Global and local imperfections develop during steel manufacturing processes due to cooling and rolling. These imperfections can cause structures to buckle and fail at lower loads than predicted. The author conducts finite element analyses to investigate the impact of initial imperfections on steel portal frame behavior. The analyses show that initial imperfections can increase or decrease stresses, bending moments, and shear forces in the frame. Imperfections also reduce the frame's buckling capacity and increase the influence of second-order effects. By modeling both ideal and initially imperfect frames, the author aims to provide guidance to designers on considering stability impacts.

1. ABSTRACT: Global and local imperfections (i.e. initial curvature and geometric imperfections) develop during various steel
manufacturing processes, mainly due to steel cooling and rolling. According to publications by the Steel Construction Institute,
these imperfections may cause premature yielding and consequent failure of a structure at lower loads than would be suggested
on the basis of a linear-elastic or plastic analysis of an ideal (i.e. without initial imperfections) portal frame structure. Buckling
of members at loads below the yield load of the members is an important consideration. The classical buckling theory initially
set out by Euler forms the basis for the Perry-Robertson formula, which in turn governs the design code formulas given in I.S
EN 1993-1-1:2005 [1]. The local, lateral, torsional and lateral torsional buckling of flexural members need to be considered
when the compression flange of a member in bending is unrestrained. Second-order order effects (i.e. effects of deflected
geometry) should be considered at design stage, as these may increase the resultant bending moments significantly.
It is recorded that the impact of initial curvature results in the increase or decrease of normal stresses and shear forces. The
considerable increase in bending moments as well as the change in the overall structural non-linearity is also established. These
findings are determined through second-order plastic finite element analysis using LUSAS finite element software; thick shell
finite elements were used for modelling. The effects of initial bow imperfections on the linear-elastic buckling capacity of the
structure and its second-order effects are also recorded.
KEY WORDS: STEEL PORTAL FRAME, IMPACT OF INITIAL IMPERFECTIONS, SECOND-ORDER EFFECTS,
BUCKLING CAPACITY, NORMAL STRESS, BENDING MOMENT, SHEAR FORCE, LUSAS, THICK SHELLS.
1 INTRODUCTION
The concept of the structural form of portal frames was
developed during the Second World War, driven by the need
to achieve a low cost building envelope, which could be built
in a reasonably short period of time. Nowadays, steel portal
frames are the most commonly used structural forms for
single-storey industrial structures. They are constructed
mainly using hot-rolled sections, supporting the roofing and
side cladding via cold formed purlins and sheeting rails.
1.1 Aims and objectives
Steel portal frames can be quite slender and are designed
using either plastic or linear-elastic analysis to determine the
distribution of moments. Stability issues are important when it
comes to design; these can be related to factors such as:
 Residual rolling stresses;
 Initial curvature (i.e. geometric imperfections) of the
individual members both in-plane and out-of-plane
and their consequent design impacts;
 Effective lengths of structural members;
 Local, lateral, torsional and lateral torsional buckling
of I-members;
 Presence of plastic hinges and their stability impacts.
The above factors as well as guidance on the design of steel
portal frames are either directly or indirectly addressed in I.S.
EN 1993-1-1:2005 [1] (Eurocode 3) and Non Contradictory
Complementary Information (NCCI) [2], [3], [4], [5], [6], [7],
[8] publications by the Steel Construction Institute. However,
when it comes to the detailed analysis of steel portal frames,
there are some aspects that need further research. Therefore,
the aim of the project is to study the structural behaviour of
ideal and then initially imperfect steel portal frames by
conducting detailed, 3-dimensional finite element analysis
(FEA) using the LUSAS v14.7 FEA [9] software package.
