difficulties in fe-modelling of an i- beam subjected to torsion ...828098/fulltext01.pdfiii...

DEGREE PROJECT, IN STEEL STRUCTURES, SECOND LEVEL

STOCKHOLM, SWEDEN 2015

Difficulties in FE-modelling of an I-beam subjected to torsion, shear and bending

MIRIAM ALEXANDROU

KTH ROYAL INSTITUTE OF TECHNOLOGY

SKOLAN FÖR ARKITEKTUR OCH SAMHÄLLSBYGGNAD

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Difficulties in FE-modelling of an I-beam

subjected to torsion, shear and bending

Miriam Alexandrou

June 2015

TRITA-BKN. Master thesis 464, KTH 2015

ISSN 1103-4297

ISRN KTH/BKN/B--464--SE

iv

©Miriam Alexandrou 2015

Royal institute of technology (KTH)

Department of Civil and Architectural engineering

Division of Structural design and Bridges

Stockholm, Sweden, 2015

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ABSTRACT

In this thesis six different models of IPE240 have been created in order to study their behavior under

shear, bending and torsion. These models simulate IPE240 but differ in the boundary conditions, in

the loading and the length of the beam and in some connections which connect certain elements. In

this study the modeling and simulation of the steel member is executed in ABAQUS Finite Element

Analysis software with the creation of input files. When developing a model for the finite element

analysis a typical analysis process is followed. All the parameters that are required to perform the

analysis are defined initially to geometry which is half the beam due to symmetry, and the material

properties of each model are defined too. Then a mesh is generated for each model, the loads of each

model are applied which are expressed as initial displacement. Subsequently, the boundary conditions

for each model are defined and finally the model is submitted to the solver when the kind of analysis

has been defined. Namely, the analysis which is performed in this thesis is static stress analysis.

When the ABAQUS has run the models, the contour plots for the von Mises stresses for each model

are studied. In these contour plots, a large concentration of stresses and problems which arise in each

one of the models are notified. As it has been observed in all models, the beam yields at the flanges

of the mid-span and collapses at the mid-span. Therefore, the failure at the mid-span is more critical

than the failure at the support. Moreover, the beams are weak in bending due to the fact that they

twist almost 60-90 degrees under a large initial displacement at the control node. Additionally, much

localized failure and buckling occurred at the mid-span, and local concentrated stresses also occurred

at the bottom flange at the support due to the boundary conditions details.

Thereafter, a verification of the results of the ABAQUS through the simple analytical hand

calculations is performed. It is concluded that the error appearing in most selected points is small.

However, in some points in the web of the mid-span the error is greater. Additionally, while

comparing the load-displacement curves of the two different plastic behaviors, it is observed that the

model with an elastic-plastic with a yielding plateau slope behavior has smaller maximum load

resistance than the model with a true stress-strain curve with strain hardening behavior.

Finally, some errors and warning messages have occurred during the creation of the input files of the

models and a way of solving them is suggested.

June 2015

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ACKNOWLEDGEMENTS

This work was carried out under the supervision of Bert Gunnar Norlin, University Lector at the

KTH Royal Institute of Technology, School of Architecture and the Built Environment. I would like

to express my deep gratitude to him for the help, expert instruction, guidance and support he has

been providing throughout the project.

Finally I will always be grateful to my family for their support throughout the course of my studies.

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CONTENTS

ABSTRACT ................................................................................................................................ v

ACKNOWLEDGEMENTS ...................................................................................................... vii

LIST OF FIGURES .................................................................................................................... xi

LIST OF TABLES ..................................................................................................................... xii

LIST OF SYMBOLS .................................................................................................................. xii

LIST OF ABBREVIATIONS .................................................................................................... xii

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

1.1. Background ................................................................................................................ 1

1.2. Scope of the study ...................................................................................................... 1

2. LITERATURE REVIEW ..................................................................................................... 3

2.1. Shear, bending and torsion ........................................................................................ 3

2.2. Stiffeners ..................................................................................................................... 5

2.3. Elastic and plastic behavior ....................................................................................... 5

2.4. FEM modeling ........................................................................................................... 7

2.4.1. Shell, beam and truss elements .......................................................................... 8

2.4.2. Multi-Point Constraints ...................................................................................... 9

3. FINITE ELEMENT MODEL DEVELOPMENT ............................................................. 11

3.1. Model definition and material properties................................................................. 11

3.2. Generated models ..................................................................................................... 14

3.3. Mesh generation ....................................................................................................... 16

3.4. Loading .................................................................................................................... 17

3.5. Boundary conditions ................................................................................................ 19

3.5.1. Mid span ........................................................................................................... 19

3.5.2. End of the beam ............................................................................................... 20

3.5.3. Load system ...................................................................................................... 21

3.6. Elastic and plastic behavior ..................................................................................... 21

3.7. Analysis .................................................................................................................... 22

4. HAND CALCULATION OF STRESSES IN THE CROSS SECTION ............................... 23

5. RESULTS AND DISCUSSION ......................................................................................... 25

x

5.1. Problems that occurred in the finite element results ............................................... 25

5.1.1. Model 1.............................................................................................................. 25

5.1.2. Model 2 ............................................................................................................. 27

5.1.3. Model 3 ............................................................................................................. 31

5.1.4. Model 4 ............................................................................................................. 33

5.1.5. Model 5 ............................................................................................................. 35

5.2. Validation of the finite element model ..................................................................... 36

5.3. Comparison of two types of plastic behaviors .......................................................... 38

5.4. Problems and errors obtained during the generation of the input file .................... 40

4. CONCLUSIONS ............................................................................................................... 43

REFERENCES ......................................................................................................................... 45

APPENDIX A – HAND CALCULATION OF THE STRESSES IN THE CROSS SECTION ... 47

APPENDIX B – INPUT FILES FOR ALL MODELS ............................................................... 59

Model 1 ................................................................................................................................. 59

Model 2 ................................................................................................................................ 69

Model 3 ................................................................................................................................ 77

Model 4 ................................................................................................................................ 82

Model 5 ................................................................................................................................ 89

Model 6 ................................................................................................................................ 96

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LIST OF FIGURES Figure 2.1 Uniform and non-uniform torsion of an I-section member ............................................. 4

Figure 2.2: Effect of cross-section on torsional behaviour1 ............................................................. 4

Figure 2.3 Stress-Strain diagram for steel ....................................................................................... 6

Figure 2.4 Elastic-plastic with a nominal yielding plateau slope........................................................ 7

Figure 2.5: True stress-strain curve with strain hardening3 ............................................................... 7

Figure 3.1 Configuration of the beam up to the symmetry line ...................................................... 11

Figure 3.2: Beam cross-section .................................................................................................... 11

Figure 3.3: Model 1 .................................................................................................................... 14

Figure 3.4: Model 2 .................................................................................................................... 14

Figure 3.5 Model 3 ..................................................................................................................... 15

Figure 3.6 Model 4 ..................................................................................................................... 15

Figure 3.7 Model 5 ..................................................................................................................... 15

Figure 3.8 Model 6 ..................................................................................................................... 15

Figure 3.9 Model 6 ..................................................................................................................... 15

Figure 3.10: Visualization of the load system for models 1 and 2 ................................................... 17

Figure 3.11 Visualization of the load system for models 4 to 6 ...................................................... 17

Figure 3.12: The load system for models 1 and 2 .......................................................................... 18

Figure 3.13: The load case of the beam ........................................................................................ 18

Figure 4.1: The selected points for the hand calculation ................................................................ 23

Figure 4.2: The corresponding section points 1 and 5 for ABAQUS .............................................. 23

Figure 5.1: Von Mises stress distribution at the support – Model 1 ................................................ 25

Figure 5.2: Von Mises stress distribution at the mid-span – Model 1.............................................. 26

Figure 5.3: Plastic strain distribution at the mid-span – Model 1 .................................................... 26

Figure 5.4: Von Mises stress distribution at the support – ............................................................. 27



Figure 5.7: Shear stress distribution at the support – ..................................................................... 29



Figure 5.10: Von Mises stress distribution – Model 2 – Boundary condition case 1 ........................ 30



Figure 5.13: Plastic strain distribution at the support – Model 2 .................................................... 31

Figure 5.14: Plastic strain distribution at the mid-span – Model 2 .................................................. 31

Figure 5.15: Von Mises stress distribution – Model 3 .................................................................... 32






xii



Figure 5.23: Load – displacement curve for the two plastic behaviors ............................................ 39

LIST OF TABLES Table 3.1 Data of the cross section .............................................................................................. 11

Table 3.2: Yield strength and ultimate tensile strength of S355 steel............................................... 12

Table 3.3: Material coefficients of steel ........................................................................................ 12

Table 3.4: SI Units ...................................................................................................................... 12

Table 3.5: Material properties of beam elements ........................................................................... 13

Table 3.6: Applied initial displacement for model 1, 2 and 4 – 6 .................................................... 19

Table 3.7: Applied initial displacement and rotation for model 3 ................................................... 19

Table 3.8: Boundary conditions at the end of the beam for models 1 and 2 .................................... 20

Table 3.9: Boundary conditions at the end of the beam for models 3 to 6 ...................................... 20

Table 3.10: True stress-strain curve with strain hardening ............................................................. 21

Table 3.11: Elastic-plastic with a nominal yielding plateau slope .................................................... 21

Table 4.1: The selected points for the hand calculation and the corresponding nodes in ABAQUS . 24

Table 5.1: Total axial stress of the selected nodes ......................................................................... 36

Table 5.2: Total shear stress of the selected nodes ........................................................................ 37

Table 5.3: Angle of twist of the nodes at the symmetry line ........................................................... 37

LIST OF SYMBOLS ε = Strain

h = Depth of the cross-section

b = Width of the cross-section

tw = Thickness of the web

tf = Thickness of the flanges

r = Root radius

A = Area of the cross-section

hi = Clear height between the flanges

Iy = Moment of inertia around y-axis Wel,y = Elastic section modulus around y-axis Wpl,y = Plastic section modulus around y-axis Iz = Moment of inertia around z-axis Wel,z = Elastic section modulus around z-axis Wpl,z = Plastic section modulus around z-axis

LIST OF ABBREVIATIONS FEM = Finite Element Method FEA = Finite Element Analysis CPU TIME = Central Processing time 3D = Three dimensional MPC =Multi-Point Constraints

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1. INTRODUCTION

1.1. Background

Steel buildings first gained popularity in the early 20th century. Their use became more widespread in

the 50’s when steel became more available. Since then steel has become the dominating material for

the construction of buildings and bridges. The less costly production process gave birth to modern

structural steel building industry and to the construction of the world’s first skyscrapers. The range

of use of steel has expanded with improved materials, products and design capabilities with the

availability of computer aided design software. Nowadays structural steel is used to build high quality,

sustainable structures such as multi-storey office buildings, industrial buildings, residential or leisure

buildings and bridges. Steel has numerous advantages namely high strength, uniformity, and elasticity.

Considering the abovementioned potentials of steel structures, it is useful to investigate structural

steel behavior.

1.2. Scope of the study

The aim of this project is to study an open steel cross section (IPE240) which is subjected to shear,

bending and torsion, and to investigate the problems which occur under this kind of loading. Several

models of the I-beam will be created and studied in linear and non-linear analysis using ABAQUS

Finite Element Analysis software.

Thereafter, the outcome results of the stresses in the finite element linear analysis will be compared

to the analytical hand calculation results of the cross section.

Finally, a load – displacement curve of two different plastic behaviors will be compared, namely:

- An elastic-plastic with a nominal yielding plateau slope behavior (less accurate plastic

behavior).

- A true stress-strain curve with strain hardening behavior (realistic behavior).

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2. LITERATURE REVIEW

2.1. Shear, bending and torsion

In most cases, structural members are required to resist numerous kinds of loading. The combined

loading causes several internal-force resultants on a section. Each kind of load produces stresses as if

each load was acting separately. Then the total stress is found by adding the stress components which

arise in the cross section.

It is known, that when a beam is loaded by transverse loads in their planes, the following two

resultants occur in the beam: the shear force and the bending moment. The bending moment is a

function of force and distance and also depends on the boundary conditions of the beam. The sign

conventions for these internal forces and moments are related to the deformation of the member.

Torsion is a consequence of direct actions (eccentric forces or moments) and indirect actions

(applied torsion forces) acting on the cross section of the member. Therefore, when a member is

subjected to torsion, it will twist about a longitudinal axis which passes through the shear center of

the cross section.

If the section is loaded in such a way that the resultant force passes through the shear center of the

cross section, the torsion will be eliminated. In many cases, the applied forces pass through the

center of gravity. In addition, if the section is double symmetric then the torsion is eliminated

because the center of gravity matches with the center of shear. Torsional effects may be influenced

by many factors such as the boundary conditions of the beam, the load arrangement, the warping

restrains and the cross section type (open or close cross section).

Generally in steel structures, torsion should be avoided as much as possible because it is not an

appropriate method of resisting loads. When torsion cannot be avoided, a use of closed sections or

box girders is suggested because they have an increased torsional resistance compared to the

torsional resistance of the open sections. Briefly, closed sections are preferred when they are

subjected to torsion.

However, there are cases when the resultant force does not pass through the shear center of the

cross section, causing torsional loading in the cross section. Moreover, the cross section might be an

open cross section which has decreased torsional resistance. Therefore, in this study, the case of

varying torque (Figure 2.1, case b), combined with shear resultant in an open double symmetric cross

section has been chosen for further analysis.

4

Figure 2.1 Uniform and non-uniform torsion of an I-section member1

In the case of non-uniform torsion the structural member is not free to warp, and the applied torque

is resisted by St. Venant's torsional shear stress and warping torsion. Therefore, the non-uniform

torsion consists of pure (St. Venant’s) torsion and warping (Vlasov) torsion. This can also be

determined by calculating the torsion parameter K. When calculating the torsion parameter K, the

kind of torsion can be distinguished as indicated below.

Figure 2.2: Effect of cross-section on torsional behaviour1

Consequently, as it is also mentioned in the Eurocode 3, EN 1993-1-1, the total torsional resistance

TEd of a cross section is considered as the summation of two components:

- Pure, plane torsion or Saint Venant torsion, Tt,Ed (uniform) which causes twist and

- Warping torsion or Vlasov torsion, Tw,Ed (non – uniform) which causes warping.

1 TRAHAIR N.S., BRADFORD M.A., NETHERCOT D.A. and GARDNER L. (1977) The Behaviour and Design of Steel Structures to EC3, 4th

ed., London and New York: Taylor and Francis group.

5

In this study, the I-beam is subjected to stresses due to torsion, shear and due to bending. The kinds

of stress components that occur and influence the resistance of cross sections are:

- Shear stresses τt,Ed due to Saint Venant torsion Tt,Ed

- Warping axial stresses σ w,Ed due to Bi-moment BEd

- Shear stresses τw,Ed due to warping torsion Tw,Ed

- Shear stress τv,Ed due to shear force VEd

- Bending axial stress σ m,Ed due to bending moment MEd

The abovementioned stress components are evaluated from an elastic analysis for open cross

sections as described in section 4 “Hand Calculations of stresses in the cross section”.

2.2. Stiffeners

Stiffeners are usually required to control buckling effects from shear stresses in steel members. They

are added to a slender girder to ensure that the web panel is able to develop its shear strength and

shear buckling resistance. Even if stiffeners are not essential, they may still be provided if desired to

increase the shear resistance of the web panel and decrease the local deformations due to the external

loading.

There are two types of stiffeners: the longitudinal stiffeners which are placed along the span direction

and the transverse stiffeners which are placed perpendicularly to the span direction of the beam. The

latter, is categorized into bearing stiffeners and intermediate stiffeners. The bearing stiffeners are

provided at the supports (above the reaction) or below the position of the concentrated loads while

the intermediate stiffeners are provided at intervals along the web.

According to Eurocode 3, EN 1993-1-5, end stiffeners can be considered as rigid end post or flexible

(non-rigid) end post stiffeners. A rigid end post is the case when more than one double-sided

transverse stiffeners are placed as close to the support. A non-rigid end post may be a single double

sided stiffener.

In this study, a flexible end post stiffener which may act as bearing stiffener resisting the reaction at

the girder support has been chosen, reducing therefore large local deformations due to the external

loading and preventing local failure. Moreover stiffeners have been used to provide more realistic

conditions for the model.

2.3. Elastic and plastic behavior

This study is carried out with linear and non-linear analysis.

The linear analysis is used to calculate the stresses and deformations of the steel member. There are

three basic assumptions that need to be valid: the steel member should be deformed with small

rotations and displacements, the loading is constant during time and the Hooke’s law is valid

(constant stress – strain relationship and the member’s stiffness never changes). In this linear range

6

the steel remains elastic and returns to its original shape on unloading. Moreover, it is not required to

update anything in the FE-program while the model is deforming.

On the other hand, the non-linear analysis is a more complicated analysis and it is used to approach

the real behavior of the steel member. If the basic assumptions of the linear analysis are invalid, the

results are more accurate than the ones in the linear analysis. Having non-linear geometry means that

the material has nonlinear behavior and therefore its stiffness changes during the deformation and

needs to be updated while the model is deforming. This occurs when the deformations are large and

cannot be neglected. Moreover, the plastic deformations at failure are larger than the elastic ones.

This procedure increases the amount of time needed to get the accurate solution.

The plastic behavior of the beam was calculated according to the Swedish standard “Boverkets

handbok om stålkonstruktioner, BSK 07” and the Eurocode 3, 1993-1-5: 2006, “Plated structural

elements” as described below:

Figure 2.3 Stress-Strain diagram for steel2

�� =��

��

(1) �� = 0.02 + 50 �� − ��

�� (3)

�� = 0.025 − 5��

�� (2) �� = 0.6�

(4)

One case with realistic plastic behavior was studied in which the true stress-strain curve (Figure 2.5)

was used whereas another case with less accurate plastic behavior was also studied in which an

elastic-plastic with yielding plateau slope (Figure 2.4) was used. According to Eurocode 3, EN 1993-

1-5, in the case with the realistic plastic behavior the true stress and strain should be calculated as

follows:

�� = �(1 + �) (5)

�� = ��(1 + �) (6)

These two cases were executed in order to be compared to each other.

2 BOVERKET (2007) Boverkets handbok om stålkonstruktioner, BSK 07, Elanders Sverige AB

7

Figure 2.4 Elastic-plastic with a nominal

yielding plateau slope3

Figure 2.5: True stress-strain curve with

strain hardening3

2.4. FEM modeling

Nowadays, the finite element method has become a great tool used by engineers worldwide in most

fields of engineering. The FEA has many advantages. It can be used for solving many types of

problems. There are no geometric, boundary conditions, loading and material properties restrictions.

Additionally, components that have different behaviors and different mathematical descriptions can

be combined. Therefore, the FEM is most suitable for increasing the success condition of this study.

There are no other known existent experimental solutions to compare rather than the numerical

solution that have been conducted in this study. Certain hand calculations of the stresses have been

performed in order to verify the results of the FEA.

It is imperative that the FEA be recognized as simulation, not as reality. Moreover, the obtained

output results from the FEA are only approximations. Namely, there is a difference between the

finite element solution and the exact solution. The type of the model should be as complex as needed

to obtain the required accuracy of the structure, but also as simple as possible to minimize the

computational time.

In this study the modeling and simulation of the steel I-beam has been executed in ABAQUS Finite

Element Analysis software (ABAQUS/standard) in order to study its behavior under a specific loading.

3Eurocode 3: Design of steel structures, EN 1993-1-5: Plated structural elements (2006)

8

2.4.1. Shell, beam and truss elements

The elements used in this study include ABAQUS’s shell, beam and truss elements, namely, S4 shell

elements, B33 beam elements and T3D2 truss elements.

Shell elements are used to model structures which have one dimension (thickness) smaller than the

other dimensions. Moreover, they are used to model structures in which the stresses in the thickness

direction are negligible.

S4 type shell elements are 4-noded general-purpose, finite-membrane-strain, quadrilateral shell

elements. These elements are conventional / displacement shell elements which discretize the

reference surface, with full integration and linear interpolation. In case of full integration more CPU

time is required than in the case of reduced integration.

