Master Thesis

DTU-TD — A User Interface for
Thin-walled Distortion
Yutong Zhu
MSc Thesis
Department of Civil Engineering
2013
DTU Civil Engineering
August 2013

I
Preface
This master thesis is written at DTU-BYG, Department of Civil Engineering in the Technical
University of Denmark, as a part of completion of the Master’s Degree in Civil Engineering. The
project corresponds to 35 ECTS points and is made during the period 01.02.2013-23.08.2013.
A user interface for thin-walled distortion has been developed and introduced in this project. The
supervisors are Professor Jeppe Jönsson and Associate Professor Michael Joachim Andreassen,
both affiliated to DTU-BYG. The work is based on the work done in the articles Sander (2013) and
Mygind (2013).
The MATLAB code of the user interface can be found in the CD-ROM together with the thesis. To
run the user interface, MATLAB environment is required. The program can be opened by run the
file “DTU_TD.m” in the main folder.
This thesis is submitted on 26th
of August 2013.
Yutong Zhu, s111373
Kgs. Lyngby
August 2013

III
Acknowledgements
Writing this thesis has been a challenging experience in my study life, at same time full of interest.
During the researching process my weakness in the theory of elastic mechanics has been
complemented in a certain level, and my ability of programing enhanced a lot which will be very
useful in the future works.
First and foremost, I would like to show my sincerely gratitude to Professor Jeppe Jönsson and
Associate Professor Michael Joachim Andreassen. I want to thank you for invested me into the
friendly research group and also thank you for the patient guidance and advice through all the way
of this project. Your passion in working and researching inspired me a lot in the study process.
Secondly, I would like to thank all the colleagues in the research group, Antonela-Flavia Achimp
Călin-Ioan Birdean, Ali Kazim Jawad Thari and Jeppe Majlund Kristensen, for the help in the past
seven months.
An extensive gratitude goes to my parents, for their unconditional support and encouragement.
Thank you for your education and love in the past twenty-four years.
Last, but not least, I am very grateful to my friend and roommate — Pengfei Lu. I want to thank you
for your disinterested help these months and all the delectable meals, which is such precious in a
foreign country thousand miles away from home.

V
Abstract
Since the use of thin-walled structural elements have been much increased during recent years
because of the significant high strength and cost-saving, more accurate calculation method is
required to deal with the complex structural behavior. In the last decades, Generalized Beam Theory
(GBT) has been developed as an effective approach in thin-walled elements analysis. In this master
thesis, a GBT-based user interface is introduced with: (i) a comprehensive explanation of the
background GBT theory. (ii) a detailed user manual and (iii) the results of application tests.
DTU-TD, which was designed to apply GBT to calculate the deformation and inner stress of
thin-wall beam elements, is mainly based on the method introduced in Sander (2013) and Mygind
(2013). In order to enable other researchers to reuse the user interface conveniently to test their
improved theory, DTU-TD was written using MATLAB Graphical User Interface Development
Environment (MATLAB GUIDE). The user interface related to this thesis is the original version
and named as DTU-TD 1.0α, which could calculate the deformations and stresses in cross-sections
of loaded homogeneous thin-walled single elements. The users are allowed to set up material
parameters, create custom cross-section profiles, identify different boundary conditions and load
conditions, save and export results in DTU-TD 1.0α. A detailed user manual is given in this thesis.
Some tests were made to verify the correctness and practicability of DTU-TD. The results have
been compared to hand calculations and results from ABAQUS. The tests improved that DTU-TD
1.0α could accomplish the calculation process for most situations and gave the users intuitive results.
It was also found that the shear contribution to deformations cannot be calculated correctly. This
problem is very obvious when dealing with short beams where shear deformations should have a
large contribution. Though distortional contributions are successfully included in this user interface,
further improvement of the background theory is required to obtain correct shear deformations and
stresses.
Disregarding the defect in shear calculation, the work with DTU-TD somehow provides a new
platform in the research area of GBT, which hopefully could make contributions to the
dissemination of thin-walled structures.

VII
Contents
Preface ................................................................................................................................... I
Acknowledgements ............................................................................................................... III
Abstract.................................................................................................................................V
List of symbols....................................................................................................................... XI
1 Introduction ....................................................................................................................... 1
1.1 Aim of work ..................................................................................................................... 2
1.2 Structure of this thesis ...................................................................................................... 2
2 Generalized Beam Theory..................................................................................................... 3
2.1 Basic Kinematic Assumptions.......................................................................................... 3
2.2 Displacement Fields ......................................................................................................... 4
2.3 Strains............................................................................................................................... 5
3 Potential Energy and Differential Equation........................................................................... 7
3.1 Potential Energy ............................................................................................................... 7
3.2 Interpolation functions...................................................................................................... 8
3.3 Local Stiffness Matrices................................................................................................... 9
3.4 Transformation from Local to Global Coordinates ........................................................ 10
3.5 Global Stiffness Matrices ............................................................................................... 11
3.6 Governing Differential Equation.................................................................................... 11
4 Solving the Differential Equation ....................................................................................... 15
4.1 Singularities in the Global Stiffness Matrices ................................................................ 15
4.2 Transformation of the Displacement Field Matrices...................................................... 15
4.3 Solving the Differential Equation................................................................................... 17
4.3.1 Step 1 – Eliminating ′′′′................................................................. 17
4.3.2 Step 2 – Doubling the Differential Equation........................................................... 18
4.3.3 Step 3 – Eliminating ′′......................................................................... 19
4.3.4 Step 4 – Solving the Eigenvalue Problem ............................................................... 19
4.3.5 Step 5 – Determining the Singularities................................................................... 20

VIII
4.3.6 Step 6 – Identification of the Two Unknown Modes.............................................. 22
4.3.7 Step 7 – Back Substitution...................................................................................... 23
5 Axial Variation Functions................................................................................................... 25
5.1 Solution for Pure Axial Displacement............................................................................ 25
5.2 Solution for Pure Bending.............................................................................................. 25
5.3 Solution for Pure Torsion............................................................................................... 26
5.4 Solution for Mode Shapes determined from ′′............................................... 27
5.5 Solution for Remaining Mode Shapes............................................................................ 27
5.6 Beam Element Stiffness Matrix...................................................................................... 27
5.6.1 Stiffness Matrix....................................................................................................... 27
5.6.2 Transformation Matrix ........................................................................................... 29
5.6.3 New Stiffness Matrix .............................................................................................. 31
6 Boundary Conditions and Generalized Displacement.......................................................... 33
6.1 Boundary Conditions...................................................................................................... 33
6.2 Generalized Displacement.............................................................................................. 34
7 Instructions of DTU-TD — A User Interface for Thin-walled Distortion................................. 37
7.1 Introduction to Matlab GUIDE....................................................................................... 37
7.2 Introduction to DTU-TD ................................................................................................ 38
7.3 User Manual of DTU-TD ............................................................................................... 38
7.3.1 Screen 1: Input Parameters.................................................................................... 39
7.3.2 Screen 2.a: Profile Topology (Typical) .................................................................... 41
7.3.3 Screen 2.b: Profile Topology (Custom)................................................................... 41
7.3.4 Screen 2.c: Profile Topology (Hand Draw).............................................................. 42
7.3.5 Screen 3: 3D-plot of Topology................................................................................ 44
7.3.6 Screen 4: Mode shapes........................................................................................... 44
7.3.7 Screen 5: Boundary Conditions .............................................................................. 45
7.3.8 Screen 6: Load on the ends .................................................................................... 46
7.3.9 Screen 7: 3D-deformation ...................................................................................... 47
7.3.10 Screen 8: Stresses................................................................................................... 48
7.4 Comments on DTU-TD 1.0α.......................................................................................... 49
8 Test of DTU-TD with Single Beam Elements........................................................................ 51

IX
8.1 Material Parameters........................................................................................................ 51
8.2 Static system................................................................................................................... 51
8.3 Shear and Bending Test with Closed Profile.................................................................. 52
8.3.1 Mode Shapes.......................................................................................................... 52
8.3.2 Displacements ........................................................................................................ 55
8.3.3 Stresses................................................................................................................... 56
8.4 Torsion Test with Closed Profile.................................................................................... 57
8.4.1 Displacements ........................................................................................................ 57
8.4.2 Stresses................................................................................................................... 58
8.5 Torsion Test with Open Profile ...................................................................................... 58
8.5.1 Mode shapes .......................................................................................................... 59
8.5.2 Displacements ........................................................................................................ 61
8.5.3 Stresses................................................................................................................... 61
8.6 Comments on the tests.................................................................................................... 62
9 Conclusion ........................................................................................................................ 63
Bibliography ........................................................................................................................ 65
Appendix I ........................................................................................................................... 67
Appendix II .......................................................................................................................... 69

XI
List of symbols
Hadarmard product
3 3 rigid body motion
* Singularities in
a Axial
A Area
A Square matrix used for boundary conditions
Topology matrix for node i and j
Cross sectional area of webs
b Width of cross section / profile
Length of element
c Centerline or a constant
c Vector containing constants used at boundary conditions
e Transformation between local and global coordinates
el Element
E Modulus of Elasticity
Plate stiffness
Unit direction vector in x-direction
Unit direction vector in y-direction
G Shear modulus
h Height of cross section / profile
H Transformation matrix
i Counter and node number
I Moment of inertia / Unit diagonal matrix
j Node number

