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7/29/2019 MSE-1112-016-Section-optimization-cold-formed-steel-columns-stiffeners.pdf
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Canadian Journal on Mechanical Sciences & Engineering Vol. 2 No. 8, December 2011
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Section optimization of cold-formed steel columns with stiffeners
Yaghob Gholipour
Engineering Optimization Research Group, College of Engineering, University of Tehran, Iran
Abstract
This work deals with the problem of optimal design of selected channel section shapes under axial force.
A design optimization of cold-formed steel columns in presence of stiffeners was investigated. In this
paper, the design goal is to minimize the weight of the structure to resist a given load. Furthermore, we
study optimal column design under pure axial compression, so the load capacity refers to the axial
compression load capacity. The elastic buckling analysis returns the elastic buckling loads of three
modes: local (Pnl), distortional (Pnd) and overall (Pne). The overall buckling mode includes flexural,
torsional and flexuraltorsional buckling. Pne is the minimum of the critical elastic column buckling loadsfor these modes.
Keywords: Cold-formed structures; Design optimization; Thin-walled columns
Introduction
Thin-walled cold-formed beams are widely used in many branches of mechanical industry and civil
engineering. Recently, an increasing interest for improving these profiles with regard to their shapes and
manufacturing can be noticed 0. The primary advantages of cold-formed steel are light weight, high
strength and stiffness, uniform quality, ease of prefabrication and mass production, economy in
transportation and handling, fast and easy erection and installation, its flexibility in forming differentcross-section shapes. However, this flexibility makes the selection of the most economical section
difficult for a particular situation. Cold-formed steel structural members may lead to an economic design
than hot-rolled members because of their superior strength to weight ratio and ease of construction. In
particular, light gauge cold-formed channels are commonly used as wall studs and chord members of
roof trusses in steel frame housing and industrial buildings 0. One of the advantages of cold-formed
steels is that the strength to weight ratio is much higher than that of common hot-rolled shapes, thus it
can reduce the total weight of structures. Therefore the cold-formed steel members are considered to
be economical for low-rise buildings when the beam spans are not too long 0. One of the conditions
required for low cost of an erected structure is the weight of the material be kept minimum, which is
associated with the maximum structural efficiency 0. The minimization of the weight of a thin-walled
beam is a difficult problem, which considers the complex and highly nonlinear constraints that govern
their design, as is shown in various design standards, for instance, the British standard 0. The
optimization of cold-formed steel sections has been studied by a number of researchers. For example,
Adeli and Karim 0 used the neural network method to optimize cold-formed steel beams; Karim and
Adeli 0 conducted the optimization for the hat section of cold-formed steel beams under uniformly
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distributed loading; Lee et al. 0 performed the optimization for channel beams using micro genetic
algorithm.
Buckling of cold-formed steel columns
A uniformly compressed channel section undergoes a shift in the line of action of the internal force
when the section locally buckles. The shift results from the asymmetric redistribution of longitudinal
stress following the development of local buckling deformations, and leads to an eccentricity of the
applied load in pin-ended channels. However, this phenomenon does not occur in fixed-ended channel
columns because the shift in the line of action of the internal force is balanced by a shift in the line of
action of the external force and consequently local buckling does not induce overall bending 0. For a
typical C-shape column under pure axial compression, the local buckling mode is the dominant mode.
However, a small change from this prototype, e.g., the addition of lip stiffeners and web stiffeners, can
markedly increase the local buckling stress and make the distortional buckling mode dominant, as
indicated by Schafer[0,0]. For members with short lip length (small d) distortional and Euler interactionseems plausible: deformations and wavelengths of the distortional mode are similar to the local mode,
which is known to interact with Euler buckling in pin-ended columns. However, for members with large
d, or with intermediate stiffeners or other modification that cause the wavelength in the distortional
mode to be significantly longer than the local mode; interaction with Euler buckling seems less plausible.
In considering local, distortional, and Euler buckling a factor not explicitly discussed is the restriction of
the distortional mode through bracing or other means. In common applications local buckling cannot be
significantly restricted because it occurs at short wavelengths 0. During collapse it is common to have
large strain demands in small regions (e.g., in the lip). The linear strain gradient in a quadratic element is
superior to the step changes in strain across meshes of linear elements 0. They have an inherent
weakness in their small torsional stiffness, which is unfavorable for columns. One way of improving the
resistance is to make the cross-section closed. To overcome to this problem, a closed section made byadding a thin cover plate connected discretely to the flange. The beneficial effect of the cover plate in
increasing the torsional stiffness of the section makes the torsionalflexural buckling load become close
to the pure flexural buckling about the weak axis. The column was modeled with 3D quadrilateral shell
elements (S4R) with sharp corners neglecting the corner radius 0. The nominal axial strength
(compressive resistance), Pn, of cold-formed steel columns under axial compression passing through the
centroid of the effective section can be calculated as follows:
n e nP A F= (1)
whereAe is effective area, and nominal compressive stress Fn is determined as follows:
( ) ( )
( )
2
2
0.658 1.5
0.8771.5
c
y
n
y
c
F
F
F
= >
(2)
where
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y
c
e
F
F =
(3)
where Fe is the least of the elastic flexural, torsional and torsionalflexural buckling stress.