Such analysis is carried out in order to examine key
objectives, which are directly related to the structural integrity
of steel portal frames:
 To investigate the impact of initial imperfections,
both in-plane and out-of-plane (i.e. lack of member
straightness), and their effects on resultant stresses,
bending moments and shear forces in the steel frame;
 To establish the impact of second-order effects (i.e.
the effects of deformed geometry) on the resultant
bending moments and the impact of the initial
curvature on second-order effects;
 To evaluate the impact of initial curvature on the
buckling capacity of the entire structure as well as on
the column and rafter separately, through first-order,
linear-elastic analysis;
 To examine the overall structural non-linearity (i.e.
examination of yielding patterns and consequent
plastic hinge formation paths under incrementally
applied load, through plastic second-order analysis).
It is hoped that the overall findings of the research will
provide more in-depth guidance to designers regarding
stability aspects of steel portal frames as well as the overall
structural behaviour in both elastic and plastic states.
Impact of Initial Imperfections on the Stability of Steel Portal Frames
Timur A. Shipilin, John J. Murphy
Department of Civil, Structural & Environmental Engineering,
Cork Institute of Technology, Rossa Avenue, Bishopstown.
email: timur.shipilin@mycit.ie, john.justinmurphy@cit.ie

2. 2 FEA OF AN IDEAL SINGLE I-MEMBER VS. FEA OF AN
INITIALLY IMPERFECT SINGLE I-MEMBER
The aim of this introductory part was to establish the impact
of initial bow imperfections, both in-plane (about y-y axis)
and out-of-plane (about z-z axis) on the lateral, torsional and
lateral-torsional buckling capacities of a single structural I-
member. 3-dimensional FE models of the IPE 200 I-beam of
different lengths (i.e. 2 m, 5 m, 8 m), were used to conduct the
FEA. The magnitudes of initial imperfections given in Table
2.1 are calculated in accordance with I.S. EN 1993-1-1:2005
[1] and are used for the buckling FEA.
Table 2.1. Initial imperfections as per I.S. EN 1993-1-1:2005
Also, the exact values of initial imperfections are evaluated
for each member using Young’s theorem of initial
imperfections [10], and then compared to limiting values
given in I.S. EN 1993-1-1:2005 [1] to assess the level of
conservativeness given in the code; calculated values are
presented in Table 2.2. It is clearly seen that values of initial
imperfections given in I.S. EN 1993-1-1:2005 (see Table 2.1)
are significantly greater and therefore more conservative.
Table 2.2. Initial imperfections as per Young’s theorem
A summary of the lateral buckling analysis of ideal I-members
without imperfections is shown in Table 2.3 and Table 2.4,
noting that differences of up to 2% in lateral buckling capacity
of initially imperfect members relative to ideal I-members
were established.
Table 2.3. Values of axial buckling loads about the minor z-z
axis, for I-beams without initial imperfections
L
(mm)
N cr 1, Euler
(kN)
N cr 2, Euler
(kN)
N cr 1,
LUSAS
(kN)
N cr 2,
LUSAS
(kN)
2000 735.97 2943.88 728.21 -
5000 117.76 471.02 117.56 467.995
8000 46.00 183.99 45.95 183.57
Table 2.4. Values of axial buckling loads about the major y-y
axis, for I-beams without initial imperfections
A summary of lateral torsional buckling results calculated by
hand using Eqn. 2.1 with C1 = 1.89 for a moment increasing
linearly from zero at one end to a maximum at the other end
(where M=Mcr), including the results obtained from the FEA
for I-beams without initial imperfections is presented in Table
2.5.
where:
E is Young’s modulus (E = 210000 N/mm2
),
G is the shear modulus (G = 80770 N/mm2
),
Iz is the second moment of area about the weak axis,
It is the torsion constant,
Iw is the warping constant,
L is the beam length between points that have lateral
restraint,
k, kw are effective length factors,
zg is the distance between the point of load application
and the shear centre; depending on the position of the
applied load it may be either positive or negative,
C1 is taken as 1,
C2 is taken as 0.
Table 2.5. Values of critical bending moments for lateral
torsional buckling, for I-beams without initial imperfections
L (mm) M cr,eqn (kNm) M cr,LUSAS (kNm)
2000 180.34 194.95
5000 53.14 57.47
8000 31.55 33.84
A summary of lateral torsional buckling results obtained from
the FEA for I-beams with initial imperfections is presented in
Table 2.6 and Table 2.7.
