitions. The user should bear in mind that not all Steel profiles of Class 4 cannot be calculated and will result
verifications prescribed in EN 1994-1-1 are carried out and in an error shown to the user. Partial encasement of the steel
some verifications may still need to be elaborated afterwards. profile cannot be taken into account as well. Four types of
concrete slabs can be entered: a bare slab, a slab supported by
II. CALCULATIONS a profiled steel sheeting, a slab supported by precast panels
Composite Beam District calculates the internal forces of and a slab supported by hollow core slabs. Haunches cannot
composite beams in a linear elastic way by using the Direct be taken into account.
Stiffness Method. For the calculations the effect of cracking of The user has to be aware that the shear buckling of the web,
the concrete is taken into account by applying an appropriate transverse forces on the web and the maximum crack width of
bending stiffness. Both simply supported and continuous the concrete are not checked.
beams can be entered. Only uniformly distributed surface Only headed stud connectors can be taken into account.
loads acting on the concrete slab, uniformly distributed line
loads and point loads can be taken into account. iii
Fire resistance calculations are not carried out and the
construction stage is only taken into account for the
verification of the deflections.
V. USER INTERFACE
Composite Beam District comes with a graphical user Figure 6 Example of a moment line graph
interface which was developed with the philosophy that
sketches and figures play an important role.
The beam geometry and the loads can be entered by an
interactive view which consists out of two steps: the definition
of the beam geometry and the addition of the loads on the
beam. The first step is depicted in figure 2.

Figure 7 Example of a shear line graph
Figure 2 Drawing of the beam geometry
The addition of the loads on the beam is depicted in figure
3.

Figure 8 Example of a deflection graph

Figure 3 Addition of the loads on the beam As in most software input definitions can be saved to the file
system and reopened afterwards.
The properties of the steel profile and the concrete slab can
be entered by a number of input fields which will update the
VI. SYSTEM REQUIREMENTS
drawings accordingly.
Composite Beam District was developed for Microsoft
The definition and the distribution of the headed stud
Windows. A version for Mac OSX and Linux can be made
connectors over the beam can be entered using an interactive
available if enough interest is provided.
view as well. The definition of the used studs is depicted in
figure 4 and the entering of the distribution is depicted in
figure 5. ACKNOWLEDGEMENTS
The author would like to thank it’s supervisor for his
guidelines and support during the development.
REFERENCES

[1] EN 1994-1-1 Eurocode 4: Design of composite steel and concrete
Figure 4 Interactive view for the definition of the headed stud structures - Part 1-1: General rules and rules for buildings. CEN. 2004
connectors. [2] EN 1993-1-1 Eurocode 3: Design of steel structures - Part 1-1: General
rules and rules for buildings. CEN. 2005
[3] R. Maquoi, R. Debruyckere, J.-F. Demonceau, L. Pyl: Staal-
betonconstructies: toepassing en berekening van staal-
betonconstructies voor gebouwen volgens Eurocode 4 bij normale
temperatuur en brand. Infosteel. 2012
[4] D. Vandepitte, Berekening van constructies : Bouwkunde en Civiele
Techniek, Boekdeel III, Story Scientia, 1981