In general, shell problems are included in one of two categories: thin shell problems (Kirchoff

elements) and thick shell problems (Mindlin elements). In this case, S4 is included in the thick shell

problems which assume that the shear deformation is important to the solution and therefore the

shear deformations are built in the solution. In addition, S4 shell elements do not have hourglass

modes neither in the membrane nor in the bending response of the element and therefore, the

element does not require hourglass control.

The resultant displacements are calculated at the nodes and the resultant stresses are calculated at the

integration points. At the location of the integrations points, there are section points through the

thickness of the shell. When ABAQUS uses numerical integration to calculate the stresses and strains

independently at each section point, it allows nonlinear material behavior. According to Simpson’s

rule, five section points through the thickness of a shell can be used, which are adequate for most

nonlinear design problems. Moreover, if Simpson’s rule is used, the section point 1 is exactly on the

bottom surface of the shell and the section points through the thickness of the shell are numbered

consecutively, starting with point 1.

Additionally, a shell element consists of the top surface (SPOS) which is the surface in the positive

normal direction, the bottom surface (SNEG) which is the surface in the negative normal direction,

and the mid-surface. The positive normal direction is defined by the connectivity of the shell

elements and the positive and negative direction can be distinguished by plotting the normals in the

model.

When defining the shell element section properties, the offset parameter in the input file was defined

as zero which indicates that the reference surface is the mid-surface of the shell element.

Another element chosen for this study is truss elements. Truss elements are used to model long,

slender structural members that can carry only tensile and compressive axial loads (loading only along

the axis or the center line of the element) but cannot carry moments (moments or forces

perpendicular to the centerline). They also have no initial stiffness to resist loading perpendicular to

their axis.

9

T3D2 type truss elements are three dimensional, 2-noded truss elements with linear displacement.

Due to the fact that the 2-noded truss elements have no bending resistance, they are useful for

modelling pin-jointed frames. They are also suitable since they allow the model to move properly in

torsion. Moreover, in order to define the section properties of the truss elements their cross-sectional

area should be defined. The cross-sectional area has been chosen to be large enough compared to the

other dimensions of the model.

Finally, beam elements are also applied. Beam elements are used to model structures which have one

dimension (length) greater than the other two dimensions of the cross section (slenderness

assumption). Moreover, they are used to model structures in which the longitudinal stresses is of

great importance.

B33 type beam elements are 3D elements and 2-noded cubic beam elements. The cubic interpolation

functions indicate that the element has 3 integration points which makes them accurate for cases

involving distributed loading along the beam. They are also beam-column elements which allow axial,

bending, and torsional deformation.

In general, beam elements are included in one of two categories: Euler Bernoulli beam elements and

Timoshenko beam elements. In this case, B33 is included in the Euler Bernoulli beam elements

which neglect and do not allow the transverse shear deformation. Therefore, this type of elements

are most effective for modeling slender beams.

In general, as previously mentioned, structural members are often subjected to torsional moments

which (torsional moments) also produce warping in the cross-section. The torsional response of

beams depends on the shape of their cross-section. The effects of torsion and warping are

considered in ABAQUS only in the three-dimensional elements and the warping calculation of

warping assumes that the warping displacements are small.

2.4.2. Multi-Point Constraints

Using multi-point constraints is an efficient way to connect the elements between them and to

impose constraints between different degrees of freedom of the model.

The MPC types that are used in the models are: MPC Linear for mesh refinement and MPC Beam,

MPC Tie, MPC Link and MPC Pin for connections and joints.

The MPC type Linear is used when a mesh refinement of first-order elements is needed. It can be

applied to all active degrees of freedom at the involved nodes.

The MPC type Beam provides a rigid beam between two nodes to constrain the displacement and

rotation at the first node to the displacement and rotation at the second node. It can also be applied

at node sets. The two nodes or node sets should be at a distance between them.

10

The MPC type Link is used keep the distance between the two nodes constant and to provide a

pinned rigid link between two nodes. The displacements of the first node are adjusted to impose this

constraint and the existing rotations at the nodes are not involved in this constraint.

The MPC type Pin is used to make the global displacements equal between two nodes but leaves the

existing rotations independent of each other and to provide pinned joint between two nodes. It can

also be applied at node sets. The two nodes or node sets should be at the same position in the model.

The MPC type Tie is used to make all the common active degrees of freedom, global displacements

and rotations of the two nodes equal. It can also be applied at node sets. The two nodes or node sets

should be at the same position in the model.

11

3. FINITE ELEMENT MODEL DEVELOPMENT

In this study the modeling and simulation of the steel I-beam has been executed in ABAQUS Finite

Element Analysis software (ABAQUS/standard) in order to study its behavior under a specific loading.

When developing a model for the finite element analysis (FEA), a typical analysis process is followed.

The finite element models were initially generated by creating input files as seen in Appendix B.

The first step of the analysis refers to the classification and identification of the steel beam that is

analyzed. All the parameters that influence the results, the most important physical phenomena

involved, the results sought from analysis and the required accuracy have been questioned. The

answers to these questions have influenced the amount of information that has been collected to

implement the analysis and the method that the problem has been modeled.

3.1. Model definition and material properties

The model is an IPE240 steel beam which is subjected to an eccentric loading that is causing torsion,

shear force and bending on the beam as shown in the Figure 3.1 and Figure 3.2.

Figure 3.1 Configuration of the beam up to

the symmetry line

Figure 3.2: Beam cross-section

The material and cross-section definitions related to the steel sections were defined in the model.

Specifically, the cross-sectional data of the IPE240 steel beam are shown in the following table:

Table 3.1 Data of the cross section

h (mm) 240 Iy (cm4) 3892

b (mm) 120 Wel,y (cm3) 324.3

tw (mm) 6.2 Wpl,y (cm3) 366.6

tf (mm) 9.8 Iz (cm4) 283.6

r (mm) 15 Wel,z (cm3) 47.27

A (cm2) 39.12 Wpl,z (cm3) 73.92

hi (mm) 220.4

12

The double symmetric section is made of structural steel of quality S355. The nominal values of yield

strength fy and ultimate tensile strength fu for this steel grade are shown at Table 3.2. Therefore, the

cross section belongs to class 1 with respect to bending and to class 2 with respect to compression.

Table 3.2: Yield strength and ultimate tensile strength of S355 steel

fy fu

S355 355 MPa 510 MPa

The material coefficients for the steel are shown in the following table.

Table 3.3: Material coefficients of steel

Modulus of elasticity E 210 GPa

Shear modulus G 81 GPa

Poisson’s ratio in elastic range ν 0.3

Density ρ 7800 kg/m3

ABAQUS has no built-in system of units. Therefore, it is important to define all the input data in

consistent units. The SI system of units is used throughout this project as shown in the following

table.

Table 3.4: SI Units

Quantity SI units

Length m

Force N

Mass kg

Time s

Stress Pa (N/m2)

Energy J

Density Kg/m3

The geometry and the loads of the model are symmetric and therefore the created model is

symmetric too. Consequently, the model has been divided to enable working with symmetric section

instead of working with the entire model. By modeling only the symmetric section, the number of

elements in the model and the CPU time are reduced.

For models 1 and 2, as presented in section 3.2 “Generated models”, the span length up to the

symmetry is 2.88 m and therefore the total length of the beam is 5.76 m. For models 3 – 6, the span

length up to the symmetry is 0.72 m and therefore the total length of the beam is 1.44 m.

The material properties of each element type are described below:

The thickness of the S4 type shell elements is equal to the thickness of the flanges, the web and the

stiffeners according to the relevant region. An offset equal to zero and Simpson’s rule with five

integration points through the shell section were used as described in section 2.4.1. “Shell, beam, and

truss elements”.

13

The material properties of the B33 type beam element are presented in the following table where the

values of the first data line in the input file are shown in Table 3.5. The values of the second data line

in the input file are the default values.

Table 3.5: Material properties of beam elements

Area, A (m2) 0.003912

Moment of inertia for bending about the 1-axis, I11 (m4) 0.00003892

Moment of inertia for cross bending, I12 (m4) 0

Moment of inertia for bending about the 2-axis, I22 (m4) 0.00003892

Torsional constant, J (m4) 0.00001892

In model 6, where truss elements are used, the area was defined as 0.0025 m2.

Thereafter, a simplification of the physical geometry into a mathematical model (idealization) and

then into a discrete model (discretization) has been done consecutively. In brief, the member has

been divided – discretized into elements.

14

3.2. Generated models

As previously mentioned, the following models have been created to examine problems emerged

when the beams are subjected to torsion, shear and bending.

The main features of each model are the following:

- Model 1 consists of three types of meshes, a load system and a stiffener at the end of the

beam.

- Model 2 consists of three types of meshes, a load system and truss elements which substitute

the stiffener of model 1 at the end of the beam.

- Model 3 consists of one type of mesh, a stiffener at the end of the beam and a concentrated

force and torsional moment at mid-span.

- Model 4 consists of one type of mesh, a load system and a stiffener at the end of the beam.

- Model 5 consists of one type of mesh, a load system and stiffeners at both the end of the

beam and the mid-span.

- Model 6 consists of one type of mesh, a load system, stiffeners at both the end of the beam

and the mid-span and truss elements connecting all the nodes of the upper and bottom

flange to the nodes.

More details about each models are described in the following sections.

Figure 3.3: Model 1

Figure 3.4: Model 2

15

Figure 3.5 Model 3

Figure 3.6 Model 4

Figure 3.7 Model 5

Figure 3.8 Model 6

Figure 3.9 Model 6

16

3.3. Mesh generation

A series of nodes and elements were used to represent the geometry. As mentioned previously, the

elements that have been used for the I-beam and stiffeners in all models are shell elements without

reduced integration (S4 type shell elements). A node and element sequence was generated in such a

way that their numbering does not coincide with each other.

For models 1 and 2, the mesh size varies along the span (longitudinal direction) depending on the

region. The mesh is divided into three sections along the beam. The area of interest at the support

possessed the smallest element size with the larger element sizes appearing further from the support.

Namely, the mesh of the first section at the support is fine, whereas the second section becomes

coarser and the third one at the mid-span even coarser. This procedure is performed so as more

accurate results are obtained at the support and the analysis-CPU time is reduced. The mesh in

models 3 – 6 is the same as the finest mesh in models 1 and 2.

In all models, each element and the mesh in general should have a square shape. To achieve this, the

following considerations for each direction have been taken into account:

The dimensions of the model

The number of nodes and elements in each row should be an integer

The increment in the node and element numbers in each row should be an integer

The length of the element should be approximately the same

In order to achieve a successful mesh refinement, the numbering of the nodes in the z –

direction should be an integer, even number and should be divided three times by two so

that the mesh will be symmetric in z-axis and get a node in the middle of the flanges in the z

– direction

The node numbering of the elements in the y – direction should be an integer and even

number

The number of elements in the x – direction should be an integer and even number so that

the evenly distributed load is created for the load system which will be explained later.

When all the nodes and elements of the I-beam and stiffeners have been generated, the nodes which

are situated at the top and bottom of the web along the span are connected to the corresponding

nodes of the flange which are situated in middle of the flange along the span by using the MPC type

Beam.

Similarly, the nodes of the perimeter of the stiffeners are connected to the corresponding nodes of

the flange by using the MPC type Beam and to the web by using the MPC type Tie.

17

3.4. Loading

In this study, the loads that are acted on the beam are placed and simulated in the most efficient way

so that they reach the desirable load case on the beam, as shown in Figure 3.13

The load for models 1, 2, 4 – 6 should be able to create shear and torsion diagrams of the beam

which vary from zero value at the mid span to maximum value at the supports. This is done to

achieve a more realistic simulation of the loading.

A load system which is applied at a distance from the neutral axis, consists of nodes which are placed

at consequent intervals. This was created in order to generate the above kind of loading. The nodes

are connected to each other with beam elements which are also connected to each other vertically,

using multi-point constraints type PIN as shown in Figure 3.10 and Figure 3.11 . The lengths of each

beam element are set in such a way that the reaction forces that are transferred to the nodes of each

level are equal.

Figure 3.11 Visualization of the load system for models 4 to 6

Figure 3.10: Visualization of the load system for models 1 and 2

18

The above figure is used exclusively to visualize the load system. In fact, all the beam elements are at

the same level as shown below.

Figure 3.12: The load system for models 1 and 2

Moreover, all nodes at the level 0 (Figure 3.10 and Figure 3.11) are connected to the middle nodes of

the upper and bottom flange using multi-point constraints type LINK. In this way, there is no

bending stiffness in the x-axis and therefore the model is free to warp. This occurs in models 1, 2, 4

and 5. It does not occur in model 6 where the connection of the middle nodes of the upper and

bottom flange to the load system has been achieved by using truss elements instead.

The load is expressed as an initial displacement which is given at the control node 20003694 for

models 1 and 2 and at the control node 20004550 for models 4 to 6. This initial displacement is

defined by using the “direct” format of the boundary conditions, by specifying the y-direction in

which the displacement is applied and by specifying the magnitude of the displacement. This is

defined in the history data in the input file.

This way of expressing the load was chosen to be able to control the reaction forces which are

transferred to the each one of the nodes of the load system.

Finally, an equal value of the reaction forces are transferred to the nodes at level 0. Because the

intervals between these reaction forces are small, the reaction forces are considered as evenly

distributed load which acts at a distance 0,31 m from the neutral axis as shown in the figure below.

Figure 3.13: The load case of the beam

19

On the other hand, the loads that are applied in model 3 are concentrated force and concentrated

torsional moment around x-axis. These two loads are acting at node 10514. All the nodes of the

upper and bottom flange and the nodes of the web are connected to the middle node of the web

(node 10514) by using multi-point constraints type BEAM. This is done to restrain warping at mid-

span.

Similarly, the loads are expressed as an initial displacement and rotation which are given at the

control node 10514. The initial displacement and rotation are defined by using the “direct” format of

the boundary conditions, by specifying their directions and magnitude. These are defined in the

history data in the input file.

The magnitude of the initial displacements which are applied in each case are summarized in the

following tables.

Table 3.6: Applied initial displacement for model 1, 2 and 4 – 6

Model Initial Displacement (m)

Elastic Analysis Plastic Analysis 1 0.3 0.5 2 0.3 0.5 4 0.3 0.5 5 0.3 0.6 6 0.3 0.6

Table 3.7: Applied initial displacement and rotation for model 3

Model Initial Displacement (m) Initial Rotation (rad) 3 0.3 0.2

3.5. Boundary conditions

Boundary conditions have also been applied to appropriate nodes throughout the analysis of each

model. Various boundary condition cases have been applied in each model. Moreover, different

boundary condition cases have been applied in the same model to study how its behavior varies

under these cases.

3.5.1. Mid span

In all the nodes which are in the symmetry line, the constraints are given directly by using the named

constraint XSYMM. This is valid for all models except model 3. XSYMM is defined as the symmetry

constraint about a plane of constant x1. Namely, the displacement in x-direction, the rotation in y-

direction and the rotation in z-direction are zero (U1= UR2=UR3=0). These restraints provide such

conditions in the symmetry line that the strong axis rotation is allowed. Therefore the model is free

to move around x-axis, thus enabling torsion and bending.

In model 3, node 10514 is restrained at x and z-directions and at the rotations around y and z-axes.

20

3.5.2. End of the beam

In models 3 and 4, the node 4000002 is constrained in y and z-directions and the node 4000502 is

constrained in z-direction. Similarly, these constraints provide such conditions that the strong axis

rotation and therefore torsion and bending are allowed.

In model 1, two cases of boundary conditions have been studied. The first case is the same as the

boundary conditions in models 3 and 4. In the second case nodes 4000002 and 4000502 are

constrained in z-direction and nodes 2 – 8000502 of the bottom flange are constrained in y-direction.

In model 2, three cases of boundary conditions have been studied. The first two cases are the same

as the two boundary condition cases in model 1. In the third case node 4000002 is constrained in y-

direction and node 4000502 and nodes 2000 – 18828 of the web are constrained in z-direction.

In model 5 and 6, nodes 4000002 and 4000502 are constrained in z-direction and nodes 2 – 8000502

of the bottom flange are constrained in y-direction.

All the boundary conditions that are applied at the end of the beam, namely the degrees of freedom

which are constrained are summarized in the following tables.

Table 3.8: Boundary conditions at the end of the beam for models 1 and 2

Model 1 Model 2

Case 1 Case 2 Case 1 Case 2 Case 3

Boundary conditions at the end of the

beam

Table 3.9: Boundary conditions at the end of the beam for models 3 to 6

Model 3 Model 4 Model 5 Model 6

Boundary conditions at the end of the

beam

21

3.5.3. Load system

At least one node of each beam element in the load system is constrained at the rotation around x-

axis in order to avoid the rotation of the beam elements and therefore not contribute to the

deformation of the I-beam. This is valid to all models except model 3 which has concentrated force

and concentrated torsional moment instead of the load system.

In models 1 and 2, control node 20003694 of the load system is constrained in x-direction in order to

avoid the movement of the load system in x-direction and to allow the load system to move in the

same way with the middle nodes of the upper flange and avoid instability.

In model 3, control node 10514 of web at the mid-span is constrained in x and z-directions and to

rotations around y and z-directions.

In models 4 to 6, control node 20004550 of the load system is constrained in x-direction in order to

avoid the movement of the load system in x-direction and to allow the load system to move in the

same way with the middle nodes of the upper flange and avoid instability.

3.6. Elastic and plastic behavior

The following values of the plastic behavior, which are used in the input file, were calculated as

described in section 2.3 “Elastic and plastic behavior”.

For models 1 to 6, a realistic behavior was used. The values used in the input file to describe this

behavior are shown below:

Table 3.10: True stress-strain curve with strain hardening

Stress (MPa) Strain Plastic Strain

0 0 0

355.600 0 0

359.564 0.001689 0.011063

539.021 0.012775 0.052778

561.000 0.055345 0.092639

Model 6 was also studied in the case with less accurate plastic behavior. Similarly, the values that are

used in the input file to describe this behavior are shown below:

Table 3.11: Elastic-plastic with a nominal yielding plateau slope

Stress (MPa) Strain Plastic Strain

0 0 0

355 0.00169 0

365 0.10000 0.09831

300 0.25000 0.24857

22

3.7. Analysis

After defining the abovementioned input data, the history data and the type of the analysis has been

defined. Then, the model has been submitted to the analysis solver. A time step has been defined and

the equilibrium equation system has been solved.

The finite element analysis that has been used is a static stress analysis, which is used for stable

problems and when inertia effects can be neglected. It may include linear or nonlinear response.

In general, in elastic analysis, the model has small deflections and no time step is required to be

defined. In this case the solution can be calculated by solving a system of linear equations. In plastic

analysis, ABAQUS/Standard uses Newton’s method to solve the nonlinear equilibrium equations. In

this case the solution cannot be calculated as in the case of elastic analysis. However, the solution can

be calculated by specifying the loading as a function of time and by using time increments to obtain

the nonlinear response and the equilibrium solution in each increment.

Therefore, in all models, the geometry in the elastic analysis has been defined as linear and no time

step has been defined. However, the geometry in the plastic analysis has been defined as non-linear

and a time step has been specified. Namely, the initial time increment has been set to 0.01 and the

time period of the step has been set to 1.0. The maximum number of increments in a step has been

set to 300 so as not to limit the analysis and cause its termination. In addition to this, in model 6, the

minimum time increment has been set to 10-9 and the maximum time increment has been set to 0.1.

23

4. HAND CALCULATION OF STRESSES IN THE

CROSS SECTION

Simple analytical calculations and handbook formulas from an elastic analysis have been used to

evaluate the stress components at specific points in the cross-section of the I–beam. The finite

element results will be compared to these hand calculations so as to check the accuracy of the finite

element results.

Specifically, the hand calculations have been executed for model 1. The model is considered as

simply supported beam with distributed load along the span of the beam with an eccentricity from

the neutral axis as shown in Figure 3.12 in section 3.4 “Loading”.

The method which was followed for the calculation of the stresses is described below:

The points 1 – 12 on the cross-section where the stresses are to be determined have been selected as

shown in Figure 4.1. The stress resultant at the selected points of the cross section has been

determined for each one of the reaction forces and moments which were caused by the external

loading. The axial stresses have been calculated at the selected points at the mid-span of the beam

and the shear stresses have been calculated at the selected points at a distance 0.24 m from the

support. Then, all the stress components of each one of the selected points have been combined

separately and therefore the total stress of the selected points has been calculated.