Table 2.6. Values of critical bending moments for lateral
torsional buckling, for I-beams with initial imperfections
about the major y-y axis
L (mm) eo, at centre (mm) M cr,LUSAS (kNm)
2000 8 192.74
5000 20 56.34
8000 32 32.40
Elastic
analysis
Plastic
analysis
L
(mm)
eo, at centre,
y-y (mm)
eo, at
centre, z-z
(mm)
eo, at
centre, y-y
(mm)
eo, at
centre, z-z
(mm)
2000 1.67 3.31 1.90 5.17
5000 8.47 9.29 9.65 14.53
8000 15.26 15.27 17.39 23.89
L
(mm)
N cr 1, Euler
(kN)
N cr 2, Euler
(kN)
N cr 1,
LUSAS
(kN)
N cr 2,
LUSAS
(kN)
2000 10070.35 40281.39 - -
5000 1611.26 6445.02 1679.56 -
8000 629.40 2571.59 665.71 2592.41
Elastic
analysis
Plastic
analysis
L
(mm)
eo, at centre,
y-y (mm)
eo, at
centre, z-z
(mm)
eo, at
centre, y-y
(mm)
eo, at
centre, z-z
(mm)
2000 6.67 8.00 10.00 10.00
5000 16.67 20.00 20.00 25.00
8000 26.67 32.00 32.00 40.00

3. Table 2.7. Values of critical bending moments for lateral
torsional buckling, for I-beams with initial imperfections
about the minor z-z axis
L (mm) eo, at centre (mm) M cr,LUSAS (kNm)
2000 8 197.79
5000 20 59.80
8000 40 35.48
Minor (within 5%), linearly increasing differences in applied
critical bending moments for initially imperfect I-beams,
compared with I-beams with no initial imperfections, are
recorded.
The impact of non-dimensional slenderness and initial
imperfections about both the weaker (i.e. z-z axis) and the
stronger (i.e. y-y axis) axes on the lateral and lateral torsional
buckling capacities of the steel strut has been established (see
Figure 2.1). This figure shows how the buckling capacity of
the structural member reduces, as non-dimensional
slenderness increases. The non-dimensional slenderness takes
account of the length of the member and the buckling curve
itself takes account of initial imperfections, both in-plane (i.e.
y-y axis) and out-of-plane (i.e. z-z axis). It is seen that the
buckling curves shown in Figure 2.1 are very similar to
buckling curves outlined in I.S. EN 1993-1-1:2005 [1] (see
Figure 2.2), where curve a takes account of initial
imperfections about the y-y axis and curve b about the z-z
axis.
Figure 2.1. Buckling curves obtained from FEA
Figure 2.2. Buckling curves given in I.S. EN 1993-1-1:2005
A summary of torsional buckling results obtained from the
FEA, for I-beams with initial imperfections is shown in
Tables 2.8 and 2.9.
Table 2.8. Values of torsional buckling loads, for I-beams
with initial imperfections about the major y-y axis
L (mm) eo, at centre (mm) N cr,T,LUSAS (kN)
2000 8 1482.82
5000 20 707.89
8000 32 647.79
Table 2.9. Values of torsional buckling loads, for I-beams
with initial imperfections about the minor z-z axis
L (mm) eo, at centre (mm) N cr,T,LUSAS (kN)
2000 8 1483.71
5000 20 724.74
8000 40 607.75
No direct relationship is found between the initial
imperfection and critical axial load that causes torsional
buckling. It is noted that the overall impact of the initial
imperfection on the critical axial load that causes torsional
buckling depends upon the slenderness of the member and its
corresponding initial bow. Further studies are recommended
in this particular field.