XII
J Matrix used at Hadarmard product
k Local stiffness matrix
K Global stiffness matrix
̃ Reformulation of stiffness terms to enable analytical integration
Element stiffness matrix
̅ Stiffness term
̿ Stiffness term
L Length of beam
Lower matrix
n Local direction normal to the element
N Interpolation function
o Other degrees of freedom
P Point load
Vector containing loads at boundaries
s Local direction parallel to the element and placed in the centerline
t Thickness of an element
T Transposed
Transformation matrix to new degrees of freedom
Transformation matrix to assemble global stiffness matrix for element j
u Displacement
Boundary displacements
U Potential energy
V Degree of freedom describing a displacement or a rotation
w In plane displacements

XIII
x Global direction in plane
x-coordinate for node i
y Global direction in plane
y-coordinate for node i
z Local and global direction in the axial direction
Greek Letters
α Angle
γ Shear strain
δ Virtual variation
ε Normal strain
η Variation of displacement in axial direction along the element
λ Eigenvalue
ν Poisson’s Ratio
Ψ Matrix containing all axial solutions
̅ Square diagonal matrix containing all axial solutions
σ Axial and transverse stresses
τ Shear stress
Ω Displacement out of plane
Functions
w(s) Displacement of an element in plane
Ψ z Variation of in plane displacement along the element
Ψ’ z Variation of displacement in axial direction along the element
Ω s Displacement of an element out of plane (warping)

Introduction 1
Chapter 1
Introduction
With the development of new materials and building technology, thin-walled structural elements
has become more and more widely used in civil, mechanical and aerospace industry in the last
decades, due to the significant high strength and cost-saving. Meanwhile, the slenderness of these
structures leads to more complex structural behavior which requires more accurate calculation
approaches.
Nowadays, the most widely used approach in thin-walled structures calculation is finite element
method (FEM). There are many commercial FEM software products applied for structural analysis
in the industry. However, finite element method is often very time-consuming and unnecessary
when dealing with some structures having regular geometric sections and simple boundary
conditions. Thus, Generalized Beam Theory (GBT) was developed as an alternative method, which
can relatively shorten the calculation process for the ordinary thin-walled structures. (Andreassen,
2012)
With respect to promote the application of thin-walled structural elements in industrial area, a group
in Civil Engineering Department of Technical University of Denmark (DTU) has been committed
to improve Generalized Beam Theory these years under the leadership of Professor J. Jönsson and
M.J. Andreassen. In 2013, M. Mygind and L.B. Sander had developed a finite element program,
which includes flexural, torsional and distorsional deformation modes, based on prior research in
Jönsson and Andreassen (2010) and Jönsson and Andreassen (2012).
Based on the Mygind and Sander’s research, the author has developed a user interface, which is
named as DTU-TD, to calculate the deformation and inner stress of thin-wall beam elements. This
user interface is designed simple and visual, which enables users to set up material parameters, build
custom elements, identify different boundary conditions and load conditions, save and export
results. Now the original version of DTU-TD can only support the calculation of single elements,
which have loads at the ends.
Figure 1.1: Examples of thin-walled structural elements

2
In this thesis, a detailed introduction of DTU-TD is presented with the background theory, a user
manual and some tests with different thin-wall elements.
1.1 Aim of work
The design work of DTU-TD mainly has two purposes:
(i) Promote the dissemination of Generalized Beam Theory. The application of GBT in
industrial area is somehow limited due to lacking of GBT-based computer software. A
tool which engineers and students could use easily to calculate the behavior of
thin-walled elements with improved Generalized Beam Theory will certainly help the
spread of this novel approach. DTU-TD offers a simple and intuitive entrance for the
new GBT contacts. Users could calculate the deformations and stresses of some simple
thin-walled elements easily, even though they don’t have any related knowledge with
GBT or distortional mechanics.
(ii) Provide a platform for further research of Generalized Beam Theory. By only changing
the backend code and keep the user interface, other researchers can check the
applicability of their theory within different shapes of beam element conveniently.
According to this aim, DTU-TD is developed using Matlab GUIDE which is more
familiar for civil engineers and easy to rewrite.
1.2 Structure of this thesis
This thesis consists of three parts. At first, the background GBT theory is introduced in Chapter
2—6. The theory is presented comprehensively from the basic kinematic assumptions to the
differential equation and then finished with generalized displacement. So the readers could get an
overall understanding of Generalized Beam Theory without reference articles. Then a detailed
introduction of DTU-TD is presented in Chapter 7, where the function of each screen is explained.
At last, three tests are made to test the correctness and practicability of DTU-TD. The results and
discussion are given in Chapter 8.

Generalized Beam Theory 3
Chapter 2
Generalized Beam Theory
In order to solve the distortional displacement problem for a thin-walled beam profile, Generalized
Beam Theory has been developed based on a series of assumptions which will be discussed in this
chapter. In this thesis semi-discretization formulation is applied and the deformations are separated
into different displacement fields. That means the displacements can be expressed by the product of
cross-section displacement functions with finite elements and the axial variation functions which
are determined exact.
2.1 Basic Kinematic Assumptions
In Figure 2.1 a thin-walled beam element is placed in a global Cartesian ( ) coordinate system,
where the z-axis is in the longitudinal direction of the beam. The local coordinate system is shown in
Figure 2.1 as well, where s is used to indicate a curve parameter which runs through the entire
cross-section and n is used to indicate the direction perpendicular to s.
The cross-section is divided into several straight elements. Each element consists of two notes at the
ends and has 8 degrees of freedom, which enables the elements do in-plane or out of plane
deformations. In Figure 2.2 we can see that the in plane displacements are denoted with , while
the out of plane displacements are denoted with .
Figure 2.1 Components of the displacement vectors of a straight cross-section element
Figure 2.1 Local and global coordinate system

4
2.2 Displacement Fields
The in plane displacements of cross-section now can be described with the combination of
cross-section function s and s , which stand for the displacement of the element in n and s
direction respectively, together with the axial variation function z . The displacements at the
center line of the element could be found as
（2.1）
（2.2）
Considering the thickness t of the element, the displacement in the s-direction will vary a little due
to the rotation of element. In this case the displacement in the n-direction is assumed to be same
thought the entire thickness since the variation is too small which can be neglected. The
displacement is shown in Figure 2.3.
Figure 2.2 Local components of displacement
The comma in subscripts means derivatives, for example, s s. Then the displace-
ment in plane in both s and n are given as
（2.3）
（2.4）
In order to get the axial deformation, some assumption and calcution has been given in this chapter.
First , the central shear strain in s-z plane is calculated as
（2.5）
Then the equation can be integrated,
∫ ∫ （2.6）
For calculation purpose, it is assumed that the shear strain is equal to ̅ , where ̅ is
the constant shear strain through the thickness of element.
∫ ̅ ̅ （2.7）
Where which indicated the axial warping displacement mode has been included.

Generalized Beam Theory 5
∫ ̅ （2.8）
As mentioned before, the out of plane deformation will vary though the thickness due to the rotation
of element as in plane displacement (see Figure 2.3). Thus the displacement of the element out of
plane is given as
（2.9）
Thereby all three displacement field are found and list below
2.3 Strains
Using the determined displacement field, strains occur in the element can be calculated. According
to J. Jönsson (1995), linear strain tensor and shear can be determined by
（2.10）
（2.11）
Where . Using s, n and z to replace the i and j in Eq. (2.10) and (2.11), all 6 strains in the
element can be determined combine with Eq. (2.3), (2.4) and (2.9)
（2.12）
（2.13）
（2.14）
( ) ( )
) （2.15）
（2.16）

6
（2.17）
Thus, it can be seen that only three strains are non-zero strain. These strains are named as the axial
strain , the cross-section distortional strain and the shear strain .
（2.18a）
( ) （2.18b）
) （2.18c）

Potential Energy and Differential Equation 7
Chapter 3
Potential Energy and Differential Equation
In this chapter the potential energy of a single deformation mode is formulated based on the
semi-discretization system and strains introduced in Chapter 2.
In the following calculation we assume that the material is linear elastic with a modulus of elasticity
E and a shear modulus G. At the same time we assume a plate type elasticity modulus
, where is the Poisson’s ratio. Notice that because of numerical errors in Matlab homogeneous
differential equation cannot be solved correctly when Poisson’s effect is included. Thus in this
project, the Poisson’s effect between deformations in the cross-section and in the axial direction is
not taken into consideration.
3.1 Potential Energy
By neglect the Poisson’s effect, simple constitutive relations can be defined.
Then the potential energy can be formulated based on Eq. (3.1).
∫
We can expand the potential energy formulation by integrate though the thickness, t, the width of
each element, , and over the length of the beam, L.
∫ [∑ ]
Inserting Eq. (2.18) into Eq. (3.3),

