Final Report

Adam Lebaigue certifies that all the material contained
within this report (including electronic information such as
the Matlab® program) is his own except where it is clearly
referenced to others.

Contents
1.0 – Abstract ----------------------------------------------------------------------------------------------------- 1
2.0 – Introduction ----------------------------------------------------------------------------------------------- 1
2.1 – Overview of Cold Formed Steel ------------------------------------------------------------- 1
2.2 – Eurocode 3 Design and Coldformed Program ------------------------------------------- 2
3.0 – Cold Formed Steel ---------------------------------------------------------------------------------------- 5
3.1 – Cold Forming Process, Fabrication and Construction ---------------------------------- 5
3.2 – Stresses in Cold Formed Members --------------------------------------------------------- 6
3.3 – Local and Distortional Buckling ------------------------------------------------------------- 7
3.3.1 – Local Buckling ----------------------------------------------------------------------- 7
3.3.2 – Distortional Buckling -------------------------------------------------------------- 8
3.4 – Overall (Column) Buckling ------------------------------------------------------------------- 9
3.5 – Shift of Effective Centroid and Second Order Moments ---------------------------- 11
4.0 – Eurocode 3: Overview of Design Approach and Methodology ------------------------------ 13
4.1 – Overview of Eurocode 3 References and Design (for Cold Formed Steel) ------ 13
4.2 – Eurocode 3 Approach to Local and Distortional Buckling --------------------------- 13
4.2.1 – Geometrical Checks -------------------------------------------------------------- 13
4.2.2 – Effective Areas for Local Buckling -------------------------------------------- 14
4.2.3 – Effective Areas for Distortional Buckling ----------------------------------- 16
4.3 – Eurocode 3 Approach to Capacity Checks under different Ultimate
Limit States ------------------------------------------------------------------------------------- 19
5.0 – Coldformed Computer Program -------------------------------------------------------------------- 20
5.1 – Program Overview and Aims --------------------------------------------------------------- 20
5.2 – Program Navigation and Error Provisions ----------------------------------------------- 21
5.2.1 – Program Flow Chart -------------------------------------------------------------- 21
5.2.2 – Error Provisions ------------------------------------------------------------------- 22
5.3 – Program Windows ---------------------------------------------------------------------------- 23
5.3.1 – Overall Geometry and Loading Window ------------------------------------ 23
5.3.2 – Local Geometry Window ------------------------------------------------------- 24
5.3.3 – Final Results Window ------------------------------------------------------------ 25
5.3.4 – Notes Window for the Overall Geometry Loading Screen -------------- 26
5.3.5 – Notes Window for the Local Geometry Screen --------------------------- 26
5.3.6 – Notes Window for the Results Screen -------------------------------------- 27
5.3.7 – Initial Input Form Window ----------------------------------------------------- 27
5.3.8 – Intermediate Error Message Window --------------------------------------- 27
5.4 – Axial Compression Programmed Solution (Overview) ------------------------------- 28
5.5 – Bending Programmed Solution (Overview)--------------------------------------------- 30
5.6 – Shift in Effective Centroid Programmed Solution ------------------------------------- 32
5.7 – Ultimate Limit States Programmed Solution ------------------------------------------ 34
5.7.1 – Axial Compression/Tension --------------------------------------------------- 34

5.7.2 – Bending ----------------------------------------------------------------------------- 35
5.7.3 – Buckling ----------------------------------------------------------------------------- 36
5.8 – Deflection Calculator ------------------------------------------------------------------------- 37
5.9 – 3D Graphics ------------------------------------------------------------------------------------- 38
5.10 – 2D Graphics ----------------------------------------------------------------------------------- 39
5.11 – Exporting Data ------------------------------------------------------------------------------- 40
5.12 – Additional Program Add-ins -------------------------------------------------------------- 40
5.13 – Future Program Improvements ---------------------------------------------------------- 41
6.0 - Conclusions and Final Remarks --------------------------------------------------------------------- 42
7.0 – Bibliography ---------------------------------------------------------------------------------------------- 43
Appendix A – Exported Data from the Coldformed Program (Example) ------------------------- 45
Appendix B – Program Notes and Information --------------------------------------------------------- 50
1.0 – Input Form -------------------------------------------------------------------------------------- 51
2.0 – Overall Geometry Notes -------------------------------------------------------------------- 51
2.1 – Overall Geometry and Input Data ---------------------------------------------- 51
2.2 – Sign Conventions and Units ------------------------------------------------------ 51
2.3 – Section Orientations --------------------------------------------------------------- 51
2.4 – Moments ----------------------------------------------------------------------------- 51
2.5 – Shift of Effective Centroid and Second Order Effects ---------------------- 52
2.6 – Mechanism and End Supports --------------------------------------------------- 52
2.7 – Beam Moment Calculator -------------------------------------------------------- 52
3.0 – Local Geometry Input Notes -------------------------------------------------------------- 52
3.1 – Input Geometry Checks ---------------------------------------------------------- 52
3.2 – Material Properties Information ----------------------------------------------- 53
3.3 – Section Thickness Information -------------------------------------------------- 53
3.4 – Partial Safety Factors -------------------------------------------------------------- 53
3.5 – Stress Plots Information ----------------------------------------------------------- 53
3.6 – Overview of Program Functionality -------------------------------------------- 53
3.6.1 – Cases Calculated (Effective Sections) ------------------------------ 54
3.7 – Self Weight Calculations ---------------------------------------------------------- 54
3.7.1 – Ignoring Self Weight --------------------------------------------------- 55
4.0 – Final Results Notes --------------------------------------------------------------------------- 55
4.1 – Final Section Capacities ----------------------------------------------------------- 55
4.2 – Exporting to Excel ------------------------------------------------------------------ 55
4.3 – Deflection Calculator --------------------------------------------------------------- 56
4.3.1 – Self Weight -------------------------------------------------------------- 56
4.3.2 – Span Limit ---------------------------------------------------------------- 56
4.3.3 – Bending Axis ------------------------------------------------------------- 56
5.0 - Lateral Torsional Buckling (Program Addition) ---------------------------------------- 56
5.1 – Calculation Process ---------------------------------------------------------------- 56
5.2 – Lateral Torsional Buckling Interaction ---------------------------------------- 57

5.3 – Additional Concerns ---------------------------------------------------------------- 57
Appendix C – Lateral Torsional Buckling ----------------------------------------------------------------- 58
1.0 – Overview of Lateral Torsional Buckling -------------------------------------------------- 59
2.0 – Eurocode 3 Design Approach to Lateral Torsional Buckling ------------------------ 59
2.1 – Reducing the Y-Y Bending Capacity --------------------------------------------- 59
2.1.1 – Shear Centre and Torsion Constant -------------------------------- 60
2.1.2 – C1 and C2 Factors ------------------------------------------------------- 61
2.1.3 – Slenderness and Calculating Mb,Rd --------------------------------- 62
2.2 – Interaction Formulas for Lateral Torsional Buckling ----------------------- 62
2.2.1 – Interaction Factors for Lateral Torsional Buckling ------------- 63
3.0 – Additional Program Features for Lateral Torsional Buckling ----------------------- 64
3.1 – Program Navigation (Including Lateral Torsional Buckling) --------------- 65
4.0 – Lateral Torsional Buckling Program Windows ----------------------------------------- 66
5.0 – Lateral Torsional Buckling Programmed Solution ------------------------------------- 67
6.0 – Bibliography ------------------------------------------------------------------------------------ 67
List of Figures
Main Report
Figure 1 – Typical Cold Formed Steel Sections ----------------------------------------------------------- 1
Figure 2 – Cold Formed Steel, Case Study Examples --------------------------------------------------- 2
Figure 3 – Delta Stud Thermal Advantages --------------------------------------------------------------- 2
Figure 4 – Yield Stress Profiles and Effective Section --------------------------------------------------- 3
Figure 5 – Yield Stress Profiles for Bending and Effective Section ----------------------------------- 4
Figure 6 – Splash Page for the Cold Formed Program -------------------------------------------------- 4
Figure 7 - Cold Formed Steel Fabrication Methods ----------------------------------------------------- 6
Figure 8 - Stress Strain Diagrams for Virgin Plate Steel and Cold Formed Steel ------------------ 7
Figure 9 - Von Kármán Stress Idealisation and Effective Section Width --------------------------- 8
Figure 10 – Local and Distortional Buckling Modes ----------------------------------------------------- 9
Figure 11 – Eurocode 3 Approach to Stiffness and Distortional Buckling ------------------------- 9
Figure 12 – Column Effective Lengths --------------------------------------------------------------------- 10
Figure 13 – Shift in Effective Centroid and Sign Conventions
for the Second Order Moments ------------------------------------------------------------ 12
Figure 14 - Calculating the Bending Stress Distribution ---------------------------------------------- 14
Figure 15 – Eurocode 3 Part 1-5 Tables for Finding Effective Length Values -------------------- 15
Figure 16 – Effective Areas used for Distortional Buckling ------------------------------------------- 16
Figure 17 – Additional Geometry for the Effective Areas in the Flanges ------------------------- 17
Figure 18 - Flow Chart Outlining the Iterative Process for Distortional Buckling in Eurocode 3
Part 1-3 -------------------------------------------------------------------------------------------- 18
Figure 19 - Examples of Final Effective Areas for Different Loading ------------------------------ 18

Figure 20 – Flow Chart Outlining Program Navigation ----------------------------------------------- 21
Figure 21 – Print Screen of Overall Geometry and Loading Window ----------------------------- 23
Figure 22 – Print Screen of Local Geometry Window ------------------------------------------------- 24
Figure 23 – Print Screen of Final Results Window ----------------------------------------------------- 25
Figure 24 – Print Screen for the Overall Geometry and Loading Notes Window -------------- 26
Figure 25 – Print Screen for the Local Geometry Notes Window --------------------------------- 26
Figure 26 – Print Screen for the Results Screen Notes Window ----------------------------------- 27
Figure 27 – Print Screen for the Input Form Data Window ----------------------------------------- 27
Figure 28 – Print Screen for the Intermediate Error Window -------------------------------------- 27
Figure 29 – Flow Chart of the Programmed Solution for Axial Compression ------------------- 28
Figure 30 – Matlab® Code for the While Loop Process ---------------------------------------------- 29
Figure 31 – Matlab® Code for the Error Checking Process ----------------------------------------- 29
Figure 32 – Flow Chart of the Programmed Solution for Bending -------------------------------- 30
Figure 33 - Matlab® Code for the Bending Cases ----------------------------------------------------- 31
Figure 34 – Flow Chart of the Programmed Solution for Centroidal Shift ----------------------- 32
Figure 35 – Matlab® Code for the Centroidal Shift --------------------------------------------------- 33
Figure 36 – Flow Chart of the Programmed Solution for the Axial Compression Limit State 34
Figure 37 – Matlab® Code for the Axial Compression Limit State --------------------------------- 34
Figure 38 – Matlab® Code for the Bending Limit State ---------------------------------------------- 35
Figure 39 – Flow Chart of the Programmed Solution for the Bending Limit State ------------ 35
Figure 40 – Matlab® Code for the Flexural Buckling Limit State ---------------------------------- 36
Figure 41 – Deflection Calculator ------------------------------------------------------------------------- 37
Figure 42 – Maximum Deflection Formulas ----------------------------------------------------------- 37
Figure 43 – 3D Plot of Channel Section with Varying Transparency ----------------------------- 38
Figure 44 – 3D Plotting Panel ------------------------------------------------------------------------------ 38
Figure 45 – 3D Plot of End Moments about Y-Y Axis (M1_y & M2_y) --------------------------- 38
Figure 46 – 3D Plot of Additional Moments about Y-Y and Z-Z Axis (M3_y and M3_z) ----- 38
Figure 47 – 3D Plot of the Axial Forces (N1 and N2) ------------------------------------------------- 39
Figure 48 – Print Screen of 2D Channel Section Stress Plots --------------------------------------- 39
Figure 49 – Print Screen of 2D Plots from the Results Window ------------------------------------ 39
Figure 50 – Setting Output File Path ---------------------------------------------------------------------- 40
Figure 51 – Print Screen of the Beam Calculator Add-in --------------------------------------------- 40