The hand calculations of the total stress of the selected points 1 – 12 are shown in Appendix A.

The selected points at the cross-section of the hand calculations are equivalent to the section points 1

and 5 of some certain nodes at the cross-section of ABAQUS as shown in Table 4.1. It is essential

that while comparing the results between the hand calculations and ABAQUS, the selected points of

the hand calculations should correspond to the proper section points of ABAQUS as shown in

Figure 4.2.

Figure 4.1: The selected points for the

hand calculation

Figure 4.2: The corresponding section

points 1 and 5 for ABAQUS

24

Table 4.1: The selected points for the hand calculation and the corresponding nodes in ABAQUS

Selected points for the hand calculations

Corresponding nodes at mid-span

Corresponding nodes at 0.24 m from the support

1 and 2 6000886 6000534 3 and 4 2000886 2000534 5 and 6 14404 14052 7 and 8 7192 6840 9 and 10 6000386 6000034 11 and 12 2000386 2000034

The results from ABAQUS are the approximated values and the results from the hand calculations

are the exact values. The error of these values is calculated as follows:

�� (%) = �

�� − ��

�� ∙ 100

(7)

25

5. RESULTS AND DISCUSSION

5.1. Problems that occurred in the finite element results

The results of the six models analyzed using ABAQUS are presented in the following figures. The

von Mises and the shear stress distributions were plotted. The contour plots of the finite element

mesh of all models provide an overview of the distribution of the stresses. Moreover, an enlarged

portion of the mesh of the models is discussed in this section.

The stresses at each point are calculated directly from the solution variables. The interpolation

functions in a displacement based finite element analysis are used to obtain the strains from the nodal

point displacements.

5.1.1. Model 1

A problem occurred in the elastic analysis of model 1 and in both boundary condition cases that were

applied in the model. In Figure 5.1, the contour plot represents the stress distribution of the von

Mises stresses at the negative surface of the shell element. An enlarged portion at the end of the

beam has been studied.

Figure 5.1: Von Mises stress distribution at the support – Model 1

26

The stress values obtained at the corners of the upper flange are too large compared to the stresses

of the region area. Specifically, the red region shows that the total stresses are approximately 2186

MPa while the stresses in the middle of the flange are approximately 34.5 MPa. This also occurs in

the bottom flange and it is visible when the stresses are plotted in the positive surface of the shell

elements. This leads to a bending failure at the corners of both flanges, fact that should not have

occurred and therefore these results were not the expected ones.

As presented in the figure below, the von Mises stress distribution at the mid-span is smaller than the

one in the support which is reasonable because the reaction forces of the load system create shear

and torsion diagrams of the beam which vary from zero value at the mid span to maximum value at

the supports.

Figure 5.2: Von Mises stress distribution at the mid-span – Model 1

Moreover, in the plastic analysis, the initial displacement (0.5 m) which is applied at the control node

of the load system causes yielding of the material at the mid-span but does not cause yielding at the

support nor at the web panel, as shown in the figure below.

Figure 5.3: Plastic strain distribution at the mid-span – Model 1

27

Comparing the results from the two boundary condition cases of the model it is observed that the

stresses of the first case are slightly smaller than the stresses of the second case of the boundary

conditions. In general, the results are similar to each other in both cases.

5.1.2. Model 2

In the elastic analysis of model 2, different results were obtained for the three different boundary

condition cases that were applied in the model. Figure 5.4, Figure 5.5 and Figure 5.6, illustrate the

contour plot of the von Mises stress distribution at the negative surface of the shell elements. The

problem in these cases occurs at the corners of the web panel.

The first case of the boundary conditions produces too large stresses at the corners of the web

compared to the stresses of the region area. Specifically, the red region shows that the total stresses

are approximately 2063 MPa while the total stresses in the middle of the web are approximately

945.95 MPa. The stresses at the upper and bottom flange at the support in the same area are much

smaller than the ones on the web.

Similar problem occurs in the third case of the boundary conditions where the total stress values

obtained at the corners of the web are approximately 4486 MPa compared to the overall area of the

support where the values of the total stresses vary between 0.314 MPa and 1122 MPa.

In the second case of the boundary conditions the large concentration of stresses at the bottom

corner of the web has disappeared because all the nodes of the bottom flange are constrained in the

y-direction, thus restraining warping of the bottom flange. On the other hand, the large

concentration of stresses at the upper corner of the web still remains. The stresses at that point are

approximately 1764 MPa. In addition, the stresses of the bottom flange are larger than the stresses of

the upper flange.

Figure 5.4: Von Mises stress distribution at the support –

Model 2 – Boundary condition case 1

28





Additionally, in all boundary condition cases the shear stresses of the web panel at the support area

are too large compared to the region area. As shown in Figure 5.7 the shear stresses in the middle of

the web vary from 386 MPa to 619 MPa. The shear stresses in case 2 (Figure 5.8) vary from 134 MPa

to 535 MPa and the stresses in case 3 (Figure 5.9) vary from 432 MPa to 560 MPa.

29

Figure 5.7: Shear stress distribution at the

support – Model 2 – Boundary condition case 1

Figure 5.8: Shear stress distribution at the

support – Model 2 – Boundary condition case 2

Figure 5.9: Shear stress distribution at the support –


The large stresses at the corners of the web and the large shear stresses at the web panel are caused

by the boundary conditions and especially by the restrained degree of freedom 2 at the bottom flange

which acts as if a force is applied upwards the support.

In the plastic analysis, the applied initial displacement causes a large torsional moment and a twisting

of almost 90 degrees of the cross section at the mid-span as shown in Figure 5.10, Figure 5.11 and

Figure 5.12. The magnitude of the initial displacement has been given the value of 0.5 m to cause

yielding at the beam. Despite this, yielding appeared only in small areas in the flange at the mid-pan

and in the web at the support as shown in Figure 5.13 and Figure 5.14.

30

Figure 5.10: Von Mises stress distribution – Model 2 – Boundary condition case 1



31

Figure 5.13: Plastic strain distribution at the

support – Model 2

Figure 5.14: Plastic strain distribution at the

mid-span – Model 2

In general, the stresses that occurred in models 1 and 2 are large. The study of these models has been

done in order to observe how the behavior of the beam varies with different boundary conditions

and support details.

The beam yields at the flanges of the mid-span and collapses at the mid-span. The resistance to bi-

moment is larger than the shear resistance of the beam. Therefore, the mid-span area is more critical

than the support area. Moreover, the beam is weak in bending due to the fact that a too long beam

twists almost 90 degrees under a large initial displacement at the control node.

5.1.3. Model 3

Model 3 was created in order to eliminate the abovementioned problems at the support and the large

torsion which occurs at mid-span of model 1 and 2. As previously mentioned, the beam in model 3 is

shorter than the beams in model 1 and 2 and has a consecrated load and torsional moment acting at

the middle node of the web at mid-span.

In the plastic analysis of model 3, the total stress distribution and the deformed shape, as shown in

Figure 5.15, represents a realistic behavior of the beam when subjected to such loading. Constraining

all the nodes of the flanges and the web at the symmetry line to the middle node of the web enables

the beam to act as a rigid beam. Therefore, the cross-section remains constant during the

deformation of the beam.

Even if the results are reasonable, there is a much localized failure at the mid-span. There is a

combined failure due to the shear force, the bending moment and the warping of the beam. In

addition to this, local buckling of the flanges and the web is occurred at the same region area.

32

The stresses at the mid-span are larger than the stresses at the support and the beam yields firstly at

the mid-span. Local concentrated stresses also occurred at the bottom flange at the support due to

the boundary conditions as shown in Figure 5.16.

Figure 5.15: Von Mises stress distribution – Model 3

Figure 5.16: Plastic strain distribution at the support – Model 3

33

5.1.4. Model 4

In order to avoid the local failure and buckling of the beam in model 3, the load system was applied

in the same model instead of the concentrated load and torsional moment to enable a more even

distribution of the load along the beam length.

In the plastic analysis of model 4, the total stress distribution of the von Mises stresses at the negative

surface of the shell element are presented in Figure 5.17. It is observed that the large local buckling

does not occur anymore, but the cross-section at mid-span does not remain constant. Namely, there

is a local buckling at the upper flange at the mid-span and a bending of the web panel.

The applied initial displacement also causes a large torsional moment and a twisting of almost 60

degrees of the cross section at the mid-span as shown in Figure 5.18.

The stresses at the mid-span are larger than the stresses at the support and the beam. Local

concentrated stresses also occurred at the bottom flange at the support due to the boundary

conditions as shown in Figure 5.19 and Figure 5.20.


34




35

5.1.5. Model 5

In model 5, another stiffener was added to the mid span so that the cross section of the beam would

remain constant during the deformation as shown in Figure 5.22. Figure 5.21, illustrates the

deformed shape of the plastic analysis and the contour plot of the von Mises stress distribution at the

negative surface of the shell elements.



36

Comparing models 4 and 5, it is observed that the stresses which occurred at the mid-span in model

5 are smaller than the stresses in model 4. Therefore, adding the stiffener to the mid-span the stresses

decrease. Specifically, the total stress values obtained at the support in model 5 vary from 299 MPa to

358.7 MPa, whereas in model 4 they vary from 314 MPa to 390 MPa. Additionally, the total stress

values obtained at the mid-span in model 5 are approximately 567 MPa whereas in model 4 they are

approximately 618.8 MPa.

Moreover, a small local buckling appears at the upper flange.

Finally, in model 5 the stresses of the stiffener at the support are larger than the stresses of the

stiffener at the mid-span.

5.2. Validation of the finite element model

The finite element results have been compared to the hand calculation results. The stresses which are

extracted from ABAQUS are taken from .dat file. In order to limit the results which are given to the

.dat file, node sets and element sets have been created which include the selected nodes and elements

that the stresses are calculated. The stresses are calculated at the integration points of the elements.

In order to get the stresses at the nodes, the stresses at the integration points are extrapolated at the

nodes. The values of the total axial and shear stresses and the values of the angle of twist are

presented in the following tables.

Table 5.1: Total axial stress of the selected nodes

Selected

Nodes

Section

Point

Total axial stress -

ABAQUS value - S11

(MPa)

Total axial stress - Hand Calculations value (MPa)

Error (%)

6000886 5 -426.550 -439.441 2.933

6000886 1 -450.750 -427.719 5.385

2000886 5 121.800 152.359 20.057

2000886 1 171.320 164.082 4.411

6000386 5 451.010 427.719 5.445

6000386 1 425.690 439.441 3.129

2000386 5 -171.060 -164.082 4.253

2000386 1 -122.660 -152.359 19.493

14404 1 -95.157 -56.494 68.437

14404 5 -24.412 -56.494 56.788

7192 1 95.885 56.494 69.726

7192 5 25.140 56.494 55.500

37

Table 5.2: Total shear stress of the selected nodes

Selected

Nodes

Section

Point

Total shear stress - ABAQUS value - S12

(MPa)

Total shear stress - Hand Calculations value (MPa)

Error (%)

6000534 5 670.050 644.391 3.829

6000534 1 -635.250 -612.233 3.623

2000534 5 664.320 644.391 3.000

2000534 1 -641.170 -612.233 4.513

6000034 5 635.890 612.233 3.720

6000034 1 -670.100 -644.391 3.837

2000034 5 640.060 612.233 4.348

2000034 1 -664.370 -644.391 3.007

14052 1 370.300 417.241 12.676

14052 5 -414.940 -377.766 8.959

6840 1 371.190 417.241 12.406

6840 5 -414.340 -377.766 8.827

Table 5.3: Angle of twist of the nodes at the symmetry line

Nodes at the

symmetry line

Angle of twist

- ABAQUS value - UR1

(rad)

Angle of twist - Hand

Calculations value

(rad)

Error (%)

386 1.428 1.494 4.410

886 1.428 1.494 4.410

4788 1.435 1.494 3.941

7192 1.440 1.494 3.607

9596 1.442 1.494 3.473

12000 1.442 1.494 3.473

14404 1.44 1.494 3.607

16808 1.435 1.494 3.941

2000386 1.428 1.494 4.410

2000886 1.428 1.494 4.410

4000386 1.429 1.494 4.343

4000886 1.429 1.494 4.343

6000386 1.428 1.494 4.410

6000886 1.428 1.494 4.410

8000386 1.428 1.494 4.410

8000886 1.428 1.494 4.410

Referring to axial stresses, it is observed that the error that occurs in most of nodes varies from 3 to

5 %. However, nodes 2000886 (section point 5) and 2000386 (section point 1) have relatively high

rate of error which reaches approximately up to 20%. In addition, nodes 14404 and 7192 of the web

in the mid-span have a very high rate of error which varies from 55-70 %.

38

This error may appear for the following reasons:

- The stresses in the web are influenced only by the axial bending stresses. Therefore, the

problem is caused by the axial bending stresses either in the hand calculations or in

ABAQUS. Consequently, the bending theory might not work properly.

- When the nodes of interest are located close to the line of the load at z-direction, the results

of the stresses show a greater influence.

- The mesh at mid-span is the coarsest mesh in the model. If the finite element mesh is too

coarse, the equilibrium conditions are satisfied poorly and the errors in the stresses can be

considerable, thus causing a discretization error.

- When linear static analysis, compatible meshes and full numerical integration are used, the

error might be caused by the fact that the equilibrium equations are not locally satisfied

everywhere.

- The boundary conditions might not have been simulated properly. At the boundaries of the

finite element model it can be observed how much calculated stresses deviate from known

stresses.

- In order to minimize the global error in the model, the finite element method allows local

inaccuracies.

Referring to shear stresses, it is observed that the error that occurs in the nodes varies from 3 to

12.5%. Similarly, the error in the angle of twist that occurs in all nodes at the symmetry line varies

from 3.5 to 4.5%. The error is relatively small and is within the acceptable limits.

In general, it is observed that in both axial and shear stresses, the nodes 14404 and 7192 of the web

have the highest rates of error. Despite this, the finite element results of the rest of the nodes and the

hand calculation results are in a good agreement. Therefore, this indicates that the finite element

model reflects fairly accurately the real behavior of the structure.

5.3. Comparison of two types of plastic behaviors

In model 6, a load – displacement curve was plotted, showing the change in the reaction force at the

y-direction acting on the control node 20004550. This is done in order to obtain extra information

about model 6.

The graph has been plotted for two cases:

- When the model has an elastic-plastic with a nominal yielding plateau slope behavior (less

accurate plastic behavior).

- When the model has a true stress-strain curve with strain hardening behavior (realistic

behavior).

39

The comparison of the load-displacement curves related to the abovementioned plastic behaviors is

displayed in the following graph.

Figure 5.23: Load – displacement curve for the two plastic behaviors

As observed in the graph, the two load-displacement curves have different max load resistance.

Comparing the load capacity of the two load-displacement curves, it is observed that the model with

the realistic plastic behavior has a higher maximum load resistance than the model with the less

accurate plastic behavior. Specifically, the load capacity of the former is 362.29 kN and occurs when

the displacement is 0.454 m. The load capacity of the latter is 85.95 kN and occurs when the

displacement is 0.176 m.

It is interesting to note that both load-displacement relationships show linear elastic response at low

reaction force values. This occurs when the reaction force extends from 0 to 25.40 kN. The behavior

then becomes non-linear in which the beam exhibits material and geometric non-linearity and may

cause the deformations to become very large. The magnitude of the deformations and the non-linear

behavior depends on the elastic modulus E and the shear modulus G.

In the case of the realistic plastic behavior, there is a sharp increase of the reaction force between

0.427 m and 0.454 m reaching the first maximum value at 0.454 m. Thereafter, there is a slight

fluctuation in the force values with a second maximum value of 364.06 kN, and then it drops quickly

to 114.18 kN. Finally, the reaction force increases up to 259.88 kN and decreases again to 238.40 kN

at the 0.6 m.

In the case of the less accurate plastic behavior, a more steady increase in the reaction force is

observed. There is a first maximum force value, which is the load resistance as mentioned previously,

and a second maximum force value of 171.73 kN at 0.531 m.

0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

400.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70

Re

acti

on

Fo

rce

(kN

)

Displacement (m)

Realistic plastic behaviour Elastic-plastic with a nominal plateau slope behaviour

40

In both cases the second maximum values of the reaction force are caused by secondary effects. In

general, the behavior of the beam depends on the initial deformation which is applied at the control

node and therefore it depends on the reaction force at the control node which is caused by this initial

deformation. Torsion members remain linear until their non-linearity becomes important.

5.4. Problems and errors obtained during the generation of the input file It is essential that the problems that occur during the generation of the input file be considered. The

most common problems, warnings and errors are discussed below.

**WARNING: ELEMENT LABEL 2260 HAS BEEN USED MORE THAN ONCE. ONLY THE

LAST DEFINITION IS CONSIDERED.

It was difficult to generate a mesh in which the numbing of the nodes and elements did not coincide.

The abovementioned message appeared frequently in various element labels. This message indicates

that there is a distortion in the element labels and that somewhere in the model two element labels

coincide. A more careful control of the node and element numbering definition has been made.

**ERROR: DEGREE OF FREEDOM 1 DOES NOT EXIST FOR NODE 2012. IT HAS

ALREADY BEEN ELIMINATED BY ANOTHER EQUATION, MPC, RIGID BODY,

KINEMATIC COUPLING CONSTRAINT, TIE CONSTRAINT OR EMBEDDED ELEMENT

CONSTRAINT. THE REQUIRED MPC (TYPE LINK) CANNOT BE FORMED.

It should be taken into account that when using any type of MPC, node numbers or node sets which

are set in the second parameter in the first data line of the MPC command in the input file, are not

considered to exist in the whole model. As a result, the non-existing nodes have not been used

further on in the input file.

**WARNING: THE STRAIN INCREMENT HAS EXCEEDED FIFTY TIMES THE STRAIN

TO CAUSE FIRST YIELD AT 32 POINTS

This indicates that the plastic yielding in the given increment exceeds the given strain to cause first

yield. Therefore, the plastic material properties, the units and the load magnitude have been double

checked to identify the error which has been corrected.

**ERROR: TOO MANY ATTEMPTS MADE FOR THIS INCREMENT

This shows that the given increment in the input file is too large. During the analysis, ABAQUS has

divided this increment over 4 for five times; this information is given at the end of the .msg file as “5

cutbacks in automatic incrementation”, and still the increment has been found too large. Therefore,

the analysis could not be completed. After several efforts, it was found that the beam elements in the

load system had unrealistic properties.

**WARNING: SOLVER NUMERICAL SINGULARITY WHEN PROCESSING NODE

20000562 D.O.F 4 RATIO=7.37280E+15

The abovementioned message appeared in various nodes. According to the message the value of the

torsion is too big. Moreover, the “numerical instability” indicates that the model is unstable, that

there are non-valid boundary conditions and that there are negative eigenvalues of the stiffness

41

matrix. Therefore, the nodes where this problem appears are restrained at the degree of freedom 4 so

that there is no rotation around x-axis.

**ERROR: THE INREMENT REQUIRED IS LESS THAN THE MINIMUM SPECIFIED

This error message occurred during the generation of the input file of model 6. The message means

that the required increment of the analysis is lower than the minimum specified time increment.

Therefore, the minimum time increment has been set to 10-9 so as not to limit the analysis. In

addition, the STABILIZE parameter was included to the STATIC command so as to use automatic

stabilization and eliminate the local instabilities.

43

4. CONCLUSIONS

A consecutive series of models have been generated. In the effort to eliminate the problems that

occurred in the first model, other models were created in which different problems appeared. Each

one of these models exposed special characteristics.

In general, it has been observed in all models that the beam yields at the flanges of the mid-span and

collapses at the mid-span. Therefore, the failure at the mid-span is more critical than the failure at the

support. Moreover, the beams are weak in bending due to the fact that they twist almost 60-90

degrees under a large initial displacement at the control node.

Much localized failure and buckling occurred at the mid-span, and local concentrated stresses also

occurred at the bottom flange at the support due to the boundary conditions details. A relatively large

initial displacement has been applied so that a yielding on the beam could be caused. However, the

beam has not yielded except in some small areas in the beam.