8
∫
[
∑
(
[ ]
[ ( ) ]
[ ( ) ( ) ]
) ]
3.2 Interpolation functions
As shown in Figure 2.2, each straight cross-section element has eight degrees of freedom. Among
all the DOFs it is assumed that the axial displacements, Ω, are interpolated linearly and the
transverse displacement in the element direction will also be interpolated linearly. And in the
transverse direction, the displacements are interpolated cubically. The displacements in a straight
cross-section element are then interpolated as follows:
Here and are linear interpolation matrices and is a cubic interpolation matrix. is the
nodal displacement and is the displacements and rotations in the cross-section.
[ ] [ ]
[ ]
[ ]
The interpolation functions are determined by apply one unit displacement or rotation for one
degree of freedom at one time, and other DOFs are set up to zero, then the function can be found by
solving the boundary conditions. Figure 3.1 shows the six interpolation modes and the
corresponding boundary conditions.
[ ]
[ ]
[ ]

( ) ( )
( )
( ) ( )
( )
3.3 Local Stiffness Matrices
By combining Eq. (3.4) and Eq. (3.5), the element stiffness contributions to the axial strain, shear
strain and transverse strain energy now can be determined and are shown in Table 3.1.
Figure 3-1 Interpolated function boundary conditions

10
Table 3-3-1 Local element stiffness matrices
∫
∫
∫ ( )
∫ ( )
∫
∫
3.4 Transformation from Local to Global Coordinates
After the stiffness matrix for each element has been determined, it needs to be transformed to
the global system. Thus a transformation must be set up which is able to combine all the
elements regardless of their positions in the global coordinate system. Eq. (3.9) gives the
formulation of transforming from local to global coordinates.
Where k indicates one of the local stiffness matrices in Table 3.1 and indicates the transform
matrices. The index j can be substitute with w or Ω, which corresponds to the DOFs in plane or out
of plane respectively.
The transform matrices is determined from the coordinates of the start note and end note. If the
coordinates in the global coordinate system for the start note is (x1, y1) and the end note is (x2, y2), the
length of the element can be given as,
√
Then the direction of the element in global coordinate system can be found by sin and cos as well.
The found transformation matrices and are shown below. Notice that is a 6×6
matrix because the 6 DOFs in while is only 2×2 due to due to the two DOFs out of plane.

[
s
s
s
s
]
[ ] [ ]
3.5 Global Stiffness Matrices
Using Eq. (3.9), all the local stiffness matrices can be transformed to global coordinated system.
The total stiffness matrix will be the sum of all the elements’ stiffness each in a correct position.
Thus, topology matrix T is introduced base on the transform matrix in Eq. (3.12).
[ ]
Here and are part matrix from in Eq. (3.12), which is a unit diagonal matrix with the
dimension 1×1 when a transformation for the axial degrees is executed. When transform in plane
degrees of freedom and are 3×3 matrices as shown in Eq. (3.12). The column numbers or and
are decided by the number of start note and end note of the element.
∑
Table 3-2 Assembly in total stiffness contributions
∑
∑
∑
∑
∑
∑
3.6 Governing Differential Equation
Eq. (3.4) which describes the potential energy can now be rewritten by using the global stiffness
matrices in Table 3.2.

12
∫ [ ]
This can be arranged with matrix form
∫
[
[ ] [ ] [ ]
[ ] [ ] [ ]
[ ] [ ] [ ]
]
The differential equation can now be taken the first derivation of the expression with Taylor
formulation. Since the displacements in plane are relate to the variation and the displacements
out of plane are relate to , the expression is arranged by create and combine like terms with
( ) and ( ). Since the complete process is too lengthy, only the first stiffness matrix’s
expansion is given below as an example in order to be brief.
∫
∫ [ ]
∫ [ ] [ ]
And the whole differential equation expands into:
∫ [
( )
( )
]
[ ] [ ] [ ]
[ ] [ ]
To find the minimum value of the potential energy the first variation must be equal to zero, which
means the integration terms in Eq. (3.18) need to be zero. Since the material used is assumed to be
linear elastic and only small strains occur, the second variation will always be positive, which
means the inner variation of the displacement fields and must be equal to zero. Notice
that the expression multiplied by is change to an expression multiplied by , for
compare reason.
( )
( )

Eq. (3.19) can be rewritten into matrix form:
[ ]
⏞
[ ] [ ]
⏞
[ ] [ ]
⏞
[ ] [ ]

Solving the Differential Equation 15
Chapter 4
Solving the Differential Equation
The homogeneous differential equation from Chapter.3 need be solved when the constraint of the
strains in and out of plane is taken into consideration. The singularities in some of the matrices have
to be identified and eliminated before the system can be solved. Also it is not possible to solve an
equation which contains 3 parts using an eigenvalue problem. In the following chapter, the
mathematical steps which Sander (2013) and Mygind (2013) performed are elaborate briefly.
4.1 Singularities in the Global Stiffness Matrices
By solving the eigenvalue problem, singularities are found in the matrices , and . The
singularities can then be identified
 – Singularity occurs when a node is an end node or between two elements where the
direction vector of these two elements are parallel.
 – Contains 3 singularities which correspond to 3 rigid in plane body motions – pure
horizontal displacement, pure vertical displacement and pure rotation.
 – Contains 1 singularity which corresponds to a pure axial displacement.
The singularities represent the corresponding displacements do not contain any energy is the
system.
4.2 Transformation of the Displacement Field Matrices
The identified singularities now can be defined in the differential equation given in Eq. (3.20) by
making a new description of the displacement field. For this purpose, a transformation matrix
is introduced. The original displacement field is equal to the transformation matrix multiplied by the
new description of the displacement field.
[ ] [ ]
where contains the 3 degree of freedom corresponding to the singularities in , contains the
degrees of freedom corresponding to the singularities in and is correspond with the
singularity in which is a pure axial displacement. and contains the remaining in and out
of plane degrees of freedom respectively. The transformation matrix is given as
[ ]

16
where contains the 3 eigenvectors found for , contains the eigenvectors found for
and contains only ones, which correspond with the singularity in . Other degrees of
freedom that have not been coupled to a singularity are expressed by and .
Then the new stiffness matrices can be given as
and Eq. (3.20) can be rewritten into
[ ]
[ ]
[ ][ ]
[ ] [ ] [ ]
In Eq. (4.4), some transformation matrices which contain singularities are multiplied by the
corresponding stiffness matrices, and then the results come out to be zero. And a new superscript of
the stiffness matrices is applied to simply Eq. (4.4), which is adding the superscript of the
transformation after the original notation of the stiffness matrices, instead writing the full symbol of
the transformation matrices. For example

Thus, the Eq. (4.4) could be rewritten into a new form.
[ ] [ ]
[ ][ ]
[ ] [ ] [ ]
4.3 Solving the Differential Equation
4.3.1 Step 1 – Eliminating
Aiming at solving the homogeneous differential equation, several steps are executed in order to
determine the mode shapes.
As can be seen in Eq. (4.5) the 4th
row in 2nd
and 3rd
matrix only consists of zeros, so is
able to eliminated from the system.
where is only a number, not a matrix. Then substitute Eq. (4.6) into Eq. (4.5) and a new
stiffness matrix can be got
[ ̅ ][ ] [ ][ ]
[ ]
[ ]
[ ]

18
where
̅
4.3.2 Step 2 – Doubling the Differential Equation
The differential equation is doubled in order to eliminate more singularities.
[
̅
̅ ][ ]
[ ̅ ][ ]
[ ]
In the above equation both two matrices has all zeros in the 6th
row, that means in the system
does not produce any energy, so this row can be removed.
[
̅
̅ ][ ]
[ ̅ ] [ ]
[ ]

4.3.3 Step 3 – Eliminating
It can be seen that the 4th
row of the second matrix in Eq. (4.9) only contains zeros, so can
be eliminated by using other stiffness matrix to replace it. By extracting the 4th
row of Eq. (4.9), the
equation below can be get
( ) ̅
̅ ( )
Then Eq. (4.10) is substituted back to Eq. (4.9), the equation can be rewritten as
[
̿ ̿ ̿ ̿
̿ ̿ ̿ ̿
̿ ̿ ̿ ̿
̿ ̿ ̿ ̿ ] [ ]
[ ̅ ][ ]
[ ]
The definition of the new stiffness matrices introduced in Step 3 are given in Table 4.1
Table 4.1: Revised stiffness terms after Step 3 without the constrain
̿
̿
̿
̿ ̅
̿ ̿
̿
̿ ̅
̿ ̿
̿ ̿
̿
̿ ̿
̿ ̿
̿ ̿
̿ ̅ ̅
̿ ̿ ̅
4.3.4 Step 4 – Solving the Eigenvalue Problem
Now we can start to solve the differential matrix equation Eq. (4.11) to find the eigenmodes. Here,
as given in normal beam theory, the axial variation function along the beam is assumed to be
exponential form
（4.12）