Appendix B
Figure 1.1 – Input Form -------------------------------------------------------------------------------------- 51
Figure 1.2 - Overall Geometry Window ----------------------------------------------------------------- 51
Figure 1.3 – Section Orientation --------------------------------------------------------------------------- 51
Figure 1.4 – Section Orientation for Moments and Axial Loads ----------------------------------- 52
Figure 1.5 – Centroid Shift ---------------------------------------------------------------------------------- 52
Figure 1.6 – Local Section Geometry Window --------------------------------------------------------- 52
Figure 1.7 – Sign Convention for Applied Stress Plots ----------------------------------------------- 53
Figure 1.8 – Final Results Window ------------------------------------------------------------------------ 55
Appendix C
Figure A1 – Lateral Torsional Buckling Visualisation ------------------------------------------------- 59
Figure A2 – C1 and C2 Factors for End Moment Loading -------------------------------------------- 61
Figure A3 – C1 and C2 Factors for Transverse Loading Cases --------------------------------------- 61
Figure A4 – Cmy , Cmz , and CmLT Factors for Lateral Torsional Buckling -------------------------- 63
Figure A5 – Program Navigation with Lateral Torsional Buckling Included -------------------- 65
Figure A6 – Print Screen of the Lateral Torsional Buckling Window ----------------------------- 66
Figure A7 – Additions to the Final Results Window -------------------------------------------------- 66
Figure A8 – Programmed Solution for Lateral Torsional Buckling -------------------------------- 67
List of Tables
Main Report
Table 1 – Eurocode 3 References and Design Process ----------------------------------------------- 13
Table 2 – Outline of Calculations in Eurocode 3 Part 1-5 Section 4.4 ---------------------------- 16
Table 3 – Outline of Calculations in Eurocode 3 Part 1-3 Section 5.5.3.1 ----------------------- 17
Table 4 – Calculation Process in Eurocode 3 Part 1-1 ------------------------------------------------ 19
Table 5 – Additional Program Features ------------------------------------------------------------------ 20
Table 6 – Outline of Error Messages Incorporated in the Program ----------------------------- 22
Table 7 – Future Program Improvements -------------------------------------------------------------- 41
Appendix B
Table 1.1 – Span Limits for Different Beam Conditions --------------------------------------------- 56
Appendix C
Table A1 – Interaction Factors for Lateral Torsional Buckling -------------------------------------- 64
Table A2 – Differences between Restrained and Unrestrained Sections ----------------------- 64

Symbols Used (listed Chronologically)
Stresses and Material Properties
fyb -- The basic yield stress (N/mm2
)
σ -- Normal Stress (N/mm2
)
M -- Applied moment (Nmm)
y -- Distance from the Neutral axis of a section to a point on the cross section (mm)
I -- The second moment of area can be Iy or Iz (mm4
)
fu -- Ultimate Stress (N/mm2
)
ϵ -- Strain
Geometry and Stiffness
be1 -- Effective width 1 (mm)
be2 -- Effective width 2 (mm)
bp -- Outside width of the flange (mm)
b1 -- Distance from the web-flange junction to the flange-stiffener junction (mm)
Cθ -- Rotational stiffness of the web-flange (N/θ)
U -- Unit load (N)
d -- Deflection (mm)
θ -- Rotational angle
K -- Spring stiffness (N/mm)
Euler Buckling
F -- The maximum or critical force for Euler buckling
E -- The modulus of elasticity (210x103
N/mm2
for steel)
K -- Column Stiffness (N/mm)
L -- Column Length (mm)
Leff -- Effective length (mm)
NEd -- Applied axial load (N)
Aeff -- Total effective area (mm2
)
fy -- Material yield strength (N/mm2
)
χ -- Reduction factor for buckling (non-dimensional)
Second Order Moments and Moment Resistances
My,Ed -- Bending resistance around the Y-Y axis (Nmm)
Mz,Ed -- Bending resistance around the Z-Z axis (Nmm)
Weff,y,min -- Elastic section modulus around the Y-Y axis (mm3
)
Weff,z,min -- Elastic section modulus around the Y-Y axis (mm3
)
γM0 -- Partial safety factor (1.0 for steel sections)
eNy -- Shift in effective centroid in y direction (mm)
eNz -- Shift in effective centroid in z direction (mm)

Geometry and Geometry Checks
tcor -- The core thickness of the steel excluding zinc or other metallic coatings (mm)
r -- The internal radius of the bent corners (mm)
bp -- The notional flat widths i.e the width of the individual flat plate areas of the cross
section measured from between the midpoints of the curved corners (mm)
bf -- Width of the flanges (mm)
c -- Width of the lipped stiffeners (mm)
hw -- Height of the web (mm)
t -- Thickness of the section (taken as t = tcor) (mm)
Centroids and Parallel Axis Theorem
NAy -- Distance to the neutral axis in the y direction (Cartesian coordinates) (mm)
NAx -- Distance to the neutral axis in the x direction (Cartesian coordinates) (mm)
An -- Area of the nth
rectangle (for calculating the centroid) (mm2
)
yn -- Distance to the centroids in y for the nth
rectangle (mm)
xn -- Distance to the centroids in x for the nth
rectangle (mm)
n -- nth
rectangle (integer)
bn -- Width of the nth
rectangle (mm)
dn -- Depth of the nth
rectangle (mm)
ybar_n -- Distance from the neutral axis to the centroid for the nth
rectangle (mm)
Local Buckling
c -- Width of plate (mm)
bbar -- Width of plate (mm)
bc -- Distance to tensile zone (mm)
bt -- Width of section in tension (mm)
σ1 -- Stress in the flat plate sections (at one end) (N/mm2
)
σ2 -- Stress in the flat plate sections (at one end) (N/mm2
)
ψ -- Psi factor (ratio of stresses)
kσ -- Buckling factor
λp -- Non dimensional slenderness
ρ -- Reduction factor for local buckling (dimensionless)
Distortional Buckling
Is -- Second moment of area of the effective area of the lipped stiffener and effective
area of the flange at the flange-stiffener junction (about major axis Y-Y) (mm4
)
K1 -- Spring stiffness of the flange (N/mm)
ν -- Poissons ratio (0.3 for steel)
b1 & b2 -- Distance from the web-flange junction to the centroid of the effective areas of the
flange and stiffener (mm)
ceff -- Effective length of the lipped stiffener (mm)
be2 -- Second effective width of the flange (mm)
As1 -- Total effective area of the top stiffener and top flange (i.e. [be2 + ceff]*t) (mm2
)
As2 -- Total effective area of the top stiffener and top flange (i.e. [be2 + ceff]*t) (mm2
)

σcr,s -- Elastic critical buckling stress (N/mm2
)
χd -- Reduction factor for distortional buckling (dimensionless)
As,red -- Reduced area for distortional buckling
Ultimate Limit States
Nc,Rd -- Axial resistance (N)
S -- Elastic section modulus
Mc,Rd -- Bending resistance (Nmm)
λbar -- Non dimensional slenderness
Nb,rd -- Buckling resistance (N)
Deflection
δmax -- Maximum deflection (mm)
w -- Total load applied (N)
Appendix A – Lateral Torsional Buckling
Mcr -- Elastic critical moment for bending about the Y-Y axis (Nmm)
It -- Torsion constant (mm4
) {sometimes denoted as J)
Iw -- Warping constant (mm4
) {sometimes denoted as Cw)
zg -- Distance from where the transverse load is applied to the location of the shear
centre (mm)
C1 & C2 -- Factors for lateral torsional buckling
τavg -- Average shear stress (N/mm2
)
F -- Applied transverse load (N)
ψ -- Ratio of moments (used for C1 and C2 factors)
χLT -- Reduction factor for lateral torsional buckling
Mb,Rd -- Bending resistance about Y-Y axis including lateral torsional buckling (Nmm)
kyy , kzy , kyz , kzz -- Interaction factors for lateral torsional buckling
Cmy , Cmz , CmLT -- Factors for lateral torsional buckling
Acknowledgements
I would like to thank all my friends and family who have supported me throughout my
time at University and also my dissertation supervisor Dr Jurgen Becque, who helped
make this report and the cold formed program a reality.

Figure 1 – Typical Cold Formed Steel Sections
1.0 - Abstract
The use of cold formed steel has historically been used in the design and implementation of
secondary structural members. However with recent advancements in structural materials
and the methods used to analyse them, cold formed steel is being used as a primary
structural member more frequently and in more diverse situations. Due to this paradigm
shift it is therefore important for designers to be able to create safe and economic
structures in cold formed steel. This report provides an in depth investigation into the
design of cold formed sections in accordance with Eurocode 3 as well as a brief overview of
the history and usage of these types of sections. The calculation and design process within
Eurocode 3 has also been analysed and included within a computer program which includes
a GUI (graphical user interface) allowing a user to define loads and geometry before
calculating the section capacity under different limit states. This computer program is also
discussed in detail for its operation, design, limitations and relevance. Future improvements
for the computer program are also discussed and again these also make appropriate
references to the calculations set out in Eurocode 3.
2.0 – Introduction
2.1 – Overview of Cold Formed Steel
Cold formed steel sections have been used in a wide range of civil engineering projects in
previous years and designers have adopted many different shapes to address a variety of
engineering problems (see figure 1). These types of sections have traditionally been used in
secondary structural elements whose strengths do not govern the overall capacity of the
structure. However in recent years the use of cold formed steel sections for primary
structural members has increased due to their benefits of reduced structural weight, higher
flexibility in manufacture and a larger range of sections to choose from (Tata Steel, 2014). As
a result cold formed steel is increasingly being used for multi-storey structures with
companies such as Metsec (UK) and Trussteel (US) providing a variety of different design
solutions (see the case studies in figure 2). Also more complex cold formed shapes are being
adopted to provide additional benefits to the designer such as increased thermal efficiency
as seen with the Deltastud sections (Steelform, 2008) (figure 3). Unfortunately the thin
1