The hand calculations have shown that the stresses at most of the selected points have a small error,

except in some points. The error at nodes 14404 and 7192 is too large and a further research is

needed to find what really caused such a big error and how it can be eliminated. The avoidance of

errors requires a critical attitude towards the results. Despite this, the finite element results of the rest

of the nodes and the hand calculation results are in a good agreement.

Finally, the comparison of the load – displacement curves of the two different plastic behaviors

showed that the model with an elastic-plastic with a yielding plateau slope behavior has smaller

maximum load resistance than the model with a true stress-strain curve with strain hardening

behavior.

45

REFERENCES TRAHAIR N.S., BRADFORD M.A., NETHERCOT D.A. and GARDNER L. (1977) The

Behaviour and Design of Steel Structures to EC3, 4th ed., London and New York: Taylor and

Francis group.

NETHERCOT D. A., SALTER P. R. and MALIK A. S. (1989) Design of Members Subject to

Combined Bending and Torsion, Berkshire: The Steel Construction Institute.

ROBERT D. COOK, DAVID S. MALKUS, MICHAEL E. PLESHA, ROBERT J. WITT

(1974) Concepts and applications of finite element analysis, 4th ed., University of Wisconsin –

Madison: John Wiley & Sons.

ΒΑΓΙΑΣ Ι., ΕΡΜΟΠΟΥΛΟΣ Ι., ΙΩΑΝΝΙΔΗΣ Γ. (2001) Εκδόσεις δομικών έργων από χάλυβα,

Αθήνα: Εκδόσεις Κλειδάριθμος.

NORLIN B.G (2013) Lecture 9, Bending and torsion of linearly elastic beams, Stockholm, Royal

Institute of Technology KTH. Available from: Bilda [13/10/10].

KAROUMI R. (2011) The finite element method, Stockholm, Royal Institute of Technology

KTH. Available from: Bilda [14/03/24].

EN 1993-1-1 (2005). Eurocode 3: Design of steel structures – Part 1.1: General rules and

rules for buildings.

EN 1993-1-5 (2006). Eurocode 3: Design of steel structures – Part 1.5: Plated structural

elements

BOVERKET (2007) Boverkets handbok om stålkonstruktioner, BSK 07, Elanders Sverige

AB

DASSAULT SYSTÈMES, (2012) Abaqus 6.12, Abaqus/CAE User’s Manual

DASSAULT SYSTÈMES, (2012) Abaqus 6.12, Getting Started with Abaqus: Interactive

Edition

DASSAULT SYSTÈMES, (2012) Abaqus 6.12, Getting Started with Abaqus: Keywords

Edition

DASSAULT SYSTÈMES, (2012) Abaqus 6.12, Keywords Reference Manual

DASSAULT SYSTÈMES, (2012) Abaqus 6.12, Analysis User’s Manual, Vol.1-5

46

SUSSMAN T., BATHE K.J. (1986) Studies of finite element procedures – stress band plots and the

evaluation of finite element meshes, Department of Mechanical Engineering, Massachusetts

Institute of Technology, Cambridge, USA [Online] Available from:

http://web.mit.edu/kjb/www/Publications_Prior_to_1998/Studies_of_Finite_Element_Pr

ocedures_Stress_Band_Plots_and_the_Evaluation_of_Finite_Element_Meshes.pdf

[Accessed: 5th June 2015]

JOHANSSON B. (2005) Module 5 Cross-sectional resistance, Luleå, SBI Swedish Institute of Steel

Construction, LTU Luleå University of Technology, KTH Royal Institute of Technology

http://web.mit.edu/kjb/www/Publications_Prior_to_1998/Studies_of_Finite_Element_Procedures_Stress_Band_Plots_and_the_Evaluation_of_Finite_Element_Meshes.pdf

http://web.mit.edu/kjb/www/Publications_Prior_to_1998/Studies_of_Finite_Element_Procedures_Stress_Band_Plots_and_the_Evaluation_of_Finite_Element_Meshes.pdf

47

APPENDIX A – HAND CALCULATION OF THE STRESSES IN

THE CROSS SECTION

59

APPENDIX B – INPUT FILES FOR ALL MODELS

Model 1 *HEADING I beam IPE240 – Model 1 ** **Model Definition *NODE, NSET=Keynodes 2, 0, 0, 0 8000002, 0, 0, -0.12 2000, 0, 0.0049, -0.06 18828, 0, 0.2253, -0.06 22000, 0, 0.0049, 0 4022000, 0, 0.0049, -0.06 20000514,-0.09, 0.2302, 0.25 20000586,-0.63, 0.2302, 0.25 20000610,-0.81, 0.2302, 0.25 20000682,-1.35, 0.2302, 0.25 20000706,-1.53, 0.2302, 0.25 20000874,-2.79, 0.2302, 0.25 ****MESH 1*********************** **BOTTOM FLANGE***** *NGEN, NSET=BottomFlangeRight 2, 8000002, 500000 *NCOPY, CHANGE NUMBER=100, OLD SET=BottomFlangeRight, SHIFT, NEW SET=BottomFlangeLeft -0.75, 0, 0 ,, *NFILL, NSET=BottomFlange BottomFlangeRight, BottomFlangeLeft, 100, 1 ** **UPPER FLANGE***** ** *NCOPY, CHANGE NUMBER=500, OLD SET=BottomFlange, SHIFT, NEW SET=UpperFlange 0, 0.2302, 0 ,, ** **WEB********** ** *NGEN, NSET=WebRight 2000, 18828, 601 *NCOPY, CHANGE NUMBER=100, OLD SET=WebRight, SHIFT, NEW SET=WebLeft -0.75, 0, 0 ,, *NFILL, NSET=Web WebRight, WebLeft, 100, 1 ************** *ELEMENT, TYPE=S4 2, 2, 500002, 500003, 3 2000, 2000, 2601, 2602, 2001 502, 502, 500502, 500503, 503 ** *ELGEN, ELSET=BFlangeElements 2, 100, 1, 1, 16, 500000, 500000 *ELGEN, ELSET=WebElements 2000, 100, 1, 1, 28, 601, 601 *ELGEN, ELSET=UFlangeElements 502, 100, 1, 1, 16, 500000, 500000 ** *ELSET, ELSET=FinerMesh

60

BFlangeElements WebElements UFlangeElements ** *SHELL SECTION, ELSET=UFlangeElements, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=BFlangeElements, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=WebElements, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0062, 5 ****MESH 2*********************** *NGEN, NSET=BottomFlangeRight2 102, 8000102, 1000000 *NCOPY, CHANGE NUMBER=100, OLD SET=BottomFlangeRight2, SHIFT, NEW SET=BottomFlangeLeft2 -0.75, 0, 0 ,, *NFILL, NSET=BottomFlange2 BottomFlangeRight2, BottomFlangeLeft2, 50, 2 ** **UPPER FLANGE***** ** *NCOPY, CHANGE NUMBER=500, OLD SET=BottomFlange2, SHIFT, NEW SET=UpperFlange2 0, 0.2302, 0 ,, **WEB********** ** *NGEN, NSET=WebRight2 2100, 18928, 1202 *NCOPY, CHANGE NUMBER=100, OLD SET=WebRight2, SHIFT, NEW SET=WebLeft2 -0.75, 0, 0 ,, *NFILL, NSET=Web2 WebRight2, WebLeft2, 50, 2 ************** *ELEMENT, TYPE=S4 102, 102, 1000102, 1000104, 104 2100, 2100, 3302, 3304, 2102 602, 602, 1000602, 1000604, 604 ** *ELGEN, ELSET=BFlangeElements2 102, 50, 2, 2, 8, 1000000, 1000000 *ELGEN, ELSET=WebElements2 2100, 50, 2, 2, 14, 1202, 1202 *ELGEN, ELSET=UFlangeElements2 602, 50, 2, 2, 8, 1000000, 1000000 ** *ELSET, ELSET=MiddleCoarserMesh BFlangeElements2 WebElements2 UFlangeElements2 ** *SHELL SECTION, ELSET=UFlangeElements2, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=BFlangeElements2, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=WebElements2, MATERIAL=STEEL, OFFSET=0, POISSON=0.3

61

0.0062, 5 ****MESH 3*********************** *NGEN, NSET=BottomFlangeRight3 202, 8000202, 2000000 *NCOPY, CHANGE NUMBER=184, OLD SET=BottomFlangeRight3, SHIFT, NEW SET=BottomFlangeLeft3 -1.38, 0, 0 ,, *NFILL, NSET=BottomFlange3 BottomFlangeRight3, BottomFlangeLeft3, 46 , 4 ** **UPPER FLANGE***** *NCOPY, CHANGE NUMBER=500, OLD SET=BottomFlange3, SHIFT, NEW SET=UpperFlange3 0, 0.2302, 0 ,, ** **WEB********** ** *NGEN, NSET=WebRight3 2200, 19028, 2404 *NCOPY, CHANGE NUMBER=184, OLD SET=WebRight3, SHIFT, NEW SET=WebLeft3 -1.38, 0, 0 ,, *NFILL, NSET=Web3 WebRight3, WebLeft3, 46, 4 ************** *ELEMENT, TYPE=S4 202, 202, 2000202, 2000206, 206 2200, 2200, 4604, 4608, 2204 702, 702, 2000702, 2000706, 706 ** *ELGEN, ELSET=BFlangeElements3 202, 46, 4, 4, 4, 2000000, 2000000 *ELGEN, ELSET=WebElements3 2200, 46, 4, 4, 7, 2404, 2404 *ELGEN, ELSET=UFlangeElements3 702, 46, 4, 4, 4, 2000000, 2000000 ** *ELSET, ELSET=MiddleEndMesh BFlangeElements3 WebElements3 UFlangeElements3 *SHELL SECTION, ELSET=UFlangeElements3, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=BFlangeElements3, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=WebElements3, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0062, 5 ****MESH REFINEMENT****************************** *NSET, NSET=BFp1, GENERATE 500102, 7500102, 1000000 *NSET, NSET=BFa1, GENERATE 102, 7000102, 1000000 *NSET, NSET=BFb1, GENERATE 1000102, 8000102, 1000000 ** *NSET, NSET=BFp2, GENERATE 1000202, 7000202, 2000000 *NSET, NSET=BFa2, GENERATE 202, 6000202, 2000000

62

*NSET, NSET=BFb2, GENERATE 2000202, 8000202, 2000000 *MPC LINEAR,BFp1, BFa1,BFb1 LINEAR,BFp2, BFa2,BFb2 *NSET, NSET=UFp1, GENERATE 500602, 7500602, 1000000 *NSET, NSET=UFa1, GENERATE 602, 7000602, 1000000 *NSET, NSET=UFb1, GENERATE 1000602, 8000602, 1000000 *NSET, NSET=UFp2, GENERATE 1000702, 7000702, 2000000 *NSET, NSET=UFa2, GENERATE 702, 6000702, 2000000 *NSET, NSET=UFb2, GENERATE 2000702, 8000702, 2000000 ** *MPC LINEAR,UFp1, UFa1,UFb1 LINEAR,UFp2, UFa2,UFb2 ** *NSET, NSET=Wp1, GENERATE 2701, 18327, 1202 *NSET, NSET=Wa1, GENERATE 2100, 17726, 1202 *NSET, NSET=Wb1, GENERATE 3302, 18928, 1202 *NSET, NSET=Wp2, GENERATE 3402, 17826, 2404 *NSET, NSET=Wa2, GENERATE 2200, 16624, 2404 *NSET, NSET=Wb2, GENERATE 4604, 19028, 2404 *MPC LINEAR,Wp1, Wa1,Wb1 LINEAR,Wp2, Wa2,Wb2 ****CONSTRAIN WEB NODES & FLANGES NODES************ *NSET, NSET=A, GENERATE 2000, 2100, 1 *NSET, NSET=A1, GENERATE 4000002, 4000102, 1 *NSET, NSET=B, GENERATE 2102, 2200, 2 *NSET, NSET=B1, GENERATE 4000104, 4000202, 2 *NSET, NSET=C, GENERATE 2204, 2384, 4 *NSET, NSET=C1, GENERATE 4000206, 4000386, 4 *NSET, NSET=D, GENERATE 18828, 18928, 1 *NSET, NSET=D1, GENERATE 4000502, 4000602, 1 *NSET, NSET=E, GENERATE 18930, 19028, 2 *NSET, NSET=E1, GENERATE 4000604, 4000702, 2 *NSET, NSET=F, GENERATE 19032, 19212, 4 *NSET, NSET=F1, GENERATE 4000706, 4000886, 4 ** *MPC

63

BEAM, A, A1 BEAM, B, B1 BEAM, C, C1 BEAM, D, D1 BEAM, E, E1 BEAM, F, F1 ****STIFFENERS************************************ *NGEN, NSET=BFStiffenerNodes 22000, 4022000, 500000 *NCOPY, CHANGE NUMBER=8428, OLD SET=BFStiffenerNodes, SHIFT, NEW SET=UFStiffenerNodes 0, 0.2204, 0 ,, *NFILL, NSET=Stiffener1 BFStiffenerNodes, UFStiffenerNodes, 28, 301 *NCOPY, CHANGE NUMBER=7500000, OLD SET=Stiffener1, SHIFT, NEW SET=Stiffener2 0, 0, -0.06 ,, *ELEMENT, TYPE=S4 22000, 22000, 522000, 522301, 22301 7522000, 7522000, 8022000, 8022301, 7522301 ** *ELGEN, ELSET=Stiffener1Elements 22000, 28, 301, 301, 8, 500000, 500000 *ELGEN, ELSET=Stiffener2Elements 7522000, 28, 301, 301, 8, 500000, 500000 ** *SHELL SECTION, ELSET=Stiffener1Elements, MATERIAL=STEEL, OFFSET=0.5, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=Stiffener2Elements, MATERIAL=STEEL, OFFSET=0.5, POISSON=0.3 0.0098, 5 ** *NSET, NSET=WebStiffenerNodes1, GENERATE 4022301, 4030127, 301 *NSET, NSET=UFnodesForStiffener1, GENERATE 502, 4000502, 500000 *NSET, NSET=BFnodesForStiffener1, GENERATE 2, 4000002, 500000 *NSET, NSET=WebNodesForStiffener1, GENERATE 2601, 18227, 601 *NSET, NSET=WebNodesForStiffener2, GENERATE 2601, 18227, 601 *NSET, NSET=WebStiffenerNodes2, GENERATE 7522301, 7530127, 301 *NSET, NSET=UFnodesForStiffener2, GENERATE 4000502, 8000502, 500000 *NSET, NSET=UFStiffenerNodes2, GENERATE 7530428, 11530428, 500000 *NSET, NSET=BFnodesForStiffener2, GENERATE 4000002, 8000002, 500000 *NSET, NSET=BFStiffenerNodes2, GENERATE 7522000, 11522000, 500000 *MPC BEAM, BFStiffenerNodes, BFnodesForStiffener1 BEAM, UFStiffenerNodes, UFnodesForStiffener1 TIE, WebStiffenerNodes1, WebNodesForStiffener1 TIE, WebStiffenerNodes2, WebNodesForStiffener2 BEAM, UFStiffenerNodes2, UFnodesForStiffener2 BEAM, BFStiffenerNodes2, BFnodesForStiffener2 **% ****LOAD NODES GENERATION - LEVEL 0********************

64

*NGEN, NSET=LoadNodes1 20000514, 20000586, 24 *NGEN, NSET=LoadNodes2 20000610, 20000682, 24 *NGEN, NSET=LoadNodes3 20000706, 20000874, 24 ** *MPC LINK, 4000514, 20000514 LINK, 4000014, 20000514 LINK, 4000538, 20000538 LINK, 4000038, 20000538 LINK, 4000562, 20000562 LINK, 4000062, 20000562 LINK, 4000586, 20000586 LINK, 4000086, 20000586 LINK, 4000610, 20000610 LINK, 4000110, 20000610 LINK, 4000634, 20000634 LINK, 4000134, 20000634 LINK, 4000658, 20000658 LINK, 4000158, 20000658 LINK, 4000682, 20000682 LINK, 4000182, 20000682 LINK, 4000706, 20000706 LINK, 4000206, 20000706 LINK, 4000730, 20000730 LINK, 4000230, 20000730 LINK, 4000754, 20000754 LINK, 4000254, 20000754 LINK, 4000778, 20000778 LINK, 4000278, 20000778 LINK, 4000802, 20000802 LINK, 4000302, 20000802 LINK, 4000826, 20000826 LINK, 4000326, 20000826 LINK, 4000850, 20000850 LINK, 4000350, 20000850 LINK, 4000874, 20000874 LINK, 4000374, 20000874 ****LOAD NODES SYSTEM******************************************* *NODE, NSET=Keynodes 20000526, -0.18, 0.2302, 0.25 20000862, -2.70, 0.2302, 0.25 20001526, -0.18, 0.2302, 0.25 20001574, -0.54, 0.2302, 0.25 20001622, -0.90, 0.2302, 0.25 20001670, -1.26, 0.2302, 0.25 20001718, -1.62, 0.2302, 0.25 20001766, -1.98, 0.2302, 0.25 20001814, -2.34, 0.2302, 0.25 20001862, -2.70, 0.2302, 0.25 20002550, -0.36, 0.2302, 0.25 20002646, -1.08, 0.2302, 0.25 20002742, -1.80, 0.2302, 0.25 20002838, -2.52, 0.2302, 0.25 20003598, -0.72, 0.2302, 0.25 20003790, -2.16, 0.2302, 0.25 **** level 0 ***************** *NGEN, NSET=MiddleLoadNodes 20000526, 20000862, 48 ***** level 1 *************** *NGEN, NSET=LoadNodes11 20001526, 20001574, 24

65

*NGEN, NSET=LoadNodes12 20001622, 20001670, 24 *NGEN, NSET=LoadNodes13 20001718, 20001766, 24 *NGEN, NSET=LoadNodes14 20001814, 20001862, 24 ***** level 2 ************** *NGEN, NSET=LoadNodes21 20002550, 20002646, 48 *NGEN, NSET=LoadNodes22 20002742, 20002838, 48 ***** level 3 ************** *NGEN, NSET=LoadNodes31 20003598, 20003790, 96 ********************************** *ELEMENT, TYPE=B33 20001526, 20001526, 20001550 20001622, 20001622, 20001646 20001718, 20001718, 20001742 20001814, 20001814, 20001838 20002550, 20002550, 20002598 20002742, 20002742, 20002790 20003598, 20003598, 20003694 ** *ELGEN, ELSET=BeamElements1 20001526, 2, 24, 24 20001622, 2, 24, 24 20001718, 2, 24, 24 20001814, 2, 24, 24 20002550, 2, 48, 48 20002742, 2, 48, 48 20003598, 2, 96, 96 ********************************** *NSET, NSET=level1MPC, GENERATE 20001526, 20001862, 48 *MPC PIN, MiddleLoadNodes, level1MPC ** *NSET, NSET=level2MPC, GENERATE 20002550, 20002838, 96 *NSET, NSET=MiddleLevel1MPC, GENERATE 20001550, 20001838, 96 *MPC PIN, MiddleLevel1MPC, level2MPC *NSET, NSET=level3MPC, GENERATE 20003598, 20003790, 192 *NSET, NSET=MiddleLevel2MPC, GENERATE 20002598, 20002790, 192 *MPC PIN, MiddleLevel2MPC, level3MPC *NSET, NSET=ControlNode 20003694 ****BEAMS LEVEL 0 ****************************** *ELEMENT, TYPE=B33 20000514, 20000514, 20000526 20000562, 20000562, 20000574 20000610, 20000610, 20000622 20000658, 20000658, 20000670 20000706, 20000706, 20000718 20000754, 20000754, 20000766 20000802, 20000802, 20000814 20000850, 20000850, 20000862 ** *ELGEN, ELSET=BeamElements2