20
where λ is an inverse length scale parameter which may be complex.
Inserting Eq. (4.12) to Eq. (4.11), it becomes the following eigenvalue problem:
[
̿ ̿ ̿ ̿
̿ ̿ ̿ ̿
̿ ̿ ̿ ̿
̿ ̿ ̿ ̿ ][ ]
[ ̅ ] [ ]
[ ]
By solving this eigenvalue problem, the corresponding eigenvalues and eigenvectors can be
determined. It can be seen that 5 eigenvalues are equal to zero. These values correspond to the
singularities in the system. Among them three are related to , and the other two values need also
to be identified and determined.
4.3.5 Step 5 – Determining the Singularities
By making a linear combination of the first 3 vectors determined from the eigenvalue problem the 3
mode shapes corresponding to the singularities in can be modeled, including two bending modes
and one torsion mode. The other two singularities are not easily found. In Sander (2013) and
Mygind (2013), they were identified as bending solutions which exactly equal to the two bending
mode shapes from , which means the bending mode shapes would appear twice.
4.3.5.1 Bending Modes
The two bending modes are indentified by using Eq. (3.20). Since a pure bending does not produce
any energy from and , these to stiffness matrices are singular for the modes. Thus
must be used for the identification of the bending modes.
The two bending shapes can be described as a unit in plane displacement in the x and y-direction.
[ ]
In order to determine , Eq. (4.10) is rewritten
(̅ ( ) )
(̅ ( ( ) ))

̅
Inserting Eq. (4.12) into upper equation, besides the eigenvalues correspond to these pure bending
shapes is equal to zero, can be get as follow
̅
( ) (̅ ( ) ( ))
In the same way, is determined as well
( ) ( )
Then
The displacement for pure bending shapes can then be represented as
[ ]
The eigenvalue problem is given as
( )
where
The two orthogonal eigenvectors corresponds to pure bending in the principle flexural direction can
be get as . By multiplying by , the eigenvectors can be transform from the subspace to the
full space. Then the two orthogonal pure bending mode shapes are given as
4.3.5.2 Torsion Modes
As mentioned before, the third singularity is assumed to be torsion. Here the same method as
determining the bending modes is used. The displacement for torsion is assumed to be

22
[ ]
However, the assumed displacement field given in Eq. (4.23) is set for a torsion mode rotates
around the point with the coordinates (0,0), which may not be the shear center of the cross-section.
Therefore the bending part needs to be subtracted in order to have the pure torsion mode.
where is a 2×2 matrix which expresses how much of each bending modes are subtracted from
. can be found by multiplying Eq. (4.24) by
( )
The new matrices given in Eq.(4.25) are defined in Table 4.2
Table 4.1: Revised stiffness terms after Step 3 without the constrains
Considering the guessed modes is pure rotation around the shear center, the coupling between the
bending modes and the rotation must be zero. Then can be found as
Inserting Eq. (4.26) back into Eq. (4.24), the pure torsion mode can then be found.
4.3.6 Step 6 – Identification of the Two Unknown Modes
4.3.6.1 Mode Shapes corresponding to
In Eq. (4.6) , which describes pure axial deformation, was eliminated. Here it is identified
ad substituted back as a solution for the differential equation.
For this mode shape transformation matrix is used, which corresponds to a unit deformation of
all nodes out of plane and the all in plane displacements are equal to zero at the same time. It can be
easily verified that the differential equation Eq. (5.20) is equal to zero after inserting this mode
shape. Therefore this mode shape can be present as one of the solution.

4.3.6.2 Mode Shapes corresponding to
In Eq. (4.10), was eliminated, here its corresponding mode shape needs to be determined
as well. It is assumed that for this mode shape there is no in plane deformation but only
deformations out of plane. And it is also assumed that is equal to zero.
Base on the above assumptions, we can get from Eq. (4.10) that
In order to obtain a subspace, the transformation matrix is used to multiple with .
The eigenvalue problem is solved for
The eigenvectors is transformed back as in Eq. (4.17) and Eq. (4.18)
The mode shapes corresponding to the elimination of then can be given as
[ ] [ ]
4.3.7 Step 7 – Back Substitution
Now all the solutions from Eq. (4.13) can be determined by making back substitution with the
identified mode shapes in the upper sections.
From Eq. (4.1) and Eq. (4.2), the in plane part of the mode shapes can be given as
Inserting Eq. (4.12) to Eq. (4.10),
̅ ( )
̅ ( )
It is mentioned above in Eq. (4.17) that

24
Then finally the warping contributions are found with Eq. (4.18)
Now all mode shapes are determined and they need to be checked whether they are orthogonal to
each other for the differential equation. This work has been done by Sander and Mygind, the
detailed process is not elaborated here. Further explanation can be found in Mygind (2013), Section
4.8 and Sander (2013), Section 6.4.

Axial Variation Functions 25
Chapter 5
Axial Variation Functions
After all the mode shapes are determined, the axial variation given as need to be determined as
well, in order to found the displacements through the whole element.
The mode shapes identified in the last section are all correspond to the singularities (i.e. ) and
the axial variation functions cannot be exponential form. In that case the axial variation functions
will be determined individually for these mode shapes. Then the axial variation functions for the
remaining modes which relate to non-zero are identified.
5.1 Solution for Pure Axial Displacement
First the axial variation functions for the mode shape corresponding to the elimination of ,
which represents a pure axial displacement of the cross-section, is determined.
In Eq. (4.6), was eliminated from the differential equation which were given
It is assumed that this mode shape only contains axial displacement and all nodes has a same
displacement, so is zero in the above equation. Which causes
Here a variation is introduced, is used instead of ,
The solution for can then be determined by integrating the expression given in Eq. (5.2) and in
matrix notation it is given as
[ ] [ ]
[ ] [ ]
5.2 Solution for Pure Bending
The axial solutions for pure bending are determined in the second. It is done by multiplying the
mode shapes of pure bending on both side of the differential equation, which is given as

26
[ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ]
Considering pure bending only produces energy in , Eq.(5.5) can be rewritten as
[ ] [ ] [ ]
The solution can then be determined by integrating the expression given in Eq. (5.6)
[ ] [ ]
where b in the subscript indicates the bending modes which correspond to mod shape 2 and 3.
5.3 Solution for Pure Torsion
The axial solution for pure torsion is determined as pure bending shapes, but the differential
equation is doubled before multiplied by the pure torsion mode shape .
[ ] [ ][ ]
[ ]
[ ]
[ ]
As the eigenvalue corresponding to pure torsion is equal to zero and pure torsion does not produce
energy in Eq. (5.8) can be rewritten into
[ ] [ ] [ ]
So the axial variation for pure torsion can be solved
[ ] [ ]

5.4 Solution for Mode Shapes determined from
The axial solutions for the mode shapes correspond to are determined using the same
method as pure torsion. By multiplying the mode shapes on both sides of the doubled differential
equation it gives
[ ] [ ][ ]
[ ]
[ ]
[ ]
As mentioned in Section 4.3.6.2 it was assumed that is equal to zero and is also zero
since this mode shape only contains displacements in the axial direction. So Eq. (5.11) can be
rewritten into
The only solutions for this mode shape are
That means the mode shapes determined from then elimination of are not real solutions
which can be used. The reason is in the differential equation (Eq. 3.20), , which is only
multiplied by zero, exists only for keep the matrices in the same order. So was exist for
mathematical requirement but do not has physical meaning.
5.5 Solution for Remaining Mode Shapes
The axial solutions for remaining modes corresponding to non-zero from solving Eq. (4.13) are
determined at last. As the eigenvalue is in the squared form of , so each mode has two
eigenvalues—a positive one and a negative one. So the axial solution for these mode shapes are
given as
[ ] [ ]
where j depends on the number of nodes in the cross-section.
5.6 Beam Element Stiffness Matrix
5.6.1 Stiffness Matrix
The solutions for the axial variation for all mode shapes given in the last sections need to be
combined now. First the solutions from Section 5.1—5.3 are collected in one matrix.

28
[ ]
[ ]
Insert Eq. (5.4), Eq. (5.7) and Eq. (5.10) into Eq. (5.14),
[ ]
[ ]
Then the solutions for the entire mode shapes determined from non-zero eigenvalue are collected
in a matrix. The number of mode shapes depends on the number of nodes in the cross-section.
[ ] [ ]
[ ]
[ ]
Then one matrix combines the 2 matrices given in Eq. (5.15) and Eq. (5.16) which includes all the
solutions for the differential equation can be get as
[ ] [ ]
Then the displacements in and out of plane can be given as

where is the constants which must be determined from the boundary conditions for the beam
element. and are matrices contain all the mode shapes determined in Chapter 4.
Inserting the displacements back into Eq. (3.16), the differential equation can be rewritten as
∫
[
[ ] [ ] [ ]
[ ] [ ] [ ]
[ ][ ] [ ]
]
Here it is simplified as
̃
where ̃ is the beam element stiffness matrix which is a assembly of all partial stiffness
contribution matrices, the identification of ̃ is given as
̃ ̃ ̃ ̃ ̃ +̃ ̃ ̃
The stiffness contribution matrices in Eq. (5.21) are defined in Table 5.1.
Table 5.1: Stiffness contribution matrices
̃ ∫ ̃ ∫
̃ ∫ ̃ ∫
̃ ∫ ̃ ∫
̃ ∫
5.6.2 Transformation Matrix
For easier calculating and programming, is transformed into a diagonal matrix which only
contains values in the diagonal. This new matrix is given as ̅ and defined by Eq. (5.22).
̅
Then the displacement field matrixes also need to transform into a new form.
̅
̅