Figure 3 – Delta Stud Thermal Advantages
Holes in the section reduce the weight and
allow for greater cooling and heat
dissipation during a fire (Steelform, 2008).
section thicknesses required for cold formed sections
can produce complex buckling mechanisms (Hancock,
2003) which can be tedious and difficult to account for
using simple hand calculations alone. Therefore the use
and development of automated software tools for the
design of cold formed steel is an important ongoing task
and can allow for faster, safer and more accurate designs
for future projects.
2.2 – Eurocode 3 Design and Cold formed Program
The provisions set out in Eurocode 3 parts 1-3 and 1-5
(British Standards Institute, 2006) account for the
localised buckling effects of thin cold formed sections.
These effects are accounted for by converting the full
cross sectional area into an effective section which has a
lower load bearing capacity. This effective section is based
around the yield stress conditions within the section and accounts for the local and
distortional buckling modes of failure. The strength under axial compression is determined
from the effective section based on the yield stress criterion, which under compression is
uniform throughout the section and is equal to the basic yield stress fyb (see figure 4). This
Whitley Bay Joint Services Centre
Stoke and Derby Schools for the Future
(Metsec, 2013)
Figure 2 – Cold Formed Steel, Case Study Examples
A composite steel and concrete framing system was used for this
project with cold formed sections used for a majority of the
framing elements. The use of cold formed sections alongside
expertise from the Metsec design team provided savings from the
volume of steel required for the project as well as the fabrication
and construction process involved.
Cold formed steel has been used as the secondary structure for
this project with a mixture of brickwork and curtain walls used for
the exteriors. The cold formed sections provided a cheaper
structural solution and allowed for a more efficient construction
regime.
(Metsec, 2013)
2

case is different for the bending situations around the major (Y-Y) and minor (Z-Z) axis. The
bending cases produce a stress gradient from tensile to compressive across the section. This
distribution of the stress, which is again based around the yield stress (a material property)
is also dependent on the geometry and second moment of area (I) through the engineers
bending equation:
where:
σ : The value of stress at a point within the cross section (N/mm2
)
M: The applied moment (although in this case this applied moment is virtual) (Nmm)
y : The distance from the neutral axis of the section to the point of the cross section under
examination (i.e. where the stress is being measured) (mm)
I : The second moment of area of the section for the given bending direction i.e. Iy or Iz
(mm4
)
This creates a situation where the basic yield stress fyb does not define the values of the
stress field at every point in the section, like in the compression case shown in figure 4.
Rather the bending situation may only have one area (for example the bottom flange)
where the yield stress is fully attained (this ideology is highlighted in more detail in figure 5).
The bending cases also include areas of the section in tension which become fully effective
because they cannot buckle in either the local or distortional mode.
The process set out in the Eurocode 3 is useful for calculating the section capacities
under different loading situations, however it is also very cumbersome. For example, the
steps to find the Y-Y bending stress field to initialise the calculation process used to find the
effective areas can be lengthy, especially for sections like that shown in figure 5.
fyb is the basic yield
stress and is a
material property
which depends on
the steel grade
adopted for the
section. The stress
distribution is
therefore not an
applied stress profile
but an idealised
stress profile based
on the yield stress
values.
Figure 4 – Yield Stress Profiles and Effective Section
Eq - 1
3

This long calculation process is then followed by an iterative approach to account for the
effects of local and distortional buckling, which involves recalculating the bending stresses
for the effective section on each iteration.
The design calculation methods outlined by the Eurocode are therefore unrealistic
for a hand calculation process, as time and accuracy are often critical factors for the modern
civil engineer. To combat these problems a Matlab® program has been developed (‘cold
formed’, figure 6) which allows the designer to specify the loading and geometry for a C-
lipped channel section in cold formed steel. The program automatically calculates the
strength of the section for the axial compression case and for bending around the Y-Y and Z-
Z axis and then uses the provisions set out in Eurocode 3 Part 1-1 (British Standards
Institute, 2004) to determine the strength of the section under various ultimate limit states.
This report aims to provide an in depth look at the calculations and design ideology for cold
formed steel in Eurocode 3 and explains the tools and methods used to incorporate these
calculations into the cold formed software tool.
Figure 5 – Yield Stress Profiles for Bending and Effective Section
In this case the
geometry causes the
maximum stress to
occur as a tensile
stress in the bottom
flange of the section.
Symmetric sections
do not exhibit these
problems as the
neutral axis falls in
the centre and causes
the y value in the
engineers bending
equation to be the
same at both the top
and bottom flanges.
Figure 6 – Splash Page for the Cold Formed Program 4

3.0 – Cold Formed Steel
3.1 – Cold Forming Process, Fabrication and Construction
Cold formed steel sections are created at room temperature using one of the following
methods:
 Cold Roll Forming
 Press Braking
 Bending Brake Operation
The most common process used for the fabrication of steel construction elements is cold
roll forming (figure 7) where a continuous sheet of steel is fed through a series of rollers
which incrementally bend the sheet into the desired shape (Yu & LaBoube, 2010). This
process allows for continuous manufacture of steel sections where the length of the
member can be easily adjusted. The other cold forming processes both involve press
machines (usually hydraulic) which bend a strip of plate steel into the required shape. These
processes are not continuous and are therefore not as suitable for the manufacture of cold
formed steel sections on a mass scale.
The fabrication process and design of cold formed sections create various advantages at the
construction stage of a project, some of these benefits are bullet pointed below.
 Freedom of design
Due to the fabrication process which relies on simple bending of the steel in the
correct places, a large degree of flexibility is available to the designer for the section
size and shape. This can benefit the construction process due to a tailored design
methodology i.e. smaller and lighter sections with simpler connections could be
used for the many purlin rails often seen in roof spaces.
 Volume of Steel
Cold formed sections are generally much thinner than their hot formed
counterparts and therefore reduce the amount of steel required for a project. This
also simplifies the construction process as the sections are easier to transport,
handle and lift into place.
 Modular Construction
The lightweight nature of cold formed sections lends itself to being integrated
within larger prefabricated units (such as walls or rooms) which can then be
5

assembled quickly on site. This has been successfully employed in many projects
such as Lille Road in Fulham and the Royal Northern College of Music in Manchester
(Lawson, et al., 2005). Also additional structural components can be added at the
fabrication stage such as intumescent paint for fire protection (Steel Construction
Organisation, 2012) which allows for greater control over the final product and
streamlines the workflow on site. (There are also many software packages (Argos
Systems, 2014) which allow for the design of this type of construction and reduce
the inherent complexities involved i.e. specifying geometry for large numbers of
panels).
3.2 - Stresses in Cold Formed Members
The cold forming process changes the mechanical properties of the steel plate due to the
large plastic deformations induced in the corners of the formed section. The strength and
yielding properties of cold formed sections rely on three factors (Karren & Winter, 1965).
 Strain Hardening
 Strain Ageing
 The Bauschinger Effect
These factors cause an increase in the yield stress of the steel and also produce important
changes to the stress strain diagram (figure 8). The cold working also results in varying yield
Cold forming
rollers
Direction of
steel through
the forming
process
Cold Roll Forming
Press Brake Forming
Hydraulic press
Bending Brake Operation
Sheet metal is clamped
Figure 7 - Cold Formed Steel Fabrication Methods
6

stress conditions which are dependent on the location within the section. A higher yield
stress and therefore capacity of the section is apparent in the corners and a lower value of
yield stress occurs in the flat plate sections (although not lower than the original yield stress
in an unformed plate). In addition to the increased yield stress, residual stresses are also
induced in the section after the cold forming process. The residual stresses produce
complex yielding patterns at failure on the outer faces of the plated steel elements and
these stresses also govern how the load and the applied stresses are distributed within the
section (Schafer & Pekoz, 1998). Furthermore residual stresses within the section are
distributed with compressive stresses on the inside face and tensile stresses on the outside
face. This difference between tensile and compressive stresses can govern the face on
which yielding occurs when load is applied to the section (Weng & Pekoz, 1990).
3.3 - Local and Distortional Buckling
3.3.1 - Local Buckling
The local buckling of thin plated steel elements is a key consideration for cold formed steel
and can govern the ultimate capacity of a section. Many methods have been proposed to
analyse the capacity of such elements, however most formulations stem from the
investigations conducted by Von Kármán where a thin plate in axial compression and simply
supported on all sides is assumed (Kármán, et al., 1932) (Shanley, 1939). The stress
Strain ϵ Strain ϵ
Stress σ Stress σ
Elastic Region
Yield
stress
fy
Failure
Failure
Proportional
Limit fpr
ft
Strain
Hardening
Yielding
Plastic
Range
Necking
Ultimate
Stress fu
Yield strength fy
The cold forming process increases the yield strength and also changes the yielding properties. With cold
formed steel the yielding is more gradual whereas with virgin plate steel the transition between the elastic
and plastic region is more sudden.
Figure 8 - Stress Strain Diagrams for Virgin Plate Steel and Cold Formed Steel
Virgin Plate Steel Cold Formed Steel
7

distribution in the plate is then idealised as two separate uniform stress blocks which are
concentrated at the edges of the plate and act over effective widths be1 and be2 (figure 9).
The sum of be1 and be2 equals the total effective width for the plated element and for the
simply supported case be1 = be2. This method can be adopted for cold form steel sections by
idealising the flat elements (i.e. the flanges and web) as individual plate elements and
calculating the effective width for each section individually. This methodology is utilised in
Eurocode 3 Parts 1-3 and 1-5 to find the initial effective area for the section. This is then
modified further by taking into account the effects of distortional buckling.
3.3.2 - Distortional Buckling
The distortional buckling mode is another important criterion to consider when designing
cold formed steel sections and can also be a governing failure mode for the ultimate
strength of a member. This failure mode involves buckling of the web combined with
rotation and translation of some of the corners of the section. This causes the flanges of the
steel section to either open up or fold inwards as highlighted in figure 10. The calculations
for this failure mode involve the assumption that the flanges are the key drivers for the
failure mode and that they rotate and bend laterally as the critical load is applied (Lau &
Hancock, 1987). With this assumption the flanges and web can be modelled independently
on a system of springs with rotational and translational stiffnesses. Then according to
(Schafer & Pekoz, 1999):
“Buckling ensues when the elastic stiffness at the web/flange juncture is eroded by the
geometric stiffness”
From this criterion a system of equations can be developed to model the rotational
Example of Thin Plate Loaded in Compression
A
A
Full width B
Be1 Be2
1
Idealised
Stress
Distribution
Actual Stress
Distribution
Effective
Width
Stress Distribution along Section A-A
Simply supported on all edges
Uniform compression applied
Figure 9 - Von Kármán Stress Idealisation and Effective Section Width
8