66

20000514, 2, 12, 12 20000562, 2, 12, 12 20000610, 2, 12, 12 20000658, 2, 12, 12 20000706, 2, 12, 12 20000754, 2, 12, 12 20000802, 2, 12, 12 20000850, 2, 12, 12 ** *ELSET, ELSET=AllLoadElements BeamElements1 BeamElements2 *BEAM GENERAL SECTION, SECTION=GENERAL, ELSET=AllLoadElements 0.003912, 3.892e-5, 0, 3.892e-5, 1.892e-5 0, 0, -1 210E9, 81E9 ****NSETS FOR THE BOUNDARY CONDITIONS FOR THE BEAM ELELMENTS************ *NSET, NSET=Level0BC, GENERATE 20000514, 20000850, 48 *NSET, NSET=Level1BC, GENERATE 20001526, 20001814, 96 *NSET, NSET=Level21BC 20002550 *NSET, NSET=Level22BC 20002742 *NSET, NSET=Level3BC 20003598 ****NSETS AND ELSETS NEEDED FOR THE RESULTS************ *NSET, NSET=BC1, GENERATE 2985, 18611, 601 *NSET, NSET=UpperFlangeLeft3, GENERATE 886, 8000886, 2000000 *NSET, NSET=ReactionNodeSets BC1 BottomFlangeLeft3 UpperFlangeLeft3 ControlNode *ELSET, ELSET=UFresults12 5500533 6000533 6000534 5500534 *ELSET, ELSET=UFresults34 1500533 2000533 2000534 1500534 *ELSET, ELSET=BFresults910 5500033 6000033 6000034 5500034 *ELSET, ELSET=BFresults1112 1500034 2000033 2000034 1500034 *ELSET, ELSET=WEBresults56 13450 14051 14052 13451 *ELSET, ELSET=WEBresults78

67

6238 6239 6840 6239 *ELSET, ELSET=MidSpanUFresults12 4000882 6000882 *ELSET, ELSET=MidSpanUFresults34 882 2000882 *ELSET, ELSET=MidSpanBFresults910 4000382 6000382 *ELSET, ELSET=MidSpanBFresults1112 382 2000382 *ELSET, ELSET=MidSpanWEBresults56 14400 11996 *ELSET, ELSET=MidSpanWEBresults78 7188 4784 *ELSET, ELSET=FlangesElsetsResults UFresults12 UFresults34 BFresults910 BFresults1112 *ELSET, ELSET=WebElsetsResults WEBresults56 WEBresults78 *ELSET, ELSET=MidSpanFlangesElsetsResults MidSpanUFresults12 MidSpanUFresults34 MidSpanBFresults910 MidSpanBFresults1112 *ELSET, ELSET=MidSpanWebElsetsResults MidSpanWEBresults56 MidSpanWEBresults78 ** ****NSETS NEEDED FOR THE BOUNDARY************ *NSET, NSET=WebNodesRightEnd , GENERATE 7522000, 7530428, 301 *NSET, NSET=RightMiddleWebNodeResults 7526214 *NSET, NSET=MiddleUFNode 4000502 *NSET, NSET=MiddleBFNode 4000002 ************************************* ** *MATERIAL, NAME=STEEL *DENSITY 7800.0 *ELASTIC 210E9, 0.3 *PLASTIC 3.55600E+08, 0 3.59564E+08, 1.1063E-02 5.39021E+08, 5.2778E-02 5.61000E+08, 9.2639E-02 ** *BOUNDARY BC1, XSYMM BottomFlangeLeft3, XSYMM

68

UpperFlangeLeft3, XSYMM ** *BOUNDARY **CASE 1 ControlNode, 1 Level0BC, 4 Level1BC, 4 Level21BC, 4 Level22BC, 4 Level3BC, 4 MiddleUFNode, 3 MiddleBFNode, 2, 3 ** **History Data ** *STEP, NLGEOM=YES, INC=300 *STATIC 0.01, 1.0 *BOUNDARY ControlNode, 2, 2, -0.5 ** *OUTPUT, FIELD *NODE OUTPUT CF, U, UR *ELEMENT OUTPUT S, E, PE *NODE OUTPUT, NSET=ReactionNodeSets RF, U , UR *NODE PRINT, NSET=ReactionNodeSets RF *NODE PRINT, NSET=ReactionNodeSets U, UR *NODE PRINT, NSET=RightMiddleWebNodeResults U, UR *NODE PRINT, NSET=WebNodesRightEnd RF3 *NODE PRINT, NSET=MiddleUFNode RF3 *NODE PRINT, NSET=MiddleBFNode RF3 *END STEP

Data lines for case 7 of the boundary conditions:

MiddleUFNode, 3

MiddleBFNode, 3

BottomFlangeRight, 2

69

Model 2 *HEADING I beam IPE240 – Model 2 ** **Model Definition *NODE, NSET=Keynodes 2, 0, 0, 0 8000002, 0, 0, -0.12 2000, 0, 0.0049, -0.06 18828, 0, 0.2253, -0.06 20000514, -0.09, 0.2302, 0.25 20000586, -0.63, 0.2302, 0.25 20000610, -0.81, 0.2302, 0.25 20000682, -1.35, 0.2302, 0.25 20000706, -1.53, 0.2302, 0.25 20000874, -2.79, 0.2302, 0.25 **% ****MESH 1*************************** **BOTTOM FLANGE***** *NGEN, NSET=BottomFlangeRight 2, 8000002, 500000 *NCOPY, CHANGE NUMBER=100, OLD SET=BottomFlangeRight, SHIFT, NEW SET=BottomFlangeLeft -0.75, 0, 0 ,, *NFILL, NSET=BottomFlange BottomFlangeRight, BottomFlangeLeft, 100, 1 ** **UPPER FLANGE***** ** *NCOPY, CHANGE NUMBER=500, OLD SET=BottomFlange, SHIFT, NEW SET=UpperFlange 0, 0.2302, 0 ,, **WEB********** *NGEN, NSET=WebRight 2000, 18828, 601 *NCOPY, CHANGE NUMBER=100, OLD SET=WebRight, SHIFT, NEW SET=WebLeft -0.75, 0, 0 ,, *NFILL, NSET=Web WebRight, WebLeft, 100, 1 ************** *ELEMENT, TYPE=S4 2, 2, 500002, 500003, 3 2000, 2000, 2601, 2602, 2001 502, 502, 500502, 500503, 503 ** *ELGEN, ELSET=BFlangeElements 2, 100, 1, 1, 16, 500000, 500000 *ELGEN, ELSET=WebElements 2000, 100, 1, 1, 28, 601, 601 *ELGEN, ELSET=UFlangeElements 502, 100, 1, 1, 16, 500000, 500000 ** *ELSET, ELSET=FinerMesh BFlangeElements WebElements UFlangeElements *SHELL SECTION, ELSET=UFlangeElements, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=BFlangeElements, MATERIAL=STEEL, OFFSET=0, POISSON=0.3

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0.0098, 5 *SHELL SECTION, ELSET=WebElements, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0062, 5 ****MESH 2*************************** *NGEN, NSET=BottomFlangeRight2 102, 8000102, 1000000 *NCOPY, CHANGE NUMBER=100, OLD SET=BottomFlangeRight2, SHIFT, NEW SET=BottomFlangeLeft2 -0.75, 0, 0 ,, *NFILL, NSET=BottomFlange2 BottomFlangeRight2, BottomFlangeLeft2, 50, 2 ** **UPPER FLANGE***** *NCOPY, CHANGE NUMBER=500, OLD SET=BottomFlange2, SHIFT, NEW SET=UpperFlange2 0, 0.2302, 0 ,, **WEB********** *NGEN, NSET=WebRight2 2100, 18928, 1202 *NCOPY, CHANGE NUMBER=100, OLD SET=WebRight2, SHIFT, NEW SET=WebLeft2 -0.75, 0, 0 ,, *NFILL, NSET=Web2 WebRight2, WebLeft2, 50, 2 ************** *ELEMENT, TYPE=S4 102, 102, 1000102, 1000104, 104 2100, 2100, 3302, 3304, 2102 602, 602, 1000602, 1000604, 604 ** *ELGEN, ELSET=BFlangeElements2 102, 50, 2, 2, 8, 1000000, 1000000 *ELGEN, ELSET=WebElements2 2100, 50, 2, 2, 14, 1202, 1202 *ELGEN, ELSET=UFlangeElements2 602, 50, 2, 2, 8, 1000000, 1000000 ** *ELSET, ELSET=MiddleCoarserMesh BFlangeElements2 WebElements2 UFlangeElements2 *SHELL SECTION, ELSET=UFlangeElements2, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=BFlangeElements2, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=WebElements2, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0062, 5 ****MESH 3*************************** *NGEN, NSET=BottomFlangeRight3 202, 8000202, 2000000 *NCOPY, CHANGE NUMBER=184, OLD SET=BottomFlangeRight3, SHIFT, NEW SET=BottomFlangeLeft3 -1.38, 0, 0 ,, *NFILL, NSET=BottomFlange3 BottomFlangeRight3, BottomFlangeLeft3, 46 , 4 **UPPER FLANGE*****

71

*NCOPY, CHANGE NUMBER=500, OLD SET=BottomFlange3, SHIFT, NEW SET=UpperFlange3 0, 0.2302, 0 ,, **WEB********** *NGEN, NSET=WebRight3 2200, 19028, 2404 *NCOPY, CHANGE NUMBER=184, OLD SET=WebRight3, SHIFT, NEW SET=WebLeft3 -1.38, 0, 0 ,, *NFILL, NSET=Web3 WebRight3, WebLeft3, 46, 4 ** ************** *ELEMENT, TYPE=S4 202, 202, 2000202, 2000206, 206 2200, 2200, 4604, 4608, 2204 702, 702, 2000702, 2000706, 706 *ELGEN, ELSET=BFlangeElements3 202, 46, 4, 4, 4, 2000000, 2000000 *ELGEN, ELSET=WebElements3 2200, 46, 4, 4, 7, 2404, 2404 *ELGEN, ELSET=UFlangeElements3 702, 46, 4, 4, 4, 2000000, 2000000 *ELSET, ELSET=MiddleEndMesh BFlangeElements3 WebElements3 UFlangeElements3 *SHELL SECTION, ELSET=UFlangeElements3, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=BFlangeElements3, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=WebElements3, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0062, 5 ****MESH REFINEMENT************************************ *NSET, NSET=BFp1, GENERATE 500102, 7500102, 1000000 *NSET, NSET=BFa1, GENERATE 102, 7000102, 1000000 *NSET, NSET=BFb1, GENERATE 1000102, 8000102, 1000000 *NSET, NSET=BFp2, GENERATE 1000202, 7000202, 2000000 *NSET, NSET=BFa2, GENERATE 202, 6000202, 2000000 *NSET, NSET=BFb2, GENERATE 2000202, 8000202, 2000000 *MPC LINEAR,BFp1, BFa1,BFb1 LINEAR,BFp2, BFa2,BFb2 ** *NSET, NSET=UFp1, GENERATE 500602, 7500602, 1000000 *NSET, NSET=UFa1, GENERATE 602, 7000602, 1000000 *NSET, NSET=UFb1, GENERATE 1000602, 8000602, 1000000 *NSET, NSET=UFp2, GENERATE 1000702, 7000702, 2000000 *NSET, NSET=UFa2, GENERATE 702, 6000702, 2000000

72

*NSET, NSET=UFb2, GENERATE 2000702, 8000702, 2000000 *MPC LINEAR,UFp1, UFa1,UFb1 LINEAR,UFp2, UFa2,UFb2 ** *NSET, NSET=Wp1, GENERATE 2701, 18327, 1202 *NSET, NSET=Wa1, GENERATE 2100, 17726, 1202 *NSET, NSET=Wb1, GENERATE 3302, 18928, 1202 *NSET, NSET=Wp2, GENERATE 3402, 17826, 2404 *NSET, NSET=Wa2, GENERATE 2200, 16624, 2404 *NSET, NSET=Wb2, GENERATE 4604, 19028, 2404 *MPC LINEAR,Wp1, Wa1,Wb1 LINEAR,Wp2, Wa2,Wb2 ****TRUSS ELEMENTS ONLY BETWEEN THE WEB NODES AT THE RIGHT END******** *ELEMENT, TYPE=T3D2 900, 2601, 3202 *ELGEN, ELSET=TrussElements 900, 26, 601, 1 *ELEMENT, TYPE=T3D2, ELSET=TrussElements2 930, 18227, 4000502 931, 2601, 4000002 *SOLID SECTION, ELSET=TrussElements, MATERIAL=STEEL 0.0025 *SOLID SECTION, ELSET=TrussElements2, MATERIAL=STEEL 0.0025 ****CONSTRAIN WEB NODES & FLANGES NODES*************** *NSET, NSET=A, GENERATE 2000, 2100, 1 *NSET, NSET=A1, GENERATE 4000002, 4000102, 1 *NSET, NSET=B, GENERATE 2102, 2200, 2 *NSET, NSET=B1, GENERATE 4000104, 4000202, 2 *NSET, NSET=C, GENERATE 2204, 2384, 4 *NSET, NSET=C1, GENERATE 4000206, 4000386, 4 *NSET, NSET=D, GENERATE 18828, 18928, 1 *NSET, NSET=D1, GENERATE 4000502, 4000602, 1 *NSET, NSET=E, GENERATE 18930, 19028, 2 *NSET, NSET=E1, GENERATE 4000604, 4000702, 2 *NSET, NSET=F, GENERATE 19032, 19212, 4 *NSET, NSET=F1, GENERATE 4000706, 4000886, 4 *MPC BEAM, A, A1 BEAM, B, B1 BEAM, C, C1 BEAM, D, D1 BEAM, E, E1

73

BEAM, F, F1 ** ****LOAD NODES GENERATION - LEVEL 0******************************* *NGEN, NSET=LoadNodes1 20000514, 20000586, 24 *NGEN, NSET=LoadNodes2 20000610, 20000682, 24 *NGEN, NSET=LoadNodes3 20000706, 20000874, 24 *MPC LINK, 4000514, 20000514 LINK, 4000014, 20000514 LINK, 4000538, 20000538 LINK, 4000038, 20000538 LINK, 4000562, 20000562 LINK, 4000062, 20000562 LINK, 4000586, 20000586 LINK, 4000086, 20000586 LINK, 4000610, 20000610 LINK, 4000110, 20000610 LINK, 4000634, 20000634 LINK, 4000134, 20000634 LINK, 4000658, 20000658 LINK, 4000158, 20000658 LINK, 4000682, 20000682 LINK, 4000182, 20000682 LINK, 4000706, 20000706 LINK, 4000206, 20000706 LINK, 4000730, 20000730 LINK, 4000230, 20000730 LINK, 4000754, 20000754 LINK, 4000254, 20000754 LINK, 4000778, 20000778 LINK, 4000278, 20000778 LINK, 4000802, 20000802 LINK, 4000302, 20000802 LINK, 4000826, 20000826 LINK, 4000326, 20000826 LINK, 4000850, 20000850 LINK, 4000350, 20000850 LINK, 4000874, 20000874 LINK, 4000374, 20000874 ** ****LOAD NODES SYSTEM****************************** *NODE, NSET=Keynodes 20000526, -0.18, 0.2302, 0.25 20000862, -2.70, 0.2302, 0.25 20001526, -0.18, 0.2302, 0.25 20001574, -0.54, 0.2302, 0.25 20001622, -0.90, 0.2302, 0.25 20001670, -1.26, 0.2302, 0.25 20001718, -1.62, 0.2302, 0.25 20001766, -1.98, 0.2302, 0.25 20001814, -2.34, 0.2302, 0.25 20001862, -2.70, 0.2302, 0.25 20002550, -0.36, 0.2302, 0.25 20002646, -1.08, 0.2302, 0.25 20002742, -1.80, 0.2302, 0.25 20002838, -2.52, 0.2302, 0.25 20003598, -0.72, 0.2302, 0.25 20003790, -2.16, 0.2302, 0.25 *** level 0 ***************** *NGEN, NSET=MiddleLoadNodes 20000526, 20000862, 48

74

***** level 1 *************** *NGEN, NSET=LoadNodes11 20001526, 20001574, 24 *NGEN, NSET=LoadNodes12 20001622, 20001670, 24 *NGEN, NSET=LoadNodes13 20001718, 20001766, 24 *NGEN, NSET=LoadNodes14 20001814, 20001862, 24 ***** level 2 ************** *NGEN, NSET=LoadNodes21 20002550, 20002646, 48 *NGEN, NSET=LoadNodes22 20002742, 20002838, 48 ***** level 3 ************* *NGEN, NSET=LoadNodes31 20003598, 20003790, 96 ********************************** *ELEMENT, TYPE=B33 20001526, 20001526, 20001550 20001622, 20001622, 20001646 20001718, 20001718, 20001742 20001814, 20001814, 20001838 20002550, 20002550, 20002598 20002742, 20002742, 20002790 20003598, 20003598, 20003694 ** *ELGEN, ELSET=BeamElements1 20001526, 2, 24, 24 20001622, 2, 24, 24 20001718, 2, 24, 24 20001814, 2, 24, 24 20002550, 2, 48, 48 20002742, 2, 48, 48 20003598, 2, 96, 96 ********************************** *NSET, NSET=level1MPC, GENERATE 20001526, 20001862, 48 *MPC PIN, MiddleLoadNodes, level1MPC *NSET, NSET=level2MPC, GENERATE 20002550, 20002838, 96 *NSET, NSET=MiddleLevel1MPC, GENERATE 20001550, 20001838, 96 *MPC PIN, MiddleLevel1MPC, level2MPC *NSET, NSET=level3MPC, GENERATE 20003598, 20003790, 192 *NSET, NSET=MiddleLevel2MPC, GENERATE 20002598, 20002790, 192 *MPC PIN, MiddleLevel2MPC, level3MPC *NSET, NSET=ControlNode 20003694 ****BEAMS LEVEL 0 ****************************** *ELEMENT, TYPE=B33 20000514, 20000514, 20000526 20000562, 20000562, 20000574 20000610, 20000610, 20000622 20000658, 20000658, 20000670 20000706, 20000706, 20000718 20000754, 20000754, 20000766 20000802, 20000802, 20000814 20000850, 20000850, 20000862

75

*ELGEN, ELSET=BeamElements2 20000514, 2, 12, 12 20000562, 2, 12, 12 20000610, 2, 12, 12 20000658, 2, 12, 12 20000706, 2, 12, 12 20000754, 2, 12, 12 20000802, 2, 12, 12 20000850, 2, 12, 12 *ELSET, ELSET=AllLoadElements BeamElements1 BeamElements2 *BEAM GENERAL SECTION, SECTION=GENERAL, ELSET=AllLoadElements 0.003912, 3.892e-5, 0, 3.892e-5, 1.892e-5 0, 0, -1 210E9, 81E9 ****NSETS FOR THE BOUNDARY CONDITIONS FOR THE BEAM ELELMENTS********* *NSET, NSET=Level0BC, GENERATE 20000514, 20000850, 48 *NSET, NSET=Level1BC, GENERATE 20001526, 20001814, 96 *NSET, NSET=Level21BC 20002550 *NSET, NSET=Level22BC 20002742 *NSET, NSET=Level3BC 20003598 ****NSETS AND ELSETS NEEDED FOR THE RESULTS*************************** *NSET, NSET=BC1, GENERATE 2985, 18611, 601 *NSET, NSET=UpperFlangeLeft3, GENERATE 886, 8000886, 2000000 ** *NSET, NSET=ReactionNodeSets BC1 BottomFlangeLeft3 UpperFlangeLeft3 ControlNode *ELSET, ELSET=UFresults12 5500533 6000533 6000534 5500534 *ELSET, ELSET=UFresults34 1500533 2000533 2000534 1500534 *ELSET, ELSET=BFresults910 5500033 6000033 6000034 5500034 *ELSET, ELSET=BFresults1112 1500034 2000033 2000034 1500034 *ELSET, ELSET=WEBresults56 13450 14051 14052 13451 *ELSET, ELSET=WEBresults78