30
Like the axial variation matrix , the transformation matrix also consists of two parts— and
, relate to the solutions for the singularities which given in Eq.(5.15) and the solutions from
eigenvalue problem which given in Eq. (5.16) respectively.
[ ]
where always is the same if the profile consists of more than 2 nodes is will be given as
[ ]
The size of depends on the number of nodes in the profile
[ ]
So Eq. (5.22) can be rewritten into
[ ] [ ] [
̅
̅ ]
When the transformation matrices are determined, the diagonal axial variation matrix can also be
determined.
̅ [
̅
̅ ] [ ]
The equation above can be split into two parts as
̅
[ ] [ ]

̅
[ ] [ ]
5.6.3 New Stiffness Matrix
With the diagonal matrix ̅ the stiffness matrices given in Table 5.1 can be rewritten to reduce
calculation work load. Since all the stiffness matrices must be integrated over the length of the
beam in the axial direction in order to find the stiffness, and only ̅ is a function of z, the stiffness
matrices can be rewritten. Here ̃ is given as an example
̃ ∫
∫ ̅ ̅
∫ ̅ ̅ ̅ ̅
̅ ̅ ∫ ̅ ̅
where is a matrix that only contains ones and has a size equal to ̅ and the mathematical symbol
represents the Hadarmard product.
All the stiffness terms given in Table 5.1 are rewritten as shown in Table 5.2
Table 5.2: Reformulated stiffness contribution matrices
̃ ̅ ̅ ∫ ̅ ̅ ̃ ̅ ̅ ∫ ̅ ̅
̃ ̅ ̅ ∫ ̅ ̅ ̃ ̅ ̅ ∫ ̅ ̅
̃ ̅ ̅ ∫ ̅ ̅ ̃ ̅ ̅ ∫ ̅ ̅
̃ ∫ ̅ ̅

Boundary Conditions and Generalized Displacement 33
Chapter 6
Boundary Conditions and Generalized
Displacement
6.1 Boundary Conditions
Having solved the eigenvalue problem and determined all the mode shapes, the constrains need to
be determined by set up the boundary conditions, and further the generalized displacement of the
beam element can be found at last.
From the first variation of the potential energy which given in Eq. (3.18), it can be seen that the
boundary displacements are consist of three parts – the axial displacement , the transverse
displacements and the axial derivative of the transverse displacements . The displacements
for all degrees along the beam can be thereby be given as
[ ] [ ]
where is used to eliminate the nodes correspond to the singularities when eliminating
from differential equation.
In order to determine the constants, we need to define the boundary displacements at both end of the
beam element, i.e. at and at where is the length of the beam element. Then the total
boundary displacement vector can be given as
[ ] [ ]
where is a an invertible square matrix. The first variation of the potential energy equation then
can be rewritten into
̃
By inserting Eq. (6.2) into Eq. (6.3), it can be get that
̃

34
Then the beam element stiffness matrix can then be defined as
̃
When apply the external load through the boundary conditions, the formulation is given as
where denotes the reaction vector for both two ends of the beam element.
6.2 Generalized Displacement
After boundary conditions and load conditions are defined, the total boundary displacement vector
and the ends’ reaction vector can be split into two parts. Because of the boundary constraints,
the displacements under the restrained condition is known and equal to zero, those restrained
displacements are assembled and defined as . While the other unrestrained displacements which
need to be determined are defined as . Then the displacement vector can be given in a new form,
[ ]
A transformation matrix is introduced to connect the new displacement vector with the original
one, which is given as
Then the new reaction vector and the new beam elements stiffness matrix can also be
given as
[ ]
[ ]
where and is the reactions relate to and respectively. is a square matrix which has
the same order with , and is a square matrix which has the same order with . Then
according to Eq. (6.6) the equation below can be get,
[ ] [ ] [ ]
As the load situation is identified, the reactions relate to unrestrained degrees of freedom at two ends
is known. From Eq. (6.11) the second equation is given as

Boundary Conditions and Generalized Displacement 35
where all parameters are known expect for the displacement . Since is an invertible matrix,
can be determined as
Then all the displacements at both ends are thereby determined and are given as
̅ [ ] [ ]
Then ̅ can given as
̅ ̅
As given in Eq. (5.18), the variation of the displacements along the beam can be express as
Combining Eq. (6.15) with Eq. (6.2), the variation of the displacements along the beam can be given
as
̅
̅
Then the deformation through the whole beam element is identified.

Instructions of DTU-TD — A User Interface for Thin-walled Distortion 37
Chapter 7
Instructions of DTU-TD — A User Interface for
Thin-walled Distortion
After introduced the background theory of beam’s distortion, this chapter is going to give an overall
introduction of the user interface – DTU-TD (acronym for “Denmark Technical University” and
“Thin-walled Distortion”), which developed with aim to let users to apply GBT conveniently to
calculate the deformation and inner stress of thin-wall beam elements when a distortion happens.
This user interface was created using Matlab GUIDE (GUI Development Environment), and mainly
based on edit the codes written by Sander (2013) and Mygind (2013), the hand-draw profile part is
adapted from a code written by Jeppe Majlund Kristensen. Those codes are rewritten and
rearranged to fulfill the functional demands.
An introduction to Matlab GUIDE is first present in this chapter. Then a user guide of DTU-TD is
introduced in detail screen by screen.
7.1 Introduction to Matlab GUIDE
A graphical user interface (GUI) is a graphical display in one or more windows containing controls,
called components, which enable a user to perform interactive tasks. The user of the GUI does not
have to write script or input commands in the command line to accomplish the tasks. So the user
does not need to understand programming or Matlab or the theory. (Matlab, 2013)
When the user manipulates a GUI, each control motion such as pressing a screen button, clicking a
mouse button, selecting a menu item, will active one or more specific written callback codes, which
could be scripts or functions, to achieve corresponding responds that the creator of the GUI defined
what the program do to react the control motion, while the user does not have to understand the
detail process.
GUIDE, the MATLAB Graphical User Interface Development Environment, provides a set of tools
for creating graphical user interfaces (GUIs). These tools greatly simplify the process of designing
and creating GUIs. GUIDE covers two kinds of tasks:
 Laying out of GUI
 GUI programming
As the traditional method to create GUIs is somehow complex and inconvenient, GUIDE is very
helpful when creating the GUIs, especially for the new starters of Matlab GUI design.

38
7.2 Introduction to DTU-TD
As mentioned in Chapter 1, for the purpose of developing the application of thin-walled structural
elements in industrial area, DTU Civil Engineering Department has been committed to improve the
general beam theory these years. J. Jönsson and M.J. Andreassen published the paper ‘Distortional
eigenmodes and homogeneous solutions for semi-discretized thin-walled beams’ and ‘Distoritional
solutions for loaded semi-discretized’ in 2011 and 2012. Based on these articles Mygind and Sander
had developed a finite element program, which includes flexural, torsional and distorsional
deformation modes, which composes the main part of the original codes of DTU-TD.
DTU-TD was born for two mainly purposes: 1. Develop a tool which engineers and students could
use easily to calculate the deformation of thin-walled elements with improved general beam theory,
even though they don’t have any related knowledge with GBT or distortional mechanics. 2. Enable
other GBT researchers to use or to rewrite this user interface conveniently, in order to check the
applicability of their theory within different shapes of beam element. Hence, Matlab GUI was
chosen to accomplish this task because of its widely used among civil engineers and the
convenience in edit process. The Matlab codes could also be compiled in to an executable file which
can be used in a computer without Matlab program.
With the aims mentioned above, DTU-TD 1.0α was developed by implements the formulations
present in Chapter 2 – Chapter 6. The program now can calculate the deformations and stresses in
cross-sections of loaded homogeneous thin-walled elements (both open cross-sections and closed
cross-sections). Users are allowed to draw profile shapes by themselves, or choose typical profiles
conveniently. So far, DTU-TD 1.0α can only apply on single element, which means the boundary
constraint and load can only be added at the two ends. The constraints can be set up for each end
node for each direction, so users can create all kinds of support conditions required, like simply
supported, fixed, or sliding supported. Same as boundary condition, load is also added to each end
node, four kinds of point load corresponding to four degrees of freedom of each node are provided
as options. The results of calculation can be saved as excel files and jpg image.
Figure 7.1: The interface of DTU-TD
7.3 User Manual of DTU-TD
DTU-TD 1.0α can be opened by run the file “DTU_TD.m” in Matlab.