Figure 11 – Eurocode 3 Approach to Stiffness and
stiffnesses for the flange and the web and
the buckling stress can be obtained. This
method is slightly different to the one given
in Eurocode 3 Parts 1-3 and 1-5 where only
the flange is modelled on springs and an
iterative process is used to move from the
effective area under local buckling to a
convergence on the final effective area for
both local and distortional buckling (figure
11). The ideas involving the rotational
stiffness of the flanges are still similar
however and allow for a more general
approach to be adopted.
3.4 – Overall (Column) Buckling
In addition to the local and distortional modes of buckling failure, a member subjected to
compressive loading can also buckle in a global mode known as Euler (or column) buckling.
This failure mode is governed by the following formula for sections responding elastically:
Where:
F : The maximum or critical force (largest axial load the column can carry N)
E : Modulus of elasticity (210x103
N/mm2
for steel – Eurocode 3 Part 1-1 Section 3.2.6)
I : The second moment of area (mm4
)
Local Buckling
Corners remain
at the same
angle, and plate
elements buckle
outwards
Corners rotate
and the section
either opens up
(left) or closes up
(right)
Figure 10 – Local and Distortional Buckling Modes
Eq – 2.0
9

Figure 12 – Column Effective Lengths
K : The stiffness of the column (relies on support conditions) (N/mm)
L : The unsupported length of the column (mm)
K*L: Effective length of the section (mm) which depends on the columns length and support
conditions. Some common effective
length parameters for different support
conditions are shown in figure 12.
The overall strength of a section under
this Euler buckling situation is
dependent on the material properties
(E), the support conditions (KL) and the
geometry of the cross section (I). The
material properties and support
conditions are not affected by whether
a section is hot or cold formed. However
the geometry and specifically the
second moment of area (I) changes
dramatically if cold formed steel is used.
For a cold formed member the effects of local and
distortional buckling result in a cross section which does not fully contribute to the strength
against axial compression. This problem can be accounted for using the following methods:
 Effective width method
 Column curve method (direct strength)
The effective width method is adopted by Eurocode 3 parts 1-3 and 1-5 and uses a reduced
cross sectional area to account for the loss in strength due to the local and distortional
buckling cases (figures 4 and 5). This reduced area also lowers the values of the second
moment of area around both the Y-Y and Z-Z axis. Conversely the column curve method
does not adopt effective areas to reduce the section strength and instead uses the gross
cross sectional properties in conjunction with column curves under the local and distortional
buckling cases to reduce the overall strength (Schafer, 2001). The final calculations for both
methods involve finding a limiting load which the column can safely carry. For the Eurocode
3 method using the effective width concept the final capacity of the section is determined
10

by reducing the axial section capacity for the effective section by the factor χ. This χ factor is
dependent on the material properties (elastic modulus E and yield strength fyb), the support
conditions (effective length Leff) and on the second moment of area (Iy and Iz).
3.5 – Shift of Effective Centroid and Second Order Moments
An additional issue to consider with the design of cold formed steel sections under uniform
compression involves the shift of the effective centroid along which the initial axial loading
force is applied. This effect occurs as the uniform longitudinal stresses redistribute during
the local buckling of the section and form a non-uniform stress distribution (Young, 2005).
This is the same phenomena highlighted in figure 9 and is the same stress distribution which
can be idealised by the effective area concept previously discussed. The shift in effective
centroid can therefore be thought of as a consequence of moving from the geometry of a
fully effective section to the effective section which deals with the local and distortional
buckling problems (see figure 13). It should also be noted that this shift in effective centroid
does not occur for every single case in cold formed sections. If the section is doubly
symmetric the centroid doesn’t move and if the section is singly symmetric the centroid only
moves in 1 direction i.e. in x or y (Cartesian coordinates) and not both. This occurs because
the stresses redistribute equally on either side of the centroid and produce effective areas
which balance each other when calculating the centroids location. Additionally if the section
has fixed support conditions at both ends the shift in effective centroid does not induce
second order moments within the section and no additional curvature is experienced.
The process in Eurocode 3 part 1-1 deals with the shift in effective centroid by
calculating the second order moments generated by the now eccentric axial load and
includes these moments in the interaction equation shown below.
Where:
NEd : Applied axial load (N)
Aeff : Total effective area (mm2
)
fy : Material yield strength (N/mm2
)
My,Ed & Mz,Ed : The bending resistances around the Y-Y and Z-Z axis (Nmm)
Eq – 3.0
11

Weff,y,min & Weff,z,min : The elastic section modulus around the Y-Y and Z-Z axis (mm3
)
γM0 : Partial safety factor (=1.0 for steel sections)
eNy & eNz : The shift in the effective centroid in the y and z direction (mm)
The moments due to the centroid shift are the NEd* eNy and NEd* eNz terms in the interaction
equation above and can be positive or negative depending on the sign convention adopted
and on the direction in which the centroid moves in relation to the applied axial load. (This
sign convention issue is also highlighted in figure 13 below).
Figure 13 – Shift in Effective Centroid and Sign Conventions for the Second Order Moments
12

4.0 – Eurocode 3: Overview of Design Approach and Methodology
4.1 – Overview of Eurocode 3 References and Design (for Cold Formed Steel)
The design process for cold formed steel sections spans over 3 different sections in
Eurocode 3 (Parts 1-1, 1-3, 1-5). A brief overview of the design process for a simple channel
section with edge stiffeners is outlined in table 1 below. (This is also the method followed
for the C-lipped channel section considered for the cold formed software).
Calculation Eurocode Reference
Partial Safety Factors
Material properties for cold formed sections
Checking the thickness and thickness tolerances
Checking the influence of rounded corners
Checking geometric proportions
Eurocode 3 Part 1-3: Section 2
Section 3.1 Tables 3.1a, 3.1b
Section 3.2.4
Section 5.1
Section 5.2 Table 5.1 & Equations 5.2a, 5.2b
Using the stress distribution to find the effective areas under
local buckling
Eurocode 3 Part 1-5: Section 4.4
Tables 4.1 & 4.2 then:
Eq 4.3, 4.2 & 4.1
Finding the effective area for distortional buckling Eurocode 3 Part 1-3: Section 5.5.3.1
Eq 5.10b, 5.15, 5.12d, 5.12c, 5.12b, 5.12a
Finding the capacity under axial loading limit state Eurocode 3 Part 1-1:
Section 6.2.3 Eq 6.5 & 6.6
Section 6.2.4 Eq 6.9 & 6.11
Finding the capacity under the bending limit state Eurocode 3 Part 1-1:
Section 6.2.5 Eq 6.12 & 6.15
Finding the capacity under the combined bending and axial limit
state
Eurocode 3 Part 1-1:
Section 6.2.9.3 Eq 6.44
Finding the capacity under the overall and flexural buckling limit
states
Eurocode 3 Part 1-1:
Section 6.3.1.1 Eq 6.46, 6.48, 6.49, 6.51
Additional calculation methods are required in the design process and these are listed below:
- Using the process of geometric decomposition to find the centroid, centroid shift and neutral axis
- Using the parallel axis theorem to find the second moment of area around Y-Y and Z-Z
- Using the engineers bending equation to find the elastic section modulus S
- Using the engineers bending equation to find the bending stress distributions
- Calculating the elastic critical buckling force Ncr
4.2 – Eurocode 3 Approach to Local and Distortional Buckling
4.2.1 – Geometrical Checks
The design process starts with Eurocode 3 Part 1-3 and involves checking the cross section
of a member against various geometrical conditions. These conditions are outlined in the
equations that follow. (The references in the following equations refer to Eurocode 3
Part 1-3)
Table 1 – Eurocode 3 References and Design Process
13

0.45mm ≤ tcor ≤ 15mm (Section 3.2.4 Thickness and thickness tolerances)
tcor : The core thickness of the steel excluding zinc or other metallic coatings (mm)
4.2.2 – Effective Areas for Local Buckling
To begin the calculations, a stress profile for the cross section is first required. This stress
profile is based on the basic yield stress (fyb – Eurocode 3 Part 1-3, Table 3.1b) and is
different depending on the type of loading. For example with axial compression the stress
distribution is uniform and equal to the basic yield stress at every point (figure 4) whereas
the bending situations around the Y-Y and Z-Z axis include a stress gradient as the section
stresses change from compressive to tensile (figure 5). The stresses for the bending
situations can be obtained by finding the neutral axis using the geometrical decomposition
method and then applying the parallel axis theorem to find the second moment of area.
Then using the engineers bending equation the stress profile can be obtained (figure 14).
(Section 5.2 Geometrical Proportions)
r ≤ 5t and r ≤ 0.1bp (Section 5.1 Influence of rounded corners)
r : The internal radius of the bent corners (all must be checked) (mm)
bp : The notional flat widths i.e. the width of the individual flat plate areas of the cross section
measured from between the midpoints of the curved corners (mm)
If corner radii are within the values given the cross section can be idealised as having sharp corners
for all further calculations (Section 5.1 figure 5.2)
bf / t ≤ 60
c / t ≤ 50
hw / t ≤ 500
bf : width of the flanges (mm)
c : width of the lipped stiffeners (mm)
hw : height of the web (mm)
t : section thickness (mm)
Distance to Centroid and Neutral Axis
NAy : distance to neutral axis in y
NAx : distance to neutral axis in x
An : Area of nth
rectangle
yn : Distance to centroids in y for nth
xn : Distance to centroids in x for nth
n : nth
rectangle
Y
X
Parallel Axis Theorem (Finding Iy & Iz)
bn : width of nth
rectangle
dn : depth of nth
rectangle
ybar_n : distance from NA to centroid for nth
rectangle
Bending stresses (finding σ)
M : Moment
y : distance from NA to
point of interest
I : second moment of area
Figure 14 – Calculating the Bending Stress Distribution 14

This stress distribution is then used in conjunction with Eurocode 3 Part 1-5 to find the
effective area of each flat element of the cross section, which for a C-lipped channel
includes the top and bottom flanges, top and bottom lipped stiffeners and the web. The
tables in Eurocode 3 Part 1-5 Section 4.4 are first used for the calculation of the buckling
factor kσ for each of these flat plated elements and included within these tables are
provisions for internal and outstand elements of which a c-lipped channel has both (this
information is outlined in figure 15). The buckling factor kσ is then used in the following
equations:
b : The appropriate width (mm)
t : The section thickness (mm)
fyb : The basic yield stress (N/mm2
)
The calculation is continued with the following conditions to find the reduction factor ρ:
Ψ = σ2 / σ1 Buckling Factor kσ
1 4
1 > Ψ > 0 8.2 / (1.05 + Ψ)
0 7.81
0 > Ψ > -1 7.81 – 6.29Ψ + 9.78Ψ2
-1 23.9
-1 > Ψ ≥ 3 5.98(1- Ψ)2
σ2 @ the support (LHS)
Ψ = σ2 / σ1 Buckling Factor kσ
1 0.43
0 0.57
-1 0.85
-1 > Ψ ≥ 3 0.57 – 0.21Ψ + 0.07Ψ2
σ1 @ the support (LHS)
1 0.43
1 > Ψ > 0 0.578/(Ψ+0.34)
0 1.70
0 > Ψ > -1 1.7 - 5Ψ + 17.1Ψ2
-1 23.8
Figure 15 – Eurocode 3 Part 1-5 Tables for Finding Effective Length Values
Eq – 4.0 Eq – 5.0
Be1 – effective length 1 (mm)
be2 – effective length 2 (mm)
bbar & c – width of plate (mm)
bc – distance to tensile zone (mm)
bt – width of section in tension (mm)
σ1 & σ2 – normal stresses (N/mm2
)
15