3 FEA OF AN IDEAL STEEL PORTAL FRAME VS. FEA OF AN
INITIALLY IMPERFECT STEEL PORTAL FRAME
Two steel portal frames with incrementally increasing vertical
UDL, one without any imperfections (see Figure 3.1) and the
second with initial in-plane imperfections (see Figure 3.2), are
considered. In order to evaluate the impact of initial in-plane
imperfections on a steel portal frame, normal stresses, shear
forces and bending moments are graphed and compared at six
different cuts (see Figure 3.3); plastic second-order analysis
was utilised while undertaking the FEA.
Figure 3.1. Steel portal frame used

4. Figure 3.2. Initial in-plane imperfections used (calculated in
accordance with I.S. EN 1993-1-1:2005)
Figure 3.3. Six critical sections at which results are observed
The comparison summary is shown in Figures 3.4 - 3.11. The
term (+) indicated within each of the legends in the presented
figures, denotes the results obtained from the steel portal
frame with initial in-plane imperfections.
Figure 3.4. Graph of applied UDL vs. normal stress, Section
1, ideal vs. initially imperfect
Figure 3.5. Graph of applied UDL vs. normal stress, Section
2, ideal vs. initially imperfect
Figure 3.6. Graph of applied UDL vs. normal stress, Section
3, ideal vs. initially imperfect
Figure 3.7. Graph of applied UDL vs. normal stress, Section
4, ideal vs. initially imperfect

5. Figure 3.8. Graph of applied UDL vs. normal stress, Section
5, ideal vs. initially imperfect
Figure 3.9. Graph of applied UDL vs. normal stress, Section
6, ideal vs. initially imperfect
Figure 3.10. Graph of applied UDL vs. shear force, all
sections, ideal vs. initially imperfect
Figure 3.11. Graph of applied UDL vs. bending moment, all
sections, ideal vs. initially imperfect
It is seen how initial structural imperfections affect the
resultant stresses, shear forces and bending moments at
different sections, by either increasing or decreasing these. In
the case of normal stresses, it is recorded that an increase of
up to 59.5% and a decrease up to 51% can be obtained relative
to stresses obtained from the portal frame structure without
initial imperfections. In terms of the shear forces, the
maximum increase of 7.1% and decrease of 13.9% relative to
the structure without initial imperfections have been noted. In
the case of resultant bending moments, a maximum increase
of 15.9% was recorded.
The impact of an initial curvature on the buckling capacity
and of the steel portal frame its second-order effects, has been
established. Two different sets of buckling analyses were
considered. The first analyses included the entire steel portal
frame structure with pinned bases (see Table 3.1). The second
analyses considered the structural I-members (i.e. rafter and
column) separately; one by one with fixed and pinned bases
(see Tables 3.2, 3.3 and Figure 3.12).
Table 3.1. Summary of assessment of second-order effects for
pinned base portal frame; ideal and initially imperfect frames
considered
LUSAS
asymmetric,
ideal
LUSAS
symmetric,
ideal
LUSAS
asymmetric,
imperfect
LUSAS
symmetric,
imperfect
α cr 9.23 16.57 6.87 15.40
Amp.
factor
(linear-
elastic)
1.121 - (˃10) 1.170 - (˃10)
Amp.
factor
(plastic)
Less critical
than linear-
elastic
- (˃15)
Less critical
than linear-
elastic
- (˃15)

6. Table 3.2. Summary of buckling analyses of ideal and initially
imperfect stanchions containing plastic hinges; pinned at
plastic hinge (top) and fixed at bottom
I-member
N cr,
ideal,
LUSAS
[kN]
N cr,
imp,
LUSAS
[kN]
Stanchion
(initial)
33 525 1.00 33 296 0.99
Left
stanchion
(at
collapse)
32 838 0.98 33 152 0.97
Right
stanchion
(at
collapse)
32 838 0.98 32 818 0.95
Table 3.3. Summary of buckling analyses of ideal and initially
imperfect stanchions containing plastic hinges; pinned at
plastic hinge (top) and pinned at bottom
I-member
N cr,
ideal,
LUSAS
[kN]
N cr,
imp,
LUSAS
[kN]
Stanchion
(initial)
16 645 0.99 16 645 0.99
Left
stanchion
(at
collapse)
15 970 0.95 15 785 0.91
Right
stanchion
(at
collapse)
15 970 0.95 15 887 0.93
Figure 3.12. Summary of buckling analyses of braced out-of-
plane rafters, deformed mesh factor x2000; ideal and initially
imperfect rafters are considered
It is noted that initial in-plane imperfections enhance the
frame’s sensitivity to second-order effects, which in turn
increase the moment amplification factor (see Table 3.1).