DTU-TD 1.0α contains eight screens, including parameter input, profile topology, plot 3D-model,
mode shapes, boundary condition, load, plot deformation and stresses result.
Figure 7.2: The structure of DTU-TD graphical user interface
The flow chart shown in Figure 7.2 illustrates the work process of DTU-TD 1.0α. The detailed
operation method is explained below screen by screen.
7.3.1Screen 1: Input Parameters
The first screen, shown in Figure 7.3, contains five data-input boxes. In this screen users are request
to input some initial parameters, which respectively are:
(i) material elastic modulus
(ii) Poisson’s ratio
(iii) the thickness of beam elements
(iv) the length of beam elements and
(v) number of parts the beam length is split into
The first two parameters are defined as “Material Characters”, while the last three are defined as
“Geometry Parameters”.
According to the explanation in Chapter 3, we assume that the material is linear elastic. After the
users set up the value of elastic modulus E and Poisson’s ratio , shear modulus G and the plate
type elasticity modulus is calculated automatically using the equation below,

40
Because of numerical errors in Matlab homogeneous differential equation cannot be solved
correctly when Poisson’s effect is included. Thus in this project, the Poisson’s effect between
deformations in the cross-section and in the axial direction is not taken into consideration.
The parameter “L_part” means how many parts the length of the element will be split into, i.e. the
level of discretization in axial direction. The bigger number users input for this parameter, the more
complex 3D-model will be built, and the more accurate results will be obtained. But too large value
will affect the aesthetic of the figures and the calculation process will be slow as well. Notice, here
the “L_part” parameter can only accept positive integer, input negative values or zero will cause
mistakes in later calculation.
Users can change all parameter values back to default numbers by press “Default” button in the
bottom. The default values of all parameters are shown in Table 7.1.
After defined all initial parameters, users should press “Next” button to open next screen.
Table 7.1 Default values of parameters in Screen 1
Parameters Default value
E 210000 MPa
0.3
t 2 mm
L 100 mm
L_part 12
Figure 7.3: DTU-TD: General view of Screen 1

7.3.2 Screen 2.a: Profile Topology (Typical)
DTU-TD 1.0α provides three methods for profile creating process—typical, custom and hand draw
profiles. By using shift buttons in the top-left corner of Screen 2.x, users can change between the
three screens to choose the appropriate method.
Figure 7.4: DTU-TD: General view of Screen 2.a
The typical profiles are given in Screen 2.a. As shown in Figure 7.4, DTU-TD 1.0α has preset six
typical profiles as options—three closed profiles and three open profiles. The geometry and the
discretization are all set in the background codes and cannot be changed directly in this user
interface. Those profiles are set in convenience of testing and comparing the calculate result with
previous data in reference paper. Users can click the radio buttons to choose profiles, and the
detailed topology will be shown directly in the right axes with right geometry.
7.3.3 Screen 2.b: Profile Topology (Custom)
Screen 2.b, depicted in Figure 7.5, enable users to create the simple custom shape of cross-section,
which is rectangle closed profiles or rectangle open profiles with flanges. The detailed steps are:
(i) Choose profile’s type in “Type” box.
(ii) Input profile’s dimension: height, width and flange length. (The flange length box is
available only when the type “Open” is chosen.)
(iii) Input the number of divisions for each part of cross-section into the little boxes around
the axes. Notice here only positive integer can be used.
(iv) Press “OK”. Then the topology will comes out in the right axes.
The topology numbers of node are defined automatically in counterclockwise order, first the corner
nodes are defined, then the intermediate nodes. The elements are numbered counterclockwise as
well.

42
Users should remember to press “OK” every time after changed the type or the numbers to get the
right topology data.
Figure 7.5: Screen 2.b—Profile Topology (Custom)
7.3.4 Screen 2.c: Profile Topology (Hand Draw)
Users can also create profile shape by using the “Hand Draw” panel, which given in Screen 2.c. In
this screen, profiles can be built arbitrarily according to user’s demands. To create a hand draw
profile, users should follow the steps below:
(i) Input numbers of main nodes and elements. (Here “main nodes and elements” means
the nodes and straight sides before the profile is split by adding intermediate nodes, e.g.
a regular rectangular profile has four main nodes and four main elements.)
(ii) Use two “Zoom” buttons to adjust the axis range in the right axes, until the axis is
suitable for the profile. Then press “Start Draw” button, the mouse pointer will turn
into full crosshair type.
(iii) Place the main nodes on the axes by click appropriate points one by one. After all main
nodes are settled, the coordinate of the main nodes will be list in the right table.
(iv) Correct the points’ coordinate by directly click the table cells and edit. Make sure that
all coordinate are correct and press “Confirm”.
(v) Now the elements need to be set by linking the main nodes. To define one element,
users should choose the start node by click one main node and then choose the end node
by click once more on another node. Notice the start node and the end node should not
be the same node. For convenience’s sake, when users click a blank space system will
automatically choose the nearest node.

(vi) After all elements are placed, a popup window will appear to let users decide the
number of sub-divisions of each side, as shown in Figure 7.6. The default numbers is
the advice value calculated by program according to the length of elements, which
concerned to lead both reasonable calculate results and good-looking 3D-plot. Press
“OK” after input appropriate number for all sides.
Figure 7.6: The popup window in Screen 2.c
After all the steps, the profile topology will be finished and shown in the axes. An example of a hand
draw profile is given in Figure 7.7.
Figure 7.7: Screen 2.c—Profile Topology (Hand Draw)
It should be noted that due to the incompleteness of the background theory, some profiles with
specific shape, like a shape with all elements inclined, might cause errors in later calculation. One
reason is that more singularities will occur in the stiffness matrices within such specific

44
cross-sections, so the differential equation cannot be solved correctly with the method explained in
Chapter 4.
7.3.5 Screen 3: 3D-plot of Topology
After defining the shape of cross-section, the 3D –model of the element is shown in Screen 3, which
provides users a more intuitive image of the element. No operation is performed in Screen 3. Users
could use this screen to check the shape, dimension or discretization of the element.
Figure 7.8: Screen 3—3D-plot of Topology
7.3.6 Screen 4: Mode shapes
As the profile of cross-section is defined, the mode shapes can be calculated by solving the
homogeneous differential equation described in Chapter 4. The mode shapes are displayed in
Screen 4 with scaled displacements in and out of plane, in order to show the forms that the cross
section can deform into. Both in plane displacements and out of plane displacements consist of two
parts—the real part and the complex part, which are shown separately in different axes.
When Screen 4 is opened, the program will start to calculate the mode shapes and then display the
first mode shape on the screen. Since the amount of calculation is very large, this process may take
a little time, until the mode shape number in the middle top turns from “0” to “1”, which means the
whole calculate process is finished.
The mode shapes are sort in the order of ascending eigenvalues. Users could use button “<<” or
“>>” to switch forward or back to check other mode shapes, or jump directly to one mode shape by
input the mode shape number into the small box in the bottom and press “Go To”. When the mode
shape number in the middle top changed, that means the plot work is finished, and the specified
mode shape is shown correctly on the screen.

All mode shapes are scaled in a same manner in order to be easy to compare with each other. The
largest displacement in the mode shape is set to be equal to 1, and then all displacements at other
nodes are scaled proportionally. When a complex displacement part is plotted, the factor that the
complex displacements part is scaled by, compared to the real displacement part is given in the
middle of the figure. If no number appears then no complex part exists in the current mode shape.
It should be noted that the first four mode shapes which corresponding eigenvalues equal to zero are
determined as the singularities. As mentioned before in section 5.4, the mode shapes determined
from then elimination of are not real solutions, so they are not plotted in this step.
Figure 7.9: Screen 4--Mode Shapes
7.3.7 Screen 5: Boundary Conditions
Before the calculation of the deformations, the boundary conditions and the load have to be defined
by users. In DTU-TD 1.0α can only add boundary constraint and load on the two ends (The start end
where z = 0 is defined as “End 1”, while the end z = 0 is defined as “End 2”). At first, the boundary
conditions are defined through Screen 5. Screen 5 consists of two sub-screens— Screen 5.a and
Screen 5.b, corresponding to two ends of the element.
In Screen 5, the boundary condition are created by add constraints one by one to the certain nodes.
Initially, the cross section is defined as totally free, which means all the nodes can move in all
directions. In order to add a constraint, users should at first choose one constraint direction from the
panel. There are four directions as options, corresponding to the four degrees of freedom of one
node. And then press the button “Add” and add the constraint to one node by click on the figure.
Then a cross mark will appear beside the node, which means the node is locked in the chosen
direction. Users can also choose the last type — “Restrict all Dofs”, to fix the node in all directions
at once.