Figure 5 Interactive view for the definition of the distribution of the
headed stud connectors over the beam.
The user interface allows to specify the partial safety factors
as well.
The results of the calculations are displayed using graphs of
the reaction forces, the moment line, the shear line and the
deflection line. The graphs are interactive such that the exact
values can be read from these graphs. A few example graphs
are depicted in figure 6, 7 and 8.
iv
Contents
1 Introduction 1
1.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 General overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Applicability & limitations . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4.1 Sign conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4.2 Zero-based indices . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 The Direct Stiffness Method 5
2.1 Using the computer at its best . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Effectuation of the Direct Stiffness Method . . . . . . . . . . . . . . . . 6
2.3 Determination of the element stiffness matrix . . . . . . . . . . . . . . . 7
3 The beam calculator module 10
3.1 The solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1.1 Definition of nodes . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1.2 Definition of elements . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.3 Assembly of the system stiffness matrix . . . . . . . . . . . . . . 12
3.1.4 Rearrangement of the system stiffness matrix . . . . . . . . . . . 13
3.1.5 Solving the system . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1.6 Sparsity of the system stiffness matrix . . . . . . . . . . . . . . . 17
3.1.7 The universal applicability of the solver . . . . . . . . . . . . . . 17
3.2 The preprocessor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.1 Definition of a beam . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.2 Parsing the elements . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 The analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3.1 Numerical errors . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3.2 Shear line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.3 Moment line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.4 Moment zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.5 Vertical deflections . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.6 Exact calculation of the deflections . . . . . . . . . . . . . . . . . 30
4 The composite module 31
4.1 Transformation of a composite beam definition . . . . . . . . . . . . . . 31
4.1.1 Cross-sectional simplifications . . . . . . . . . . . . . . . . . . . . 31
4.1.2 Load transformations . . . . . . . . . . . . . . . . . . . . . . . . 32
v
4.1.3 Load configurations . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1.4 Effective width of a concrete flange . . . . . . . . . . . . . . . . . 35
4.1.5 Bending stiffness EI of a composite beam . . . . . . . . . . . . . 36
4.1.6 Type of analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 Processing of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2.1 Filtering of insignificant load configurations . . . . . . . . . . . . 38
4.2.2 Verifications in ultimate limit state . . . . . . . . . . . . . . . . . 41
4.2.3 Verifications in serviceability limit state . . . . . . . . . . . . . . 41
5 Ultimate limit state 42
5.1 Critical sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.2 Vertical shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.3 Bending resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3.1 Plastic resistance moment . . . . . . . . . . . . . . . . . . . . . . 44
5.3.2 Elastic resistance moment . . . . . . . . . . . . . . . . . . . . . . 46
5.3.3 Bending and vertical shear . . . . . . . . . . . . . . . . . . . . . 46
5.4 Longitudinal shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.4.1 Degree of shear connection η . . . . . . . . . . . . . . . . . . . . 47
5.4.2 Limitation on the use of partial shear connections . . . . . . . . 53
5.4.3 Class 3 profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.5 Lateral-torsional buckling . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.5.1 Note on the construction stage . . . . . . . . . . . . . . . . . . . 57
6 Serviceability limit state 58
6.1 Verification of the maximum vertical deflection . . . . . . . . . . . . . . 58
6.2 Calculation of the deflections . . . . . . . . . . . . . . . . . . . . . . . . 58
6.2.1 Initial calculation of v (x) . . . . . . . . . . . . . . . . . . . . . . 59
g
6.2.2 Calculation of v (x) . . . . . . . . . . . . . . . . . . . . . . . . 60
g+q
6.2.3 Recalculation of v (x) . . . . . . . . . . . . . . . . . . . . . . . . 61
g
6.2.4 Remark on the calculation of v (x) . . . . . . . . . . . . . . . . . 62
q
6.2.5 Numerical errors . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.3 Filtering of insignificant load configurations . . . . . . . . . . . . . . . . 63
7 Graphical User Interface 65
7.1 First run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.2 Welcome screen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7.3 Tab “General” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7.4 Tab “Geometry & Loads” . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7.4.1 Definition of the beam geometry . . . . . . . . . . . . . . . . . . 67
7.4.2 Definition of the loads . . . . . . . . . . . . . . . . . . . . . . . . 68
7.5 Tab “Cross section” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.5.1 Sub tab “Steel profile” . . . . . . . . . . . . . . . . . . . . . . . . 69
7.5.2 Sub tab “Concrete slab” . . . . . . . . . . . . . . . . . . . . . . . 70
7.6 Tab “Shear” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7.7 Tab “Safety factors” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.8 Tab “Results” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.8.1 Results overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.8.2 ULS details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.8.3 Critical sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
vi
7.8.4 SLS details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
8 Calculation examples 79
8.1 M+ of an IPE 300 profile . . . . . . . . . . . . . . . . . . . . . . . . . 79