Also, it is recorded that the buckling capacity of the initially
imperfect portal frame I-member (i.e. stanchion or rafter)
reduces relative to the ideal portal frame I-member (see
Tables 3.2, 3.3 and Figure 3.12).
It is recorded that the maximum moment amplification factor
obtained from the linear-elastic buckling analysis of the
imperfect steel portal frame, including the second-order
effects, is 1.170, this increase in resultant bending moments
can be expressed in percentage terms as 17.0% (see Table
3.1), and the maximum impact of the initial imperfection on
the magnitude of the elastic bending moment (i.e. moment
which is less than a plastic moment at which the plastic hinge
forms) that was recorded from the second-order plastic
analysis, also including the second-order effects, was 15.9%.
Therefore, the structure would reach the plastic moment and
consequently yield, at a lesser load.
The impact of initial bow imperfections on second-order
effects in terms of the increase in resultant bending moment is
evaluated to be 1.049 (i.e. 1.170 – 1.121 = 1.049, see Table
3.1), which in percentage terms can be expressed as 4.9%.
4 CONCLUSIONS
The following impacts of initial in-plane imperfections (i.e.
initial curvature) on the stability of a particular steel portal
frame structure are established:

7.  It reduces the linear-elastic lateral buckling capacity
of a single I-member by 2%.
 It reduces the linear-elastic lateral torsional buckling
capacity of a single I-member by 5%.
 It increases the resultant bending moments by up to
17.0%, of which 12.1% is the increase due to second-
order effects.
 It changes the patterns of the overall structural non-
linearity at which yielding of the structure (i.e. initial
local yielding, formation of elasto-plastic and plastic
hinges) occurs.
Also, the magnitudes of initial in-plane and out-of-plane
imperfections for the steel structural members given in I.S.
EN 1993-1-1:2005 [1], are found to be conservative.
ACKNOWLEDGEMENTS
I would like to thank Mr. John Justin Murphy for his advice
throughout the course of the research.
The research reported in this paper was conducted as part of
the taught MEng Structural Engineering programme at Cork
Institute of Technology.
REFERENCES
[1] National Standards Authority of Ireland, I.S. EN 1993-1-1:2005.
Eurocode 3: Design of Steel Structures - Part 1-1: General rules and
rules for buildings, Dublin: NSAI, 2005.
[2] King, C.M., Elastic Design of Single-Span Steel Portal Frames
Buildings to Eurocode 3, Berkshire: SCI - P397, 2012.
[3] King, C. M., Design of Steel Portal Frames for Europe, Berkshire: SCI -
P164, 2001.
[4] King, C. M., Plastic Design of Single Storey Pitched-Roof Portal
Frames to Eurocode 3, Berkshire: SCI - P147, 1995.
[5] King, C.M., In-Plane Stability of Steel Portal Frames, Berkshire: SCI -
P292, 2001.
[6] Koschmidder, D. M., Brown, D. G., Elastic Design of Steel Portal
Frames to Eurocode 3, Berkshire: SCI - P252, 2012.
[7] Steel Alliance, Single-Storey Steel Buildings Part 4: Detailed Design of
Portal Frames, Berkshire: SCI, 2005.
[8]
[9]
Gardner, L., Stability of Beams and Columns, Berkshire: SCI - P360,
2011.
Finite Element Analysis Ltd. (www.lusas.com/usrcheck.html), LUSAS
Engineering Analysis Software, Surrey: LUSAS Distributors Worldwide,
1982 - 2014.
[10] Young, T., A Course of Lectures on Natural Philosophy and the
Mechanical Arts, Berkeley: University of California, 1807.