46
The button “Fix End” is used to add constraints on all nodes in all direction, which makes this end
totally fixed.
Users could eliminate all the constraints by click “Clean Chart” button.
The general view of Screen 5 is shown in Figure 7.10.
Figure 7.10: Screen 5—Boundary Conditions
7.3.8 Screen 6: Load on the ends
Screen 6 is quite similar to Screen 5, but instead of adding boundary restraints, users are required to
define the load conditions in these two screens. So the only difference in operation is, after choosing
the direction of load, users should input a value in the editable box, and then press “Add” button to
add the load on the node. Users can add a load in a counter direction of coordinate arrow by
inputting a negative value. Remember the unit here used for force is “N”, and for moment is “N∙m”
Figure 7.11: Screen 6—Load on the Ends

7.3.9 Screen 7: 3D-deformation
Since the boundary conditions and the loads are identified in Screen 5 and 6, the deformation of the
elements can then be calculated with the method explained in Chapter 6. Screen 7, which depicted
in Figure 7.12, is designed to display the result of the element’s deformation and output the
deformation data.
Figure 7.13: Screen 7 — 3D-deformation
The 3D-plot of element’s deformation is shown automatically when the screen opens, users can use
the six buttons around the figure to rotate the view. It should be noticed that the displacements
shown in the 3D figure are not the real values but scaled in order to observe easily. The scale factor
is shown in the editable box below the figure and can be changed by the users to obtain an
appropriate view.
Figure 7.13: “save as” window in Screen 7

48
The displacement data for each node in one cross-section are listed in the right table. Users can
change the position of the cross-section with the slider below, e.g. users can slide the square to the
place where z = 0.5L, and the displacement data of the cross section at middle point of the element
will be shown in the table.
DTU-TD 1.0α allows users to output the data of deformations and stresses into “.xls” files. When
press the button “Save”, a popup window will appear which let users to choose the destination
folder and input the file name. In the drop-down menu of “Save as type”, users can also choose
“.jpg” (see Figure 7.13), then the 3D-deformation figure will be saved as a jpg picture.
7.3.10Screen 8: Stresses
The last screen is used to output the result of stresses. Same as the deformations, the stresses are also
displayed within a chosen cross-section. And users could change the position of the cross-section
with the slider and change the plot scale by using the editable box.
The stresses are determined by using the expressions for the strains given by Eq. (2.18) and
assuming that the beam is linear elastic.
( ) （7.1a）
( ) （7.1b）
) （7.1c）
By inserting Eq. (3.5) and the solutions for the axial variation into the equations above, the
stresses can be given as
( ) （7.2a）
( ) （7.2b）
) （7.2c）
The stresses are calculated in the outer fibers of the cross section, i.e. for . While
and are calculated in the middle of the beam where .
It should be noticed that some differences in the stresses for the same node may occur in the result.
That is because the element formulation states that the degrees of freedom for an element must be
continuous and the different element must have the same displacement in the node they are
connected to, but the derivation of the degrees of freedom are not forced to be continuous so the
stresses are not ensured to be continuous.
The stresses data for each element in the chosen cross-section are listed in the left table. Users could
also save the stresses data and figures, with the same method as saving displacements result.

Figure 7.14: Screen 8 – Stresses
7.4 Comments on DTU-TD 1.0α
This chapter introduced the user interface DTU-TD 1.0α, which performs deformation and stresses
calculation for thin-walled elements by implements the theory described in previous chapters. The
first version need to work under Matlab environment and allows the users to create single elements
with custom profiles and define the boundary constraints and loads condition at the ends. The whole
calculated process is accomplished through eight screens. All the screens are designed to be very
simple to use and the main functions are fulfilled, which enables the new starters for GBT to apply
the theory to calculate some simple models.
Due to the limited time, the modeling for multi-elements, which is highly desired, is not included in
this version. Hope that this function will be added into the program in the near future.
In the next chapter, some examples with different shapes of cross-section and different loads are
modeled to test the correctness of this program.

Test of DTU-TD with Single Beam Elements 51
Chapter 8
Test of DTU-TD with Single Beam Elements
In the last chapter, the user interface for thin-walled distortion — DTU-TD has been introduced
with a detailed user manual. Then some examples are made to test the correctness and practicability.
The results calculated by DTU-TD are compared with the results from Sander (2013) and Mygind
(2013), the ABAQUS results from Nielsen (2012). In this chapter, the results of three tests are given,
including displacements and the stresses, followed with comments and discussion.
The examples are all single elements with point loads placed at the end of the elements because then
only the homogeneous differential equation needs to be solved. First, a beam element with closed
profile is test under vertical loads condition. The beam is test with different lengths since according
to normal beam theory for short beams shear deformations will be domination while for longer
beams bending deformations will be dominant. The next two tests are torsion test with a closed
profile and an open profile respectively.
8.1 Material Parameters
In the flowing tests the material parameters used are all the same as given in Table 8.1. The
thickness of the elements is also same for easier comparison. However the other geometry
parameters may vary among the tests, these parameters will be given in the respective test later.
Table 8.1: Material parameters used for the tests
E
[MPa]
Es
[MPa]
G
[MPa]
ν
[─]
t
[mm]
210∙103
231∙103
80.8∙103
0.3 2
8.2 Static system
The static system for the beam elements applied in the tests is shown in Figure 8.1. The beam will be
considerate as a cantilever, which is restrained in the end z = 0 (defined as End 1) and free in the end
z = L (End 2). The loads will be added on the free end.
Figure 8.1: Static system for the test

52
8.3 Shear and Bending Test with Closed Profile
First a shear and bending test are made for a closed profile. The profile geometry, the number of
nodes and elements, and the load applied can be seen in Figure 8.2.
Figure 8.2: Bending and shear test for closed profile
The total load applied in this case is 40 kN. In order to observe global deformation, the point load is
distributed to all the vertical side notes. The loads for the 4 corner nodes are equal to 1 kN, while the
loads for the other side nodes are equal to 2 kN.
This profile is tested twice with two different element lengths, 50 mm and 200mm respectively.
8.3.1Mode Shapes
First the mode shapes are determined by DTU-TD. In total 182 mode shapes are determined for this
example, in Figure 8.3 and 8.4 the 16 mode shapes with the smallest eigenvalues can be seen for
respectively the displacements in and out of plane.
As can be seen from Figure 8.3 and 8.4, the first 4 modes are seen to be conventional beam modes,
among them mode 1 is corresponding to the axial mode, mode 2 and 3 corresponds to the bending
modes and mode 4 is the rotational mode. The remaining 12 are either global or local distortional
modes.

(a) Real part
(b) Complex part
Figure 8.3: In plane displacements for closed profile

54
(a) Real part
(b) Complex part
Figure 8.4: Out of plane displacements for closed profile

8.3.2Displacements
The results of deformation calculation of the beam can be seen in Figure 8.5 for the two different
lengths. The displacements are scaled differently in order to see the shape of the deformations
clearly. The scale factor for deformation of short beam is 250, while in long beam case it is 50.
(a) L=50 mm (b) L=200 mm
Figure8.5: Deformations for bending and shear test with closed profile
In Figure 8.5, it can be seen obviously that the deformation shape of the short beam forms a s-shape
in the free end, while the long beam mainly performs bending deformations. It is as expected since
the short beam should contain more shear deformations. The deformations are compared with the
results in Sander (2013) and Nielsen (2012), and the shapes match very well.
Then the detailed displacement values are taken into comparison. The vertical displacements of the
middle node of the right side (Node 13 in Figure 8.2) is picked out to do the comparison with the
hand calculated result in Sander and Mygind (2013) and the ABAQUS result in Nielsen (2012). The
results are given in Table 8.2.
Table 8.2: Deformations for shear and bending test with closed profile
Node Number
Beam length L
[mm]
uy,b
[mm]
uy,s
[mm]
uy,t
[mm]
uy,D
[mm]
uy,A
[mm]
13 50 0.0095 0.0619 0.0714 0.0078 0.0747
13 200 0.610 0.248 0.858 0.554 0.869
In Table 8.2, uy,b represents the deformation due to bending and uy,s is shear deformation. The total
deformation is given as uy,t which is the sum of the former two deformations. uy,D and uy,A are the
deformation results calculated by DTU-TD and ABAQUS respectively.
From the comparison it can be seen immediately that for both situations the deformation determined
by DTU-TD is close to the theoretical deformations due to bending. And the difference between
DTU-TD and ABAQUS result is almost equal to the shear deformation. However, the results
calculated with GBT in Nielsen (2012) is almost agree with the ABAQUS results, so it can be

56
concluded that something is missing in the present formulation which is most likely to be the shear
contribution.
According to Sander and Mygind (2013), the reason for the large deviations probably is the
identification procedure of conventional beam mode in Section 4. In Nielsen (2012), two more
shear modes are determined besides the bending modes, which take more shear deformations into
account.
8.3.3Stresses
Lastly, the stresses results for both length cases are given in Figure 8.6. The stresses in determined
at a distance of 0.1L from the free end.
(a) L = 50 mm
(b) L = 200 mm
Figure 8.6: Stresses for bending and shear test with closed profile
The σ -stresses are determined in the outer fibers where , in the results it can be found that
the curve for both length situations shows obvious discontinuities. Besides the reason mentioned
before in Section 7.3.10 which is imperfect element formulation, another reason could be the
singularities caused by the applied loads.
Overall, the shape of stress distribution for σ and σ is seen to be as expected. The negative
stresses almost counterbalance the positive stresses, and in the middle height of the cross-section the
stress are equal to zero. That is because the top flange and the upper half of the webs are in tension
condition and there is compression in the lower flange and lower half of the webs.

Like the shear deformation, the shear stresses seem to be not correct as well. The value is much
smaller than expected, and the shape shows significant discontinuities. This supports the conclusion
got from deformation results that shear is formulated incorrect in the background theory.
These stresses are given in neither Sander and Mygind (2013) nor Nielsen (2012), so they cannot be
compared.
8.4 Torsion Test with Closed Profile
A torsion test in now made for a closed profile to see how DTU-TD works in torsion situations. The
profile geometry, the number of nodes and elements, and the load applied can be seen in Figure 8.7.
Figure 8.7: Torsion test for closed profile
In this test load is added to every node in order to present a pure torsion situation. The loads applied
on the corner nodes are 20 kN and for other nodes the value is 40 kN.
The mode shapes determined for this test is similar to the result in Section 8.3.1, so it is not repeated
here.
8.4.1Displacements
The deformation of the beam is illustrated in Figure 8.8.
Figure 8.8: Deformation for torsion test with closed profile
The shape of deformation of the beam given in Figure 8.8 is seen to be in accordance with
expectation. All sides of the cross section deforms in the same manner. The displacements of four
corner nodes are given in Table 8.3 together with the ABAQUS result from Nielsen (2012).