76

6238 6239 6840 6239 *ELSET, ELSET=MidSpanUFresults12 4000882 6000882 *ELSET, ELSET=MidSpanUFresults34 882 2000882 *ELSET, ELSET=MidSpanBFresults910 4000382 6000382 *ELSET, ELSET=MidSpanBFresults1112 382 2000382 *ELSET, ELSET=MidSpanWEBresults56 14400 11996 *ELSET, ELSET=MidSpanWEBresults78 7188 4784 *ELSET, ELSET=FlangesElsetsResults UFresults12 UFresults34 BFresults910 BFresults1112 *ELSET, ELSET=WebElsetsResults WEBresults56 WEBresults78 *ELSET, ELSET=MidSpanFlangesElsetsResults MidSpanUFresults12 MidSpanUFresults34 MidSpanBFresults910 MidSpanBFresults1112 *ELSET, ELSET=MidSpanWebElsetsResults MidSpanWEBresults56 MidSpanWEBresults78 ****NSETS NEEDED FOR THE BOUNDARY ******************** *NSET, NSET=WebNodesRightEnd , GENERATE 2601, 18227, 601 *NSET, NSET=RightMiddleWebNodeResults 10414 *NSET, NSET=MiddleUFNode 4000502 *NSET, NSET=MiddleBFNode 4000002 ******************************************************* *MATERIAL, NAME=STEEL *DENSITY 7800.0 *ELASTIC 210E9, 0.3 *PLASTIC 3.55600E+08, 0 3.59564E+08, 1.1063E-02 5.39021E+08, 5.2778E-02 5.61000E+08, 9.2639E-02 ** *BOUNDARY BC1, XSYMM BottomFlangeLeft3, XSYMM UpperFlangeLeft3, XSYMM *BOUNDARY

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**CASE 1 ControlNode, 1 Level0BC, 4 Level1BC, 4 Level21BC, 4 Level22BC, 4 Level3BC, 4 MiddleUFNode, 3 MiddleBFNode, 2, 3 ** **History Data ** *STEP, NLGEOM=YES, INC=300 *STATIC 0.01, 1.0 *BOUNDARY ControlNode, 2, 2, -0.5 ** *OUTPUT, FIELD *NODE OUTPUT CF, U, UR *ELEMENT OUTPUT S, E, PE *NODE OUTPUT, NSET=ReactionNodeSets RF, U , UR *NODE PRINT, NSET=ReactionNodeSets RF *NODE PRINT, NSET=ReactionNodeSets U, UR *NODE PRINT, NSET=RightMiddleWebNodeResults U, UR *NODE PRINT, NSET=WebNodesRightEnd RF3 *NODE PRINT, NSET=MiddleUFNode RF3 *NODE PRINT, NSET=MiddleBFNode RF3 *END STEP

Model 3 *HEADING I beam IPE240 – Model 3 **Model Definition *NODE, NSET=Keynodes 2, 0, 0, 0 8000002, 0, 0, -0.12 2000, 0, 0.0049, -0.06 18828, 0, 0.2253, -0.06 22000, 0, 0.0049, 0 4022000, 0, 0.0049, -0.06 **% ****SHELL FINER************* **BOTTOM FLANGE***** *NGEN, NSET=BottomFlangeRight 2, 8000002, 500000 *NCOPY, CHANGE NUMBER=100, OLD SET=BottomFlangeRight, SHIFT, NEW SET=BottomFlangeLeft -0.75, 0, 0 ,, *NFILL, NSET=BottomFlange BottomFlangeRight, BottomFlangeLeft, 100, 1 **UPPER FLANGE*****

Data lines for case 2:

MiddleUFNode, 3

MiddleBFNode, 3

BottomFlangeRight, 2

Data lines for case 3:

WebNodesRightEnd, 3

MiddleUFNode, 3

MiddleBFNode, 2, 3

78

*NCOPY, CHANGE NUMBER=500, OLD SET=BottomFlange, SHIFT, NEW SET=UpperFlange 0, 0.2302, 0 ,, ** **WEB********** *NGEN, NSET=WebRight 2000, 18828, 601 *NCOPY, CHANGE NUMBER=100, OLD SET=WebRight, SHIFT, NEW SET=WebLeft -0.75, 0, 0 ,, *NFILL, NSET=Web WebRight, WebLeft, 100, 1 ************** *ELEMENT, TYPE=S4 2, 2, 500002, 500003, 3 2000, 2000, 2601, 2602, 2001 502, 502, 500502, 500503, 503 ** *ELGEN, ELSET=BFlangeElements 2, 100, 1, 1, 16, 500000, 500000 *ELGEN, ELSET=WebElements 2000, 100, 1, 1, 28, 601, 601 *ELGEN, ELSET=UFlangeElements 502, 100, 1, 1, 16, 500000, 500000 *ELSET, ELSET=FinerMesh BFlangeElements WebElements UFlangeElements *SHELL SECTION, ELSET=UFlangeElements, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=BFlangeElements, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=WebElements, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0062, 5 ****CONSTRAIN WEB NODES & FLANGES NODES***************** *NSET, NSET=A, GENERATE 2000, 2100, 1 *NSET, NSET=A1, GENERATE 4000002, 4000102, 1 *NSET, NSET=D, GENERATE 18828, 18928, 1 *NSET, NSET=D1, GENERATE 4000502, 4000602, 1 *MPC BEAM, A, A1 BEAM, D, D1 ****STIFFENERS******************************************************* *NGEN, NSET=BFStiffenerNodes 22000, 4022000, 500000 *NCOPY, CHANGE NUMBER=8428, OLD SET=BFStiffenerNodes, SHIFT, NEW SET=UFStiffenerNodes 0, 0.2204, 0 ,, *NFILL, NSET=Stiffener1 BFStiffenerNodes, UFStiffenerNodes, 28, 301 ** *NCOPY, CHANGE NUMBER=7500000, OLD SET=Stiffener1, SHIFT, NEW SET=Stiffener2 0, 0, -0.06 ,,

79

*ELEMENT, TYPE=S4 22000, 22000, 522000, 522301, 22301 7522000, 7522000, 8022000, 8022301, 7522301 *ELGEN, ELSET=Stiffener1Elements 22000, 28, 301, 301, 8, 500000, 500000 *ELGEN, ELSET=Stiffener2Elements 7522000, 28, 301, 301, 8, 500000, 500000 *SHELL SECTION, ELSET=Stiffener1Elements, MATERIAL=STEEL, OFFSET=0.5, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=Stiffener2Elements, MATERIAL=STEEL, OFFSET=0.5, POISSON=0.3 0.0098, 5 *NSET, NSET=WebStiffenerNodes1, GENERATE 4022301, 4030127, 301 *NSET, NSET=UFnodesForStiffener1, GENERATE 502, 4000502, 500000 *NSET, NSET=BFnodesForStiffener1, GENERATE 2, 4000002, 500000 *NSET, NSET=WebNodesForStiffener1, GENERATE 2601, 18227, 601 *NSET, NSET=WebNodesForStiffener2, GENERATE 2601, 18227, 601 *NSET, NSET=WebStiffenerNodes2, GENERATE 7522301, 7530127, 301 *NSET, NSET=UFnodesForStiffener2, GENERATE 4000502, 8000502, 500000 *NSET, NSET=UFStiffenerNodes2, GENERATE 7530428, 11530428, 500000 *NSET, NSET=BFnodesForStiffener2, GENERATE 4000002, 8000002, 500000 *NSET, NSET=BFStiffenerNodes2, GENERATE 7522000, 11522000, 500000 *MPC BEAM, BFStiffenerNodes, BFnodesForStiffener1 BEAM, UFStiffenerNodes, UFnodesForStiffener1 TIE, WebStiffenerNodes1, WebNodesForStiffener1 TIE, WebStiffenerNodes2, WebNodesForStiffener2 BEAM, UFStiffenerNodes2, UFnodesForStiffener2 BEAM, BFStiffenerNodes2, BFnodesForStiffener2 ****NSETS AND ELSETS NEEDED FOR THE RESULTS*********************** *NSET, NSET=BC1, GENERATE 2701, 18327, 601 *NSET, NSET=UpperFlangeLeft, GENERATE 602, 8000602, 500000 *NSET, NSET=ReactionNodeSets BC1 BottomFlangeLeft UpperFlangeLeft *ELSET, ELSET=UFresults12 5500533 6000533 6000534 5500534 *ELSET, ELSET=UFresults34 1500533 2000533 2000534 1500534 *ELSET, ELSET=BFresults910 5500033 6000033 6000034 5500034

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*ELSET, ELSET=BFresults1112 1500034 2000033 2000034 1500034 *ELSET, ELSET=WEBresults56 13450 14051 14052 13451 *ELSET, ELSET=WEBresults78 6238 6239 6840 6239 *ELSET, ELSET=MidSpanUFresults12 5500601 6000601 *ELSET, ELSET=MidSpanUFresults34 1500601 2000601 *ELSET, ELSET=MidSpanBFresults910 5500101 6000101 *ELSET, ELSET=MidSpanBFresults1112 1500101 2000101 *ELSET, ELSET=MidSpanWEBresults56 13518 14119 *ELSET, ELSET=MidSpanWEBresults78 6306 6907 *ELSET, ELSET=FlangesElsetsResults UFresults12 UFresults34 BFresults910 BFresults1112 *ELSET, ELSET=WebElsetsResults WEBresults56 WEBresults78 *ELSET, ELSET=MidSpanFlangesElsetsResults MidSpanUFresults12 MidSpanUFresults34 MidSpanBFresults910 MidSpanBFresults1112 *ELSET, ELSET=MidSpanWebElsetsResults MidSpanWEBresults56 MidSpanWEBresults78 *NSET, NSET=ControlNode 10514 ****NSETS NEEDED FOR THE BOUNDARY CONDITIONS - RIGHT END******** *NSET, NSET=WebNodesRightEnd , GENERATE 7522000, 7530428, 301 *NSET, NSET=RightMiddleWebNodeResults 7526214 *NSET, NSET=MiddleUFNode 4000502 *NSET, NSET=MiddleBFNode 4000002 *****MPC - ALL SUMMETRY NODES AT THE CONTROL NODE********************** *MPC BEAM, 602, 10514 BEAM,500602, 10514

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BEAM,1000602, 10514 BEAM,1500602, 10514 BEAM,2000602, 10514 BEAM,2500602, 10514 BEAM,3000602, 10514 BEAM,3500602, 10514 BEAM,4000602, 10514 BEAM,4500602, 10514 BEAM,5000602, 10514 BEAM,5500602, 10514 BEAM,6000602, 10514 BEAM,6500602, 10514 BEAM,7000602, 10514 BEAM,7500602, 10514 BEAM,8000602, 10514 *MPC BEAM, 102, 10514 BEAM, 500102, 10514 BEAM, 1000102, 10514 BEAM, 1500102, 10514 BEAM, 2000102, 10514 BEAM, 2500102, 10514 BEAM, 3000102, 10514 BEAM, 3500102, 10514 BEAM, 4000102, 10514 BEAM, 4500102, 10514 BEAM, 5000102, 10514 BEAM, 5500102, 10514 BEAM, 6000102, 10514 BEAM, 6500102, 10514 BEAM, 7000102, 10514 BEAM, 7500102, 10514 BEAM, 8000102, 10514 *MPC BEAM, 2701, 10514 BEAM, 3302, 10514 BEAM, 3903, 10514 BEAM, 4504, 10514 BEAM, 5105, 10514 BEAM, 5706, 10514 BEAM, 6307, 10514 BEAM, 6908, 10514 BEAM, 7509, 10514 BEAM, 8110, 10514 BEAM, 8711, 10514 BEAM, 9312, 10514 BEAM, 9913, 10514 BEAM, 11115, 10514 BEAM, 11716, 10514 BEAM, 12317, 10514 BEAM, 12918, 10514 BEAM, 13519, 10514 BEAM, 14120, 10514 BEAM, 14721, 10514 BEAM, 15322, 10514 BEAM, 15923, 10514 BEAM, 16524, 10514 BEAM, 17125, 10514 BEAM, 17726, 10514 BEAM, 18327, 10514 ***************************************** *MATERIAL, NAME=STEEL *DENSITY 7800.0

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*ELASTIC 210E9, 0.3 *PLASTIC 3.55600E+08, 0 3.59564E+08, 1.1063E-02 5.39021E+08, 5.2778E-02 5.61000E+08, 9.2639E-02 *BOUNDARY MiddleUFNode, 3 MiddleBFNode, 2, 3 ControlNode, 1 ControlNode, 3 ControlNode, 5 ControlNode, 6 **History Data *STEP, NLGEOM=YES, INC=300 *STATIC 0.01, 1.0 *BOUNDARY ControlNode, 2, 2, -0.3 ControlNode, 4, 4, 0.2 ** *OUTPUT, FIELD *NODE OUTPUT CF, U, UR *ELEMENT OUTPUT S, E, PE *NODE OUTPUT, NSET=ReactionNodeSets RF, U , UR *NODE PRINT, NSET=ReactionNodeSets RF *NODE PRINT, NSET=ReactionNodeSets U, UR *NODE PRINT, NSET=RightMiddleWebNodeResults U, UR *NODE PRINT, NSET=WebNodesRightEnd RF3 *NODE PRINT, NSET=MiddleUFNode RF3 *NODE PRINT, NSET=MiddleBFNode RF3 *END STEP

Model 4 *HEADING I beam IPE240 – Model 4 **Model Definition *NODE, NSET=Keynodes1 2, 0, 0, 0 8000002, 0, 0, -0.12 2000, 0, 0.0049, -0.06 18828, 0, 0.2253, -0.06 22000, 0, 0.0049, 0 4022000, 0, 0.0049, -0.06 20000506, -0.03, 0.2302, 0.25 20000594, -0.69, 0.2302, 0.25 ****SHELL FINER *************************************************** **BOTTOM FLANGE***** *NGEN, NSET=BottomFlangeRight 2, 8000002, 500000 *NCOPY, CHANGE NUMBER=96, OLD SET=BottomFlangeRight, SHIFT, NEW SET=BottomFlangeLeft -0.72, 0, 0 ,,

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*NFILL, NSET=BottomFlange BottomFlangeRight, BottomFlangeLeft, 96, 1 **UPPER FLANGE***** *NCOPY, CHANGE NUMBER=500, OLD SET=BottomFlange, SHIFT, NEW SET=UpperFlange 0, 0.2302, 0 ,, **WEB********** *NGEN, NSET=WebRight 2000, 18828, 601 *NCOPY, CHANGE NUMBER=96, OLD SET=WebRight, SHIFT, NEW SET=WebLeft -0.72, 0, 0 ,, *NFILL, NSET=Web WebRight, WebLeft, 96, 1 ************** *ELEMENT, TYPE=S4 2, 2, 500002, 500003, 3 2000, 2000, 2601, 2602, 2001 502, 502, 500502, 500503, 503 *ELGEN, ELSET=BFlangeElements 2, 96, 1, 1, 16, 500000, 500000 *ELGEN, ELSET=WebElements 2000, 96, 1, 1, 28, 601, 601 *ELGEN, ELSET=UFlangeElements 502, 96, 1, 1, 16, 500000, 500000 *ELSET, ELSET=FinerMesh BFlangeElements WebElements UFlangeElements *SHELL SECTION, ELSET=UFlangeElements, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=BFlangeElements, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=WebElements, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0062, 5 ****CONSTRAIN WEB NODES & FLANGES NODES***************** *NSET, NSET=A, GENERATE 2000, 2096, 1 *NSET, NSET=A1, GENERATE 4000002, 4000098, 1 *NSET, NSET=D, GENERATE 18828, 18924, 1 *NSET, NSET=D1, GENERATE 4000502, 4000598, 1 *MPC BEAM, A, A1 BEAM, D, D1 ****STIFFENERS********************************************************* *NGEN, NSET=BFStiffenerNodes 22000, 4022000, 500000 *NCOPY, CHANGE NUMBER=8428, OLD SET=BFStiffenerNodes, SHIFT, NEW SET=UFStiffenerNodes 0, 0.2204, 0 ,, *NFILL, NSET=Stiffener1 BFStiffenerNodes, UFStiffenerNodes, 28, 301 *NCOPY, CHANGE NUMBER=7500000, OLD SET=Stiffener1, SHIFT, NEW SET=Stiffener2 0, 0, -0.06 ,,

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*ELEMENT, TYPE=S4 22000, 22000, 522000, 522301, 22301 7522000, 7522000, 8022000, 8022301, 7522301 ** *ELGEN, ELSET=Stiffener1Elements 22000, 28, 301, 301, 8, 500000, 500000 *ELGEN, ELSET=Stiffener2Elements 7522000, 28, 301, 301, 8, 500000, 500000 *SHELL SECTION, ELSET=Stiffener1Elements, MATERIAL=STEEL, OFFSET=0.5, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=Stiffener2Elements, MATERIAL=STEEL, OFFSET=0.5, POISSON=0.3 0.0098, 5 *NSET, NSET=WebStiffenerNodes1, GENERATE 4022301, 4030127, 301 *NSET, NSET=UFnodesForStiffener1, GENERATE 502, 4000502, 500000 *NSET, NSET=BFnodesForStiffener1, GENERATE 2, 4000002, 500000 *NSET, NSET=WebNodesForStiffener1, GENERATE 2601, 18227, 601 *NSET, NSET=WebNodesForStiffener2, GENERATE 2601, 18227, 601 *NSET, NSET=WebStiffenerNodes2, GENERATE 7522301, 7530127, 301 *NSET, NSET=UFnodesForStiffener2, GENERATE 4000502, 8000502, 500000 *NSET, NSET=UFStiffenerNodes2, GENERATE 7530428, 11530428, 500000 *NSET, NSET=BFnodesForStiffener2, GENERATE 4000002, 8000002, 500000 *NSET, NSET=BFStiffenerNodes2, GENERATE 7522000, 11522000, 500000 *MPC BEAM, BFStiffenerNodes, BFnodesForStiffener1 BEAM, UFStiffenerNodes, UFnodesForStiffener1 TIE, WebStiffenerNodes1, WebNodesForStiffener1 TIE, WebStiffenerNodes2, WebNodesForStiffener2 BEAM, UFStiffenerNodes2, UFnodesForStiffener2 BEAM, BFStiffenerNodes2, BFnodesForStiffener2 ****LOAD NODES GENERATION - LEVEL 0 ****************************** *NGEN, NSET=LoadNodes 20000506, 20000594, 8 *MPC LINK, 4000506, 20000506 LINK, 4000006, 20000506 LINK, 4000514, 20000514 LINK, 4000014, 20000514 LINK, 4000522, 20000522 LINK, 4000022, 20000522 LINK, 4000530, 20000530 LINK, 4000030, 20000530 LINK, 4000538, 20000538 LINK, 4000038, 20000538 LINK, 4000546, 20000546 LINK, 4000046, 20000546 LINK, 4000554, 20000554 LINK, 4000054, 20000554 LINK, 4000562, 20000562 LINK, 4000062, 20000562 LINK, 4000570, 20000570 LINK, 4000070, 20000570 LINK, 4000578, 20000578

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LINK, 4000078, 20000578 LINK, 4000586, 20000586 LINK, 4000086, 20000586 LINK, 4000594, 20000594 LINK, 4000094, 20000594 ****LOAD NODES SYSTEM************************************************ ****NODE GENARATION - FOR THE LOAD NODES **************************** *NODE, NSET=Keynodes2 20001506, -0.03, 0.2302, 0.25 20001514, -0.09, 0.2302, 0.25 20001522, -0.15, 0.2302, 0.25 20001530, -0.21, 0.2302, 0.25 20001538, -0.27, 0.2302, 0.25 20001546, -0.33, 0.2302, 0.25 20001554, -0.39, 0.2302, 0.25 20001562, -0.45, 0.2302, 0.25 20001570, -0.51, 0.2302, 0.25 20001578, -0.57, 0.2302, 0.25 20001586, -0.63, 0.2302, 0.25 20001594, -0.69, 0.2302, 0.25 20002510, -0.06, 0.2302, 0.25 20002526, -0.18, 0.2302, 0.25 20002542, -0.3, 0.2302, 0.25 20002558, -0.42, 0.2302, 0.25 20002574, -0.54, 0.2302, 0.25 20002590, -0.66, 0.2302, 0.25 20003518, -0.12, 0.2302, 0.25 20003534, -0.2, 0.2302, 0.25 20003550, -0.36, 0.2302, 0.25 20003566, -0.52, 0.2302, 0.25 20003582, -0.6, 0.2302, 0.25 20004534, -0.2, 0.2302, 0.25 20004550, -0.36, 0.2302, 0.25 20004566, -0.52, 0.2302, 0.25 ***** level 1 *************** *NGEN, NSET=LoadNodes11 20001506, 20001514, 4 *NGEN, NSET=LoadNodes12 20001522, 20001530, 4 *NGEN, NSET=LoadNodes13 20001538, 20001546, 4 *NGEN, NSET=LoadNodes14 20001554, 20001562, 4 *NGEN, NSET=LoadNodes15 20001570, 20001578, 4 *NGEN, NSET=LoadNodes16 20001586, 20001594, 4 ******** level 2 ************** *NGEN, NSET=LoadNodes21 20002510, 20002526, 8 *NGEN, NSET=LoadNodes22 20002542, 20002558, 8 *NGEN, NSET=LoadNodes23 20002574, 20002590, 8 ****ELEMENT GENARATION - FOR THE LOAD NODES************************ *ELEMENT, TYPE=B33, ELSET=BeamElements1 20001506, 20001506, 20001510 20001522, 20001522, 20001526 20001538, 20001538, 20001542 20001554, 20001554, 20001558 20001570, 20001570, 20001574 20001586, 20001586, 20001590 20002510, 20002510, 20002518 20002542, 20002542, 20002550