Sections which can be shown not to be subjected to torsional or torsionalflexural buckling, the elasticflexural buckling stress, Fe, can be determined as follows:
( )
2
2e
EF
KL r
=
(4)
where Kis the effective length factor, L the laterally unbraced length of member, rthe radius of gyration
of full unreduced cross section about axis of buckling. For singly symmetric sections subject to torsional
flexural buckling, Fe shall be taken as the smaller ofFe calculated according to above equation and Fe
calculated as follows:
( ) ( )21
42
e ex t ex t ex t F
= + + (5)
where
( )2
0 01 x r =
(6)
( )
2
2ex
x x x
E
K L r
=
(7)
( )
2
2 2
0
1 wt
t t
ECGJ
A r K L
= +
(8)
2 2 2
0 0x yr r r x= + + (9)
wherex0 is the distance form shear center to centroid along principal x-axis, taken as negative, r0 the
polar radius of gyration of cross section about shear center. rx,ry is the radii of gyration of cross section
about centroidal principal axes, Kx,Ktthe effective length factor for bending about x-axis and for twisting,
Lx,Ltthe unbraced length of member for bending about x-axis and for twisting,A the full unreduced
cross-sectional area, G the shear modulus,J the Saint-Venant torsional constant of cross section, Cwthe
torsional warping constant of cross section 0.
In the pin-ended tests, the load was generally applied with a small eccentricity from the geometric
centroid which was adjusted such that the initial eccentricity at mid-length of the line of action of the
force was nominally equal to half of the measured overall geometric imperfection about the minor axis
at mid-length of the fixed-ended specimen of the same effective length 0.
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Fig. 1. Deformed position obtained using measurements of buckling
Ant colony optimization cold-formed column design algorithm
The ant colony system (ACS) is an ACO algorithm based upon the original work of Gambardella and
Dorigo Error! Reference source not found.. The advantages of applying ACO to the design of structures
are similar to the advantages of other evolutionary algorithms. In particular, ACO shares the advantages
of a GA, which include discrete design variables and open format for constraint statements. Like a GA,
an ACO algorithm does not require an explicit relationship between the objective function and the
constraints. Instead, the objective function for a set of design variables is penalized to reflect any
violation of the design constraints. In optimizing a cold-formed column, the objective is typically to
minimize the cost of fabrication while satisfying the AISI standard specifications. The cost of such a
member is a function of the volume of the material required for construction. When the geometry ofthe column is given, the cost is directly related to the cross-sectional properties of each element
(column and stiffeners) in the structure.
To apply an ACO algorithm to structural design, the concept of a tour developed for a TSP is slightly
redefined. Recall that the objective of a TSP is to find the shortest tour that connects all the cities in a
particular problem. In cold-formed column design, the objective is to select a dimension for each
parameter so that the weight is minimized and the AISI specifications are satisfied. The design of a
column may be mapped into the form of a TSP by considering the following modifications:
There are multiple paths from one node in the frame to another (in a traditional TSP, there is one path
between the city, iandj);
The order in which the members of a frame are visited by an ant is not important, (in a traditional TSP,
the order in which the cities are visited is the solution); and
A design developed by an ant is not necessarily feasible (feasibility in a TSP is guaranteed by a tabu list).