58
Table 8.3: Deformations for torsion test with closed profile
Node
ux,D
[mm]
uy,D
[mm]
ux,A
[mm]
uy,A
[mm]
Corner nodes 12.33 12.33 12.38 12.38
The comparison result comes out to be quite well. The deviation between displacement result from
DTU-TD and ABAQUS is 0.4% which means that the performance of DTU-TD is reasonable when
subjected to pure torsion loads. The missing contribution from the shear forces does not have effect
in torsion deformations.
8.4.2Stresses
Figure 8.9: Stresses for torsion test with closed profile
The stresses calculated by DTU-TD are shown in Figure 8.9. From the result it can be seen that in
this case σ and σ is almost zero compare to the shear stress τ. This result is reasonable since only
torsion deformation occurs in this test.
8.5 Torsion Test with Open Profile
In the third test, a cross section which is an open c-profile with flanges is subjected to two horizontal
forces in the flanges, to check distortional performance of DTU-TD. The profile geometry, the
number of nodes and elements, and the load applied can be seen in Figure 8.10.
It can be seen form Figure 8.10 that two horizontal loads are applied on the flanges in opposite
directions. Each load is 0.1 kN.
Figure 8.10: Torsion test for open profile

8.5.1Mode shapes
In total 128 mode shapes are determined for this example, in Figure 8.11 and 8.12 the 16 mode
shapes with the smallest eigenvalues can be seen for respectively the displacements in and out of
plane.
(a) Real part
(b) Complex part
Figure 8.11: In plane displacements for open profile

60
(a) Real part
(b) Complex part
Figure 8.12: Out of plane displacements for open profile

8.5.2Displacements
The deformation of the beam is illustrated in Figure 8.13.
Figure 8.13: Deformation for torsion test with open profile
The shape of deformation of the beam given in Figure 8.13 is seen to be in accordance with
expectation. Both two flanges tilts outwards and the deformations are seen to decrease along the
beam towards the restrained end. The displacements of four nodes (Node 1, 5, 6, 10) are given in
Table 8.4 together with the ABAQUS result from Nielsen (2012).
Table 8.3: Deformations for torsion test with open profile
Node
ux,D
[mm]
uy,D
[mm]
ux,A
[mm]
uy,A
[mm]
1 0.000 0.115 0.000 0.119
5 -0.666 0.455 -0.686 0.469
6 -0.665 0.115 -0.685 0.119
10 0.000 0.352 0.000 0.386
The comparison result comes out to be reasonable. The deformations of all 4 nodes determined by
DTU-TD are very close to those given by ABAQUS. The largest deviation occurs in the middle
point of bottom plane in y-direction which is 8.8%. The reason of this deviation can be the fact that
is used instead of and thereby makes the beam stiffer. Another reason could be the missing
contributions of shear as mentioned before.
8.5.3Stresses
The stresses calculated by DTU-TD are shown in Figure 8.14. The stresses are determined at a
distance of 0.1L from the free end. From the result it can be seen that all the stresses show symmetry
about midline as expected. The largest σ s ss occurs in the bottom flange where element
bends due to the moment. While the largest σ -stress happens in the end of flanges where also
performs the biggest displacement.

62
Figure 8.14: Stresses for torsion test with closed profile
8.6 Comments on the tests
In this chapter, three tests were made to test the correctness and practicability of DTU-TD. Different
load cases were applied with both open and closed profiles to test the performance of this program
under bending and shear situations and torsion situations. The mode shapes, deformations and
stresses results were given for each test. The results of the two torsion tests showed highly similarity
with the ABAQUS results in Nielsen (2012). But the deformations result of bending and shear test
came out to be too small than expected. It was found out that the contribution to the deformation due
to shear was missing, which would have more influence in the short beam where shear deformation
should have more contribution. The stresses were also determined and for most cases the results
were reasonable, if the discontinuities are ignored. Again the shear stresses were found presented
incorrect because the defect in formulation.
Another two torsion tests with open profile were also made for comparison, the results can be found
in Appendix 2.

Conclusion 63
Chapter 9
Conclusion
Based on the Generalized Beam Theory presented in Mygind (2013) and Sander (2013),
DTU-TD— a user interface for thin-walled distortion was developed and introduced in this thesis.
DTU-TD was designed using Matlab GUI with respect to dissemination of Generalized Beam
Theory and at the same time offering a tool helping the researchers to test their theory conveniently.
The version presented in this thesis was named as DTU-TD 1.0α and enabled to calculate the
deformations and stresses in cross-sections of loaded homogeneous thin-walled single elements.
DTU-TD 1.0α contains eight screens, including parameter input, profile topology, plot 3D-model,
mode shapes, boundary condition, load, plot deformation and stresses result. All the screens were
designed in conciseness so can be understood by new users rapidly. Tests are made with different
load models and improved that DTU-TD 1.0α could accomplish the calculation process and gave
the users intuitive results. But the results for the deformations not always were as expected when
they were compared to hand calculations and results found in the commercial FEM-program
ABAQUS. The reason was found to be the defect of the background theory in representation of
shear contribution. This drawback is very obvious when dealing with short beams where shear
deformations should have a large contribution.
Due to the inaccurate representation of shear contribution, the based theory is considered
incomplete. Meanwhile this original version is also lacking a lot of functions such as the modeling
for multiple elements and distributed loads. However, DTU-TD 1.0α still provides a good start of
the application of Generalized Beam Theory and a ideal platform for further development.
For further research, the flowing questions deserve more attention:
(i) Including the Poisson effect into consideration is highly desired. In order to include the
Poisson effect, new displacement field and new differential equation need to be
identified.
(ii) The problem of incorrect representation of shear contribution to deformation need to be
solved. A start point may be trying to improve the identification procedure when
solving the differential equation.
(iii) In DTU-TD 1.0 α only single elements can be calculated so the loads can only be added
at the ends of beam elements. However, multiple elements modeling is very necessary
which will enable the program to apply loads in the middle cross-section, thereby
expending the capabilities of the program.
(iv) Distributed loads need to be added into the program besides point loads. To do this, the
inhomogeneous differential equation should be solved as well and the determined
solution should then be added to the homogeneous solution.

64
(v) Due to limitation of the present version of Matlab, DTU-TD 1.0a cannot be compiled to
an executable file. Anyway, in the future this work should be done in order to attract
more users to apply Generalized Beam Theory during their works.
(vi) The backend code and the user interface can never be good enough for users. The
screens need to be designed more user friendly and the code could also be improve to
enhance the running speed.

Bibliography 65
Bibliography
[1] Jönsson, Jeppe. Continuum Mechanics of Beam and Plate Flexure. Aalborg University, 1995.
[2] Andreassen, Michael J. Distortional Mechanics of Thin-Walled Structural Elements. DTU,
2012.
[3] Nielsen, Michael Teilmann. Avancerede bjælkeelmenter med tværsnitsdeformation. Master’s
thesis, Technical University of Denmark, 2012.
[4] Sander, Lotte Braad. Beam Theory and Modeling of Distortion. Master’s thesis, Technical
University of Denmark, 2013.
[5] Mygind, Martin. Advanced Beam Elements with Distorting Cross Sections. Master’s thesis,
Technical University of Denmark, 2013.
[6] MATLAB®
Creating Graphical User Interfaces. The MathWorks, Inc. 2013

67
Appendix I
Local Stiffness Matrices
[ ]
[ ]
[ ]
[ ]
[ ]

69
Appendix II
Torsion Tests with Open Profile
Test 1
Figure II.1: Torsion test 1 with open profile
Figure II.2: Deformation of torsion test 1
Table II.1: Deformations for torsion test 1
Node
ux,D
[mm]
uy,D
[mm]
ux,A
[mm]
uy,A
[mm]
1 -0.066 -0.144 -0.071 -0.152
5 0.120 -0.560 0.116 -0.575
6 0.120 -0.145 0.116 -0.153
10 -0.066 0.000 -0.070 0.000

70
Figure II.3: Stresses of torsion test 1
Test 2
Figure II.4: Torsion test 2 with open profile
Figure II.5: Deformation of torsion test 2

71
Table II.2: Deformations for torsion test 2
Node
ux,D
[mm]
uy,D
[mm]
ux,A
[mm]
uy,A
[mm]
1 -0.000 -0.245 0.001 -0.272
5 0.455 -0.754 0.469 -0.822
6 0.455 -0.246 0.469 -0.272
10 0.000 -0.370 0.000 -0.408
Figure II.6: Stresses of torsion test 2

DTU Civil Engineering
Department of Civil Engineering
Technical University of Denmark
Brovej, Building 118
2800 Kgs. Lyngby
Telephone 45 25 17 00
www.byg.dtu.dk

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