Outstand Elements (i.e.
Lipped stiffeners)
Internal Elements (i.e. webs and flanges whose
edges are supported with lipped stiffeners)
Once the reduction factor ρ is known, the effective lengths for the different parts of the
cross section can be obtained by returning to tables 4.1 and 4.2 in Eurocode 3 Part 1-5
(figure 15). These tables provide the final formulas and diagrams required for both the value
of the effective lengths and also their locations. This process for local buckling within
Eurocode 3 is essentially an adaptation of the idealisation made by Von Kármán (figure 9)
except that additional provisions are required to adequately model the behaviour of the flat
plates when pinned supports are not provided at each end and when a stress gradient is
experienced.
4.2.3 – Effective Areas for Distortional Buckling
Once the effective area for local buckling is obtained the calculation process can then
progress and take into account the distortional buckling mode. This mode of buckling is
more complex and varies depending on the stiffening elements included within the cross
section. The method explained in this section focuses on the distortional buckling mode for
a flange with a lipped stiffener however the same process is adopted for intermediate
stiffening elements.
The calculation process begins by finding the second moment of area for the system
of effective areas for the flanges (Is). More specifically these areas are the effective area of
the stiffener and the second effective area of the flange (be2) (see figure 16). The next
For: For:
For:
Table 2 – Outline of Calculations in Eurocode 3 Part 1-5 Section 4.4
For:
Areas in orange are the effective areas for which the second
moment of area must be determined using the parallel axis
theorem (the second moment of area (Is) is taken around the
major Y-Y axis).
be2 : second effective width of the flange (mm)
ceff : effective width of the lipped stiffener (mm)
NA : neutral axis
Figure 16 – Effective Areas used for Distortional Buckling
16

Figure 17 – Additional Geometry for
the Effective Areas in the Flanges
parameter required is the spring stiffness for the flange which is determined using the
following formula (from Eurocode 3 Part 1-3 Eq 5.10b):
Where:
E : Modulus of Elasticity (210x103
N/mm2
for steel)
b1 : Distance from where the flange meets the web to the
centroid for the effective areas for the top flange (mm) (see
figure 17)
b2 : Distance from where the flange meets the web to the
centroid for the effective areas for the bottom flange (mm)
(see figure 17)
hw : height of the web (mm)
kf = 0 if the second flange is fully effective i.e. in tension
kf = As2 / As1 if the second flange is in axial compression where
As2 and As1 are the total effective areas for the top and bottom
flanges i.e. the total areas in orange in figure 17)
kf = 1 for a section in axial compression which is also symmetric about the Y-Y axis.
The values of spring stiffness for the flange (K1) and second moment of area (Is) are
then used to find the elastic critical buckling stress σcr,s using the following formula
(Eurocode 3 Part 1-3 Eq 5.15):
Where:
As : The total effective area of the flange stiffener system (total area in orange for the top
flange and stiffener in figure 17).
Finally this value for the critical buckling stress is used to find the reduction factor for
distortional buckling (χd) using the following conditions.
Eq – 6.0
Eq 7.0
Eq 8.0
For:
For:
For:
Table 3 – Outline of Calculations in Eurocode 3 Part 1-3 Section 5.5.3.1
17

Figure 18 - Flow Chart Outlining the
Iterative Process for Distortional Buckling
in Eurocode 3 Part 1-3
Figure 19 – Examples of Final Effective Areas for Different Loading
Cases
The provisions in Eurocode 3 Part 1-3 suggest iterating on
the reduction factor to gain more accurate results under
distortional buckling. This iterative process involves
modifying the initial stress distribution using the reduction
factor (χd) and then re-evaluating the effective areas for
local buckling under this new stress profile. This looped
calculation process is outlined in the flow chart in figure 18.
The final reduction factor value is then used to reduce the
thickness of the effective areas for the lipped stiffeners and
flanges (figure 16). This is done according to the following
formulas (from Eurocode 3 Part 1-3 Eq 5.17):
Where:
As,red : The reduced area for distortional buckling (mm2
)
As : The total effective area for this distortional buckling
system (mm2
)
σcom,Ed : The value of compressive
stress along the centreline of
the stiffener (N/mm2
)
γM0 : Partial safety factor (=1.0)
tred : The reduced thickness for
distortional buckling (mm)
t : The section thickness (mm)
An example of the final effective areas
for the axial compression and bending
cases around the Y-Y and Z-Z axis are
shown in figure 19.
Eq 9.0
Eq 10.0
18

4.3 – Eurocode 3 Approach to Capacity Checks under different Ultimate Limit States
The final effective area for local and distortional buckling is used in conjunction with the
provisions in Eurocode 3 Part 1-1 to calculate the members strength under different
ultimate limit states. A brief overview of the calculation process is given in table 4 below:
Limit State Calculation Process
Axial
Loading
Capacity
Using the effective area for local and distortional buckling under compression the axial loading
capacity can be found using (Eurocode 3 Part 1-1 Eq 6.11):
This value of Nc,rd can then be checked against the loading to see if the section is adequate.
Bending
Capacity
Using the effective area for local and distortional buckling under bending around the Y-Y and Z-Z
axis the bending capacity can be found by first finding the second moment of area (Iy & Iz) of this
effective cross section. The elastic section modulus (S) can then be calculated by dividing the
second moment of area by the distance from the neutral axis to the maximum compressive fibre
of the material (S = I / y). The section modulus can then be used in the equation below to work
out the bending capacity in both the Y-Y and Z-Z direction (Eurocode 3 Part 1-1 Eq 6.15):
This value for Mc,rd can then be checked against the loading to see if the section is adequate.
Buckling
Capacity
(Flexural)
Using the effective area for local and distortional buckling under compression and the gross
properties of the section the buckling capacity can be obtained. The capacity for buckling
depends on the slenderness of the section which is reliant on the overall restraint conditions as
well as the effective area through the following relationship (Eurocode 3 Part 1-3 Eq 6.51)
The value for slenderness is used to find a reduction factor (χ) which is used to reduce the axial
capacity from its value determined previously. This is done using the following equation
(Eurocode 3 Part 1-1 Eq 6.48):
Aeff : total effective area (mm2
)
γM0 : partial safety factor (=1.0)
Weff,min : elastic section modulus (mm3
)
fyb : Basic yield stress N/mm2
fyb : Basic yield stress
N/mm2
Lcr : effective length (mm)
i : radius of gyration (mm)
Aeff : effective area (mm2
)
A : total area (mm2
)
λ1 : 93.9ϵ where ϵ = √235/fyb
χ : reduction factor for buckling
Aeff : effective area (mm2
)
fyb : basic yield stress
Table 4 – Calculation Process in Eurocode 3 Part 1-1
19

5.0 – Cold formed Computer Program
5.1 – Program Overview and Aims
The overall aim for the cold formed computer program was to create a simple intuitive tool
which allows a user to design and check the suitability of a cross section in cold formed steel
accurately and quickly. The final program uses many different techniques to achieve this
goal however one of the most important aspects involves the use of a GUI (graphical user
interface). The GUI allows the user to easily navigate between different program functions
as well as to input data for the design calculations. The use of a GUI also allows for greater
clarity in regards to sign conventions and units for the design process. In regards to the
design and calculation aspect however, the objective has been to create a tool which is
accurate, quick and automated. This part of the program runs behind the GUI and uses a
combination of different programmatic statements within Matlab® to replicate the
calculation process outlined in Eurocode 3 and explained in section 4 of this report.
Additional program functions were also considered at the outset and some of these are
discussed in table 5 below.
Program Aim Solution Adopted
Incorporating the ability to output the calculation and
input data into a user friendly format (for commercial
and practical uses).
Two excel template files were created which can
receive exported data from the final program window.
This file can then be edited and saved accordingly.
Adding resources within the program to explain the
functionality and limitations of the program to the
user.
A separate graphical popup window has been included
for all of the main GUI sections with informative notes
and diagrams. A PDF readme file has been included
within the programs folder to further inform the user.
To allow for future improvements and maintenance
of the programs source code.
The source code includes numerous text fields which
explain the different parts of the program. Also the
assignments within the program are consistently and
sensibly labelled (i.e. eNx the centroidal shift has the
assignment eN_x within the script).
To provide visual aids for the user so that the section
dimensions and stresses are better understood.
A 2D plot has been included in 2 of the graphical
windows allowing the user to plot the C-Channel
section with stresses and with its effective areas.
Table 5 – Additional Program Features
20

5.2 – Program Navigation and Error Provisions
5.2.1 – Program Flow Chart
The cold formed program
has been designed with a
linear work flow where
the user first adds
company information
then loading and then
geometry for the channel
section. This linear
workflow allows
backtracking through the
program and also
prevents the user from
continuing through the
design if errors are
present (i.e. intersecting
geometry, fails under
loading). The navigational
system is outlined in
figure 20 and highlights
the complexities and
additional popup windows
also included within the
program.
Figure 20 – Flow Chart Outlining Program Navigation 21

5.2.2 - Error Provisions
To prevent errors from occurring with the final design for a section a series of ‘if’ statements
have been included within the program which prevent progress through to subsequent
graphical windows. These ‘if’ statements when true, also output information about why the
error has occurred to the graphical windows to better inform the user. The different error
checks included within the program are outlined in table 6 below.
Error Statement Outputted String
Overall Geometry and Loading Window
If the section end connections are:
pinned – free | free – pinned | free – free
A mechanism is induced
***Error with Section End Connections –
Mechanism Induced***
If the end moments (M1_y, M2_y, M1_z, M2_z) are non-zero
around a pin this creates a contradiction as pins cannot carry
moments.
***Error with Section Moments – Top/Bottom of
Section Cannot Carry Moments***
If the end moments are non-zero and the additional moments
M3_y and M3_z are non-zero this creates a problem as the
additional moments are only included so that a pinned-pinned
section can have bending.
***Error with Moments M1_y and M2_y cannot
be non-zero when M3_y is also non-zero***
Local Geometry Window
Geometry intersects or is unrealistic - lipped stiffeners are longer than the web depth
- top or bottom flanges are too small
- the corner radii are too large
Section thickness is outside of the range stated in Eurocode 3
Part 1-3
0.45mm ≤ tcor ≤ 15mm (Section 3.2.4 Thickness and thickness
tolerances)
Section thickness is outside of suggested values
and additional strength tests may be required
Ratios of thickness to different parts of the cross section
(Eurocode 3 Part 1-3 Section 5.2)
Width of top flange to steel thickness is too large,
Width of web to steel thickness is too large etc.
The size of the rounded corners (Eurocode 3 Part 1-3 Section
5.1)
Effect of rounded corner 1 must be assessed
Size of the lipped stiffeners (Eurocode 3 Part 1-3 Section 5.2 Eq
5.2a)
Top lipped stiffener is TOO small in relation to the
flange
The section fails under one of the ultimate limit states Section FAILS in {name of limit state}
The error output to the graphical window also includes references to relevant sections of
Eurocode 3 and uses titles and line breaks to make the error list easier to read.
Additionally if the section passes all of the checks a summary of the passing criteria is
outputted to the graphical window for the results page and again this includes line breaks
and Eurocode 3 references for easier reading. (There are also checks and provisions to
make sure zeroes are not entered for input data in certain places to avoid problems that
occur from dividing by zero).
Table 6 – Outline of Error Messages Incorporated in the Program
22

