MSE-1112-016-Section-optimization-cold-formed-steel-columns-stiffeners.pdf

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

199

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.

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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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MSE-1112-016-Section-optimization-cold-formed-steel-columns-stiffeners.pdf