pl,Rd
8.2 M+ of a HE 360A profile . . . . . . . . . . . . . . . . . . . . . . . . . 79
pl,Rd
8.3 M− of an IPE 300 profile . . . . . . . . . . . . . . . . . . . . . . . . . 79
pl,Rd
9 Used technologies 81
vii
List of Figures
1.1 Typical cross-sections of composite beams, extracted from EN 1994-1-
1:2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Sign convention for the shear force and the bending moment. . . . . . . 3
1.3 Sign convention for loads . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 Equilibrium of a node . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Free beam element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Relative and absolute rotation of a node . . . . . . . . . . . . . . . . . . 8
2.4 Clamped beam with uniformly distributed load . . . . . . . . . . . . . . 8
3.1 Degrees of freedom of a beam node . . . . . . . . . . . . . . . . . . . . . 10
3.2 Once statically indeterminate beam . . . . . . . . . . . . . . . . . . . . . 13
3.3 System stiffness matrix rearrangement algorithm . . . . . . . . . . . . . 16
3.4 Simple thermal network . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5 Isolated element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.6 Three spring system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.7 Two-span beam with a cantilever . . . . . . . . . . . . . . . . . . . . . . 21
3.8 Moments acting on the edges of elements e and e . . . . . . . . . . . . 25
1 2
3.9 Moment line for an element . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1 Base cross-section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Three-span continuous beam . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3 Two-span continuous beam . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4 Effective width b of a concrete flange, extracted from EN 1994-1-1:2004 35
eff
4.5 Cantilever with a point load . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.6 Moment lines for a point load of 1 kN . . . . . . . . . . . . . . . . . . . 39
4.7 Two-span continuous beam carrying a point load and a line load . . . . 40
4.8 Moment lines for all load configurations of figure 4.7 . . . . . . . . . . . 40
5.1 Example plastic stress distribution, extracted from EN 1994-1-1:2004 . 44
5.2 Reduction factor β, extracted from EN 1994-1-1:2004 . . . . . . . . . . 45
5.3 Influence of vertical shear on the resistance moment, extracted from EN
1994-1-1:2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.4 Relation between M and η, extracted from EN 1994-1-1:2004 . . . . 47
Rd
5.5 Profiled steel sheeting with stud connectors, extracted from EN 1994-1-
1:2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.6 Spreading of V (Maquoi et al., 2012) . . . . . . . . . . . . . . . . . . 49
l,Rd
5.7 Freed zone of longitudinal shear . . . . . . . . . . . . . . . . . . . . . . . 49
5.8 Load configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.9 Moment and shear line for example 5.4.1 . . . . . . . . . . . . . . . . . . 50
viii
5.10 Example continuous composite beam . . . . . . . . . . . . . . . . . . . . 51
5.11 Moment and shear line for example 5.4.2 . . . . . . . . . . . . . . . . . . 51
5.12 Load configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.13 Moment and shear line for example 5.4.3 . . . . . . . . . . . . . . . . . . 52
5.14 Load configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.15 Moment and shear line for example 5.4.4 . . . . . . . . . . . . . . . . . . 53
5.16 Inverted U-frame model, extracted from EN 1994-1-1:2004 . . . . . . . 55
5.17 Buckling length l to take into account . . . . . . . . . . . . . . . . . . . 56
6.1 Symmetric two-span continuous beam . . . . . . . . . . . . . . . . . . . 62
6.2 Two-span continuous beam carrying two point loads . . . . . . . . . . . 64
6.3 Deflections for all load configurations of figure 6.2 . . . . . . . . . . . . . 64
7.1 Start screen at the first run of Composite Beam District . . . . . . . . 65
7.2 Info on the association of .cbd file extensions . . . . . . . . . . . . . . . 66
7.3 Request for administrator rights . . . . . . . . . . . . . . . . . . . . . . 66
7.4 Welcome screen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7.5 Controls menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7.6 Dragging of the heart line . . . . . . . . . . . . . . . . . . . . . . . . . . 68
7.7 Entering of the span lengths . . . . . . . . . . . . . . . . . . . . . . . . . 68
7.8 Addition of a point load . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
7.9 Addition of a surface or line load . . . . . . . . . . . . . . . . . . . . . . 69
7.10 Parameters of the steel profile . . . . . . . . . . . . . . . . . . . . . . . . 70
7.11 Sub tab “concrete slab” . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.12 Entering of profiled steel sheeting . . . . . . . . . . . . . . . . . . . . . . 71
7.13 Entering of precast panels . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.14 Entering of a hollow core slab . . . . . . . . . . . . . . . . . . . . . . . . 72
7.15 Entering of the used stud connectors . . . . . . . . . . . . . . . . . . . . 72
7.16 Addition of stud zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.17 Modifying stud zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.18 Geometric limitations for the stud zones (Maquoi et al., 2012) . . . . . . 73
7.19 Overview of the verifications in ultimate limit state . . . . . . . . . . . . 74
7.20 Overview of the verifications in serviceability limit state . . . . . . . . . 75
7.21 Details of a load configurations in ULS . . . . . . . . . . . . . . . . . . . 76
7.22 Details of a critical section . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.23 Details of a load configuration in SLS . . . . . . . . . . . . . . . . . . . 78
7.24 Verification of δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2
9.1 Node.js . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
9.2 Old logo of node-webkit . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
9.3 GitHub’s mascotte Octocat . . . . . . . . . . . . . . . . . . . . . . . . . 82
9.4 Commit density during development . . . . . . . . . . . . . . . . . . . . 82
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