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Fig. 2. Mapping the concept of tour for cold-formed optimization
Fig. 2 is an illustration of the virtual paths between two nodes in a problem. Each virtual paths length is
determined by the cost of fabrication. An artificial ant will travel from node ito nodejvia any one of the
virtual paths. The length of each one of these virtual paths represents different value from the value set
of each variable. Fig. 2 indicates that the shortest virtual path (highlighted lines) leads to the lowest
fabrication cost. In this way, the concept of ACO optimization can be applied to the cost optimization of
cold-formed columns. Such value set, not only should satisfy the AISI standard criterion (see Section 0),
but also should lead to the minimum fabrication cost. In this paper, the influence of stiffeners has been
considered. As the number of stiffeners increases, the sustainable axial load will increase (Fig. 3). The
model has been built using w=200, b=100, d=20, L=3000, h=20 (dimensions are in mm). Although the
number of stiffeners has a significant effect on critical buckling load, yet it raises fabrication cost and
causes the objective function to far away from the minimum value. Therefore, the objective is to reach
to an optimum combination of stiffeners and column fabrication cost. A modification to the ACO
algorithm when applied to column optimization is to check the feasibility of the design. In the ACO
application to the TSP, every complete tour is a feasible solution to the problem. In column design, a
tour is defined by a set of values selected for each variable in the column geometry. The values
developed in a tour may define a column in which the dimensions and critical loads violate allowable
limits, resulting in an infeasible design. To account for design infeasibility, a penalty function is applied
to the fabrication cost. The penalized term helps focus the ACO search on designs with the smallest
structural cost that satisfies the design constraints.
Fig. 3. Influence of number of stiffeners on critical buckling load
Ant colony optimization of cold-formed steel column
0
0.1
0.2
0.3
0.4
0.5
0.6
2 3 4 5 6 7 8 9 10
Pcr/Py
Ns
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Cold-formed steel member cross-section shapes are difficult to optimize because of the non-linear
behavior of such members under buckling loads. Traditional gradient-based optimization schemes,
employing deterministic design specifications for the objective function, are inefficient and severely
limited in their ability to search the full solution space of member cross-sections. Ironically, in practice,
only limited cross-sections are adopted. Among them, in the US, the C shape is the most widely used
(Fig. 4). Optimizing the cross-section shape of a cold-formed steel member is interesting from a
structural mechanics viewpoint, and due to the vast geometric possibilities in the design, the problem is
also challenging from an optimization viewpoint. The highly non-linear nature of the strength of thin-
walled cold-formed steel members is due to the fact that member strength is controlled by a complex
combination of overall, distortional and local buckling modes and material strength. Common gradient-
based optimization methods tend to be unreliable for such highly non-linear objective functions. The
objective is to fold the steel sheet into a shape with a cross-section geometry so that the column can
withstand the given axial compression load with the lowest total weight that satisfies the design
requirements.
Fig. 4. C shape cross-sections (plain and lipped)
In the training data set for long columns, overall buckling is dominant. In intermediate-length columns,
both local and overall buckling modes are expected to be important. Thus, the sub region of feature
space that correspond to cross-sections that can resist both buckling modes would be the most logical
place to look for good designs 0. Parametric studies are carried out for span length and loading intensity
0. The material properties of the steel were assumed to be elasto-plastic with a modulus of elasticity of
210 GPa, Poisson ratio 0.3, yield strength 370 MPa and ultimate strength 460 MPa at 6% of engineering
strain 0. According to 2004 edition of the AISI cold-formed steel design manual 0, maximum flat width-
to-thickness ratios, b/t, of unstiffened compression elements such as compression flange of cthe
channel ection or stiffened compression eelementshaving one longitudinal edge connected to a web
and the other edge is stiffened by a simple lip such as compression flange of lipped channel section is
limited to 159, maximum flat depth-to-thickness ratios, h/t, of sthe stiffened ompression eelementswith
both longitudinal edges connected to other stiffened elements such as compression web of channel and
lipped channel section is limited to 472 and maximum flat-width-to-thickness ratios of lip, dd/t, islimited
to 33. In addition, the slenderness ratio, KL/r, of all compression members should not exceed 200,
except that during construction only, KL/rshould not exceed 300. Therefore, maximum slenderness
ratio, 200, is applied to the optimization of cold-formed steel column sections in this study 0. The
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objective function is fabrication cost of column consisted of design variables, overall depth
(w={100,150,,500}), overall width (b={20,40,,200}), base metal thickness (t={1,1.5,,3}), lip depth
(d={0,5,,50}), length of column (L=3000), number of stiffeners (ns={2,3,,10}), stiffener width
(h={10,20,,200}) and fabrication cost per unit weight (C) as shown in Fig. 4. Eq. (10) represents the
objective function of channel column with stiffener.
( ) ( ). . 2 2 1.05 . . .
: 1
472
159
33
200
s
n
Minimiz e f L t C w b d C n t w h
PSubject to
P
w
t
b
t
d
t
KL
r
= + + +
(10)
The fabrication cost of each stiffener has a surcharge equal to 0.05Cto join it to the column. Assuming
d>0, transforms the plain channel to lipped channel. A parametric finite element model has been made
in ABAQUS containing the parameters mentioned above. The first constraint (P/Pn)1, will be considered
as a penalty function and the other constraints will be satisfied as a tabu list. The first step in the
application of ACO to column design is to set an initial trail value, 0, which is defined as
0
min
1
Cost =
(11)
Where Costmin= fabrication cost of the column resulting from assigning the smallest feasible value to
each geometric parameter due to the objective function (Eq. (10)). Each ant in the colony is randomly
assigned a parameter, which serves as the starting point of its tour. The first ant then selects a value or
path for the parameter, using the following decision process. The ant decision table at time t, aij(t) is
( )( )
( ) [ ]1
.