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20002574, 20002574, 20002582 *ELEMENT, TYPE=B33, ELSET=BeamElements2 20003518, 20003518, 20003534 20003534, 20003534, 20003550 20003550, 20003550, 20003566 20003566, 20003566, 20003582 20004534, 20004534, 20004550 20004550, 20004550, 20004566 *ELGEN, ELSET=BeamElements3 20001506, 2, 4, 4 20001522, 2, 4, 4 20001538, 2, 4, 4 20001554, 2, 4, 4 20001570, 2, 4, 4 20001586, 2, 4, 4 20002510, 2, 8, 8 20002542, 2, 8, 8 20002574, 2, 8, 8 ****MPC PIN - FOR THE LOAD NODES******************************* *MPC PIN, 20000506, 20001506 PIN, 20000514, 20001514 PIN, 20000522, 20001522 PIN, 20000530, 20001530 PIN, 20000538, 20001538 PIN, 20000546, 20001546 PIN, 20000554, 20001554 PIN, 20000562, 20001562 PIN, 20000570, 20001570 PIN, 20000578, 20001578 PIN, 20000586, 20001586 PIN, 20000594, 20001594 *NSET, NSET=MiddleNodesLevel1, GENERATE 20001510, 20001590, 16 *NSET, NSET=NodesLevel2, GENERATE 20002510, 20002590, 16 *MPC PIN, MiddleNodesLevel1, NodesLevel2 *NSET, NSET=MiddleNodesLevel2, GENERATE 20002518, 20002582, 32 *NSET, NSET=NodesLevel3, GENERATE 20003518, 20003582 ,32 *MPC PIN, MiddleNodesLevel2, NodesLevel3 *MPC PIN, 20003534, 20004534 PIN, 20003566, 20004566 *********************************** *NSET, NSET=LoadControlNode 20004550 *ELSET, ELSET=AllLoadElements BeamElements1 BeamElements2 BeamElements3 *BEAM GENERAL SECTION, SECTION=GENERAL, ELSET=AllLoadElements 0.003912, 3.892e-5, 0, 3.892e-5, 1.892e-5 0, 0, -1 210E9, 81E9 ****NSETS FOR THE BOUNDARY CONDITIONS FOR THE BEAM ELELMENTS****** *NSET, NSET=Level1BC, GENERATE 20001506, 20001586, 16 *NSET, NSET=Level2BC, GENERATE 20002510, 20002574, 32 *NSET, NSET=Level3BC, GENERATE

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20003518, 20003550, 32 *NSET, NSET=Level4BC 20004534 ****NSETS AND ELSETS NEEDED FOR THE RESULTS*************************** *NSET, NSET=BC1, GENERATE 2697, 18323, 601 *NSET, NSET=UpperFlangeLeft, GENERATE 598, 8000598, 500000 *NSET, NSET=ReactionNodeSets BC1 BottomFlangeLeft UpperFlangeLeft *NSET, NSET=LoadControlNode 20004550 *ELSET, ELSET=UFresults12 5500533 6000533 6000534 5500534 *ELSET, ELSET=UFresults34 1500533 2000533 2000534 1500534 *ELSET, ELSET=BFresults910 5500033 6000033 6000034 5500034 *ELSET, ELSET=BFresults1112 1500033 2000033 2000034 1500034 *ELSET, ELSET=WEBresults56 13450 14051 14052 13451 *ELSET, ELSET=WEBresults78 6238 6239 6840 6839 *ELSET, ELSET=MidSpanUFresults12 5500597 6000597 *ELSET, ELSET=MidSpanUFresults34 1500597 2000597 *ELSET, ELSET=MidSpanBFresults910 5500097 6000097 *ELSET, ELSET=MidSpanBFresults1112 1500097 2000097 *ELSET, ELSET=MidSpanWEBresults56 13514 14115 *ELSET, ELSET=MidSpanWEBresults78 6302 6903 *ELSET, ELSET=FlangesElsetsResults UFresults12

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UFresults34 BFresults910 BFresults1112 *ELSET, ELSET=WebElsetsResults WEBresults56 WEBresults78 *ELSET, ELSET=MidSpanFlangesElsetsResults MidSpanUFresults12 MidSpanUFresults34 MidSpanBFresults910 MidSpanBFresults1112 *ELSET, ELSET=MidSpanWebElsetsResults MidSpanWEBresults56 MidSpanWEBresults78 ****NSETS NEEDED FOR THE BOUNDARY CONDITIONS - RIGHT END******** *NSET, NSET=WebNodesRightEnd , GENERATE 7522000, 7530428, 301 *NSET, NSET=RightMiddleWebNodeResults 7526214 *NSET, NSET=MiddleUFNode 4000502 *NSET, NSET=MiddleBFNode 4000002 *NSET, NSET=WebControlNode 10510 ****************************************** *MATERIAL, NAME=STEEL *DENSITY 7800.0 *ELASTIC 210E9, 0.3 *PLASTIC 3.55600E+08, 0 3.59564E+08, 1.1063E-02 5.39021E+08, 5.2778E-02 5.61000E+08, 9.2639E-02 *BOUNDARY BC1, XSYMM BottomFlangeLeft, XSYMM UpperFlangeLeft, XSYMM *BOUNDARY Level1BC, 4 Level2BC, 4 Level3BC, 4 Level4BC, 4 LoadControlNode, 1 MiddleUFNode, 3 MiddleBFNode, 2, 3 **History Data ** *STEP, NLGEOM=YES, INC=300 *STATIC 0.01, 1.0 *BOUNDARY LoadControlNode, 2, 2, -0.3 *OUTPUT, FIELD *NODE OUTPUT CF, U, UR *ELEMENT OUTPUT S, E, PE *NODE OUTPUT, NSET=ReactionNodeSets RF, U , UR *NODE PRINT, NSET=ReactionNodeSets RF

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*NODE PRINT, NSET=ReactionNodeSets U, UR *NODE PRINT, NSET=RightMiddleWebNodeResults U, UR *NODE PRINT, NSET=WebNodesRightEnd RF3 *NODE PRINT, NSET=MiddleUFNode RF3 *NODE PRINT, NSET=MiddleBFNode RF3 *EL PRINT, POSITION=AVERAGED AT NODES, ELSET=FlangesElsetsResults S *EL PRINT, POSITION=AVERAGED AT NODES, ELSET=WebElsetsResults S *EL PRINT, POSITION=AVERAGED AT NODES, ELSET=MidSpanFlangesElsetsResults S *EL PRINT, POSITION=AVERAGED AT NODES, ELSET=MidSpanWebElsetsResults S *END STEP

Model 5 *HEADING I beam IPE240 – Model 5 **Model Definition *NODE, NSET=Keynodes1 2, 0, 0, 0 8000002, 0, 0, -0.12 2000, 0, 0.0049, -0.06 18828, 0, 0.2253, -0.06 22000, 0, 0.0049, 0 4022000, 0, 0.0049, -0.06 20000506, -0.03, 0.2302, 0.25 20000594, -0.69, 0.2302, 0.25 **BOTTOM FLANGE***** *NGEN, NSET=BottomFlangeRight 2, 8000002, 500000 *NCOPY, CHANGE NUMBER=96, OLD SET=BottomFlangeRight, SHIFT, NEW SET=BottomFlangeLeft -0.72, 0, 0 ,, *NFILL, NSET=BottomFlange BottomFlangeRight, BottomFlangeLeft, 96, 1 **UPPER FLANGE***** *NCOPY, CHANGE NUMBER=500, OLD SET=BottomFlange, SHIFT, NEW SET=UpperFlange 0, 0.2302, 0 ,, **WEB********** *NGEN, NSET=WebRight 2000, 18828, 601 *NCOPY, CHANGE NUMBER=96, OLD SET=WebRight, SHIFT, NEW SET=WebLeft -0.72, 0, 0 ,, *NFILL, NSET=Web WebRight, WebLeft, 96, 1 ************** *ELEMENT, TYPE=S4 2, 2, 500002, 500003, 3 2000, 2000, 2601, 2602, 2001 502, 502, 500502, 500503, 503 ** *ELGEN, ELSET=BFlangeElements 2, 96, 1, 1, 16, 500000, 500000

90

*ELGEN, ELSET=WebElements 2000, 96, 1, 1, 28, 601, 601 *ELGEN, ELSET=UFlangeElements 502, 96, 1, 1, 16, 500000, 500000 *ELSET, ELSET=FinerMesh BFlangeElements WebElements UFlangeElements *SHELL SECTION, ELSET=UFlangeElements, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=BFlangeElements, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=WebElements, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0062, 5 ****CONSTRAIN WEB NODES & FLANGES NODES***************** *NSET, NSET=A, GENERATE 2000, 2096, 1 *NSET, NSET=A1, GENERATE 4000002, 4000098, 1 *NSET, NSET=B, GENERATE 18828, 18924, 1 *NSET, NSET=B1, GENERATE 4000502, 4000598, 1 *MPC BEAM, A, A1 BEAM, B, B1 ** ****STIFFENERS****************************************** *NGEN, NSET=BFStiffenerNodes 22000, 4022000, 500000 *NCOPY, CHANGE NUMBER=8428, OLD SET=BFStiffenerNodes, SHIFT, NEW SET=UFStiffenerNodes 0, 0.2204, 0 ,, *NFILL, NSET=Stiffener1 BFStiffenerNodes, UFStiffenerNodes, 28, 301 ** *NCOPY, CHANGE NUMBER=7500000, OLD SET=Stiffener1, SHIFT, NEW SET=Stiffener2 0, 0, -0.06 ,, *ELEMENT, TYPE=S4 22000, 22000, 522000, 522301, 22301 7522000, 7522000, 8022000, 8022301, 7522301 *ELGEN, ELSET=Stiffener1Elements 22000, 28, 301, 301, 8, 500000, 500000 *ELGEN, ELSET=Stiffener2Elements 7522000, 28, 301, 301, 8, 500000, 500000 *NCOPY, CHANGE NUMBER=98, OLD SET=Stiffener1, SHIFT, NEW SET=Stiffener3 -0.72, 0, 0 ,, *NCOPY, CHANGE NUMBER=7500000, OLD SET=Stiffener3, SHIFT, NEW SET=Stiffener4 0, 0, -0.06 ,, *ELCOPY, ELEMENT SHIFT=98, OLD SET=Stiffener1Elements, SHIFT NODES=98 , NEW SET=Stiffener3Elements *ELCOPY, ELEMENT SHIFT=98, OLD SET=Stiffener2Elements, SHIFT NODES=98 , NEW SET=Stiffener4Elements *SHELL SECTION, ELSET=Stiffener1Elements, MATERIAL=STEEL, OFFSET=0.5, POISSON=0.3

91

0.0098, 5 *SHELL SECTION, ELSET=Stiffener2Elements, MATERIAL=STEEL, OFFSET=0.5, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=Stiffener3Elements, MATERIAL=STEEL, OFFSET=-0.5, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=Stiffener4Elements, MATERIAL=STEEL, OFFSET=-0.5, POISSON=0.3 0.0098, 5 *NSET, NSET=C, GENERATE 4022301, 4030127, 301 *NSET, NSET=C1, GENERATE 2601, 18227, 601 *NSET, NSET=D, GENERATE 7522301, 7530127, 301 *NSET, NSET=D1, GENERATE 2601, 18227, 601 *NSET, NSET=E1, GENERATE 2, 4000002, 500000 *NSET, NSET=F, GENERATE 7522000, 11522000, 500000 *NSET, NSET=F1, GENERATE 4000002, 8000002, 500000 *NSET, NSET=G1, GENERATE 502, 4000502, 500000 *NSET, NSET=H, GENERATE 7530428, 11530428, 500000 *NSET, NSET=H1, GENERATE 4000502, 8000502, 500000 *MPC TIE, C, C1 TIE, D, D1 BEAM, BFStiffenerNodes, E1 BEAM, F, F1 BEAM, UFStiffenerNodes, G1 BEAM, H, H1 ** *NSET, NSET=I, GENERATE 4022399, 4030225, 301 *NSET, NSET=I1, GENERATE 2697, 18323, 601 *NSET, NSET=J, GENERATE 7522399, 7530225, 301 *NSET, NSET=J1, GENERATE 2697, 18323, 601 *NSET, NSET=K, GENERATE 22098, 4022098, 500000 *NSET, NSET=K1, GENERATE 98, 4000098, 500000 *NSET, NSET=L, GENERATE 7522098, 11522098, 500000 *NSET, NSET=L1, GENERATE 4000098, 8000098, 500000 *NSET, NSET=M, GENERATE 30526, 4030526, 500000 *NSET, NSET=M1, GENERATE 598, 4000598, 500000 *NSET, NSET=N, GENERATE 7530526, 11530526, 500000 *NSET, NSET=N1, GENERATE 4000598, 8000598, 500000 *MPC TIE, I, I1

92

TIE, J, J1 BEAM, K, K1 BEAM, L, L1 BEAM, M, M1 BEAM, N, N1 ****LOAD NODES GENERATION - LEVEL 0 ******************************* *NGEN, NSET=LoadNodes 20000506, 20000594, 8 *MPC LINK, 4000506, 20000506 LINK, 4000006, 20000506 LINK, 4000514, 20000514 LINK, 4000014, 20000514 LINK, 4000522, 20000522 LINK, 4000022, 20000522 LINK, 4000530, 20000530 LINK, 4000030, 20000530 LINK, 4000538, 20000538 LINK, 4000038, 20000538 LINK, 4000546, 20000546 LINK, 4000046, 20000546 LINK, 4000554, 20000554 LINK, 4000054, 20000554 LINK, 4000562, 20000562 LINK, 4000062, 20000562 LINK, 4000570, 20000570 LINK, 4000070, 20000570 LINK, 4000578, 20000578 LINK, 4000078, 20000578 LINK, 4000586, 20000586 LINK, 4000086, 20000586 LINK, 4000594, 20000594 LINK, 4000094, 20000594 ****LOAD NODES SYSTEM************************************************* ****NODE GENARATION - FOR THE LOAD NODES ***************************** *NODE, NSET=Keynodes2 20001506, -0.03, 0.2302, 0.25 20001514, -0.09, 0.2302, 0.25 20001522, -0.15, 0.2302, 0.25 20001530, -0.21, 0.2302, 0.25 20001538, -0.27, 0.2302, 0.25 20001546, -0.33, 0.2302, 0.25 20001554, -0.39, 0.2302, 0.25 20001562, -0.45, 0.2302, 0.25 20001570, -0.51, 0.2302, 0.25 20001578, -0.57, 0.2302, 0.25 20001586, -0.63, 0.2302, 0.25 20001594, -0.69, 0.2302, 0.25 20002510, -0.06, 0.2302, 0.25 20002526, -0.18, 0.2302, 0.25 20002542, -0.3, 0.2302, 0.25 20002558, -0.42, 0.2302, 0.25 20002574, -0.54, 0.2302, 0.25 20002590, -0.66, 0.2302, 0.25 20003518, -0.12, 0.2302, 0.25 20003534, -0.2, 0.2302, 0.25 20003550, -0.36, 0.2302, 0.25 20003566, -0.52, 0.2302, 0.25 20003582, -0.6, 0.2302, 0.25 20004534, -0.2, 0.2302, 0.25 20004550, -0.36, 0.2302, 0.25 20004566, -0.52, 0.2302, 0.25 ***** level 1 *************** *NGEN, NSET=LoadNodes11

93

20001506, 20001514, 4 *NGEN, NSET=LoadNodes12 20001522, 20001530, 4 *NGEN, NSET=LoadNodes13 20001538, 20001546, 4 *NGEN, NSET=LoadNodes14 20001554, 20001562, 4 *NGEN, NSET=LoadNodes15 20001570, 20001578, 4 *NGEN, NSET=LoadNodes16 20001586, 20001594, 4 ******** level 2 ************** *NGEN, NSET=LoadNodes21 20002510, 20002526, 8 *NGEN, NSET=LoadNodes22 20002542, 20002558, 8 *NGEN, NSET=LoadNodes23 20002574, 20002590, 8 ****ELEMENT GENARATION - FOR THE LOAD NODES************************* *ELEMENT, TYPE=B33, ELSET=BeamElements1 20001506, 20001506, 20001510 20001522, 20001522, 20001526 20001538, 20001538, 20001542 20001554, 20001554, 20001558 20001570, 20001570, 20001574 20001586, 20001586, 20001590 20002510, 20002510, 20002518 20002542, 20002542, 20002550 20002574, 20002574, 20002582 *ELEMENT, TYPE=B33, ELSET=BeamElements2 20003518, 20003518, 20003534 20003534, 20003534, 20003550 20003550, 20003550, 20003566 20003566, 20003566, 20003582 20004534, 20004534, 20004550 20004550, 20004550, 20004566 *ELGEN, ELSET=BeamElements3 20001506, 2, 4, 4 20001522, 2, 4, 4 20001538, 2, 4, 4 20001554, 2, 4, 4 20001570, 2, 4, 4 20001586, 2, 4, 4 20002510, 2, 8, 8 20002542, 2, 8, 8 20002574, 2, 8, 8 ****MPC PIN - FOR THE LOAD NODES******************************* *MPC PIN, 20000506, 20001506 PIN, 20000514, 20001514 PIN, 20000522, 20001522 PIN, 20000530, 20001530 PIN, 20000538, 20001538 PIN, 20000546, 20001546 PIN, 20000554, 20001554 PIN, 20000562, 20001562 PIN, 20000570, 20001570 PIN, 20000578, 20001578 PIN, 20000586, 20001586 PIN, 20000594, 20001594 ** *NSET, NSET=MiddleNodesLevel1, GENERATE 20001510, 20001590, 16 *NSET, NSET=NodesLevel2, GENERATE

94

20002510, 20002590, 16 *MPC PIN, MiddleNodesLevel1, NodesLevel2 *NSET, NSET=MiddleNodesLevel2, GENERATE 20002518, 20002582, 32 *NSET, NSET=NodesLevel3, GENERATE 20003518, 20003582 ,32 *MPC PIN, MiddleNodesLevel2, NodesLevel3 *MPC PIN, 20003534, 20004534 PIN, 20003566, 20004566 *********************************** *NSET, NSET=LoadControlNode 20004550 *ELSET, ELSET=AllLoadElements BeamElements1 BeamElements2 BeamElements3 *BEAM GENERAL SECTION, SECTION=GENERAL, ELSET=AllLoadElements 0.003912, 3.892e-5, 0, 3.892e-5, 1.892e-5 0, 0, -1 210E9, 81E9 ****NSETS FOR THE BOUNDARY CONDITIONS FOR THE BEAM ELELMENTS element********** *NSET, NSET=Level1BC, GENERATE 20001506, 20001586, 16 *NSET, NSET=Level2BC, GENERATE 20002510, 20002574, 32 *NSET, NSET=Level3BC, GENERATE 20003518, 20003550, 32 *NSET, NSET=Level4BC 20004534 ****NSETS AND ELSETS NEEDED FOR THE RESULTS************************** *NSET, NSET=BC1, GENERATE 2697, 18323, 601 *NSET, NSET=UpperFlangeLeft, GENERATE 598, 8000598, 500000 *NSET, NSET=ReactionNodeSets BC1 BottomFlangeLeft UpperFlangeLeft LoadControlNode *NSET, NSET=LoadControlNode 20004550 *ELSET, ELSET=UFresults12 5500533 6000533 6000534 5500534 *ELSET, ELSET=UFresults34 1500533 2000533 2000534 1500534 *ELSET, ELSET=BFresults910 5500033 6000033 6000034 5500034 *ELSET, ELSET=BFresults1112 1500033 2000033 2000034