5.3 – Program Windows
5.3.1 – Overall Geometry and Loading Window
(1) - 3D Section Plot: This space allows for the section and sign
conventions to be plotted.
(2) – Input Data: The user can specify the loading and column length in
this panel. The end supports can also be selected from a drop down list.
(3) – Errors: This list box displays any errors which have occurred when
the user tries to continue to the next program window.
(4) – 3D Plotting Functions: This panel allows the user to view
the sign conventions for the different loading conditions as
part of the 3D plot. The shaded graphics can also be adjusted
in this panel.
(5) – Beam Moment Calculator: This opens a separate popup
window which allows the user to find the maximum moment
for a series of standard loading cases.
(6) – Notes Window: This opens a
separate popup window with
information about the overall loading
window (instructional pictures are also
included).
(7) – Arrows: Allows the user to plot
additional sign convention images.
Figure 21 - Print Screen of Overall Geometry and Loading Window
23

5.3.2 – Local Geometry Window
Figure 22 – Print Screen of Local Geometry Window
(1) – Diagram of Channel Section: Provides
information regarding the geometry of the
channel section which can be inputted
using the geometry panel (5).
(2) – 2D Plot: The channel section and the
axial and bending stresses can be plotted
here using the plotting panel (7).
(3) – Input Errors: This window shows any
input errors when the user selects the
calculate function to continue.
(4) – Section Capacity Errors: This window
shows the utilisations and failure
mechanisms of the cross section under the
different limit states.
(5) – Geometry Panel: The channel geometry
and Poisson’s ratio can be added using this
panel.
(6) – Material Properties: The steel grade
and modulus of elasticity can be added using
this panel.
(7) – Plotting Panel: Allows user to plot the channel
geometry and stress in the 2D plot space (2).
(8) – Self Weight: The self-weight and safety factor
can be added to the calculation process.
(9) – Cost: The cost per kg and density can be added
using this panel.
(10) – Notes Window: Opens a separate popup
window with notes and diagrams explaining some of
the functions for the local geometry window.
24

5.3.3 – Final Results Window
(6) – Exporting: The excel template files
location can be specified using windows
explorer. The results from the program can
then be exported to this file in either a full
or summarised format.
(1) – Checks and References: A list of the checks and
passing criteria for the cross section is outputted to
this list box.
(2) – 2D Plot: The idealised channel section,
centroids and effective areas can be plotted here
using the plotting panel in (4).
(3) – Final Section Capacities: This panel shows the
final capacities under different ultimate limit states
as well as the amount of utilisation in each limit
state.
(4) – Plotting Panel: Allows channel section,
centroids and effective areas to be plotted in
the 2D plotting space in (2).
(5) – Deflection Calculator: Allows for the
deflection to be calculated for some standard
loading conditions. The second moments of
area are also outputted here so more complex
deflections can be determined using hand
calculations.
Figure 23 – Print Screen of Final Results Window
25

5.3.4 – Notes Window for Overall Geometry Loading Screen
(1) – Notes List Box: This part of the window is a scrollable list box
with information explaining the functions of the overall geometry
window. The following topics are discussed:
- Sign conventions and units
- Section orientations
- Moments
- Shift of effective centroid and second order effects
(2) – Diagrams: The interactive push buttons on this side of the
window allow the user to plot several different diagrams with
additional information. These diagrams cover the following topics:
- The sign conventions of moments and axial forces
- The centroid shift and the sign conventions involved
- The type of mechanism (unstable structure) which can occur from
different end support conditions
- The orientation of the section i.e. the different bending axis
Figure 24 – Print Screen for the Overall Geometry and Loading Notes Window
Figure 25 – Print Screen for the Local Geometry Notes Window
5.3.5 – Notes Window for Local Geometry Screen
(1) – Notes List Box: This part of the window is a scrollable list box
with information explaining the functions of the local geometry
window. The following topics are discussed:
- Input geometry checks
- Material properties information (i.e. the default values used)
- Section thickness information (i.e. core thickness tcor)
- Partial safety factors
- Stress plots information
- Overview of program functionality
- Self weight calculations
(2) – Diagrams: The interactive push buttons on this side of the
window allow the user to plot several different diagrams with
additional information. These diagrams cover the following topics:
- Stress plot sign convention
- An example of contraflexure in a beam
- The axial compression limit state formula
- The bending limit state formula
- The buckling limit state formula
- The combined bending and axial force limit state 26

5.3.6 – Notes Window for the Results Screen
(1) – Notes List Box: This part of the window is a
scrollable list box with information explaining the
functions of the local geometry window. The
following topics are discussed:
- The exporting to excel functionality and how to
ensure that the correct file names and file paths
are adopted (i.e. the excel file and the excel
sheet name must both be labelled correctly and
the correct folder location must be located and
inputted as a string within the ‘Set working
directory’ box (figure 23)).
- The deflection calculation and how this aspect
of the program works as well as standard
deflection limits for different real life situations. Figure 26 – Print Screen for the Results Screen Notes Window
Figure 27 – Print Screen for the Input Form Data Window
5.3.7 – Initial Input Form Window
This window appears at the beginning of the
program and allows the user to add job specific
information to the final output data i.e. this
information appears in the outputted excel files
automatically. The following fields are covered by
the input form:
- Company name
- Project Name
- Calculation Reference
- Engineer Responsible
- Date (automatically set as the current date by the
Matlab® code but can be manually altered)
- Checked by
- Additional Notes
Figure 28 – Print Screen for the Intermediate
Error Window
5.3.8 – Intermediate Error Message Window
This window appears between the overall geometry and
local geometry windows and only opens when the user
selects a fixed top restraint and a free bottom restraint. The
window simply warns the user that this arrangement of
supports is ok for a beam but unsuitable for a column (the
column should always be attached at the base).
27

5.4 – Axial Compression Programmed Solution
(Overview)
The programmed process for the axial compressive
strength of a cold formed section is outlined in the
flow chart in figure 29. The solution involves
automating the process for the local and
distortional buckling calculations. This is achieved
through steps 3  9 in figure 29. Additionally the
iterative process for determining the χd factor for
distortional buckling is achieved by setting a ‘while’
loop within the program. This can be seen with the
back arrow from step 9 to step 3 in figure 29. (The
program code for the while loop is shown in figure
30). The ‘while’ loop works by finding the difference
between 2 consecutive values for χd and finding the
absolute difference between them. If this difference
is less than the tolerance value (0.001) the loop
ends and the most recent χd value is outputted.
The back end of the calculation process
involves using the effective areas for this
compression case to find the strength of the cross
section under different limit states (step 10 onwards
in figure 29). The effective area under compression
impacts the strength of the cross section under
compression, buckling resistance and the combined
axial and bending limit state.
Figure 29 – Flow Chart of the Programmed Solution for Axial Compression 28

%starting values for the while loop
tolerance=0.001;
initial_value1=10000;
initial_value2=10000;
chi_d_top=1; %initial chi value
chi_d_bottom=1; %initial chi value
while (initial_value1-chi_d_top)>tolerance &&
(initial_value2-chi_d_bottom)>tolerance;
%-----calculations for local and distortional
buckling are carried out-----
%stress field values are altered using the
calculated chi_d factors
sigma_f1_1=chi_d_top*sigma_f1_1/gammaM0;
sigma_f1_2=chi_d_top*sigma_f1_2/gammaM0;
sigma_f2_1=chi_d_bottom*sigma_f2_1/gammaM0;
sigma_f2_2=chi_d_bottom*sigma_f2_2/gammaM0;
%if the chi_d value has converged end the while
loop
if (initial_value1-
chi_d_top)>tolerance|(initial_value2-
chi_d_bottom)>tolerance;
initial_value1=chi_d_top;
initial_value2=chi_d_bottom;
end
end
Sigma_f1_1: Stress in the top flange
Sigma_f1_2: Stress in the top flange
Sigma_f2_1: Stress in the bottom flange
Sigma_f2_2: Stress in the bottom flange
gammaM0 : Partial safety factor
Figure 30 – Matlab® Code for the While Loop
Process
A system of error checking and error output
is also adopted within the Matlab® script.
This is achieved by using an ‘if’ function to
check a certain parameter, for example the
section thickness. If the thickness is within
acceptable values an assignment of zero is
given to an error variable whereas if the
thickness is not acceptable the assignment
given to the error variable is equal to 1. At
the end of the code all the error
assignments are added together and if this
value is non-zero an error screen is
outputted to a list box in the graphical
window. This process is adopted for both
the overall geometry window and the local
geometry window. However for the local
geometry window the errors are divided
between section errors and limit state
errors (this can be seen in figure 22). (An
example of the coded solution for error
checking is given in figure 31).
%----Rounded Corner Check 1-----
rounded_corner_ref={'BS EN 1993-1-3: 2006 Eurocode 3 - 5.1 Influence of rounded
corners'};
bp_mat1=[bp_f1,bp_s1,bp_hw]; %[flange width, stiffener width, web width]
rounded_corner1_check1={'Effect of Rounded Corner 1 Must be assessed'};
rounded_corner1_check2={'Effect of Rounded Corner 1 may be ignored and effective
section adopted'};
if r1<=(5*t) && r1<=(0.1*min(bp_mat1));
display11='------------------------------';%string to display in graphical window
comment10=(rounded_corner1_check2(1,1)); %string to display in graphical window
error_counter10=0; %error assignment (0 = PASS)
else
display11=(rounded_corner1_check1(1,1)); %string to display in graphical window
error_counter10=1; %error assignment (1 = FAIL)
end
%------------------------------
%-----Rest of the program------
%------------------------------
error_counter_total=error_counter1+ error_counter2+. . .;
if error_counter_total==0
close(CchannelGUI); %closes current window
results1; %opens the next program window
elseif error_counter_total>0
display_final=[display1;display2. . .]; %saves the error strings in a matrix
set(handles.checks_errors,'string',display_final); %sets the check_errors list box
%equal to the error string matrix
Figure 31 – Matlab® Code for the Error Checking 29

Figure 32 – Flow Chart of the Programmed Solution for Bending
5.5 – Bending Programmed Solution (Overview)
The programmed solution for the bending
strength of a cold formed section is outlined in
the flow chart in figure 32. The process again
involves the use of a ‘while’ loop to allow for the
convergence of the χd value however additional
provisions are required to account for the
redistribution of the bending stresses with an
effective section.
The differences for this programmed
solution can be seen in steps 2, 3 and after the
‘NO’ option at step 8 in figure 32. The initial
stress distribution is determined from the
idealised cross section (step 2 & 3) and this
allows for the calculations for local and
distortional buckling to be initialised. The stress
distribution must then be recalculated for the
effective section under local and distortional
buckling for each iteration of the ‘while’ loop
(after step 8). Finally after the ‘while’ loop has
determined the reduction factor (χd) the effective
section must be used to find the second moment
of area and thus the elastic section modulus
(step 9). (A brief example of the coded solution
under bending is given in figure 33).
30