.v
ij ij
ij N
il il
l
ta t
t
=
=
(12)
Wherej=path (value) assigned to the parameter i, and Nv=number of possible values. The probability
that ant k (k=1,2,...,m) will assign the valuejto the member iat the time t, ( )k
ijp t is
1
( )( )
( )v
ijk
ij N
il
l
a tp t
a t=
=
(13)
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The ACO process begins when the first ant selects a value for its parameter ibased upon these
probabilities. After the decision is made and the value assigned, the intensity of the trail on this path is
lowered in order to promote exploration in the search using the following local update rule:
( ) ( ).newij ijt t = (14)
Where =adjustable parameter representing the persistence of the trail. The local update technique,
given in Eq. (14), is different from the update used in ACS in that it does not rely on the initial trail value
0. Since it would be difficult to determine a column design based on the nearest-neighbor heuristic. It
was desirable to create a local update technique that did not depend upon 0. The second ant then
assigns a value to its parameter i, and the local update rule is again applied. The ant value-selection
process continues until all the ants in the colony have assigned a value to their starting parameters of
the column, completing the first iteration of the tour. Each ant then progresses to its next parameter
and assigns a value to its parameter i+1, applying the local update rule each time, and then proceeding
to the next parameter in the sequence. The ant decision mechanism continues until all the values have
been assigned by all ants. Because the ants always move about the parameters in sequential order, atabu mechanism is not needed to prevent the ants from visiting a parameter more than once. Each
column generated by each ant is then analyzed to determine the critical load and AISI specification
limits. This combination is then compared with the design constraints to determine if the column is a
feasible solution to the problem. The columns generated by the ants are ranked by their penalized costs.
The elitist ant is the ant that built the column with the smallest penalized costs found in all cycles. The
values selected by the top ranked ants receive an update according to the following process. The
number of ants receiving this update,, can be any number less than m (the number of ants in the
colony). Let m represent a tour receiving a rank between 1 and, and ij
be the amount of trail to be
added to this tour. The penalized cost of the column generated by ant k, COSTk , is
k k kCOST Cost = (15)
( )1
ijCOST
=
(16)
( ) ( ) ( ) ( )1 . . rij ij ij ijt n t + = + + (17)
Where k is the total cost penalty for the column generated by ant k. The change in trail is computed by
Eq. (16); otherwise0ij
=. Global trail update is computed by Eq. (17) where the (1-) represents the
evaporation rate.
At this point, a cycle has been completed, and a new one begins. When the penalized cost of the best
solution in each cycle has not changed for some number of consecutive cycles, the ant colony is
considered to have converged to a solution. In all applications of the ACO algorithm to column design
presented in this study, the adjustable parameters defined in the ACO algorithm remained constant. In
Eq. (12), as (which controls the visibility of the ants) increases in value, the ants are more likely to
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choose the shorter paths or, in structural optimization, smaller parameter values. A high level of
visibility is a desirable property when solving a TSP; however, in column design smaller values often
produce infeasible solutions (according to the AISI standard specifications). Computational results
showed that a value of=2 helped enforce some level of feasibility in column designs. In the local
update, given in Eq. (14), a value of=1 provides a good balance between exploration and exploitation.
The value of the penalty function exponent is important in that it governs the rate of increase in the cost
of infeasible designs, which directly effects the exploration of the ant colony by adjusting the trail
values, Eq. (17) and selection probabilities, Eq. (13).
Fig. 5. Cold-formed column optimization convergence diagram
Computational results showed that a value of=2 leads to better results and helps the algorithm to be
converged. Computational experiments applying ACO to column design indicate that approximately 5
ants consistently generated the best designs. The value of is equal to 0.1. Fig. 5 and Error! Reference
source not found. show the ant colony optimization convergence diagram and optimum solutions
respectively. Results of the optimized thickness, web depth, flange width, lip length, stiffener width andnumber of stiffeners, obtained by the ant colony algorithm under axial load ofP=104.
Conclusions
This paper has presented an optimum design of cold-formed channel columns with the presence of
stiffeners subjected to a pure axial load. The design variables include the web depth, flange width, lip
length, thickness of section, stiffener width and number of stiffeners, which are determined by using the
ant colony optimization method based on the AISI specifications. Numerical examples have
demonstrated the cost effectiveness of stiffeners in column with respect to surcharge cost of fabrication
for stiffeners. The optimum design section result and convergence diagram are shown in Error!