95

1500034 *ELSET, ELSET=WEBresults56 13450 14051 14052 13451 *ELSET, ELSET=WEBresults78 6238 6239 6840 6839 *ELSET, ELSET=MidSpanUFresults12 5500597 6000597 *ELSET, ELSET=MidSpanUFresults34 1500597 2000597 *ELSET, ELSET=MidSpanBFresults910 5500097 6000097 *ELSET, ELSET=MidSpanBFresults1112 1500097 2000097 *ELSET, ELSET=MidSpanWEBresults56 13514 14115 *ELSET, ELSET=MidSpanWEBresults78 6302 6903 *ELSET, ELSET=FlangesElsetsResults UFresults12 UFresults34 BFresults910 BFresults1112 *ELSET, ELSET=WebElsetsResults WEBresults56 WEBresults78 *ELSET, ELSET=MidSpanFlangesElsetsResults MidSpanUFresults12 MidSpanUFresults34 MidSpanBFresults910 MidSpanBFresults1112 *ELSET, ELSET=MidSpanWebElsetsResults MidSpanWEBresults56 MidSpanWEBresults78 ****NSETS NEEDED FOR THE BOUNDARY CONDITIONS - RIGHT END******** *NSET, NSET=RightMiddleWebNodeResults 10414 *NSET, NSET=MiddleUFNode 4000502 *NSET, NSET=MiddleBFNode 4000002 *NSET, NSET=WebControlNode 10510 *********************************************************************** *MATERIAL, NAME=STEEL *DENSITY 7800.0 *ELASTIC 210E9, 0.3 *PLASTIC 3.55600E+08, 0 3.59564E+08, 1.1063E-02 5.39021E+08, 5.2778E-02

96

5.61000E+08, 9.2639E-02 *BOUNDARY BC1, XSYMM BottomFlangeLeft, XSYMM UpperFlangeLeft, XSYMM *BOUNDARY Level1BC, 4 Level2BC, 4 Level3BC, 4 Level4BC, 4 LoadControlNode, 1 E1, 2 F1, 2 MiddleUFNode, 3 MiddleBFNode, 3 **History Data *STEP, NLGEOM=YES, INC=300 *STATIC 0.01, 1.0 *BOUNDARY LoadControlNode, 2, 2, -0.3 ** *OUTPUT, FIELD *NODE OUTPUT CF, U, UR *ELEMENT OUTPUT S, E, PE *NODE OUTPUT, NSET=ReactionNodeSets RF, U , UR *NODE PRINT, NSET=ReactionNodeSets RF *NODE PRINT, NSET=ReactionNodeSets U, UR *NODE PRINT, NSET=RightMiddleWebNodeResults U, UR *NODE PRINT, NSET=D1 RF3 *NODE PRINT, NSET=MiddleUFNode RF3 *NODE PRINT, NSET=MiddleBFNode RF3 *EL PRINT, POSITION=AVERAGED AT NODES, ELSET=FlangesElsetsResults S *EL PRINT, POSITION=AVERAGED AT NODES, ELSET=WebElsetsResults S *EL PRINT, POSITION=AVERAGED AT NODES, ELSET=MidSpanFlangesElsetsResults S *EL PRINT, POSITION=AVERAGED AT NODES, ELSET=MidSpanWebElsetsResults S *END STEP

Model 6 *HEADING I beam IPE240 – shell & beam elements **Model Definition *NODE, NSET=Keynodes1 2, 0, 0, 0 8000002, 0, 0, -0.12 2000, 0, 0.0049, -0.06 18828, 0, 0.2253, -0.06 22000, 0, 0.0049, 0 4022000, 0, 0.0049, -0.06 20000506, -0.03, 0.2302, 0.25

97

20000594, -0.69, 0.2302, 0.25 ****SHELL FINER ***************************************************** **BOTTOM FLANGE***** ** *NGEN, NSET=BottomFlangeRight 2, 8000002, 500000 *NCOPY, CHANGE NUMBER=96, OLD SET=BottomFlangeRight, SHIFT, NEW SET=BottomFlangeLeft -0.72, 0, 0 ,, *NFILL, NSET=BottomFlange BottomFlangeRight, BottomFlangeLeft, 96, 1 **UPPER FLANGE***** *NCOPY, CHANGE NUMBER=500, OLD SET=BottomFlange, SHIFT, NEW SET=UpperFlange 0, 0.2302, 0 ,, **WEB********** *NGEN, NSET=WebRight 2000, 18828, 601 *NCOPY, CHANGE NUMBER=96, OLD SET=WebRight, SHIFT, NEW SET=WebLeft -0.72, 0, 0 ,, *NFILL, NSET=Web WebRight, WebLeft, 96, 1 ************** *ELEMENT, TYPE=S4 2, 2, 500002, 500003, 3 2000, 2000, 2601, 2602, 2001 502, 502, 500502, 500503, 503 ** *ELGEN, ELSET=BFlangeElements 2, 96, 1, 1, 16, 500000, 500000 *ELGEN, ELSET=WebElements 2000, 96, 1, 1, 28, 601, 601 *ELGEN, ELSET=UFlangeElements 502, 96, 1, 1, 16, 500000, 500000 *ELSET, ELSET=FinerMesh BFlangeElements WebElements UFlangeElements *SHELL SECTION, ELSET=UFlangeElements, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=BFlangeElements, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=WebElements, MATERIAL=STEEL, OFFSET=0, POISSON=0.3 0.0062, 5 ****MPC BEAM, A, A1, CONSTRAIN WEB NODES & FLANGES NODES************** *NSET, NSET=A, GENERATE 2000, 2096, 1 *NSET, NSET=A1, GENERATE 4000002, 4000098, 1 *NSET, NSET=B, GENERATE 18828, 18924, 1 *NSET, NSET=B1, GENERATE 4000502, 4000598, 1 *MPC BEAM, A, A1 BEAM, B, B1 ****STIFFENERS********************************************************* *NGEN, NSET=BFStiffenerNodes

98

22000, 4022000, 500000 *NCOPY, CHANGE NUMBER=8428, OLD SET=BFStiffenerNodes, SHIFT, NEW SET=UFStiffenerNodes 0, 0.2204, 0 ,, *NFILL, NSET=Stiffener1 BFStiffenerNodes, UFStiffenerNodes, 28, 301 *NCOPY, CHANGE NUMBER=7500000, OLD SET=Stiffener1, SHIFT, NEW SET=Stiffener2 0, 0, -0.06 ,, *ELEMENT, TYPE=S4 22000, 22000, 522000, 522301, 22301 7522000, 7522000, 8022000, 8022301, 7522301 *ELGEN, ELSET=Stiffener1Elements 22000, 28, 301, 301, 8, 500000, 500000 *ELGEN, ELSET=Stiffener2Elements 7522000, 28, 301, 301, 8, 500000, 500000 *NCOPY, CHANGE NUMBER=98, OLD SET=Stiffener1, SHIFT, NEW SET=Stiffener3 -0.72, 0, 0 ,, *NCOPY, CHANGE NUMBER=7500000, OLD SET=Stiffener3, SHIFT, NEW SET=Stiffener4 0, 0, -0.06 ,, *ELCOPY, ELEMENT SHIFT=98, OLD SET=Stiffener1Elements, SHIFT NODES=98 , NEW SET=Stiffener3Elements *ELCOPY, ELEMENT SHIFT=98, OLD SET=Stiffener2Elements, SHIFT NODES=98 , NEW SET=Stiffener4Elements *SHELL SECTION, ELSET=Stiffener1Elements, MATERIAL=STEEL, OFFSET=0.5, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=Stiffener2Elements, MATERIAL=STEEL, OFFSET=0.5, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=Stiffener3Elements, MATERIAL=STEEL, OFFSET=-0.5, POISSON=0.3 0.0098, 5 *SHELL SECTION, ELSET=Stiffener4Elements, MATERIAL=STEEL, OFFSET=-0.5, POISSON=0.3 0.0098, 5 *NSET, NSET=C, GENERATE 4022301, 4030127, 301 *NSET, NSET=C1, GENERATE 2601, 18227, 601 *NSET, NSET=D, GENERATE 7522301, 7530127, 301 *NSET, NSET=D1, GENERATE 2601, 18227, 601 *NSET, NSET=E1, GENERATE 2, 4000002, 500000 *NSET, NSET=F, GENERATE 7522000, 11522000, 500000 *NSET, NSET=F1, GENERATE 4000002, 8000002, 500000 *NSET, NSET=G1, GENERATE 502, 4000502, 500000 *NSET, NSET=H, GENERATE 7530428, 11530428, 500000 *NSET, NSET=H1, GENERATE 4000502, 8000502, 500000 *MPC TIE, C, C1 TIE, D, D1

99

BEAM, BFStiffenerNodes, E1 BEAM, F, F1 BEAM, UFStiffenerNodes, G1 BEAM, H, H1 *NSET, NSET=I, GENERATE 4022399, 4030225, 301 *NSET, NSET=I1, GENERATE 2697, 18323, 601 *NSET, NSET=J, GENERATE 7522399, 7530225, 301 *NSET, NSET=J1, GENERATE 2697, 18323, 601 *NSET, NSET=K, GENERATE 22098, 4022098, 500000 *NSET, NSET=K1, GENERATE 98, 4000098, 500000 *NSET, NSET=L, GENERATE 7522098, 11522098, 500000 *NSET, NSET=L1, GENERATE 4000098, 8000098, 500000 *NSET, NSET=M, GENERATE 30526, 4030526, 500000 *NSET, NSET=M1, GENERATE 598, 4000598, 500000 *NSET, NSET=N, GENERATE 7530526, 11530526, 500000 *NSET, NSET=N1, GENERATE 4000598, 8000598, 500000 *MPC TIE, I, I1 TIE, J, J1 BEAM, K, K1 BEAM, L, L1 BEAM, M, M1 BEAM, N, N1 ****LOAD NODES GENERATION - LEVEL 0 & TRUSS ELEMENTS**************** *NGEN, NSET=LoadNodes 20000506, 20000594, 8 *ELEMENT, TYPE=T3D2, ELSET=TrussElements 900, 4000506, 20000506 901, 4000006, 20000506 902, 4000514, 20000514 903, 4000014, 20000514 904, 4000522, 20000522 905, 4000022, 20000522 906, 4000530, 20000530 907, 4000030, 20000530 908, 4000538, 20000538 909, 4000038, 20000538 910, 4000546, 20000546 911, 4000046, 20000546 912, 4000554, 20000554 913, 4000054, 20000554 914, 4000562, 20000562 915, 4000062, 20000562 916, 4000570, 20000570 917, 4000070, 20000570 918, 4000578, 20000578 919, 4000078, 20000578 920, 4000586, 20000586 921, 4000086, 20000586 922, 4000594, 20000594 923, 4000094, 20000594 **

100

*SOLID SECTION, ELSET=TrussElements, MATERIAL=STEEL 0.0025 ****LOAD NODES SYSTEM************************************************** ****NODE GENARATION - FOR THE LOAD NODES ***************************** *NODE, NSET=Keynodes2 20001506, -0.03, 0.2302, 0.25 20001514, -0.09, 0.2302, 0.25 20001522, -0.15, 0.2302, 0.25 20001530, -0.21, 0.2302, 0.25 20001538, -0.27, 0.2302, 0.25 20001546, -0.33, 0.2302, 0.25 20001554, -0.39, 0.2302, 0.25 20001562, -0.45, 0.2302, 0.25 20001570, -0.51, 0.2302, 0.25 20001578, -0.57, 0.2302, 0.25 20001586, -0.63, 0.2302, 0.25 20001594, -0.69, 0.2302, 0.25 20002510, -0.06, 0.2302, 0.25 20002526, -0.18, 0.2302, 0.25 20002542, -0.3, 0.2302, 0.25 20002558, -0.42, 0.2302, 0.25 20002574, -0.54, 0.2302, 0.25 20002590, -0.66, 0.2302, 0.25 20003518, -0.12, 0.2302, 0.25 20003534, -0.2, 0.2302, 0.25 20003550, -0.36, 0.2302, 0.25 20003566, -0.52, 0.2302, 0.25 20003582, -0.6, 0.2302, 0.25 20004534, -0.2, 0.2302, 0.25 20004550, -0.36, 0.2302, 0.25 20004566, -0.52, 0.2302, 0.25 ***** level 1 *************** *NGEN, NSET=LoadNodes11 20001506, 20001514, 4 *NGEN, NSET=LoadNodes12 20001522, 20001530, 4 *NGEN, NSET=LoadNodes13 20001538, 20001546, 4 *NGEN, NSET=LoadNodes14 20001554, 20001562, 4 *NGEN, NSET=LoadNodes15 20001570, 20001578, 4 *NGEN, NSET=LoadNodes16 20001586, 20001594, 4 ******** level 2 ************** *NGEN, NSET=LoadNodes21 20002510, 20002526, 8 *NGEN, NSET=LoadNodes22 20002542, 20002558, 8 *NGEN, NSET=LoadNodes23 20002574, 20002590, 8 ****ELEMENT GENARATION - FOR THE LOAD NODES************************* *ELEMENT, TYPE=B33, ELSET=BeamElements1 20001506, 20001506, 20001510 20001522, 20001522, 20001526 20001538, 20001538, 20001542 20001554, 20001554, 20001558 20001570, 20001570, 20001574 20001586, 20001586, 20001590 20002510, 20002510, 20002518 20002542, 20002542, 20002550 20002574, 20002574, 20002582 *ELEMENT, TYPE=B33, ELSET=BeamElements2 20003518, 20003518, 20003534

101

20003534, 20003534, 20003550 20003550, 20003550, 20003566 20003566, 20003566, 20003582 20004534, 20004534, 20004550 20004550, 20004550, 20004566 *ELGEN, ELSET=BeamElements3 20001506, 2, 4, 4 20001522, 2, 4, 4 20001538, 2, 4, 4 20001554, 2, 4, 4 20001570, 2, 4, 4 20001586, 2, 4, 4 20002510, 2, 8, 8 20002542, 2, 8, 8 20002574, 2, 8, 8 ****MPC PIN - FOR THE LOAD NODES******************************* *MPC PIN, 20000506, 20001506 PIN, 20000514, 20001514 PIN, 20000522, 20001522 PIN, 20000530, 20001530 PIN, 20000538, 20001538 PIN, 20000546, 20001546 PIN, 20000554, 20001554 PIN, 20000562, 20001562 PIN, 20000570, 20001570 PIN, 20000578, 20001578 PIN, 20000586, 20001586 PIN, 20000594, 20001594 *NSET, NSET=MiddleNodesLevel1, GENERATE 20001510, 20001590, 16 *NSET, NSET=NodesLevel2, GENERATE 20002510, 20002590, 16 *MPC PIN, MiddleNodesLevel1, NodesLevel2 *NSET, NSET=MiddleNodesLevel2, GENERATE 20002518, 20002582, 32 *NSET, NSET=NodesLevel3, GENERATE 20003518, 20003582 ,32 *MPC PIN, MiddleNodesLevel2, NodesLevel3 *MPC PIN, 20003534, 20004534 PIN, 20003566, 20004566 *********************************** *NSET, NSET=LoadControlNode 20004550 *ELSET, ELSET=AllLoadElements BeamElements1 BeamElements2 BeamElements3 *BEAM GENERAL SECTION, SECTION=GENERAL, ELSET=AllLoadElements 0.003912, 3.892e-5, 0, 3.892e-5, 1.892e-5 0, 0, -1 210E9, 81E9 ********************************** ****NSETS FOR THE BOUNDARY CONDITIONS FOR THE BEAM ELELMENTS********* *NSET, NSET=Level1BC, GENERATE 20001506, 20001586, 16 *NSET, NSET=Level2BC, GENERATE 20002510, 20002574, 32 *NSET, NSET=Level3BC, GENERATE 20003518, 20003550, 32 *NSET, NSET=Level4BC

102

20004534 ****NSETS AND ELSETS NEEDED FOR THE RESULTS**************************** *NSET, NSET=BC1, GENERATE 2697, 18323, 601 *NSET, NSET=UpperFlangeLeft, GENERATE 598, 8000598, 500000 *NSET, NSET=ReactionNodeSets BC1 BottomFlangeLeft UpperFlangeLeft LoadControlNode *NSET, NSET=LoadControlNode 20004550 *ELSET, ELSET=UFresults12 5500533 6000533 6000534 5500534 ** *ELSET, ELSET=UFresults34 1500533 2000533 2000534 1500534 *ELSET, ELSET=BFresults910 5500033 6000033 6000034 5500034 *ELSET, ELSET=BFresults1112 1500033 2000033 2000034 1500034 *ELSET, ELSET=WEBresults56 13450 14051 14052 13451 *ELSET, ELSET=WEBresults78 6238 6239 6840 6839 *ELSET, ELSET=MidSpanUFresults12 5500597 6000597 *ELSET, ELSET=MidSpanUFresults34 1500597 2000597 *ELSET, ELSET=MidSpanBFresults910 5500097 6000097 *ELSET, ELSET=MidSpanBFresults1112 1500097 2000097 *ELSET, ELSET=MidSpanWEBresults56 13514 14115 *ELSET, ELSET=MidSpanWEBresults78 6302 6903 *ELSET, ELSET=FlangesElsetsResults UFresults12

103

UFresults34 BFresults910 BFresults1112 *ELSET, ELSET=WebElsetsResults WEBresults56 WEBresults78 *ELSET, ELSET=MidSpanFlangesElsetsResults MidSpanUFresults12 MidSpanUFresults34 MidSpanBFresults910 MidSpanBFresults1112 *ELSET, ELSET=MidSpanWebElsetsResults MidSpanWEBresults56 MidSpanWEBresults78 ****NSETS NEEDED FOR THE BOUNDARY CONDITIONS - RIGHT END******** *NSET, NSET=RightMiddleWebNodeResults 10414 *NSET, NSET=MiddleUFNode 4000502 *NSET, NSET=MiddleBFNode 4000002 *NSET, NSET=WebControlNode 10510 *********************************************************************** *MATERIAL, NAME=STEEL *DENSITY 7800.0 *ELASTIC 210E9, 0.3 **Realistic behaviour *PLASTIC 3.55600E+08, 0 3.59564E+08, 1.1063E-02 5.39021E+08, 5.2778E-02 5.61000E+08, 9.2639E-02 *BOUNDARY BC1, XSYMM BottomFlangeLeft, XSYMM UpperFlangeLeft, XSYMM *BOUNDARY Level1BC, 4 Level2BC, 4 Level3BC, 4 Level4BC, 4 LoadControlNode, 1 E1, 2 F1, 2 MiddleUFNode, 3 MiddleBFNode, 3 **History Data *STEP,NLGEOM=YES, INC=300 *STATIC, STABILIZE 0.01, 1.0, 10E-9, 0.1 *BOUNDARY LoadControlNode, 2, 2, -0.6 ** *OUTPUT, FIELD *NODE OUTPUT CF, U, UR *ELEMENT OUTPUT S, E, PE *NODE OUTPUT, NSET=ReactionNodeSets RF, U , UR *NODE PRINT, NSET=ReactionNodeSets

Data lines for less accurate plastic behaviour:

*PLASTIC

3.55000E+08, 0

3.65000E+08, 9.8310E-02

3.00000E+08, 2.4857E-01

104

RF *NODE PRINT, NSET=ReactionNodeSets U, UR *NODE PRINT, NSET=RightMiddleWebNodeResults U, UR *NODE PRINT, NSET=D1 RF3 *NODE PRINT, NSET=MiddleUFNode RF3 *NODE PRINT, NSET=MiddleBFNode RF3 *END STEP

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TRITA -BKN. MASTER THESIS 464, 2015

ISSN 1103-4297

ISRN KTH/BKN/EX-431-SE

difficulties in fe-modelling of an i- beam subjected to torsion ...828098/fulltext01.pdfiii...

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