%----Initial Second Moment of Area for Idealised Section----
Iyy=BD312_yy_total+Ay2_bar_total; %second moment of area, major axis mm4
%BD312_yy_total : sum of all the BD3
/12 values
%Ay2_bar_total : sum of all the y2
* A values
%The Iyy value above is worked out using the parallel axis theorem
%------------------------------------------------------------
%Initial Stress Distribution
%’If’ function to work out the maximum distance from the NA to the extreme fibre so
a virtual moment M_equiv_yy can be calculated
if (bp_hw+t)/2>=y_NA_bar;
M_equiv_yy=sigma_com_ed*Iyy/((bp_hw+t)-y_NA_bar);
elseif (bp_hw+t)/2<y_NA_bar;
M_equiv_yy=sigma_com_ed*Iyy/(y_NA_bar);
End
%bp_hw : depth of the web (mm)
%t : section thickness (mm)
%y_NA_bar : distance to the neutral axis (mm)
sigma_f1_1_bendingyy1=((M_equiv_yy)/Iyy)*((bp_hw+t)-y_NA_bar);
%sigma_f1_1_bendingyy1 is the stress in the top flange from using the engineers
bending equation (σ=My/I)
%------------------------------------------------------------
%****WHILE LOOP STARTS****
%calculations for local and distortional buckling and reduction factor for
distortional buckling are found
I_Eff_yy1=(sum(I_EffSect1_yy1)+sum(I_EffSect2_yy1)+
(ybar2_EffSect1_yy1.^2*Area2_EffSect1_yy1)+. . .
%I_EffSect1_yy1 : Row vector with BD3
/12 values
%ybar2_EffSect1_yy1 : Row vector with ybar values
%Area2_EffSect_yy1 : Column vector with area values
%The I_Eff_yy1 value above is worked out using the parallel axis theorem by for the
effective section
%’If’ function to work out the maximum distance from the NA to the extreme fibre so
a virtual moment M_equiv_yy can be calculated
if (bp_hw+t)/2>=ybar_EffSect_yy1;
M_equiv_yy1=sigma_com_ed*I_Eff_yy1/((bp_hw+t)-ybar_EffSect_yy1);
elseif (bp_hw+t)/2<ybar_EffSect_yy1;
M_equiv_yy1=sigma_com_ed*I_Eff_yy1/(ybar_EffSect_yy1);
end
%****WHILE LOOP ENDS****
%------------------------------------------------------------
I_Eff_yy1=(sum(I_EffSect1_yy1)+(ybar2_EffSect1_yy1.^2*Area2_EffSect1_yy1)+. . .
%Same assignments as before but for the final effective section
%------------------------------------------------------------
%Elastic Section Modulus
S_modulus_yy1=I_Eff_yy1/((bp_hw+t)-ybar_EffSect_yy1); %mm3
%(bp_hw+t)-ybar_EffSect_yy1 is the distance from the NA to the maximum compressive
fibre of the section
%------------------------------------------------------------
%Bending Resistance
Mc_rd_yy1=((S_modulus_yy1*fyb)/gammaM0)/1e6; %kNm
%from EC 1993-1-1 formula 6.15 for class 4 sections
%fyb : Basic yield stress
%gammaM0 : partial safety factor
Figure 33 – Matlab® Code for the Bending Cases
It is also important to note that for each bending case (Y-Y & Z-Z) there are two sub cases for hogging and sagging.
This occurs because the section may not be symmetric which results in differing stress distributions depending on
the direction of bending. To account for this, 4 effective sections under bending are required and these are
checked accordingly depending on the applied loading cases. (The four bending cases: yy1, yy2, zz1, zz2).
31

5.6 – Shift in Effective Centroid
Programmed Solution
The programmed solution for
the centroidal shift of a cold
formed section is outlined in
the figures 34 and 35. The
calculation process takes place
after the effective section
under axial compression is
calculated and involves working
out the centroid location for
both the idealised section and
effective section. The
difference between these two
locations is then used to find
the second order moments
which are used in the
interaction formula shown in
step 4 in figure 34. It should be
noted that beneficial second
order moments are ignored by
the program, for instance a
second order hogging moment
applied to an overall sagging
moment case is ignored.
Figure 34 – Flow Chart of the Programmed Solution for Centroidal Shift 32

%---------------Centroids for Idealised Section---------------
%Column vector with the individual areas (rectangles) for the idealised section
Area_MAT=[(bp_f2+t)*t;(bp_hw-t)*t;(bp_f1+t)*t;(bp_s1-(t/2))*t;(bp_s2-(t/2))*t];
%Row vector with the distances to the centroids of the rectangles in the y direction
y_NA_MAT=[t/2,t+((bp_hw-t)/2),bp_hw+(t/2),bp_hw-((bp_s1-(t/2))*0.5),t+((bp_s2-
(t/2))*0.5)];
%Row vector with the distances to the centroids of the rectangles in the x direction
x_NA_MAT=[(bp_f2+t)/2,t/2,(bp_f1+t)/2,bp_f1+(t/2),bp_f2+(t/2)];
%using sum(Area*distance)/sum(Area) to find the distances to the centroid in x and y
x_NA_bar=(sum(Area_MAT*x_NA_MAT))/sum(Area_MAT);
y_NA_bar=(sum(Area_MAT*y_NA_MAT))/sum(Area_MAT);
%---------------Centroid for Effective Section---------------
%Column vector with the individual areas (rectangles) for the effective section
Area_MAT_compression=[be1_f1*t;be2_f1*t_top_reduced;be1_hw*t;be2_hw*t;be1_f2*t;
be2_f2*t_bottom_reduced;be1_s1*t_top_reduced;be1_s2*t_bottom_reduced];
%Row vector with the distances to the centroids of the rectangles in the y direction
y_NA_MAT_compression=[bp_hw+(t/2),bp_hw+(t/2),(bp_hw+(t/2))-
(be1_hw/2),(t/2)+(be2_hw/2),(t/2),(t/2),((t/2)+bp_hw)-(be1_s1/2),(t/2)+(be1_s2/2)];
%Row vector with the distances to the centroids of the rectangles in the x direction
x_NA_MAT_compression=[(t/2)+(be1_f1/2),((t/2)+bp_f1)-
(be2_f1/2),(t/2),(t/2),(t/2)+(be1_f2/2),((t/2)+bp_f2)-
(be2_f2/2),bp_f1+(t/2),bp_f2+(t/2)];
%using sum(Area*distance)/sum(Area) to find the distances to the centroid in x and y
y_NA_bar_compression=(y_NA_MAT_compression*Area_MAT_compression)/sum(Area_MAT_compre
ssion);
x_NA_bar_compression=(x_NA_MAT_compression*Area_MAT_compression)/sum(Area_MAT_compre
ssion);
%---------------Centroid Shift---------------
eN_x=x_NA_bar_compression-x_NA_bar; %mm
eN_y=y_NA_bar_compression-y_NA_bar; %mm
%---------------Second Order Moments---------------
%Y-Y moments
delta_My_ed1=(eN_y/1000)*(N1+N2); %kNm
delta_My_ed2=(eN_y/1000)*(N1+N2)*-1; %kNm
%Z-Z moments
delta_Mz_ed1=(eN_x/1000)*(N1+N2)*-1; %kNm
delta_Mz_ed2=(eN_x/1000)*(N1+N2); %kNm
%These second order moments are then used in the interaction equation for the
bending and axial force limit state check.
Idealised Section Geometry
bp_f1 : width of top flange (mm)
bp_f2 : width of bottom flange (mm)
bp_hw : depth of the web (mm)
bp_s1 : depth of top lipped stiffener
(mm)
bp_s2 : depth of the bottom lipped
stiffener (mm)
t : thickness of the section (mm)
Effective Section Geometry
be1_f1 : effective width 1 of top flange (mm)
be2_f1 : effective width 2 of top flange (mm)
be1_f2 : effective width 1 of bottom flange (mm)
be2_f2 : effective width 2 of bottom flange (mm)
be1_hw : effective width 1 of web (mm)
be2_hw : effective width 2 of web (mm)
be1_s1 : effective width of the top lipped stiffener (mm)
be1_s2 : effective width of the bottom lipped stiffener
(mm)
t_top_reduced : reduced thickness of section (mm)
t_bottom_reduced : reduced thickness of section (mm)
Loading
N1 : Axial load 1 (kN)
N2 : Axial load 2 (kN)
Figure 35 – Matlab® Code for the Centroidal Shift 33

Figure 36 – Flow Chart of the Programmed Solution for
the Axial Compression Limit State
5.7 – Ultimate Limit States Programmed Solution
5.7.1 – Axial Compression/Tension
The programmed solution for the axial
compression limit state check is outlined in figure
36 and figure 37. The process starts by
determining whether the applied loading creates
a tensile or compressive situation. If the member
is subjected to tensile loading the section cannot
buckle and the cross section becomes fully
effective. Therefore for the strength calculation
the full area of the cross section can be used. For
compressive loading the member is able to
buckle and the effective section for axial
compression must be used in determining the
strength.
Additional provisions are also included to
account for the self-weight of the member. The
self-weight is calculated by finding the total
volume of the section (Area * Length) and
multiplying by the density of steel which can be
specified by the user (default is 7850kg/m3
). The
self-weight is only included for the axial loading
situation if the member is a column. If the
member is a beam the self-weight creates
moments which are added to the bending limit
state calculations instead.
%-----Compression Checks-----
if axial_final>0; %axial_final : resultant axial force (including self-weight)
checks1={'Section is in Compression'}; %Outputted string
utilisation1=axial_final/Nrk; %Checking the strength (Load/Resistance)
if utilisation1<=1; %Utilisation is less than 100%
checks2={'Section PASSES in PURE COMPRESSION'}; %Outputted string
error_counter17=0;
elseif utilisation1>1; %Utilisation is more than 100%
checks2={'Section FAILS in PURE COMPRESSION'}; %Outputted string
error_counter17=1;
end
elseif axial_final<0; %Tensile Loading (-ve), the strength checks are repeated
Figure 37 – Matlab® Code for the Axial Compression Limit 34