Reference source not found. and Fig. 5 respectively.
References
R.J. Kasperska,K. Magnucki,M. Ostwald. Bicriteria optimization of cold-formed thin-walled beams with
monosymmetrical open cross sections under pure bending. Thin-Walled Structures. (2007)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 2 4 6 8 10 12 14
Fabrication
Cost
10^6
Iteration
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Canadian Journal on Mechanical Sciences & Engineering Vol. 2 No. 8, December 2011
208
Ben Young,Ju Chen. Column tests of cold-formed steel non-symmetric lipped angle sections. Journal of
Constructional Steel Research. (2008)
Yu WW. Cold-formed steel design. John Wiley and Sons Inc. 2000
Ben Young,Kim J.R. Rasmussen. Behaviour of cold-formed singly symmetric columns. Thin-Walled
Structures. (1999)
A.Ghersi, R.Landolfo and F.M.Mazzolani. Design of metallic cold-formed thin-walled members. Spon
Press. 2002
H. Liu, T. Igusa, B.W. Schafer. Knowledge-based global optimization of cold-formed steel columns. Thin-
Walled Structures. 2004
Gregory J. Hancock,Thomas M. Murray,Duane S. Ellifritt. Cold-formed steel structures to the AISI
specification. Marcel Dekker. Inc. 2001
Tuan Tran, Long-yuan Li. Global optimization of cold-formed steel channel sections. Thin-Walled
Structures. (2006)
Schafer, B.W. Distortional buckling of cold-formed steel columns. American Iron and Steel Institute.
2000
B. W. Schafer. Local, distortional, and euler buckling of thin-walled columns. Journal of structural
engineering. 2002
Long-yuan Li, Jian-kang Chen. An analytical model for analyzing distortional buckling of cold-formed
steel sections. Thin-Walled Structures. 2008
B.W. Schafer. Computational modeling of cold-formed steel. Fifth International conference on coupled
inastabilities in metal structures. 2008
Milan Veljkovic, Bernt Johansson. Thin-walled steel columns with partially closed cross-section: Tests
and computer simulations. Journal of Constructional Steel Research. 2008
Jaehong Lee, Sun-Myung Kim, Hyo Seon Park. Optimum design of cold-formed steel columns by using
micro genetic algorithms. Thin-Walled Structures. 2006
Osama Bedair. A cost-effective design procedure for cold-formed lipped channels under uniform
compression. Thin-Walled Structures. 2009
T. Mocker, P. Linde,S. Kraschin,F. Goetz,J. Marsolek,W. Wohlers. ABAQUS FEM analysis of the
postbuckling behaviour of composite shell structures.
B.W. Schafer, T. Pekoz. Computational modeling of cold-formed steel:characterizing geometric
imperfections and residual stresses. Journal of Constructional Steel Research. 1998.
7/29/2019 MSE-1112-016-Section-optimization-cold-formed-steel-columns-stiffeners.pdf
11/11
Canadian Journal on Mechanical Sciences & Engineering Vol. 2 No. 8, December 2011
209
Cristopher D. Moen, B.W. Schafer. Elastic buckling of cold-formed steel columns and beams with holes.
Engineering Structures. 2009.
G. Kiymaz. FE based mode interaction analysis of thin-walled steel box columns under axial compression.
Thin-Walled Structures. 2005.
P. Borges Dinis, Dinar Camotim, Nuno Silvestre. FEM-based analysis of the local-plate/distortional mode
interaction in cold-formed steel lipped channel columns. Computers and Structures. 2007
Mahmud Ashraf, Leroy Gardner, David A. Nethercot. Finite element modelling of structural stainless
steel cross-sections. Thin-Walled Structures. 2004.
Cristopher D. Moen, B.W. Schafer. Experiments on cold-formed steel columns with holes. Thin-Walled
Structures. 2008.
British Standard Institution. BS5950: structural use of steel in building: part 5: code of practice for design
of cold formed sections. 1998.
Adeli H, Karim A. Neural dynamic model for optimization of cold formed steel beams. J Struct Eng ASCE
1997
Karim A, Adeli H. Global optimum design of cold-formed steel hat-shape beams. Thin-walled Struct 1999
Lee J, Kim SM, Park HS, Woo BH. Optimization design of cold-formed steel channel beams using micro
genetic algorithm. Eng Struct. 2005
AISI. Specification for the design of cold-formed steel structural members. Washington, DC: American
Iron and Steel Institute; 2004 Edition
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