5.7.2 – Bending
The programmed solution for the bending limit state checks is
outlined in figure 38 and figure 39. This process first involves
calculating the effective sections for both the Y-Y and Z-Z bending
situations in both the hogging and sagging directions. The program
then uses a series of ‘if’ statements to determine the type of loading
applied i.e. whether the moments are hogging, sagging or cause
contraflexure. It is at this point that the moments due to the self-
weight are added to the overall moments if the section under design
is a beam. The final applied moments are then compared to the
appropriate resistances for the cross section (i.e. sagging moment 
sagging resistance) and checked to determine whether the member
is safe.
Figure 39 – Flow Chart of the Programmed Solution for the Bending Limit State
%----Hogging around Y-Y axis----
if (M2<=0 && M1>0)|(M2<0 && M1>=0) && M3==0;
Myy_max1=0;
Myy_max2=max(abs(M2)+Mmax_SW2,M1+Mmax_SW1);
%Finding the max moment with self-weight included
checks3={'Section is subjected to HOGGING around
YY axis'};
%Output string
utilisation2=Myy_max2/Mc_rd_yy2;
%Checking the capacity Load/Resistance
if utilisation2<=1; %Section Passes
error_counter18=0;
checks4={'Section PASSES for YY Bending'};
elseif utilisation2>1; %Section Fails
error_counter18=1;
checks4={'Section FAILS for YY Bending'};
end
elseif . . .
%This layout is then repeated for all applied moment
cases i.e. for every case shown in figure 39
Figure 38 – Matlab® Code for the Bending Limit 35

5.7.3 Buckling
The ultimate limit state for buckling is similar to the process for axial compression. The
differences involve the reduction factor χ which is multiplied by the pure axial resistance
(previously calculated) to find the resistance to buckling. The reduction factor relies on the
support conditions and the cross sectional properties. This limit state is handled
programmatically by firstly using ‘if’ statements to determine the correct support
conditions. The buckling resistance around both the major and minor axis of the section are
then determined by using the geometry of an effective section for axial compression. Finally
another ‘if’ statement is used to check the sections capacity and utilisation in this limit state.
(The coded solution for this limit state is shown in figure 40 below)
%---------------------Flexural Buckling----------------------------
%------------Finding Effective Length for Different Support Conditons------------
lambda1=pi*sqrt(E/fyb);
if top_support==1 && bottom_support==1; %fixed-fixed support conditions
Lcr=0.7*L; %Effective length
elseif top_support==2 && bottom_support==2; %pinned-pinned support conditions
Lcr=L; %Effective length
elseif (top_support==2 && bottom_support==1)|(top_support==1 &&
bottom_support==2); %fixed-pinned support conditions
Lcr=0.85*L; %Effective length
elseif (top_support==1 && bottom_support==3)|(top_support==3 &&
bottom_support==1); %fixed-free support conditions
Lcr=2*L; %Effective length
end
%--------------------Flexural Buckling around Y-Y axis----------------
i_yy=sqrt(Iyy_compression/Aeff); %radius of gyration I (mm)
lambdabar_flex_yy=(Lcr/i_yy)*(sqrt(Aeff/Area_total)/lambda1); %slenderness
PHI_flex_yy=0.5*(1+(alpha*(lambdabar_flex_yy-0.2))+lambdabar_flex_yy^2);
CHI_flex_yy=1/(PHI_flex_yy+sqrt(PHI_flex_yy^2-lambdabar_flex_yy^2));
if CHI_flex_yy>1;
CHI_flex_yy=1;
end
Nbrd_flex_yy=(CHI_flex_yy*Aeff*fyb)/(gammaM1*1000); %Buckling resistance (kN)
%----------------Flexural Buckling around Z-Z axis--------------------
i_zz=sqrt(Izz_compression/Aeff); %radius of gyration i (mm)
lambdabar_flex_zz=(Lcr/i_zz)*(sqrt(Aeff/Area_total)/lambda1);
PHI_flex_zz=0.5*(1+(alpha*(lambdabar_flex_zz-0.2))+lambdabar_flex_zz^2);
CHI_flex_zz=1/(PHI_flex_zz+sqrt(PHI_flex_zz^2-lambdabar_flex_zz^2));
if CHI_flex_zz>1;
CHI_flex_zz=1;
end
Nbrd_flex_zz=(CHI_flex_zz*Aeff*fyb)/(gammaM1*1000); %Buckling resistance (kN)
%----------------Final Flexural Buckling Capacity--------------------
Nbrd_flex=min(Nbrd_flex_yy,Nbrd_flex_zz); %Final buckling resistance (kN)
%The value of Nbrd_flex is then checked against the applied loading to determine
whether or not the section passes for flexural buckling.
Figure 40 – Matlab® Code for the Flexural Buckling Limit State
36

5.8 – Deflection Calculator
The deflection calculator (figure 41) is an additional toolbox included within the final results
window of the program. It allows the user to quickly determine the maximum deflection for
the channel section when subjected to a uniformly distributed load. The user can enter a
value for the distributed load and can also include the self-weight whose loading is
calculated automatically by the program. Additionally the second moments of area for the 4
different bending scenarios (hogging and sagging in Y-Y and Z-Z) are also outputted in this
window to allow the user to calculate deflections for different situations (i.e. where the
loading isn’t a UDL). The formulas used for the maximum deflections are shown in figure 42
below.
Built in Beam (Fixed – Fixed) Propped Cantilever (Fixed – Pinned)
Simply Supported (Pinned – Pinned) Simply Supported (Pinned – Pinned)
w : Total load applied (N)
L : Length of the member (mm)
E : Modulus of elasticity (N/mm2
)
I : Second moment of area (mm4
)
Figure 42 – Maximum Deflection Formulas
Input data
The user can specify the loading for
the section as well as the partial
safety factor which is applied to both
the specified loading and the self-
weight. The deflection limit also
allows for the maximum allowable
deflection value to be computed and
outputted.
Automated Data
The second moment of areas
and modulus of elasticity for
the section are provided so
more complex deflections
can be determined using
hand calculations.
Automated Results
These results provide the user with
the maximum deflection for a
series of standard cases. The
deflection limit is also provided.
Figure 41 – Deflection Calculator
37

Figure 43 – 3D Plot of Channel Section with Varying
Transparency
Figure 44 – 3D Plotting Panel
5.9 – 3D Graphics
A 3D plot of the channel section is included within
the program to allow for greater clarity of the sign
conventions and orientation of the different axis.
The channel section can be plotted with shaded
graphics of different transparencies (specified by
the user) or as a wireframe model (figure 43). A
wide range of options are also available within the
3D plotting panel and these are both listed below
and shown in figure 44.
- The bending axis (X-X, Y-Y and Z-Z) (figure 43)
- The top and bottom support locations (included
in all plots)
- The sign convention for the Y-Y moments (M1_y
and M2_y) (figure 45)
- The sign convention for the Z-Z moments (M1_z
and M2_z
- The tensile side of the section for the M3 Y-Y
moments (figure 46)
- The tensile side of the section for the M3 Z-Z
moments (figure 46)
- The axial forces (figure 47)
The 3D plots were created
using the plot3 and fill3
functions in Matlab® which
allow lines and planes to be
plotted in a 3D workspace.
Figure 45 – 3D Plot of End Moments about Y-Y Axis
(M1_y & M2_y)
Figure 46 – 3D Plot of Additional Moments about
Y-Y and Z-Z Axis (M3_y & M3_z) 38

Axial Compression Bending around Y-Y
Axis
Bending around Z-Z
Axis
Figure 48 – Print Screen of 2D Channel Section Stress Plots
5.10 - 2D Graphics
A series of different 2D
plots are included within
the program to illustrate
the geometry of the c-
lipped channel section. On
the local geometry
program window, the user
specified input geometry
can be plotted with the
stress distributions as an additional
overlay (figure 48). The stress plots are based on the idealised cross section (sharp corners)
and are provided as a sanity check so the user can verify the loading conditions. An
additional plot is available for the results window and allows the user to show:
- The idealised section
- The centroid and centroid shift
- The effective section in axial compression
- The effective sections in bending about the Y-Y and Z-Z axis
(The plots for the results window are also shown in figure 49 below).
The 3D arrows for the axial loading plot were
created using a freely available Matlab® script
called draw_line3.m (Sivapalan, 2012). This
script plots a 3D cone and cylinder to mimic
the shape of an arrow and saved valuable
time when coding the 3D graphics toolbox.
Figure 47 – 3D Plot of the Axial Forces (N1 and N2)
Idealised Section &
Centroid Locations
Effective Section for
Axial Compression
Y-Y Bending
Z-Z Bending
Figure 49 – Print Screen of 2D Plots from the Results Window
39

Figure 50 – Setting Output File Path
5.11 – Exporting Data
The cold formed program also includes exporting
functionality which allows the user to transfer the
design data and computed results to excel
templates. In order for this to work the user has
to specify the file location using the windows
explorer popup window shown in figure 50. Once
the file path has been set the user can specify a
full output of the results or a summary. The full
output includes in depth information on all the
geometry and ultimate limit states and fits on 3
A4 pages. Conversely the summary output fits
onto 1 A4 page and only includes the input
geometry and final utilisation factors for the
different limit states. An example of the full and
summarised excel spreadsheets is shown in
Appendix A.
5.12 – Additional Program Add-ins
The cold formed program also includes a beam moment calculator (figure 51). This
calculator appears as a popup window when selected from the overall geometry section of
the program. The popup window contains a list of standard loading cases for a beam with
differing support conditions. The user can add the loading and basic geometry and calculate
the maximum resulting bending moment for the given case.
Figure 51 – Print Screen of the Beam Calculator Add-in 40

5.13 - Future Program Improvements
There were many aspects of the cold formed program which due to imposed time
constraints were unable to be implemented. Some of these future improvements are
outlined in table 7 below:
Additional Program Feature Implementation Strategy
Exporting Results as a PDF report:
Although the program includes an exporting
feature to an excel spreadsheet the PDF format is
more widely used and could provide a neater and
more detailed output file.
This could be implemented by using the report
generator tools within Matlab®. This aspect of Matlab®
allows for data within a script (m-file) to be exported in
a range of different formats including PDF. However
due to time constraints and compatibility issues with
the Matlab® console this aspect was not attempted.
Allowing the user to add intermediate stiffeners:
The program already allows for the use of lipped
stiffeners at the end of the flanges however rolled
in stiffeners are also common on cold formed
sections.
This is a very complex addition to the program and
would require a significant programming effort. Every
additional stiffening feature within the cross section
must be checked for local and distortional buckling and
as this is done within an iterative process the looping
structure within the Matlab® script would become
complex. Additionally the 2D plots and input geometry
would need to include provisions to allow the location
of the stiffeners to be added and plotted.
Allowing the user to select different cold formed
sections
The program allows the user to design a c-lipped
channel section however many other shapes are
used in construction (i.e. Z and Σ sections)
This would require an additional program window
where the user can select the preferred cold formed
section for design. For some of the cold formed sections
such as the ‘hat’ section large sections of the program
code could be reused however the plotting functions
and geometry inputs would have to be altered
completely. (For the Z and Σ sections the previous
program code would be mostly inapplicable).
Including Torsion
This would allow the user to specify moments
around the X-X axis for when loading is applied
eccentric to the shear centre of the cross section.
This involves calculating the total direct stress and the
total shear stress using Eurocode 3 Part 1-3 Section
6.1.6 and applying formulas 6.11a, 6.11b and 6.11c.
Table 7 – Future Program Improvements
41

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